ANALYTICAL ESTIMATION OF THERMAL PROPERTIES AND VARIATION OF TEMPERATURE IN ASPHALTIC PAVEMENTS Thesis for the Degree of Ph. D. MICHIGAN STATE UNlVERSITY AKBAR KAVIANIPOUR 1971 Luann I]. l ogan S! ‘2‘" ': Univcl if}? This is to certify that the thesis entitled "Analytical Estimation of Thermal Properties and Variation of Temperature in Asphaltic Pavements presented by Akbar Kavi anipour has been accepted towards fulfillment of the requirements for Ph.D. degree in Civil Engineering ‘ 7: Major professor WWW 0-7639 ...— ”4"". A Lg.\ ‘ ‘ =5 amomo av ‘5 NIIAG & “'3' MN BINDERY Ill: LIBRARY BINDERS ABSTRACT ANALYTICAL ESTIMATION OF THERMAL PROPERTIES AND VARIATION OF TEMPERATURE IN ASPHALTIC PAVEMENTS BY Akbar Kavianipour The recent developments of full-depth asphaltic pavement for use in heavy duty highways and airports requires a more comprehensive study of thermal behavior and variation of temperature in the pavement structure. The consideration of variation of temperature at various depths of the pavement is necessary in the study of pave- ment deflection, load capacity, and rheological charac- teristics of asphaltic pavements during the winter and summer months. The variation of temperature at various depths of a 19-inch full-depth pavement of BishOp Airport, Flint, Michigan, was recorded over a period of two years (1969- 70). Similar data from a 7-inch asphaltic pavement was also employed in this study. Daily and monthly variations of temperature at varying depths, due to the change in ambient temperature, are obtained for different seasons of the year. The Akbar Kavianipour nonlinear estimation procedure, which is an extension of least-square analysis, is applied to calculate thermal properties of pavement material from recorded temperature data. Laboratory experiments were performed on samples of asphaltic concrete and the measured temperature recorded as the function of time and position. A criterion is developed to determine the optimum experiment and the location of the thermocouple. A simple method is derived for calculating the thermal diffusivity of pavement material from recorded as well as experimentally measured temperature data. A new method is proposed for directly calculating the thermal conductivity, K, and specific heat from tempera— ture and heat flux boundary conditions. Calibrations of pavement temperature, as well as prediction of frost penetration, requires the knowledge of surface pavement temperature from air temperature which is related to the heat transfer coefficient hC between the air and the pave- ment surface. A method of obtaining this heat transfer coefficient, is given, for the case of negligible radiant heat transfer. Thermal properties of asphaltic concrete are calculated by the nonlinear estimation method,as well as the simplified method, and the results are compared with each other. Akbar Kavianipour Various methods of calculating temperature in the pavement are introduced. Temperature in the soil-pavement system is calculated by using a modified Crank-Nicolson difference approximation (49) applied to the heat— conduction equation. A comparison between the calcu— lated and the measured temperatures at various depths of the pavement is given. Recommendations are made for the extension of this research. ANALYTICAL ESTIMATION OF THERMAL PROPERTIES AND VARIATION OF TEMPERATURE IN ASPHALTIC PAVEMENTS BY Akbar Kavianipour A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Civil Engineering 1971 ACKNOWLEDGMENTS The writer is greatly indebted to his major advisor, Dr. G. C. Blomquist, for his encouragement and guidance. The writer wishes to express his deepest gratitude to Dr. J. V. Beck, for his supervision and suggestion of the basic concept for measurement of thermal diffusivity using the simplified method. The writer is also indebted to the other members of his guidance committee: Dr. 0. B. Andersland, Dr. M. M. Mortland, and Dr. J. W. Trow. Appreciation must also be expressed to Mr. G. P. Manz, and the Asphalt Institute Association, whose sponsorv ship in providing the raw data made the investigation possible. The writer also wishes to thank Maryellen Jones for her cOOperation and assistance in typing the preliminary manuscript of this dissertation. ii LIST OF LIST OF INTRODU PURPOSE Chapter I. TABLE TABLES . . . . FIGURES. . . . CTION. O O O Q S O O O O O 0 LITERATURE REVIEW. 1.1 Introduction. OF CONTENTS 1.2 Effect of Climate on Pavement Temperature and Frost Penetration. . 1.2.1 Solar Radiation . . . . . 1.2.2 Freezing Index. . . . . . 1.2.3 Correlation Factor "n" . . . 1.2.4 Pavement and Air Temperature . 1.3 Thermal Properties of Soils and Pavement Materials. . . . 1.4 Estimated Thermal Properties Based on Published Data and Existing Prediction Equations . . . . . . 1.4.1 Thermal Properties of Soils and Granular Systems. . . . 1.4.2 Thermal Properties of Asphaltic Concrete . . . . 1.4.2-1 1.4.2-2 Factors Influencing Thermal Properties of Asphaltic Concrete . Changes in Volume and Density Due to Temperature. . . . iii Page vi viii 10 11 12 13 15 16 16 27 28 3O Chapter Page 1.4.2-3 Thermal Conductivity of Asphaltic Mixture . . 32 1.4.2-4 Specific Heat Capacity of Asphaltic Mixture . . 35 1.5 Estimated Thermal Properties of Pavement Materials Based on Field Temperature Measurement . . . . 37 1.6 Temperature and Rheological Character- istics of Asphaltic Concrete. . . . . 43 II. DATA COLLECTION. . . . . . . . . . . 47 2.1 Site Location and Climate. . . . . . 47 2.2 A Comparison Between the Recorded and the Long-Term Average Temperature . . . 47 2.3 Observation Period and Auxiliary Data. . 49 2.4 Installation . . . . . . . . . . 52 2.5 Thermocouples. . . . . . . . . 53 2.6 Data Analysis. . . . . . 53 2.7 Daily Temperature Distribution . . . . 57 2.8 Monthly Temperature Distribution . . . 63 2.9 The Effect of Snow . . . . . . . . 69 2.10 Freezing Indices. . . . . . . . . 70 2.11 Correlation Factor "n". . . . . . 73 2.12 Air and Surface Temperature . . . . . 75 III. EXPERIMENTATION. . . . . . . . . . . 77 3.1 Purpose of the Experiment. . . . . 77 3.2 Preparation of Test Specimen. . . . . 79 3.3 Facilities and Equipment . . . . . 81 3.4 Procedure . . . . . . . . . 83 3.5 Heat Flux Calculation . . . . . . . 85 3.6 Results. . . . . . . . . . . . 86 IV. NONLINEAR ESTIMATION METHOD. . . . . . . 96 4.1 Nonlinear Estimation Method of Measuring Thermal PrOperties . . . . . . . . 96 4.2 Nonlinear Estimation Procedure . . . . 97 4.3 Computer Program. . . . . . . . . 100 4.3.1 Data Input . . . . . 101 4.3.2 Temperature Boundary Conditions . 101 4.3.3 Heat Flux Boundary Condition . . 102 4.3.4 Other Variable Input . . . . . 103 4.4 Output and Results . . . . . . . . 103 iv Chapter V. SIMPLIFIED METHOD . . . . . . . 5.1 U101UIU1UIU'IUIUIU'I o o o o o o o o o H‘qua‘mhww Simplified Analysis for Estimation of Thermal Diffusivity Using the Laplace Transform . . Simplified Laplace Transform Method Laplace Transform Criteria The Optimum Experiment. . Optimum Location (x) . . Optimum "s" . . . . Heat Flux Condition. . . . Heat Transfer Coefficient. . . . Correction for Initial Condition Computer Program. . . . . 5.10.1 Input. . . . . 5.10.2 Output and Results . . . VI. PAVEMENT TEMPERATURE . . . . . . . Temperature Distribution in Pavement Exact Solution Based on Periodic Surface Temperature. . . The Integration Method (Volterra Integral Equation) . . . . . . Finite Difference Method . . . . Computer Program. . . . . . . 6.5.1 Input . . . . . . 6. 5.2 Output and Results. . . VII. RESULTS AND DISCUSSION . . . . . . 7.1 7.2 7.3 CONCLUSIONS Recorded Temperature Data. . . . Thermal Properties of Asphaltic Concrete . . . . . . . . Pavement Temperature . . . . . Recommendations for Further Research. BIBLIOGRAPHY . . . . . . . . . . . . APPENDIX 1 APPENDIX 2 Page 112 112 113 118 122 123 124 129 131 135 148 151 151 156 156 158 163 164 169 170 171 176 176 180 190 192 194 198 203 210 Table 1.6.1 3.6.3-A 3.6.3-B LIST OF TABLES Dynamic Modulus E* and Phase Lag ¢ for BishOp Airport Asphalt Concrete Wear- ing and Binder Courses, and Hot-Mix Asphalt Base Course, After The Asphalt Institute Laboratory . . . Annual Climatological Data of Flint, Michigan. . . . . . . . . . Duration of Warmest Temperature Levels at Various Depths for a 19-inch Asphalt Concrete Pavement During the Summer of 1969. . . . . . . . Duration of Coldest Temperature Levels at Various Depths of a 19-inch Asphalt Concrete Pavement During the Winter of 1969-70. . . . . . . Composition and Density of Dynamic Modulus Test Specimens (Bishop Airport Asphalt Concrete Wearing and Binder Courses, and Hot-Mix Asphalt Base Course) After The Asphalt Institute Laboratory, July, 1970 . . . . . Calculated Thermal Diffusivity of Asphaltic Concrete from Laboratory Data (Nonlinear Estimation Method) . Calculated Thermal Conductivity and Specific Heat of Asphaltic Concrete from Experimental Data (Nonlinear Estimation Method) . . . . . Analysis of the Variance for the Repre- sentative Case. . . . . . . . Calculated Values of Root-Mean-Square and Thermal Diffusivity for the Representative Case . . . . . . vi Page 46 50 67 68 78 89 90 93 93 Table 4.4.1 5.9.4 6.5.1 Page Calculated Thermal Diffusivity of Asphaltic Concrete (Entire Pavement) from Recorded Temperature Data (Nonlinear Estimation Method) . . . . 110 Calculated Thermal Diffusivity of Wearing and Binder Courses from Recorded Temperature Data (Nonlinear Estimation Method) . . . . . . . . . . . 111 Correction Factor and Combined Effect of Temperature Pulses. . . . . . . . 149 Calculated Thermal Diffusivity of Wearing and Binding Courses from Recorded ' Temperature Data (8 = 4/tmax), Simpli— fied Method . . . . . . . . . . 153 Calculated Thermal Diffusivity of Asphaltic Concrete (Entire Pavement) from Recorded Temperature Data (3 = 4/t ) Simplified Method. . . . 154 max Calculated Thermal Diffusivity of Wearing and Binder Courses from Laboratory Data (Pmax.=,120 seconds and s = 4/tmax), S1mp11f1ed Method . . . . . . . . 155 Calculated Temperature at Various Depths of the Pavement During a Few Winter Days 0 O O O O O O O O O O O 172 Calculated Temperature at Various Depths of the Pavement During a Few Summer Days 0 O I O O O O O O I O O 173 Statistical Comparison Between the Calcu- lated Thermal Diffusivity of Asphaltic Concrete (Analysis of the Variance and F-Test) . . . . . . . . . . . 188 vii Figure 1.1.1 ‘LIST OF FIGURES Intrinsic Factors Which Influence Frost Action (27) . . . . . Extrinsic Factors Which Influence Frost Action (27) . . . . . . Moist-Soil Models. . . . . . . Thermal Conductivity of Soils After Kersten (31). . . . . . . . Thermal Properties of Sandy Soils After Makowski and Mochlinski (33) BishOp Airport Temperature Study. Average Monthly Temperatures During 1969-1970 Compared with Mean Temperatures of 1942-1969 . . . Full-Depth Asphaltic Pavement Temperature During a Summer Day (Sunny) . . . . . . . . . Pavement Temperature During a Winter Day (1/5/70) 0 o o o o o o 0 Temperature Distribution in 19" Full- Depth Asphaltic Pavement and a Gravel Section at Selected Times During a Winter Day . . . . . . . . Pavement Temperature During a Spring Day 0 O O O O O O O O O 0 Temperature Distribution in a 7" Asphaltic Pavement and a Gravel Pavement During a Winter Day (1/13/65). . . . . . . . viii Page 18 24 25 48 51 54 55 56 59 60 Figure 2.7.6 2.10.1 2.10.2 4.4.1 Pavement Temperature During a Winter Day 0 O O O O O O O I O O Pavement Temperature During a Summer Day (7/27/64). . . . . . . . . Full-Depth Asphaltic Pavement Temperature During the July and August of 1969 . . . . . Subgrade Temperatures Beneath the 19" Full-Depth Asphaltic Pavement and a Gravel Section (February 1970) . . . Average Daily Temperature and Cumulative Degree Days (February 1970) . . . . Cumulative Degree-Days Above and Below 32°F at BishOp Airport, Flint, Michigan . . . . . . . . . . Location of Thermocouples in the Asphaltic Sample. . . . . . . . Experimentally Measured Temperature at the Calorimeter Surface and Various Depths of the Asphaltic Sample (Solid and Dashed Lines Refer to First and Second Thermocouple Sets) . Difference Between Calculated and Measured Temperature (Residuals) at l/2-Inch and 1-Inch Below the Surface of the Specimen (Representative Case). A Comparison Between Calculated and Measured Temperature at 1/2—Inch and l—Inch Below the Surface of the Specimen (Representative Case) . . . Variation of Thermal Diffusivity During One Iteration Process (Representative Case) . . . . . . . . . . Summation of Root-Mean-Square for the Representative Case. . . . . ix Page 61 62 64 65 72 74 80 87 95 105 106 107 Figure Page 5.3.1 Laplace Transform Criteria . . . . . 119 5.6.1 Calculated Values of [(g)2 + 1]_1. . . 127 5.9.1 Temperature Distribution in 19-Inch Full-Depth Asphaltic Pavement at Selected Times During a Winter Day. . 136 5.9.2 Temperature Profile in a l9-Inch Full-Depth Asphaltic Pavement During Winter and Summer . . . . . . . 137 5.9.3 Transient Temperature Distribution for Rectangular-Shaped Temperature Pulses. 142 5.9.4 Combination of Temperature Pulses and Correction Temperatures . . . . . 143 5.9.5 Correction Factor for a Single Rectangular-Shaped Temperature Pulse (y/a = 3.0) . . . . . . . . . 146 6.2.1 Daily Variation of Temperature (Exact Solution) . . . . . . . . . . 160 6.2.2 Annual Variation of Temperature (Exact Solution) . . . . . . . . . . 161 6.4.1 Soil-Pavement System . . . . . . . 165 6.4.2 Location of Thermocouples, Regions and Nodes in the Soil-Asphalt System . . 166 6.5.1 Calculated Pavement Compared with Measured Temperature During the Summer Days . . . . . . . . . 174 6.5.2 Calculated Pavement Temperature During Summer (Variable Initial Temperature). 175 7.2.1 Calculated Thermal Conductivity and Volumetric Specific Heat from Laboratory Data . . . . . . . . 182 7.2.2 Calculated Thermal Diffusivity of Asphaltic Concrete from Recorded Field Data. . . . . . . . . . 183 INTRODUCTION Frost is a major source of damage to the structure of highway and airfield pavements in the northern United States and the countries in the higher latitudes of the northern hemisphere. In order to withstand the detri- mental effect of frost action, some estimate of frost penetration under the pavements is necessary. The current approach to the problem of determining the depth of frost penetration is based on a rational formula named "modified Berggren formula" (38), which is rooted in the theory of heat transfer. The data necessary for the solution of such an analytical equation must be realistic and accurate in order to properly predict the depth of frost pene- tration. It is possible to utilize the analytical procedure for calculating temperature at various depths of the pave- ment and predict the depth of frost penetration provided the thermal properties of pavement elements and the ambient condition are known. It was in recognition of the need for a reliable value of thermal properties that the major portion of this research was devoted to determining estimates of the thermal preperties of asphaltic concrete. The thermal properties of soils have been the subject of numerous investigators,whi1e little effort has been directed at evaluating the thermal properties of asphaltic concrete. Literature contains few references to the ther- mal preperty of asphaltic concrete. The traditional approach to preperty measurement is to create experimental conditions for which the system behavior is approximated by a limited form of a fabricated model in the laboratory. In practice, the application of the thermal preperties obtained under the controlled condition may not be capable of predicting accurate thermal behavior and temperature distribution in the pavements under natural conditions. For these reasons, a method of estimating thermal property from recorded field temperature data is needed. The digital computer has made it possible to model complex physical systems and include numerous coupled phenomena, which are difficult to combine when conducting laboratory experiments, to measure individual properties. In this study a review of various methods of preperty measurements in the laboratory or from field data are made and a method is introduced which embraces both the new concept and traditional methods. Extensive available temperature data is utilized to evaluate the thermal preperties of the asphaltic concrete and results are compared with laboratory values of thermal properties. The recorded temperature data utilized in this research were obtained from an existing portion of pavement at Bishop Airport, Flint, Michigan. The thermo- couples were installed to record the air temperature and the temperature in the pavement surface, base, subbase, and subgrade soils. Temperatures were recorded over a period of two years (1969-1970). Additional data employed in this study gave the temperature variation of a 7-inch thick asphalt pavement in Alger Road, Gratiot County, Michigan, recorded during the years 1964-1965 (34). Data for evaluation of climatic factor and ambient condition was obtained from United States Weather Bureau records. The method of data collection and analysis is described, and the results of this study are presented in tabular and graphical form. Current knowledge contained in the published litera- ture is presented in the form of a brief literature review. The following: 1. PURPOSES primary purposes of this research are the To obtain the temperature variations at various depths of the l9-inch full-depth asphaltic pavement and underlying subgrade soils. The available temperature data provide information verifying the range of temperature that an actual asphaltic pavement has in the central Michigan climate during various seasons of the year. To approximate the actual pavement condition with those of the laboratory test and to deter- mine whether the 140° F standard testing tempera- ture of asphaltic concrete is applicable to the central Michigan climate. To determine the ranges of temperature in various layers of the pavement (wearing course, binder course, and hot mix base course) which are necessary to consider in selecting the mechani- cal properties of asphaltic concrete. To determine the minimum temperature in the subgrades beneath a l9-inch full-depth asphalt pavement, and a comparison between the tempera- ture beneath this pavement and a gravel section during the freezing season. To advance a method for determining thermal preperties of pavement material through the field as well as the laboratory temperature data, with particular regard to asphaltic concrete. To calculate temperature distribution at various depths of the pavement and compare the calculated temperature with the measured temperature. CHAPTER I LITERATURE REVIEW 1.1 Introduction In studying the pavement temperature and frost action, many influencing factors must be considered. The various factors which influence the pavement temperature are divided into two groups: extrinsic and intrinsic factors (27). Extrinsic factors are those which deter- mine the ambient conditions. These factors are sum- marized in the block diagram of Figure 1.1.1. Intrinsic factors are those inherent to the pavement material. They include various components of a pavement system such as: paving materials, base, subbase, and subgrade soils, as well as physical and chemical preperties of each component. More specifically, intrinsic factors include thermal properties of pavement elements such as thermal con- ductivity, K, specific heat, C, and thermal diffusivity, a. These factors are given in Figure 1.1.2. It is beyond the scepe of this study to discuss each one of these factors separately.' However, a syste- matic approach to increase the knowledge of the subject, 333339820 Ahmv coaeom amouw wocwcamcfi gowns muouomm camcfiuucH H Season .38. coins." no 2832. fianahm “.858 HHOW dough Ho :pmuofim H o a) u a) :2: T _ 956036;] :ofiusom paw—56: 8330: 5.38.8qu 63A 3 0333:“— 2330909 8325.» 2 .3: 833 SSBESS 9:. 3:88 3% 598 .H.H munmnm 53.5.5.8 25 unopcoo £3.30: noaahonok ions EOE 25 330% and: 30m 05 no uofiuoaoum H8355 r ESE Guise #33390” _mod$uoaloo 3.33. pnoum 8:335” 5.2.3 :30: mmoaom cam thcmEdm#1 EH hocmz no moaoom 8. .3? USE eofiumpfinwoonm ocsaaaa< ouzpmpoesoa ufi< moopaumq _:oapmoog _ 0:43.011...” "U coauo< enema oonosququ sown: muopoum 989.33 0 H9 gum 382m 25m 96: hm: uaoa mcwbo: «acoam>am no 2305 omnmzouam _ UMOA _ the separation of the major factors, and the determination of the relative influence of each one separately'has been made. The effect of various extrinsic factors has been the subject of numerous investigators, but, among the intrinsic variables, little effort has been directed toward evaluating thermal preperties of pavement materials, especially asphaltic concrete. Because of the complexity of the subject matter, an extensive review is given of the literature dealing with the thermal properties of pavement material. The material in this chapter presents first, a brief discussion and description of governing factors influencing pavement temperature and finally, the thermal properties of pave- ment material will be presented in detail. 1.2 Effect of Climate on Pavement Temperature and Frost Penetration Among the climate factors, solar radiation, freezing indices, duration of freeze, and the relationship between air temperature and pavement temperature are of major concern in determining pavement temperature and frost penetration. The above mentioned factors are related to the climatic characteristic of an area. A brief dis- cussion of the factors is presented in the following paragraphs. 10 1.2.1 Solar Radiation Solar radiation consists of two types: (a) short— wave, and (b) longwave radiation. The shortwave type makes up the majority of received waves. Solar radi- ation has a greater heating potential than the air temperature on varying pavement temperature. Scott (45) introduced an equivalent air temperature v . T a by. T' = T + T (1.2.1) where Ta and Tr are air temperature and equivalent radi- ation temperature respectively. The influence of radi- ation effects on the pavement surface temperature is given by (45): . 2n T — M + Br ° Ar Sln §€§ t (1.2.2) where Mr and Ar are the mean value and amplitude of the equivalent air temperature, respectively. Br is the amplitude reduction factor defined by equation (1.2.4). The amount of radiation received at the surface of the pavement depends upon the amount of water vapor, time of year, and location of the area (45). Methods of com- puting radiation quantities at the earth's surface are complicated by the presence of the atmosphere which tends to scatter, absorb, and reflect the radiation as it passes through. 11 Fowle (21) and Kimball (30) conducted studies on the relationship between the depletion of the incoming radi- ation and the quantities of agents in the atmosphere. It is assumed that asphaltic pavement is a black mass and 10 per cent of the solar radiation is emitted from the pavement (43). Scott (45) presented methods of calculation and tabular solar radiation data from which the total radiation quantity could be determined for average atmospheric conditions for a black mass for any given latitude. 1.2.2 Freezing Index The general method of calculating frost penetration uses the air freezing index as a measure of the cold quantity to which the pavement is subjected (38). The freezing index is the difference between the maximum and minimum points of the cumulative degree-day plotted for one year (23). Index values employed in design are the average air-freezing indices of the three coldest winters in the last thirty years of record (23). The design freezing index should be verified at least once every five years unless more recent temperature records indicate a signifi- cant change in thickness design requirements for frost (23). 12 The freezing index calculated from the recorded temperature data for Flint, Michigan, is given in Chapter II. A value of the freezing index equal to 1263 degree- days has been obtained by Oosterbaan gt 31. (41) for Midland, Michigan. Similar data, taken from the Tri-City Airport, Saginaw, Michigan, was equal to 1622 degree-days. 1.2.3 Correlation Factor n A correlation factor "n" which is the ratio of pavement surface freezing index to the air freezing index is referred to as the "n" factor (38): surface freezing index air freezing index n: The freezing index determined for air temperature about five feet above the ground is commonly designated as the air—freezing index, while that determined for temperature immediately below a surface is known as the surface freezing index (23). Carlson (12), in a study of data obtained from a Corps of Engineer installation in Alaska, determined a correlation factor equal to 0.6, for both bituminous and portland cement concrete pavements. Kersten (31) obtained a range of correlation of 0.6 to 0.8 for a bituminous pavement in Minnesota. Aldrich and Paynter (3) concluded that a value of the air-surface correlation factor "n", of 0.9 should be adapted for all pavements until further research has clarified the problem. 13 A value of "n"-factor approximately equal to 1.0 is given by Oosterbaan gt_al. (41) for Midland, Michigan. The "n"-factor calculated from the recorded temperature data for Flint, Michigan is given for comparison purposes in Chapter II. 1.2.4 Pavement and Air Temperature The relatively wide range of observed "n"—factors from references illustrates the complexity of the inter- relationship between the air and pavement surface tempera- ture. The depth of frost penetration, and all subsurface temperature, is governed by variation in the pavement surface temperature. The pavement surface temperature variations are generally unknown, however, while the air temperatures are available. In studies by Aldrich (2) and Carlson (12) it was found that the surface temperature was generally warmer than the air temperature during both summer and winter. The relationship between air and pavement surfaces may only be obtained through careful theoretical and experimental consideration of the controlling variables. Scott (45) conducted a theoretical investigation into the factors affecting heat transfer at the air-earth interface with limited results. He assumed a sinusoidal variation of air temperature and derived the pavement surface temperature as: 14 Tp = M + - A ° Sin(wt - 6) (1.2.3) ’7? 2 V/(S + ml) + ml where S w __l.o = we — —1__l—.o S _ K, ml 2a, O - tan (O + (.01), and M and A are the mean value and amplitude of air tem- perature fluctuation, respectively. 81 is the heat transfer coefficient between the air and the pavement surface. Equation (1.2.3) indicates that variation of pavement surface temperature is also sinusoidal with a phase lag equal to: B = S (1.2.4) 2 2 /QS + ml) + wl In order to use Equation (1.2.3) the thermal properties of the paving material and the coefficient of heat trans- fer,s, must be known. The value of S1 and the thermal properties are assumed to be constant. Fewell (19) em— ployed regression analysis to study the relationship between air and pavement surface temperature. The result- ing equation describing the pavement surface temperature is given by the following expression: T = C Sin(wt +0) (1.2.5) + BT + C p l a 2 15 where C1 = (1 - B)M +-—— and C = B —— qm and qa are the mean value and amplitude of the radi- ative heat flux, respectively, and 0 is the phase angle difference between the air temperature and radiation waves. The results of the studies by Scott (45) and Fewel (19) indicate that some progress has been made in relating air and pavement temperature surfaces. However, most designers still employ the empirical correlation factor, "n," discussed earlier, to convert air freezing index to pavement surface freezing index. 1.3 Thermal Properties of Soils and Pavement Materials Over the past few decades, many investigators have presented various techniques both in the laboratory and field to evaluate thermal properties of soils and pavement materials. These techniques may be grouped in two cate- gories: (l) estimated thermal properties based on pub- lished data, laboratory and existing prediction equations; and (2) estimated thermal properties based on field temperature measurements. In the following, we review the literatures and researches dealing with each technique separately. The list is by no means complete, but points out that an extensive number of methods have been proposed 16 in the area of thermal properties of soils and pavement materials. 1.4 Estimated Thermal Properties Based on Published Data and Existing Prediction Equations 1.4.1 Thermal Properties of Soils and Granular Systems Estimated thermal properties based on published data can only be used with caution. This is because the ther- mal properties of soils and pavement materials have a strong dependency on density, mineral type, grain size, and moisture content of the materials (31). The analysis of the published data provides a general guide to the range of value of thermal properties. Accepted existing equations for prediction of the thermal properties frequently include questionable restrictions or assumptions, and hence the indiscriminate use of these equations leads to erroneous conclusions (see references 10, 3S, and 50). Many investigators have presented various techniques which eliminate one or more of these restrictions, but most of their contributions have been to present a new method of obtaining specific properties (e.g., thermal con— ductivity) rather than all the thermal properties (50). Birth gt 31. (10) for calculating thermal resis- tivity of mixed aggregates assumed a series arrangement 17 (see Fig. 1.4.1A) for heat flow and effective conductivity K calculated from: (1.4.1) Nil-4 II (relax + 21sz + (m g. N m The effective conductivity for parallel treatment results in the equation: K = K X + K X + K X (1.4.2) 9 9 w w s s where K and X are the conductivity and volume fraction of each component and 9, w, and s, referred to gaseous matter, water, and solids, respectively. Mickley (35) introduced a method of determining thermal conductivity of a soil mass for moist, saturated, and dry conditions. Figure (1.4.lB) shows the picture of a three-component system which is assumed for the soil components. The thermal property of soils has been related to its mechanical properties by a number of investigators. The experimental data obtained by Farouki (18) indicate that thermal conductivity of cemented granular materials increases linearly with the compressive strength. For a granular material Farauki used the following equation for conductivity, K, (ca1/°C cm sec) K = B(13,555-3,407f + 0.443 RC) (1.4.3) 18 I. . .I '- t. I.- . 90-'. ' o a 3"" .0... . . . I .I I... , . ... ‘ «.0 Q I ...Q l . \‘o ., . - 'n C - . ° _ '0- - e . .° ‘\‘o. . .. . .. _. ‘ .. -.¢ ’ O . ' ' I ’ '..o "l u . . 0! KS SOLID WATER GAS SOLID Ike-fl In— xs—aI —-Ix.I+.Ix‘:Ia_ T THERMAL THERN‘L FLow FLOW SERIES TREATMENT PARALLEL TREATMENT . o . o o. _' |" . \ ' . . D . . " I. . THERuAtZ:;§_ FLOW j~, ' DRY SOIL MOIST SOIL o I . O ; l I I A r - fl ' 7 W E "'I I] THERMAL FLOW WATER [Elana “i ‘ . g .a . , ‘ a o ' M-_—--'—-P—1>- ‘ ' l ‘v . I .‘I ..'c " '.c 0 . I no———|-a, ——b-(h-n . I" l MICKLEY'S MOIST-SOIL MODEL (8.) Figure 1.4.1 Moist Soil Models 19 where Rc = compression strength of the soil (PSI) f = volume of clay/volume of granular material, and 8 depends on density, specific heat, and other mechanical properties of soil and varies from 0.9 x 10-5 to 3.4 x 10"6 gm/cm2 sec per °C/cm. Farouki (48) introduced a comprehensive equation for the thermal conductivity of a generalized soil in terms of partial volume of its components which has also included the effect of moisture migration in both the film and the vapor phase. The overall thermal conductivity, K, of un- saturated soils is given by: _ '_ -5 K — sts + Kw(vw 10 S) + Kava + hL + hv + hw + 10'5 5K0 (1.4.4) where Vs’ Vw' and Va are the partial volumes of the solids, water, and air in a given soil; S cmZ/cm3 is the specific surface of the soils; K is the thermal conduc- 0 tivity of oriented water; hL, hV’ are the heat trans— ferred by migrated moisture in liquids and vapor phase; and hw is the amount of heat conducted by static moisture in liquid phase. Equation (1.4.4) is written in a simple form as: _ l K - K + hL + hv + hw (1.4.5) 20 where K' contains the terms which consider the effect of oriented water and volumic fraction of soil components. Farouki (18) studied the effect of moisture content on thermal resistivity of a back-fill material and con- cluded that saturated materials have lower thermal resis- tivity than those which have been mixed and densified in the air-dry state. He concluded that, when the material dries out, the thermal resistivity increases linearly as the logarithm of the moisture content decreases. However, the thermal conductivity increases linearly as the logarithm of the moisture content increases. Farouki (18) obtained the following relation between the thermal resistivity, p = l/K, and the moisture content 0): lag w==a - U‘I‘O (1.4.6) where a, and b are positive constants for a given soil. The binding effect between soil particles has been also mentioned by Farouki (18) as a cause for variations of K (Equation 1.4.5) at low moisture content. Farouki (18) concluded that at high moisture con- tents, binding effect does not exist while as the material decreases its moisture content this effect increases, reaching a maximum in the dry state. The oriented water around the soil particles has also been considered as an additional contributor by 21 Farouki (18) in obtaining a relation between the thermal conductivity of soils at various moisture contents. This review shows the relative significance of mineral constituents and the strong dependency of phase composition on the thermal property of a soil system. The density of the mass system also has a signifi- cance and governing effect on the thermal preperties of pavement materials. Kersten (31) concluded for a con- stant moisture content the rate of increase in thermal conductivity varies from 1.2 to 4.4 per cent for each one pound per cubic foot of density increase in unfrozen soils. For the frozen soils, the values varied from 1.6 to 4.6 per cent increase per additional pound in density. The relation between thermal conductivity K, and density is given by equation: K = A(10)B ° DenSitY (1.4.7) A and B are constant for a particular soil at a constant moisture content (31). Kersten (31) found reliable empirical equations which express the thermal conductivity of a soil as a function of its water content for a given density. For unfrozen soils, he recommended the following equation: K = A(1ag10 moisture content) + B (1.4.8) 22 where A is the difference in conductivity between the l per cent and 10 per cent moisture content, and B is the value of thermal conductivity at l per cent moisture con- tent. For frozen soils, Kersten (31) derived the following relationship: K = A + B(moisture content) (1.4.9) where A and B are constants for the soil at a given density. Kersten (31) combined Equations (1.4.7), (1.4.8), and (1.4.9), to derive expressions defining thermal con- ductivity of soils in terms of soil type, moisture con- tent, and dry density. For fine-grained soils the follow- ing equations relate: (0.9 laglo w — 0.2)1o°°01 yd (1.4.10) unfrozen K 0.01(10)°'°22 yd frozen K 0.008 + 0.085(10) yd (w) (1.4.11) For coarse-grained soils the resulting equations were as follows: _ 0.01 unfrozen K - (0.7 laglow + 0.4)10 yd (1.4.12) frozen K = (0.076(10)0°013 yd 0.0146 + 0.032(10) yd (w) (1.4.13) 23 The expressions for fine—grained soils are valid for moisture contents of 7 per cent or more, while Equation (1.4.12) and (1.4.13) are valid for moisture contents of l per cent or more. These equations are presented graphically in Figure 1.4.2. Gemant (22) derived an equation to predict thermal conductivity of soils in terms of water content, thermal conductivity of water and solid components, and the dry density of the soil particles. Makowski and Mochlinski (33) compared the basic equations of Kerten (31) and Gemant (22) and presented a nomograph (Fig. 1.4.3) for rapid solution of determin- ing thermal conductivity and resistivity of sandy soils. This nomograph is based on the following equations: K = (a laglo w + b) x 10 y/100 watts/cm2 - c/cm w = moisture content (1.4.14) Y = dry density The values of a and b are defined differently for Kersten (31) and Germant (22) equation as following: For the Kersten (31) equation: a for sand = 1.03; (clay = 1.29) b for sand 0.565; (clay = -0.283) 24 — SANDY SOILS -- SILT O CLAY SOILS [unraozefl] 2.0 -— SANDY SOILS -- SILT S CLAY SOILS 90 o “m no 1. ' Figure 1.4.2 Thermal conductivity of soils after Kersten (31) 25 Ammv fixmcflacooz cam fixmzoxmz Houmm mHHom mpGMm mo mmfiuuomoum Hmfiuoce m.e.H madman 11$ 0 M oo o H. < om “#1.: on ”on 0 on m . 11 m H a a m. m. on 8 31.- a o m .2 3.7% o om o: H. Hr «JIM. 8 a . o m. D 02 3.1 o.I m. mm. mm fl J 0 4” OH on." :JIMT OVA/V 0- o w I” m m 8 GAO J o .38 ea mu m 0 some“ “fall. “0 “I? O O m o elm m m m m 0.7. 03 re .3322 :38 t. 9m l 3 I m 0. MM mg” no“ 1 m to Wm m . - 0.: no w. m a n- H .528 N we u T 1 0. In H "u ONH Q O O.“ r h- “ no." mm m I on." fix 3 M.“ «0+ m.mn A copmu ,tu Imm 0»; a S nfilhw ma}: ax 32:8 Ea 26 For the Gemant (22) equation: a = 1.424 - 0.00465 P b = 0.419 - 0.00313 P and "U ll percentage of clay in the soil In the nomograph points C and 8 correspond to the Kersten (31) data for clay and sand respectively. Heat capacity and latent heat of fusion: The volumetric heat capacity of soils may be ex- pressed as follows: (16) For dry soil C = Cyd (1.4.15) For moist, w unfrozen 3011s Cu = yd(C + 1.0 —66) (1.4.16) For moist, w frozen 30113 Cf = yd(c + 0.5 100) (1.4.17) The average value w for mOist SOllS Cavg = Yd(C + 0.75 100) (1.4.18) where c = specific heat of the dry soil Yd = dry unit weight w = moisture content of the soil in the percentage of dry weight. The value of c equal to 0.17 BTU/1b °F has been recommended by most of the investigators including Kersten (31) and Aldrich (2). 27 The latent heat of fusion of water is assumed to be 144 BTU/lb; and it follows that the latent heat of fusion of a soil, L, may be given by the following expression (16): L = 144 yd 1%,)- (1.4.19) 1.4.2 Thermal Properties of Asphaltic Concrete The thermal properties of soils have been the subject of numerous laboratory and field studies, while little effort has been directed at evaluating the thermal proper- ties of bituminous concretes. Although relatively little information is available in the literature with respect to the thermal properties of bituminous concrete mixtures, it appears that there is less variation in magnitude of the thermal properties of asphalt concrete than for soils (20). The purpose of this section is to introduce and describe the various factors which have the strongest influence on the thermal properties of asphaltic concrete, and finally a review of literature and research will be presented which has been conducted both in the laboratory and field in the study of thermal properties of asphaltic concrete. 28 1.4.2-1 Factors Influencing Thermal Properties of Asphaltic Concrete Bituminous concrete may be defined as a system com- posed of the solid, semi-solid or liquid, and gaseous phases, with a temperature dependent thermal properties. The solid phase is the mineral material consisting of sand, gravel, crushed stone, slag, and mineral filler. The semi-solid phase is the visco-elastic asphaltic material produced from petroleum (or bitumen) in a variety of types and grades ranging from hard brittle solid to almost water-thin liquids. The gaseous phase is the air which fills the voids. The mineral materials are surrounded and bound together by asphaltic material. Thermal properties of bituminous concrete mixtures vary depending upon the temperature, types of aggregates, asphalt content, density, porosity, and the presence of moisture in the voids available in the mixture. Moisture: Water in either vapor or liquid phase, enter the bituminous concrete through the voids available in the mixture. The presence of accessible pores, cre- vices, and capillary forces results in the penetration of water into the pores. Bituminous aggregate coatings are believed to have very small pin holes through which water may penetrate. In high void ratio mixes, such as most base course material, water may freely circulate through the voids. Well compacted hot-rolled asphalt mix, 29 has a very low permeability to water, while in open- graded mixtures water penetrates easily into the asphalt concrete. The results of studies by Heley (24) showed that high moisture content associated with relatively open- graded bituminous concrete mixed used on many highways in West Virginia have been responsible for obtaining higher thermal conductivity of pavement materials. Aggregates: An increase in thermal conductivity may be expected depending upon the types of aggregates. Most of the rock-forming minerals exhibit the conductivity- temperature behavior which is characteristic of dielectric solids at high temperature; that is, the thermal conduc- tivity follows roughly the law K = (AT + B), where K is thermal conductivity, T is temperature, and A and B are constants (10). Thermal property of the rocks is affected by the type rock-forming minerals, porosity, and the direction of heat flow. Quartz has the highest conduc- tivity in the direction parallel to its optic axis, while mica has the lowest value in the direction perpendicular to the cleavage (10). Thermal properties of aggregates depend on their parent materials and the state of the pore spaces inside the aggregate. Thermal conductivity of aggregates is estimated by Saal (44) and a value of K = 0.7 to 1.4 BUT Ft/ft2,°F, hr has been suggested for most aggregates. 30 Little reliable information is available concerning specific heat-capacity of aggregates. Hogbin (59) con- ducted experiments on five common road aggregates, and a filler, from sources in Great Britain. A value of C = 0.2 BTU/lb °F is suggested by Hogbin (59) for most aggregates. 1.4.2-2 Changes in Volume and Density Due to Temperature A material like asphaltic concrete undergoes volume change with change in temperature. This volume change has a significant effect on the thermal property of asphaltic concrete. A dry mixture of asphaltic concrete may be assumed as a three-phase system composed of mineral, asphaltic, and gaseous matter. For a given temperature, gaseous matter expands more than asphaltic material and asphaltic material expands more than the mineral materials. The cubical coefficient of expansion gives a change in volume of material over a particular temperature range and can be defined by: (1.4.20) in which Vo = volume at the same reference temperature, and AV = change in volume due to temperature change, At, from reference temperature. 31 This coefficient for a variety of asphalts is given in tabulated form by Saal (44) and Abraham (1) with an average volume change of B = 1.08 x 10-3 per °F. Thermal expansion of asphalts and other forms of bituminous matter may be calculated from the equation (1): _ 2 VT - V60[1 + A(T - 60) + B(T - 60) ] (1.4.21) where A = 3.41 x 10'4 B = 1.0 x 10"7 V and V are the volumes of asphalt at 60°F and 60 T at a temperature of T°F,respective1y. The values of V60/VT are given in a tabulated form for various tempera- ture ranges. Abraham (1) established a value of B = (0.4 - 0.43) x 10"4 per °F for asphaltic material. For aggregates, the average values of the linear coefficient of expansion given by Troxell and Davis (47) and Mitchell (36) is B = 0.135 x 10-4 per °F. Com- paring the values of thermal expansion for aggregates with those of asphalts, it can be seen that there is at least an order of magnitude difference in the eXpansion characteristics of these materials. The cubical coefficient of expansion of asphaltic mixture may be obtained by including the effect of aggregate type, asphalt hardness, and asphalt content. 32 Disregarding the volume of air voids in a volume of asphaltic concrete, we may write, AVmix = Avasph + Avagg (1.4.22) where AV . =AV 8 mix mix AT (1.4.23) mix Let n be the percentage of asphalt content by volume of aggregate, then from Equation (1.4.23), the expansion coefficient of asphaltic mixture will be obtained equal to: 8mix = [n Basph + (100 - n) Bagg] (1.4.24) The coefficient of linear and cubical expansion of asphaltic concrete has been investigated also by Hook (26) and Goetz (26). They suggested an average value of Bmix = 1.26 x 10.4 per °F for the cubical coefficient of expansion. The small value of cubical coefficient of expansion indicates that the density of asphaltic con- crete is much more insensitive to temperature change than either thermal conductivity or specific heat; it is thus assumed to be constant in this study. 1.4.2-3 Thermal Conductivity of Asphaltic Mixture Thermal conductivity of soils have been the subject of numerous laboratory and field studies while little 33 effort has been directed at evaluating the thermal con- ductivity of asphaltic concrete. The United States Army Corps of Engineers reference gave a thermal conductivity value of 0.82 BTU/hr ft °F for dry asphaltic concrete. Aldrich (2) used a value of 0.84 BTU/hr ft °F in calcu- lation of frost penetration and Barber (5) has suggested a value of 0.7 BTU/hr ft °F in computation of pavement temperature. The significance of asphalt content within the mix- ture was evaluated by Saal (44) in his empirical equation for thermal conductivity of the mix: log Kmix = x log Kasph + (l - x) log Kaggr (1.4.25) where Kmix = thermal conductivity of the mixture K = thermal conductivity of the asphalt asph Kaggr = thermal conductivity of the aggregate, and x = fraction by volume of asphalt. The thermal conductivity of various asphalt is ob- tained by Saal (44). A value of 0.09 BTU/ft °F hr is sug- gested for most asphalts. In general, the thermal con- ductivity of asphalts decreases as the temperature in- creases and varies linearly in the range of 0 to 110 °F (32). 34 K = Ko(l - CZAT) (1.4.26) where C2 is constant for a particular asphalt, and K and K0 are thermal conductivity of asphalt at particular temperatures and reference temperature, respectively. A comparison between obtained values of, K, for rock and aggregate by Saal (44) and Birch gt_§l (10) indi- cates that the conductivity of aggregate is approximately one order of magnitude larger than that of the asphalt. In deriving Equation (1.4.25), Saal (44) assumed a value of asphalt content equal to 5 per cent by weight of mixture. The bituminous concrete is assumed to be dry and the thermal conductivities of aggregates and bitumen are assumed to be equal to 1.50 and 0.44, respectively. The calculation of thermal conductivity of bitumi- nous concrete, using Equation (l.4.25), have a value of 1.29 BTU/hr ft °F, which is substantially higher than commonly accepted values. O'Blenis (40) and Heley (24) attempted to utilize field temperature data to compute the thermal conductivity of highway pavement materials. Values of conductivity obtained by O'Blenis (40) varied from 0.49 to 1.34 BTU/hr ft °F. Heley (24) used field data to calculate the conductivity of bituminous concrete for various seasons and concluded that thermal conductivity varied considerably with the season,with the highest values occurring during the freezing period. 35 1.4.2-4 Specific Heat Capacity of Asphaltic Mixture The specific heat capacity of asphaltic mixtures, like other engineering materials, is not a constant but is a function of temperature. The literature contains few references to the specific heat of bituminous con- crete mixture. Abraham (1) has suggested the following relationship: 0 I mix — 0.0l[(100 - x) Casph + xcaggr] (1.4.27) in which x per cent by weight of aggregate; and Cm' 1x' C and C represent the specific heat capacity asph' aggr of asphaltic mixture, asphalt, and aggregates,respectively. Values for Ca for a series of asphalts over the sph temperature range 32 to 570°F were obtained by Saal (44). These values indicate that the specific heat increases linearly with temperature by the following equation sug- gested by Mack (32): C = C0 + ClAT (1.4.28) where C = specific heat at a particular temperature, Co = specific heat at reference temperature (e.g., 32°F), and C = constant for a particular asphalt. 36 Abraham (1) suggested the following relationship for specific heat capacity of asphalt: _ _1_ o Casph - /E_(0.388 + 0.00045 T) BTU/lb F (1.4.29) where d = specific gravity of asphalt, and T = temperature in °F. A value of 0.4 BTU/lb °F is suggested for most asphalts. In cases where asphalt is mixed with various amounts of solids such as sand, crushed rock, etc., the following equation is suggested by Abraham (1) for specific heat of the mixture: Cn = 0.01[(100 - x)Ca + x Cs] (1.4.30) where x = percentage by weight of solids, and the sub- scripts n, a, and 5 refer to mixture, asphalt, and solid, respectively. Hogbin (25) employed a method based on Equation (1.3.27) to obtain the specific heat capacity of a series of course materials. An average specific heat value of C = 0.2 BTU/lb °F was obtained for wearing and base courses with various compositions. 37 1.5 Estimated Thermal Properties of Pavement Materials Based on Field Temperature Measurement Field temperature measurement in conjunction with classical periodic wave analysis (13) has been used to evaluate thermal properties of soils and pavement materials. Jumikis (28) introduced a method to calculate temperature distribution at various depths below the ground surface. The analysis is based on periodic sur- face temperature variation. He assumed that surface temperatures vary diurnally and annually about a mean temperature in a manner which approximates a sine wave. It is further assumed that the soil beneath the surface has a constant thermal property and follows the same temperature function as the surface, but is reduced in amplitude and delayed in time. In a given cycle the temperature amplitude at two different depths, x1, and x2, are given by the two equations: — . _'"_. "X _ . 2 .11. where a is the thermal diffusivity of the material under consideration in BTU/hr ft °F and P is the period of the temperature wave in hour. Solving these two equations simultaneously for a gives: 38 w x1 - x2 2 0!. = '5' W (1.5.3) The general wave equation may also be solved in terms of time which yields: 1r(x2 - x1)2 a = (1.5.4) 2 P(62 - 91) where (62 - 61) is the phase difference between the two temperature curves at the two depths. If t1 and t2 are the time of temperature maximum at depth x1 and x2 in hour, Equation (1.5.4) may be written in the following form: . = £2.43? 2 :1,” 2 l (1.5.5) Equations 1.5.3 and 1.5.5 may be used with either the diurnal or annual temperature cycle to calculate theoretically thermal prOperties of the pavement materials. [Most investigators agree, however, that the annual cycle could lead to errors in conclusions, because of the wide range of weather phenomena experienced during an entire year.] O'Blenis (40) used the recorded temperature data obtained at twelve locations in the state of West Virginia to calculate thermal properties of pavement materials. He found out that the thermal property obtained by Equations (1.5.3) and (1.5.5) could be equal if the comparative 39 variation between the depths follows a true sinusoidal pattern. He selected those days which most closely follow a sine wave temperature variation. Assuming a sinusoidal and d temperature variation at any two depths d the l' 2’ Equation 1.5.3 and 1.5.5, for a period of twenty-four hours is written in the following form: d - d 2 a = .23: ____1 2 (1.5.6) 1n Tl/TZ 2 (d - d ) a = 2:1. 1 2 (1.5.7) 4w (t _ t )2 l 2 O'Blenis (40) equates these two equations with the resultant expression: Tl/T2 = e°°262(t1 ' t2) (1.5.8) which is the relationship between the amplitude ratio and time lag for any two depths. Since Equation (1.5.8) represented the conditions resulting from a sinusoidal variation, O'Blenis (40) selected days on which the amplitude ratio and time lag satisfies this equation, and then used Equations (1.5.6) and (1.5.7) to calculate thermal diffusivity. Voluminous primary data was processed to determine a few days which approximate a sine wave and satisfies the criterion given by Equation (1.5.8). Since the daily 4O heating cycle is normally other than the simple sine function, Carson (14) proposed higher harmonics and the Fourier analysis for daily temperature wave. He recom— mended a second or third harmonic for a daily temperature cycle and a single harmonic for the annualtemperature cycle. Heley (24) programmed the O'Blenis (40) method and reduced the manual calculation to a great extent. O'Blenis (40) and Heley (24) concluded that the magnitude of the daily temperature wave becomes inappreciable at depths below 18 inches and the application of this method to layers below the l8-inch depth produces extremely large values of thermal conductivity. It was concluded that either the assumptions of the method did not satisfy the natural conditions or that the method itself was in error. Chudnovski (15) introduced a method which elimi- nates the assumption of sinusoidal surface temperature, but requires a linear temperature distribution with depth. This method, denoted as "generalized wave method" offered a definite advantage over the periodic method employed by O'Brien (40) and Heley (24). The analysis of Chudnovski (15) is based on the solution of the general heat equation in a semi-infinite body with a time variable surface temperature and con- stant thermal properties which is given by Duhamel's integral (15): 41 e-x2/4a(t-1) 3/2 T(x,t) = X It T(0.A) 2/F2 o (t - A) d1 + T1 (x) (1.5.9) Where 1 is a dummy time variable of integration, and initial temperature T1 is assumed to be uniform at t = 0. Thermal diffusivity may be obtained by using this equation for a given temperature variation at a certain depth. However, it is not practical to determine 0 directly from this equation. Chudnovski (15) transformed this equation by multi- plying by dx and integrating between the limits X = 0 and X = w. The result may be written: n f00 (T(x,t) - T(x,0))dx a = °t (1.5.10) f O T(0:}\i : 3(010) (1X The numerator of Equation 1.5.10 represents the area between the temperature-depth curve at the instant (t), and the initial time t = 0 which is determined with a planimeter. To evaluate the integral in the denominator, the integral is divided into sub-intervals in each of these T(0,1) - T(0,0) can be regarded as linear. Finally the value of diffusivity is determined by the following expression: 42 n fm(T(x,t) - T(x,0))dx a = ° (1.5.11) 2/E[alsl + ... + bnt(2/3 - on_1)] Where ai and bi are the initial temperature ordinates and the slope of each time-temperature curve in the sub- interval,respectively. The values of si and Oi are computed for various values of dummy time variable (15). The generalized wave method is utilized to calculate thermal diffusivity of pavement material subjected to any surface heat pulse. The requirement for initial condition is met by selecting the time period to begin just as the daily heat pulse reaches the upper interface of the layer under con- sideration and is continued until the temperature at the lower interface begins to rise. This process provides a linear initial temperature profile. O'Blemis (40) and Heley (24) employed this method to calculate thermal properties of pavement materials from field temperature data. They indicated that the generalized wave method is best used on those layers having little or no moisture present and to those layers which are waterproof, such as the wearing course and waterproof base. The finite differ- ence method also was employed by Heley (24) to evaluate thermal conductivity of deeper layers and subgrade soils in West Virginia. Of all the methods presented, the numerical method required the least time, but most precise 43 raw data. In this method the one dimensional heat equation approximated by finite difference method and finally was solved directly for the thermal conductivity K. i,j+1 _ i,j-l . Ax 2At T i+i,j ' i,j i-l,j (1.5.12) where i and j represent location and time.respectively. It is observed that when the temperature in the data point location were such as to have the sum of the upper and lower values equal to two times the center value, the denominator of second term in Equation (1.5.12) becomes zero and the corresponding value of K approached infinity. Despite the simplicity of the entire procedure the limited applicability and sensitivity of the method to the form of data has made it impractical. The thermal proper- ties obtained by this method are questionable. 1.6 Temperature and Rheological Characteristics of Asphaltic Concrete Temperature has a significant influence on the mechanical and rheological characteristics of asphaltic concrete. A great deal of laboratory work has been accomplished concerning temperature dependent properties of asphalt concrete pavement. These properties include deflection, stresses and strains, stiffness, and viscosity. 44 The load-carrying capacity and stiffness of the asphaltic mixtures are much lower during the summer months than the winter months. There is a possibility of the pavement rutting under the heavy loads during the summer months. On the other hand, during the winter months the pavement does not deflect under heavy loads and cold temperatures may cause cracking in the asphaltic concrete. Stress-strain characteristics of asphaltic material are defined by various coefficients such as dynamic modulus E*, relaxation modulus, and relation modulus E(t). The dynamic modulus E*, in compression, tension, and tension—compression can be determined by (20), Q .9. E O |E*l = where 00 is the applied stress and so is the resultant strain. The dynamic tension-compression modulus of asphaltic concrete may be obtained through the application of a sinusoidal loading to a specimen. If the material is viscoelastic, the deformation resulting from the load will have the same sinusoidal variation with time but will lag behind the stress by a time represented by (20) ti ¢ = IT (360°) where ti is the time lag between the sinusoidal cycle of 45 stress and strain in second, and w is the frequency of loading per second. The absolute value of the dynamic modulus |E*| and phase lag 0 for the wearing course, binder course, and hot mix asphaltic base course similar to the existing asphaltic concrete pavement at Bishop Airport in Flint, Michigan, are given in Table 1.6.1. The absolute values of the dynamic modulus and phase lag over a range in frequencies and various tempera- tures of 10°, 40°, 70°, and 100°F indicate the visco- elastic response of the asphaltic concrete. 46 m: mm.H «a oo.H an mm.o ooa «n on.“ ms nu.“ an o~.m 05 amazon an :.mH AH :.wa Hm m.~a o: comm cannons N H.3m n n.~m m m.m~ ca Maxnuom a: so.a mm mm.o mm o:.o cos on mn.o an on.: on mm.~ on amazon Nevada AH :.o~ ma m.AH mm A.nH o: oponouoo uHannm< o H.0m o b.3n n w.Hm 0H m: om.o mm ma.o mm mm.o ooa mm «3.: H: mm.~ n: :m.a on onnsoo wcaumos ma m.ma Hm ~.nH «m «.0H 0: opouosoo pflasau< « :.on n «.mm m ~.n~ 0H A.mouv Awoa N «adv A.modv Amoa H «may A.mocv Anoa H «any nomv oonsoo B km B tn up ..u 0959690959 anon—chum one on one a one H nozmnammm 02Hn4 H.N.N ousmflm onoa on: no: ua< no: pom use can >02 900 new ms< ash nose nonmaumomav .nsoa haspco: .mba macaw mm 9664 go unsusnonaoe coo: coma 994 o OH ON on c: on ow on ow oo (05) oanqaaadmam 52 underlying subgrade. During winter months, results ob- tained from the recorder were confused and nearly indis- tinguishable. For this reason, it was felt necessary to compare these data with previously studied temperature data. The data used for comparison was concerned with temperature variation in a 7-inch thick asphalt pavement in Alger Road, Gratiot County, Michigan, which was con- ducted and reported by Glenn P. Manz (34). 2.4 Installation The facilities utilized for recording temperature data consisted of two basic components: (1) a recorder and supporting equipment, and (2) the thermocouples and connecting leads. A Minneapolis Honeywell-Brown lZ-channel tempera- ture recorder was utilized to compile the data. The recorder was furnished by Leonard Refineries, of Alma, Michigan. The recorder was capable of recording twelve thermocouples, one each thirty seconds, hence, the same thermocouple once every six minutes. The temperature range of the recorder was 0°F to 160°F with an accuracy of plus or minus one-half degree F. An ice bath was utilized for checking the recorder for accuracy during the testing period. One of the probes attached to the automatic recorder was an ambient air temperature probe, and was located in the shade of the recorder. A glass, mercury-filled, air thermometer 53 was placed beside this probe to check the air temperature thermocouple. A time synchronized check was made periodically. 2.5 Thermocouples Temperature probes were placed at various critical depths in the pavement and subgrade to record the neces- sary data. Probes were also placed in an untreated base and subgrade section outside the paved area. All probes were 20 gauge iron-constantan thermocouple probes. 2.6 Data Analysis The data compiled by the multi-point recorder indi- cated a temperature reading for each probe every six minutes. This collection procedure resulted in a voluminous amount of temperature data which was later selectively reduced, to represent hourly temperature values. These hourly values were extracted from the recorder chart and transferred to data sheets. The high and low temperatures for each pavement probe and subgrade probe were recorded to establish the extreme temperatures for a given twenty-four period (see Figures 2.7.1 through 2.7.7). From the twenty—four hour period cycles of data an average temperature for each period was also calculated for each depth (see Tables 2.8.1 and 2.8.2). .Amccsmv moo umfifism m mcfluso ousumuomfiou ucoem>om oeuaonmmo numoolaaom H.>.N musmflm Ansonv made _ . om . as . .8 ..om 100H .OHH oomunsm IONH (cg) aangsaadmam 55 NH 0H .Aow\m\av moo Hopcfiz m mcfluso ououmuomfimu ucoEm>mm N.n.m madman Em NH 0H m o a N u - q q - so.on|n ..‘I..I.:l..|..l..|-.l..|.i!il--l \\- -- - - - segmd OH- 0 OH I a m d a I D. m ON 1 a m. on o: .mmo umucH3 m mcHHso meHu vmuomem um coHuomm Hw>mnm m can ucwEm>om UHuHmcmmm anmUIHHSM =mH cH GOHuDQHHumHo musumumafima Sm NH Ed OH — . A . _ a . a H ’ acoaobcn onHsnans no oanHsu newshound - ? m.n.~ magmas - 0: on 0N OH 0: Aomv ousumnmnfime 0 Ln 0 O O N M d’ (seuaur) undaa O H O :3 O n o» c: (saqour) qqdeq O H 57 To facilitate this study some data relative to solar radiation of the Flint area was desired. Upon searching for such data, it was found that none was available for the Flint area, however, similar data for the East Lansing, Michigan area was available. Knowing the climatalogical data of the Flint and East Lansing areas, a correlation of the solar radiation for the Flint area could be ob- tained provided the amount of cloud cover and temperatures were considered equal. 2.7 Daily Temperature Distribution Daily temperature distribution graphs were plotted based on hourly data collected from recorder charts. During a continuous sunshiny day in summer the temperature cycles generally would be represented as is given in Figures 2.7.1 and 2.7.7. The air and surface temperature curves are regular with only one peak, occurring during the middle of the afternoon. Cloud cover will, however, effect the surface temperature resulting in several peaks and irregular air and surface temperature curves. During the cloudy and the rainy days the temperature within the pavement was uniform and nearly equal at all depths, consequently, distinguishment between the various thermo- couple records was difficult. Analysis of such collected field data required a comprehensive information concerning the local climato- logical data, which provided necessary information 58 concerning cloud cover, wind velocity, and precipitation. The above information was obtained from the National Weather Service which is located less than a mile from the study site location. All readings of cloud cover were recorded by physical observation. Additional information such as hourly precipitation in the form of rain and snow, direction and intensity of wind speed may be obtained from the local climatological data sheets. Temperature distribution in a l9-inch Bishop Air- port asphalt concrete pavement and the underlying soil during a summer day is shown in Figure 2.7.1. Variation of temperature at different depths, down to 42 inches below the surface of the full-depth asphaltic pavement for a winter day are shown in Figures 2.7.2 and 2.7.3. A similar curve for a spring day is plotted in Figure 2.7.4. Daily temperature profile in a 7—inch asphalt concrete pavement and subgrade on Alger Road, Gratiot, Michigan, during a winter day are shown in Figure 2.7.5. The temperature distribution curves shown in Figure 2.7.6 illustrate variation of temperature in the Alger Road pavement and subgrade during a few days in winter. Figure 2.7.7 shows temperature distribution for a few summer days in a 7-inch pavement layer. 59 and moannm s weausu ohspnhodaou ucoaobcm _:.m. Ansonv mafia NH OH m m a N 3N NN 0N N shaman 0H 0H q _ a. H a _ 2 . _ _ \\\III.I'I’I \ \ ..omé , 82 \ \\ \l/ I IIIIIIIIIII \\ \II/ \\ \ ll lllll IIIII I. ”I’ l \\ /' fill] I II, [N [I'll-III, ...... 0 ll """" \ MOI-'0 II I, cm: 1 0H ON on 0.: on ow (ad) eanqaaedmem Depth (inch) Depth (inch) 10- 20 30 #0 50 I V F 60J 60 Asphaltic pavement t \ /‘ Gravel \ \ \ \ Pavement 1 ‘ 1 | O *-‘ 9:00 AM 12:00 AH \ 1 I I I l 1 I 3:00PM 6:00 PM 9:00 PM Figure 2.7.5 Temperature distribution in a 7" asphaltic pavement and a gravel pavement during a winter day (1/13/65) .hsc hound: c wcausu unapsponsop psoaobsm 0.5.N ouswam Ansonv oaaa ON mH OH OH NH OH 0 w a N 3N NN ON mH OH OH NH 777: \\1 II In a I. \x. II sMNoo IIIII x \ (VII-ll", 1111111 I so.N Ill \ 88 -(u-(. // .. lol- /0 l/ \ \ 8 o a , I IOIOI/sls IOIII‘I ‘ |\Jn OH nH ON mm on no O3 n: (04) oanqaaodmom A:O\mm\uv and possum e mcHnsd ouspsnoaaos usosobcm n.n.N onstm AnsonvoaHs om OH OH 3H NH OH O O a N 2N NN ON OH OH 3H NH OH 1 q u u u a 4 u a a u a J H J u a s O m 0 \O O N O 00 O 0‘ (03) eznuexadmel OOH OHH ONH OMH fl mason 3N A JOOH 63 2.8 Monthly Temperature Distribution Monthly temperature distribution graphs were plotted based on hourly data collected from each depth (1/2-, 2—, 4-, 12-, 19-, 30-, and 42-inch) in the airport pavement and subgrade. Figure 2.8.1 shows the temperature distri- bution of wearing, binder, and mix base course of a 19- inch asphalt concrete pavement and soils beneath the pave- ment during the summer of 1969. Minimum daily temperature of the subgrade beneath the l9-inch full-depth asphaltic pavement and a gravel test section during February, 1970 are shown in Figure 2.8.2. Average daily temperatures of air and pavement surface are given in Figure 2.10.1. Discontinuous curves of temperature distributions, are resultant of incomplete temperature data during these periods. During the summer,variations of temperature in top layers of asphalt concrete were of primary interest, while during the winter months the subgrade temperature was more relevant. Duration of a given temperature as well as maximum, minimum, and average temperature was useful in a long-term study of pavement performance. Trott (46) intro- duced an apparatus for recording the duration of various temperatures in roads. Determination of the percentage of time that a depth remained at an individual temperature during a specific interval was deemed important. 64 :33 2.2. mOmH mo umsmoa 0cm HHSO may mcHusO wusumuomEmu ucmew>mm UHHHmnmmm nummOIHHsm 0 4... up“ O 6" mm P u ur- c—Ip J- IEI'II 030809‘08 “d4 oosuusm IOHom son HHounse ens ocosobsc seasons oosuuoch 88.28 :28 .2 oceans» IOHom u: _I||Inl oomunam :OHom 3N ousunuoqaos ooahusm aesms< -rH —iE- mH ul-m H.m.~ musmHm on OO on OOH OHH ONH 22. onH (cg) oanqsaedme; 65 Aoan unwound OHuHmsmmm I my coHuowm Hm>mum m summo HHom zmH on» condemn mmusumummwwm wmwmwmmm ~.m.~ mousse mm «N 5N nN nN H Aehsev osHa 1) . . . N 0H NH mH nH H 1 . j) 1 T .mH HO N.) M . p O > - \. OH x” \/ \J 1 _ 7 . x _ _ a _ .z \ 2 _. _ a _ . x a s \\3 _ _ 1 H . l , l ,\ . _ MH . . , . _ . a _ l I a . _ l _ 1 ” O . _ _ . . . l . _ . H l _ . _ x . _ _ . _ H . _ . 1 fl 1 . \ . _ _ 1 O. . ._ _ \ .— _ _ ON a . . _ a . _ . _ . _ . . a _ l . l _ _ _ N l . \ . . ” x A _ . fl _ _ H A _ l _ x w . u s .. __ .. a A _ __ ... MN . __ _. _ __ l . . _ _ . _ fl . _ e40 —— . ..r. (OJ) oanqsaeduom 66 The percentage of time may be determined for a period of one day, one month, or one year. Due to incom- plete and discontinuous annual data, only daily and monthly duration of temperature are possible to calculate. A 10°F increment of temperature was utilized as a standard for temperature retention in the pavement. This made best use of the available field data. The result of the above calculations, which are the duration of tempera— ture levels at various depths of l9-inch asphalt concrete during the summer of 1969 and the winter of 1970, are tabulated in Tables 2.8.1 and 2.8.2. Temperatures ranging between 130°F and 140°F at the pavement surface occurred only 1 per cent of the time dur- ing the summer months. During these periods, temperatures at 4 inches depth beneath the pavement surface did not exceed 107°F and at a depth of 12 inches beneath the pave- ment surface temperatures reached a maximum of 90°F. Similar results were obtained by B. K. Kallas (29) in studying variation of temperature in an asphalt concrete pavement in College Park, Maryland. The duration temperature data given in Table 2.8.1 and the results of previous field studies, support the commonly accepted standard of 140°F for paving mixture stability, and for asphalt consistency testing. The data in Table 2.8.1 indicates that laboratory testing tempera- tures below 110°F should be considered in designing paving mixtures for the lower courses of the pavement. 67 .. .. - n u .. we Nu . .. .3 u .. u u .. .. om om . .. .NH .. .. .. .. R m: 9 N .. .. .3 u .. .. mm mm 3 m m .. . ..~ .. H m m: on m N m - .. .m .. u - u . mm 3 m H .. .3... 3H mnH oNH oHH 00H me am as 3 mm ...... cm. .W. ...... ...“. a. m... a. m... m... cassava as: buy ousunuocaou on» :ngs wsHusv OoHpoa on» yo accouom .mOmH mo HmEESm on» mcHudp ucme>mm mumuocoo uHonmmm nocHImH c How msummp msoHum> um mHm>oH ousumuomSwu ummEHm3 mo soHuoHso H.m.N mHnme 68 - . mm me u u . .Ns - . OH so - - u .on u u . mm mm - - .aH - - - Om 5H . - .N n - oH mH o: mm OH .m - m n AH mm mm um 823 3 mm m: mm mm 3 m a. a. m... ... a. u. N Sassoon as: Amy ouspsnonsou on» :oHs: mcHnsu OoHuon 029 no accouom .onanmH mo umucH3 may mcHHsO ucoam>mm mumuocoo uHmnmmm socHumH o no mnummo msoHum> um mHo>oH ousumuomsmu umwOHoo mo QOHumnso N.m.N oHnma 69 Duration of temperature, as well as temperature variation in wearing course, binder course, and hot mix asphalt base should be considered in the selection of dynamic modulus, E*, and phase lag 0 in design procedures. The dynamic modulus, E*, and phase lag, o, for temperatures of 40°, 70°, and 100°F and different loading cycles are given in Table 1.6.1. 2.9 The Effect of Snow A comparison between the temperature distribution at various depths of the gravel test section and the actual pavement at Bishop Airport pointed out that during the winter months the temperature at various depths of the gravel section is generally higher than those of the actual pavement. The opposite was observed from the Alger Road data. Figures 2.7.3 and 2.7.6 show the temperature distribution at various depths of the Bishop Airport test section and the Alger Road pavement respectively. The observation of temperature distribution during the winter pointed out that even when the air temperature dropped below 10°F, the temperature at various depths of the gravel section did not fall below 30°F. The higher temperature beneath the gravel sections was mainly due to the insulation effect of the covering layer of snow and ice that had not been removed from the surface of the gravel section. 70 According to Beskaw (9), the effect of the layer of snow covering the surface may be taken as an equivalent increase in the thickness (Sf) of the frozen soil. k _ .2 Sf-ssk s where S5 = thickness of snow, kf and k8 = thermal con- ductivity of frozen soil and snow respectively. Since the thermal conductivity of the snow, ks, is about 10 to 15 per cent of that of ordinary pavement material (Beskaw [9]), the thermal resistance of snow is equivalent to a soil layer approximately seven to ten times thicker than the thickness of the snow cover. The result is that the insulating effect of this equi- valent layer has protected the lower layers of the gravel section from freezing. 2.10 Freezing Indices The depth of frost penetration beneath the pavement is influenced by many factors, such as: thermal proper- ties of pavement material, pavement structure, and sur- face conditions. The depth to which freezing penetrates is principally dependent upon the amount of water available within the subgrade soils and the magnitude and duration of below freezing temperature. The freezing index is used as a measure of combined duration and magnitude of below freezing temperature. Freezing index is defined as the number of degree-days between the highest and lowest 71 point on a curve of cumulative degree-days versus time for one freezing season (16). The degree-days for any one day equals the differ- ence between the average daily air temperature and 32°F (Tavg - 32). Degree-days are negative when the average daily temperature is below 32°F and positive when above (see Figure 2.10.1). These are referred to as freezing degree-days and thawing degree-days, respectively. The freezing index may also be calculated by deter- mining the area between the 32°F line and the curve of the average monthly temperature and time (16). The air freez— ing index may be calculated on a daily or a monthly basis from either the monthly or annual summary of the climato- logical data published by the National Weather Service. Monthly climatological data includes maximum, minimum, and average daily temperature while the annual summary includes the average monthly air temperature. Freezing index isograms are available for the continental United States. However, it is preferable to compute the local freezing index from the recorded daily temperature data. This is because freezing conditions may vary widely over short distances due to the differences in elevations, topographic position, bodies of water or other sources of heat. This fact has been observed from the calculated freezing index for the Midland and Tri-City airports, Michigan, which are located less than fifty 72 Aoan mumsunomv wasp mmummp m>HuMHsEso cam musumumofiwu MHHMO mmmuo>¢ H.OH.N ousmHm Ahsvv oaHB #:8888888882211 TL .Oom mu surnames—on. oomuusm ufiaobom llllll n .83 m. announce—.09 .34 m -oom a m a ..OON a ... lllllll\\|\\ II..\ I/ (1 I, 1 OOH m // I \\|lll|lll 8 o 7 .3 a / \|/ H 1 I x / O / om .m \l/ \\/ I..\ / I O \ / \ / \u / 1 I L: . in). a) a, )1 - on m. z .I(\ / .\\\ n / \‘\ / x ( u < / i . 3 f u z 7.. .3 73 miles from the study site location. A difference of 360 degree-days has been observed between the freezing indices of these two locations, probably due to the influence of the Tittabawassee River which moderates the temperature of the Midland area (41). Figure 2.10.2 shows the cumulative degree-day curves obtained from the recorded temperature data at Bishop Airport during the winter of 1969-1970. The average daily temperature, based on hourly temperature, would be slightly more accurate, but such precision is not usually warranted. The air freezing index, as well as the surface freezing index, are given in Figure 2.10.2. The air freezing indices obtained from the annual climatological data since 1950 at Flint, Michigan, are given in Table 2.2.1. An average air freezing index of the three coldest winters since 1950 must be used for design purposes (41). 2.11 Correlation Factor n The existing method of calculating pavement tempera- ture and predicting the depth of frost penetration requires a correlation between the air and surface indices. The ratio of the surface index to the air index is designed as the "n-factor" for the purpose of correlation (38). Most designers still employ the empirical correlation factor, n, to convert the air freezing index to the surface freezing index. #00 300 200 100 0 -100 0) >9 “5 e -200 (D 0) H g~-3oo Q Q) _g -400 JJ “5 '3 500 5 U -600 -700 -800 -900 -1000 74 / “’fi l /.A \\ . o 'd c H a surface 5 curve N oxn new hxn a. o o t. g s 03 Air Freezing Index = 940 I l. Period in which Freezing Conditions ‘J occur J: *1 "1 Spring frost melting ‘35 i1t5i3‘r H Nov.69 Dcc.69 Jan.70 Feb.70 Mar.70 Apr.70 Figure 2.10.2 Cumulative degree-days above and below 32°F at BishOp Airport, Flint, Michigan 75 A correlation factor, n, of 0.51 is obtained from the freezing indices shown in Figure 2.10.2. An accurate value of n-factor requires concurrent observation of air and surface temperature throughout a number of complete freezing and thawing seasons. However, the temperature data at Bishop Airport is recorded discontinuously during one freezing period. Consequently, the obtained value of the correlation factor is questionable. 2.12. Air and Surface Temperature As indicated in the literature review, a number of mathematical relationships between the air and pavement surface temperature have been developed. However, there has not been any significant progress in the application of these methods to the determination of cumulative pave- ment surface cold quantities from basic meteorological data. Usually air temperature is available, but surface temperature is not. In order to obtain a correlation between the air temperature and surface temperature, the combined effects of radiative, convective, and conductive exchange at the surface must be considered. The difference between the air and surface temperature at any specific time is influenced by the latitude, cloud cover, time of year, time of day, surface characteristic, and finally, the thermal properties of pavement materials and the heat transfer coefficient. Surface temperature is also 76 influenced by ambient conditions such as wind velocity, temperature level, and precipitation (45). Realistic values of the pavement thermal property are necessary in order to obtain a correlation between air and surface temperature. A major portion of this thesis is devoted to the estimation of thermal property of paving material with particular regard to asphaltic concrete, in recognition of this need. A method of calcu- lating the coefficient of air-heat transfer from the thermal conductivity of pavement is given in Chapter V. CHAPTER III EXPERIMENTATION 3.1 Purpose of the Experiment The purpose for obtaining experimental data is to provide a means of comparison of thermal properties ob- tained from available field pavement studies to those obtained by simulated laboratory model tests under con- trolled conditions. For determining thermal conductivity and specific heat of asphalt concrete, the knowledge of the combination of temperature and heat flux histories is necessary. In the field pavement data, no information concerning the heat flux was available. Hence, a simulated model was prepared in the laboratory in order to obtain required data for evaluation of both thermal properties. The experimentation consisted of: (l) measuring temperature as a function of time and at several locations of the specimen, and (2) calculating heat flux from measured temperatures of the calorimeter. 77 78 oensoo omen o.nsH o.s N.n 8 NH mN an m: Nu me am Na ooH pHasaua quuuom onnsco Houch m.meH n.: 3.0 H m aH AN mm on a: an no ooH enouocoo pHmzam< ounsoo wcHusos m.osH o.O 3.: A HN an Nn NO Hm oOH oamuocao pHanau< »N\6nq announce ooN* ooH* one on* OHH m* a .chxn .cHNxH .cHe\m .:H H ouunoo owwmuum aHusau< oz mm at each depth 4' 3' Flat surface ! l‘8'4 ( thermocouples I ’ *" 1: 9 . ’rfi 1.02 I 1'25'1‘ \\ cylindrical face 1.5" l . \ [Bur thermocouples Figure 3.2.1 Location of thermocouples in the asphaltic sample 81 thermocouples were connected to produce an average of these; hence, eight temperature histories were measured, two at each of the four depths. 3.3 Facilities and Equipment The facilities used in this experiment may be grouped as following: 1. Hardware and data acquisition equipment. 2. Data analysis facilities. Hardware and data acquisition facilities consist of the heat transfer frame, hydraulic system, controls, signal amplifiers, and an IBM 1800 to digitize the data. The heat transfer frame retains the specimen and a standard plate calorimeter in two separately insulated containers. In this experiment, sixteen thermocouples at four levels and at varied distances from the surface of the specimen were utilized. One thermocouple near the heated surface of the calorimeter was utilized to measure the surface temperature during the test. Temperatures were obtained in terms of voltage by the quadratic equation (6) T = AiV2 + BiV + Ci' where i = l to 10, and T is temperature in degrees Farenheit, and V is voltage in volt. Coefficients Ai' B. I and C., 1 J. are found by a least-square fit. The temperature rises in the experiments produced voltage changes on the order of several millivolts. 82 The IBM 1800 was used to digitize the analog signals produced by the thermocouples. It can take analog inputs between t 10 volts; therefore, the amplifiers were utili- zed to raise the millivolt thermocouple signals to the order of several volts (for the details, see reference [6]). The temperature data were recorded by the IBM 1800. There are two programs for printing and punching the data. One is the off-line which stores the data until the test is completed, and the other is a non-line program which prints and punches during the test. In this experiment, the off-line program was employed. For analyzing data, a high speed digital computer, such as the CDC 3600 or 6500 is used. The properties were calculated by the "Property Program" which will be described in the next chapter. The facilities utilized in this experiment are capable of simultaneously measuring thermal conductivity, specific heat, thermal diffusivity, and a number of properties not used in this study. The method may be used to measure thermal properties of metals, porous materials, and most of the materials which change in composition when heated. Organic material and soils which are a multiphase-system composed of solids, liquids, and air are also included in this group. 83 3.4 Procedure For one-dimensional heat flow, the specimen should be insulated on all surfaces except the interface with the calorimeter. The interface of the specimen and the calori- meter should be as smooth as possible and should provide intimate thermal contact. The opposite surfaces acquire an insultation condition by attaching the calorimeter and the specimen to insulating solids. For high thermal con— ducting materials, it was easy to approximate an insul- ation condition by attaching the specimen to a low con- ductivity material, while for very low thermal conductivity materials, such as asphaltic concrete, the specimen in effect, insulates itself. The specimen was made thick enough (1 1/2 inches in this case) so that the temperature does not rise signifi- cantly at the "insultated" surface during the test; the specimen is then said to be effectively semi-infinite in thickness. The cylindrical surfaces of the calorimeter and specimen were insulated using fiberglass. The required duration of the test, as well as side heat losses, can be reduced by decreasing the thickness of the specimen and the calorimeter. The thickness of the calorimeter was l-inch and the asphalt sample had a thickness of 1.5 inches. Both the specimen and the calorimeter were 3 inches in diameter. There are a variety of methods which may be utilized to heat the 84 calorimeter, including an electric heater which was used in this test. The sample of asphaltic concrete was de- formed at a temperature beyond 150°F. Hence, the 150°F temperature was utilized as the limit in heating the specimen. In one test, the temperature of the sample was decreased to -30°F by employing dry ice (C02). The copper calorimeter with accurately known specific heat and high thermal conductivity, was heated to a uniform temperature of about 120°F and then brought suddenly into contact with the specimen which was at another uniform initial tempera— ture (-30°F in one test). A series of tests at several initial temperature levels of the specimen were performed. The initial tem- peratures represented various seasonal temperatures with the laboratory tests to simulate the natural conditions. In the tests described above, each successive test was started with a higher uniform initial temperature and the specimen was heated. In another series of tests, the specimen was at a relatively high initial temperature (about 120°F) and the calorimeter cooled. The specimen and the initial temperature of the specimen decreased from test to test. In order to evaluate the effect of moisture on thermal properties of asphaltic concrete, tests were performed on a sample before and after it was soaked in water and the obtained temperature data were compared. Before initiating the 85 series of tests, the thermocouples were calibrated. (For the details and further information, see reference 6.) 3.5 Heat Flux Calculation One of the main objectives of the experimentation was to compensate for the lack of information on heat flux in field data. In order to obtain the values of thermal conductivityIK, and specific heat,Cp, the heat flux as well as temperature data must be known (8). (Also see Figure 6.4.1 and related heat transfer equation.) Experimentally measured temperature at the surface of the calorimeter may be employed to calculate the heat flux at the interface of the calorimeter and the specimen. There is usually a significant resistance to heat flow at the interface due to an imperfect contact. Regardless of the magnitude of the resistance, the same heat flow leav- ing the calorimeter at the common interface must enter the specimen. The heat flux q(t) may be calculated by employing the first law of thermodynamics for the calorimeter to obtain: _ dT th) — pCp L dt (3.5.1) where %%-is the gradient of temperature/time curve over time interval dt at the surface and pCp and L are the volumic specific heat and thickness of standard,respec- tively. 86 This equation is valid if: 1. There is a negligible temperature difference across the calorimeter at any time, and 2. All surfaces are insulated except the common interface. Experimentally measured temperatures at the surface of the standard are given as the function of time in Figure 3.5.1. Curves of temperature versus time, similar to the one shown in Figure 3.5.1 are plotted for each test. The most sensitive part of the evaluation process is deter- mining the temperature gradient %%. Since the temperature gradient is not constant over the entire time period, it was determined by taking the differences of temperature at different time intervals. The heat flux has been calculated by utilizing the ob- tained temperature gradient for different time intervals in Equation 3.5.1. The calculated heat flux will be used as a boundary condition in the mathematical model, used while estimating the thermal conductivity and specific heat of asphalt concrete. 3.6 Results The experimentally measured temperature at various depths of a sample of asphaltic concrete as the function of time are shown in Figure 3.5.1. Two sets of tempera- tures were measured for each level of the specimen. It 87 Temperature (F’) 8 2. c... on 3 on ON 6H o ONH 4 a d a J u 1 .\ .mNHun IIIIIIII \I ......IIIIII IIIII II IIIIII IIMHIMI IIIIWIIII‘IHHU. (14s. In» nNH 1.) I. .II. ::.....I.|.\I. .II.\.\\.\ .\.\.\. III. \\\ .o.Huu \.\.\ \.\. \ ONH V ‘.\. \\ I .\\. \ \.\ \ (I. .\ \ . o \\ II.I!\\ \ . \ '\ l OMH xx \ \\ \\\ \\ I 03H l \\\\\ 1 .IIIIIIII .naop oonuHSm HouoaHHOHmm. 3H . (III--- .86.... Amuse mHmsoooanosu pcoomm was umHHm ou Hommu mocHH common use OHHOmO OHmEpm OHHHmnmmm may no mnpmmp msoHHm> one moomuom HouoEHHOHso on» us mnsumuwmfimu Omusmmma mHHmucmEHHmmxm H.m.m onsmHm Accooomv oaHs Om OO ON om om Temperature (F') 88 was observed that error in measuring temperatures at various levels was mainly a systematic error rather than an independent random variable. This may be verified by comparing each two columns of temperature data or by the curves of temperature versus time, similar to the one shown in Figure 3.5.1. Systematic error in measuring temperature was due mainly to the error in measuring the location of the thermocouples, bad contact between the thermocouples and the specimen, as well as error due to the presence of the thermocouples themselves. The observed systematic error in measuring tempera- ture could also have been due to the inhomogeneity of the asphaltic concrete. In calculating thermal properties by the nonlinear estimation method from these transient temperature data, the correction for the biased error caused by the systematic errors should be made if possible. The aver— age measured temperature as well as a separate set of data at various depths of the specimen was utilized to calculate thermal conductivity and volumetric specific heat of the asphaltic concrete specimens. The obtained values of thermal properties are given in Tables 3.6.1, 3.6.2, and 5.8.4. The results indicated that the average of the thermal properties obtained by using each set of data separately was equal to the result obtained when 89 Table 3.6.1 Calculated thermal diffusivity of asphaltic concrete from laboratory data (nonlinear estimation method) Sample Temp Boot— Thermal Case Condition (F °) nean- Diffusivity Ha: Inn Square (ftz/hr) 1.1 dry 106 118 1.76 0.000 1.2 ' 1M6 118 0.79 0.0“1 1.3 ' 146 118 0.93 0.002 1.“ ' 106 118 1.17 0.0h1 2 ” 130 97 1.72 0.045 3 ' 12“ 80 2.82 0.0“3 u ' 117 75 2.99 0.039 5.1 ' 80 1.33 0.0“? 5.2 not 75 2.91 0.061 503 dry 85 -1 3.12 0.055 6.1 dry 26 -27 2.57 0.052 6.2 net 25 -25 0.91 0.067 6.3 dry 32 -15 0.78 0.058 90 Hm.on Oe.H mo.o OH- NO use n.O mH.Hm Oo.N Nk.H ON- ON no: N.O Ho.on oO.H OO.H AN- ON sue H.O oo.mN NO.H O8.N H- mm sue m.O NO.oa HH.N Nm.N N me you N.O Om.mN os.H Oo.H m cm . H.O OH.aN O3.H Nk.N we NHH . a Oo.aN Om.H mm.n om HNH . m Nn.ON mN.H NH.N um onH . N om.ON NN.H No.n OHH OHH . s.H OO.NN ON.H Om.n OHH OHH . n.H on.mN ON.H OO.N OHH OHH . N.H On.mN ON.H NO.H OHH OHH Hue H.H ..N\aaso\:»mv Ha.»c hexane. apogee nH: new use: oHaHooam huH>Hposecoo undo: H.NO caHuHeaou ammo oHuooasHob Hosanna. Iaoom neon. oHnflcm Aponume soHumeHumm HMOCHHcocO sump HoucmEHuomxm scum cuouocoo OHUHmnmmm mo upon OHHHoQO can auH>Huoapcoo Hashes» pmustono N.O.m mHnme 91 employing the average of the temperature for each depth (see cases 1.1 through 1.4 in tables). The data obtained on the moist specimen differed from that of the dry ones due to the existence of liquids in the pore space and the fact that thermal conductivity of water is approximately twenty times that of air (11) (see cases 7 and 8 in Tables 3.6.1 and 5.8.4). Thermal diffusivity of the wet samples were approximately 10 per cent higher than the dry samples. Although in this study only a few tests were conducted, their results pointed out the effect of moisture on thermal properties of asphaltic concrete. In order to see if the two sets of measured tempera- ture data agree reasonably with each other, a case was considered as the representative of all the cases, and the analysis of variance was performed for each set of data. Since there are two sets of thermocouples at each depth of the specimen, with a resultant of two average temperatures for each sample level, the sample variance is obtained as (48): where subscript 1 refers to the time and Ti1 and Ti2 are the average temperatures measured by each set of thermocouples at the same depth of the specimen. The 92 calculated values of the variance are given in Table 3.6.3A. The low values of the variance which were a measure of variation or the spread of the measured tempera- ture near the mean indicates a good agreement between the two sets of data. The adequacy of the model which was utilized in calculating thermal properties was confirmed by comparing the calculated average values of variance from the data alone with the values of root-mean-square (RMS) obtained in the process of iteration in estimating thermal proper- ties by the nonlinear estimation method. The calculated values of RMS for the representative case are given in Table 3.6.3B. An accepted procedure for determining the adequacy of the model is to calculate the ratio of the calculated variances and root-mean-squares to determine whether or not the values of RMS are significantly greater than the calculated variance. Average values of 0.94 and 1.16 were obtained for calculated values of variance and the value of RMS for the representative case, respectively. It is observed that the calculated values of variance are in good agreement with the values of root-mean-squares and there is not a significant difference between these values. With this rational, there appears to be no reason to doubt the adequacy of the model. 93 Table 3.6.3-A Analysis of the variance for the repre- sentative case Thermocouple level Calculated Variance Standard Deviation (inch) (P )2 (F ) 0.25 0.70 0.84 0.50 0.70 0.84 1.00 1.35 1.16 1.25 1.00 1.00 Table 3.6.3-B Calculated values of root-mean-square and thermal diffusivity for the representative case Calculated Case Root-mean-square Thermocouples Thermal D ffusivity (ans) ft /hr 101 0093 1930507 OOOuzu 1.2 1.76 2.4.6.8 0.0397 1.3 0.79 (1&2).(3&4). 0.0412 (avg) (5&6).(7&8) 94 The close agreement of the calculated variances and root-mean-squares leads to the conclusion that the model has fully described the relevant physical process and there is a reasonable agreement between the measured and calculated temperatures at various depths of the specimen. Even though the model has been found adequate, as a standard procedure in regression analysis, the residuals are plotted in Figure 3.6.1. Several ways of examining the residuals in order to check the model are given in reference (48). Discussion concerning examination of residuals is beyond the scope of this study. 95 Hommo m>Humuc0monoHO coEHocmm on» no commune on» onmo nocHIH Opp nocHIN\H um HmHmsOHmmHv musumuomE0p Omusmmma Ode OOHMHDono com3umn wocmHmMMHo H.O.m musmHm Aucooouv osHa ONH OHH OOH OG cm on O6 OM Od OM ON OH H H H H H _ H H 1 4 H . 1 - I. .m? HI - I L 1. : sWoOIN ‘ — _. = A scoHlH . gL m.HI O.H+ .N+ (as) 1V7 CHAPTER IV NONLINEAR ESTIMATION METHOD 4.1 Nonlinear Estimation Method of Measuring Thermal Properties The conventional methods of thermal property measurement determine the properties utilizing a simple mathematical model for a simple geometric and simplified experimental conditions. Each of these methods have some restrictions and generally are capable of measuring a single thermal property (i.e., a guarded hot-plate has been used frequently to measure thermal conductivity). The guarded hot-plate method requires the existence of a steady-state and a controlled constant temperature- boundary condition for obtaining accurate values of the thermal conductivity. In estimating thermal properties by classical periodic wave analysis, it is assumed that surface temperature varies about a mean temperature in a manner which approximates a sine wave (28). Recent procedures using the nonlinear estimation method to measure thermal prOperties is not subjected to any of the above mentioned restrictions. The digital computer has made it possible to model a complex physical system accurately. 96 97 The models may describe numerous phenomena which are difficult to separate when designing experiments to measure individual properties. The models in this disser— tation are not extremely complex, but nonlinear estimation must be used. 4.2 Nonlinear Estimation Procedure A mathematical procedure referred to as nonlinear regression, nonlinear least-square, or nonlinear esti— mation, has been known for a century before the high- speed digital computer made it practical to use routinely. Beck (8) developed this method to determine the thermal properties of solids. One of his interests has been to design analytically an optimum experiment for the simul- taneous determination of thermal conductivity and specific heat. The use of this method for determining the value of thermal diffusivity requires minimizing the sum-of- square function n n . . . 2 J _ J i T(Cal)i(a) T(exp)i (4.2.1) with respect to thermal diffusivity, a. The expression T(cal) represents the calculated temperature at position i . x. and t.. TJ is the ex erimental t eratur at 1 J (exp)i p emp e xi and tj. The weighting factor W2 is used to give greater weight for accurate measurement and less for 98 inaccurate ones. However, in most of this work W2 was set equal to a constant. In this procedure, the best value of the thermal diffusivity is the one which minimizes the difference in a least-square sense between theory and experiment for all of the available data. In other words, by varying c, T(cal)i is made to agree in a least-square sense with the 1' (eXp)i discrete times. The summation is over all the thermo- measured temperature T for n thermocouples and m couple measurements except the first that serves as a boundary condition (see Figure 6.4.2). The limits of m depend on the time interval for which a is to be found and on the time step chosen. Minimizing the sum-of-square function F(a) with respect to thermal diffusivity (a) implies differentiation and setting the resultant equal to zero; or n m . . 8T(cal)i =2 2w3 3 _ e T - T - 0 i=1 j=l 1 (cal)i (exp)i 3a (4.2.2) At each iteration step, the calculated temperature is approximated by the Taylor series as follows: 8T(a) cal 2+1 cal fl 3a (c)£ 2 where 99 (Au)2 = ((1),;+1 - (a)£ The derivative is approximated by: 2,—1.- ~ T(0I)£+l - T(0I)£ (4 2 4) 30 2’ ~ (Aa)p gg-is called the "sensitivity coefficient." The T's on the right-hand side of Equation 4.2.4 are calculated using a finite-difference method (see Chapter VI). The initial value of diffusivity do is estimated, and the iterative procedure begins with an estimated value for do. The value of thermal diffusivity (a) at the 2th iteration would be equal to: (0)2 = (OUR-l + (Aa)2_l (4.2.5) Substituting (4.2.3) into (4.2.2) gives: n m . . . 0T 1 > r— [T Tca1(0)£l i=1 j=1 °°(a)g 9*? 2H3” )2 (Aa)£ = (a)2 - (“)1-1 = i=l j=l 30New}. (4.2.6) and Equation (4.2.5) becomes: n m .2 .2 [Texp'- Tcal(a)£] %E = a + “11:1 “1”“ (4.2.7) 22 z 2 i=1 j=1 0{new old n m [ 2 (6') 9.] 100 The iteration procedure continues until all the (a)2 - (01)‘L_1 satisfy some predetermined criterion such as: (a). - (“)1—1 (6)2-1 i 0.001 (4.2.8) With this process,the original thermal diffusivity would be replaced by that determined by Equation 4.2.5. In the analysis of the data, it was found that the thermal diffusivity, a, converged to the correct value without significant oscillation. The number of iterations depended on the estimated value of thermal diffusivity. For a well-designed experiment and with the initial esti- mation of a about 10 per cent in error, only a few iter- ations are usually necessary to satisfy Equation 4.2.8. 4.3 Computer Program The nonlinear estimation method is well adapted for use with the digital computer. A basic computer program for estimation of thermal properties has been developed by J. V. Beck (7). This program, referred to as "Property Program," is on magnetic tape in the Computer Center at Michigan State University and may be called into use as needed. This program has been utilized to calculate thermal properties of asphalt concrete, as well as to predict the temperature at various depths of pavement and underlying soils. A finite difference method is employed 101 for the solution temperature as the boundary condition to the model and calculated the temperature at desired locations in the pavement. The properties are found by making the calculated temperature match the measured temperature in a least-square sense. (See the descrip- tions following Equation 4.2.1.) 4.3.1 Data Input In order to determine the thermal properties by "Property Program," some of the variables or environmental conditions that effect the transfer of thermal energy in asphalt pavement may be used directly as data input. Measured temperature and heat flux are two basic inputs which serve as the boundary conditions for the model. In the following paragraphs, these boundary conditions are briefly described. 4.3.2 Temperature Boundary Conditions In order to determine thermal diffusivity only the measured temperatures are necessary. This is evident from Figure 6.4.1 and the describing equations. The property program uses this temperature as a boundary condition to the model and calculates the properties. Variation of temperature, taken at the surface and at various depths of both a 7-inch and a l9-inch asphalt pavement have been used as the temperature boundary condition to calculate thermal diffusivity of asphalt concrete. In order to get 102 some estimate of the thermal properties at different seasonal temperatures, measured temperatures during winter, spring, fall, and summer have been utilized. These data provide an extensive range of temperature which is neces- sary in determining temperature-dependent thermal proper- ties of the asphalt concrete. Similar temperature data obtained in the laboratory from the sample of asphalt concrete has also been used as the temperature boundary condition to determine the thermal properties under the controlled condition in comparison to the field condition. 4.3.3 Heat Flux Boundary Condition To determine the values of thermal conductivity (K)’ and specific heat (C0)' heat flux and temperature must be known. Here the program uses the heat flux and an insul- ation condition as the boundary conditions to the model and calculates the properties (see Figure 6.4.1). Since there was no information relative to the heat flux in the field data, it was necessary to perform laboratory experiments and employ the results to calcu- late thermal conductivity and specific heat of asphalt concrete. (The details of this experimentation was dis- cussed in Chapter III.) 103 4.3.4 Other Variable Input Other variable input included length of regions, the number of nodes in each region, the number of regions, and the time for given data. The pavement is broken into a number of regions (see Figure 6.4.2) and in turn, each region is divided into equal thicknesses which are equal to An = %. Where L‘is the length of the region and specified by the dis- tance between two consecutive thermocouples and n is the number of subdivisions in the region. Each one of these subdivisions represents a finite control volume which may be denoted as a nodal point at the center of the volume. It is conceptually advantageous to imagine that the nodes are located at the center of the volume. The interface of each two regions is at the location of a thermocouple. The first and the last nodes are used as the bound- ary condition, while the internal nodes are located be- tween the thermocouples to improve the finite difference approximation. 4.4 Output and Results The program outputs the time step, the calculated temperature "CAL TEMP," measured temperature "ETEMP," and their differences denoted as residual "DTETC." As a standard procedure in regression analysis, the tempera- ture differences (residuals) for various positions are plotted versus time in Figure 3.6.1. Printed output also 104 includes the sum of square of residuals "RMS," and sensi- tivity coefficients symbolized by "BDB." The various output quantities for a representative case are plotted in Figures 4.4.1 through 4.4.3. The values of root-mean-square, which is a direct measure of agreement between the model and the experiment, are given in Tables 4.4.1 and 4.4.2. For perfect agree- ment between the calculated and the measured temperature, the value of "RMS," (i.e., root-mean-square), approaches zero. However, "RMS" will never reach zero since there is never perfect agreement between calculated and measured temperatures. The sources of the errors may be ascribed to experimental measurement error, imperfect model and finite difference calculation using the numerical model. The sum-of-square function F(a) is related directly to the root-mean-square. The program seeks out a minimum in the sum-of-square of the residual by using an iterative search procedure. Since the procedure is iterative, the initial estimate of the thermal diffusivity is very im— portant. In many cases, solution of the problem occurs even for poor initial estimates of the parameters. In difficult cases, however, the convergence is strongly dependent on the use of good estimates of the thermal properties. In performing the regression analysis, it is assumed that the error is independent, has a zero mean, a constant variance, and follows the normal distribution. 105 Ammmo m>HumucwmeQmHO cmEHowmm on» no mommusm ecu sonn nocHIH Ocn SUGHIN\H up musumumafimu Omusmmme was OcuMHooHso cmmzumn camHHmmEoo N H.e.v ousmHm Accooomv oaHe ONH OHH OOH OO OO ON OO on OH on ON OH N! 5 q d u u in d d! a u d xO.HnH as use» consume: \. em.Ouu as 960» OOOGHOOHGO an.0hu as use» concede: L L l c: on OO on (03) eanqsasdmom 106 Ammmo o>HumucwmmHmmHO mmmooum coHuwHouH mco OCHHDO wuH>Hmsmme Hcshmzu mo coHumHHm> N.O.e musmHm 3:033 05H. OOH OO OO ON OO on OH OH ON OH 4 p p p p p p p P -o O H0.0 . . H.O N0.0 L . N.O V p H.O V JUN. n I... v 0 m3... no . 3.0 .m.o 107 mmmo w>HumucwmeQmH can How mumcwmlcmmsnuoou mo coHumEEsm Hpsooesv oaHa H.O.H ONOOHH ONH OHH OOH OO am on 00 Oh Oi ON ON OH “I! A q 11 H (H q HI 7 H 4 HI .0 as sooHflH W A Q O..H shOoOIN . LOH O.N (snags) scents-neou-qooa JO uotqsmmns 108 The residuals should exhibit tendencies to confirm these assumptions, or at least, should not exhibit a denial of them. To a limited extent, the residuals appear to be random. The nonlinear estimation method yields smaller errors in the calculated property when the errors are random. For the case of systematic errors, corrections for biased errors may be made. Figure 4.4.2 illustrates the change in thermal diffusivity (a) as the function of time steps during one iteration process for the given case. These values are given in the list of the output. The change in a at the 2th iteration [(Aa)£ = (a)£ - (c)£_l] is an iteration variable for which an optimum value is sought. The value of Ad continuously varies with time during each iteration process. The current estimates of the thermal properties including the number of iterations, are printed at the end of each iteration. When convergence occurs, or if the maximum number of iterations has been exceeded, the final values of the calculated thermal properties are printed. The analysis of field data, as well as laboratory experiments, has been accomplished with as few as two iterations for convergence. On the average, the number of iterations was equal to three. Calculated thermal diffusivity of asphaltic con- crete for various seasons and under different ambient 109 conditions are given in Tables 4.4.1 and 4.4.2. The calculated thermal diffusivity of asphaltic concrete from experimental temperature data are given in Table 3.6.1. The values of thermal conductivity and volumetric specific heat of bituminous concrete calculated from experimental data are given in Table 3.6.2. 110 Table 4.4.1 Calculated thermal diffusivity of asphaltic concrete (entire pavement) from recorded temperature data (nonlinear estimation method) Surface Temp Root Thermal W .2282. ... (:8 2.2 ”85):?” ha: Min ‘ (ans) 1 7.69 Sunny 127 65 120 2.62 0.038 2 8.69 ' 115 64 21 2.39 0.047 3 6.69 ' 120 60 16 1.42 0.037 4 4.69 ' 112 41 16 1.94 0.051 5 3.69 ' 56 37 70 0.97 0.057 6 1.70 n 34 5 16 1.18 0.058 7 1.70 ' 28 -2 118 1.92 0.054 8 8.69 Cloudy 135 70 16 1.39 0.048 9 4.69 ' 116 68 35 2.08 0.051 10 1.70 * 28 -3 16 1.09 0.051 11 1.70 ' 36 10 118 273 0.055 12 7.69 Rainy 118 73 16 2.10 0.049 13 3.69 ' 33 20 16 2.29 0.058 14 7.65 Sunny 131 68 70 0.97 0.041 (Alger 15 7.65 Road) 123 69 68 1.69 0.046 16 1.65 ' 29 6 96 2.34 0.053 17 1.64 s 40 -5 62 1.88 0.057 111 Table 4.4.2 Calculated thermal diffusivity of wearing and binder courses from recorded temperature data (nonlinear estimation method) Ambient Surface Temp t Root- Thermal Case Date Condition (F°) max Hean- Diffugivity Max Min (hr) Square (ft /hr) 1 7.69 sunny 127 65 120 2.43 0.049 2 8.69 ' 115 64 21 3.00 0.046 3 6.69 ' 120 60 16 1.13 0.040 4 4.69 ' 112 41 16 0.63 0.058 5 3669 ' 56 37 70 0.95 0.053 6 1.70 ” 34 5 16 1.30 0.050 7 1.70 ' 28 -2 118 2.33 0.057 8 8.69 cloudy 135 70 16 1.08 0.051 9 4.69 ' 116 68 35 0.93 0.054 10 1.70 ' 28 -3 16 0.96 0.059 11 1.70 ' 36 10 118 3.10 0.065 12 7.69 rainy 118 73 16 1.02 0.064 13 3.69 ' 33 20 16 0.89 0.060 14 7.65 s 131 68 70 0.68 0.052 (Alger) 15 7.65 ' 123 69 68 1.49 0.053 16 1.65 ' 29 6 96 1.25 0.052 17 1.64 ' 40 -5 62 1.76 0.061 CHAPTER V SIMPLIFIED METHOD 5.1 Simplified Analysis for Estimation of Thermal DiffusivitypUsing the Laplace Transform The traditional approach to property measurement is to create an experimental condition for which system be- havior is approximated in the laboratory. The obtained thermal properties under the controlled conditions of the laboratory may not be capable of predicting accurate thermal behavior and temperature distribution in the pave- ment under the natural conditions. Estimated thermal properties from the periodic heat analysis was based on the assumption that the surface temperature varies according to a sine wave. Since the daily heating-cycle is normally other than the simple sine function, higher harmonics and the Fourier analysis have been introduced for the daily temperature variation (14). The mathematical treatment for approximating such a natural boundary condition is usually time consuming and impractical. 112 113 The results of the analysis undertaken by O'Blenis (40) and Heley (24) dealing with the periodic heat method concluded that either the periodic assumption of the sur- face temperature did not satisfy the natural conditions or that the method itself was in error. The estimation of thermal properties by the non- linear estimation method is an iterative search procedure which requires a considerable amount of computer time. In comparison with the nonlinear estimation method, the simplified method described herein saves on computer time, since it is not an iterative procedure. For these reasons, a method of estimating thermal properties from natural recorded surface temperature is needed. The method described in this chapter, eliminates the assumption of sinusoidal surface temperature. This method may be utilized to estimate thermal diffusivity of pavement materials subjected to any surface heat pulse. 5.2 Simplified Laplace Transform Method Consider a semi-finite body to be initially at a constant uniform temperature and at the surface (x = 0) the temperature varies with time in an arbitrary but known manner: T(o,t) = To(t) (5.2.1) The partial differential equation describing one- dimensional heat conduction in such a system, assuming 114 constant properties and no internal heat generation may be written: 2 3T(x,t) _ 3 T(x,t) where T is temperature, t is time, and a is the thermal diffusivity of the conducting medium which is equal to K/pCp. K is the thermal conductivity, 0 is density, and CD is specific heat. The initial temperature is known and is T(x,o) = T (5.2.3) initial and the temperature x = w may be considered to be T(”,t) but with the assumption of uniform initial temperature, we may write: T(w,t) = Tinitial (5.2.4) Let T'(x,t) be defined as: I — _ T (x,t) - T(x,t) Tinitial (5.2.5) then the mathematical problem may be written as: 8T'(x t) 32T'(x t) T= O. 2' (5.2.6) 3x T'(o,t) = [T(o,t) - T ] = T' (5.2.7a) initial 0 115 T'(x,o) [T(x,0) - T ] = 0 (5.2.7b) initial T'(m,t) [T(w,t) - T ] = O (5.2.7C) initial Taking the Laplace transform of Equation 5.2.6, gives: 2 I I «Ilene T “2"“ =.{[E—3(-’§LE-)-] (5.2.8) 8x or 33(T'(x t) c 2' = sIT'(x,t) (5.2.9) 3x Similarly, by taking the Laplace transform of Equation 5.2.7, we get: 2((T'(o,t)) .(vr'o) (5.2.10) ll 0 ..((T(oo,t)) (5.2.11) Solving Equation 5.2.9 gives: 0((11' (x)) = Cle + C2e (5.2.12) To find C and C in Equation 5.2.12, we know that as x 1 2 approaches infinity,.("(T'(x)) approaches zero rather than infinity, therefore, C2 = 0. From Equation 5.2.10, Ciis evaluated and equation 5.2.12 becomes: S _ _ X . X(T'(x)) =«((T'o)e/; (5.2.13) 116 Solving 5.2.13 for a gives: I 1n[ T (X) = -/§x (5.2.14) .11T O) a and 2 a = 5" (5.2.15) 2 T'(x) 1. [KL-H .> J x where s is the Laplace transform parameter, ..((T(t)) = f e’St T(t)dt (5.2.16) 5 has the units of reciprocal time, and it may have any real value between zero and infinity and may also be com- plex. We shall choose a number of real values of s and obtain a value of a for each one. An averaging scheme is needed to determine the best value of a in some sense. Let us use weight least squares: 2 n s .(T'.) n 2 213 3(a) = Z X W.. a - s. x.//{n _—T—Tl— (5.2.17) j=1 3:51 13‘: i 3 xi T 0)] where a is the actual value of thermal diffusivity and the second term in the brackets is the estimated value of thermal diffusivity defined by Equation 5.2.15. The subscripts of i and j relate to the s parameter and depth x respectively. 117 The best value of a in the least-squares sense is the one for which the sum of square function 5(a) for n values of s and m discrete depths is minimized with respect to a. In other words, by varying the values of a the estimated value of a is made to agree in a least- squares sense with the actual thermal diffusivity. Mathematically minimizing 3(a) with respect to a implies differentiation and setting the resultant equal to zero; we then obtain: n s m s gfl¢”,) a 21 2n wi' = 2 2n Six'z Wi' 1n2 _TTT17 j=1 s=si J j=1 s=si 3 3 08’ 0 (5.2.18) or the estimated value of a, designated a, is: m s T'.) n 2 Z‘XK Z Z 5.x. W.. 1n —T—TJ—- j=1 s=s. l J l] ‘K,T O) a = 1 n 8 (5.2.19) 2: z“ wi. j=1 s=si 3 As noted in the nonlinear estimation method, the process of making actual and estimated thermal diffusivity agree in the least-square sense, requires a number of iterations. There are no iterations involved in the above procedure because Equation 5.2.15 is linear in a. The criteria for optimum experiments, as well as the limitations in the Laplace transform, leads to some 118 conclusions regarding the best value of s, which will be discussed in the next section. Assuming equal experimental conditions, equal un- certainties in measuring the temperature and uncorrelated errors, the weighting factor, Wij,may be given the value of one. These assumptions are not always there, but are used herein to simplify the choice of Wij' 5.3 Laplace Transform Criteria The Laplace transform of function T'(x,t) is given by: (X) ot(T'(x,t)) = f e-St[T'x(t)]dt (5.3.1) 0 where T'(x,t) is given by: T'(x't’ = T(X't’ ' Tinitial Geometrically, the value of this integral repre- sents the area bounded by the curve e-StlT'(x,t)], the t axis and ordinates at t = 0 and t = w (see Figure 5.3.1C). The values of T'(x,t) are selected from the hourly tempera- ture distribution curves such as shown in Figure 2.7.1. St is plotted versus (st) in Figure 5.3.1A. The function e- -st . . . Curves of e versus t for various real, p031t1ve values of s are represented in Figure 5.3.lB. 119 1.01 E 005- 8-813 .9 m I O 0.0 I o 1' 2' 3' n 5' at (A) 1.0 E 4; 005‘ 3:0.05 '0 J‘ 5:0.10 \\Q\\0,\O ’3 S=O.¢ o ' ' T 10 20 50 no 50 Time (hour) (3) IOJ ,‘ Time (hour) 5; o 10 29 yg____;9 up ¥f__ 50 .9 - “'"V m I 0 -10_ (C) Figure 5.3.1 Laplace transform criteria 120 The calculated values of e—StlT'(x,t)] for two values of 5 equal to 0.20 and 0.05 are shown by curves A and B in Figure 5.3.1C. It may be observed that the magnitude of e-St as st well as e_ [T'(x,t)] decreases with increasing the value of st, and for st 3 4.6, the effect of T'(x,t) in Equation 5.2.19 is insignificant. The above criterion may be used in estimating the parameter 5. Therefore, real 3 values should be chosen such that: 4.6 S >’ (50302) 1 t max Complex values of "s" have not been investigated. Let tmax = NAt, where N is equal to the number of time steps and At is the equally spaced time interval between the measurements. As tm becomes larger, a ax smaller value of 51 should be chosen. For a period of twenty-four hours, tmax = 24 hours and si = 1/6 per hour. Possible si values might then be, si = l/6,l/3,l,2,3,... . However, the more temperature data effectively utilized in Equation 5.2.19, that is for the smaller si values, the more accurately thermal diffusivity would be obtained. Equation 5.3.1 represents the area bounded by any curves such as A and B and t axis, Figure 5.3.lC, which are employed in estimating the thermal diffusivity from Equation 5.2.19. The calculated value of a corresponding 121 to curve B usually would be more accurate than that of curve A because more temperature data is used for curve A. As more equal time intervals are used in the pro- gram, the smaller the value of 3 might be. In order to obtain an accurate value for a, the number of time steps should exceed some minimum number. Since tmax = NAt, then the combined values of N and At define a time inter- val for which a is to be found. The optimum period of the experiment as well as the number of time steps will be discussed in the next section. It is assumed that the value of e-St after one time step drops no smaller than about 0.95. Hence, for e-S(At) = 0.95 this leads to another limit for the parameter s, such as: s(At) : 0.051 (5.3.3) hence: 0.051 _ 0.051N _ 0.051N SET-W-——t (53-4) max If "s" is made equal to 4.6/tmax and N z 100 (and thus t = tmax/IOO) both conditions on "s" (Equations 5.3.2 and 5.3.4) are simultaneously made the equalities rather than inequalities. (These conditions have proven to be good ones.) 122 5.4 The Optimum Experiment To determine the optimum experiment, a criterion must be developed to obtain the Optimum value of x, i.e., the location of a thermocouple, and the optimum value of the Laplace transform parameter s. The optimum location of x and the parameter s are found when the Laplace transform of an interior temperature contains the maximum information regarding a. This occurs when the quantity 0(x,a,s) defined by: 00 — _ -st _ 0(x,a,s) — of e [T(x,t,a) Tinitia11dt (5.4.1) is most sensitive to changes in a. Now from Equation 5.2.13 fl - — x O. 0(x,a,s) = 0(0,s)e (5.4.2) where 6(x,a,s) =X’[T'(x,a,t)) (5.4.3) €(o,s) = .(IT'(o,t)) (5.4.4) Note that the temperature at x = 0 is a prescribed function and may have arbitrary time-dependence which is completely independent of the property a. Given an interior location x and some arbitrary surface temperature history, the temperature at x is a function of a. Both §|x and §|x=0 are dependents upon 5. 123 We can find how sensitive 0(x,a,s) is with respect to a by differentiating Equation 5.4.1 with respect to a or 80(x,a,s) _ — //§ //E x a 3d - 0(o,s)[l/2 a xe a (5.4.5) where SE/Ba is called the "sensitivity coefficient." The larger Equation 5.4.5 is in magnitude for a given surface temperature history with respect to x and s, the more accurately a may be estimated. 5.5 Optimum Location (x) The optimum thickness and location of thermocouples may be investigated by differentiating Equation 5.4.5 with respect to x and setting the resultant equal to zero. 32‘ — 1 E '//§ X 1 s E '//§’x “3755:9‘0’2/39 +2/;X[va]e '0 (5:5.1) or //§.xopt = 1 (5.5.2a) 0r — 2 xopt — //; (5.5.2b) 1.24 5.6 thimum "s" To determine the optimum value of s, we differenti- ate Equation 5.4.5 with respect to 5 while using the identity given by Equation 5.5.2a, or azé-(XIO-IS) l = — e 383a xOpt 2 38 (5.6.1) Equation 5.6.1 requires the knowledge of surface tempera- ture B(o,t). Two types of time variable surface tempera- tures will be discussed. Consider first a step change in surface temperature or B(o,t) = 00 (5.6.2) then - m —st -st 1 m 0o 0(o,s) = f e 00 dt = o 00(- E) = 1; (5.6.3) o o For this case Equation 5.6.1 gives 3251:: a s) e a Opt, ' = i e'1 —°— (5 6 4) 353d 2 ~32 ' ’ Equation 5.6.4 approaches zero as s+W. However, s=co corresponds to an infinitesimal duration of the experiment. 125 Substitution of Equation 5.6.3 into Equation 5.4.5 indicates that the values of 0%; are positive, since the values of 3 should reside in [o,+w). The values of the sensitivity coefficient given by Equation 5.4.5 approach zero as 8*”. Thus as 5+“, the sensitivity coefficient is not a maximum. Equation 5.4.5 indicates that the maximum value of the sensitivity coefficient is obtained as s+o, for 0(0,s) 28:?- of the experiment. We now observe that given the restric- . This corresponds to an infinite duration tion (5.3.2), the optimum value of s is then: _ 4.6 Sopt _ t (5.6.6) max We also observe from Equation 5.5.2b that the optimum location is l Xopt = ? vat (5.6.7) which corresponds to 2 _. atma§//; opt — 4 (5.6.8) Consider next a periodic surface temperature as given by: 0(o,t) = Oosin(wt) (5.6.9) for which 00 we 5(o,s) = f 0(o,t)e-St dt 0 (5.6.10) 0 s + w 126 Note that we we (25) a o o __ _______ = _ (5.6.11) as[52 + w2 (52 + w2)2 and then Equation 5.6.1 becomes: 1 s 2- ——(-) 3 9(x .a,s) _ 2 w a a§§§ = - e 1 60 g 2 2 (5.6.12) ((5) +1] which is equal to zero only when s/w = 0. It is instructive to note the loss one must take for more realistic values of 5. Consider the "sensitivity coefficient " for the periodic surface temperature as given by: 36(x ,a,s) _ 0L ()Bt = 6 e 11 1 1 8a 0 (5.6.13) Figure 5.6.1 shows values of [(3)2 + lJ-l. Depend- ing upon the "loss" in effectiveness acceptable, a choice of several reasonable values of s/w may be made. A possible value is 0.2, hence, or st = 0.2 wt (5.6.14) max max sun n 0 E6 Combining this result with Equation 5.5.2b, gives x = V a (5.6.15a) 127 HI- 3+ 3 NW. VH HO mQDHM> mumflflflganMU H.w.m QHDOHK 128 or a __§___.= 0,2 (5.6.15b) wx opt then atmax —2—— = 0.2 wt = st (5.6.150) max max x opt Note that with this choice of s, the sensitivity coef- ficient is reduced only 4 per cent compared to that for 5+0. For measurements in soils and pavements which are exposed to the daily ambient temperature _ 2_w__1_ w — 24 hr (5.6.16) and thus 5 = 0.2 1L2 ” 0.0524 l/hr (5.6.17) Associated with this value is the optimum period of experiment, t = i = 76 3 hr (5 6 18) max 3 ' ’ ' or approximately three days. Since tmax = NAt, then the optimum number of time steps with a duration of one hour would be equal to N = 76. 129 The optimum distance corresponding to a = 0.05 ftZ/hr d. atmax /atmax Xopt = 0.2“) = E— = T- 7‘ 1.0 ft. (5.6.19) max is: Therefore, the optimum location of thermocouples in a semi-infinite body such as a pavement, is at x = 1.0 ft. 5.7 Heat Flux Condition Assuming that the heat flux were known at x = 0 as a function of time or 3T 5;, q(t) (5.7.1) x=0 Take the Laplace transform of Equation (5.7.1) to obtain =q—(s) (5.7.2) The time dependence of q(t) need not be stored. All that is needed is E which could be found sequentially, that is, all the detailed structure of q(t) need not be stored but only the current approximation to the integral, ” -st. 3(8) = f e-St q(t)dt 2 2” e 1 q(ti)At (5.7.3) o i 1 From Equation 5.2.13 0(x,s,a) = Cl e (5.7.4) 130 and also - — .. E q(s) — -KC1[ y/g] (5.7.5) or = g—(S) cl K//§ (5.7.6) (1 Note: 1 _ 0C _ 7 K//; _ K —E3 _ /'EEE; (5.7. ) 6(o,s,Kpc ) = —§1§L (5.7.8) 9 JKpCs Then —( ) 2 1 KpC = [3—5—] — (5.7.9) 0 g S 0 where 90 = 6(0.s.Kpcp) (5.7.10) Thus, we have a means of calculating Kpcp and a if there are means of calculating temperature and heat flow in a semi-infinite body. If a, and Kpcp are measured we may obtain K and pcp directly by: 131 K = VKpCpa (5.7.11) and KpC pC = .9 (5.7.12) 0 a 5.8 Heat Transfer Coefficient The value of the heat transfer coefficient, hc’ should be known in order to evaluate the quantity of heat flow between the air and the pavement surface. The value of hc is required in the analysis of the air and pavement surface temperature. The pavement surface temperature may be determined from climatological data if a realistic value of the heat transfer coefficient, hc' is available. Scott (44) developed a procedure for estimating the heat transfer coefficient, hc' between the air and ground surface. Scott‘s method is based on the ambient condi- tions such as wind velocity, atmospheric stability, and ground surface roughness. A lot of uncertainty and errors are involved in obtaining hC from this method, since it depends on the ambient conditions which are random vari— ables. The heat transfer coefficient is also obtained by a "back computing" process to Equation 1.2.3. This pro- cedure is also tedious and impractical. For these reasons, a method of estimating hC from recorded surface 132 temperature is needed. The method introduced herein may be used to calculate hc if the thermal properties of pave- ment material are known. Assume the heated surface at x = 0 is exposed to a medium which has no heat capacity, but possesses a con- ductance, or heat transfer coefficient, hc' such as air. The temperature of air Tm(t) may be assumed to be known, but the temperature of the surface need not be known. The boundary condition would be hCITm(t) - T(o,t)] (5.8.1) Subtracting the initial temperature Ti from Equation 5.8.1 in the manner: 8[T(o,t) - Ti) (5.8.2) Taking the Laplace transform of Equation 5.8.2 gives 3'60 _ _ From Equation 5.2.13 e (5.8.4) 133 Assume the thermocouple nearest the heated surface is at x1. Then, using Equation 5.8.4: 5 - _ x _ //; l _ Cle 01 (5.8.5) and hence: s — /a—'X1 Cl — ele (5.8.6) or — -— @Xl ’/§X 0 = ele e (5.8.7) Introduce this result into Equation 5.8.3 to obtain: 5 //§ — //E x — — //;-x1 [- a] -K01e a 1 - hC 00° - ele (5.8.8) Solving for hC/K gives: fix 5.1801 h K = ‘7 E (5.8.9) o _ //E x a -61 e a 1 + 0” or [é] (5.8.10a) 134 or v’stCp hC = '5 _flx (5.8.1013) m a l :— e " 1 a1 If in addition to the temperature at x1 and of the air, a and KpCp are known, hC can be estimated from this equation. The surface thermocouple in the pavement was located at a depth of l/2-inch from the surface of the pavement, and it is assumed that this thermocouple gives representative temperature for the surface of the pavement. Since the uppermost l/2-inch of the pavement has a large temperature gradient, surface thermocouples should be located such that one side of the thermocouple be exposed to the air and the other sides be surrounded by asphaltic concrete. In this case, x1 = 0 and Equation 5.8.10b becomes: hc = (5.8.10c) which is independent of thermal diffusivity unlike Equation 5.8.10b. This method lends itself to computer solution and the value of hc may be estimated directly for the thermal 135 conductivity installation sites. The ALPHA program (dis- cussed in Section 5.10) may be expanded to calculate hC from Equation 5.8.10c. 5.9 Correction for Initial Condition One assumption in the simplified method for esti- mation of thermal diffusivity is that the initial tempera- ture distribution is uniform throughout the semi-infinite body. In reality, this is not usually true for pavement under natural conditions. The temperature of the upper- most layer of the pavement is approximately uniform before sunrise. A uniform temperature may also be provided by placing an insulating blanket over the surface of the pavement. In this section, we seek an optimum initial condition and a method will be introduced for the cor— rection of non-uniform initial conditions. Examples of hourly temperature profiles of the pave- ment and underlying soil during winter, spring, and summer, are given in Figures 2.7.1, 5.9.1, and 5.9.2. Figure 5.9.1 shows a temperature profile which has been used as an initial condition during a winter day. It is observed that the temperature is relatively uniform for the first few inches of pavement thickness, but as the depth increases, the temperature deviates further from the uniform temperature condition. Pavement at lower depths has a higher temperature than near the surface. Similar data for a summer day shows the reverse, 136 onuamnmmm auamuuflasm aucnuma an coausnwuuman 06556668209 H.m.m musmflm Auv mac Hmucfl3 m mcansc mosey pmuomamm um ucmEm>mm rm NH on :4 NH :m m Id a Adv any Adv ON) as ON ON :m o mm m on He «9 ’ "”//7//////’ ' . . . .< “.m. onsaauonloa any “no on on c: on ON OH (99q0u1) HQdOO (souout) undoa Hmaasm was umucfi3 msflusp ucmsm>mm owuamcmmm anmUIHHSM nocHImH m 2H wawmoum musumummEme N.m.m musmflm AHOEMV Ahmusasv 137 mm m «a P P P n P oaa and ONH OHH OOH ca om on cm on Aomv casuanmAan .89 mm on (Beneat) undoa 138 and the temperature is rarely uniform throughout the pavement at any one time. Observation of temperature profiles at different seasons indicate that ground temperature profile is affected by the combined daily and seasonal changes, and the temperature at a point can be represented by: Tmeas(x,t) = Tm + Tl(x,t) + T2(x,t) (5.9.1) where T0° is the temperature of underlying soils subjected to seasonal change as it is shown in Figure 5.9.2. Tl(x,t) is the difference between the average daily temperature of the top few feet and T0° (see Figure 5.9.2). T2(x,t) is the difference between Tl(x,t) and the measured temperature at the initial time. Since we are interested in evaluation of the thermal prOperties of asphalt concrete, only variations of Tl(x,t) and T2(x,t) need be considered in the correction of the initial temperature. It is thus assumed that the measured temperature is composed of the two basic parts discussed next. 1. One part is a uniform temperature, Tl(x,t) equal to the average temperature of the tOp few feet of soil-pavement and with a time variable boundary condition at x = 0. For this uniform temperature, we have: 139 2 3 Tl(x,t) 8T1(x,t) O. 2 = T (5.9.2) 3x and the boundary and initial conditions are: T(o,t) = To(t) ll 0 at x (5.9.3) ll 0 Tl(x,9) = Ti(x) = constant t (5.9.4) 2. The other part is the departure or deviation of the temperature at a given depth from the assumed uniform temperature at t = 0. This temperature deviation is called T2(x,t) and the associated problem is mathematically given by: 2 3 T2(x,t) 3T2(x,t) 01 2 = T (5.9.5) 8x T2(o,t) = T2(o) = Ti(x) = constant at x = 0 (5.9.6) T2(m,t) 3 T2(o) at x + w (5.9.7) T2(x,o) = Ti(x) t = 0 (5.9.8) Then, the measured field temperature is written in the form: Tmeas(x,t) = Tl(x,t) + T2(x,t) (5.9.9) Mathematically, Tl(x,t) can be represented by: 140 1 t x Tl(x,t) = E— f f T(x,t), dx,dt (5.9.10) x o o where t is the time during which the average temperature has been taken and x is the depth at which the effect of daily variations of temperature vanishes. This is repre- sented graphically in Figure 5.9.2. Usually the initial time has been chosen to be a time between 6:00 and 9:00 A.M. Figure 5.9.lb shows the selected initial time as well as T1(x,t) and T2(x,t). Assuming an average value for T1(x,t), we can define T2(x,t) as the deviation temperature from T1(x,t) so that it has a value of zero at x = 0 and a non-uniform initial temperature. T2(x,t) can be approximated as the temperature in a region of which part is initially at constant temperatures Ta’ Bb' Tc' etc. . . . and the remainder at zero. This approximation is illustrated in Figure 5.9.3A which shows that the area between curves of non-uniform temperature profiles is assumed to be composed of a number of rectangular-shaped temperature pulses, Sa’ Sb’ and SC. (Note that these pulses are the initial temperature distribution over position and not over time.) Assigning to 5T2 the area between these two curves, we can write for the case of three non-uniform initial temperature regions: ST = S + S a 2 + sC (5.9.11) b 141 Each one of these temperature regions can be assumed to be an individual transient temperature problem. Thus, the transient temperature distribution as the function of time and depth should be used as the transient tempera- ture distribution of 8T2. Consider a heat pulse similar to each sub-area of ST2 such as shown in Figure 5.9.33, where the region -a < x < a is initially at a constant temperature Tmax and the region |x| > a is initially at zero. The transient temperature distribution in an infinitely thick plate such as pavement at time t, due to this heat source, can be obtained by Carslaw and Jaeger (13): T - T _______l_.= % erf 1§_2_§1 + erf §_i_§. (5.9.12) 2¢at 2/at where "erf" refers to the error function defined by: x 2 erf(x) = f e't dt (5.9.13) 0 .2. /F In order to provide a temperature distribution similar to Figure 5.9.3A with a zero temperature at x = 0, a combination of a positive and a negative heat pulse such as illustrated in Figure 5.9.4A is necessary. The temperature distribution in the infinite body due to such a combination is obtained by: 142 “. mmmasm mucumummsmu commnm unmasmsmuomn mom coausnfiuumflc musumummewu ucmflmcmne m.m.m musmflm (BOHOUI) undaa A95 mnsumummsma Temperature (P') Tenperature (F’) 143 0 1 2 3 u 5 6 Tine (hour) Correction Temperature for Initial Condition 70 r (B) .<:f:::eoted for initial condition 60 - I”’ leaeured temp. so I 1 1 1 AL n J 1- 2 3 h 5 6 7 Time (hour) (C) Figure 5.9.4 Combination of temperature pulses and correction temperatures Ia - le a + x2 - erf -———————-- erf -—-——— (5.9.14) 2/at 2/at where x1 and x2 are the distances of a point from the center of the pulses,respectively. There would be a point in the domain such as point 0 in Figure 5.9.4A, with a zero temperature due to the combined effect of the sources. The pavement may be assumed as a semi-infinite body with a surface at point 0. The values of X1 and x2 are defined in terms of the variables a, b, and y, and Equation 5.9.14 is brought into a convenient form for tabulation such as: T-T 1-Y’b'al 1+Y'b’a T -1T =29“ c11/2 ‘+erf 511/2 max 1 21 2T a a ll _ y7+ 2 + b 1 + y_f 2 + b - erf: » -erf 21 l/2 21 1/2 a a (5.9.16) where T = SE a 2 145 Equation 5.9.15 may be written simply as: = f[Ta, fall, 2] (5.9.16) which is a function of three variables Ta, y/a, and b/a. This equation was programmed for numerical solution on the IBM 1800. The values of the error function are calculated by a sub-routine denoted as "Subroutine ERF." The input are the values of the variables Ta, y/a, and b/a. This pro- gram calculates the values of T - Tl/Tmax - T1 called "correction factor" which are printed as the output. Figure 5.9.5 shows the values of correction factor for y/a = 3.0. Measured temperature correction is plotted as a function of time for various depths of the pavement in Figure 5.9.4B. In order to assume a uniform initial temperature, the first few measured temperatures should be corrected. Figure 5.9.4C illustrates measured,as well as corrected temperature distribution,due to a single temperature pulse. Usually the area 8T2 is composed of several tem- perature pulses, such as shown in Figure 5.9.3A. Thus, the combined effect of these temperature pulses should be used as the correction for initial conditions. According to Equation 5.9.11, the combined effect of temperature pulses can be obtained by: Ao.m u «\sc mmadm wusumuwmfimu wommnmnnmasmcmuowu wamcam a How Na\uvo “MN. 0.H :.H N.H o.H w.o N.O H.O who _ . ua\n .1111 \\1.o o m.onm\p )o.mnm\n m.Hnu\n uouumw cofluomunou m.m.m mnnmfim TL-xamL/IL-L Joqoad uotqoailoo 147 a b T2(X.t) = Tmax ¢a(X.t) + Tmax ¢b(x.t) c + Tmax ¢c(x,t) + ... (5.9.17) or more simply “ i T2(x,t) ~§ Tmax ¢i(x,t) 1 = a,b,c,...n (5.9.18) 1—a where ¢i(x,t) 18 the reduction factor T - Tl/Tmax - Tl obtained by Equation 5.9.15 and Tmax is the maximum amplitude of temperature pulses. Sometimes the temperature pulses are located on both sides of uniform temperature Tl‘ In these cases, the algebraic sum of Tmax ¢i(x,t) should be considered. A temperature pulse located below the uniform temperature Tl(x,t), is treated as negative and above T1(x,t), as positive. Each pulse is specified by three variables: b/a, y/a, and at/az. The variable b/a = B is constant while y/a and'uxare functions of position and time respectively. For each value of y/a which represents Specific position in the pavement, various values of uxshould be calculated. A program has been written to calculate'uxas well as ¢i(x,t) for various values of a and t. A listing of the computer program and the input is given in Appendix 2. This program uses the previous program called PHI (¢) as 148 a subroutine to calculate the T2(x,t) defined by Equation 5.9.17. The program outputs all the values of Ta, T2,(x,t), ¢i(x,t), and b/a for various values of y/a. A sample of output is tabulated and is given in Table 5.9.1. 5.10 Computer Program It was necessary to write a program utilizing Equation 5.2.19 to calculate thermal diffusivity. The program is written in FORTRAN IV and can be run on most computers although it is written to be run on the IBM 1800. If another system is used, it may be necessary to change the input and output statements. Equation 5.2.19 is brought into a form suitable for electronic computation. Since temperature measure- ments are not made continuously with time, but at discrete times (which is the most common case), then the numerical integration in the Laplace transform is replaced by a sum- mation over time. This summation is denoted by sum (ITC, IS) in the program. For the first thermocouple which is used as a driving boundary condition, sum (l,IS) represents the summation in the Laplace transform. The variable names used in the program were abbreviations of words employed in the discussion of the method. These variables were utilized in the program within the limitation of the computer language. Measured temperatures were denoted by T(JJ,N) and defined by 149 Table 5.9.1 Correction factor and combined effect of temperature pulses 33;. I" ”32 *‘x't)=T'T1/TM——L—'T + my...) b/a=0.7 b/as3.0 b/a=0.7 b/a-3.0 (Po) y/a-0.7 1 0.64 0.76 0.35 0.03 -1.83 2 1.28 1.51 0.22 0.07 -1.24 3 1.92 2.27 0.15 0.08 -0.91 4 2.56 3.02 0.11 0.08 -0.71 5 3.20 3.78 0.08 0.07 -0.57 y/asloBB 1 0.64 0.75 0.56 0.08 -2.98 2 1.28 1.51 0.36 0.14 -2.09 3 1.92 2.26 0.25 0.15 -1.57 4 2.56 3.02 0.19 0.15 -1.24 5 3.20 3.78 0.15 0.13 -1.01 6 3.84 4.53 0.12 0.12 -0.84 y/a-4.0 1 0.64 0.75 0.12 0.58 -1.79 2 1.28 1.51 0.19 0.43 -1.80 3 1.92 2.26 0.20 0.36 -1.72 4 2.56 3.02 0.19 0.31 -1.60 5 3.20 3.78 0.18 0.27 -1.45 6 3.84 4.54 0.17 0.25 -1.33 7 4.48 5.29 0.15 0.23 -1.21 8 5.12 6.05 0.14 0.21 -1.11 9 5.76 6.80 0.13 0.19 -1.01 10 6.40 7.56 0.11 0.18 -0.94 150 Equation 5.2.5 represented by T(IA,IB). S(J) and x(J) are the Laplace transform parameter and depth dimensions respectively. The symbols E, ZH, ASUM, and AA, are clearly defined in the list of the program. The values of these variables are calculated based on input data and are printed as the output. Subsequent operations produced values for the numerator and denominator of the Equation 5.2.19, and finally,"ALPHA," the thermal diffusivity was calculated. This program may be used to calculate the air heat transfer coefficient of (be) defined by Equations 5.8.10b and 5.8.10c. The heat transfer coefficient (he) may also be obtained by some additional operations which utilized the obtained value of thermal diffusivity as the input Equation 5.8.10b or 5.8.10c and calculated (hc)' This process may be done by writing the mentioned equations in a proper form acceptable for the computer and using the result as the subroutine of the main program. Thermal conductivity of asphalt concrete may also be obtained by an additional operation which multiplies the volumic specific heat by "ALPHA" to produce the re- sulting thermal conductivity in BTU/ft hr °F. Output from the program is in tabular form. Both the program and a sample of the output are included in Appendix 1. 151 5.10.1 Input The environmental condition,as well as temperature data and a number of variables,are utilized as the data input for the program. The temperature data obtained both in the field and laboratory are utilized as the boundary conditions in the program. The list of the variables which are used in a suitable form for the program are given in Appendix 1. Daily, as well as data cumulative at the surface and at various depths of both a 7-inch and a l9-inch asphalt pavement,have been used as the tempera- ture boundary condition to calculate thermal diffusivity of asphalt concrete under the field condition. Recorded temperature data during winter, Spring, fall, and summer have been utilized to calculate thermal diffusivity relat- ing to different seasons. Experimental temperature data have also been employed to obtain values of (a) under the closed laboratory conditions. Both sets of temperature data provide an extensive range of temperature which is required in evaluating the value of thermal diffusivity as the function of temperature. 5.10.2 Output and Results The program outputs all the values of e-St, Ze-StT1x,t)dt, and T'(x,t) for all the time steps and various parameters 5, and thermocouples. The program also calculates and prints the values of.47T'j)/41T'O), as well as logarithms and square logrithms values of 152 6f(T'j)Ax7T'o). Printed output also includes time steps, the sequence of time steps, the values of Laplace trans- form parameters, the thermocouple number, and the values of the numerator and the denominator of Equation 5.2.19. The result oijubsequent operations which calculate 2 T . Jgfifflr and its summation for the o parameter s and x, and finally the value of thermal the values sx2 ln diffusivity (a) are given sequentially in the list of output. Calculated thermal diffusivity of asphaltic concrete for various seasons and under different ambient conditions are given in Tables 5.9.2 and 5.9.3. The calculated thermal diffusivity of asphaltic concrete from experimental temperature data are given in Table 5.9.4. 153 Table 5.9.2 Calculated thermal diffusivity of wearing and binding courses from recorded temperature data (s = 4/tmax), simplified method Ambient Surface Temp tmax Thermal Case Date Condition na£F° Min (hr) D(;:E;:;)ty l 7.69 sunny 127 65 120 0.042 2 8.69 ' 115 64 21 0.044 3 6.69 ' 120 60 16 0.043 4 4.69 ' 112 41 16 0.051 5 3.69 ' 56 37 70 0.049 6 1.70 ' 34 5 16 0.053 7 1.70 ' 28 -2 118 0.054 8 8.69 cloudy 135 70 16 0.050 9 4.69 ' 116 68 35 0.048 10 1.70 ' 28 -3 16 0.062 11 1.70 ' 36 10 118 0.059 12 7.69 rainy 118 73 16 0.060 13 3.69 ' 33 20 16 0.055 14 7.65 s 131 68 70 0.052 (Alger) 15 7.65 ' 123 69 68 0.052 16 1.65 ' 29 6 96 0.056 17 1.64 ' 40 -5 62 0.069 154 Table 5.9.3 Calculated thermal diffusivity of asphaltic con— crete (entire pavement) from recorded temperature data (s = 4/tmax) simplified method Case Date Surigg; Temp tnax n053:1;ginni66:21:661e1ft66gigcted Max Min (hr) I.T. I.T. I.T. 1 7.69 127 65 120 0.041 0.032 0.039 2 8.69 115 64 21 0.033 0.041 0.044 3 6.69 120 60 16 0.040 0.036 0.040 4 4.69 112 40 16 0.039 0.043 0.050 5 3.69 56 37 70 0.050 0.051 0.055 6 1.70 34 5 16 0.040 0.049 0.053 7 1.70 28 -2 118 0.080 0.072 0.060 8 8.69 135 70 16 0.050 0.061 0.050 9 4.69 116 68 35 0.049 0.058 0.047 10 1.70 28 -3 16 0.047 0.041 0.05? 11 1.70 36 10 118 0.050 0.065 0.059 12 7.69 118 73 16 03063 0.059 0.053 13 3.69 33 20 16 0.049 0.049 0.055 14 7.65 131 68 70 0.054 0.049 0.039 15 7.65 123 69 68 0.041 0.043 0.049 16 1.65 29 6 96 0.059 0.052 0.060 17 1.64 40 -5 62 0.059 0.078 0.063 155 Table 5.9.4 Calculated thermal diffusivity of wearing and binder courses from laboratory data (tmax = 120 seconds and s = 4/tmax), simplified method Case Condition surfgg) Temp Diggagtsity Max Min (rtZ/hr) 1.1 dry 146 118 0.039 1.2 ' 146 118 0.042 1.3 ' 146 118 0.039 1.4 ' 146 118 0.038 2 * 130 97 0.041 3 ' 124 80 0.042 4 ' 117 75 0.051 5.1 ' 80 5 0.048 5.2 wet 75 2 0.064 5.3 dry 85 -1 0.050 6.1 dry 26 -27 0.054 6.2 wet 25 -25 0.071 6.3 dry 32 -15 0.062 CHAPTER VI PAVEMENT TEMPERATURE 6.1 Temperature Distribution in Pavement The main objective in the collection of the re— corded field temperature data was to obtain temperature variation at various depths of the asphaltic pavement. It was realized that this temperature data was not ade- quate for this study. Necessary additional information had to be obtained in order to conduct a more comprehen— sive analysis of the simulated model. This additional data consisted of thermal properties of pavement materials, surface convection coefficient, solar radiation, and other environmental knowledge. The lack of information on ther- mal properties has been emphasized in earlier chapters. In order to obtain a correlation between the ob- served and calculated temperature variation, a simulation model was required. This simulated model should be based on the basic heat transfer equations for radiation, con- duction, and convection. The problem in this study is a one-dimensional time-dependent boundary value problem of heat conduction 156 157 with termperature—variable thermal properties. The problem is nonlinear because of the temperature-dependent thermal prOperties of pavement material. The recorded temperature of pavement may be utilized as the boundary condition to the simulated model to obtain the calculated temperature variation at various depths of the pavement. One of the most important objectives of the simu- lation model was to predict pavement temperatures at various depths, in the simulated pavement, under the actual and the extreme conditions which the pavement might experience during its lifetime. The reasoning be- hind this condition is that during the period of data collection the coldest or hottest condition may not be reached; thus, the extreme temperature will be missed. For this reason a simulated model with a variable range temperature was required. Most of the studies on the subject of temperature distribution in pavement have dealt with steady, stepped, or strictly sinusoidal boundary values (37),whereas as pavement exposed to daily change in temperature does not experience such convenient surface variation. The purpose of this study was to define methods which may be utilized to analyze a realistic surface temperature history as the boundary condition to a simu- lated model for calculating theoretical values of tempera- ture in a pavement. 158 6.2 Exact Solution Based on Periodic Surface Temperature Daily, and annual variation of pavement surface temperature may be approximately represented by a steady periodic wave function according to the expression (13): T(o,t) - Ti = (TAd - TMd)cos wdt (6.2.1) where T(o,t) is the pavement surface temperature at the time t, and Ti’ TM, are the initial and mean daily tem- perature at x = 0,respectively. Amplitude and angular frequency of surface temperature is denoted by (TA - TM d d) and w,respectively. The subscript "d" refers to daily temperature. It is assumed that pavement is a semi-finite body with a sinusoidal surface temperature change; temperature at a depth x below the surface of pavement may be ob- tained by (13): _ _ t-/9E T(x,t) - Ti — (TAd TMd)e cos[wd Za]x (6.2.2) which is also periodic and its amplitude decays in an exponential manner by a damping factor e- wdlZd x. Further, the temperature at the depth x below the surface lags behind the temperature at the surface, and in pro— portion to /wd/2a x. 159 In order to compare measured temperature data with the results of Equation 6.2.2, a possible value of a = 0.04 ftz/hr for asphaltic concrete given in Table 4.4.1 should be selected and Equation 6.2.2 is solved for the temperature at a depth x below the surface. 1 Equation 6.2.2 is solved for annual and daily periods and surface temperature, and the calculated tem- perature distributions are plotted in Figures 6.2.1 and 6.2.2. Since the temperature at a depth x below the surface is the function of daily and seasonal change, Equations 6.2.1 and 6.2.2 may be written in the following form to account for daily and seasonal changes: [T(o,t) - Ti] = (TAa - TMa)cos mat + (TAd - TMd)cos wdt (6.2.3) and _ wa x [T(x t)-T.] = (TA ~TM )e 5; cos(w t-)/93-x).+ ' 1 a a a _ 20 _ 24(365)‘ - % 2a x + (TAd-TMd)e cos(wd t- 2a :)x _21 24 160 fiscausaom uomxmv musucummamu mo coaucflum> maflco H.N.m musmflm “agony cede o: cm on om V) \\ , co ooa (ad) ednaezedmom OHH c809 occunsm 2 62 new - and 161 mn< Acoflusaom uomxmv musumummEmu mo coflumflum> amazed ~.N.w musmfim hash cash was HA4 an: Dom. can one boa uoo anew ms< hash 0:55 he: 1% d u q q q u q u q d u u u - d L 1:111. \\ II/ .. \A\\:|.::./.// \.\\ .. ./ I x ..\ /.o . I . .. .\U /. / \\.. \ \ x . z .. x. N / /) I \ c . / . ..\ .\ \ MU".IM/x \\\\ \\ \\ 1 no.» 3 //... ..\.. .\ .\ x . \\ \ 668 m .\. x .l.\ \ \ I. 000.“ N I \\ ousucnonaoa oocuusm OH ON on c: on ow on om om 00H (03) oanqsxadmam 162 where TMd and TMa are the mean daily and annual surface temperatures,respectively. Ti is the initial temperature and TAd and TAa are the daily and annual temperature amplitude at the surface, respectively. For the annual variation of the temperature, the angular frequency of the fluctuation is ma = Zn/P 20/24(365) per hour for a period of one year or 365 days, while wd = Zn/P = 20/24 per hour for a period of one day. It may be observed that the first term on the right- hand side of Equation 6.2.4 represents the effect of seasonal temperature variation and the second term stands for daily fluctuation of temperature. For small values of x up to three feet, the second term dominates, while for large values of x, the effect of second term is negligible and the first term characterizes the tempera- ture. Neglecting the effect of seasonal temperature vari- ation, daily variation of temperature lies within an envelope determined approximately by i e -/Ud723 x. These envelopes are actually the two exterior temperature distri- bution curves. It was assumed in the foregoing that the surface temperature varies according to a simple harmonic sine curve. Such, however, is not the case in reality, in- stead, the surface temperature follows a complicated 163 temperature process which may be characterized as the sum of a series of partial oscillations. The Fourier-series analysis has been proposed by Carson (14) in describing this phenomena and the surface temperature is assumed to vary according to the expression: T(o,t) = F(t) = T + ):°° (A cos —— nt + BnSin —— nt) (6.2.5) where An and Bn are the Fourier coefficients determined through standard Fourier techniques. Tm and P are the mean temperature and the period of the cycle respectively, and n is the ordinal of partial oscillation. 6.3 The Integration Method (Volterra Integral Equation) For a semi-infinite body such as a soil-pavement system with a time-variable surface temperature, the nonlinear heat conduction equation is transformed into an integral equation in the form (13): t -X2/4a f T(A-Ti)e (t-Al-dl(6.3.l) 2/FE o (t-M372 X T(x,t) = Ti + where T(A) is the known variable surface temperature and A is a dummy time variable of integration. Equation 6.3.1 may be solved for T(x,t) by a number of analytical methods (4). Monismith gt 31. (37) employed 164 this equation to determine the temperature distribution in a slab with sinusoidal surface temperature. In this study we employed the Runge-Kutta method (17) for inte- gration and the Equation 6.3.1 was written in FORTRAN form and programmed. The recorded surface temperature was used as T(A) in Equation 6.3.1 to calculate the temperature T(x,t), at any depth x below the surface. Difficulty was encountered using this method and the obtained value of the calculated temperature was not satisfactory. The conclusion is that this equation cannot be employed in this study and the application of the analytical methods given by Ames (4) for the solution of Equation 6.3.1 is not practical. 6.4 Finite Difference Method In this analysis it may be assumed that the pavement is a semi-infinite body with a time variable surface tem- perature and temperature-variable thermal properties. Nonlinear boundary conditions may also be treated. Figure 6.4.la illustrates the geometry and the boundary condition of the soil-pavement system which was selected for the model. The related one-dimensional heat transfer equations and the prescribed boundary conditions are the following: 8 8T _ 9.“: _ '3‘}?[K1'3T{'] " (pcp)3t I T(XIO) - TO(X) O < X < L (6.4.1) 165 4 gr); x_= thT -To(t) ) = q(t) \\\ Pgwment§fluihce_\L# 2 T 3T: a ~§4£ T(o’t)=T°(t) Paveggnt 3t: 1 3x2 T(x,0)=To(1) Thickness Inhndhce—\\‘ 1 'K1 3:3 K2 $14 01‘ 321? I L 37; ... 0‘2 33:5 T(L,t)=¢(L,t) Bngzt) =0 (A) | I ld+lu__1?l1s 1:141 1-_1i__ (L— tj.fl_ tame (B) Representation of the nodal points in finite-difference method. Figure 6.4.1 Soil-pavement system 166 3 h n-2 n-1 n 151* L L L. L9 L2 "'2 ”'1 r‘ e V r‘ I I '1‘ W Nodes l and n used as boundary conditions Nodes 2 through n-l used to improve difference approximation \ (4) Interface of Asphaltic Pavement Surface and Subgrade Soils of \- “ Pavement ‘~ T=To(t) gm (3) Figure 6.4.2 Location of thermocouples, regions and nodes in the soil-asphalt system 167 _ 8T(x,t) 22 - _ + _ K1 3X _ K2 3x + T(L ,t) — T(L ,t) x — L L ' (6.4.2) EL ‘22 _ §3_ 3T(x,t) _ m 8x (K23x) ' (“C792 t ' '—ax— ’0 L < X < x+oo (6.4.3) The surface temperature T(o,t) is assumed to be known and the heat flux q(t) has been calculated using experimental temperature data. The convective and radi- ative boundary condition may be employed. The pavement was divided into a number of equal thicknesses such as shown in Figure 6.4.2. Since the temperature fluctuates more in the upper layers of the pavement than in the lower layers, the regions in the upper layers are divided into smaller nodes than the lower regions. Each one of these subdivisions represents a finite control volume which may be denoted as a nodal point at (or near) the center of the volume. The first and the last nodes (1 and n in Figure 6.4.2A) are utilized as the boundary con- ditions, while the internal nodes (nodes no 2,3,4, ...,n—l) are employed to improve the finite difference approxi- mation. The temperature at the nodal point xi at the time tj is denoted by: = j T(xi,tj) Ti (6.4.4) 168 Guided by the indexing scheme shown in Figure 6.4.lb, an approx1mation to Equation 6.4.1 at nodal p01nt (Xi’tj+l/2) is shown by: BT j+l/2 (K _— p 5? 3x = pC 8 5;' (6.4.5) i Following the finite difference procedure, the resulting equation in finite difference form may be written as (48): 5+1 _ 5 5 5+1 _ 5 5 5+1 _ 5+1 (0C )j Ti Ti = n Ki-1/2(Ti—l Ti) I K1+1/2(T1+1 Ti ) D 1 At (Ax)2 5 5+1- 5 5 5' _ 5 + (1-n) Ki-1(Ti-l Ti) + Ki+l/2(Ti+l Ti) (4:02 (6.4.6) where i indicates position and j indicates the time step. K1-l/2 is the thermal conductivity at a temperature equal to [TE-1 + T2]/2 ° A forward difference, backward difference, or Crank-Nicolson technique is specified by setting the vari- able n to be equal to 0.0, 1.0, or 0.5,respectively (48). Thermal properties of asphalt concrete are given in Tables 4.4.1 and 4.4.2. It can be assumed that the properties are constant for each seasonal temperature range. Equation 6.4.6 for constant K and 0C may be 0 written as: 169 Aj Ti = r[n0§ Ti+l + (1-n) 6% Ti] (6.4.7) where r = aAt/(Ax)2 and, Aj and 01 are the forward and central difference operators,respectively. Forward difference and second central difference of Ti are here defined as following: 3: j+1_j Aj Ti Ti Ti (6.4.8) 512 T3 = T3_l - 2 Ti + Ti+l (6.4.9) Equation 6.4.7 reduces to the solution of a set of algebraic equations which should be solved simultaneously. The result would be the calculated temperature at each nodal point as a function of time. 6.5 Computer Program The process of simultaneously solving the set of equations 6.4.7, has been employed in the Property Program which may be used to obtain temperature at various depths of the pavement and underlying subgrade soils. (The com- puter program has been described in Chapter IV.) This program was run on the IBM 3600. The program utilizes the pavement surface tempera- ture as the time-dependent boundary condition to calcu- late temperature at various depths of the pavement. This program may be used to calculate temperature in a homogeneous as well as composite media and multilayered 170 bodies such as a soil-pavement system. In order to calculate temperature at various depths of a multilayer pavement, the thermal properties and the thickness of each layer should be given as the input to the program. The latent heat of fusion was not taken into account, thus, the temperature is calculated if no change of state takes place; there is a method that can be used with the program to simulate phase-change, however. 6.5.1 Input The program requires two sets of main input; the initial and the boundary conditions. The initial tempera— ture condition may be found by plotting the measured temperature of the pavement and subgrade soils versus depth at t = 0. However, no information can be found for initial pavement temperature unless the pavement is monitored. The measured surface temperatures are utilized as the boundary condition to the model. The thermal properties of pavement and subgrade soils should be included in the input. As seen in Tables 4.4.1 and 4.4.2, the thermal properties of wearing course, binder course, and base course, are different. Consequently, the pavement should be divided into several regions with various thermal prOperties. The thermal properties of subgrade soils are assumed to be uniform in depth. The length of each region, the number of nodes in each region, and other 171 variables which were utilized as input in estimating properties are also included in the input. 6.5.2 Output and Results The program outputs the time step and calculated temperatures, for various locations. A sample of output which contains time steps, the measured surface tempera- ture, and the calculated temperature at 2, 4, 8, 12, 15.5, and 19 inches below the surface of the pavement’is given in Tables 6.5.1 and 6.5.2. The calculated and measured temperatures are compared in Figure 6.5.1 for a few summer days. Calculated temperature at various depths of the pavement during a few summer days are plotted in Figure 6.5.2 (see the results and discussion in Chapter VII) 0 Table 6.5.1 Calculated temperature at various depths of the 172 pavement during a few winter days Measured Location of Calculated Temperature Time Surface , 5 Steps Temp 2' 4' 8' 12' 15.5'. 1.5000 17.9500 20.5366 22.6745 25.7646 28.9998 31.0893 __24S0nn___l6435flfl__.1811130___Zl;&431__1254é958M_.28.4946 30432821 3.5000 14.7500 17.3411 20.3233 24.8091 27.9655 29.8928 4.5000 13.2500 15.9891 19.1631 23.9960 27. 3590 29.3786 __5A50nfl___12A2500___14;8110___18101201”“2361210-1,26.69997M,28.84231 6.5000 11.5000 14.0815 17.1817 22.2750 26. 0123 28.2985 7.5000 10.6000 13.1731 16.3153 21.4934 25. 3332 27.7352 .__8sinnQ___lQ41flflfl___124&l§6___1§4§31&_ 20.7322_-wgid駧1_m_21517101 9.5000 9.8000 12.0912 14.9799 20.0422 24.0213 26.6105 10.5000 9.4500 11.6594 14.4711 19.4375 23. 4132 26.0624 11.5000. 9.1500, 11.2928 14.0205“”418,8849I_f22._ 8458 25.5359 12.5000 8.9500 10.9787 13.6141 18.3790 22. 3150 25.0336 13.5000 8.5500 10.6734 13.2718 17.9208 21.8193 24.5567 114.5000 7.8500 10.0573 12.7634 17.4715 21.3551“- 29-1059, 15.5000 7.3500 9.4862 12.2074 16.9843 20.9005 23. 6753 16.5000 7.1000 9.1437 11.7731 16.5025 20.4463 23.2565 17.5000 7.3500_ 9.0458 11.4685er3630709 20.0069 .022.8479 18.5000 7.9000 9.4068 11.5086 15.7498 19.6035 22.4563 19.5000 8.9500 9.9152 11.6689 15.5649 19.2662 22.0949 27 11595481-.-1969199 “El-7785 21.5000 12.8500 12.5806 13.1772 15.7466 18.8693 21.5232 22.5000 15.2500 14.8076 14.7050 16. 2101 18.8648 21.3482 23.5000 16.7500 16.1957 15.9482‘m_1668906 719.0280 21.2742 24.5000 17.2500 17.1702 16.9969 17.5811 19.3117 21.3082 25.5000 16.5000 16.9407 17.2962 18.1389 19.6443 21.4267 26.5000 15.0000, 316.1139 17.0605 18.4128 _ 1912229”- 21.5899” 27.5000 13.7500 14.9178 16.3252 18.3865 20.1126 21. 7440 28.5000 13.5000 14.5238 15.8754 18.1847 20.1584 21.8521 2935000 13.7500 14.5370 15.7169“. 1860191 ~20. 1281 21.9027 30.5000 13.5000 14.5889 15.7590 17.9479 20. 0880 21.9166 31.5000 11.7500 13.5831 15.3107 17.8615 20. 0590 21.9168 32.5000 9.0000 11.5760 14.1233 17.5259_”Jl9.981} m 21_. 9021 33.5000 6.7500 9.4331 12.5147 16.8578 19.7674 21. 8385 34.5000 5.5000 8.1076 11.2258 16.0232 19.3944 21.6891 35.5000 4.7500 7.1934 10.2373__ 1503299- ‘18_. 9274 21.4502 36.5000 4.5000 6.6905 9.5479 14.5282 18. 4347 21.1478 37.5000 4.7500 6.6073 9.1757 13.9680 17.9607 20. 8123 38.5000 5.0000 6.7656 9.0889 13-5113-15121§394 . 20. 4718 39.5000 5.2500 6.8047 8.9940 13.2978 17.1888 20.1496 40.5000 5.7500 7.1207 9. 0766 13.1013 16.8978 19.8576 41.5000 6.2500 7.4774 9. 251_0 ,113- 0024 16.6662 19.5996 42.5000 7.2500 8.0425 9 5243 12. 9805 16.4948 19.3802 43.5000 9.5000 9.4547 10.2607 13.0962 16.3882 19.2029 44.5000 12.5000 11.7768 11-71421_11335050 rul663918 19.0804 173 mm¢0.~m mmmc.mm 0606.0m >m~m.¢m mmmm.cm mmcm.c> cooc.ms oocm.¢m 6666.66- 666666.. ---...666. 66. 6.66.64 66 6666.66 66666.66.-- 6666.66 6666.66 mmcm._w cmmc. 0m “smm. or 060m. hm ¢m00.mm ~¢~m.mm ooom.0~ oocm.mm mo:m.~m ¢m¢x.mm 00mm. cm 00cm. 0m c0m¢.mm >0co.m¢ oomh.mm oocm.~m mmimm.$ 6636.6, 1660.66.66.99 $5. 610---! mapmtmbui. M66160- 83.; oocmém -m0.om ~mm~.mw mmmo.mm >000. o0 ~m~m. o0 -co.>0 oooo.m0 cocm.0m mosh.cm ¢~¢~.Nm m0m¢.¢m cmos.0m ccmm. m0 cmom.ncn coco.mo~ oocm.wm bcuw.om comm.~m omm~.mm ~m-.>m wo¢—.>0 N0ho.mc~ oooo.-~ oocm.>m Nouo.~m ~0cm.~m cm_¢.mm hmo~.¢m 0__m.~0 o¢c~.oow adom.m06 ooqmuom ~60~._m mocm.mm m0mn.mm ~0m—.~m m¢-.mm 666m.60 ooom.m0 ooom.mm cnmm.—m ~¢~>.mw ocmc.mm oom5.~m ~m~0.0h 0660.om coco.mm oocm.¢m mwnp. _m mo-.mm _o6o.nm 6666. 66 u m~m~19~sls9qwa.ms 6666.66 adem.mm cmoh. um ~m~o.mm N¢mm.¢m @600. 0a smho. om N~o~.hh ooom.m> oocm.mm go6~6.~6 6666.66 6666.66 6666.66 6666.66 6666.66 6666.66 6666.66 Noc¢.~m m~nm.mm hcc¢.om ~ms~.mm m~om.mm comm.mm ooom.ms odcm.om oo~o.—m ~>0¢.mm cmoc.om msco.o0 mnc>.0w cmw0.cm coom.mm oocm.0~ -.wwww. 66 10666-66-1.606W466--1 _6m6666 6666.66 6666.66 6666.66 6666.66 00mm. cm chmo. mm m¢h~.mm enhm._0 m00¢.00 momo._o~ ooom.00 odum.s~ m~00.0> 0mcm.~m 0mmc.mm 000m.0m smm¢.oo~ mm0h.oon oooo.o- oocm.o~ -16666.66 6666.66 666w. 66 6666.66 6666.66 6666.666 6666.666 6666.66 «moo.cm mmcc.cm 000m.~m ~0o¢.mm 09cm.~0 «00¢.ocu oooo.o- ooom.¢fi ¢o0~.cm msmm.cm cocc.~m omm~.~m mmm¢.¢m ~mmm.~0 ocom.00 cocm.m~ 660m.mmsliwwmew@-2.MrNM1mmn1:mmwm.66 6666.66 6666.66 6666.66 6666.66 m0mc. cm m>¢o. ~m mmmm. mm ¢~>¢._m cmsm.>> m00m.m> coco.ms occm.- 05>m.cm sooo.~m ¢>-.mm 00>0.~m aosm.0h ¢omm.oh coco.m> oocm.o~ ¢m>~.cm N060.~m 0m0~.mm hmmm.¢m ~000.—m 00mm.m~ ooom.mh oocm.0 0005.05 seem.~m 00om.¢m mmwm.om 00o>.¢m mmm~.~m ooom.¢~ oocm.m 6666.66 mmom._6 6666.66 6666.66 6_c~.66 6666.66 6666.66 6666.6 mrrwumrr 6666. pm:;.66mm..mmei-6mmw;wm.s 6666.66 666~.~6 6666.66 6666.6 ¢0-.¢> cmm0. 05 ~>0m. mm omm0. mm ¢~m¢.>0 m~h~.00 coom.m0 ccom.m 00mp.>h hcm¢.mh cmhm.o¢ 0005.0m «www.m0 0mm0.mo~ oocm.0o~ occm.¢ .;Mhmwwsb mm0>.>> -mmmm.m> moom.mm 0m-.m0 oomm.mo~ oooo.¢- ooom.m c0om.#> 0mmm.>s ~m¢h.h> N¢Nm.0h mmmc.0m omcm.00 oooc.c- coam.m occm.>> Loom.>~ mocm.>b mooc.>> 0c0¢.0h 0mm0.¢m ocoo.m0 occm.~ . N _ ale sad 3% “H BNH em a: a OOGHW5m Inovm ounpeuoaaoa uouuasoauo ho noduaooq _ consume: 0:69 mmmo umaesm 366 6 026656 ucmEm>mm may 90 mzummv msowum> um mnsumummEmu omumasoamu m.m.m magma mama HmEEdm ms» mcfluac mnsumummemu vaSmme nuaz wwummfioo musumummewu gamew>ma nmumasoamo H.m.m musmflm Auaosv onus :4 :4 ON 0H 0H 3H NH OH .0 c 3 N #N NN ON 0H 0H 3H NH OH Q.“ E _ u + u u u q d . fl - q u q d u u c LOW \\\\ IIIIIIIIIIIIIIII AHaOV .NH \\\ \\\\\\ I. O“ \\ o o ’olollllllll \\\ \x \ . \I.|.l.ll”.’. IIIIII Ammo—5 INH \ - \ - /.//..l/H.H.l/. z// \A \ . cm 1":\ \\\\ \ o\\\ -.I/ 11:: \\\\\ \ \ o n .A/ s . om / /:§< \\ \\ . -\\ o \. Add»: ...: If ..nl..|\.\ \\\ lOCH Ammoav .3 VI \\\. flqh.\. fl AHGOV ..N A0.7—n oomuhfiu AQQOBV 8N .ONH .onH 03H (0;) alnaazadmom 175 an.“ AmusumummEmu HMfluwcw mHQMfiHm>V HmEEdm mcwusv musumuwmfimu ucmew>mm cmumasoamu m.m.m musmflm “nanny ends on cm on c: on om ca d - d u a 1 a CO 1 ca 1 // .2 a! Ajnzur. , .. :.zuzur ONH \l‘!‘ -- .\ // \ oomunsm 1 ILONH T .2 am ‘° A“ // (OJ) oanzazoduo; CHAPTER VII RESULTS AND DISCUSSION The results and discussion are summarized under three headings: (1) Recorded Temperature Data, (2) Ther- mal Properties of Asphaltic Concrete, and (3) Pavement Temperature. 7.1 Recorded Temperature Data The analysis of temperature distribution data leads to the following observations. During the summer months, the average temperature of the pavement decreases with increasing depth. However, during the winter months, this reverses. This may be ob- served from daily temperature distribution, as well as temperature profiles such as shown in Figure 5.9.2. Temperature profiles were plotted from temperature dis- tribution for several days during the summer, spring, and winter months. Available data in the section of literature review indicate that the soil beneath the asphalt pavement has approximately the same thermal properties as asphaltic concrete, thus it may be assumed that profile curves are 176 177 continuously smooth at the interface of soil-asphalt concrete. During the summer months, temperatures ranging between 130°F and 140°F at the pavement surface occurred only 1 per cent of the time. During this period, the maximum temperature at a depth of 4 inches was llO°F and at 12 inches below the surface, 90°F. It was observed that the surface and the few upper— most inches of pavement were greatly influenced by air temperature. However, at lower depths in the pavement, variations in temperature decreased, until a depth was reached at which the temperature was approximately con- stant. The deeper layers of the pavement remained fairly constant for a long period of time, while the surface layers fluctuated with the daily temperature cycles. This phenomena produced the crossing and overprinting of individual points. The recorded temperature data were usually congested especially during the cloudy and rainy days, since frequently the entire profile was at nearly the same temperature. Frequently it was impossible to determine the time of peak temperature at a given depth. Time lag between maximum and minimum temperatures at various pavement depths was another interesting phenomenon which may be observed from daily temperature distribution curves. This time lag is not constant but increases with the depth of the recording in the pavement. 178 On a sunny summer day the maximum temperature occurred on the pavement surface at approximately 4:00 P.M., while the maximum temperature at a depth of 12 inches below the sur- face of the pavement was at 12:00 P.M., and at a depth of 19 inches maximum temperature occurred at 6:00 A.M. of the following day. The time lag of temperature was also true for winter days and minimum pavement temperature at each depth. Observation of daily temperature distri— bution supports the validity of the assumption of sinu- soidal variation of daily air and surface temperature, which has been adapted by many investigators (28, 40). The daily variation of the air and pavement surface temperature are graphically presented in Figures 2.7.6 and 2.7.7 for various seasons. These figures indicate that pavement surface temperature was generally warmer than air temperature during both summer and winter. This was due to solar radiation which has a greater heating poten- tial than does the air in the summer and due to warmer pavement with negligible radiation input in the winter. This fact is also observed from curves of cumulative degree-days based on both air and pavement surface temperature (see Figures 2.10.1 and 2.10.2). The observation of the temperature distribution in a 7-inch and a l9-inch pavement indicates that under the same surface temperature there is no significant differ- ence between the temperature at depths of 2, 4, or 7 inches, and both sections behaved in the same manner. From the 179 observation of the Alger Road temperature data, we con- cluded that during the freezing season the temperature of subgrade soils beneath the asphaltic pavement was higher than in the underlying gravel section. The higher temperature of the subgrade under the asphaltic pavement was mainly due to the absorption of solar radiation. A comparison between recorded tempera- ture data at Bishop Airport and Alger Road, indicates that during the freezing seasons, under the same surface tem— perature, the temperature beneath the 19-inch full-depth pavement is higher than the 7-inch pavement. The result of the evaluation of air and surface freezing index at the thermocouple installation site are illustrated in Figure 2.10.2. The values of freezing index derived from the annual climatological data for the Flint area during the last twenty years varies from year to year, but generally freezing indexes were higher than the value obtained from the recorded temperature data at Bishop Airport during 1969-70. The pavement surface freezing index was much less than the air freezing index. As a result, a small value of correlation factor "n" equal to 0.51 is obtained. The comparison between this value and the values recommended by Aldrich 23.2l- (3) and Oosterbaan 33 El. (41) lead to the conclusion that the adOption of the obtained correlation factor would be entirely unjustified for the purpose of frost 180 depth prediction beneath the flexible pavements for central Michigan. The significant difference between the calculated air freezing index and the average of the air freezing index derived from the National Weather Service's records indicates the need for a more extensive and accur- ate study of pavement temperature variation for design purposes. 7.2 Thermal Properties of Asphaltic Concrete Despite the vast quantities of recorded tempera- ture data, very little were of use in thermal property determination. This was because not all of the recorded temperature data was discernible. The values of thermal property calculated from both the field and laboratory experimental temperature data by the nonlinear estimation method and simplified method discussed earlier, are pre- sented in Tables 4.4.1, 4.4.2, 5.9.2, and 5.9.3. The property calculated most independently was the thermal diffusivity a since the heat flux need not be given to obtain a. It should be pointed out that the pavement was not homogeneous but consists of several layers of asphaltic concrete with various aggregate sizes and asphaltic contents. The thermal properties of each layer were obtained by utilizing the temperature history of the upper and the lower interfaces. The temperature history of the upper boundary was used as the driving boundary condition and that of the lower one, as the data. 181 The estimated thermal property of the upper two layers (referred to as the wearing and the binder courses) are shown in Tables 4.4.2 and 5.9.2. Tables 4.4.1 and 5.9.3 give the thermal properties of the entire depth of the pavement obtained from temperature field data. Calculated thermal properties for upper layers of the pavement, estimated from recorded temperature data, are more reliable than those of the deeper layers. This is because both methods used in the estimation of the thermal properties require a discernible temperature change within a given layer. For the top layers, the daily temperature cycle usually produces sufficient fluctuation, while this change does not occur daily or even monthly for deeper layers. The values of thermal properties for upper layers of the pavement were more consistent than those for the lower layers, while these values for deeper layers exhibit a wider variation in magnitude. Generally the thermal diffusivity of the wearing and binder courses were higher than those of the base courses. The calculated values from field data appear to vary widely from day to day. However, certain of these variations are expected, due to various ambient conditions and seasonal changes. Calcu— lated thermal properties are plotted versus temperature in Figures 7.2.1 and 7.2.2. A comparison of the calculated thermal diffusivity from field and experimental data indicates the calculated 182 Volumetric Specific Heat (C)(Btu/cuft/F) Odd and ONH OHH 00H om o.u~ o.mN o.a~ coon o.an o.~n comm mumo huoumuonma Eoum ummc owmaommm canumfisHo> cam muw>fluoscooo Hmeumnu omumasoamo H.~.h musmflm A.MV onsumnooaoe om o5 ow on c: on ON 0H 0 d 1“ fi Id 4 d w.o m.o o.H H.H mod ( J a; xq/nza) (X) KntAtaonpuoo tamaaum 183 mumo camwm Umonooou Eoum mumuocoo cauamnmmm mo muw>HmDMMHo Hmaumnu Umumasoamu m.m.n musmflm A.MV ouspmnoasoa 0nd oNH 0HH 00H om om on 00 on 0: on om OH 0 0H: ‘1 I di 0 1 d d d I d 1 1 venues uoficdaaadm canoes :Oauwadumo nmccaacoc J u 1 No.0 no.0 :o.o no.0 00.0 no.0 no.0 00.0 (an/34:) KatAIBnJJta Ismaaqm 184 thermal diffusivity from field temperature data are slightly higher than those calculated from experimental data. The discrepancy between the laboratory and field data may exist because the laboratory data was collected under a controlled condition while the recorded tempera- ture data was obtained under the natural ambient con— dition. Laboratory conditions may be controlled to measure the effect of a single variable (e.g., moisture), under a specified form of heat transform,while in nature the pro- cesses of the inflow and outflow heat transfer do not lend themselves readily to theoretical heat transfer analysis. It should be pointed out, however, that the laboratory experimentations gave an explanation of the phenomena and thus provide a basis for interpreting field results. Moreover, the theory of heat transfer deals with a dry homogeneous substance subjected to a known heat source. A comparison between the contents of Tables 4.4.1, 4.4.2, 5.9.2, and 5.9.3 indicates that there is not a significant difference between the calculated thermal properties by the two methods of calculation. The properties obtained by the nonlinear estimation method appear to be much more consistent than those calculated by the simplified method. The nonlinear estimation method is an iterative search procedure which requires an initial estimation of 185 thermal prOperties as well as a considerable amount of computer time, while the simplified method does not. In terms of computing time, the simplified method is somewhat more advantageous than the nonlinear estimation method. The most important feature of nonlinear estimation method is its application to materials which have temperature- dependence thermal properties (7). Both methods eliminate the assumption of periodic surface temperature in esti- mating the thermal property of pavement materials. Pave- ment temperature under natural conditions may be utilized in both methods to estimate the thermal properties of pavement material. As a result, both methods have prefer- ence over the previously introduced method of estimating thermal properties which are limited to the periodic boundary condition (28, 40). The estimation of thermal diffusivity by the simplified method is based on the assumption that the initial temperature is uniform. The discrepancy in the estimated thermal properties is due to the fact that the simplified method is quite sensitive to initial tempera- ture. The lack of this condition being uniform necessi— tates the process of correcting the initial condition. The accurate correction for the initial temperature has been made, but a satisfactory mathematical treatment of the problem is time consuming and sometimes impractical. 186 The calculated thermal diffusivity of asphaltic concrete for the corrected and uncorrected initial con- dition is given in Table 5.9.3. A comparison between these two sets of values indicates that the correction for initial temperature is efficient. A criterion is derived for finding the optimum experiment as well as the location of the thermocouple and Laplace transform parameter "s" which is employed in the proposed method of estimation of thermal diffusivity. Further research is needed to consider the complex form of the Laplace transform parameter which reduces the effect of the initial condition in the estimation of thermal diffusivity. In order to determine statistically whether the two sets of calculated thermal diffusivity differ signifi- cantly, their means and standard deviations should be com- pared. Since there were only two sets of data, the vari- ance ratio test, known as the F-test, may be applied. The F value is calculated as (48), F = ($1)2/(52)2' where 51 and 32 are the estimated standard errors (variance) of sample 1 and 2, respectively. (Note that (sl)2 should be greater than (sz)2.) The critical values of F for various degrees of freedom and probability are given in the tabular form (pp. 306, ref. 48). The calculated variance ratio should not exceed the tabulated values for an insignificant 187 difference between two sets of data. The mean, as well as the variance of each set of data,are calculated and are given in Table 7.2.1. The calculated variance ratios are compared with the tabulated values at the 5 per cent level of significance in Table 7.2.1. It is observed that for each set of data, the calculated F values were generally smaller than the tabu- lated values. The conclusion is that the two sets of calculated thermal properties do not differ significantly and they are in good agreement with a 95 per cent proba- bility, though their mean values are different. In the case of the obtained thermal properties from the laboratory temperature data, neither the mean values nor the results of the F-test show a significant difference between two sets of data calculated by two methods. Similar conclusions as above were taken from the results of the F-test in comparing the values of thermal pr0perties obtained by two methods under various ambient conditions and at two different locations. The results of the F-test are given in Table 7.2.1. Statistically, we can state that the data belong to the same population with a 95 per cent probability. The thermal property values were determined for each test and the average value for all tests was used as the most realistic estimate. The average values of a 188 “0.0.0 0 H.0.0 nuancav «can Havana 0a.: 00.H 00.0HH 00.00 000.0 030.0 huopcnonaa nanonuu .vm m~.0 00.N 00.0NH oo.H0 000.0 020.0 noma< 0.H0H 00.00 00.0 00.H0 000.0 000.0 madam A0.0.m d H.: 00.0 00.0 00.0w 00.0 000.0 a00.0 sesoflo moanuso acoaobum 0~.d 00.H 00.00 00.00 000.0 030.0 mans» oudpnu . .om m~.0 00.0 00.00 00.0H 000.0 000.0 nomH< “0.0.0 0 «.0 0.HOH 00.H 00.0H 00.0 000.0 «00.0 0:000 moapmav onusoo 0N.0 0~.H 00.0: 00.00 000.0 000.0 hosoao noocdm a meanuox 0N.d 00.H 00.0: 0.~0 000.0 N0o.o henna osHu> mozaab NAHmVamumoum NAHmvaanwoum Hannahm acnmonm m Hmoapano m doucasoaco annH< huncnopm asaH< hmHoQOHWI macaudonoo onhdoo ~0noH H ~00 oocdaum> moaad> adv: pcodna< acolobmm Aummulm 0cm mundaum> on» no mammamcmv mumuosoo owuamsmmm mo muw>flm50000 Hmeumsu nonmasoamo may cmmzumn somfiummfioo Hmowgmflumum H.~.0 manna 189 calculated by the nonlinear estimation method from re- corded temperature data are 0.045 and 0.055 for summer and winter, respectively. The obtained values from the simplified estimation method are 0.042 and 0.051. The calculated values of thermal properties by the estimation method were approxi- mately 8 per cent higher than those of the simplified method. The calculated values of thermal conductivity, K, and volumetric specific heat, C = pCp from experimental temperature and heat flux data are given in Table 3.6.2. The specific heat values Cp may be found from volumetric specific heat by utilizing the value of the density of asphaltic concrete. The density is relatively easy to measure and is much more insensitive to temperature change than either K or CD, as mentioned in the section regard- ing the expansion coefficient of asphaltic concrete. Though the small discrepancy is observed between the calculated thermal properties of asphaltic concrete by two methods, the obtained values are considerably higher than the assumed values (28, 2, 16). The accuracy of the values may be shown by comparing the calculated and measured pavement temperature distributions (see Figure 6.5.1). This result suggests the adoption of ther- mal diffusivity values considerably higher than the values which are generally in use in predicting the depth of frost penetration beneath flexible pavements for Central Michigan. 190 7.3 Pavement Temperature The calculated temperature at various depths of the pavements are given in Tables 6.5.1 and 6.5.2 for summer and winter, respectively. A good agreement was observed between the calcu* lated pavement temperature by the finite difference method and the recorded temperature data both in summer and winter. For the purpose of comparison, the calculated and recorded pavement temperatures are plotted in Figure 6.5.1. The application of the finite difference method in calculating pavement temperature requires a realistic value of thermal properties of pavement material. The results of the exact solution of Equation 6.2.2 indicate that the sine wave model for air and pavement surface temperature variation is a reasonable approxi— mation in estimating subsoil temperature changes, as long as a prOper sine wave frequency is selected. It was ob- served that the maximum depth at which daily variation of pavement surface temperature fluctuates is approximately between two to three feet, while the maximum depth of the annual variation does not exceed fifty to sixty feet. Below a depth of approximately three feet, temperature remains practically constant from day to day and is not subject to alternation due to change at the surface. With the aid of the exact solution and finite difference methods, we may investigate the penetration 191 of periodic temperature waves in the pavement and under- lying subgrades, the range or variation of temperature at various depths for the daily and annual changes, as well, as the rate of penetration and the time at which the maximum or minimum temperature may be. CONCLUSIONS A full-depth pavement has the advantage of being a better insulator than asphaltic pavement. The asphaltic pavements are good insulators that can stabilize the subgrade at a higher temperature than that of a gravel pavement during winter. Under the same surface temperature, the freezing line (32°F isotherm), generally penetrates to greater depths beneath the gravel section than the paved section. There is little temperature variation at lower levels in the subgrade during a given twenty-four hour period, but several days of zero temperature does have a definite influence at all levels down to the subgrade. The thickness of the pavement does not have a significant effect on the temperature of the upper-most inches of the pavement. Snow has a significant insulating effect on the penetration of the frost depth and the subgrade 192 10. 11. 193 temperature remains at a higher temperature under a snow covered pavement. The l40°F testing temperature of the standard Marshall Stability and Consistency Test is on the conservative side for the Central Michigan environment. The temperature of llO°F should be considered for the laboratory testing temperature of lower course paving mixtures. A simple method for calculating thermal conduc- tivity, K, and specific heat, C, from the tempera— ture and heat flux boundary conditions, is pro- posed. A method is introduced to calculate the heat transfer coefficient hC from recorded temperature data and thermal properties of pavement materials. The nonlinear estimation procedure and the simpli- fied method of calculating thermal properties of asphaltic concrete are satisfactory. Both methods are superior to the conventional methods which are limited to periodic surface temperature. A good agreement is found between the calculated values of thermal diffusivity from field and laboratory temperature data. 12. 13. 194 It was found that thermal properties of asphaltic concrete are not constant, but vary with temperature. The thermal diffusivity of bituminous concretes obtained by two methods, was nearly equal and considerably higher than commonly assumed. For design purposes, values of thermal diffusivity of asphaltic pavement equal to 0.043 and 0.055 BTU/hr should be adapted for summer and winter, respectively. A value of 29.0 BTU/cu ft/°F should be adapted for volumetric specific heat of asphaltic concrete for both winter and summer. The temperature in pavement and underlying soil subgrades may be accurately calculated using a modified Crank-Nicolson difference approximation applied to the heat conduction equation. A good agreement was found between the measured and calculated pavement temperatures. Recommendations for Further Research This study is the first step in evaluating the factors influencing the depth of the frost pene- tration beneath the pavement. Further research is needed in order to utilize the result of this study in developing a reliable method for pre- dicting the depth of frost penetration for pave- ment design purposes. 195 The geographic area of this study was limited to two locations in central Michigan. In order to determine the validity of the result, this study should be extended to other locations and areas other than the state of Michigan. The measurement of temperature data should be accompanied by the measurement of the moisture content of subgrade soils. It would be possible to install a continuous moisture content monitor of a resistivity type along with the thermocouples in the pavement and underlying subsoil to record the variation of moisture content at various depths. Variation of soil moisture is required in the estimation of thermal properties of subgrade soils as well as the study of the migration of moisture beneath pavements. The process of data collection should be accom— panied by the measurement of the solar radiation which is a governing factor in increasing the pavement surface temperature. Solar radiation data are required in studying the relationship between surface and air temperature. The effect of latent heat of fusion is not taken into account, thus, the temperature is calculated if no change of state takes place. In practice, the problem involves the change of state due to 196 either melting or freezing. This study should be extended to include the effect of the latent heat of fusion in determining pavement tempera- ture in freezing seasons. The studies of the thermal properties of pavement material should be continued with emphasis being placed on the determination of the thermal properties of granular base and subbase materials. The simplified method of estimating thermal properties is quite flexible and can be extended or modified with little difficulty. A few possible improvements and extensions are as follows: a. The accuracy of the procedure may be improved by weighting the measured temperature data which are utilized as the boundary condition and data. The assumed value of weighting factor equal to one was selected for simplification. b. The procedure should be extended to eliminate the assumption of uniform initial temperature. c. In deriving a criterion for finding the optimum Laplace transform parameter, utilized in the simplified method, only positive and non-imaginary values of "s" are considered. 197 Further research is needed to consider negative, as well as complex forms of the Laplace transform parameters. BIBLIOGRAPHY BIBLIOGRAPHY Abraham, H., Asphalt and Allied Substances, vol. 2, D. Van Nostrand Company, New York, N.Y. Aldrich, Hart, P., Jr., "Frost Penetration Below Highway and Airport Pavements," Bulletin 135, Highway Research Board, 1956. Aldrich, H. P. and Paynter, H. M., Analytical Studies of Freezing and Thawing of Soils, First Interim Report, Artic Construction and Frost Effects Laboratory, Corps of Engineers, 1953. Ames, W. F., Nonlinear Partial Differential Equation In Engineering, Academic Press, New York, 1965. Barber, E. S., "Calculations of Maximum Temperatures from Weather Reports," Bulletin 168, Highway Research Board, 1957. Beck, J. V., "Optimum Transient Thermal Properties," Unpublished paper. Beck, J. V.,'The Optimum Analytical Design of Transient Experiments for Simultaneous Determi— natiOns of Thermal Conductivity and Specific Heat," Ph.D. Dissertation, Mechanical Engineering Dept., Michigan State University, East Lansing, Mich., June 1964. Beck, J. V. and Dhank, A. M., "Simultaneous Determi- nations of Thermal Conductivity and Specific Heat," ASME Paper No. 65-HT-l4. Presented at the ASME-AICHE Heat Transfer Conference, Los Angeles, Cal., August 1965. Beskow, G., "Scandinavian Soil Frost Research of the Past," Highway Research Board, Proceedings, 1947. 198 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 199 Birch, F. and Clark, H., "The Thermal Conductivity of Rocks and Dependence Upon Temperature and Composition," Amer. Jour. Sci., 238, 1940. Callendar, H. L. and McLeod, C. H., "Observations of Soil Temperature with Electrical Resistance Thermometers," Transactions, Royal Society of Canada, Second Series, vol. 2, sec. III. Carlson, H. and Kersten, M. 8., "Calculation of Depth of Freezing and Thawing Under Pavements," Bulletin 71, Highway Research Board, 1953. Carslaw, H. S. and Jaeger, J. C., Conduction of Heat in Solids, Clarendon Press, Oxford, 1959. Carson, J. E., "Analysis of Soil and Air Temperatures by Fourier Techniques," Journal of Geophysical Research, vol. 68. Chudnoviskii, A. F., Heat Transfer in the Soil (translated from the Russian), IsraéI’Program for Scientific Translations, Jerusalem. Departments of the Army and the Air Force, Calculation Methods for Determination of Depths of Freeze and Thaw in Soils, Department of the Army Technical Manual TM 5-852-6, Department of the Air Force Manual AFM 88-19, 1966. Erwin Kreyszig, Advanced Engineering Mathematics, John Wiley and Sons, New York, N.Y., 1962. Farouke, P. T., "Physical Properties of Granular Materials with Reference to Thermal Resistivity," Highway Research Record, 128, Highway Research Board. Fewell, R. B., "Cold Quantities in West Virginia," Master of Science Thesis, Civil Engineering Dept., West Virginia University. Finn, F. N., "Factors Involved in the Design of Asphaltic Pavement Surface," National Cooperative Highway Research Program, Report 39, Highway Research Board, 1967. Fowle, F. E., Journal of Astrophysics, vol. 42, 1915. Gemant, A., "How to Compute Thermal Soil Conductivity," Heating, Piping, and Air Conditioning, January, 1952. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 200 Gilman, G. D., "The Freezing Index in New England," Special Report 63, Corps of Engineers, U.S. Army, 1964. Heley, W., "A Field Evaluation of the Thermal Proper- ties of West Virginia Highway Materials," unpub- lished report, 1967. Hogbin, L. B., "The Heat Capacity of Some Road Materials," Res. Note No. RN/4082/LBR, Brit. Road Res. Lab., November 1961. Hooks, C. C. and Goetz, W. H., "Laboratory Thermal Expansion Measuring Techniques Applied to Bituminous Concrete," Report to U.S. Army Engi- neers, Waterways Experiment Station, Vicksburg, Miss., by Purdue University, August 1964. Johnson, A. W. and Lovell, C. W., "Frost Action Research Needs," Bulletin 71, Highway Research Board. Jumikis, A. R., The Frost Penetration Problem in Highway Engineering, Rutgers University Press, New Brunswick, New Jersey, 1955. Kallas, B. F., "Asphalt Pavement Temperature," Highway Research Record No. 150, Highway Research Board. Kimball, H. H., Monthly Weather Review, vol. 55, 1927; vol. 56, 1928; vol. 59, 1931. Kersten, Miles 8., "Thermal Properties of Soils," Experiment Station Bulletin No. 28, University of Minnesota, Institute of Technology, 1949. Mack, C., "Physical Chemistry," in Bituminous Materials: AsphaltsLTars, and Pitches, vol. I, A. J. Hoiberg, ed., Interscience, 1964. Makowski, M. W. and Mochlinski, K., "An Evaluation of Two Rapid Methods of Assessing the Thermal Resistivity of Soil," Proceedings, Institution of Electrical Engineers, vol. 103, part A, 1956. Manz, G. P., "Study of Temperature Variation in Hot Mix Asphalt Base, Surface Course and Subgrade," Highway Research Record, 150, Highway Research Board. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 201 Mickley, A. S., "The Thermal Conductivity of Moist Soil," Trans. American I.E.E. Tech. Paper 51-326. Mitchell, L. J., "Thermal Expansion Tests on Aggregates, Neat Cement and Concrete," Proc. ASTM, vol. 53, 1953. Monismith, C. L., Secor, G. A. and Secor, K. E., "Temperature Induced Stresses and Deformation in Asphalt Concrete," Proceedings of the Association of Asphalt Paving Technologists, vol. 34, 1965. Moulton, L. K., "Prediction of the Depth of Frost Penetration in West Virginia for Pavement Design Purposes," Ph.D. Dissertation, West Virginia University, 1968. Necati, Ozisik M., Boundary Value Problems of Heat Conduction, International Textbook Company, Scranton, 1968. O'Blenis, J. D., "Thermal Properties of West Virginia Highway Materials," Master of Science Thesis, Civil Engineering Dept., West Virginia University. Oosterbaan, M. D. and Leonards, G. A., "Use of Insulating Layers to Attenuate Frost Action in Highway Pavements," Highway Research Record No. 101, Highway Research Board, 1965. Penner, E., Canadian Journal of Earth Science, 7.982, 1970. Przybycien, F. E., "Bituminous Pavement Temperature Related to Climate," Highway Research Record No. 256, Highway Research Board, 1968. Scott, R. F., Heat Exchange at the Ground Surface, Cold Regions Research and Engineering Laboratory, U.S. Army Material Command, Report ll—Al, 1964. Saal, R. N. J., "Physical Properties of Asphaltic Bitumen," Chapter 2 in The Properties of Asphaltic Bitumen, ed. by J. P. H. Pfeiffer, Elsevier Publishing Company, Inc., New York. Trott, J. J., "An Apparatus for Recording the Duration of Various Temperatures in Roads," Road and Road Construction, vol. 41, no. 491, pp. 342— 345, 1963. 47. 48. 49. 50. 202 Troxell, G. E. and Davis, G. E., Composition and Properties of Concrete, McGraw-Hill, New York, 1956. Draper, N. R. and Smith, H., Applied Regression Analysis, John Wiley and Sons, Inc., New York, 1966. Ozisik, M. Nicati, Boundary Value Problems of Heat Conduction, International Textbook Co., New York, 1968. Martinus, VanRooyen and Winterkorn, Hans F., "Theoretical and Practical Aspects of the Thermal Conductivity of Soils and Similar Granular System," Highway Research Board Bulletin 168, 1957. APPENDIX 1 APPENDIX 1 LIST OF INPUT AND DEFINITION OF TERMS USED IN "ALPHA" PROGRAM Card 1 FORMAT (3 F 10.0, 3 I 10) Case, Date, DT, NTIME, NTC, NS CASE—Case number, any number desired example 2.0 DATE—date or any number example 6.24 DT-Time steps of measuring data example 1.0 hour NTIME-Number of times of data For example, we might have measurements at 0.0, 1.0, 2.0, 3.0, hours, hence, NTIME=4 NTC-Number of thermocouples when data is given example: for two intermal thermocouples, NTC=2 NS-Number of Laplace transform parameters For example: parameter 3 might have values of 0.1, 0.2, 0.3, hence, NS=3 203 Card 2 Card 3 Card 4 Card 5 204 FORMAT (8 F 10.5) x(I), I=1, NTC x(1) = depth dimension which is denoted in the vertical distances between the thermo- couples. For example, if thermocouples are located at the surface and at 2", 4", and 8" below the surface of the pavement, X(I)=0.0 0.166 0.333 0.666 FORMAT (8 F 10.5) S(J), J=l, NS S(J) = values of Laplace transform parameter. For example, parameter 5 might have values of 0.1, 0.2, 0.3, S(J) = 0.1 0.2 0.3 FORMAT (8 F 10.5) T (JJ,N), JJ=1, NTC, N=1, NTIME T (JJ,N) = represents the temperature measured by JJ thermocouples at N different times. Each set of thermocouples data should be stacked together and placed one behind the other FORMAT (8 F 10.5) TI (IT), IT=1, NTC TI = initial temperature distribution It is assumed that the temperature is uni- form throughout the pavement. If it is not so, it should be corrected for initial 205 condition according to section 5.8. For example: TI(IT) = 45.0 45.0 45.0 206 « .vm. o 7 ZDKWZQEF “hinZ.HnZ.nz.oovkv “RH".mvutmo} A.:.....Hh.Hu./H.?....)..,0F.X0. F.0fVCU. amfifl.m0whnfli AthHV.fluionzooookvaofl~omofiqu0 UHZ.HufiD on" on .ur.fin7.afivm0 “um".moukHWB “.mzofinfioafivm.xqvhq.xF~>HWDPLTNQ Jd_,_n/..m1.r WMFq¥ cw“ 0" up“ Owu “(a (Ca U UU z... UZC* FUHJ* .wcow* 0020......x. ECU \\ 0C0 \\ 207 “Pom.m0LFHGE AA..H.LL.H0..L...MVOOJ,..HAU.H.UFHC....).,.(,. “UH0UFH0.:;<¥ owx \\ APPENDIX 2 210 7; ‘I .,< “au.cfiu.xu0m. lfivhcznon “30>.nyvn.fiujc.k..¥0_.L.VMM¢.HM1&.“VCU.A?V>.AV;Z.Hu3 mom on Paum3Hk “0.0Humv quuou .yz.~u¥ .ny.Z.~ufi .010>v AOC~.N.Pmou >Z.¥Z “wom.mv qufl .Ofiv<.ACHVQZ<.ACHOE..OH.>..CNV uc Omm232u>z.xz quDQ UQDFdOmOPUF uC IFO~3H< UMJDQ UQDFFH>~u3uuHO J<¥ non \\ cow \\ 211 UUtQLFZH (D0? L70$ CCU \\ U7U ZQDFUQ HHIC «RCU.NVLHHDJ ACCIUUIDU+<< ”*UQCH— ~10 Dc .mom.m0uk~03 A>ZDD.CC.QNVuflw JJdU >>\+C.Qu¢X UU nucm.flvwF~U? A>§?C.UU.mNVLGm JJQU >>\0Xumh .m+>.lumx QC nmcm.mvmkufli A>:30.E$.NNvu0w JJ>\NXHNN ml>uNX «amocuu.xmv.lfiv F<§aou << “mom.m0MFHU3 “>SDC.<<.HNVUQm JJdU >>\fiXuflN A> m+>lo.mu~x O. C [(1 .>z.yz.HHIu.a.>.d¥ OUX \\ 02m zankma xuam¢xommu0h.+x*a«Immmficmmmmq.+x*.mlmfi noooouxmmwmmormo. +x*nmlmmafiommfi.+x*Anummromonm.+x*qlmmmocm¢.......u HNNNN v ococcuam norm nu.¥vwF~&$ 0 0 03m q.q.maqg+umm.+xvu0 m mcoocumu m.m.naqc+mmm.lxvu_ «coccuam “comm quk sz<> mFDJomsq z_ Jahango .waq maurqq.xovk<2aou a . wDZHonu m. o.«uy< a" m" OF 00 o.~lu¥< XlnX (— -.-.0~ Axum" mu"? “coccuam auuom.xuam.x. uaw wzuksoamam 2