COMPUTER ‘ ASSISTED EXPERIMENTAL AND ANALYTICAL STUDY OF TIME / TEMPERATURE - DEPENDENT THERMAL PROPERTIES OF THE ALUMINUM ALLOY 2024 ~ T351 Dissertation for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY KHOSROW FARNIA 1976 _ d A- This is to certify that the thesis entitled COMPUTER-ASSISTED EXPERIMENTAL AND ANALYTICAL STUDY OF TIME/TEMPERATURE-DEPENDENT THERMAL PROPERTIES OF THE ALUMINUM ALLOY 2024-1351 V presented by '“’°Uw Khosrow Farnia has been accepted towards fulfillment of the requirements for Ph.D.‘ Mechanical Engineering degree in 7/,” Major professor / \ Datefifi 13/, MM / 037639 ABSTRACT COMPUTER—ASSISTED EXPERIMENTAL AND ANALYTICAL STUDY OF TIME/TEMPERATURE-DEPENDENT THERMAL PROPERTIES OF THE ALUMINUM ALLOY 2024-T351 By Khosrow Farnia The aluminum alloy 2024-T3Sl (Al-2024-T35l) belongs to a family of metals with a so-called precipitation-hardenable or heat- treatable characteristic. This alloy, when solution heat-treated and subjected to a temperature ranging from 300-500°F, undergoes changes in the microstructure which, in turn, influence the proper- ties of the alloy. In this investigation a transient thermal properties measurement facility was develOped to study the thermal property changes of as received aluminum alloy 2024-T351 under the influence of isothermal precipitation hardening. The developed transient measurement facility utilized the IBM 1800 computer for both the acquisition of transient temperature measurements and the analysis of this data. The recent analytical method developed by J. V. Beck and S. Al-Araji ("Investigation of a New Simple Transient Method of Thermal Property Measurement,“ Journal of Heat Transfer, Trans., ASME 96 (1974) Series C:59-64) was used to determine the values of thermal conductivity (k) and specific heat (cp) utilizing the Khosrow Farnia transient temperature measurements resulting from the IBM 1800 com- puter. The transient thermal properties measurement technique was tested using Armco iron, as received Al-2024-T351 with no precipita- tion (fast measurementcycles), and the annealed Al-2024-T35l materials. For these three cases the k and cp values are only temperature-dependent. The results of experiments are presented in both tabulated and equation form. The functional relations were found using the least-squares technique. The tested transient thermal properties measurement facility was also used to obtain k and cp values of as received Al-2024-T35l with precipitation under isothermal conditions. Values of k (in this case, time- and temperature-dependent) were mathematically modeled and the associated linear and nonlinear parameters were determined using the CDC 6500 computer. The mathematical model of k involves the volume fraction of precipitation. A differential equation is proposed which can be used to predict the instantaneous values of volume fraction of precipitation under arbitrary changes in temperature. In the solution of these equations theinstantaneous values of k are calculated using the thermal conductivity- precipitation relationship. It was found that the variation of cp for as received Al-2024-T351 with ageing time at any fixed temperature was insig- nificant. However, the k values at any ageing temperature increase with ageing time to a maximum value. When precipitation is com- pleted the values of k remain unchanged as ageing time increases. The increase in k values due to precipitation at isothermal ageing Khosrow Farnia temperature 350°F is 20.7%, while this increase for isothermal ageing temperature 425°F is only ll.6%. Relationships are given by which the increase in k and the maximum volume fraction of precipi- tation, at any ageing temperature, can be predicted. A FORTRAN computer program was developed to solve numer- ically the partial differential equation of heat conduction and the prOposed differential equation of precipitation. From values of precipitation, the time- and temperature-dependent values of thermal conductivity are calculated. The influence of precipitation on the temperature history, volume fraction of precipitate versus time, and the thermal conductivity history of the Al-2024-T35l material are presented graphically for a one-dimensional case with a step increase in surface temperature. COMPUTER-ASSISTED EXPERIMENTAL AND ANALYTICAL STUDY OF TIME/TEMPERATURE-DEPENDENT THERMAL PROPERTIES OF THE ALUMINUM ALLOY 2024-T351 By Khosrow Farnia A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mechanical Engineering 1976 ACKNOWLEDGMENTS A debt of gratitude is owed to the writer's major professor, Dr. James V. Beck, for his extensive and always perceptive comments during the period of research and for his detailed review of all chapters and valuable suggestions during the preparation of this dissertation. The writer is also grateful to the other members of his com- mittee, Professors Kenneth J. Arnold, Norman L. Hills, Mahlon C. Smith, and Denton D. McGrady, for their guidance and discussions. The writer's thanks go out to very many couperative and helpful personnel, in the Division of Engineering Research for pro- viding the testing materials and eqipment, in the machine shop for being so patient and careful in machining the specimens and main- taining the equipment, and at the IBM l800 computer center for making every effort to keep the computer running without failure during the ageing experiments. Gratitude is also extended to the National Science Founda- tion and the Division of Engineering Research, for their financial support. Finally, the writer is grateful to his wife, Cecelia, for her understanding and support during the graduate study and research. Her encouragement and sacrifice have been an invaluable contribution. ii TABLE OF CONTENTS LIST OF TABLES . LIST OF FIGURES LIST OF SYMBOLS Chapter l. INTRODUCTION 1-1 Objectives and Importance of This Research . 1-2 Literature Review ‘ . . . . . . l- 3 ransient Methods . . 3.1 Modified Angstrfim Method . Line Source and Probe Methods . The Flash Method . Nonlinear Estimation Method to Determine the Thermal Properties . . Linear Finite Rod and Radial Methods T l- l- l- l- 9° (1003(1) 01 #00“) '|- APPARATUS AND TRANSIENT METHOD OF DETERMINING THERMAL PROPERTIES. APPLIED TO ARMCO IRON . . . 2- l Development of Working Equations . 2- 2 Deve10pment and Description of Transient Thermal Properties Measurement Facility . 2-2.1 Hydraulic System . . 2- 2. Temperature- -Controlled Housing 2-2. Temperature Controller . 2- 2. Computer Signal Conditioner 2- 2. Thermocouple Panel . 2- 2. The IBM 1800 Computer . 2- 2. DC Power Supply Equipment . . 2-3 Method of Transient Temperature Measurements to Det Mat 2- 3. 2- 3. 2- 3. chnmiawiv ermine the Thermal Properties of Reference erial . . . . . . . 1 Reference Material . 2 Thermocouple Type and Installation 3 Electric Heater Specification and Heat Input Measurement . . Page vi ix xii Chapter 2-4 2-5 System Calibration Transient Temperature Measurements Using the IBM 1800 Computer . . Calculation of Thermal Conductivity and Specific Heat . . . . . Computer Program CALIBR. Analysis . . Errors Due to Convection and Radiation Heat Losses . . . Determination of Overall Heat Transfer Film Coefficient Regression Analysis of Heat Transfer Film Coefficient . . . Computer Program NLINA . . Errors Due to Energy Absorbed by the Heater Assembly and Heat Sink Compound (Silicone Grease). . 2-4. 6 Errors Due to Unknown Location of the Thermocouple Junctions . . . 2- 4. 7 Errors Due to Thermal Expansion Comparison and Discussion . . mmmmrrmwmmm p999¢1wwww 014th I 1 I -"'5\I Os 01-h 3. ALUMINUM 2024-T351 AND EXPERIMENTAL RESULTS woo III I WV 0301 wa-J now (‘00) 0000 The Sample Composition . . Metallurgy of Precipitation . Sequence of Precipitation . Different Techniques to Study Precipitation Hardening. . . Experimental Procedure . . Description of the Experimental Data and. Cor- responding Thermal Conductivity- Time Curve . Error Analysis Comparison and Discussion 4. MODELING . 4-1 4-2 4-3 4-4 4-5 Time and Temperature Dependence of Thermal Con- ductivity for Isothermal Ageing Condition Isothermal Experimental Data for Specific Heat of As Received A1- 2024- T351 . . . Comparison Between Dimensionless Thermal Conduc- tivity and Volume Fraction of Precipitation Assumptions Regarding the Nature of Precipita- tion During Thermal Cycling. Proposed Differential Equations of Precipitation During Arbitrary Thermal Cycling . iv Page 42 44 48 56 57 59 61 66 66 68 73 80 80 84 87 88 92 113 117 128 128 132 135 138 140 Chapter Page 4-5.1 Equation of Precipitation During Up- Cycling Thermal Process (Step Rise in Temperature) . . . . 141 4-5.2 Equation of Precipitation During Down- Cycling Thermal Process (Step Drop in Temperature) . . . . . . . . . . 144 4-6 Comparison and Discussion . . . 146 4-7 Use of the Proposed Mathematical Models in an Engineering Problem . . . . . . . . . . 154 4-7.1 Thermal Properties . . . . . . . . 155 4-7.2 Method of Solution . . . . . . . . 156 5. SUMMARY AND CONCLUSIONS . . . . . . . . . . . 168 5-1 Recommendations for Further Study . . . . . . 171 REFERENCES . . . . . . . . . . . . . . . . . 175 2-4.4 2-4.5 2-5.1 2-5.2 2-5.3 2-5.4 3-5.l 3-6.1 LIST OF TABLES Values of k and E and their respective standard deviations as apfunction of temperature for Armco iron Selected guard temperatures and corresponding heat transfer film coefficient . . . F-test criteria to determine the optimum value of order of polynomial n for a good fit Correction data for a pair of Armco iron specimens . . . . . Correctional multipliers for a pair of Armco iron specimens, low heat input (about 272 watts) Correctional multipliers for a pair of Armco iron specimens, high heat input (about 540 watts) . Thermal conductivity of Armco iron as given by various observers and present study . Specific heat of Armco iron as given by TPRC and present study . . . . . . . . The percentage difference of corrected R and E of present data and TPRC as a function of p temperature for Armco iron . . . . Percenta e error of k and cp , as defined in Equa- tions T2- 5. 7) and (2- 5. 8), as a function of temperature . . Summary of tests arrangement for A1-2024-T351 specimens . Isothermal ageing test at 350°F, _Test No.1 for Al- 2024-T351, values of k and cp and their respective standard deviations vi Page 54 62 65 69 71 72 75 75 76 78 90 94 Table 3-6. 3-6. 3-6. 3-6. 3-7. 10 .ll .12 .13 Isothermal ageing test at 350°F, _Test No.2 for A1- 2024- T351, values of R and Ep and their respective standard deviations . . Isothermal ageing test at 375°F, _Test No.1 for A1-2024-T351, values of R and E and their respective standard deviations Isothermal ageing test at 375°F,_Test No. 2 for Al-2024-T351, values of k and E and their respective standard deviations p . . . Isothermal ageing test at 400°F, _Test No.1 for A1-2024-T351, values of k and E and their respective standard deviations p . . . Isothermal ageing test at 400°F,_Test No. 2 for A1-2024-T351, values of E and E and their respective standard deviations . . Isothermal ageing test at 425°F,=Test No. 1 for A1-2024-T351, values of E and c and their respective standard deviations . . Isothermal ageing test at 425°F, _Test No.2 for A1- 2024- T351, values of R and E and their respective standard deviations p. Values of k and E and their respective standard deviations for 55 received A1- 2024- T351 at low indicated temperatures . . . Values of R and Ep and their respective standard deviations for aged Al-2024-T351 . Values of E and E and their respective standard deviations as a function of temperature for as received A1-2024-T351 . . . Values of R and E and their respective standard deviations as a function of temperature for Al-2024-T351, annealed from 575°F . Values of k and Ep and their respective standard deviations as a function of temperature for Al-2024-T351, annealed from 925°F . Correction data for a pair of A1-2024-T351 specimens . . . . . . . . vii Page 95 98 99 102 103 106 107 111 112 114 115 116 117 Table 3-7.2 3-8.1 3-8.2 3-8.3 3-8.4 Correctional multipliers for a pair of A1-2024-T351 specimens Comparison of present Ra and ref. [24] and [21] for as received A1 -2024- T351 and their respec- tive percentage differences . Comparison of values ofk of present study and ref. [24] for annealed AT-2024-T351 . . Comparison of values of c of present study and ref. [24] for Al-2024- 1351 . . . Typical values of percentage error, as defined in Equations (25 .7) and (2- 5. 8), for R and cp Of A1- 2024- T351 . . . . viii Page 118 120 121 122 127 2-3. 2-3. 2-4. 2-4. .1a .1b .1c LIST OF FIGURES Infinite flat plate, insulated at the rear face . . . Transient thermal properties measurement facility . . . . Side view of transient thermal properties measurement facility . . . . . Temperature-controlled housing of the transient thermal properties measurement facility Temperature-controlled housing (schematic) Schematic diagram of DC power supply equipment . Thermocouple installation on the flat surface of the Specimen Two types of electric heaters and two specimens schematically arranged for an experiment A typical heated and insulated transient surface temp. . . Thermocouple arrangement and their respective locations in the flat surface of one specimen A plot of step-wise calculated thermal conduc- tivity . . . . . . . A configuration using idea of superposition to determine the effect of the heat losses at the rear face of the specimen . . Configuration to determine the overall heat transfer coefficient . A plot of heat-transfer coefficient u as a function of guard temperature . ix Page 25 28 29 30 32 36 39 41 47 49 53 57 60 67 Figure 2-4.4 3-2.1 4-1.2 4-5.1 4-5.2 Configuration to show the location of the thermocouple junctions . . Partial equilibrium diagram for aluminum side of aluminum-copper alloys, with temperature range for heat treating operations . . . Thermal conductivity as a function of time at 350°F for Aluminum 2024-T351 Specific heat as a function of time at 350°F for Aluminum 2024-T351 . Thermal conductivity as a function of time at 375°F for Aluminum 2024-T351 . Specific heat as a function of time at 375°F for Aluminum 2024-T351 . . . . Thermal conductivity as a function of time at 400°F for Aluminum 2024-T351 . . Specific heat as a function of time at 400°F for Aluminum 2024-T351 . . . . . . Thermal conductivity as a function of time at 425°F for Aluminum 2024-T351 . . Specific heat as a function of time at 425°F for Aluminum 2024-T351 . . . Thermal conductivity of A1- 2024- T351 as a func- tion of temperature for annealed, aged, and as received conditions . . . . Thermal conductivity of A1-2024-T351 as a func- tion of ageing time . . Time constant versus inverse absolute temp. Precipitation during up- cycling thermal process . . . . . . Precipitation during down- cycling thermal process . . . . . Volume fraction of precipitate as a function of ageing time for as received A1-2024-T351 Page 70 82 96 97 100 101 104 105 108 109 125 133 134 143 143 145 Figure 4-6.1 4-6.2 4-6.3 4-7.1 4-7.2 4-7.3 4-7.4 4-7.5 4-7.6 A plot of thermal conductivity as a function of ageing time for A1- 2024- T351 of present study and reference [1].. . A plot of thermal conductivity as a function of ageing time for Al- 2024-1351 of present study and reference [1]. . A plot of electrical conductance as a function of ageing time for Al-2024-T351 . Boundary conditions, initial condition, and nodal arrangement for A1-2024-T351 bar . Crank-Nicolson finite difference solution of conduction equation for A1-2024-T351 bar in cases of as received properties (no precipi- tation), as received with precipitation, and annealed conditions . . . . . Temperature history of the insulated surface of the Al-2024-T351 bar in cases of as received properties (no precipiation), as received with precipitation, and annealed conditions . Thermal conductivity history of the A1-2024-T351 bar in cases of as received pr0perties (no precipitation) and the annealed condition . Thermal conductivity history of as received A1-2024-T351 bar with precipitation . Volume fraction of precipitate of as received Al-2024-T351 bar . . . . . . xi Page 147 148 151 157 161 162 163 164 165 7711 K1 3' LIST OF SYMBOLS Cross-sectional area of specimen. Linear parameters. Total exposed area of the specimens Linear parameters. Linear parameter. Nonlinear parameter. Degree Celsius. Correction multiplier for thermal conductivity. Correction multiplier for specific heat. Average specific heat. Average of four Ep values. Average of sixteen Ep values. Specific heat of annealed specimen. Specific heat of as received specimen. Statistical F distribution. Degree Fahrenheit. Heat loss or heat gain. Instantaneous heat loss or heat gain. Kelvin, the scale of absolute temperature. Thermal conductivity. Average of four k values. Average of 16 E values. xii kan kia Thermal conductivity value of as received specimen. Thermal conductivity of annealed specimen. Thermal conductivity values at isothermal ageing condi- tions. Dimensionless thermal conductivity at isothermal ageing conditions. Thermal conductivity value of aged specimen. Thermal conductivity value of aged specimen measured at room temperature. Thermal conductivity value on location i and at the time j. Specimen thickness. Mean free path. Number of observation. Number of parameter. Power input. Cumulative heat added. Heat absorbed by heater assembly and silicone grease. Rate of heat input. Resistance of electric heater. Void radius in the heat sink. Universal gas constant. Radius. Defined in Equation (4-7.8). Standard deviation. Temperature. Initial'temperature. xiii _1 t—I —1-—1—1—+ S-h-hI -4—1—-1 a m OI _q CD a: (D d- —1 d0 (.1. :3 Temperature at the insulated surface. Temperature at heated surface. Final temperature. Maximum front surface temperature. Maximum back surface temperature. Limiting temperature. Average specimen temperature. Ageing temperature. Guard temperature. Temperature at location i and time j. Time. Heat transfer film coefficient. Volume of the specimen. Millivolt output of thermocouple. Average particle velocity. Cartesian coordinate system. Sensitivity matrix, defined in Equation (2-5.4). Thermal diffusivity, units. Aluminum + copper in solution. Defined in Equation (4-7.6). Euler's constant = 0.5772. Volume fraction of precipitate Maximum volume fraction of precipitate. Volume fraction of precipitate at location i and time j. Maximum volume fraction of precipitate at location i and time j. xiv Heating time. Intermetallic compound (CuAlz) Density. Time constant. Time constant at location i and time j. Covariance matrix of the measurement errors. XV ill I T- I Till! I T i 11 CHAPTER 1 INTRODUCTION As received aluminum alloys, such as the 2024-T351 series, when subjected to temperatures ranging from 300-500°F undergo cer- tain microstructural changes in which thermal, mechanical, and electrical properties are affected. Changes in the mocrostructure of as received aluminum 2024-T351, in the above temperature range, are called "precipitation hardening.", The changes in mechanical and electrical properties of as received A1-2024-T351, due to precipitation hardening, have been reported by many investigators. With the exception of one study initiated at Michigan State Univer- sity by Al-Araji [1], no attempts have been made to determine the effects of precipitation hardening on thermal properties of as received Al-2024-T35l. One objective of this investigation is to determine the thermal properties of A1-2024-T351 under the influence of the pre- cipitation hardening which is an extension of the study performed by Al-Araji [l]. A further objective is to develop a more appro- priate relationship for thermal conductivity, which is time- and temperature-dependent, and to relate this with the percent of pre- cipitation for as received A1-2024-T351. More detailed objectives are given in the next section. The measurement of thermal properties of as received A1-2024-T351 under isothermal ageing conditions was first reported by Al-Araji [1]. Temperature and time dependent values of thermal conductivity k(T,t) and specific heat cp(T,t) were given. An empirical mathematical model was also developed to predict the thermal conductivity under restricted conditions. Values of temperature and time dependence of thermal con- ductivity of as received A1-2024-T351 obtained experimentally by Al-Araji [1] did not provide adequate information to correlate the thermal conductivity and percent of precipitation, and also, in some cases inconsistencies existed (see Chapter 4). Therefore, it necessitated the use of the same bar of A1-2024-T351 to verify the work that had already been accomplished and to obtain additional information needed to fulfill the objectives of this investigation. 1-1 Objectives and Importance of This Research The following are the primary objectives of this research: 1. Modify previous equipment and develop additional units to accommodate the necessary transient thermal properties measurement facility for this investigation. 2. Develop the associated measurement techniques. 3. Test the modified facility and new procedures by using a reference material such as Armco Magnetic Ingot Iron for DC Applications. 4. Compare the results obtained by the present method and those reported by other investigators. 5. Obtain values of thermal properties as a function of temperature for as received Al-2024-T351 before precipitation occurred (fast measurement cycles, so that it can be assumed that the amount of precipitation is negligible). 6. Obtain values of thermal properties as a function of temperature for annealed A1-2024-T351. 7. Perform isothermal ageing tests in the precipitation heat treating temperature range (300-500°F) to obtain values of temperature- and time-dependent thermal properties for as received A1-2024-T351 in the following manner: a. obtain adequate and repeatable data in each selected isothermal ageing temperature; b. hold the specimens in each selected isothermal ageing temperature for a sufficiently long ageing time so that it can be assumed that precipitation is completed at that temperature; and c. choose the smallest possible time interval between the experiments, in the initial stage of precipitation and at high ageing temperatures, in order to obtain as many data points in relatively short measurement time as is possible. 8. Mathematically model the results of experiments and determine the relationship between k(T,t) and volume fraction of precipitation. This research is important for several reasons: 1. The aluminum alloys of 2024-T351 series are commercially available and are used in various segments of the space industry and in domestic heating equipment. To design the equipment to operate safely, a precise knowledge of the properties may be required. 2. In heat transfer applications, a knowledge of thermal properties are needed to solve the equation of conduction. The mathematical model of thermal properties (time and temperature dependent) resulting from this investigation may be used in equa- tion of conduction to determine the temperature history of Al-2024-T351 accurately. 3. The change in thermal conductivity of A1-2024-T351 due to precipitation hardening is postulated to depend upon the amount of precipitation. Assuming this is true, the relationship involving thermal conductivity can also be used to determine the volume fraction of precipitation. The mathematical model of thermal conductivity and subsequently, the precipitation relation- ships, are important and useful not only in heat transfer applica- tions but also in metallurgy and material science to study the microstructural changes of heat treatable alloys under the influence of ageing temperatures. 4. Accurate measurements of thermal properties using transient methods are attractive to scientists and related experts because of short measurement time, reduced cost of Operations, and reduced heat losses. In addition, the transient methods can be used for materials whose thermal pr0perties change rapidly with time. The transient procedure and the newly developed measurement techniques proved to be accurate (see Chapter 2) and rapid (approxi- mately 35 seconds test duration for A1-2024-T351). The method can be used for any heat treatable alloys, non-heat treatable solids, and biological materials to determine the thermal conductivity, specific heat, and thermal diffusivity simultaneously from a single test. 1-2 Literature Review The kinetic theory of gases suggests the simple expression for the thermal conductivity pcp v 2 (1-2.1) k: col—I where v is an average particle velocity, A is mean free path, and pcp is heat capacity per unit volume. In solids, usually more than one kind of excitation is present. Since the energy of various excitations (lattice wave, electronics excitations, and in some cases electromagnetic waves, spin waves, etc.) gives rise to the thermal conductivity, an equation analogous to (1-2.l) can be written by including the con- tributions of each kind of excitation to the thermal conductivity. A generalized form of expression for the thermal conductivity of solids is given by [2] _ l_ _ k - 3 T (pcp)i v.i £1 (1 2.2) where the subscript i denotes the kind of excitation, and the sum- ' mation is over all kinds. Unfortunately, difficulty arises in estimating the compo- nents of equation (1-2.2). For some materials in particular, there is no adequate theory available to predict the values of mean free path 2 [2], and hence this method for determining the thermal conductivity of all solids cannot be used at the present time. The determination of thermal conductivity with an experi- mental arrangement is accomplished by using a solution of the differential equation of conduction. The transient one-dimensional differential equation of conduction for homogeneous isotropic plates can be written as: 3 3T _ gl 5? [k 5;] " pcp at _ (1'2'3) A steady state solution of Equation (1-2.3) with no radial heat losses and constant thermal conductivity leads to the well- known expression: q = _k 1(0)L;L1(Li = M TET‘ (14.4) of 7‘— where q is the rate of energy input, A is the area normal to heat flow, and AT is the steady state temperature drop across the thickness L. For heat flow in a circular cylinder of infinite length and with heat flow between radii r1 and r2 (r2:>r]) the equation analogous to (1-2.4) is 1(r1) - 1(r2) AT q = _k r = 211k _T— (1-2.5) M. m [.2] r W "l 2 By measuring the energy input rate q and the steady state temperature dr0p AT, Equations (l-2.4) and (1-2.5) can be used to calculate the thermal conductivity. Based upon these equations, various suitable experimental devices have been developed to deter- mine the thermal conductivity under steady state conditions. The present investigation cannot use steady state methods because thermal properties of as received A1-2024-T351 are time and temperature dependent and most steady methods need a relatively large time for equilibrium. Since the present investigation requires a transient method to determine the thermal properties, a detailed description of steady state methods are not given, although the names of a few methods and corresponding references are mentioned. One of the most accurate methods of steady state is the so-called Guarded Hot Plate Method. The National Bureau of Standards has used this method for the measurement of the thermal conductivity of low thermal conductivity materials [2, 3]. McElroy and Moore [4] investigated five different classes of radial heat flow methods to obtain the thermal conductivity of solid materials. It is reported that their configurations are most suit- able for measurements on powders and loose fill insulations. Watson and Robinson [5] modified Equation (l-2.4) to deter- mine the thermal conductivity of solids and heat losses simultane- ously from the results of two experiments. 1-3 Transient Methods For each different set of boundary conditions, there is a unique solution to the transient equation of conduction (1-2.3) and hence, a possible new transient method to measure the thermal diffusivity, thermal conductivity, and/or both. Based on these solutions, various transient experimental equipment has been designed to determine the thermal properties. Each transient method, in general, differs from the Others in the choice of the boundary conditions, heat source, and/or geometry. In this section, the advantages of the transient methods over steady state methods are given briefly and in the remaining part of this chapter several transient methods are described. The advantages of transient methods for measuring thermal- transport properties over the steady state methods are: First, the value of thermal diffusivity cannot be measured directly by steady state methods. However, by measuring thermal conductivity, Specific heat, and rate of energy input, the value of thermal diffusivity can then be determined. In transient methods of measuring thermal diffusivity the rate of energy input is not needed. Second, the transient methods are faster than steady state, therefore the heat losses have a lesser influence when measurement times are short. Third, in most transient methods, accurate and rapid-response instruments are essential. Fortunately, great advances in the field of instrumentation have been made in the past decade and are continuing. These advances have improved transient property measurements a great deal. Unfortunately, steady state methods cannot benefit to the same extent from these advances. 1-3.1 Modified Angstrfim Method The Angstrdm method is the oldest periodic temperature method developed to determine thermal diffusivity of solids. The describing differential equation for this method is a modification of that given in (1-2.3). The one-dimensional equation of conduc- tion with surface heat losses (radiation, conduction, and convec- tion) and constant thermal conductivity may be written as: 32(1 - 1,) 3(1 - 1,) d 2 = -——-—————-+ u (1 — 1,) (1-3.1) 8x 3t where T - Ti representS'Uwetransient temperature change in the sample, and u is a coefficient of surface heat losses (which takes into account heat losses by radiation, conduction, and convection). Equation (l-3.1) can be applied to various geometries with a periodic or sinusoidal heat source [6]. Sidles and Danielson [7] applied Equation (1-3.1) to a semi-infinite radiating rod with a heat source located at one end, whose temperature varies sinusoidally with time. The detailed mathematical derivations of the modified Angstrdm method and measure- ment procedures are given in [6, 7]. A brief derivation of the 10 working equations is also described by [l]. A more complete description of the method, modifications, and application of Equation (l-3.1) to the other geometries can also be found in [6]. The mathematical derivations of this method are not repeated here. However, a brief description of the differences between the present method and modified Angstrdm method is given below. The following are the major differences between the present method of determining the thermal properties and the modified Angstrdm method: 1. Although a similar transient partial differential equa- tion like that of (l-2.3) is applied to a flat plate with one side heated and the opposite side insulated, the mathematical approach of the present method is quite different from the modified Angstrdm method (see Chapter 2 for mathematical derivations of working equa- tions of the present method). 2. The modified Angstrdm method is quasi-steady state while the present method is transient. 3. In the modified Angstrdm method a heat source whose temperature varies sinusoidally is needed. The present method does not have such a restriction. 4. The geometries are different (semi-infinite for the modified Angstrom method and finite for the present method). 5. The modified Angstrdm method yields directly only the thermal diffusivity from a single test, while the present method is a multi-property method. 11 1-3.2 Line Source and Probe Methods The solution of the transient differential equation of con- duction (l-2.3) for an infinitely long, continuous, thin heat source, embedded in an infinite, homogeneous medium, initially at equilibrium is given [8] as: 21 r ' 40.1: (II-3.5) .. :-_g_' T T1 4nk 5‘ where T - Ti is the temperature rise in the medium at a distance r from the line heat source, q is a constant heat rate per unit length liberated from this source at the start of t = O, and o and k are thermal diffusivity and thermal conductivity of the medium, respectively. -Ei(-z) is the exponential integral which for small values of 2 may be approximated by the series expansion, Ei (-z) = 1n (2) - z + %_22 + - ° ° + v (1-3.6) where v is Euler's constant = .5772156. For sufficiently large values of t, z = y2/4qt becomes small and then Equation (l-3.6) can be used in (l-3.5) to obtain _ z_9_ 1e - - 1 11. m [1n t1.L ln [r2] y] (1 3.7) For a time interval t2 - t], the temperature rise AT at a point in the medium is t ._9_ _2_ _ AT 4nk 1n [t1] (1 3.8) 12 A plot of AT versus 1n (tZ/t1) is a straight line with a slope of ~q/4nk. From the measured temperatures and corresponding times, the slope of the plotted line can be determined. Using the known values of q the value of k can be calculated. Taylor and Underwood [9] modified Equation (l-3.8) to be t - t i—E} (1-3.9) =.Jl_ AT 4nk 1" [t] - tC where tc is a time constant factor which is used to correct the data obtained. The value of tc has to be found experimentally for each particular material to take into account the uncertainties introduced by the effects of contact resistance and the presence of temperature sensor. The determination of thermal conductivity by the probe method is an extension of the line source method. Wechsler and Kritz [10] used Equation (l-3.9) and evaluated the performance of 18 laboratory probes for soil and insulation material at various low temperatures. The line source and probe methods are reported to be suit- able for low thermal conductivity materials and plastic foam, powders, rock, etc. The disadvantages of these methods are the uncertainties introduced by the effects of contact resistance and material dependence of the value of tC in Equation (l-3.9). See [2], pp. 376-388. Another disadvantage is that these methods are quasi-steady state and accurate values of thermal property cannot be obtained if the pr0perties of the material are time dependent. 13 The major differences between the present method and the line source and probe methods are the following: 1. Same as 1 in Section l-3.1. 2. Same as 2 in Section l-3.l. 3. For the line source and probe methods a constant heat source is needed. The present method does not have such a restriction. 4. The geometries are different. 5. The line source and probe methods are single prop- erty methods. With these methods only the values of thermal conductivity can be determined. The present method is a multi-property method. l-3.3 The Flash Method The flash method [11] is a transient procedure for which the heat source is flash lamp or some other almost instantaneous heat source. The method can be used to determine thermal diffu- sivity, heat capacity, and thermal conductivity. Small thermally insulated specimens of uniform thickness are mounted in a ceramic holder. The front surface of the specimen is blackened with camphor black to increase the absorptivity of the surface. The energy input to the sample comes from a commercial flash tube. The speci- men assembly is installed a short distance from the envelope of the flash lamp. Chromel-alumel thermocouple leads are pressed against the back surface and the signal from the thermocouple is recorded using an oscillosc0pe. The derivation of the equations utilized in this method are found in [1, 3, 11]. It is appropriate to mention the working equations to clarify the differences between the flash method and the present method. 14 Parker and coworkers [11] assumed that the pulse radiant energy is absorbed instantaneously and uniformly in a small depth at the front surface of the thermally insulated solid of thickness L. For this case and for opaque materials the temperature history at the insulated surface is given [3, 11] as: T(I’ti ;.T1 = 1 + 2 T (-1)n e‘"2“ (1-3.11) m 1 n=1 where w = nzot/Lz, a = thermal diffusivity, Ti = initial temperature, T(L,t) = temperature at the rear surface, and Tm = the maximum temperature at the rear surface. Parker et a1. [11] plotted [T(L/t)-Ti]/(Tm-Ti) versus w (Figure l in [11]) and suggested two conditions from which thermal diffusivity could be calculated: 1. When [T(L’t)'Ti]/(Tm'Ti) = .5, w is equal to 1.38, and therefore, a = 1.38 LZ/nzt1 (1-3.12) /2 where t.”2 is the time needed for the back surface to reach half the maximum temperature rise. 2. It is also suggested that an extrapolation of the "straight line" portion of the curve given by Equation (l-3.1l) may be employed to determine the thermal diffusivity a. It is rather difficult to determine what portion of the curve can be considered 15 to be the ideal straight line; it is also reported that a small error in the slope determination of this method can lead to a con- siderable error in the value of thermal diffusivity. Nevertheless, it is reported that the w = .48 corresponds to a time in which the extrapolated straight line portion of the curve intercepts the w axis and hence: a = .48 LZ/nztx (1-3.13) where tX is the time axis intercept of the temperature versus time curve. Beck et a1. [12] analytically investigated the optimum heat-pulse experiment for determining thermal pr0perties. The investigation indicates the choice of the one-half time method, that is the first method using Equation (1-3.12), is superior to any other arbitrary time (such as the 1/4 time), providing that there are no heat losses in the system. On the other hand, if it is not con- venient to record the back face temperature for a sufficient time to determine Tm or the system cannot be well insulated, then the second method [Equation (l-3.l3)] is more appropriate [3, 11]. If there are no heat losses and if the amount of heat absorbed by the front surface is measured, the product of the density and the specific heat of the material can be calculated from: pcp=AL(ng_ T1) (1-3.14) 16 where Q is the total energy input and A is the cross-sectional area of the specimen. The thermal conductivity can be found by the relationship: k = a pep (I‘3.IS) The maximum front surface temperature rise for a flash tube radiation source is given [11] by: _ 1/2 ' Tf - Ti - 38 L (Tm - Ti)/o (1-3.16) where the constant 38 depends on the characteristic of the flash lamp, L is in centimeters, a is in centimeters squared per second, and Tm, Tf, and Ti are in °C. The advantages of the flash method are the following: 1. It needs a simple energy source for a very short time interval. The flash tubes, and recently the laser, are two such energy sources which are readily available. 2. The use of a small-sized specimen can be considered to be an advantage of the flash method due to compact testing equip- ment. However, too small specimens may cause other difficulties in regard to thermocouple size, availability of surface area for thermocouple placement, and.difficulties in thermocouple installa- tion and handling, unless well-designed equipment is provided. The disadvantages of the flash method are: 1. Both methods, that is Equations (1-3.12) and (1-3.l3), of determining thermal diffusivity depend on the shape of the curve resulting from the working Equation (1-3.ll). Small heat losses 17 and slightly inaccurate temperature measurements may result in considerable error in o. 2. Despite the relatively high temperature rise near the heated surface, the flash method neglects the variation of thermal diffusivity with temperature. 3. The effect of the relatively high temperature rise near the heated surface appears to be critical for heat treatable alloys in which time and temperature are both important. It is demonstrated for heat treatable alloys, such as received Al-2024- T351, that higher temperatures result in shorter ageing times (see Figures 3-6.l through 3-6.8 in Chapter 3). In fact, no time is needed for alloys to reach a state of complete precipitation when ageing temperatures exceed their annealing temperatures. The maximum front surface temperature can be calculated using Equation (1.3.16) given by Parker et a1. [11]. The value of the maximum front surface temperature for as received A1-2024-T351 specimen, when ageing temperature is 425°F, indicates that the portion of the specimen near the heated surface has reached the annealing tempera- ture during the first ageing test. Therefore, no additional informa- tion for this portion of the specimen can be obtained from additional tests. Differences between the present and the flash methods are the following: 1. As mentioned in Section l-3.1, these methods differ in their mathematical derivations for obtaining the working equa- tions. l8 2. In the present method, values of thermal properties are directly determined using the measured transient temperatures. The transient temperatures obtained by the flash method are used to determine t],2 for Equation (1-3.12) or tx for Equation (1-3.13) by curve fitting or by visual inspection, etc. A small error in determining the values of t”2 or tx may be compounded to lead to a considerable error in the values of thermal properties. 3. The heat losses by the present method are believed to be less than by the flash method and can be treated with less diffi- culty (see Chapter 2). The heater for the present method is sandwiched between two identical mating specimens. The heat source for the flash method is the radiant type and since there is no direct contact between the heat source and the Specimen, the heat losses of the flash method may be much more than the present method. 4. The results of ageing tests for as received Al-2024-T351 at high temperatures (above 400°F), using the flash method, may be inaccurate for two reasons. First, the temperature rise near the heated surface in the flash method, unlike the present one, is relatively large. Second, the precipitation rate at high tempera— tures is greater (see Chapter 4). For these two reasons, a signifi- cant nonhomogeneity in the specimens occurs during ageing tests, and therefore can give inaccurate results. 19 1-3.4 Nonlinear Estimation Method to Determine the Thermal Properties The nonlinear estimation method is a procedure to estimate thermal properties by making calculated temperatures agree in a least-squares sense with corresponding measured temperatures. In this transient procedure, generally a finite-difference method and a digital computer are used to reduce the solution of the equation of conduction (l-2.3) to a set of algebraic equations which are solved simultaneously. In addition to the prescription of the thermal histories of the boundaries of the sample, a transient measured temperature at a location of the sample is also needed. The measured and corresponding calculated temperatures of the pre- scribed location are made to agree in a least-squares sense by minimizing a sum of squares. The computer program PROPERTY developed by Beck [13] incorporates the nonlinear estimation method. Program PROPERTY uses the Crank-Nicolson finite-difference approximation with two sets of boundary conditions to determine the thermal diffusivity or thermal conductivity and specific heat. To calculate the thermal diffusivity only, time variable heated surface temperatures and initial conditions are needed. An insulation condition could also be used for the other boundary condition to solve the partial differential equation of conduction. In addition, a measured transient temperature is needed at some location other than where the surface temperature is prescribed. A convenient location for 20 this additional thermocouple, in this investigation, is on the insulated surface [12, 14] opposite the heated surface. To determine the thermal conductivity and specific heat simultaneously, the surface heat flux history must be known instead of the surface temperature. A transient temperature history mea- sured at any location in the specimen is needed for the sum of squares function. Optinnmilocations for measuring transient tempera- tures are at the heated and/or insulated surfaces [15]. The experimental data collected by the present method can be introduced as an input to program PROPERTY to determine the thermal diffusivity or thermal conductivity and Specific heat. In fact, a few sets of experimental data generated in the present research have been analyzed using program PROPERTY. PROPERTY is a multi-purpose program with several built-in options. It can be used in composite geometry to determine the thermal properties of a single material. PROPERTY can also be used to determine the temperature variable properties which are given sequentially. This program provides much useful information including the sum of squares of residuals (i.e., a degree of agree- ment between the calculated and measured temperatures). The pro- gram, due to its versatility, requires a relatively large comPUter capacity; unfortunately, it cannot be run on the IBM 1800 computer which is readily accessible to the Department of Engineering Research. The working equations for this investigation (derived in Chapter 2) are such that a simple FORTRAN program can make the 21 necessary calculations efficiently and effectively. The computer program COND (also see Chapter 2) was developed to process the experimental data obtained for this investigation using the IBM 1800 computer. Property values obtained by program COND are nearly the same as those obtained by program PROPERTY for the purpose of this investigation. l-3.5 Linear Finite Rod and Radial Methods The mathematical principles for linear finite rod and radial methods are similar to the nonlinear estimation method (see Section 1-3.4). Klein et a1. [16] applied the linear finite rod method to determine the thermal diffusivity of Armco iron. The transient temperature measurements were performed at three different locations along the length of the finite rod with a coaxial radiation guard of the same material. A heater was attached to the sample at one end to provide the energy input. In each test, the measurement of transient temperatures at the first and third locations of the rod were used as empirical boundary conditions. Assuming a constant thermal diffusivity for each test (small temperature rise) and estimating an initial value of a, the transient differential equa- tion of conduction (l-2.3) was solved by a finite-difference method. Using the estimated a, empirical boundary conditions, and initial temperature, a set of values of temperatures as a function of time were calculated for the second locations. The calculated and corresponding measured temperatures then were 22 compared and minimized in a least-square sense to obtain the thermal diffusivity [16]. Carter et a1. [17] used a three cylindrically symmetric stack of disks with an axial heater. This method because of its geometrical configuration is called the “radial method." Three thermocouples were embedded in the midplane at three radii to record the temperature as a function of time. A calcula- tion procedure similar to the linear finite rod method then follows and thermal diffusivity of the sample can be determined. CHAPTER 2 APPARATUS AND TRANSIENT METHOD OF DETERMINING THERMAL PROPERTIES: APPLIED TO ARMCO IRON This chapter is divided into several sections, each dealing with an important part of the transient method for determining the thermal pr0perties of solids. The first section concerns the analysis developed by Beck [18]. Since detailed derivations of the method are given in [1, 18, 19], only the working equations and the related assumptions are described in the first section of this chapter. The second section of this chapter deals with the adapta- tion of the previous experimental apparatus to a thermofoil electric heater as a heat source, and development of several new components to accommodate the necessary transient measuring equip- ment for determining the thermal properties for this investigation. In the third section a new procedure for transient tempera- ture measurements is described. The thermal conductivity and specific heat of the reference material, Armco Magnetic Ingot Iron, as a function of temperature are determined. To perform the transient temperature measurements and carry out the necessary calculations, numerous computer programs are developed. In the fourth section a detailed error analysis is given and the possible errors are investigated. In this study the 23 24 working equations are modified to take into account the estimated combined radiation, conduction, and convection heat losses. The values of thermal conductivity and specific heat are compared with literature values. Recommended values of thermal conductivity and specific heat of Armco iron as a function of temperature are given in the last section. 2-l Development of Working Equations The general partial differential equation of heat conduc- tion with no internal heat generation can be written as: ifliflifl=fl _ 3X [k 3X] + By [k 3y] + 32 [k 32] DCp 3t (2 1.1) where k, p, cp, T, and t are thermal conductivity, density, specific heat, temperature, and time, respectively. Beck [18] introduced a transient method for the determina- tion of thermal conductivity and specific heat which can be applied to various geometries [1, 19]. The detailed derivations can be found in [19]. The thermal conductivity (k) of a flat plate with thickness L heated on one side and insulated on the opposite side (Figure 2-1.l) can be calculated using: 2 2 MHz-1111 A 2 L L k: (2-1.2) {:tT(X2,t) ' T(X1,t)] dt 25 l O l r insulated Figure 2-1 1 Infinite flat plate, insulated at the rear face. In developing Equation (2-1.2), the following assumptions are made: 1. Thermal conductivity (k) remains constant for the short heating time and small temperature rise across the specimen; 2. the cumulative heat added (0) during heating period is finite; 3. the density of specimen (p) is constant; 4. there are no heat losses; and 5. the solid is homogenous. The upper limit of integration in Equation (2-1.2) corre- sponds to a time elapsed in which the specimen attains its equilib- rium temperature. The cumulative heat flux (Q/A) added to the plate is given [19] as: Tf Q: .. A LT] pcp dT (2 1.3) and. 26 where Ti and Tf are the initial and final uniform temperatures of the specimen, respectively. If the temperature rise due to a short heating time is small, then the temperature dependency of (p) in Equation (2-1.3) can be ignored. Let us define 1 Ep = f f cp d1/(1f - Ti), with this assumption and definition, it 11. 1 follows that o fo cp d1 = Q/AL or 1 ‘ _ Q/A Cp - DL (Tf _ Ti) (2'1.4) The assumptions 2, 3, 4, and 5 made in Equation (2-l.2) also prevail in Equation (2-1.4). Utilizing the required measurements,Equations (2-l.2) and (2-l.4) are used to calculate the thermal conductivity and specific heat, respectively. During the experiments the transient temperatures are measured in order to evaluate 5[T(x2,t)-T(x],t)]dt. Also, the heat input (0). initial temperature (Ti)’ and final temperature (Tf) are measured. By also substituting the numerical values of x1, x2, L, p, and A into the working Equations (2-l.2) and (2-1.4), thermal conductivity (k) and average specific heat (Ep) are determined. 2-2 Develo ment and Description of Transient Thermal Properties Measurement Facility Part of the facility used in this investigation is a modi- fication of that described in [1]. Other parts were developed to make up the required facility for transient thermal properties 27 measurement. The combined equipment consists of (l) a hydraulic system, (2) temperature-controlled housing, (3) temperature con- trollers, (4) computer signal conditioner, (5) thermocouple panel, (6) the IBM 1800 computer, and (7) a DC power supply. For this investigation, the previously available equipment (numbers 1 through 4) were modified and the remaining (number 5 through 7) were developed, adapted, and added to the previous equipment. The combined equipment (except the IBM 1800 computer) is shown in Figure 2-2.1. A brief description of this equipment and its function for this investigation is given in this section. 2-2.l Hydraulic System The hydraulic system consists of (1) a hydraulic pump, (2) a non-shock cylinder-piston assembly, (3) a hand gate valve, and (4) a load frame. The hydraulic pump is a 20 gpm (150-1500) psi, of the pressure-compensated type, and can produce adequate force to move the piston without shocks at both ends of the stroke. The load frame assembly itself consists of two steel flat plates with four adjustable connecting steel rods. The speed of the piston can be regulated by the hand gate valve. The non-shock cylinder-piston assembly is installed on the lower steel plate of the load frame which is attached to a steel stand. A--Computer signal conditioner. The computer cables are connected from back face. B--Amplifier digital display (DVM). C--Ten DC DANA amplifiers. U--Temperature controllers. E--Thermocouples panel. F--Control panel. G--Three dualled NJE DC power supplies. Figure 2-2.1a Transient thermal properties neasurewent facility. A--Hydraulic pump. B--Insulation box containing the thermocouples terminal board. C--Cylinder-piston assembly. D—-Load frame. E--Timer switch and relay contact. F--Electric timer. G--AC/DC digital multi-meter. Figure 2-2.1b Side view of transient thermal properties measurement facility. The hydraulic system and the associated piping are also shown. A--Upper stationary temperature-controlled specimen housing. Upper specinen not shown. B--Lower movable temperature-controlled specimen housing. C--The lower encased specimen. D--Auxiliary specimen's heater. Not used in this investiga- tion. Figure 2-2.lc Temperature-controlled housing of the transient thermal properties measurement facility. 31 2-2.2 Temperature-Controlled Housing The temperature-controlled housing is composed of two identical mating cylinders. The upper cylinder is fastened to the upper steel plate of the load frame while the lower one is connected to the piston of the hydraulic pump. The schematic diagram of the housing is shown in Figure 2-2.2. Each cylinder consists of an outer shell and a guard heater. The space between the guard and outer shell is filled with high temperature Fiberglas insulation. The outer shell is made of sheet metal, while the guard is a copper cup with 7/16 inch thick walls. The height of the guard is 2.25 inches and its diameter is 4.875 inches. 0n the outer circumference of the guard is a helical groove containing an electrical heating coil. One end of the guard is fastened to a transite disk attached to a circular steel plate. The specimen is supported inside the guard by three 0.138 inch diameter, hollow screws. One end of each screw is fastened to the specimen while the other end is into the transite backing. Par- tially hollow, supporting screws were used to minimize the heat loss from the specimen. The temperature of the guard is maintained by a temperature controller (see next section). 2-2.3 Temperature Controller The temperature of each guard heater was controlled by an Electromax Mark III Leeds and Northrup temperature controller. This temperature controller maintained the copper guard temperature 32 Supporting screws \ \\ \\\ \\\\\ Fastened to the upper load frame l l Insulated surface XXXXIXIXXII Upper stationary specimen S\\\\\\\ L DC power supply Heated surface leads Guard heating c0115 10 ohms heater element Lower movable encased specimen s\ Connected to the piston of the pump F“\\...\.\.\. High temp. Fiber— T Transite glas insulation backing Figure 2-2.2 Temperature-controlled housing (schematic). 33 within 1.5°F about the set point temperature during the ageing tests. In all experiments, the upper and the lower guard had the same set point temperature. 2-2.4 Computer Signal Conditioner The computer signal conditioner is equipped with (l) ten DC amplifiers, (2) a built-in amplifier digital display (DVM), (3) an intercom system, and (4) a computer interrupt switch. The ten amplifiers are DANA model 3400. The output of nine of these amplifiers can be transmitted by computer cables to the data acqui- sition station (IBM 1800 computer). The maximum output of the amplifiers is i 10 volts with a nominal 20 percent over range capability. The amplifier gain was selected to be 750. Before each series of tests, the amplifiers and system were calibrated with a constant electrical DC signal. 2-2.5 Thermocouple Panel Thermocouple leads for the heaters and specimens were brought to two terminal boards, one for the upper assembly and the other for the lower assembly. The metal strips of the terminal boards were made of similar thermocouple metals to minimize the effects of dissimilar metal junctions. The terminal boards were placed in a closed insulation box to reduce the effects of ambient air tempera- ture fluctuations. Copper extension leads were used to connect the terminal boards to the taper-pin terminal blocks which were located adjacent to the control panel. 34 2-2.6 The IBM 1800 Computer The IBM 1800 equipment consists of an analog and a digital computer (internally connected to form a unit), a 1442 card read/punch, a 1443 line printer, an 1816 typewriter/keyboard, and a 563 call comp. The analog and digital computers can operate inde- pendently or can be used as a unit in experimental applications. For this investigation, the analog computer received the amplified thermocouple signals resulting from the heated and insulated surfaces of the specimens. Then the corresponding digitized signals were introduced to the digital computer for storing in a process disk or for anydesired mathematical opera- tions with appropriate conversion coefficients (i.e., millivolt- temperature conversion; see also Section 2-3.4). At the end of each experiment, the digital computer of the IBM 1800 is used to calculate the thermal conductivity and specific heat using the punched, transient, digitized thermocouple's result (see Section 2-3.6). The 563 Calcomp is used to plot and tabulate the results of experiments obtained for this investigation. 2-2.7 DC Power Supply Equipment The energy input to the specimens was obtained from three identical regulated DC power supplies (NJE Corporation, Model QR 160-3, 50-160 DC volts). Since a single power supply could not provide sufficient current for the low resistance thermofoil electric heater (see Section 2-3.3), three identical units were dualled in a parallel 35 fashion according to the manufacturer's specifications [20]. By having a three dualled power supply, the input energy to the speci- mens can be varied from 270 to 600 watts. The power output was controlled and timed by a triple-pole- double-throw relay contact, a timer switch, a toggle switch, and a timer (see Figure 2-2.3). Components involved in the controlling aspects of power output are the two poles of the relay contact, toggle switch, and timer switch. Two poles of the relay contact are placed in series to carry the power input to the thermofoil electric heater. The reason for using two combined poles for power was to prevent high current arcing across the poles. The duration of an experiment was nominally made 15 seconds using the timer switch. The experiment was initiated by manually pressing the toggle switch. Components involved in the timing aspects of power output are the third pole of the relay contact and a timer capable of measuring the heating time within 0.001 minute. When the toggle switch is pressed, the timer is simultaneously activated by the third pole. When the timer switch disconnects the power input to the heater, it also disconnects the power to the electric timer. 2-3 Method of Transient Temperature Measurements to Determine the Thermal Properties of Reference MateriaTT 2-3.1 Reference Material In order to test a new technique, it is customary to compare values obtained using the new method with the known values of a reference material. For several reasons, the reference material in 36 Three dualled NJE QRP 160-3 Two poles relay contact in series Made by Electric Time Corp. S-6, Inst. No. Power supply 21943 toggle r‘““1 Line cord r ————————— n 115 V AC 115 V AC ‘ I I l l I l l l Clock clutch I —-1 —————— i. 11] lg 2:. _._.__J Relay contact KRP I rk -3 Normally closed 14AG made by tAird pole Industrial Time Corp. Motor of contact switch Figure 2—2.3 Schematic diagram of DC power supply equipment. 37 this investigation was Selected to be Armco Magnetic Ingot Iron for DC Applications. These reasons include its high purity, homogeneity, stability over the testing temperature range, ready availability, and well-known property values [2, 21, 22, 23, 24]. The reference material provided by Steel Corporation had the density of 7.86 gr/cm3 (490.70 1b m/ft3) and had a typical analysis, in weight % of C = .015%, Mg = .028%, Ph = .005%, S = 0.25%, Si = .003%, and the balance (99.924%) Fe. 2-3.2 Thermocouple Type and Installation The thermocouple type "J" (iron and constantan) with 30 gage (0.010 inch) wire diameter was selected. Each thermocouple lead was electrically insulated with Fiberglas and the assembly was protected from moisture penetration by another layer of Fiberglas impregnated with wax. The diameter of the thermocouple assembly ranged from 0.048 to 0.050 inch. The size selection was a compromise between minimum tem- perature perturbation in the specimen and the installation plus handling durability of the wires. The millivolt output of each thermocouple was amplified, transmitted to the IBM 1800 computer, and then this calibrated transient data was converted directly to temperature. For type "J" thermocouples the temperature-voltage relaion- 2 ship can be accurately approximated by the relation T = A + BV + CV for the temperature range of 80-500°F. 38 Every experiment required two specimens. Each specimen used four thermocouples on each flat surface. Thermocouples were installed on the surface 3/4 inch from the outer edge. Each thermo- couple assembly was brought in through a 0.055 inch slanted hole which was drilled from the outer edge of the specimen, 1/4 inch below the flat surface and 90° apart (see Figure 2-3.l). Note that the thermocouple assembly diameter is approximately 0.050 inch and the diameter of each slanted hole is made to be no larger than 0.055 inch to minimize the error caused by drilling holes into the body of the specimen. Four elliptical cavities were created on each flat surface by the slanted holes. Along the minor axis and away from the cavi- ties, two slots, 0.25 inch in length, 0.010 inch in depth, and 0.010 inch in width were machined. The thermocouple assembly was brought in through the slanted hole and approximately 0.40 inch of thermocouple insulation was stripped back. The hot junction was wade by placing the circular thermocouple leads into the rectangular machined grooves and peening 0.125 inch of the thermocouple ele- nents (Figure 2-3.1). Hot junctions of this type are called separated junctions because the surface becomes an electrical con- ductor between the elements of each thermocouple. It is shown [25] that the output of thermocouple in a separated junction is a weighted mean of the two individual junction temperatures plus some error. The error in a separated junction may be positive or negative, depending upon the junction temperature and the corresponding Seebeck coefficient of the thermocouple elements. 39 Not to scale l 1 Cut out and enlarged below _ Flat surface of 3'0 the specimen I Thermocouple cavity and 1' slanted hole Peened portion of thermocouple on the flat surface Enlarged view of the thermo- couple cavity, thermocouple, and slots on the flat sur- faces of the specimen 7 * * V4 Sectional view of the slanted hole without thermocouple Figure 2-3.l Thermocouple installation on the flat surface of the specimen. 40 The error in a separated junction is zero if the two indi- vidual junction temperatures are the same. Since the test materials have relatively high conductivities and since the leads are installed close together, the errors due to this effect are very small. 2-3.3 Electric Heater Specification and Heat Input Measurement The thermofoil heater element is a flexible surface heater custom-made by Minco Products, Inc. Each heater is a thin, flexible. etched-element providing approximately uniform heat flux over a 3-inch diameter region. The nominal heater resistance is about 10 ohms. Two types of heater elements were used in this investigation. One had Kapton film insulating material (with a maximum allowable temperature of 450°F), having a thickness of 0.008 inch. The other type of heater had silicone rubber insulation, with a thickness of approximately 0.016 inch, and could be used up to a temperature of 500°F (see Figure 2-3.2). In order to increase the temperature response of the thermo- couples on the heated surfaces, a 0.015 inch thick layer of the heat sink compound (silicone grease) was uniformly applied to both heated surfaces by using a special comb. The power input to the specimens was calculated by P = Vz/R, where V is the DC voltage drop across the heater element and R is its electrical resistance. The DC voltage drop was measured by an AC/DC digital multi-meter, Keithley Model 171. A comparison of this multi-meter with a standard digital multi-meter, available in 41 u’v’uv’qr' an." ‘hh-‘A 112:...1'121" A--Thermofoil electric surface heater with silicone rubber insulation. B--Thermofoil electric surface heater with Kapton film insulating material. C--Two identical mating specinens with thermofoil surface heater between them. Figure 2-3.2 Two types of electric heaters and two specimens schematically arranged for an experiment. 42 the Department of Electrical Engineering at Michigan State Univer- sity, gave nearly identical readings in the voltage range of the present investigation. The resistance-temperature relationship of the surface heater was obtained in the following manner. The specimens were heated in steps by a heater up to 450°F. The specimens' surface temperatures and corresponding heater resistance were measured. At the end of each step, the resistance was measured by a Wheatstone- Bridge made by Industrial Instruments, Inc., Model No. RNl, Serial No. 6712, which was checked with a 10 ohm standard resistor. With this device the resistance can be measured within 0.001 ohm (0.01%). The thermocouple output corresponding to the average surface temperature was measured by the multi-meter with a temperature read- ing to within 0.03°F. Using the method of least-squares a temperature-resistance relationship was found. For a given thermofoil heater, a typical relationship found was R = 9.26920 + .00054 T where T is in °F. As can be seen, the resistance is a weak function of temperature. 2-3.4 System Calibration It is absolutely necessary to calibrate the equipment that amplifies and transmits the thermocouple's signal from the laboratory to the computer. This process makes the necessary adjustments for the amplifiers and proper corrections for cables and other parts of the system. 43 Several methods of calibration were tried to obtain a calibration relation between the voltage output of a thermocouple, v, and corresponding temperature, T. A second order polynomial, T = A + BV + CV2, was found to be adequate. From knowledge of the testing temperature range, the type of thermocouples, and the reference junction temperature, the maximum amplifier gain can be specified. The ambient temperature, about 80°F, was selected to be the reference junction temperature. During tests the ambient temperature was also recorded and necessary corrections were made for ambient temperatures other than 80°F. Note that, in the calculations of k and ED, a reference temperature correction is not needed because both calculations depend only on temperature differences. The gain of the amplifiers was adjusted in the following manner. The millivolt corresponding to the midpoint of the testing temperature was introduced to each amplifier by a parallel input connection. The output of each amplifier was observed on the digital display of the computer signal conditioner (see Section 2-2.4). The predetermined gain then was adjusted by a gain adjust- ment screw. Shorting out the amplifier input, a zero reading should appear on the DVM. This reading was adjusted to read zero using the zero adjustment screw. This procedure was repeated for each amplifier until no change in the zero and output readings was observed on the DVM display. After the amplifiers were adjusted, the testing temperature span was divided into the two groups of 80 to 300°F and 300 to 500°F. 44 The millivolts corresponding to each subgroup temperature were found in [25] and a calibration relation was found for each group using five different temperatures. The DC source used to obtain the millivolt correSponding to a temperature was Leeds and Northrup Potentiometer, Cat. No. 8687. The voltage corresponding to each subgroup temperature was measured by the multi-meter, and then introduced to the nine amplifiers. The output of these nine amplifiers was transmitted to the IBM 1800 com- puter. Computer program TAITA was used to obtain punched data cards. Computer program CALIBR uses the data resulting from the TAITA and applies the least-squares technique to determine the calibration relation. Descriptions of programs TAITA and CALIBR are given in Sections 2-3.5 and 2-3.6, respectively. 2-3.5 Transient Temperature Measurements Using the IBM 1800 Computer The transient temperature measurements were made using the computer program TAITA which is a multi-purpose program stored in a process disk. This program receives the output of nine thermo- couples simultaneously. Then the calibrated millivolt output is directly converted to temperature using the calibration relation, T = A + BV + CV2, where A, B, and C are different for each thermo- couple. The temperature output of each thermocouple can be optionally punched into computer cards or printed on computer paper, and/or punched and printed for the three consecutive periods of 45 pretest, test, and after test. The experimental data obtained in the pretest and after test periods are not stored in the disk. In this investigation, these two periods are basically for inspections. A visual inspection can be made regarding the nonequilibrium speci- mens' temperature, accuracy of the calibration relation, and the time required for specimens to reach equilibrium. These periods are also useful to check the uniformity of initial and final tempera- tures of the specimens, proper thermocouple installations, etc. In the test period 250 x 9 data points, with a prescribed time interval between the data points, were stored in the process disk. This period was started by the laboratory located interrupt switch and was terminated automatically by the computer after storing the 250 x 9 data points with the prescribed time interval between them. At the end of the test duration, any portion of the stored data can be punched into computer cards or all can be destroyed, if it is desired. The desired portion of the test period data (which was punched on computer cards) was used to calcu- late the thermal conductivity and specific heat using the computer program COND and the IBM 1800 computer. A brief description of the computer program COND is given in Section 2-3.6. In this investigation, Armco iron Specimens, 3 inches in diameter and 1 inch in thickness (see Figure 2-3.4 on page 49) were used. Tests were run with constant heat inputs between 272 and 600 watts. The heating time was fixed at 15 seconds nominally. The value of the Fourier modulus, eo/LZ, associated with this heat- ing time is 0.475 (e is heating time,|.o By using the First Law of Thermodynamics for Case l of Figure 2-4.l, the average value of specific heat can be determined as: 59 EP = OLE$;H1/$—i7 (2-4.4) where H in equations (2-4.3) and (2-4.4) is the heat loss (or gain) due to total exposed surface area. For the calculation of thermal conductivity we assume that total heat loss occurs at the rear face of the specimens. An objective is to evaluate the value of H and then determine the corresponding correction. 2-4.2 Determination of Overall Heat Transfer Film Coefficient One method of calculating H is to determine the natural convection and radiation heat transfer in an air space [29]. The schematic geometry can be seen in Figure 2-4.2. The heat transfer calculations should be performed for the rear surfaces and the circumferencescfiithe specimens. The calculations are time con- suming and in some cases uncertain because the emissivities are not accurately known. Another method of evaluating H is to determine experimentally the overall heat transfer film coefficient and then calculate the heat loss using this coefficient. Referring to Figure 2-4.2, the rate of heat loss from specimens to guards, for time larger than zero, is approximately given by: h = - pC V at' = uAt(T - TG) (2-4.5) where p, cp, V, and At are the specific weight, specific heat, total volume, and total exposed area of the specimens. Let us make the following assumptions: 6O .13/J’Afl/1/1/All/g’.l'z’l/AKA/ ./ j h / f 1’ . / h Upper spec1men h ,1 T A i h Lowerr specimen h / / l I“ 1’ I? I 7 Heater element TG Figure 2-4.2 Configuration to determine the overall heattnansfercoefficient. l. the temperature T of the specimen is uniform at any time t; 2. the overall film coefficient (u) is constant with time; and 3. the surrounding temperature (T6) is uniform with time and position. For these conditions the solution to Equation (2-4.5) is T - TG uAt ._______T - T6 = exp - pC V t (2'4.6) 1 P where T1. is the initial specimen's temperature. 6] TG, Ti’ At’ p, cp and V are known. The objective is to determine the overall film coefficient (u) using measurements of T as a function of t. To obtain measurements of T versus t, the guards and speci- mens were heated to a predetermined temperature TG. Then the specimens alone were heated l5 seconds. After an elapsed time of approximately 45 seconds, temperatures versus time were recorded. Note that a one-cycle heating (l5 seconds) of the specimen creates the same temperature differential between the specimens and the guards as during the tests. After one cycle of heating, the specimens gradually cool and approach the temperature of the guards. Experiments of this type were repeated for the selected guard temperatures in the testing temperature range and the film coefficient u was determined. The values of film coefficient for high and low heat inputs, as a function of guard temperature and specimens' temperature rise (Ti - Tf) are given in Table 2-4.l. 2-4.3 Regression Analysis of Heat Transfer Film Coefficient For the lower temperatures and smaller temperature differ- ences between the specimens and guards, it can be assumed that u varies primarily with guards' temperature. For two heat inputs the values of u as a function of guards' temperature is shown in Table 2-4.l. 62 TABLE 2-4.l. Selected guard temperatures and corresponding heat transfer film coefficient. Temp. of Ti - Tf u Run No. Guards 0 o Btu F F 2 (hr-ft -°F) Low heat input (272 watts) 1 81.8 7.46 1.63 2 156.9 7.38 3.34 3 204.6 8.48 3.58 4. 255.1 8.17 3.67 5 303.6 8.67 5.21 6 352.5 6.51 6.41 7 405.2 7.12 8.63 High heat input (540 watts) 1 83.0 15.17 1.66 2 155.8 13.06 2.62 3 205.1 15.62 2.87 4 255.6 15.28 3.50 5 307.5 16.28 4.29 6 353.1 13.88 4.95 A least-squares technique can be performed to obtain the functional relationship between u and guards' temperature. For a given heat input this relationship can be written as a polynomial of degree n u = 2 A T" (2-4.7) 63 where An are parameters and will be determined with a set of experimental data. The unbiased sample variance of u is defined as: 2 l q A 2 S = fi:fi:T' i2] (ui - ui) (2-4.8) where N is the number of observations, "i is data given in Table 2-4.l, and 0i is the corresponding estimated value from Equation (2-4.7). In performing the least-squares method with a statistical justification, it is assumed that errors are independent (indepen- dent observation), and have a zero mean, constant variance, and normal or Gaussian distributions. The question is how high an order of the polynomial must be used to obtain a reasonably good fit. There are some possible arbitrary criteria [30] and a statistical method for an optimum value of n. An arbitrary criterion states that a value of n is optimum if the sample unbiased standard deviation, S, obtained from Equation (2-4.8) is minimum, or the next higher order polynomial causes a 10 percent or less reduction in the value of S. The statistical method of obtaining an optimum value of n is more complicated. This method will be applied to the data of Table 2-4.l and an optimum number of parameters with their numerical values will be determined. The method is based on statistical F distribution [3l]. The form of a statistic Fx is defined as: 64 2 2 _ S (n-l) - S (n) - x s2(n)/ 3mzv mmucmwum Facemxga mg» so» memxwmc< Lossu use cowpuaumm upon .copm:w>mm .1 .ms« .Am.e-Nv =o_ee=am e? emceeaae o.—mp oc.m voo. Nmo. coo. P m a m.m~ NN._ mpo. NNo. omo. N a m _.o_ mN.N mpo. meo. mmo. m m N PN.N mo.mNN mNo. NNm.m Pop. e N P mNo.N m P o page? paw; nae; .mpcmomlmumu Faucmewgmaxm u o n z m.mp mm.N mmo. NON. NNP. N m a F.o_ Nm.o . «NP. mym. «um. m a m _N.N mp.m nmN. mNm.P mm_.F a m N Fo.o mm.¢m Nam. m¢.mN NPN.N m N P NmF.Nm o F o page? “mm: zop .mucwoa mwmu _mucmewsmaxm u N u z x Aav Amv Piciz msmuwsmsma _mweocx_oa No ..a a\. u a .a.a\1evm Aevm-1_-evm .Aevm mafimwmena.wwa to .02 aeemeo " e .uwm voom a sow c Powsocxpog No amuse $0 mapm> Easwpao mg» mewsgmuou op mesmuwso “moan; N.¢iN m4m . OH I Sumo: .155 wan—2H Eu: zogim B Qzqumwmxuv SNZH Em: 13:0 4 . Duos: .255 SNZH FD: 1311‘” ON I CM 1 fl ow I . 1 N 8...: .08 \2 Ci: .amimIV \_.:.m .. om baa mummzcfiuwcm: D $.58 mun—«EELS: .0 Home: £22. _ _ . _ _ mum m: mg 2. m. _ mum 68 contained a layer of approximately 0.015 inch of silicone grease. The silicone grease was spread by a special comb to provide uni- fbrm distribution throughout the heated surfaces. Much of the silicone grease squeezes out whenever the two heated surfaces, with the electric heater in between, are pressed together. We assume that one-half of the silicone grease squeezes out. Then the thickness LSh of the silicone grease and heater element is 0.0l5" + 0.008" = 0.023". Then the amount of heat absorbed by the silicone grease and the heater element can be calculated by: 06 = pep LSh A (if - Ti) (2-4.14) where 0a is the amount of heat absorbed by the silicone grease and the heater element. The pcp product for most solids (excluding porous materials) is between 40 to 55 Btu/(ft3-F). Since this product is not known for silicone grease or Kapton and further, since LSh is uncertain, the value of pcp = 52.0 Btu/(ft3-F) was arbitrarily chosen. Using Equation (2-4.l4) the values of 0a are calculated for low and high heat inputs and are listed in Table 2—4.3. 2-4.6 Errors Due to Unknown Loca- tion of the Thermocouple Junctions The thermal conductivity was determined using Equation (2-l.2). For the Armco iron specimens the locations of the thermo- couples, x1 and x2, were set equal to zero and one, respectively. There is some uncertainty in these values. When thermocouples are 69 TBBLE 2-4.3 CORRECTION 0810 FOR B PHIR OF BRHCO IRON SPECIMENS. GUBRD SHHPLE HEBT HEHT OH 8 H DL/Lo TEMP. TEHP. INPUT INPUT (2-4.14) (2-4.13) DEG.E. DEG.E. NBTTS BTU BTU BTU --- LON HEHT INPUT(BBOUT 272 NBTTS) 80. 86.0 276.31 3.9292 .042 .020 .00036 80. 90.0 276.35 3.9290 .040 .033 .00039 150. 154.5 286.61 4.0757 .043 .031 .00081 200. 205.8 274.18 3.8990 .039 .055 .00114 250. 247.0 273.97 3.8959 .040 -.036 .00140 250. 251.0 275.73 3.9210 .040 .012 .00143 300. 305.0 272.90 3.8807 .039 .073 .00178 350. 352.5 272.53 3.8754 .037 .043 .00209 350. 361.5 272.18 3.8705 .037 .199 .00215 HIGH HEBT INPUT(BBOUT 540 ”0118) 80. 97.0 535.38 7.6130 .079 .070 .00043 80. 100.0 545.03 7.7500 .079 .082 .00045 150. 165.0 530.11 7.5380 .078 .094 .00087 200. 208.0 527.31 7.4980 .077 .063 .00115 200. 227.0 538.35 7.6550 .077 .212 .00127 250. 261.0 525.40 7.4710 .076 .103 .00150 250. 270.0 550.07 7.8220 .078 .188 .00155 300. 313.5 538.30 7.6540 .075 .148 .00184 350. 362.5 522.21 7.4260 .071 .156 .00215 350. 367.5 533.86 7.5910 .073 .219 .00219 . NEGATIVE ntni Loss ncnns chi SHIN TO THE SPECIHENS. 70 attached to the specimen, the center of the thermocouples are about 0.005 from the surface of interest (see Figure 2-4.4). Not to scale Thermocouple lead Thermocouple slot ' ....- 3‘ 1.000" 'eened Figure 2-4.4. Configuration to show the location of the thermocouple junctions. If the center of the thermocouples best measures the local tempera- ture, the x1 and x2 values in Equation (2-l.2) ought to be 0.005 inch and 0.995 inch, respectively. The respective correction multi- pliers are shown in Tables 2-4.4 and 2-4.5. Note that no comparable correction is needed for Specific heat as can be seen from Equation (2-l.4). 2-4.7 Errors Due to Thermal Expansion There is no thermal expansion correction needed for the specific heat because pAL in Equation (2-4.4) is a constant. The term L/A in Equation (2-4.3) for k should be corrected for thermal expansion. The coefficient of thermal expansion 71 TBBLE 2-4.4 CORRECTIONHL HULTIPLIERS F OR B PHIR OE BRHCO IRON $PECIHENS.LON HEHT INPUT (BBOUT 272 NHTTS). GUBRD snMPLE cKH cKTC 0KE cKH 0K TEMP. TEMP. (2-4.15) DEBIFI DEBIFI "’ "’ -" --- ”-- 30. 36.0 .9393 .9900 .9993 1.005 0.934 30. 90.0 .9396 .9900 .9992 1.003 0.937 150. 149.0 .9396 .9900 .9935 0.993 0.977 150. 154.5 .9395 .9900 .9934 1.003 0.936 200. 205.3 .9399 .9900 .9977 1.014 0.992 250. 247.0 .9393 .9900 .9972 0.990 0.963 250. 251.0 .9393 .9900 .9971 1.003 0.930 300. 305.0 .9901 .9900 .9964 1.019 0.995 350. 352.5 .9905 .9900 .9953 1.011 0.937 350. 361.5 .9905 .9900 .9957 1.051 1.027 6WD WP“: °an c0PTc ccPE ccPH CCP TEMP. TEMP. (2—4. 16) DEB.F. 0E6.F. -—- -—— --- -—— --- 30. 36.0 .9396 1. 1. 0.995 .9343 30. 90.0 .9393 1. 1. 0.992 .9312 150. 149.0 .9397 1. 1. 1.002 .9914 150. 154.5 .9396 1. 1. 0.992 .9319 200. 205.3 .9900 1. 1. 0.937 .9759 250. 247.0 .9393 1. 1. 1.009 .9991 250. 251.0 .9393 1. 1. 0.997 .9363 300. 305.0 .9900 1. 1. 0.931 .9714 350. 352.5 .9900 1. 1. 0.939 .9794 350. 361.5 .9904 1. 1. 0.949 .9397 TBBLE 2-4.5 CORRECTIONBL HULTIPLIERS FOR 0 PBIR OF HRHCO IRON SPECIMENS. HIGH HEBT INPUT (BBOUT 540 NHTTS). 72 TEHPn TEMP: (2‘4. 15) 0E6.F. DEG.F. --- --— —-- --- --- 30. 97.0 .9394 .9900 .9991 1.009 0.933 30. 100.0 .9396 .9900 .9991 1.010 0.939 150. 165.0 .9396 .9900 .9932 1.012 0.990 200. 203.0 .9397 .9900 .9977 1.003 0.936 200. 227.0 .9399 .9900 .9974 1.027 1.005 250. 261.0 .9399 .9900 .9970 1.013 0.991 250. 270.0 .9901 .9900 .9969 1.024 1.001 300. 313.5 .9902 .9900 .9963 1.019 0.996 350. 362.5 .9905 .9900 .9957 1.021 0.997 350. 367.5 .9905 .9900 .9956 1.023 1.005 GURRD SfiflPLE °0Pn c0PTc ccPE ccPH ccP TEMP. TEMP. (2-4.16J 0E6.F. DEG.F. --- --- —-- --- --— 30. 97.0 .9396 1. 1. 0.991 .9303 30. 100.0 .9393 1. 1.‘ 0.939 .9791 150. 165.0 .9397 1. 1. 0.933 .9772 200. 203.0 .9393 1. 1. 0.992 .9314 200. 227.0 .9900 1. 1. 0.972 .9625 250. 261.0 .9399 1. 1. 0.936 .9762 250. 270.0 .9901 1. 1. 0.976 .9563 300. 313.5 .9902 1. 1. 0.931 .9711 350. 362.5 .9904 1. 1. 0.979 .9697 350. 367.5 .9904 1. 1. 0.971 .9619 73 [AL/L0 = (L-L0)/L0 where L is the length at room temperature and 0 L is the corresponding length at higher temperature] is given in Table 2-4.3 and corresponding correction multipliers are shown in Table 2-4.4 and 2-4.5. The following notation is used in Tables 2-4.4 and 2-4.5. Subscript k stands for average thermal conductivity and subscript cp stands for average Specific heat. A correction multiplier of unity means that no correction is needed. 1. Cka = correction multiplier for k due to the energy absorbed by the heater assembly (Section 2-4.5). 2. thc= correction multiplier for 12 due to the thermocouple placement (Section 2-4.6). 3. Cke = correction multiplier for k due to thermal expansion (Section 2-4.7). 4. Ckh = correction multiplier for k due to heat loss or gain (Section 2-4.l). CCpa’ CCptC’ Ccpe’ and Ccph are Similar correction multipliers for E . p The total effective corrections for k and 6p are: - C x C x C (2-4.15) ktc ke x Ckh O I ka C = C x C x C (2-4.16) Cp opa cptc x CCpe cph 2-5 Comparison and Discussion The least-squares technique was applied to the thermal con- ductivity and specific heat values given in Table 2-3.l. The F-test 74 criteria (Section 2-4.3) was used to determine the optimum order of polynomial. It was found that the linear relationships for k and 2p versus temperature are a reasonably good fit. The equation for thermal conductivity is similar to the one given in [21] and [23]. The values obtained for three temperature levels are tabulated in Table 2-5.l, which also shows results of TPRC [24], reference 22, and reference 23. Table 2-5.2 shows the specific heat values of the present data and the results given by TPRC. The Armco iron material selected for this investigation is similar to that of [22]. Its thermal conductivity at 100°C, deter- mined by the present method and apparatus, is only 1.90 percent lower than in [22]. l The material composition of the recommended curve given by TPRC is not specified and it is questionable whether or not a cor- rection is made in regard to the thermal expansion. The percentage differences of corrected k and 2p obtained by the present method and those given by TPRC are listed in Table 2-5.3. As temperature increases, the percentage differences for corrected thermal con- ductivity is increasing and for corrected specific heat is decreasing. The values of E obtained by the present method, in the limited temperature range (30-180°C), is in agreement with the values given by TPRC within 2 percent while 5p of the present method is 5.03 percent higher than TPRC at 30°C and only 1.79 per- cent higher at 150°C. The Armco iron material used in [23] is similar to the material of the present investigation with the exception that the 75 TABLE 2-5.l Thermal conductivity of Armco iron as given by various observers and present study. Thermal Conductivity, k, in watts/(m-C) T Chang and TPRC Powell Present Study Present Study °C Blair [22]* [24]** etefl. [23]T UncorrectedTl Correctedl”1L 30 72.0 72.13 72.35 71.12 100 67.0 66.9 67.54 66.42 65.76 150 63.20 64.26 62.18 61.93 *H. Chang and M. G. Blair, "A Longitudinal Symmetrical Heat Flow Apparatus for the Determination of the Thermal Conductivity of Metals: on Armco Iron." Thermal Conductivity--Proceedings of Eighth Conference, 1969, pp. 689-698. **Y. S. Touloukian, ed., Thermophysical Properties of High Temperature Solid Materials, Vol.’1, 1966, pp.’583-585 for Armco iron, and Vol. 2, 1966, pp. 735-737 for aluminum alloys. +D. C. Larson, R. N. Powell, and D. P. Dewitt, "The Thermal Conductivity and Electrical Resistivity of a Round-Robin Armco Iron Sample, Initial Measurements from 501x>300°C." ThermalConduc- tivity--Proceedings of Eighth Conference, 1969, pp. 675-587. ++Tab1e 2-3.l. TABLE 2-5.2 Specific heat of Armco iron as given by TPRC and present study. Specific Heat, 5p, in J/(Kg,C) T °C TPRC* Present Study Present Study Uncorrected ** Corrected** 30 .451 .4821 .4737 100 .472 .5061 .4955 150 .502 .5233 .5110 *Y. S. Touloukian, ed., Thermophysical Properties of High Temperature Solid Materials, Vol. 1, 1966, pp. 583-585 for Armco Iron, and Vol. 2, 1966, pp. 735-737 for aluminum alloys. **Table 2-3.l. 76 TABLE 2-5.3 The percentage difference of corrected R and 3p of present data and TPRC as a function of temperature for ARMCO iron. T % Difference % Difference 0C for E for Cp 30 1.23 5.03 100 1.73 4.97 150 2.05 1.79 material used in [23] contains an addition of .083 percent copper (see [22]). The uncorrected value of thermal conductivity of the present method at 30°C is only .30 percent higher than the value given by [23], while at 150°C the value of k obtained by the present method is 3.34 percent lower than that of [23]. The least-squares results far k and 5 using the corrected P values of Armco iron given in Table 2-3.1 are: 73.42 - .0766 T (2-5.1) 75') ll Ep .4644 + .000377 T (2-5.2) which may be used between 30 and 180°C. Units for k are watts/(m-C) and for ED are J/(Kg-C). The unbiased estimated sample variance of k or 5p is given as: (2-5.3) 77 where N = number of data points = 20; Yi = measured k or 5p, cor- rected values of Table 2-3.l; vi = calculated k or 8p by Equations (2-5.l) or (2-5.2), respectively; and S is the estimated standard deviation of k or 5p. The covariance matrix of i_is given [33]: ( T.9'] 5)" xT (2-5.4) covd) = x where the components of §_matrix can be determined by the tempera- ture values given in Table 2-3.1, superscript T stands for matrix transpose, and superscript -1 stands for matrTx inverse. With independent and constant variance errors, replacing y by $2£_where S2 is a sample variance of k or 5? estimated by Equation (2-5.3) and }_is the identity matrix, the diagonal terms of Equation (2-5.4) give the variance of 7 for any given temperature level. The estimated standard error of T for any temperature level is found by taking the square root of the diagonal terms of Equation (2-5.4). For two-parameter cases, such as Equations (2-5.1) and (2-5.2), Equation (2-5.4) for an arbitrary temperature T is expanded and necessary calculations are carried out. The results are: 2 1/2 est. s.e. (R) = (.908 - 7.35 x 70‘ T + 6.79 x 70'5 72) (2-5.5) 1/2 5 7 -9 2) est. s.e. (8p)== (7.39 x 70' - 2.06 x 70' T + o x 70 T (2-5.6) 78 The estimated percentage error for k and cp as a function of temperature [33] is given as: for k: est. s.e. (k) x 100 (2_5.7) E A est. s.e. (E‘) x 100 'mrc: AP (253) P Cp For the three temperatures of 30, 100, and 150°C, the per- centage error of corrected thermal conductivity and specific heat, as defined in Equations (2-5.7) and (2-5.8) are given in Table 2-5.4. TABLE 2-5.4 Percentage error of k and 6 , as defined in Equations (2-5.7) and (2-5.8), as a function of temperature. T Percentage Percentage °C Error of Error of cp 30 1.05 .62 700 ' .64 .33 150 .84 .40 The values of thermal conductivity and specific heat can be calculated by the linear Equations (2-5.l) and (2-5.2), respectively. The corresponding error can be determined by Equations (2-5.7) and (2-5.8). 79 It is believed that the values of thermal conductivity and specific heat calculated by these equations are accurate, within the corresponding percentage errors defined in Equations (2-5.7) and (2-5.8), and the percentage differences obtained with the results of other investigators are within tolerance of measurement errors . 1‘ CHAPTER 3 ALUMINUM 2024-T351 AND EXPERIMENTAL RESULTS This chapter deals with the application of the method developed in Chapter 2. Various experiments were performed on the _T“" A1-2024-T351 specimens. Thermal property values for the isothermal ageing, as received, and annealed conditions have been determined. The least-squares method is used to obtain the thermal conductivity and specific heat versus temperature relationship for the data associated with as received and annealed conditions. A mathematical model for the thermal conductivity versus time is pro- posed and the associated parameters are found using the computer program NLINA. An error analysis similar to that described in Chapter 2 is made. The thermal conductivity and specific heat values of A1-2024- T351 are compared with the available literature values. The recom- mended values of thermal properties as a function of temperature, for the as received and annealed conditions, can be found in Section 3-8. 3-1 The Sample Composition The aluminum alloy (Al-2024-T351) selected for the experi- ment was provided by Kaiser Aluminum Company. This alloy is solution heat treated and naturally aged to a substantially stable 80 81 condition. The typical analysis in weight % is: 3.8-4.9% copper, 0.50% maximum silicon, 0.50% maximum iron, O.30-O.90% manganese, l.2-l.8% magnesium, 0.10% maximum chromium, 0.25% maximum zinc, 0.20% maximum zirconium plus titanium, 0.15% maximum titanium, 0.05% maximum others each, 0.15% maximum other total, and the bal- ance of the remaining is aluminum. The density of the alloy at 68°F is given as 773 lbm/ft3 (2.77 gr/cm3) and its mechanical properties at various conditions can be found in [1, 24, 34]. 3-2 Metallurgy of Precipitation The principal alloying element in the aluminum 2024-T351 is copper. This alloy is precipitation-hardenable or "heat treatable" because the presence of copper, under certain conditions, makes the alloy susceptible to the heat treating operations in which the microstructure and consequently the properties are altered in the solid state. The theoretical aspects of precipitation hardening, due to the thermal heat treatment, for all heat treatable alloys are not, as yet, fully developed. However, it is known that precipi- tation hardening changes the mechanical and electrical as well as the thermal properties of Al-2024-T351 and as a consequence, these properties are time and temperature dependent. Figure 3-2.l represents a partial equilibrium phase diagram for the aluminum side of aluminum-copper alloys. Since the experi- mental measurement procedures of the present investigation are based on the information obtained from this phase diagram, a brief 82 Max. soluble copper at eutectic, 5.65% - L I U I D utectic temp. 1018°F a + Liquid\ * 800 700 AC2322222;?7 Temperature range for c: u. : Asolution heat treating '7 So on, . on 1 1 : 00.1 L - L. (U _. O «U 33 _ Temperature range m 8 3 - /for annealing E” Q) Q) '— 7— / / ‘ . O ' Q' 84 - % ' V1 7‘6 Temperature range for precipitation heat ‘ 8 . m 8; _ treatmg l | O -1 a + 6 : : 7 ‘163 o 1 T. ' £3 1 1 1 £1 7 7 1 O 8 Figure 3-2.l Partial equilibrium diagram for aluminum side of aluminum-copper alloys, with temperature range for heat treating operations. 83 description of the phase diagram is given in the remaining part of this section. The eutectic point is a point in which two metals are completely soluble in the liquid state and completely insoluble in each other in the solid state. The corresponding temperature and percent of copper composition at this point for Al-2024-T351, as shown in the diagram, are 1018°F and 5.65%, respectively. The region of the solid solution of aluminum (A1 or a region) is a single phase homogenous region which differs from the liquid solution region only in its physical condition. According to the typical analysis given in Section 3-1, the percent of the copper composition varies from 3.8 to 4.9. The corresponding solu- tion heat treating temperature (see Figure 3-2.1) varies from 900 to 980°F, respectively [34]. The solid solution of aluminum plus the intermetallic compound region is a two phase region. This sometimes is called the a + a region where arefers to aluminum plus copper in solution and 6 refers to the intermetallic compound (CuAlZ). The microstructure and the stability of the alloys in this region depend on the condition of the intermetallic compound (a). With sufficiently slow cooling from the solid solution heat treating temperature (see Figure 3-2.l), CuAl2 precipitates from the a solid solution in the inter- mediate temperature range. The intermetallic compound CuAlz, sur- rounding the solid solution aluminum, appears as segregate particles and is stable in its microstructure. This equilibrium state can be obtained by a prolonged annealing process, just below the solution 84 heat treating temperature [35]. On the other hand, when water quenching from the a solid solution region, the homogeneous solid solution is retained near room temperature. In this state the solid solution is supersaturated and the alloys are unstable. At any temperature level, the unstable supersaturated alloys undergo a microstructural change to attain stability. The property changes due to changes himicrostructure at low temperature are called "natural ageing." Other names, such as "pre-precipitation" or "cold hardening," are also associated with low temperature precipitation hardening [36]. Low temperature precipitation hardening is iden- tified as the first stage of decomposition of the supersaturatured solid solution. The pre-precipitation process ultimately approaches a stable condition and may require a few hours, a few days, or a few years, depending upon the nature of the alloys. Reheating the alloys to the precipitation heat treating temperature range (300- 500°F), the CuAlz (e) is precipitated which is the second stage of precipitation and is sometimes called "warm hardening" or "artificial ageing." It is believed that artificial ageing corresponds to a true precipitation in which the particles (6) can be made visible by the electron microscope or by X-ray diffraction techniques. 3-3 Sequence of Precipitation There has been a half century of work and research by many investigators to present a model which can describe the stages of age-hardening. The pioneers in this area were Guinier [37] and Preston [38]. They presented the idea of a zone, a small region in 85 the matrix enriched with solute atoms. The differences between zones and precipitates are that zones do not have well-defined boundaries and lattice structure; however, they are perfectly coherent with the lattice structure of the matrix. The approximate diameters of the zones, after rapid quenching from solid solution temperature and measuring at room temperature, are about 30-50 Angstrfims. The zones of this nature, called GP[1] (Guinier- Preston Zones 1), consist of copper-rich regions of disk-like shape, formed on {100} planes of the aluminum matrix ({100} desig- nates a face of cubic lattice structure). The sequence of precipitation for the Al-Cu alloys is given as: GP[2] . I or 0" T'e'—_T 6' Quenched supersaturated-—-+ GP[1] + solid solution The description of the zone formation at various stages of precipi- tation has been modified many times since its introduction. Despite development of sophisticated electronic equipment which can provide detailed information regarding the atomic structure of the zones, discrepancies exist about the sequence of the age-hardening process and the nature of the microstructure of the precipitates. It is believed that the first stage of precipitation, immediately after quenching, corresponds to pre-precipitation or cold hardening, which does not concern the present investi- gation. It is also found that at about 100°C or higher the GP[1] zones are replaced by coherent zones of GP[2] and 86 subsequently, due to higher temperatures or longer time, to coherent 6'. The difference in microstructure between 6' and 0 is the state of coherency. When particles grow, their coherency strain decreases and at the over-aged condition (whether with high temperature or low temperature and longer time), the noncoherent 0 will be segregated. The coexistence of 0 precipitates with the aluminum matrix in a Lu stable two-phase structure is the indication of completion of true precipitation hardening. E: The four stages of the decomposition (sequence of pre- cipitation)hH>HHoao2oo chxuzh H.win uxnmmg .wmIIMZHH o: QNN 03 8 8 2. ON 0 QET._._.1._.A._ ,8 fl .NMQQz .Ih¢21 17 m .~.®z pmuhi my m 8N T 452 5:- D e .1 .aNo one Nae N ozc N ewue . 8 8H 7 . N U 1 mo nN nu T NV 13 NV ea . QNH r7 .. n. n_ .0 Nu.» m u n. 7 03 03 T moa (J-lJ-HH1/018 AlIAIIOHGNOO TUW83H1 97 (O-OX)/P IU3H OIJIOHdS 00. mo. oo.fi mo.fi . ammp7rNoN :22H224x mom N omm kc utHh No onpozng c «c ham: QHNHouNm N.o7m umaoHN .mmziqup 2; CNN 03 8 8 2. ON o N. . _ 1 _ . _ 4. a . _ 4 _ . T .Naz 57 D . .H.®2 hmuhi mu .7 NN. .N .wmo on» mam N ozc H hawk .7 ”N. . up» .u D D D D DD @m b D b 00 300990 60> monontmvm Gun. 7 n .5 .N. G r L .7 mN. 0N. (J-WBTIIUIB lUHH 0IJI03dS THBLE 3-6.3 ISOTHERHBL BOEING TEST HT 375 DEG. F. TEST NO. 1 FOR HL-2024-T351. 98 VALUES 0E E AND E. AND THEIR RESPECTIVE STBNDHRD DEVIBTIONS. RUN TIME K sK 0P SCP NO. BTU BTU --- HRS. HR-FT-F 1 .00 83.00 1.08 .2397 .00301 2 2.05 85.23 1.64 .2378 .00365 4 6.03 93.19 3.38 .2367 .00421 5 8.33 96.48 5.70 .2362 .00525 6 11.66 94.43 3.40 .2382 .00378 7 14.33 95.83 3.34 .2349 .00419 8 17.66 94.93 5.22 .2360 .00246 9 23.93 93.08 2.31 .2366 .00446 10 28.62 98.01 3.43 .2357 .00481 11 33.50 94.64 4.26 .2356 .00345 12 38.05 98.24 4.07 .2357 .00281 13 41.76 96.66 2.06 .2370 .00472 14 48.53 99.19 0.44 .2351 .00545 15 52.70 97.79 6.63 .2360 .00474 16 57.95 98.67 2.70 .2345 .00471 17 63.70 100.19 3.76 .2354 .00522 18 74.53 96.12 3.76 .2362 .00463 19 84.23 100.23 5.35 .2375 .00470 20 88.76 96.47 3.41 .2340 .00564 21 96.93 100.62 4.54 .2363 .00651 22 101.00 97.85 3.78 .2369 .00618 99 THBLE 3-6.4 ISOTHERHBL BGEING TEST HT 375 DEG. F. TEST NO. 2 FOR BL-2024-T351. VALUES OF E AND E. AND THEIR RESPEcTIVE STHNDBRD DEVIBTIONS. 97.86 RUN TIME K SK 0. SCP NO. BTU "' HRS: HR-FT-F 1 .00 82.28 2.91 .2424 .00627 2 2.06 84.04 2.47 .2415 .00657 3 3.10 88.63 2.13 .2394 .00423 4 4.20 90.78 3.16 .2399 .00302 5 5.63 94.54 2.23 .2394 .00455 6 6.76 91.90 2.13 .2365 .00510 7 8.56 93.43 3.38 .2373 .00341 8 10.75 93.04 2.10 .2376 .00394 9 12.96 93.25 1.14 .2378 .00491 10 15.96 93.87 1.87 .2365 .00273 11 18.96 94.48 3.79 .2364 .00177 12 21.96 95.71 2.71 .2387 .00504 13 24.96 95.24 3.21 .2357 .00262 14 28.96 95.40 4.32 .2358 .00461 15 32.96 95.98 2.88 .2357 .00277 16 37.10 94.48 5.52 .2374 .00255 17 41.15 98.72 2.35 .2367 .00502 18 46.15 96.14 3.22 .2347 .00085 19 51.15 93.40 3.00 .2354 .00441 20 55.95 95.34 5.98 .2334 .00458 21 61.23 95.80 4.08 .2371 .00434 22 65.61 95.95 2.68 .2362 .00427 23 71.73 98.34 2.18 .2350 .00376 24 77.38 96.60 4.72 .2373 .00518 25 83.00 95.06 5.69 .2363 .00447 26 89.10 93.94 0.85 .2347 .00369 27 94.95 97.84 3.89 .2345 .00263 28 100.23 97.41 8.55 .2358 .00552 29 106.05 3.54 .2358 .00175 100 (O-H)/M AIIAIIOHONOO TUWHBHI . 33738 222226 moN N E.» .E NEH No 28:sz m 2 $550828 .5235 «.61» mag: .ON:-N=NN OTN ONN OON O. OO O. ON O O37. N . N . _ . _ 4. . .NNOOz .INOz- .. .N.Oz NOON- NV OON .. .N.O2 NONN- NO . .N .ONO ONO NON N OZO N NOON OON .. .D D i .D D DD .966 b... .D U . 0:1 6% b n. Ubu D r U D D .l ONN .. .o» m» 00 no can now (J-IJ-UH) ”118 AlIAIUTONOO TUW83H1 TOT (O-OX1/P IU3H OIJIOBdS oo. mo. oo.H mo.N . NmmhirNoN =22H224¢ NQN N mum kc quN No zaHNozzN c ac pcuz QHNHouNm v.07m NN:OHN .wxziqup OVN ONN OON ON OO ON. ON O . . _ . _ . _ 4 _ . _ . N 7. NN T .N.Oz NONN- D . .H .62 Hwy—.7 D 77 NN. .N .ouo on» maN N ozc H pmuh .7 MN. . D D .D D.D DD 0 ONE 0660 606 Q. Owen??? T 6 $3 3. 65 r . I 7 ON. ON. (J-W811/018 lUBH OIJIOHdS 102 TABLE 3-6.5 IsoTHERMAL AGEING TEST AT 400 DEG. F. TEST NO. 1 FOR fiL-2024-T351. VALUES OF E AND E. AND THEIR RESPEcTIVE STANDARD DEVIA710NS. RUN TIME K SK 0. ScP NO. BTU --- HRS. HR-FT-F 1 .00 36.11 2.23 .2416 .00232 2 .53 91.56 2.97 .2436 .00650 3 1.20 94.53 3.50 .2494 .00325 4 1.31 94.11 5.41 .2430 .00337 5 2.35 93.60 1.53 .2394 .00290 6 2.91 95.37 0.41 .2416 .00343 7 3.46 96.90 1.02 .2417 .00130 . 3 4.33 94.97 3.77 .2411 .00604 9 5.21 93.13 6.02 .2404 .00124 10 6.30 101.93 5.00 .2402 .00253 11 7.63 96.54 6.07 .2335 .00332 12 3.36 100.23 2.79 .2331 .00199 13 10.15 97.23 2.64 .2391 .00107 14 12.13 101.61 4.16 .2377 .00356 15 14.13 99.24 9.79 .2372 .00321 16 16.60 93.21 2.11 .2395 .00247 17 22.65 93.39 2.07 .2371 .00255 13 23.75 93.54 2.30 .2377 .00156 19 34.91 97.25 1.03 .2374 .00050 20 40.00 96.15 3.45 .2335 .00154 21 49.20 101.36 1.30 .2331 .00103 703 THBLE 3-6.6 ISOTHERHHL HGEING TEST HT 400 DEG. F. TEST NO. 2 FOR HL-2024-T351. VALUES OF E AND E. AND THEIR RESPECTIVE STHNDBRD DEVIHTIONS. RUN TIME K SK 0. SCP NO. BTU --- HRS. HR-FT-F LBM-F 1 .00 36.30 2.57 .2543 .00330 2 .13 36.17 1.73 .2511 .00440 3 .53 36.22 4.15 .2430 .00233 4 .93 91.43 1.31 .2477 .00263 5 1.46 91.76 2.03 .2435 .00379 6 1.96 95.25 2.33 .2407 .00495 7 2.53 94.74 3.11 .2435 .00214 3 3.03 95.64 1.07 .2426 .00323 9 3.66 95.55 3.03 .2429 .00421 10 4.20 95.93 3.01 .2420 .00370 11 4.90 94.56 0.33 .2331 .00262 12 5.93 96.30 7.47 .2339 .00397 13 7.14 95.60 2.53 .2410 .00401 14 3.63 100.66 2.75 .2400 .00332 15 9.91 93.59 3.79 .2391 .00272 16 11.96 99.04 2.44 .2392 .00399 17 14.13 93.73 4.39 .2396 .00321 13 16.21 97.60 2.24 .2335 .00454 19 13.21 97.54 2.67 .2396 .00131 20 21.21 97.31 6.12 .2363 .00499 21 24.33 99.44 4.09 .2396 .00427 22 27.65 93.54 2.53 .2402 .00449 23 31.13 93.44 3.33 .2336 .00330 24 34.55 100.41 2.16 .2371 .00365 25 33.53 99.43 3.73 .2376 .00310 26 42.31 93.99 4.21 .2396 .00426 27 47.33 93.37 2.09 .2357 .00301 23 52.93 97.77 1.73 .2360 .00365 29 53.63 99.53 4.03 .2337 .00369 30 63.16 100.12 3.19 .2364 .00350 104 (0-N1/M ALIAIIOHGNOO 10W83H1 ova 0 ma ooa ONN ONN hi 1531': . ammN-vNON 2:2HzaOc NON N oov Nc quN No onNoan m we NNH>Hpozozou chxNIN m.c-m NxzomN .ON:-O=NN ON OO OO O_. OO ON ON O . _ . _ 4 _ . _ . _ . fl . ow . . m. .OOOOz .INOO- I m .N.Oz NOON- D .H I .N.Oz NOON- D m N . .O .OOO 8.. OOO N OZO N NOON m a n m T 1 IA 8 . m a w H I. .6 _b nO _b um .9 DD l 73 b D .D .9 DE 4.. - b _w 7. OOH U D D D mofi 105 00. mo. (O-OX)/P IUBH OIJIOEdS oo.a mo.H 0N . HmNNIVNON :22szNc moN N oov kc mzH» Na onpoan c me New: QHNHQNNO 0.01» NN=OHN .mmziquN on on ov on ON ca 0 . _ . N . _ _ . _ _ I .N.Oz NOON- D . .H .az NOE..- D I .N .wuo oov on N 52¢ H hawk L .. D b b b b .1 b a D D U U 1 b >0 b 66 6%56 I HN. NN. mN. VN. mN. 0N. (J-WBTI/fllfl IUIH OIJIOEdS 106 THBLE 3-6.7 ISOTHERHHL HGEING TEST HT 425 DEG. F. TEST NO. 1 FOR HL-2024-T351. VALUES 0r 2 AND CP AND THEIR RESPECTIVE STANDARD DEVIATICNS. RUN TIME K sK CP SCP NO. BTU BTU --- HRS. HR-FT-F LBM-F 1 .00 37.54 3.64 .2421 .00226 2 .56 96.15 3.36 .2449 .00214 3 1.01 93.24 2.04 .2473 .00449 4 1.51 96.73 3.31 .2461 .00326 5 1.93 97.34 3.36 .2463 .00607 6 2.43 93.65 3.53 .2454 .00569 7 3.43 95.99 2.31 .2437 .00510 3 4.73 97.53 3.54 .2393 .00329 9 6.20 93.22 3.35 .2403 .00429 10 7.70 99.94 6.39 .2412 .00213 11 9.36 102.07 7.33 .2420 .00460 12 11.36 97.75 2.13 .2390 .00197 13 13.36 93.67 2.77 .2363 .00253 14 13.30 93.73 4.42 .2339 .00343 15 25.93 99.33 6.39 .2351 .00406 16 33.10 101.04 4.05 .2400 .00412 TABLE 3-6.3 ISCTHERMAL AGEING TEST AT 425 DEG. F. TEST NO. 2 FOR HL-2024-T351. 107 VALUES OF R AND E. AND THEIR RESPECTIVE STRNDBRD DEVIHTIONS. RUN TIME K SK CP ScP NO. BTU BTU --- HRS. HR-FT-F 1 .00 33.53 4.09 .2520 .00394 2 .43 95.61 2.30 .2452 .00519 3 .33 93.93 2.72 .2424 .00413 4 1.26 95.94 3.03 .2450 .00233 5 1.73 93.52 5.44 .2450 .00374 6 2.20 96.21 3.30 .2452 .00301 7 2.63 93.31 2.17 .2432 .00233 3 3.15 93.93 1.56 .2416 .00292 9 3.33 99.04 2.13 .2400 .00330 10 4.50 99.94 5.33 .2417 .00403 11 5.33 102.43 4.79 .2337 .00470 12 6.91 99.31 4199 .2336 .00349 13 3.11 101.13 2.73 .2394 .00424 14 9.30 100.55 7.43 .2405 .00275 15 10.31 99.59 4.06 .2399 .00223 16 12.33 101.07 2.49 .2403 .00232 17 14.33 93.39 2.43 .2405 .00170 13 16.43 99.33 2.23 .2330 .00391 19 13.50 101.30 1.77 .2396 .00145 20 20.63 99.02 3.15 .2396 .00209 21 23.56 103.76 2.99 .2405 .00331 22 27.90 100.46 3.00 .2374 .00260 23 32.43 101.53 1.61 .2330 .00323 24 36.63 103.07 6.50 .2392 .00443 25 40.60 102.23 4.20 .2330 .00435 26 44.60 101.96 2.30 .2332 .00275 27 43.63 100.69 2.39 .2333 .00300 23 52.76 102.44 2.63 .2399 .00277 29 56.76 100.35 4.70 .2407 .00315 108 (O-WJ/M AlIAIIOUONOO lUW83H1 ova oma ova ONN ONN . flank-VNON :zsznqc aaN N va kc quN No zaHNozzN c mc >NH>Hhozozao chmuzh N.o-m “NZOHN .mmzuquN ON on on or on ON ON 0 I.N.N.N.N._._. . .Ouooz .Ich- .- .N.O2NOON-> 7 I .N.O2NOON-D . .N .ONo va NON N ozc N NONN n. N. D D @ 7P .6 .D .9 D T D 0» LC) 00 oo mo (J-lj-UH)/018 AlIAIlOHONOO TUHUEHI O O H mow 109 00. mo. (0-921/0 lUIH OIJIOIdS oo.fi mo.a ow . NmNNIVNON :22H224c moN N va Na quN No 2aHNoan c we hcu: oHNHouNm «.07m umawHN .wxziquN on on or on ON ON 0 . _ . _ . _ . _ . _ . _ . I .N.O2 NOON- D . .N.oz NwNN- mu .- .N .ouo mNr moN N ozc N NON» .D .b .6 NO .D_u .o _u D 6%. E NN. NN. MN. VN. mN. oN. CJ-W81)/018 lUIH OIJIOJdS 110 figure of thermal conductivity versus time is a plot of the mathematical model Equation (3-6.1). The values of thermal conductivity and specific heat for all isothermal ageing tests at about room temperature are shown in Table 3-6.9. The last row of Table 3-6.9 contains the average temperature, average value of thermal conductivity, average value of specific heat, and estimated standard deviations of conduc- tivity and specific heat. The values of standard deviation for 16 values of thermal conductivity and specific heat given in Table 3-6.9 were determined by: 76 ”2 __ 7 - = 2 Sk - [Eg-igl (k1 - k):] (3-6.2) s [1 1161: 2721/2 < ) = -—- c - E 3-6.3 where R and 6p are the average of 16 values of thermal conductivity and specific heat, respectively, given in Table 3-6.9. Table 3-6.10 Shows the values of the thermal conductivity and specific heat after completion of precipitation for every iso- thermal ageing temperature, measured at about room temperature. The last row of Table 3-6.10 shows the average temperature, average specific heat, and estimated standard deviation of Specific heat calculated by using Equation (3-6.3). 111 TABLE 3-6.9 VALUES or R AND E. AND THEIR RESPECTIVE STAN- DARD DEVIHTIONS FOR AS RECEIVED AL-2024-7351 AT LOW INDICATED TEMPERATURES.THE SPECIMENS ARE THEN USED FOR IsoTHERMAL AGEING TESTS. TEST AGEING SAMPLE K SK CP J ScP NO. TEMP. TEMP. BTU BTU --- 0E6.F. DEG.F. HR-FT-F LBM—F 1 350. 85.9 71.96 2.39 .2226 .00239 - ---- 86.6 72.74 2.75 .2200 .00543 2 ---- 86.7 73.02 1.22 .2210 .00290 - "" 93.6 73.00 1.48 .2215 .00471 1 375. 86.5 72.67 2.78 .2210 .00479 - ---- 93.4 71.79 4.41 .2195 .00447 2 -"' 86.2 71.61 1028 12211 .00255 ‘ "” 03.4 72081 2:23 .2235 .00474 1 400. 101.5 72.76 0.43 .2248 .00283 - ---- 105.5 72.94 2.98 .2247 .00060 2 ---- 86.7 71.42 1.87 .2208 .00086 ’ "" 87.1 70.83 2.73 £2214 £00128 1 425. 85.0 71.66 2.52 .2206 .00268 ’ "" 03.0 72060 1.25 02227 100263 2 -'-- 85.5 73.00 1.55 .2213 000305 - ---- 92.6 73.61 2.16 .2190 .00519 K fl BVERBGE 90.6 72.41 .76 . .2216 .00170 * SEE EQUBTIONS (3-6.2) 0ND (3-6.3). .1 _ 112 TABLE 3-6.10 VALUES OF E AND E. AND THEIR RESPECTIVE STAN- DARD DEVIATICNS FOR AGED AL-2024-7351.THE SPE- CIMENS ARE AGED AT HIGH TEMP.AND THEN COOLED AT BBOUT ROOM TEMP. AND VALUES ARE DBTAINED. TEST AGEING SAMPLE K SK CP J ScP NO. TEMP. TEMP. BTU BTU --- DEG.F. DEG.F. HR-FT-F LBM-F 1 350. 85.8 92.02 3.61 .2220 .00173 - "" 92.8 02.10 1.46 .2231 000242 2 ---- 86.2 93.40 1.63 .2208 .00307 - ---- 03-2 02-50 3.14 .2232 .00210 1 375. 86.3 94.60 3.37 .2180 .00354 - ---- 93.2 02-86 3.53 .2204 .0036? 2 ---- 86.6 93.16 2.44 .2228 .00147 - ---- 93.7 92.89 2.97 .2219 .00245 1' 400. 93.5 93.66 2.56 .2219 .00235 ‘ "" 9806 05.40 5.35 02233 000515 2 ---- 86.4 95.43 1.84 .2209 .00288 ‘ "" 03-6 05-32 1.40 .2200 300381 1 425. 86.1 95.97 2.79 .2205 .00257 - ---- 86.2 97.46 3.12 .2208 .00300 2 "" 86.1 07-55 2.66 .2200 .00175 ‘ "" 03-4 05-77 5:47 .2206 100024 fl AVERHGE 0°07 """ "" 02213 .00130 8 SEE EQUBTION (3-6.3). 113 Table 3-6.11 gives the values of thermal conductivity and Specific heat of as received Al-2024-T351 as a function of tempera- ture. The experiment was carried out with a pair of fresh specimens and it is assumed that due to the relatively low temperatures and fast measurements that the amount of precipitation is negligible. The values of thermal conductivity and Specific heat for the initial ageing time of isothermal ageing tests are also given in Table 3-6.11. Several pairs of specimens were heated to the annealing temperature range (see Figure 3-l.1) and were held at that tempera- ture for 24 hours, then cooled slowly to room temperature, and thermal conductivity and specific heat were determined as a function of temperature. Tables 3-6.12 and 3-6.13 show the thermal conduc- tivity and specific heat of annealed A1-2024-T351 as a function of temperature. Data in Table 3-6.12 correspondsixithe specimens which were heated to 575°F and cooled slowly in 24 hours, while data in Table 3-6.13 was obtained from a pair of specimen heated to 925°F and cooled slowly in several days. 3-7 Error Analysis The procedure to estimate the errors is analogous to Sec- tion 2-4 of Chapter 2, with the following exceptions: 1. All experiments for Al-2024-T351 were performed with the low heat input. 2. The heater element was of the silicone rubber type. 3. The specimen thickness was 1.5 inches. 114 TABLE S-6.11 VALUES OF E AND E. AND THEIR RESPECTIVE STAN- DARD DEVIATIoNS AS A FUNCTION OF TEMPERATURE FOR AS RECEIVED AL—2024-7351. UNCORRECTED , CORRECTED SAMPLE E SK C. Sc 2 C. P TEMP. BTU BTU ' . BTU BTU DEG.F. HR-FT-F LBM—F THR-FT-F LBM-F RUN NO. OWNO‘GAQNH 86.3 93.8 ,151.4 159.0 199. 7 207.0 247.5 254. 5 308.7 73.01 73.39 73.64 73.37 76. 81 75.77 76.31 77.97 80.59 2.06 1.52 2.02 1.97 1.59 2.63 1.04 3.48 1.45 .2209 .2237 .2209 .2215 .2242 .2223 .2250 .2264 .2362 .00233 .00068 .00212 .00205 .00078 .00106 .00135 .00149 .00292 71.75 72.18 72.56 72.29 75.80 74.78 75. 42 77.07 79.77 .2168 .2192 .2160 .2166 .2187 .2169 .2190 .2204 .2294 DBTH FROM INITIBL HGEING TIME OF ISOTHERMBL RGEING TESTS 10 11 12 13 14 15 16 17 350.0 350.0 375.0 375.0 400.0 400.0 425. 0 425. 0 81.92 82.06 83.00 82.28 86. 11 86.30 87.54 88.58 6.46 3.38 1.08 2.91 2.28 2.57 3.64 4.09 .2436 .2396 .2397 .2424 .2416 .2543 .2421 .2520 .00617 .00471 .00301 .00627 .00232 .00138 .00226 .00394 81.18 81.32 82.30 81.59 85. 43 85.62 86. 89 87.92 .2361 .2322 .2321 .2347 .2337 .2460 .2339 .2435 115 TABLE 3-6.12 VALUES OF R AND E. AND THEIR RESPECTIVE STAN- DARD DEVIATleNS AS A FUNCTION OF TEMPERATURE FOR AL—2024-TS51. ANNEALED FROM 575 DEG. F. UNCORRECTED C0RRECTED P BTU BTU BTU HR-FT-F THR-ET—E LBM-F SAMPLE E TEMP. DEG.F. RUN NO. BTU LBM-F OWNOIGAWNH 85.2 86.1 86.2 87.9 92.2 93.4 149.6 150.9 156. 4 158.9 198.7 203.8 205.7 210. 5 250.4 255.9 257.4 261.8 299.3 300.3 305.3 306.6 348.7 349.5 355.5 355.7 397.7 398.3 404.3 404. 7 102.25 103.60 104.05 100.97 103.77 101.55 103.30 102.33 100.13 104.67 103.60 101.15 101.04 100.93 105.68 104.18 103.88 105.86 104.92 107.49 105.32 109.37 110.09 103.84 105.07 106.39 108.73 106.55 106.61 109.05 1.02 1.14 1.12 2.35 2.47 1.09 2.11 2.90 3.63 0.97 4.29 1.50 3.33 2.32 2.69 2.23 3.44 3.60 4.30 6.36 5.35 4.27 6.07 5.29 4.40 1.23 6.72 4.03 3.33 4.35 .2225 .2198 .2183 .2225 .2209 .2211 .2205 .2197 .2195 .2189 .2190 .2212 .2193 .2210 .2223 .2227 .2227 .2233 .2303 .2334 .2333 .2322 .2320 .2326 .2343 .2347 .2363 .2365 .2367 .2385 .00191 .00163 .00116 .00130 .00428 .00291 .00190 .00110 .00216 .00440 .00216 .00177 .00106 .00249 .00209 .00118 .00298 .00346 .00396 .00505 .00376 .00621 .00469 .00344 .00493 .00444 .00466 .00522 .00422 .00224 100.49 101.82 102.26 99. 23 102.06 99.87 101.78 100.82 98.66 103.13 102.24 99. 82 99.71 99.61 104.26 102.78 102.49 104.44 103.85 106.39 104.25 108.25 109.10 102.91 104.12 105.43 107.87 105.77 105.76 108.19 .2184 .2158 .2143 .2184 .2166 .2168 .2156 .2149 .2147 .2141 .2137 .2158 .2139 .2156 .2164 .2168 .2168 .2173 .2236 .2267 .2266 .2255 .2248 .2254 .2271 .2274 .2285 .2287 .2289 .2307 116 TABLE 3-6.13 VALUES OF R AND E. AND THEIR RESPECTIVE STAN- DARD DEVIATIGNS AS A FUNCTION OF TEMPERATURE FOR AL-2024-T351. ANNEALED FROM 925 DEG. F. UNCORRECTED CORRECTED SK C. Sc 2 CP , P BTU BTU BTU LBM-F IHR-TT-E LBM—r RUN NO. SBMPLE K TEMP. DESI Fl BTU HR-FT-F NNHHHHHHHHHH HOOVNOGAQNHOOWNOWAWNH 86.3 93.1 93.9 99.5 150.7 157.9 198.1 205.1 211.3 218.1 253.3 259.8 300.7 306.8 307.3 312.6 348.8 355.1 400.5 405.7 406.5 101.45 102.17 105.84 103.60 102.99 103.13 102.70 102.28 100.29 103.62 100.76 102.59 106.45 106.81 103.39 106.89 108.36 106.76 104.07 105.85 109.33 2.35 5.82 8.58 3.34 4.62 5.36 4.79 2.32 2.75 7.77 4.35 4.67 1.71 1.61 3.65 4.86 2.92 4.60 2.93 4.18 5.87 .2221 .2244 .2231 .2228 .2212 .2208 .2206 .2238 .2202 .2266 .2266 .2265 .2324 .2351 .2365 .2353 .2381 .2359 .2415 .2396 .2446 .00361 .00286 .00395 .00452 .00122 .00192 .00183 .00294 .00375 .00440 .00278 .00214 .00245 .00539 .00347 .00665 .00107 .00670 .00128 .00202 .00149 99. 71 100.48 104.09 101.89 101.48 101.61 101.35 100.94 98.97 102.26 99.59 101.25 105.36 105.72 102.34 105.80 107.38 105.80 103.25 105.90 108.35 .2180 .2200 .2188 .2185 .2163 .2159 .2152 .2183 .2148 .2211 .2205 .2205 .2257 .2283 .2297 .2285 .2307 .2286 .2336 .2317 .2366 117 After making the necessary calculations, the results are shown in TabTes 3-7.1 and 3-7.2. Using Table 3-7.2, the measured values of k and 5p can be corrected for any given specimen temperature. 3-8 Comparison and Discussion Analogous to Section 2-5 of Chapter 2, the least-squares technique was used for the corrected values of thermal conductivity and specific heat given in Tables 3-6.11, 3-6.12, and 3-6.13. The statistical F-test criteria (Section 2-4.3) was also used to deter- mine the usefulness of the additional term. The following relation- ships for k and 6p of Al-2024-T351 versus temperature, using the THBLE 3-7.1 CORRECTION DRTH FOR B PBIR OF BL-2024-T351 SPECIMENS. GUARD SAMPLE HEAT HEAT 0H H DL/L0 TEMP. TEMP. INPUT INPUT 12-4.14) (2-4.13) DEG.F. DEG.F. NATTS BTU BTU BTU --— 30. 34.2 272.35 3.3300 .055 .076 .00009 100. 104.2 272.52 3.3750 .054 .020 .00037 150. 154.1 271.70 3.3630 .053 .030 .00107 200. 204.0 270.33 3.3520 .052 .042 .00176 250. 253.9 270.03 3.3400 .051 .052 .00246 300. 303.3 269.25 3.3230 .050 .060 .00316 350. 353.7 263.43 3.3170 .049 .070 .00336 375. 373.6 263.02 3.3110 .049 .074 .00421 400. 403.6 267.62 3.3050 .043 .073 .00456 425. 423.5 267.21 3.7990 .043 .032 .00491 H8 THBLE 3-7.2 CORRECTIONRL HULTIPLIERS FOR H PHIR OF HL-2024-T351 SPECIMENS. GUHRD SRMPLE CKR CKTC CKE CKH CK TEMP. TEHPI (2" 4. 15) DEG.F. DEG.F. --- --- --- —-- --- 80. 84.2 .9856 .9933 .9998 1.004 .9828 100. 104.2 .9857 .9933 .9992 1.005 .9835 150. 154.1 .9860 .9933 .9979 1.008 .9853 200. 204. 0 . 9863 . 9933 . 9965 1. 011 . 9869 250. 253.9 .9866 .9933 .9951 1.014 .9884 300. 303.8 .9869 .9933 .9937 1.016 .9898 350. 353.7 .9872 .9933 .9923 1.018 .9910 375. 378.6 .9874 .9933 .9916 1.019 .9916 400. 403.6 .9875 .9933 .9906 1.021 .9921 425. 428.5 .9877 .9933 .9826 1.022 .9926 TEHP. TEMP. (2-4.16) 0E0.r. DEG.F. --- -—- --- —-— -—- 80. 84.2 .9856 1. 1. .9959 .9816 100. 104.2 .9857 1. 1. .9948 .9806 150. 154. 1 . 9860 1. 1. . 9919 . 9780 200. 204.0 .9863 1. 1. .9891 .9756 250. 253.9 .9866 1. 1. .9865 .9733 300. 303.8 .9869 1. 1. .9839 .9711 350. 353.7 .9872 1. 1. .9816 .9691 375. 378.6 .9874 1. 1. .9805 .9682 400. 403.6 .9875 1. 1. .9794 .9672 425. 428.5 .9877 1. 1. .9784 .9663 119 data in Tables 3-6.ll, 3-6.l2, and 3-6.l3, are obtained. For the as received case: Ea = 124.05 - 3.74 x 10’3 T + 5.7 x 10‘4 72 (3-8.l) Epa = 0.8704-+5.8l x 70'4 T (3-8.2) For the annealed case (annealed from 575°F): Ea" = l7l.l7 + 6.49 x 70‘2 T (3-8.3) E = 0.8827 + 3.38 x 70'4 T (3-8.4) pan For the annealed case (annealed from 925°F): Ea“ = l7l.82 + 5.45 x 10'2 T (3-8.5) e = 0.8856 + 4.72 x 70'4 T (3-8.6) pan The units for E and 6p are w/(m-C) and J/(Kg-C), respectively. The temperature T is in the range of 30-225°C. The values obtained for E and 8p for three temperatures are listed in Tables 3-8.l, 3-8.2, and 3-8.3. Table 3-8.3 is for thermal conductivity of as received Al-2024-T35l, and also shows the results<2f [21,24] and the percentage differences between the values given by [21, 24] and present Ea“ The values of thermal conductivity of annealed Al-2024-T3Sl obtained by the present inves- tigation and that of TPRC are given in Table 3-8.2, which also shows the percentage differences between the values of TPRC and the 120 .emmumqm .ag .wom_ .vcmpxgmz .mcznmgmsuwmw .mucmcm$cou zucm>mm on“ $0 mmcwuwmuogauuxpw>wuu=v -cou Peacock =.mpmumz chm>mm $0 xuw>wposucou Fmecmgp one: .E:_m .< .1 use mamw_P_z .m .933 .maop_. s==_s=_. 103 emfi-mMN .aa .hmmmpv N ._o> we. .cocv ousg< toe mmm-mmm .aa .Aoom_v _ ._o> .mpmwgmpmz uwpom mgzumgmasmh saw: mo mmwpcmaoga quwwxgmoscmch ..vm .cmwx:o_=oh .m .>3 mm.m- mm._ mm.mm~ ¢~.NNF m.mm_ ¢.mm mo. mm.v om.mmp cm.wmp o.vmp N.Fm mm._ mw.m . nm.omp om.mmp m.m¢_ m.m¢_ m m x vcm HPNH .wmm m vcm memg .mmm m . < x . mqmg mmm mo :mmzpmm :wmzpmm < mPNg mam . mocmcmvmwo & mucmgmwwwo x pcmmmga game game .Ap.m-mv 5010.351 ma swampaoFmo .Au-ev\3 cw .mu .mwocmgw333u wmmpcwocma m>wuumammg gwmzp new _mmp-¢mom-_< em>wmome m. ace 4.m_mg as. .memu .481 8:. mm 8:88.18 to cem_eeaeou _.m-m msm use .cocw ousc< com mmmummm .ga .Aommpv F ._o> .mFchwpmz twpom mgaumgmqsmh saw: 30 mmmpcmqoca quwmxmmoscmch ..cm.cmwx:ofizop.mw.>3 81.5 em.¢~_ om.¢ “5.5“, F.Nm1 «.mm mm.m mm.ou_ mp.m ~_.NNF m.om_ ~._m nm.¢ mo.m~1 o¢»5 m¢.om_ «.mw_ m.m¢_ cm cm . . . 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we. .cocw ooseq .04 mmm-mmm .aa .Aommpv F ._o> .mpmwcmpmz uwFom mgaamLmQSmh saw: mo mmwugmaoca ”mowwxsaosgmgh ..um .cmmx:o_:op .m .>3 om.m- mom. mm.~- _om. 0N.N- mom. mnm. ¢.mm m_.P- mmm. op. - mFm. mN.P- «No. Npm. N._m me._- mem. mm. - _mm. N~.N- 5mm. mum. m.mep :mau cane mnw momma soc» um_.mec< domem soc; um_mace< ua>wmomm m< Damp .8589 .mmucmcmmwwo mmmpcmogwa m>wuomamwg mezp use .Ao.m-mv new .A¢.m-mv .Am.m-mv mcowpmacm an mmumpsupwo .u°-mx\n cw .am .Pmmh-¢mom-_< com .flemu .301 new 35:». pemmmca 10 w 40 mm=_5> 9° seaweaasoo m.m-m m4m<fi 123 present Ean- Table 3-8.3 shows the values cpa and cpan of this study and values given by TPRC and their percentage differences. By comparing the present values of E and 8p and those given by [2l, 24], the following conclusions can be drawn: l. In the as received condition (Table 3-8.l), the Ea values of this study generally agree within 2.4 percent of those #5 given in [21]. At low temperatures the Ra of the present study is within 1 percent of the TPRC values. However, as the temperature 1' increases, the TPRC k values become somewhat higher than Ea' This may be due to the higher precipitation rate at the higher tempera- tures. 2. In the annealed condition (Table 3-8.2), the annealing temperature range of the a1uminum alloys, as indicated in Figure 3-2.l, is between 500 to 800°F. Heating the alloys to this tempera- ture range and with sufficiently slow cooling (annealing), a stable microstructure will be formed. The properties of two annealed aluminum specimens at two different annealing temperatures, with equal cooling processes, should be equal within the measurement errors. Data for aluminum specimens at two different annealing temperatures which are in agreement within less than 0.5 percent are shown in Table 3-8.2. The values of kan given by TPRC for the annealing temperature of 575°F is about 5 percent higher than Ean of the present investigation. 3. From Tables 3-6.l0 and 3-6.ll, the specific heat of as received Al-2024-T35l at room temperature is 0.9272 J/(kg-C) and the average value of Specific heat for four isothermal ageing 124 temperatures, after completion of precipitation, measured at room temperature is 0.9259 J/(Kg-C). It seems that the specific heat values are unaffected by the precipitation processes. The values of 8p in the cases of as received, annealed from 575°F, and annealed from 925°F are shown in Table 3-8.3. By comparing these values at three different temperatures, support is given to the above mentioned discussion. The values of 8p obtained in this investigation are within 3.3 percent of the TPRC values for all three cases. A plot in the cases of as received, annealed from 575°F, and aged values of thermal conductivity, as a function of temperature, is shown in Figure 3-8.l. The aged values of thermal conductivity are obtained from thermal conductivity-time data when ageing time is maximum. Using the least-squares technique, a linear relation for aged values of thermal conductivity of Al-2024-T351 as a func- tion of ageing temperature is found as: Rag = 147.71 + 0.1137 T (3-8.7) where Rag is the thermal conductivity of aged Al-2024-T35l, in w/(m-C), in the ageing temperature range l75-220°C and T is the ageing temperature in °C. Figure 3-8.l shows the differences in thermal conductivity obtained for Al-2024-T351 in the cases of as received (zero precipi- tation), aged at a given temperature (partial precipitation), and annealed (complete precipitation). Holding the as received speci- men at a given temperature fora sufficiently long time, the as received curve eventually meets the aged one. Then, increasing 125 6295328 S>Uou¢ «a ozc .88 55522». «8 umzhcmmmzup no 2352.: c ”c SmTvmomLc .3 iguana—zoo .5251... 19.” 53: d duo: £23 08 m8 8,. £8 08 ms 03 9% q _ . _ _ . q . . . _ . 5 ms 1 . .ou>Huuu¢ we-» . ma .. .831 . 55622.11. -8 a: 1 1 mma I m 1 oo 83 .. \ . m: r N \ 08 mg .. H 1 . 8-5 \2 21:5 35 .. on ma 1 528 #255 528 65.2: - .o .89. .12”: . p _ — . — p L . _ m5 mum m3 m2” Q. 303 126 the ageing temperature to the annealed temperature, the aged curve will reach the annealed one. A procedure similar to that described in Section 2-5 has been devised to obtain coefficients of the estimated standard error equations for two and three parameter cases. For the as received case: 2 est. s.e. (Ea) (2.98 - 9.99 x 10‘ T + 1.32 x 10- The results are: 3 T2 - 7.24 x 10‘6 T3 + 1.41 x 10'8 T4)"2 (3-8.8) 4 6 est. s.e. (8p ) = (1.23 x 10' + l.47 x lO' T a + 5 x 10'9 12)”2 For the annealed case (annealed from 575°F): A _ -2 est. s.e. (kan) - (1.64 - 2.23 x 10 T + 9.64 x 10'5 T2)“2 A _ -5 -7 est. s.e. (c ) — (2.73 x 10 - 3.73 x 10 T pan + l x 10'9 T2) 1/2 For the annealed case (annealed from 925°F): est. s.e. (flan) = (3.04 - 4.13 x 10'2 T + 1.73 x 10'4 T2)”2 _ -5 -7 est. s.e. (c ) — (5 x 10 - 6.79 x 10 T pan + 2 x 10'9 T2)”2 (3-8.9) (3-8.10) (3-8.11) (3-8.12) (3-8.13) 127 Using Equations (3-8.8) through (3-8.l3) with corresponding error Equations (2-5.7) and (2-5.8), the percentage error, as defined in Equations (2-5.7) and (2-5.8), can be evaluated as a function of temperature in the range of 30-225°C. Typical values of percentage error are shown in Table 3-8.4. No reference data with the exception of [l] was found to compare the thermal conductivity and specific heat of Al-2024-T35l as a function of time. The thermal conductivity of Al-2024-T351 obtained by [l] are generally higher than the k values of the present investigation, and in some cases certain inconsistencies exist which are discussed in Chapter 4. In order to determine the accuracy of data, the experiment at every thermal ageing temperature was repeated at least once to ascertain the repeatability (as can be seen in Figures 3—6.l through 3-6.8). The experimental results are generally repeatable. In each ageing test, the value of k increases to a maximum (aged) and then fluctuates about a mean value of k as ageing time increases. TABLE 3-8.4 Typical values of percentage error, as defined in Equations (2-5.7) and (2-5.8), for E and c of Al-2024-T35l. p T As Received Annealed From 575°F Annealed from 925°F emp. A A A C For Ea For cpa For Ean For cpan For Ean For cPan % % % % 4 % % 143.6 .43 .48 .36 .28 .46 .36 9l.7 .43 .62 .36 .28 .48 .37 55.4 .50 .84 .48 .38 .65 .51 CHAPTER 4 MODELING In this chapter the experimental data of Chapter 3 is utilized to develop a mathematical relationship for thermal conduc- tivity which is time and temperature dependent. A correlation between dimensionless thermal conductivity and volume fraction of precipitation is found. The mathematical model for thermal conduc- tivity is further extended to a more general case of transient thermal cycling. A differential equation is hypothesized to deal with thermal cycling. A practical example is given utilizing the mathematical model developed in this chapter. 4-l Time and Temperature Dependence of Thermal Conductivity fOr Isothermal Ageing Condition For each isothermal ageing temperature, a mathematical model was found using isothermal experimental data obtained from two age- ing tests [see Equation (3-6.l)]. An analogous expression can be used to obtain a mathematical model covering any isothermal ageing temperature in the temperature range 175-225°C. Using l95 data points, given in Tables 3-6.l through 3-6.8, the mathematical model for isothermal ageing condition is written as: kia(Tag,t) = kia(Tag,0) + Akia(Tag) {1 — exp [-t/T(Tag)]n} (4-1.1) 128 129 ) - k (T ,0) and k. (T ,0) and where Ak. (T 1 = k- (T ia ag Ta 39 t 7a ag Ta ag’ max k (T ) are the values of thermal conductivity obtained for . t Ta ag’ max initial (t = O) and final (t = tmax) ageing times, respectively. Tag is the ageing temperature, t is the corresponding ageing time, . (T and T(Tag) lS the time constant. Note that values of k1a ag,0), Ak(Tag), and T(Tag) in Equation (4-l.l) depend on ageing tempera- ture. Equation (4-l.l) is chosen because (l) the experimental thermal conductivity-time plots given in Figures 3-6.l, 3-6.3, 3-6.5, and 3-6.7 show the exponential form, and (2) the dimension- .0)]/ less thermal conductivity {k1 (T ,t) = [kia(Tag’t)"k (t 1a ag ia ag [Akia(Tag)1}is;in accordance with the form given for the kinetic law of precipitation [40, 41]. The values of kia(Tag’0)’ Akia(Tag ), and T(Tag) in Equation (4-l.l) have been determined by the NLINA computer program using all l95 data points, obtained from 16 ageing tests, given in Tables 3-6.1 through 3-6.8. The time exponent n in Equation (4-l.l) was calculated for every ageing temperature and the average value of 0.85 was obtained. The suggested value of n from other investigators varies from l/2 to 5/2 depending upon the ageing temperature, the nature of the alloys, and the type of property. For rod-like precipitates a value of unity has been suggested [4l]. For further discussion and experimental confirmation refer to [40, 4l, 43, and 46]. 130 In this investigation the time exponent n was chosen to be unity because the average value of 0.85 is near unity and the l95 data points do not contain sufficient information to accurately determine n (the estimation of n along with the other parameters requires extremely accurate measurements because n is correlated with the other parameters). Furthermore, with n = l, a simple dif- ferential equation that leads to Equation (4-l.l) can be proposed. By using the data of Tables 3-6.l through 3-6.8, eight val- ues of the time constant (T) as a function of temperature were determined. Various forms of models were tried to obtain T versus T relationships. The following three models each produced satisfac- tory results in the testing temperature range (l75-220°C): Q 1 T = exp 1‘00 ‘1' T—Zi'l-C] (4-1.2) Q 2. T = exp {-00 +-—%J (4-l.3) Q 3. T = exp {-00 ‘1’ fig] (4-1.4) where T is in °C; QO’ Q], and C are parameters estimated by the NLINA computer program. In this investigation the form of the model Equation (4-l.4) is preferred because of several reasons. First, the form itself is in accordance with the form of character- istic time (time constant) given by [40]. Second, the self-diffusion coefficient of precipitates (diffusion of copper into aluminum) and 131 characteristic time (T) can be correlated [40, 47]. In this correla- tion the term 01 x R, where R is the gas constant [R = 1.987 ca1./ (gram-mole-°K)], is called the overall activation energy or heat of diffusion. The overall activation energy for Al-Cu alloys was calcula- ted using two methods [48]. The values obtained by these two methods are 31,400 and 34,900 cal. per gram-mole (see [48], p. 337). A value of 1.4 i 0.1 ev (l ev = 23,047 cal. per gram-mole) has also been suggested by [43]. The value of 01 in Equation (4-1.4) calcu- lated for A1-2024-T351 is 15,700°K, and therefore the overall activation energy can be determined as Q1 x R = 15,700 x 1.987 3 31,200 cal. per gram-mole (1.35 ev). Third, the extrapolated values of T at both ends of testing temperature range is consistent with the expectation of a precipitation phenomenon. For each isothermal ageing test a value of kia(Tag’O) and Akia(Tag) was obtained using the NLINA computer program. These values are also temperature dependent. Applying the least-squares technique, considering the F-test criteria (Section 2-4.3), and employing the third model for T, the following relationships were found: kia(Tag,t) = kia(Tag,0) + Akia(Tag) 1 - exp [7113311 (4-1.5a) kia(Tag,0) = 73.24 + .3725 Tag (4-1.55) Akia(Tag) = 74.94 - 0.2610 Tag (4-1.5c) map) = exp -3193 + 15.7 $439373] (44.54) 132 These relationships are valid for Tag between l75-220°C. Units of kia(Tag’t)’ kia(Tag’0)’ and Akia(Tag) are w/(m-C). A semi-log plot of kia(Tag’t) versus ageing time is shown in Figure 4-l.1. Figure 4-l.2 is a plot of the time constant T versus the inverse of absolute temperature. The value of overall activa- tion energy can be calculated from the slope of the curve in Figure 4-1.2. 4-2 Isothermal Experimental Data for Specific Heat of As Received A1-2024-T351 The average value of specific heat (6p) of as received Al-2024-T351 (Table 3-6.9) measured at room temperature, and the average value of 6p after completion of precipitation (Table 3-6.10) also measured at room temperature, indicates that the precipita- tion processes did not alter the value of 6p significantly (0.13 percent). However, isothermal ageing data (Tables 3-6.l through 3-6.8 and corresponding plots in Chapter 3) shows a slight increase in the value of 2p in the initial stages of precipitation. In par- ticular, this increase in the value of 2p in the first five hours of ageing is moderate for low ageing temperatures. As ageing temperature increases, the increase in the values of 5p is more noticeable. After this stage, there is a decrease in the value of 6p to a range in the vicinity of the initial value. As the time of precipitation increases, the values of Ep fluctuate within the range of initial values. No attempt was made to find a mathematical model for iso- thermal experimental data of specific heat, for as received 133 .w 6..” .3235. «.613 3.35. :2: Eco 45555me 3... 0283530 £23.. Eco m3 HE. .8...“an .3 E 822.53 83% 3... ozamwumxao 85:9 3.3» $2: 3238 .3 23.82:..— c we ammpnvwowfic .3 P3232328 .3255. 17+ ”32on D” II .08 .03 .3 x: uzfi A a. 0TH _.____d_ — ___~_—_ fl _ ___q_q_ _ J .0” w. €83 8. $9 858.3 a B has 8.. :2”: 838.0 a . 6 w .33 8: $9 .238.» > m» m on; 1.1.83 88 £5: ezmofu n . o o m . m a .. m f . 6 -8 H. a o o I“ 00“ .l. D D p D 1 U D DD 2. 9 D D00 0 o o ”_H . D D. DDD o . % 0 Two .9 p 1 D p 4 4 q q 0 . . . : a Do .4. a. Iluom 11!“ I .. 0 o 0 <0 0 q o 4 4 n 0:- p..-...... a... a. . .9... . 4° .4. 4 D s p 4. 4 4 4 44 4 DD bqw @144 4 é4 q I OOH 4444 4 a 4 e4 . 03 r 4 m3 (J-iJ-HH) IFIIG AIIAIIOHGNOO 1UHUBH1 134 10000 / (T=DE60 Kn ) 2.00 2.08 2.16 1004_ 1 1 1 ' r . *T 1 ' 1 )— TIME 00N3TANT ° “' HRS. 10. : 1. :- p— .1 l l l l 1 L # J 1 l 1 l L 1.12 1.16 1.20 1.24 10009 / (T=DEGI Ra ) FIGURE 4-1.2 TIME CONSTHNT VERSUS INVERSE HBSO- LUTE TEMP. THE SLOPE OF THE CURVE IS THE OVERHLL RCTIVRTION ENERGY. THE 8 EXP. DHTH CORRESPOND TO VHLUES OBTHINED FOR TIME CONSTHNT BY E0.(3-6.1). 135 A1-2024-T351, because irregular vairations in ED with ageing time are negligible and, at this time, are unexplainable. For each isothermal ageing temperature (Tables 3-6.1 through 3-6.8) an average value of specific heat is best at the present time. 4-3 Comparison Between Dimensionless Thermal Conductivity and Volume Fraction of Precipitation The kinetic law of precipitation for the isothermal age- ing condition (with time exponent n = 1) has the form [40, 41]): x = 1 - e't/T V (4-3.1) We postulate the relation x s —fi—%%—-7- (4-3.2) where n(Tag,t) is the volume fraction of precipitate at ageing temperature and time Ta and t, respectively; nm(Tag) is the maximum 9 volume fraction of precipitate at isothermal ageing temperature Tag“ The value of X in Equation (4-3.2) is the volume fraction of precipitation at isothermal ageing temperature Ta when ageing time 9 is t. The dimensionless thermal conductivity (kTa) for isothermal conditions can be obtained from Equation (4-l.5) to be: 136 k? (T t) = kia(T39’t) - kia(Tag’0) = 1 - e't/T(Tag) Ta ag’ Akia(Tag) (4-3.3) A comparison between Equations (4-3.1) and (4-3.3) yields: 1 n(Ta .t) kia(Tag,t) — W (4-3.4) which relates the thermal conductivity and volume fraction of pre- cipitation. In order to determine the denominator of Equation (4-3.4), let us first define the following: 1. Let ka(T) be the thermal conductivity of as received Al-2024-T351 as a function of temperature only. The value of ka(T) is determined for relatively low temperatures (below 300°F) where the amount of precipitation is negligible, or at high temperatures (BOO-500°F) when the ageing time is approximately zero; see Table 3-6.11 or recommended Equation (3-8.l). 2. Let kan(T) be the thermal conductivity of annealed A1-2024-T351. The annealing temperature range as shown in Figure 3-2.1 is given between SOD-800°F. The value of thermal conductivity after annealing, at any temperature level, is also only temperature dependent; see Tables 3-6.12 and 3-6.13 and recommended Equations (3-8.3) and (3-8.5). 3. Let k (T ) be the thermal conductivity of Al-2024- agr ag T351, aged at ageing temperature Tag and determined at about room temperature. The value of kagr(Tag) is determined after holding 137 Al-2024-T351 specimens for a sufficient long time at the ageing tem- (300-500°F) and then the value is obtained at about room (T perature Tag temperature. Itis shown (Table 3-6.10) that the values of kagr ag) increase as Tag (the temperature at which the Al-2024-T351 specimens were aged) increases, indicating that the complete precipitation at any ageing temperature depends on the ageing temperature. It is pos- tulated that a limiting ageing temperature, Tm, exists at which age- ing and annealing temperatures coincide. At this limiting temperature, Akia(Tag)==0. The value of limiting temperature can be determined from Equat1on (4-l.5) by equating Akia(Tag) to zero or Tm = Tag = 74.94/0.2610 = 287.1°C. At this temperature the complete precipita- tion occurs instantaneously. (For more discussion see next section.) Now, we pr0pose the following equation for determining ”m(Tag)’ k n (T )= 34,: a (4-3.5) m ag where nm(Tag), as indicated in Equation (4-3.2), is the maximum volume fraction of precipitate at an ageing temperature Tag and depends on Ta Note that, if as received Al-2024-T351 was aged at 9' about room temperature, then k (Tag) = ka and nm(Tag) becomes agr zero. Note also, 1f Tag = Tm then kagr(Tag) = kan and "m(Tag) becomes 1. The average room temperature at which the values of kagr(Tag) after ageing are determined is 32.6°C (90.7°F). 138 (see Table 3-6.10); using the least-squares technique, considering F-test criteria (see Section 2-4.3), a linear model is found as: g) = 128.76 + 0.1751 Ta (4-3.6) kagrna g is the ageing temperature in °C and k where Ta (Tag) is the g agr thermal conductivity value of aged Al-2024-T351 determined at 32.6°C in w/(m-C). At the average temperature of 32.6°C, the values of ka and kan are determined using the recommended Equations (3-8.l) and (3-8.3), respectively. These values are 124.53 w/(m-C) for ka and 173.29 w/(m-C) for kan' Substituting these values into Equation (4-3.5), the nm(Tag) is determined as 2 3 nm(Tag) = 8.68 x 10 + 3.59 x 10 Ta (4-3.7) 9 Equation (4-3.7) is valid for ageing temperatures between l75-220°C and ”m varies between .715 for 175°C: to .877 for 220°C. 4-4 Assumptions Regarding the Nature of Precipitation During Thermal Cycling In practical applications, as received Al-2024-T351 may be subjected to ageing temperatures which vary arbitrarily with time. In such cases the isothermal ageing relationships developed thus far would not be applicable. To develop mathematical relationships for predicting values of volume fraction of precipitation for arbitrary temperature histories, certain assumptions are needed. The change in volume fraction of precipitation during the thermal processes is related to the growth and redistribution of 139 solute atoms within solid-solution lattice structure. Generally, the enriched solutes or precipitates which are called GP zones (see Chapter 3) are assumed as an ideal physical model with a spherical shape having radius r. Gerold et a1. [49] theorized that the pre- cipitates, at a certain moment of time, have spherical zones with identical radii r and density 2 which is defined as the number of zones per unit volume. At any ageing temperature level (below limiting temperature Tm) z and r change with time continuously; r increases and 2 decreases in such a way as to move toward a new system having lower free energy. The system with minimum free energy is the one with stable precipitates. The kinetic theory of the precipitation processes [41] suggests the disappearance of the zones at the higher temperature levels. From this theory it is con- cluded that, at some limiting temperature Tm, all smaller zones will be absorbed into a single large zone. The size distribution of the spherical zones are therefore time and temperature dependent. In regard to the Gerold theory and the idealized physical model, the following is assumed in order to relate the change in physical model with the change in volume fraction of precipitation during the thermal processes: 1. At any ageing temperature, precipitation occurs when Spherical zones' radii increase with time. This coincides with Equation (4-3.l). See also [43, p. 252] for plot of zone radius as a function of ageing time. 2. At any isothermal ageing temperature, the radius of spherical zones reaches a maximum and remains unchanged as ageing 140 time increases. This condition coincides with nm(Tag) defined in Equation (4-3.2). 3. A limiting temperature Tm exists, in which the zones become a single large zone with a maximum radius. The system at this temperature contains minimum free energy and precipitates become stable. In this condition, the maximum possible change in microstructure occurs and nm(Tag) = l. 4. A sudden change in ageing temperature (below Tm) alters the size distribution of spherical zones only. At the up-cycling ageing temperature (step increase in ageing temperature), a certain number of zones dissolve to allow the remaining to grow. Conversely, if the ageing temperature is lowered, two possibilities exist. First, the zones' growth (volume fraction of precipitate) at the pre-aged temperature is less than the maximum possible growth of zones (4-3.7) at the down-cycling temperature (step drop in tempera- ture); in this case, the up-cycling pattern of precipitation will be continued with a decrease in precipitation rate. Second, if zones' growth at the pre-aged temperature is equal to, or larger than, the maximum volume fraction of precipitate (4-3.7) at the down-cycling temperature, no change will occur in the zones' structure to con- tribute to a variation in precipitation. For more infbrmation, see [36, 40, 41, 42, 43, 50, 51, 52, 53, 54]. 4-5 Proposed Differential Equations of Precipitation Durinngrbitrary Thermal Cycling In order to predict the instantaneous values of volume fraction of precipitation of as received Al-2024-T351 at any temperature and time, the following two differential equations are proposed: 141 3n(T3t)/3t RITT [nm(T) — n(T,t)] if pmm > n(T.t) (4-5.1.) 3n(T,t)/3t 0 if nm(T) 5.n(T,t) (4-5.lb) where n(T,t) is the volume fraction of precipitate at temperature T and time t; nm(T) is the maximum volume fraction of precipitate at temperature T, given in Equation (4-3.7); and T(T) is the time con- stant, given in Equation (4-l.5d). Equations (4-5. la) and (4-5.lb) are differential equations describing the rate of volume fraction of precipitate during arbitrary thermal cycling. These equations to the best of our knowledge are origi- nal for the precipitation ofAl-2024-T351. An equation with 11m being equal to lhas been previously suggested, however,fbr the rate of transformation [40]. The values ofn(T,t) , determined from Equations (4-5.la) and (4-5.lb) , are related to kia(T’t) by Equation (4-3.4). There is no known model in the literature to relate the volume fraction of precipitate to the thermal conductivity of Al-2024-T351 material. The differential Equation (4-5.la) can also be used to obtain isothermal ageing volume fraction of precipitation. Solving Equation (4-5.la) with the initial condition t = D, n(T,O) = D, the isothermal ageing volume fraction of precipitation, Equations (4-3.l) and (4-3.2), can be obtained. Subsequently, using the relation given in Equation (4-3.4), the isothermal ageing relationship for thermal conductivity of as received Al-2024-T351 can be determined. 4-5.l Equation of Precipitation During Up-Cycling_Thermal Proc- ess (Step Rise in Temperature) From Equations (4-3.l) and (4-3.2), the equation of iso- thermal precipitation for ageing temperature T1 is: 142 n(T],t) = nm(Tl) {l - exp [-t/T(T])]} (4-5.2) Note that T1 equals Ta because the ageing process starts at T1 = 9 T . Prior to time zero, it is assumed that the as received a9 Al-2024-T35l specimen was held at low temperatures (about room temperature). At time t, there is a step increase (up-cycling) in tempera- ture as shown in Figure 4-5.l. Since no time has passed during this abrupt change in temperature, the volume fraction of precipitate is constant during this change of temperature level or n(T].t]) = p(Tz.t1) (4-5.3) Integrating Equation (4-5.la) for temperature T2 results in: 't/T(T2) n(T2,t) = nm(72) +Ce (4-5.4) The constant of integration can be determined by the condition given in Equation (4-5.3). For t > t] one can obtain: -(t-t])/T(Tz) ”(Tzst) : ”(T19t])e '(t't])/T(T2) +nm(T2) [l - e ] for t > t1 (4-5.5) Thermal conductivity can be evaluated using Equations (4-3.4) and (4-5.5). For t > t] when T = T2, the thermal conductivity is found using: 143 .mmmuoca Pesemzu mc__u»u-czou m:_czv cowumpwawumga N.m14 mczmwm _ 13 as?» P h um ummmmgm mcwpuxu -ezee up Au.mpvc (1‘1)u 9424 -1d1394d Io UOL1DPJJ awnloA A3.thc |'||'ll 3.5: 2:... < _e as.“ .n 1 . _ _ _ 1 a a w 1 n m P» .mmmuoea _mELmsu mcw_uxo-a= mcwgau cowumpwnwumga _ as?“ .p chFUxu-a: um Au.N»vc Pb um vmmmmca _.m-4 pe=m_e A mu n w a “an. Ae._Hve m. \x m. \ u 0 \ J 41“"11111\l.| 1111111111 \\\ 2:... m In...“ D. 1111 111 111' 11111111 l d 0 E lo A. why: Amev c 4‘ _ F ms. .1.” .8 w _ _ _ I. _ a _ .w _ a w P m. h w NF 144 kia(T2’t) = kia(T2’0) + Akia(T2) x n(T2,t)/nm(T2) (4-5.6) 4-5.2 Equation of Precipitation Durin Down-Cycling Thermal Proc- ess Step Drop in Temperature)_ According to assumption 4 given in Section 4-4, two cases can be considered. In Case 1 if nm(T2) > n(T],t]), then precipita- tion during down-cycling and thermal conductivity can be calculated by the Equations (4-5.5) and (4-5.6) given for the up-cycling thermal processes. In Case 2 if nm(12) _<_ n(T],t]), then by Equation (4-5.11>) fi(T,t) == 0 or n(T,t) = C. For t = t1 and T = T], and from Equations (4-3.l) and (4-3.2), -t]/T(T1) c=pupq1=%upI1-e 1 (map For t > t1 and T = T2 -t1/T(T]) 6112.t) = nm(T]) [1 - e 1 (4-5.3) For t > t1 the thermal conductivity can be evaluated using k.a(72.t1 = 31,112.01 + Akia(T2) x n(12.t)/nm(T2) (4.5.91 The precipitation process in Case 2 is shown in Figure 4-5.2. Figure 4-5.3 shows the volume fraction of precipitate for four isothermal ageing temperatures as a function of ageing time. It also illustrates an up and down thermal cycling at t = 2 hours. 145 .3 .013 .3 >0 85.50.30 mac 0800”... 025051228 02¢ A... 3.. 0.5.2; ”.5. .2320 03¢ «H 0300”... 02503 .0257... 0 40213313 833%. «c 3.. M23. 02500 .3 20:02.... 0 00 5523“”... .3 20:002.. u2=.3> 0.01.. ”.220: .08 .84 . .8 2110:: A H .. _d‘___4_ q _qdqa_1_ a —_____m 0 1111.111 .0 .80 3. .2“: 258... e. .03 8.. .023 0230010 __.. .08 2.0 .02”... 025003 >0 0.83 8... .8: .29.... . .. .0>0 4. “EEG“... 0e - . e zerocm. u==0e> e _e . A 146 In this illustration the pre-aged temperature is selected to be 375°F (l90.6°C) and precipitation is completed at 400°F (204.4°C) or 350°F (l76.7°C) for1ur-and down-cycling, respectively. The thermal conductivity at t = t1 and when T2 > T > T1 (Figure 4-5.l) can be found using: kia(T,t]) = kia(T,0) + Akia(T) x n11],11)/nm(1) (4-5.10) Since no time has elapsed in this abrupt change in temperature, the value of volume fraction of perecipitate remained unchanged. Since the temperature is changing, the value of thermal conductivity changes accordingly. 4-6 Comparison and Discussion No reference data, with the exception of the data given by Al-Araji [l], was found to compare with the isothermal ageing data of the present investigation. The results of isothermal ageing tests of the present study and the ageing experimental data of Al-Araji [l] are shown in Figures 4-6.l and 4-6.2. The following conclusions are drawn from these experimental results obtained from the same bar of Al-2024-T35l material at the same ageing conditions: l. From Figures 4-6.l and 4-6.2 at zero time, the value of thermal conductivity kia(Tag’0) and the as received value of thermal conductivity ka(T) coincide (see Table 3-6.ll). Correspond- ing values obtained by Al-Araji [l] for ageing temperatures 350, 375, 400, and 425°F are given as 88.5, 99.6, l00.2, and 96.0 Btu/ (hr-ft-°F), respectively. The values determined by the present 147 (3*H)/N AlIAIlDflONOO THNHJHI . H u0zuxu0um 02¢ >o=»0 hzumumm 00 H00~1r~0~10¢ «02 quh 02Hu0¢ 00 20Hp0220 c 00 >hH>H~0=oz00 Aczauxh 00 #000 c H.01r um=0H0 .««I-U=Hh ova ONH oo. o. oo 0. ON 0 QTH.. . 0 . _ . _ . _ . _ . q . . . a uozuxuaum .a.oua mp” .¢:Uh QZHuocxy .>o=hw azumuaa .a.ouo a.» .020. 02Huw¢-~ _ omfi... H mozumuaua .a.¢uo om» .a:u~ ozH0o¢nu 0 ..»o=hw azuwuma .a.muo on” .azup 02Huac-fi wm:_ om: I 0.0.2. . . N—v..m .. ONH 00H ooa 00 00H 00H OHH mHH (J-lJ-dHl/HLB ALIAILOHGNOO lUHlel 148 . H uozuam0u¢ 02¢ #0:#0 #200000 00 H00#1v~o~10¢ 000 u:## 02Hu0¢ 00 20H#02=0 ¢ 0¢ ##H>H#0:oz00 0¢=002# 00 #040 ¢ «.01r 00:0H0 51.0th 3 «H OH H o _. H 0 8H T . _ . H . _ . H . _ . H . 8 M . . H 8206?. .33 3. .09 @238» . w .32” :08”... 0.93 3. $5: @232-.. 1 a m 8H 1 . H 3205:”. 0.85 8H. 0:“: .02:me . m use” #208”: .38 8.. 0:”: ozmwcH 8 0 m 8H 1 . I . mm . 1.00 v «LT 8H H.“ .. M...» .. 8H 1 1 8H 8H 1 1 3H mHH (J-lJ-SH)/fl18 AlIAILOHGNGO lUHHHl 149 method for the same ageing temperatures, shown in Tables 3-6.l through 3-6.8, are: 8l.99, 82.64, 86.20, and 88.06 Btu/(hr-ft-°F), respectively. Note that the experimental ageing data given by Al-Araji [l] tends to take an abrupt increase from ageing tempera- ture 350°F to 375°F. The present experimental results are considered to be more accurate because if we assume that the k values for Armco iron and Al-2024-T351 at about room temperature, given by TPRC [24], repre- sent the best values found in the literature at the present time, then a comparison of the present results and those given by Al-Araji [l] with the results of TPRC supports the above argument. For Armco iron, the present value of k (see Chapter 2) is l.23 percent lower than the k values given by TPRC while the value of k obtained by Al-Araji [l] is 2.5 percent higher than the result of TPRC. For as received Al-2024-T35l, the value of k (see Chapter 3) obtained by the present method is 2.7 percent higher than the value of TPRC, andk value given by Al -Araji [l] is ll.ll percent higher than the value of k presented by TPRC. From these comparisons, it is concluded that the four values of k, given by Al-Araji [l], for zero time of the four ageing temperatures, may not be accurate. 2. Figure 4-6.2 shows thermal conductivity-time curve for ageing temperatures 400 and 425°F reported by Al-Araji [l] and the present investigation. The plots given by Al-Araji [l] are ques- tionable because the values of k at the initial ageing time, and the maximum value of k for ageing temperature 425°F, are less than the corresponding values of k for ageing temperature 400°F. The 150 present experimental data (Tables 3-6.l through 3-6.8) and corresponding mathematical model do not indicate a pattern follow- ing that of Al-Araji's [l] (as shown in Figure 4-6.2). 3. The values of specific heat obtained by Al-Araji [l] are more oscillatory and generally higher than the values determined by the present study; his values are also higher than the generally accepted literature values. 4. Al-Araji [l] concluded from his experimental data that at a given ageing temperature, the values of thermal conductivity increase to a maximum (aged) and subsequently decrease (overaged). This pattern can not be detected from the present experimental data (Figures 3-6.l, 3-6.3, 3-6.5, and 3-5.7) in which the ageing times are much longer than ageing times given by Al-Araji [l]. In order to confirm that the isothermal ageing data for thermal conductivity of as received Al-2024-T351 is exponential with time [the isothermal values of thermal conductivity of as received Al-2024-T35l increase with time to a maximum (aged) and remain unchanged as ageing time increases], two sets of unreported experimental ageing data of electrical conductance (inverse of electrical resistance) on the same bar of as received Al-2024-T35l, as a function of ageing time, are shown in Figure 4-6.3. These data were not reported because refinement of experimental equip- ment and procedure is underway to modify the temperature-controlled specimen housing. It is interesting to note that these experi- mental data, shown in Figure 4-6.3, appear to be exponential with time and a mathematical model like that of EqUation (4-l.5) .Hmn#-v~o~-4¢ «.0 wth 02H0o¢ 2. zeHhozaa e we uezehoaozoo 0¢0H¢H000 0. 0.40 c 0.0-. u¢=QHO .«¢:-qu. 8 mm 8 DH 8 m o 151 - _i q — u — d — - fl - — q CHOOI 45022000200 .0 .000 00v .020# 02H00¢D . 400002.155 0.000 00—. .020# 02H00¢1~ .n#20zH000x00 .0 .000 H00 .020# 02H00¢D 1.260. .3000: .I#¢20 .0 .000 «00 .020# 02H00¢1H N w A ' 210\ H .0200 0¢0Hm#00._0 080 . 152 fits reasonably well. It is also interesting to note that the time constant found for electrical conductance for Tag = 400°F is 2.095 hours. The corresponding time constant for k is 2.230 hours. Therefore, for the aging temperature 400°F, the shapes of curves obtained for thermal conductivity and electrical conductance versus time are almost identical. A theoretical relationship between thermal and electrical conductivity is the Weideman-Franz law, which states that the ratio of thermal conductivity to electrical con- ductivity at any temperature is proportional to that temperature [2, pp. 36-39]. Because thermal and electrical conductivity at each isothermal ageing temperature are theoretically proportional, and because the experimental ageing data obtained for thermal and electrical conductivity are both exponential, it is concluded that the thermal conductivity-time curve is an exponential form rather than any other characteristic curve given by Al-Araji [l]. Numerous reports [34, 40, 54) indicate that the hardness- time curve of solution-heat treated aluminum alloys reach a maxi- mum and subsequently the values of hardness decrease to lower values as ageing time increases. The difference between solution heat- treated and as received Al-2024-T35l is in heat treatment prior to ageing tests. In all hardness ageing tests the Specimens were heated to a temperature between 900-975°F (i.e., 48 hours in a salt bath at 965°F), followed by quenching in ice-water and immediately starting the ageing tests. Such heat treatment was not given to as received Al-2024-T35l neither in this investigation nor the investigation carried out by Al-Araji [l] using the same bar. 153 5. Al-Araji [1] reported the values of thermal conductivity, determined at room temperature, for aged specimens after completion of precipitation. These data indicate that holding the specimens at any ageing temperature for sufficiently long time periods allows complete precipitation to occur, or the value of nm(Tag) in Equa- tion (4-3.5) is equal to unity. The present experimental data given in Table 3-6.l0 appear to invalidate Al-Araji's experimental findings. 6. Different procedures are applied to analyze the experi- mental ageing data. These include: First, the mathematical model given by Al-Araji [l] is proposed only until the time of the maximum kia(T’t) values (tmax). The model demonstrates a relationship between dimensionless kia’ Tag’ and dimensionless time t+ (t+ = t/max). The relationship is a form of a polynomial with linear parameters. The present model covers the entire ageing time and is exponential (4-l.5) with linear and nonlinear parameters. The time constant (4-l.4) obtained by the present method can be related to the overall coefficient of diffusion [47, 48]. Second, the present method of analysis relates the dimensionless thermal conductivity to volume fraction of precipitation. Third, for thermal cycling processes, a first-order differential equation is postulated to obtain the instantaneous volume fraction of precipitation and subsequently determine the value of thermal conductivity which is time and temperature dependent. In this respect, Al-Araji [l] introduced an algebraic procedure to 154 determine only values of thermal conductivity until t+ = l (t+ is a time when kia is maximum for each ageing temperature). 4-7 Use of the Proposed Mathematical Models in an Engineering;Problem In the design of heat transfer equipment, sometimes it is desirable to predict temperature history of its components. The Al-2024-T35l material is one of the most versatile families of metal available to the metalworking industry for fabrication of various components of heat transfer equipment. This alloy, when solution heat-treated and subjected to elevated temperatures, undergoes considerable changes in mechanical, electrical, and thermal properties which influence the design performance. The objective of this section is to provide a method for determining temperature history of Al-2024-T351 material under temperature- and time-dependent thermal properties conditions. This is accom- plished by proposing a numerical method of solution of the partial differential equation of conduction and pertaining equations for thermal conductivity simultaneously. The method is illustrated for the particular example of a step increase in surface temperature. The problem is to solve the one-dimensional partial dif- ferential equation of conduction: LE: 11. - [k 3x] pcp 3t (4 7.1) for the aluminim alloy 2024-T35l material with the boundary condi- tions of T(0,t) = 450°F and 8T(L,t)/ax = 0 and the initial condition of T(x,0) = 350°F. 155 Equation (4-7.l) is solved for the Al-2024-T35l material under the three cases of as received properties (no precipitation), as received with precipitation, and the annealed condition. 4-7.l Thermal Properties For all three cases, the values of cp are considered to vary only with temperature. The cp values are determined using one of the recommended equations given in Chapter 3. Values of p are found in [24]. Knowing the values of p and cp at any temperature, a least- squares technique is applied to obtain: pcp = 35.8l26 + 0.0ll6 T (4-7.2) The values of thermal conductivity k for the cases of as received properties (no precipitation) and the annealed condition are also considered to vary only with temperature. The associated values of k are obtained using the recommended Equations (3-8.l) and (3-8.3). The k values of as received Al-2024-T351 with precipitation depend on the amount of precipitation; consequently, the values are time- and temperature-dependent. To calculate the instantaneous values of thermal conductivity, the amount of precipitation must be calculated. The equation of volume fraction of recipitate is described by: Aug; = H177 [nm(T) - n(T,t)] for nm(T) > n(T.t) (44.3) where T = T(x,t), T(T) is given in Equation (4-l.5), and nm(T) is determined using Equation (4-3.7). 156 In the case of as received with precipitation, prior to a step increase in temperature at location x = 0, the as received Al-2024- T35l was preaged at 350°F for 25 hours. Before preageing it is assumed that the Al-2024—T3Sl was held at low temperatures (i.e., about room temperature) so that no recipitation had occurred. , Equation (4-7.3) was solved numerically to determine the value of volume fraction of precipitate n(T,t). Then the instan- taneous value of thermal conductivity for as received Al-2024-T35l with precipitation was calculated using: k(T,t) = kia(T’0) + Akia(T) X 0(T.t)/nm(t) (4'7-4) where kia(T’0) and Akia(T) are defined in Equation (4-l.5). 4-7.2 Method of Solution The partial differential equation of conduction was solved using finite differences. An energy balance about node i (see Figure 4-7.l) can be written as: qi-1,i ' qi,i+1 = E.- (4-7.5) where q. is heat flow rate from node i-l to i; q is heat 1-1,i i,i+1 flow rate from node i to i+l; andEfi is stored energy at node i for a time interval j and j+l (see Figure 4-7.l). Equation (4-7.5) for node i and for time interval j and j+l may be approximated by: 157 110,01 450°F 'o 12' Crank-Nicolson method Figure 4-7.l Boundary conditions, initial condition, and nodal arrangement for Al-2024-T351 bar. . T? - T? . T? - T? - J _1:l____1._ J _;L___;Ltl (1 B) {éki-l/Z Ax Aki+1/2 Ax :] + - Tifi-Ti” .- riftiiii B Ak? - Ak. 1-1/2 Ax 1+1/2 Ax . T4+1 - T? (pep)? AXA —J—-Tt—1_ (4'7.6) where kg_]/2 is the thermal conductivity evaluated at temperature i j . - - j - (T1._1 + Ti)/2’ Similarly, k1“,2 15 k value evaluated at i j (Ti + Ti+l)/2' Forward difference, backward difference, and Crank-Nicolson approximation can be specified by setting the value of 8 equal to 0.0, 1.0, and 0.5, respectively. Equation (4-7.6) can be rearranged as: 158 . . Ax)2 . j+l J J (” pcp)i( J+l Ti+l = BkL l/2Ti- 1 [:Bki-l/2"Bki+l/2 At Ti Bki+1/2T 1+1 ' 2 . (pc )9(Ax) . J P 1 J ' (1 B)ki- 1/2T i- -1 (1'8)ki-l/2 (1 B)N i+l/2 At T1' j j . . . _ , The values of k1._]/2 and ki+l/2 are determ1ned u51ng the relat1on g ship given in Equation (4-7.4). The value of ng_]/2 (for simplicity the functional T and t notations are omitted; instead the subscript i and superscript j are used) for node i-l/2 and time j is deter- mined using the fourth-order Runge-Kutta formulas [30]: j+l _ J ni’ _1/2 - 0i_ 1/2 + (RKO + ZRK 1 + 2RK2 + RK3)/6 = j ' j j where RK0 4t(0m,1_1/2 ni-l/2)/Ti-1/2 = 3+1/2 _ j _ 3+1/2 _ = 3+1/2 _ 3‘ 3+1/2 RK2 At(nm,i-l/2 ni-1/2 RK1/2)/ L -1/2 _ 3+1 _ j RK3 At(”111,1-1/2 nL1/2 RK2)/TL 1/2 For "i+l/2 a similar expression is written with subscript i+l/2. Note that the subscript i-l/2 or i+l/2 indicates that the components are calculated at a temperature (T1._1 + Ti)/2 or (Ti + Ti+l)/2’ respectively. 159 Equation (4-7.8) was used to calculate the volume fraction of precipitates at time j, and subsequently the values of thermal conductivity are calculated by the relationship given in Equation (4-7.4). These values were then utilized in Equation (4-7.7) to obtain the temperature of the corresponding node at the time j+l. In this procedure the time step At = 0.005 hours and the effect of the past k(T,t), nm(T),and 1(T) used for one future temperature calculation is negligible. For a total of m nodes, m linear equations like that of (4-7.7) were written. These equations were solved simultaneously to obtain the temperatures of all nodal points at each time step. Knowing all grid points' temperatures, the thermal properties were evaluated and this procedure was repeated for the next time step. These algebraic calculations were continued until the maximum speci- fied time or specified temperature had been reached. A FORTRAN computer program was developed to solve this problem; it contains the main program and three subroutines. In the main program inputs and outputs are specified. In one sub- routine the temperature or temperature- and time-dependent thermal properties are calculated. In the case of as received with pre- cipitation, the volume fraction of precipitate for all gridpoints is determined using the fourth-order Runge-Kutta formulas. The next subroutine uses the proper thermal properties to generate the coefficients of Tgt}. Tg+], and Tgi}, given in the left-hand side of Equation (4-7.7) and the value of the right-hand side of this 160 equation. Finally, the last subroutine uses the matrix of coeffi- cients and determines the nodal temperatures. The results of Crank-Nicolson finite difference approxima- tion for the as received with no precipitation, as received with precipitation, and annealed conditions are shown in Figures 4-7.2 through 4-7.6. L The numerical results using the Crank-Nicolson method are ' shown in Figures 4-7.2 and 4-7.3. Figure 4-7.2 shows temperature .... ”7_ as 3.12;. x/L for the three cases of as received proper- g as a function of x+ ties (no precipitation), as received with precipitation, and annealed conditions for times 0.05, 0.2, 0.5, and 0.9 hours. The three aforementioned cases generate three distinct curves (temperature versus x+) for any time t due to the differences in thermal con- ductivity k. The largest temperature differences for the three cases are obtained at x+ = l (insulated surface) when the time is between 0.8 and l.6 hours. The temperature history of the insulated surface for the three cases is shown in Figure 4-7.3. The thermal conductivity history of the Al—2024-T35l in the cases of as received with no precipitation and the annealed condition is shown in Figure 4-7.4. For time = 0, the value of k for the as received Al-2024-T35l bar is 80.3 Btu/(hr-ft-°F) while the corresponding value of k for the annealed condition is l05.5 Btu/(hr-ft-°F). Due to a sudden change in temperature at the heated surface, there is an abrupt change in the values of k for as received with no precipitation and the annealed condition, as 161 .mcowuwucou um#mm:=m use .cowampwamumea 50L: um>wmumg mm .AcoLHmpwawumgg ocv mmwugmaoga vw>pmomc we we momma cm can #mm# 1¢Nom1#< com covumzcm cowuuaucou 0o cowu:#om mucmgmwmvu wwwcwm com#ou#z-xcec0 N.#1¢ 002000 1_\X M +X . . . . . 0 S m m a N am ow# J 000 om# 1. com a. a 1 m m 08 m a. o I .d 3 EN 2; as 1 e S. 000 1 :o_umu#0#umga o: .7 1 00¢ cowpmpwawumga m mecca SN 1 a H . p L — one 162 mm mo mmmmo cw Lma #mm#1000~1#< 0:» Lo wuo0gam umum#:mc# may we agopmwc 0030000050# 30 aaneuadwal .mcowuwvcou um#mmccm new .comumpwawumca ;u#3 um>wmumg mm .Acowumuwamumga ocv mmwugmaoca um>wmumc .m.N-. eeamwm ..N o.~ e.H N.H.mee-eeHe m. a. o mN- a H H 41 H 0 000 00#.. 00.0020 «#mum mew“ mumwczm umum#=m:# mg» 0a .050# 1 000 00#r OONI. Uw—wwccT 1 0mm 0_N.. cowumpwawumc. 1.0#0 :o_0m000_umca oz 000 l 1. 000 0.0 u +x pm .050# 000 C. H . c . 000 go aunqeuadma] 163 (oa-w)/M K11A11onpu00 Lewuaul .mgao; mm com 0.000 0m ummmmcq mm; Lon Hmm#1¢~0~1#< m;# .cowuwucou um#mmccm 0;» 00m AcowpmpwaHowga ocv mmpp 1000000 um>wmuog mo 00 mummu :L can #mm#1¢~0~1#< 0;» mo xgoumH; xuw>wuusucou Hmsgmc# 0.010 00:0H0 m; 1meH ..~ o.~ e.H ~.H e .H m. a. o m~-o~ H 4 1 H H H J _ ummcmzu _ 0 com mew» 00H 1 0.H u +x p< _ H _ 00 omH 1 o.o . +x e<.11 1 00 0 Acowwmpw0wumcn 0:0 mmwpcmaoga um>mmumc m< o.H . +x H< 11 (da'lJ‘JH)/"48 K11A113npuoo [PWJBHL _ _ _ 0## 1 _ _ 1 00# . _ _ o u x 84 4 + (H _ omH 1 aaHaaee< “ - oHH _ _ _ _ _ + P r _ H b H GNP 164 (ac-w)/M K11A11onpuoa [PWJBHL .mgzo; mm Low moomm Hm ummmmga mm; Lag _mmh-¢~o~-_< mg» .cowpmHHQHumgg saw; Lag HmMH-H~om-_< um>Hmumg mm H0 Hgoumwg HHH>HHu=u=ou _wengh m.H-H mgzmwm . . . msguwepu . . . F . m 0 mm- _ ummcmzu omH 1 " opmum oswu _ _ . _ o H +x u<.11 _ cm. 1 _ . . o u +x H< cup 1 ,rhwlu‘ 1 oa~ i ummmmgm.1L 1 1L _ _ _1 b _ _ om cop o—P cup (jo-1;-Jq)/n19 £11A113npuoa lewaaql 165 .mgso; mm Lam “comm Hm ummmmgn powsgmcuomm mm: Lug _mmp1HNo~-H< mgh .Lma _mm»-¢~om-H< um>wmumg mm 10 mumpwqwumga Ho cowuumgH maz_o> m.~-¢ mgsmwm ¢.~ o.~ c._ N._ mg;-mswu w. H. o mm.. o. T 4 A H 1 1 JA 0 _ _ comcmsu _ m—mUW ms_p _ _ N. T H IN. H u +x H<.11 _ e. 1 1.:. m. .1a. m. 1m. mcwmmm HmsgmsuomH o.F . 1P1 p . 1% L . o.— (1‘1)u 31231d1334d JO uogqaeJ; awnlon 166 shown in Figure 4-7.4. These values for as received with no pre- cipitation and the annealed condition at 450°F are 92.3 and 107.6 Btu/(hr—ft-°F), respectively. The values of k at the insulated surface (for both cases) increase with time. The annealed Al-2024- T351 bar, due to a higher thermal conductivity, attains the final uniform temperature much faster than the as received Al-2024-T35l bar with no precipitation. Figure 4-7.5 shows the thermal conductivity history of the Al-2024-T35l bar with precipitation. The volume fraction of pre— cipitate of this bar is given in Figure 4-7.6. Prior to a step increase in temperature, the Al-2024-T3Sl bar was held at 350°F for 25 hours and before this time it is assumed that the bar was held at low temperatures (zero precipitation). For 25 hours of isothermal ageing at 350°F, the values of thermal conductivity (k) and the values of volume fraction of precipitate (n) increase with ageing time exponentially. After this period, due to an abrupt change in temperature at the heated surface, there is also an abrupt change in the values of k and a rapid increase in the values of n which are shown in Figures 4-7.5 and 4—7.6, respectively. After this abrupt change, the values of k at the heated surface, unlike the annealed and as received with no precipitation, change with time because of the influence of the precipitation. When precipitation at any location of the bar is completed, the corresponding k and n values remain unchanged thereafter as time increases. For 25 hours the k and n values given in Figures 4-7.5 and 4-7.6, 167 respectively, represent the thermal conductivity and volume frac- tion of precipitate of the entire Al-2024-T35l bar. After this period, the k and n values of two locations (heated and insulated surfaces) are shown in Figures 4-7.5 and 4-7.6. It is important to note that the precipitation at the heated surface, due to a higher temperature, is completed more rapidly than the precipita- tion at the insulated surface. Hence the k values calculated for the heated surface approach a constant value faster than the cor- responding values of k obtained for the insulated surface. CHAPTER 5 SUMMARY AND CONCLUSIONS One of the most important objectives of the present study F1 was to investigate the thermal property changes of aluminum alloy 2024-T35l (Al-2024-T35l) under the influence of precipitation age- hardening. In the course of acheiving this objective, a number of modifications in the previously available equipment were made and several new components were developed and added. The feasibility of using a thin thermofoil electric surface heater was studied. A search was made to determine the best location for the placement of thermocouples on the flat surfaces of the Specimens. Numerous trial experiments were performed to evaluate the effects of various thermocouple arrangements, heating times, time intervals between data points, test durations, and power input levels. A procedure was developed to calibrate the amplifiers and the associated system which transmits the thermocouples' output from the laboratory to the IBM l800 computer. A FORTRAN computer program (Chapter 2) was developed to calculate the calibration coefficients to convert the millivolt output of thermocouples to temperature. An analytical equation for estimating thermal properties [19] utilized the transient data generated by the calibrated equip- ment to determine values of thermal conductivity (k) and specific heat (cp). The newly developed transient facility was used to 168 169 determine k and cp values of Armco iron (reference material) in the temperature range of 80-400°F (Chapter 2). The measured k and cp values of Armco iron were compared with values reported by the other investigators. These comparisons indicated that the present transient method can produce accurate thermal properties values. In addition, the present method is transient with short measurement time period (approximately 40 sec- onds in the present tests). Advantages of a transient method with short measurement times are that the heat losses have less influence on the values of thermal properties and that the method can be used for materials whose thermal properties are time, or temperature, and time/temperatureédependent. This transient method was also used to determine the thermal properties of Al-2024-T35l before precipitation occurred (fast measurement cycles) and the annealed material in the temperature range 80-425°F (Chapter 3). Values of k and cp of as received Al-2024-T351 with precipi- tation for isothermal ageing temperatures of 350, 375, 400, and 425°F were also measured using the developed method (Chapter 3). For each isothermal ageing temperature, the k values increase with ageing time to a maximum value. For higher ageing temperatures, the k values approach maximum more rapidly than the maximum k values for lower ageing temperatures. The increase in k values due to precipitation becomes zero at a limiting temperature of about 287°C (550°F). 170 The experimental ageing data obtained for k was mathematically modeled and the linear and nonlinear parameters were determined using the NLINA computer program (Chapter 2). At each ageing temperature, the experimental values of cp tended to increase slightly at the initial stage of ageing and sub- sequently decreased to the vicinity of the initial values. Because of the small and irregular variation in the values of c , no attempt P was made to model the experimental ageing data of specific heat. The isothermal mathematical model, obtained for k values versus ageing time, is related to the volume fraction of precipita- tion. The relation found for the volume fraction of precipitation and the associated time constant are in accordance with the general form given for the kinetic laws of precipitation and diffusion, respectively (Chapter 4). Relationships were also found by which the maximum values of k and the maximum value of volume fraction of precipitation, at any ageing temperature, can be determined. In practical applications, the as received Al-2024-T35l alloy nay be subjected to nonisothermal ageing temperatures. In this case, two differential equations are proposed which are believed to be original. The solution of these differential equations gives the volume function of precipitate (n) under any arbitrary ageing temperature and time and subsequently by a k - n relationship the values of k are determined. The numerical solution of the one-dimensional partial dif- ferential equation of conduction for an Al-2024-T351 plate was given to demonstrate the influence of precipitation on temperature history 171 of the plate. Three cases were considered of the material being in its as received condition with no precipitation permitted, as received condition with precipitation permitted, and the annealed condition. The k and pc values used in the partial differential P equation of conduction for the cases of the as received with no precipitation and the annealed conditions are only temperature- dependent. The k values in the condition of as received with precipitation depend on the amount of precipitation and conse- quently are time/temperature-dependent. In this case the differ- ential equation of precipitation is also solved numerically to determine the volume fraction of precipitate (n) as a function of time and temperature. The simultaneous solution of temperature and n is required because k is a function of n. The partial differential equation of conduction was approximated using the Crank-Nicolson method. The differential equation of precipitation was solved using the fourth-order Runge-Kutta formulas. The effect of precipitation on temperature history, the thermal conductivity history, and the volume fraction of precipitate versus ageing time of the Al-2024-T35l material are given graphically in Chapter 4. S-l Recommendations for Further Research The following are some suggestions for further research: l. From theisothermal volume fraction of precipitation two differential equations are postulated which are unique and quite useful not only for the isothermal ageing conditions but also for nonisothermal cases which happen frequently in practical 172 applications. The validity of the differential equations of precipi- tation for isothermal conditions was verified. To apply the differ- ential equations to nonisothermal ageing cases, certain assumptions are made. Further research is needed to justify these assumptions and verify the differential equations for nonisothermal ageing cases; these cases include up- or down-cycling thermal processes or so- )1 called double ageing. 2. Modifications of the present temperature—controlled , specimens' housing are needed to more rapidly cool the specimens to I the set isothermal ageing temperature after the end of each experi- ment. This will reduce the time interval between consecutive ageing experiments and consequently enable the performance of more ageing experiments in a short time period. The minimum time interval between consecutive ageing experiments for the present investigation was about l5 minutes. A shorter time interval is needed to obtain more information, particularly when ageing temperature is larger than 400°F or when experiments are being performed to study precipi- tation during up- or down-cycling the thermal processes. 3. A minicomputer is also needed in the laboratory to aid in reducing the time interval between consecutive ageing experiments. 4. The determination of the electrical resistivity of the as received Al-2024-T35l under influence of precipitation age- hardening was initiated during this investigation. The results of two sets of ageing tests for electrical conductance (inverse of electrical resistance) are also given in Chapter 4. A mathematical model like that given for thermal conductivity fitted reasonably 173 well to the experimental ageing data of electrical conductance. Preliminary investigation indicates that the ageing experimental results of electrical conductivity can aid in providing quantita- tive information in an efficient manner in regard to the precipi- tation model proposed in this investigation. To determine the electrical conductivity of as received Al-2024-T35l, a well-designed temperature-controlled specimens' housing is needed. The design should provide a method of fast heating of the specimens to a set isothermal ageing temperature and maintaining the specimen and its surroundings within about l.5°F of the desired set isothermal ageing temperature. The developed equipment and measurement procedure should be tested and calibrated first using the Armco iron material which has fairly stable and known electrical conductivity. For accurate and detailed measurements, a digital data acquisition sys- tem should be used. 5. Other heat treatable alloys such as Al-Zn and Al-Mg alloys undergo similar property changes (like that of Al-Cu alloy) when solution heat-treated and subjected to a precipitation heat- treating temperature. Further study is needed to determine the behavior of the thermal property changes of such alloys under the influence of precipitation age-hardening. 6. This investigation uses a certain method to attach the thermocouples on the flat surfaces of the specimen. An attempt was also made to determine the disturbances created by these thermo- couples using this method of thermocouple installation. A consistent conclusion could not be reached. It was assumed that the 174 disturbances created by the thermocouple itself in this investiga- tion are insignificant. Further study is needed, however, to evaluate more precisely the disturbances created by the presence of the thermocouple. This is important not only for this investigation but also for other cases involving the transient temperature measure- ments with thermocouple installed in a similar manner. REFERENCES 175 [l] [2] [3] [4] [5] [6] [7] [8] [9] [10] REFERENCES Al-Araji, S. R. "Experimental Investigation of Transient Thermal Property Changes of Aluminum 2024-T351." Ph.D. dissertation, Michigan State University, l973. Tye, R. P. Thermal Conductivit . Vol. l. New York: Academic Press, l969. Tye, R. P. Thermal Conductivity. Vol. 2. New York: Academic Press, l969. McElroy, D. 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