TEEIIFERATURE DISTRIBUTIONS AND EFFECTS OF HEAT APPLIED T0 PLANT STEMS Thesis for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY CARL HENRY THOMAS 1969 f THESIS LIBR‘4 ” Y Michigan 23.1w University This is to certify that the thesis entitled TEMPERATURE DISTRIBUTIONS AND EFFECTS OF HEAT APPLIED TO PLANT STEMS presented by Carl H. Thomas has been accepted towards fulfillment of the requirements for PhoDo degree in Agr. Eng. xZ/Za/%, i/M Major professor Date August 19, 1969 0-169 Y alumna w L: “0A8 & SflNS' IIIIIIK BIIIlI'LlY INC. ABSTRACT TEMPERATURE DISTRIBUTIONS AND EFFECTS OF HEAT APPLIED T0 PLANT STEMS by Carl H. Thomas The objectives of this investigation were the determination of the response of a plant stem exposed to a high temperature environment and the development of an expression for the prediction of the tempera- ture distribution in the stem. The findings will be useful for improving recommendations involving the use of flame and electric heating for such processes as the control of weeds in crops, the burning of leaves from sugar cane and foliage from potatoes, the defoliating of cotton plants, and the rapid drying of biological materials. Expressions for predicting the temperature distribution in plant stems exposed to a suddenly changing temperature environment were developed and solved numerically by finite-differences using a digital computer. the analysis accounted for the change in the physical properties due to a change in the temperature and moisture content. It was found that the cell tissue of corn stems is killed by approximately 60 degree-seconds of heat exposure above 130°F. There- fore, a high temperature heat for a short time does not necessarily result in tissue damage at a critical depth. A parametric study showed that the diameter of the stem is the most important factor to be considered when applying heat to that part Carl Henry Thomas of a plant. This offers the possibility of selectivity for killing weeds of small diameter without causing critical damage to the crops with stems of larger diameter. The stem of a living plant has a very complex structure with properties that are difficult to describe for use in engineering analyses. The properties change with changes in their environment and are not reversible. Wide variations in densities and moisture contents were measured among positions on the corn stems. Approved fl% 9%,z/ Major Professor 9§2_ b9 Approved @( M Department Chairman TEMPERATURE DISTRIBUTIONS AND EFFECTS OF HEAT APPLIED TO PLANT STEMS BY Carl Henry Thomas A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Agricultural Engineering 1969 PLEASE NOTE: Appendix pages are not original copy.Print is indistinct on some pages. Filmed in the best possible way. UNIVERSITY MICROFILM. To: My Wife, Chris and children, Jill Steve Dan Mike ACKNOWLEDGMENTS The author wishes to extend his most sincere appreciation to Dr. F. W. Bakker-Arkema for his untiring guidance and advice through— out the author's graduate program at Michigan State University and to Dr. B A. Stout for his many helpful suggestions for getting this study under way. To Dr. J. V. Beck, Mechanical Engineering Department, and Dr. S. K. Ries, Horticulture Department, appreciation is extended for assisting with guidance of this study and for serving on the graduate committee. The author also wishes to thank Dr. C. W. Hall, Chairman of the Agricultural Engineering Department, for his assistance in so many ways. Also, the author thanks Louisiana State University and the National Science Foundation for allowed leave and financial assis- tance to complete the graduate program. Finally, appreciation is extended to Gloria Allain, Sue George, and Lynra Gieger for typing the first capies of the manuscript and to Pamela Lombard for the final typing. “*mflmm TABLE OF CONTENTS I. IMRODUCTIONIOOOC.00...OOOOOOOOOOOOOOOOOOOO...OOOOOOOOOOOOOCOOOO II. LITERATURE REVIEWOOOOO00.000.000.000...0......OOOOOOOOOOOOOOOOOO III. IV. VI. VII. Thermal Conductivity, Specific Heat and Thermal Diffus- ivity... ................................................... Heat Transfer Equations ...................................... Evaporative Cooling from'Mass Transfer ...................... Technique for Testing Plant Cells for Evidence of Life ....... Discussion of Literature Cited ............................... STATEMENT OF THE PROBLEM........................................ EQUIPMENT....................................................... PROCEDURES...................................................... NUMERICAL SOLUTION OF THE MATHEMATICAL'MODEL.................... DISCUSSION OF RESMTSOOOOOOO0.0.0COOOOCOOOOOOOOOO...00.0.0000... Discussion of Comparisons Between Experimental and Cal- culated Temperature Distributions.... ...................... Discussion of Results from Experiments with Heat Expos- ure of Corn Stem Segments .................................. Discussion of Parametric Effects ............................. Discussion of Errors ......................................... SWRYOOOOCOCOOOOOOOOOOO0....OOOOOOOOOOOOOOOOOOOO..0.0000...... SWGESTIONS FOR FURTER mRKOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO APPEme A...’...°....OCOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOQO List of Computer Symbols Used ................................ APPEmm BOOOOOCCOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO0.0.0... Program.Tempcyl.. ............................................ Program Tempwal .............................................. iv Page 1 5 ll 12 13 l4 16 17 19 31 38 38 42 56 75 77 78 80 8O 82 83 88 Page APPEme C - (sample compliter ontput8)00oooooooooooooooooooooooo 93 Cylinder ............................................ . ........ 94 Wall.. ........................... .. .......................... 100 APPEme DOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOIOOOOOOOOOOOOOOO 106 Temperature Distribution Curves for Parameter Study .......... 107 LIST OF SEIECTE MERENCESOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 137 LIST OF TABLES TABLE PAGE 1. Temperature 32 time of exposure of stem tissues of corn plants.. 43 2. Surface temperatures obtained with stem.segments of corn plants exposed to 2000°F for two seconds. . . . . . . . . . . . . . . . 49 vi LIST OF FIGURES Figure l. 2. 3. 4. 10. 11. 12. 13. 14. 15. 16. 17. Surface of corn plant (1 inch diameter) ........................... Surface of cotton plant (k-inch diameter, 2 months old) ........... Surface of cotton plant (k-inch diameter, 3 weeks old) ............ Cross-section of l-inch diameter corn stalk ....................... . Cross-section of g-inch diameter cotton stalk ..................... . Electric furnace and holder used to expose plant samples to heat.. Equipment used to inject plant samples into electric furnace ...... . Thermocouple probes being inserted into the corn stalk segment.... . The corn stalk segment was exposed to flames from four Stone- ville type burners.. ..... . ..................... ........... ....... . NOmenclature of nodal points for cylinder and wall. ............... Temperature at the surface of silicone rubber specimen (cylinder). Temperature at a depth of 0.004 ft. in silicone rubber specimen (CYlinder)oooo0000000000.coo00000000000000.coo-0.90.00.00.00. 00000 Temperature at 0.008 ft. for experimental and predicted standard.. Effects of temperature !§_time of exposure on cell tissue of corn stalks......................................... ...... . ..... ....... The surface temperature obtained on segments of corn stems exposed to 2000° F for two seconds as affected by the height on the stalk at which the sample is taken ...................... . ............... The surface temperature obtained on segments of corn stems exposed to 2000° F for two seconds as affected by the diameter of the stem The surface temperature obtained on segments of corn stems exposed to 2000° F for two seconds as affected by the density of the stem. vii Page 21 21 24 24 26 26 3O 39 4O 41 46 50 52 FIGURE . 18. The relationship between the density of a corn stem and the height 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. on the stalk at which the sample is taken . . . . . . . . . . The relationship between the moisture content of a corn stem and the height on the stalk at which the sample is taken. . . . . . . Predicted temperature distribution in long cylinder . . . . . . . Predicted temperature at the surface as affected by heat transfer coefficient for heating (cylinder). . . . . . . . . . . . . . . . Predicted temperature at the surface as affected by heat transfer coefficient for cooling (cylinder). . . . . . . . . . . . . . . . Predicted temperature at the surface as affected by thermal con- ductivity for r > rc (cylinder) . . . . . . . . . . . . . . . . . Predicted temperature at a depth of 0.004 ft. as affected by the thermal conductivity for r < rc (cylinder). . . . . . . . . . . . Predicted temperature at a depth of 0.004 ft. as affected by the thermal conductivity for r > rc (cylinder). . . . . . . . . . . Predicted surface temperature as affected r > rc (cylinder) . . . . . . . . . . . . Predicted temperature at a depth of 0.004 specific heat for r > rc (cylinder) . . . Predicted surface temperature as affected (CYIi-nder) O O O O O O O O O O I O O O O 0 Predicted temperature at a depth of 0.004 density for r > rc (cylinder) . . . . . . Predicted surface temperature as affected Predicted temperature at a depth of 0.002 diameter (cylinder and wall). . . . . . . Predicted surface temperature as affected by specific heat for by diameter (cylinder). ft. as affected by by surrounding fluid temperature (cylinder). . . . . . . . . . . . . . . . . . . . . . I Predicted surface temperature with constant XE varying heat source (cy11nder) O O O O O O O O O O O O O O O 0 viii PAGE .54 .55 .57 .58 .60 .63 .64 .65 .66 .68 .69 .70 .71 73 NOMENCLATURE thermal diffusivity (Eta/Hr). specific heat (BTU/Lb °F). mass diffusivity (Fta /Hr). overall heat transfer coefficient, 8 > 8 (BTU/Hr Ft2 °F). 1 overall heat transfer coefficient, 01< 8 S 9 (BTU/Hr Ft? 9F). 1 thermal conductivity (BTU/Hr Ft °F). number of time step. number of node. node at surface. rate of heat lost during cooling, 9 > 61 (BTU/Hr Eta). rate of heat gained during heating, 0 < 6 S 81 (BTU/Hr Ft2). radius of cylinder (Ft). critical radius at point of property changes, (Ft). outside radius of cylinder (Ft). dimensionless radius (r/rs). density' (Lb/Fta). temperature (°F). initial temperature (°F). surface temperature (°F). fluid temperature (°F). dimensionless temperature (t/ts) dimensionless surface temperature (ts/ts). dimensionless fluid temperature (tfllts)' time (Hr). total time for heating (Hr). ix T = dimensionless time «me/r52)- T = dimensionless time for heating «19 /rse). 1 1 NOte: subscript 1 implies r < rc. subscript 2 implies r 2 rc I. INTRODUCTION Flame has been used effectively for the control of weeds in crops for the past decade. The principle use has been in row crops with plants tall enough to-allow passage of the flame underneath the foliage. The plant stem is exposed to the intense heat for a short period of time during each flame application. In cotton, corn and soybeans the flame applications supplement the uses of chemical and mechanical cultivation for the control of weeds. The normal flaming operation utilizes two or more burners for each row directing the flames toward the base of the crop plant and allowing the flames to pass over the seedbed surface. The flamer normally travels at speeds between 3 and 6 miles per hour. At 3 miles per hour with two conventional burners, 7.5 inches wide, the flame contacts the main stem of a plant a maximum of 0.28 second. Due to the short exposure of the plant stem to the flame, the high temper- ature can be tolerated by the stem. Watson (1961) measured the flame temperature to be approximately 2000 degrees Fahrenheit in the zone Where the flame contacts the plant stems. This high temperature prevails at the base of the plant and near the soil surface on both sides of the seedbed. The tolerance of a plant to the flame is affected by the surface structure of the plant stem as well as internal characteristics. The characteristics of the surface of the stem may be influenced by the stage of plant maturity. For example, a young fast-growing cotton plant has a smooth waxy stem, while a mature cotton plant has a rough and corky stem. The stem of a corn plant is smooth for all stages of maturity. See Figures 1, 2 and 3. The stems of corn plants are high in moisture (55 pounds per cubic foot) near the ground and low in total moisture (25 pounds per cubic foot) near the top of the plant. The stem density is approximately 63 pounds per cubic foot near the ground and may reduce to approximately 30 pounds per cubic foot near the top of the corn plant. The use of flame for weed control in crops offers several distinct advantages over chemical and mechanical methods. Flame leaves no harmful residue in the soil and does not contaminate the crop, while chemicals may cause crop contamination and leave harmful residues in the soil. The hazards of handling flame are not as great as they are for handling chemicals. The disadvantages of using flame include the need for precise placement and timing of the application and the low efficiency for con- trolling weeds. The efficiency may be improved as a result of increased knowledge of the effects of applied heat on plant tissue. Data on physical and thermal properties are essential to an analysis of the influence of applied heat on plants. There is not much information presently available relative to these properties. The physical pro- perties such as surface roughness, moisture content and bending strength of the stem influence the timing and techniques of operations such as mechanical cultivations, chemical applications and flaming for weed control. Thermal properties such as specific heat and thermal con- ductivity influence the heat tolerance as well as the movement of heat into the stem. With information on these factors, more precise timing and techniques of applications of flame could be developed. Figure 1: Surface of corn plant (one inch diameter). ‘Figure 2: Surface of cotton stem Figure 3: Surface of cotton stem (one-half inch diameter, (one-fourth inch dia- two months old). meter, three weeks old). II. LITERATURE REVIEW Very few investigations in the heat transfer literature have been concerned with living plants. However, the techniques developed for studying other biological products may have some application to the study of living plants. Physical and thermal properties of biological products are very difficult to define. As stated by Lentz (1964):"Biological materials are much more complicated than other types of materials and they may vary considerably from one sample to another in both composition and structure." Many of the properties will change with changes in time, temperature and surroundings. These changes may not be consistent and they are probably not reproducible or reversible. Many references are available in the literature dealing with the measurement of the thermal properties of biological products (foods). Woodams §£_gl.(l968) accumulated a list of literature values of thermal conductivities of foods. The methods used for measuring these values were not evaluated by Woodams. However, Reidy (1968) made an exhaustive study of the methods for determining thermal conductivity and thermal diffusivity of foods. Reidy's accumulation of values for thermal proper- ties of foods gathered from the literature is probably the most complete list of these data available. His most prevalent comment regarding methods used by researchers to determine thermal properties was that the procedures used and the existing conditions were generally not adequately described. This lack of information reduces the confidence with which others can use the data. A few of the investigations from the literature are included in the following review as they may relate to this investigation. As most of the thermal properties reported are for foods, the values are not readily applicable to the plant stem model used. THERMAL CONDUCTIVITY, SPECIFIC HEAT AND THERMAL DIFFUSIVITY Gurney g£_gl.(l923) recognized that "the temperature-time relations in the interior of solid bodies which are either being heated or cooled may be empirically determined by inserted thermometers, thermocouples or other temperature measuring apparatus, or may be calculated from the assumed conditions in conjunction with the physical constants of the materials, the surrounding contours, shapes and media." Thermal physical data are not sufficient and technical conditions can rarely be controlled to coincide with the theoretical prototypes. "Where empirical observations and theoretical calculations may be made concurrently, new physical constants may be determined which will be found to be more reliable in predicting other time-temperature ' The curves of relations under different, though similar conditions.’ Gurney were obtained by converting some of the more common formulas for heat diffusion into expressions containing pure ratios or nondimen- sional variables onlx,thereby enormously reducing the necessary basic calculations as well as extending their field of applicability. Itethley et al.(1950) determined the average thermal conductivities of some fruits and vegetables over the range of 0° to 80° F by immersing solid objects of food and canned foods suddenly into a cold constant- temperature medium. The temperature history of the food pieces and canned foods made possible the calculation of the thermal diffusivities using the graphical method of Gurney. For the specific heat of a fruit or Vegetable, these investigators used an average apparent specific heat which was defined as "the quotient obtained by dividing by 80 the total BTU's required to raise the temperature of one pound of the substance from 0° to 80° F." For the temperature interval of 32° to 80° F Kethley found that the average thermal conductivities of strawberries, Irish potato flesh, English peas and peach flesh ranged from 0.61 to 0.78 BTU/Hr Ft °F. Eucken (1940) applied a formula, derived by Maxwell (1904) and published in (1954) by Dover Publications, Inc., that expressed the thermal conductivity of a material as a function of the relative volumes and the conductivities of the different particles of which the material is composed. Lentz (1961) applied the Maxwell—Eucken equation to 6%, 12% and 20% gelatin gel solutions and obtained a theoretical thermal conductivity. The theoretical and experimental values are in good agreement (-2.4% to 0.4% difference) for the 6% and 12% gel solutions but not for the 20% gel solution (4.3% to 14.1% difference). Lentz used a guarded hot plate apparatus to determine the experimental thermal conductivity of the concentrations of gelatin gels from -25° t0.5° C. Using this method for determining the thermal conductivity of ice:, Lentz obtained results with the guarded hot plate that are about 1% 1O'Wer'than the most reliable values available. In experiments with heat flow both parallel and perpendicular to the grain of meat, Lentz also determined the thermal conductivities of several kinds of meat for -25° C to 10° C. The thermal conductivities of all the meats are about equal and about 10% below the established values for water. For temperatures below -10° C, thermal conductivity curves for meats are a linear function of temperature with the thermal conductivity increasing as the temperature decreases. Lentz did not find a direct correlation between the thermal conductivity and the moisture content or fat content of the product. Heat conduction was found to be 15% to 30% higher along the fibers than across the fibers. A simple method for calculating the thermal diffusivity, a, involves the use of the equation<1 2:; = éE. This equation is a 66 put in finite-difference form using temperatures, t, at three positions, x, in a one-dimensional body at two times, 6. The thermal diffusivity, a, is then calculated from the finite-difference equation. The thermal diffusivity is found as a function of time and temperature. Beck (1963), however, developed a calculational method which is superior to the above method with respect to errors in the thermal diffusivity caused by differences between measured temperatures and those cal- culated by the finite-difference equations. The calculated temperatures are determined using a finite-difference approximation of the heat conduction equation. An iteration procedure is used to find the thermal diffusivity, beginning with an initially estimated value. After the thermal diffusivity is found for one time interval, it can be found for succeeding time intervals and expressed as a function of temper- atures (provided the changes in temperature with respect to position and the time intervals are not too large). The accuracy of the thermal diffusivity value depends, therefore, on the number of measured and calculational positions as well as upon the accuracy of the measure- ments. Matthews(1966) determined the thermal diffusivity of potatoes from temperature measurements with a transient heat source using a numerical method of finite-difference approximations. The tem- perature change was measured within a finite thickness of potato resulting from an applied heat at one face. The finite-difference approximations used by Matthews were based on the procedure developed by Beck (1963). Parker g£_§1. (1967) made measurements of the density, specific heat, and thermal diffusivity of cherry flesh between 80° F and 40° F from which they calculated the thermal conductivity." They arrived at the following empirical equation by multiple correlation analysis: K = -0.275 - 0.000935 + 0.280%E1 + 0.327c f1 f1 Where: Kfl = estimated thermal conductivity (BTU/Hr Ft °F) SS -= soluble solids content of the flesh (%) pf1 = density of the flesh (gm/cc) cf1 = specific heat of the flesh (BTU/Lb °F) Beck (1964) developed a technique for the simultaneous deter- mination of specific heat and thermal conductivity of solids from transient temperature measurements. The properties are found by making 10 the calculated temperatures match the measured temperatures through a nonlinear least-squares analysis. Finite-difference approximation to the heat conduction equation is used to calculate the temperatures. The thermal conductivity is determined using steady-state conditions while specific heat and thermal diffusivity require transient conditions. The heat flux and temperatures alone can be used to determine thermal diffusivity. With the use of a digital computer, these properties can be readily calculated by this method with less than 0.1% error as a result of approximations in the numerical procedure. Errors in the properties considerably greater than this are usually found due to errors in the temperature and flux measurements. Kopelman (1966) suggested the relation: cp = M.C. (1.0) - (1.0 - M.C.) P Where: c = specific heat of a substance (BTU/Lb m) P M.C. moisture content (w.b.) expressed as a decimal. Factor relating the contribution of the solids to the specific heat. P This relation is of the general form: Number of components of the material considered. § (0 H (D 5 II Factor relating the contribution of each component to the specific heat. '11 II For potatoes and carrots, which contain components quite similar to those of corn stems, Kopelman's equation agrees with the most reliable values of specific heats obtained by others and compiled by Reidy (1968) with less than 1% error. 11 Many other literature sources are available on the determination of specific heats of foods; however, most of these report specific heats for temperatures below 80° F, except for some meats. These results have little or no application to this investigation with the stems of plants. HEAT TRANSFER EQUATIONS Numerous papers have been written on the solution of heat trans- fer equations for various solids of different shapes. Some of the theories were developed many years ago. A great number of solutions of the heat conduction equation for many different initial and boundary conditions and changing thermal properties have been collected by Carslaw g£_§l.(l965). To solve the temperature-time history analytically at a large number of locations is, however, a very tedious operation because of the infinite series, Bessel functions and other complicated terms in the solution. Kreith (1964) pointed out that the graphical method of analysis known as the "Schmidt plot" can be applied to long solid or hollow cylinders. The accuracy of this method depends on the number of approx- bmations used in the solution. The Schmidt method is flexible and yields practical solutions to problems which have such complicated boundary conditions that they cannot be handled conveniently by analytical methods. In the past, the graphical method for solving unsteady heat conduction problems was widely used in industry as it gave a continuous record of the changing temperature distribution and its 12 details could be delegated to relatively untrained personnel. However, for precise computations, especially when variations of physical proper- ties are important, numerical methods are preferred. Because of striking engineering advances involving complex shapes of bodies, Dusinberre (1961) explained that it might be best to use numerical methods to predict temperature distributions. Even though the details of such calculations are often simple enough, the size of the job might be appalling. This handicap has largely been elflminated by the rapid development and widespread availability of digital and analog computers. EVAPORATIVE COOLING FROM MASS TRANSFER Luikov (1964) stated that the effect of mass transfer with evaporation of liquid from capillary-porous bodies mainly results in the change of heat and mass transfer mechanisms due to deepening of the evaporation surface into the interior of the body. When the surface of evaporation deepens, the heat transfer coefficient is higher than that with evaporation on the surface. At drying with deepening of the surface of evaporation, the heat transfer coefficient will be higher than that of a dry body. Luikov (1964) further discussed experiments by Mironov (1962) who showed that with porous cooling the heat-transfer Nusselt number is larger than Nusselt numbers for pure heat transfer. Also, Nusselt numbers with porous cooling are larger than those with drying. This 13 difference increases with Reynolds number. Mass transfer Nusselt numbers with porous cooling are smaller than those with drying. This difference decreases with increase of Reynolds number. "With porous cooling, evaporation occurs at a body surface or in a layer close to it. With drying of a moist body, evaporation 'can' take place at a certain depth even at constant rate of drying. Increase in air velocity is known to move the evaporation zone into the interior of the body." The amount of heat spent for evaporation is relative to the amount of moisture evaporated. Heat required for this evaporation is transferred to the evaporation zone not only through a boundary layer at the surface, but also through a very thin layer of the body. Heat and mass are transferred through this layer of the body by conduction and diffusion. External heat and mass transfer depends on heat and mass transfer inside the body. TECHNIQUE FOR TESTING PLANT CELLS FOR EVIDENCE OF LIFE Attempts have been made to develop a quick and simple technique for detecting evidences of life in a plant cell. The use of a micro- scope for this purpose requires the skill of a rarely found individual. Moore (1960) has discussed a technique using tetrazolium (2, 3, 5 — triphenyl tetrazolium chloride) which gives a normal red color when hydrogen from respiration processes of each living cell combines with the absorbed solution. Peculiar purplish red colors are produced by chemical reactions other than that between hydrogen and tetrazolium. 14 DISCUSSION OF THE LITERATURE CITED The unrestricted use of values for thermal properties of biological materials as compiled in the literature is questionable. The method by which the properties were measured apparently plays an important part in the magnitude of the values reported. The materials and the conditions under which the data were obtained are often poorly described. This limits the field of the applicability of the data. For this author's study, no data on thermal and physical properties of plant stems were found. It was apparent that these properties must be measured or assumed. From the study of the literature, it appeared that the methods of measuring the thermal diffusivity used by Matthews (1966) and the specific heat used by Kopelman (1966) were quite reliable. By measuring the density of the corn stems used as models, the thermal conductivity could then be calculated by the relation: _ k a - --pc P where: a = thermal diffusivity (1/Ft?) k = thermal conductivity (BTU/Hr Ft F) p = density (me/Fts) cpf specific heat (BTU/me) The evaluations of the literature regarding methods of measuring thermal properties as made by authors such as Matthews (1966), Parker, ££_al.(l966), Kopelman (1966) and Reidy (1968) were very helpful. The numerical approximation to the solution of the heat conduction equation as described by Dusinberre (1961) appeared to be the best available means of approaching this problem. Exact analytical solutions 15 as compiled by Carslaw §E_al.(l965) were not applicable to this problem, as a description of the temperature distribution at the immediate end of a very short heating period was needed to express the temperature distribution at various times during cooling. A search of the literature revealed very little information on methods of determining whether or not cell tissue of plant stems was killed with applied heat Also, no information was found relative to the temperature and time of exposure required to kill cell tissue of plant stems. The method described by Moore (1960) using tetrazolium (2, 3, 5 - diphenyl tetrazolium chloride) appeared to have some possible application to this problem although the technique was developed for determining the viability of seeds. 16 III. STATEMENT OF THE PROBLEM The purposes of this investigation were to determine the responses of a plant stem to high temperature heat exposures and to develop expressions indicating the temperature history of the exposed stem. The objectives were the following: 1) To determine the temperature and time required to kill 2) 3) the cells of a plant stem by applied heat. To measure the temperature history and depth at which cells in a plant stem are killed by a short exposure to heat. To demonstrate by numerical approximations the temperature history for a plant stem as affected by the following variables: a) b) e) d) e) Temperature of the heat source Time of exposure to heat Diameter of the plant stem Density of the plant stem Thermal properties of the plant stem 17 IV. EQUIPMENT The following apparatus and instruments were used: 1) Potentiometers - a) Servo Riter II, Manufactured by Texas Instrument Company, b) e) d) 2 pens, single chart, continuous recording, accuracy i 0.25% of full scale. Model G, Manufactured by Leeds and Northrup Company, 2 pens, single chart, continuous recording, accuracy i 0.3% of full scale. Visicorder, Model 906C, Manufactured by Honneywell, Inc., 12 channel, single chart, continuous recording, accuracy i0.3% of full scale. Precision Potentiometer, Model 8686, Manufactured by Leeds and Northrup Company, accuracy i 0.03% reading, + 3 micro- volts without a reference junction, for calibrating recording potentiometers. 2. Thermocouples - a) b) e) Copper-constantan, 0.012 inch diameter (30 gauge) wire, in 1/6 inch diameter stainless steel probe, for internal tem- peratures when using flame as the heat source. Iron-constantan, 0.012 inch diameter (30 gauge) wire, for sur- face temperature of silicone rubber model using flame as the heat source. Chromel-alumel, 0.003 inch diameter wire, closed junction in 3) 4) 5) 6) 7) 18 a 0.02 inch diameter stainless steel probe, used for sur- face and internal temperatures of the samples when using the electric furnace as a heat source. Heat Exposure Apparatus - a) Flame burner equipment, illustrated later. b) Electric furnace, Type FH305, Manufactured by Hoskins Mfg. Company, 20 volts, 50 amperes. c) Hot water bath with stirrer and heaters. Sample Holders - a) Holder for stem segments when using flame as a heat source, illustrated later. b) Holder for stem segments when using the electronic oven as a heat source, illustrated later. Hand Microtomes - a) 3/8 inch diameter tube b) 1/2 inch diameter tube c) 3/4 inch diameter tube Thermal Diffusivity Apparatus - This apparatus was designed and built by Matthews (1966) and modified to handle corn stem sections. Miscellaneous equipment used for measuring the density, spec- ific heat and moisture content of the corn segments - a) Torsion balance, Manufactured by Torbal, 0.1 gram graduation. b) Electric ovens, For determining the moisture content of corn stem segments. 8) 9) 10) 11) 19 c) Hand refractometer, Type 25B, Manufactured by American Optical Company. d) Constant temperature reference junction box, Type 106, Manufactured by Thermo-Electric Company. e) Graduated cylinders to measure water displacement. Electric timer - Manufactured by Cramer, 0.01 second graduation. Computational Services - CDC 3600 and IBM 7040 Digital Computers Corn stalks - a) MSU 100 variety - near stage of maturity, starch kernels. b) Aristogold Sweet - at milk stage, developed kernels. 2, 3, 5 - triphenyl tetrazolium chloride 20 V- PROCEDURES Stem.segments from living corn stalks were selected as the plant model because of the possible homogeneity of the physical pro- perties and the close relationship of the corn plant to the existing problem. Corn and cotton are two of the main crops on which flame is used for weeding. 0n comparing the cross sections of stems of corn and cotton plants as shown in Figures 4 and 5, the corn stem.was selected as the model for this study. Corn plants were selected from the field and stem segments were selected from the plants on the basis of size, shape and location on the stalks. l) The stem segment was trimmed with a cork borer to fit the selected hand microtome. The trimmed segment was placed in the microtome tube and thin sections, approx- imately 0.035 inch thick, were cut with a razor blade. Although thinner sections could have been cut by this method, they were more difficult to handle and did not improve the results. The thin sections were exposed to heat using a stirred water bath at the selected temperature. To establish the desired ranges of temperatures and times of exposure, initial studies included the following treatments: 21 Figure 4: Cross-section of one inch diameter corn stem. Figure 5: Cross-section of one-half inch diameter cotton stem. 22 Water Bath Exposure Temperature (°F) Time (Seconds) 120 1, 2, 4, 10 140 l, 2, 4, 10 160 l, 2, 4, 10 180 l, 2, 4, 10 210 l 2, 4, 10 The exposed sections of cell tissue were placed in small vials containing a 0.25% solution of tetrazolium (2, 3, 5 - triphenyl tetrazolium chloride). The solu- tion was prepared by adding one gram of the tetrazolium powder to one pint of distilled water. Staining was normally apparent within four hours after being treated and stored at approximately 85° F. The distinct pink coloring was evidence that the cell tissue was not killed by the applied heat. Cell tissue that remained white or pale green was dead. From the results of the above initial experiment, additional experiments were planned, which included the following treatments: Water Bath Exposure Temperature (°F) Time (Seconds) 120 4, 6, 8, 10 160 1, 2, 3, 4 180 1, 2, 3, 4 210 s. 1. 1%. 2 2) 23 Additionally, the above indicated treatments were repeated to include samples of plant tissue from four locations on the stems of corn plants. These locations were the first, third, fifth and seventh internodes above the surface of the seedbed on which the plants were grown. Heat exposed sections were compared with tissue sections that were not exposed to heat. Photographs of the results were taken and used in the discussions. To simulate the exposure of plant stems to high temperature heat as in flaming for weed control, an elec- tric furnace as illustrated in Figure 6 was used. The temperature inside the furnace was regulated with a rheostat. Temperatures up to 2200° F could be obtained with this equipment. Stem segments were obtained from corn plants at selected internode locations on the plant. The inter- nodes were numbered, beginning at the ground level. Selected stem segments were trimmed at the node on each end and carefully weighed. The volume was measured by displacement of water in a graduated cylinder. A thermocouple. 0.02 inch diameter probe, was inserted from the end of the stem segment and the prepared sample was placed in the holder, which is also illustrated in Figure 6. The sample was exposed to heat inside the furnace 24 Figure 6: Electric furnace and sample holder used to expose plant samples to heat. Figure 7: Equipment used to inject plant samples into electric furnace. 25 with the aid of the apparatus illustrated in Figure 7. Time of exposure to heat was recorded by an electric timer actuated by a micro-switch. Temperatures sensed by the thermocouple were recorded on a chart by the Visicorder recording potentiometer equipped with an amplifier. After the run was completed, the sample was weighed and sectioned to determine moisture losses and the depth of the thermocouple. Microtome sections were placed in vials containing tetrazolium solution and studied as in- dicated in procedure 1). Photographs of these results were made for use in discussions that follow later. Experimental runs were made also using flame as the heat source. The following procedures were used for these runs: Stem segments were obtained from selected positions on corn plants. The segments were prepared by trimming the ends smoothly to give a sample seven inches long. The leaves were removed and the weight and dimensions of the sample obtained before the run. Thermocouples were inserted from the end at approx- imately 1/16 inch and 1/4 inch depths from the surface of the sample. The prepared samples were held in the specimen holder shown in Figure 8. The heat exposure apparatus consisted of four Stone- ville type agricultural burners mounted on a turntable 26 Figure 8: Thermocouple probes being inserted into the corn stem segment. M ' "AFL- .. ~-fl»”«~w "~11.th ‘ . 40‘ w m a", Figure 9: The corn stem segment was exposed to flames from four Stoneville type burners. 3) 27 equipped with a liquified petroleum gas system. The burner frame was mounted so that the center of the frame Passed the stem sample at 51.5 inches from the center of rotation of the turntable. This apparatus is illustrated in Figure 9. After the run was completed, the sample was weighed and sectioned to determine more precisely the depths of the thermocouples. The results from the experimental runs using flame as the heat source were used in the discussion comparing experimental and theoretical temperature histories. A numerical solution of the conduction heat trans- fer equation was developed to demonstrate temperature histories for an infinitely long cylinder. The solution was written and computer programs were prepared. The contributing effect of each selected parameter was studied by varying the magnitude of one parameter at a time. Graphs of the temperature histories were prepared and used in the discussions. The study of the parameters affecting the temperature distribution in an infinitely long cylinder with uniform heating of the surface for a short period of time included the following values (standard values are underlined): a) b) C) d) e) f) g) h) i) J') k) 1) m) n) to approximate those a) b) C) d) e) 28 hH (BTU/Hr/ F13 F) = 2., 4., 10., 20., 40., 100. AC (BTU/Hrth F)=1.,_2__, 5 , 10,20 k.1 (BTU/Hr Ft F) = .1, Lg, 3, .6, 1 0 k; (BTU/Hr Ft F) = 1, .2, 4;, 6, 1 0 = .87, ;_1, 1.00. b cpl (BTU/L F) cp2 (BTU/Lb F) = ,gz, .91, 1.00. p (Lb/Ft3) = 30., 39;, 60., 65 1 p (Lb/Ft3) = 30., 40 , 60., 65 2 rs (Ft) = 01, .02, 03, 493, .05, .06, .08, .16, .32, .64. r (Ft) = .035,r - 0.005 c ‘--- s tfl (F) = 500., 1000., 1500., 1641., 2000. (Heating). tfl (F) = 77. (Cooling). A0 (Hr) = .00004. Ar (Ft) = _._99_2. N =.ZQ’ rS/O.002 (Integer). The values of the standard parameters were selected for a corn stem specimen as follows: hH = 40.0 BTU/Hr th F -- approximated experimentally with the aid of a silicone rubber model. hC = 2.0 BTU/Hr Ft2 F -- approximated for free con- nection from a horizontal cylinder, Kreith (1958). k = 0.2 BTU/Hr Ft F -- measured 1 k = 0.3 BTU/Hr Ft F -- measured 2 c = 0.21 BTU/Lb F -- measured P1 f) g) h) i) j) k) 1) m) n) 0) 29 cps = 0.87 BTU/Lb F -- measured p = 40.0 Lb/Fta -- measured 0 = 60-0 Lb/Ft3 -- measured e _rs = 0.040 Ft -- measured diameter of the lower stem segments of a corn stalk. rc = 0.035 Ft -- assumed point of change of properties in a corn stem. tf1 = 1641.0 F -- Patin (1967), 0 < 8 S 0. t = 77.0 F -- normal atmospheric temperature, f1 -———- e > 6. A0 = 0.00004 Hr -- assumed time increment small enough to allow several time steps during the heating period. Ar = 0.002 Ft -- rS[N. N = g_ -- convenient number of depth increments as illustrated in Figure 10. The nodal system used for the cylinder as well as a semi- infinite wall are shown. A solution of the heat equations for a semi-infinite wall was obtained and a sample output from the pre- pared computer program is shown in the Appendix (B and C). 30 Figure 10 NOMENCLATURE OF NODAL POINTS USED FOR FINHTE-DIFFERENCE APPROXIMATIONS OF THE TEMPERATURE DISTRIBUTIONS SEM I- INF‘IN ITE WALL 31 VI. NUMERICAL SOLUTION OF THE MATHEMATICAL MODEL The rapidly changing boundary conditions of the mathematical model make this problem difficult to handle analytically. Exact analytical solutions of the governing heat transfer equations require accurate descriptions of the initial conditions for all phases of heating and cooling. As heat is applied rapidly to the surface of the model for a very short period of time (less than 0.5 second) and then the model is allowed to cool at ambient conditions, the temperature of the surface and points below the surface is changing rapidly. Therefore, the intital condition for the cooling phase is not well defined. The numerical treatment of the heat conduction equation using finite-difference approximations by the Crank-Nicholson (1947) method, however, gives reliable indications of the temperature dis- tribution at any time. This application of finite-difference approx- imations demonstrates a uniqueness not normally seen in the use of heat transfer equations. The mathematical formulation of this problem involves rates of change of the dependent variable, temperature, with respect to two independent variables, radius and time. The computational development of the numerical method involves a large amount of arithmetic; therefore, whenever possible, terms are arranged for one solution to suffice for a variety of different problems. 32 Several simplifying assumptions are made to facilitate the solution to the problem. The assumptions include the following: 1) 2) 3) 4) 5) 6) 7) 8) The material is homogeneous in radial segments. Non-homogeneity may exist among radial segments. The specimen is cylindrical in shape. The specimen is infinitely long. Heat is applied evenly around the circumference and over the length of the specimen. Physical and thermal properties are constant for each radial segment of the specimen. The diameter is constant over the entire length of the specimen. No mass transfer is assumed to occur. Considering the above listed assumptions, the basic differential conduction heat transfer equation reduces to: .9. 5r 6: at (kr 3;) = pcpr '35 (6-1) Since k is assumed to be constant for each radial segment, equation (6-1) may be written in the form: 021: at g; kr—+k-—=pc Or: 6 5r pr Be (6-2) §2t+15t 1a: a}? :6: 636 (6-3) 33 Equation (6-3) can be put in the dimensionless form: 621?. 112.2: a? R R br (6-4) Where: __t ..._t_ __9_ R - r ’ T - t ’ T — r 2 S S S The initial and boundary conditions are assumed to be: T(R,O) = Ti ; T = 0, for all R (6-5) %%(O’T) = 0 ; for all T (6-6) dH = - k'%§ r = l = hH (Ts - Tfl) ; for 0 < T S T1 (6-7) 9: = ' k %% r = 1 = hc (Ts - Tfl) ; for T >rT1 (6-8) Now, consider the finite-difference approximation to equation (6-4) using the Taylor series expansion: _ s12 93 T ' T(n.m) T AR BRI (mm) + “We are ‘ (n.m) 1 baT *6 (“>3 5113' (mm) + (6'9) - 91 .1. ii: Ton-hm) - T(um) ' AR 611' (mm) + 2 (AR)? am I (W) 1 9:11 - 6 (A103 5R3 I (mm) + (6-10) Adding equations (6-9) and (6-10), neglecting fourth and . . BQT . . higher order terms and solv1ng for ER? l(n,m) gives. gig ‘ = TL'lsm) + T(n+1:ml - ZTanm) 5R2 (mm) (AR)2 (6'11) 34 Which is of order (AR)2. With small AT, the approximation, = T + T T(n,m+%) (nlm) 2 (nlm+1) (6-12) can be used in equation (6-11), ear _ 1 _ 5135' (n,m+1/2> ’ 2 (AR)? [T(n-1.m) + T -2T(n,m) - 2T(n,m+1) + T(n+1,m) + T(n+1,m+l):I (6‘13) By subtracting equation (6-10 from (6-9) four times Tm n . and Tm+l’ o e can obtain 91‘. = _1__ - - aR' (ma) 423R [T(n+1,m> + T T T(n-I.m+1>J With an error of order (AR)2. It can be shown also that, _ T(n.m+1> ' T(n.m> 9?- I at (nut) AT (6-15) With error of order (AT)3. With the substitution of equations (6-13), (6-14) and (6-15) and rearranging terms, the approximation of the solution to equation (6-4) becomes, 1 1 l 1 (Tax)? + '4m' '11) T(n+1,m+1> ‘ (Iii)? + Z?) T "3616? ‘ 4m) TOR-1,!!!) (6-16) For the case of k and pcp varying with temperature, one can derive similar to equation (6-16) for the temperature distribution in a cylinder, (6-14) - IL 35 A.- [(kr)( W5) +31%] t(n+1’m+1) - [(kr)(n+,é) + (kr)(n_3é) T 2“” Pr)(n) (23% ‘6. ,m+l) WWW.) " who-hm“) = ' [(kr) + kn 2 r3 t(n+1,m) + “MN as.) + “‘“(n-F) ‘2‘“ Pr)(n) 3152‘3t(n,m> "Ukr’m-s) ‘lziA’r'Jtm- 1 m) (6-17> Equation (6-17) permits temperature variable properties. For small AT's, k and pcp can be evaluated at Tm' Equation (6-17) is stable and valid for any values of A0 and Ar. For the temperature at the center of the gylindrical model, R = 0, =0 and ER = 0, Carslaw and Jaeger (1965) expressed the result as 1 8T _ 4 [5122+ R 6713 R=0 (AR)2 (Ti ' To) (6 18) When evaluated at time, (m + g). 1 6T _ _ 2 _ [3? + '13 3'13] R=0 ’ (AR)? [T(l,m+1) + T(1,61) T(0,1141) 'T(0,m)] (6-19) correspondingly, _| = Tflmfl) ' Term) Substituting equations (6-18) and (6-19) into (6-4), —-—2— ['12 + T - T - T J (AR)2 (1.m+1) (1.m) (0,m+1) (0.m) - —1 [T<0.m+1> ‘ T(0.m>J (6‘2” _ AT and rearranging terms, gives - SLRE .__ (AR)?— (1 + 2m )T(0,m+1) + T(1.urx+1) (1 ' 2m ) T(0,61) T<1.m) (6-22) 36 Putting equation (6-22) in dimensional terms and multiplying by k results in pc (Ar)a - __P_____ 0‘05) + 2A9 ) ”<0,m+1) + pc (Ar)2 - __E_____ ‘k 2(16) ) t<0,m> T k t<1.m) (6-23) (%) k0.) t<1.m+1> Equation (6-23) is valid for determining the temperature at the center of the cylinder. At the surface n = N, r = rs, with the conditions (6-7) and (6-8) and the approximations, at _ h S?"(N,m) "E [tfl ' t(N,m)J = term) ' 3mm) <6-24) Ar _ “1N.m+1) ' fawn) 9—P— I Be (m+%) A0 (6-25) 1 girgl (MW) = 5555 [t(N-l,m) + t(N-1,m+1) '2t(N,m) '2t(N,m+1) + t(N+l,m) + t(N+1,m+1)] (6-26) From equation (6-26), = 2A3 _ t(N+1,m) k [tfl t(N,m)J + t(N,m) (6-27) Or, _ 11.0.: - _ t(n+1,m+1) ’ k [tfl t(N,m+1)] + t(N,m+1) (6 28) 37 Therefore, est _ 1 STEI (N,m+%) - 2(Ar)2 [t(N-1,m) + t(N-1,m+1) - 2t(N,m) .. ME _ RAE 2t(N',m+1) + k t(N-1.m+1) S N_1 + (1- + A—:E) hAr 6 rs t(N.m+1> = ’ (k)n-s t + [(k) - ZhAr (1 +-%3) t (6-32) 8 .. SALE 2pcp 130 Jt(N,m) H for calculating the temperature at the surface of the cylinder. 38 VII. DISCUSSION OF RESULTS DISCUSSION OF COMPARISONS BETWEEN EXPERIMENTAL AND CALCULATED TEMPERATURE DISTRIBUTIONS In the attempt to obtain a reasonable estimate of the overall heat transfer coefficient for the surface of a cylinder of silastic silicone, several values of the coefficient for heating were used. The value giving the closest prediction of the temperatures actually measured was selected as the standard overall heat transfer coefficient for heating. In a similar manner, the coefficient for cooling was chosen. These results are illustrated in Figure 11 for the surface temperatures measured and predicted with two times of exposure using flame for the heat source as was illustrated in Figures 8 and 9. There was a time response lag in the recording potentiometer used. As the response of the recorder was one second for full scale response, the actual peak temperature of the surface was never reached with the recorder. This response lag was not important when measuring the temperature at a depth of 0.004 foot in the cylinder; therefore, a relatively good prediction resulted as can be seen in Figure 12. Temperature measurements at a depth of 0.008 foot in corn stems were consistently higher than the predicted temperatures as illus- trated in Figure 13. Since the surface temperature of the corn stalk specimen was not measured, there was no indication that the predicted surface temperature was correct. The assumed source temperature can be too low. The accuracy of these predictions can be greatly improved 560 480 400 320 TEMPERATURE (°F) T (3 39 Figure 11 TEMPERATURE AT THE SURFACE OF SILICONE RUBBER SPECIMEN (CYLINDER) -- - EXPERIMENTAL PREDICTED o. . 3.2 x 10“»: l I l l 0 IO 20 3O 4O 5O exuo‘ (m) .JLLhATLRL (of) '1‘. ' 40 Figure 12 TEMPERATURE AT A DEPTH OF 0. 004 FT. IN SILICONE RUBBER SPECIMEN (CYLINDER) 1001 ---- Experimental Predicted 110 BTU hr 1‘ng 2 BTU I) 2 x 10' hr ’9’ 0 .0110 Ft ’ “TI 1 3% r8 90‘ 804 0 1'0 2'0 30 A0 0 x 10“ (hr) TEMPERATURE (°F) 41 Figure 13 TEMPERATURE AT 0.008 FT. FOR EXPERIMENTAL PREDICTED 100 .. ---- Experimental (Corn Stem) __ Predicted (Stemdard) ,‘f’ I ’0’ p’ 80 + ’0’ 1 70 ' L O V I t v I t— O 32 6h. 96 128 160 0:: 101+ (hr) 42 when more complete and accurate descriptions are available for the properties of the bodies considered. gIscussmN OF RESULTS FROM HPERmENgs wrm HEAT EXPOSURE 0F CORN its»: sams Initial studies were made to establish the general ranges of temperatures and times of exposure required to kill the cell tissue from corn stems. The observations from these initial studies using the water bath as the source of applied heat indicated the following results of whether or not cell tissue was killed: Temperature Time of Exposure (Seconds) (°F) 1 2 3 4 120 No No No No 140 No No No Yes 160 No Yes Yes Yes 180 Yes Yes Yes Yes 210 Yes Yes Yes Yes It was obvious that there was a temperature 115, time of exposure relationship existing. Therefore, further studies were planned with narrower ranges of times of exposure. The same general temperatures were used as in the initial studies except that til'ie 120° F temperature was eliminated. A complete list of treatments, data and observations for each run was accumulated and shown in Table 1. The visual observa- tions are illustrated in Figure 14. Although it was difficult to arrive 43 N m q ooH mm N one mw.om m m 00H RN N m N OOH om N nmq oo.om m H OOH mm N n q ooH em N mom om.N¢ m m 00H mm N m N ooH mm N «we mm.H¢ m H ooH Hm N H q ooH om N moo No.mm H m 00H mH o H N ooH wH o mmq om.¢o H H ooH NH 0 s OH aqH 0H 0 «mm No.00 n w qu mH o m o qu ¢H o mqw oo.o~ n d oeH mH N m .oH qu NH N Nmo mN.om m w qu HH 0 m o qu OH 0 «RN No.0N n q qu m N m 0H qu m N qu NH.mm m w qu m N m o qu o o “mm mm.Hm n q qu m N H oH oeH e N can mH.mc H m qu m o H o qu N o NHm O¢.¢o H q qu H Hemmnux .cmmn uozuOV H.m.n NV Haum\mnHv canons m>op< Hmeaoommv Away .02 mammHH Boom uomucoo onEmm Eoum mo ouswoaxm oHaEmm numm noun: ucmsumoua pom mooHum>uomno musuwHoz mo muHmcwa wwocuoucH mo oEHB mo musumummsms mess is .3 flaw: as ,3 558% 8 55 mm ”2542258 H «Hams 44 N m N oHN on N m NH oHN mm N m H oHN «m o m w OHN Mn N H N oHN Nm N H NH oHN Hm N H H oHN on o H w oHN as N N e owH we N mom mN.oa N m omH Ne N N N omH as N mom No.0N N H omH me N m a owH as N can No.oq m m omH ma N n N owH Ne N «Na mN.mN n H owH He N . m e owH oq N wNm NH.ws m m omH on N m N omH mm N mNm om.Nm m H owH Nm N H e omH on N 0mm mN.mm H m omH mm N H N omH an N Non oo.oo H H owH mm N N s 00H Nn N owe mN.Hm N m 00H Hm N N N ooH on 0 non om.NN N H ooH mN HemmauN .emmn oozro N.m.n NV NNUNNmpHV wagons m>on< Hmucoommv Amov .oz mammHe swam ucmucoo NHQENm amum mo mpsmoqu QHQENM summ Nmuma unmauwmus Mom mcoHumzomno 9:59.82 mo ~33ch mwoaumucH .uHo mEHH mo musumummsoa AcmzaHucoov H mHnme 45 N N N oHN «o N N wHH oHN no N N H oHN No N N w oHN Ho N m N oHN 00 N m NH oHN an N m H oHN mm N m w oHN Nn AmmonuN .vmon uozu0v A.m.n NV AnuM\mnHv vasouw m>on< Amwaoommv Amov .oz mswwHH Ewum uaoucoo oHaEmm Baum mo muamomxm deamm :umm Houmx unmaummua pom mcoHuw>uowno musumHoz mo mufimcwa muocumucH mo oeHH mo uuauwuomEmH AmwscHucoov H oHan 46 Figure 14 EFFECTS OF TEMPERATURE }I_8 TIME OF EXPOSURE ON CELL TISSUE OF CORN STEMS 220 . X = Heat damaged tissue O = No apparent heat damage ¥< )< >< >( )( )( t 200 4 “ \ \ ,r \ on“ \ .H 180 . \ x x 2 \ 60 Degree-seconds above 130°F g Critical level approximated m experimentally n. a 160 1 o x [-4 E5 :5 o: m 1J+o . o x Q :3 120 d o 0 O o 0 . . . . . - 0 2 lg. 6 8 10 EXPOSURE TIME (sec) 47 at specific levels of heat exposure at which the plant tissue was killed, the results indicated that the lower portions of the stems of the corn plants were more heat tolerant than the upper portions. The moisture content per unit volume of stem tissue was higher for the lower portion of the plant; therefore, the total heat capacity was higher. Also, moisture was more readily available at the sur- face which probably resulted in evaporative cooling. In Figure 14, a broken line was drawn connecting points immed- iately below the lowest points of cell kill for each exposure time. This dotted line represents the approximate critical level for expos- ure of corn stem segments to heat. The line very closely approximates a level of 60 degree-seconds of exposure above a base line of 130° F. The critical level for heat exposure is an important factor for most plants and plant products. This factor would be important in devel- oping flame weeding recommendations, crop processing techniques including curing and drying, and other recommendations related to heat exposure. Measurement of the surface temperatures obtained when segments of corn stems were exposed to heat was attempted with a small thermo- couple (.02 inch o.d.). These temperatures were not consistent with any particular characteristic of the stem segments. Considerable diffi- culty was encountered with the placement of the thermocouple on the surfaces of the samples. An exposure time of two seconds with the electric furnace was used for these studies. Relatively uniform heating over the surface of the samples was obtained with this system. As this heat source was primarily radiation, the time of two seconds was used 48 to reach a surface temperature comparable to that reported earlier and illustrated in Figure 11. A chromel-alumel thermocouple probe, 0.02 inch diameter, was imbedded in a knife slit on the surface of the sample. It was attempted to place the level of the thermocouple as near the level of the surface of the sample as possible. Large errors would have been introduced with the thermocouple attached to the outside of the surface of the sample. This would produce a fin effect resulting in indications of the temperature of the heat source rather than the sample. Attempts to use temperature indicating lacquers were unsuccessful, as it was very difficult to interpret the zone of actual temperature indication. Surface radiometers could not be used with flame as a heat source because radiation of the flaming particles introduce large radiation errors. They could not be used with the electric furnace because the resolution was not small enough to sense only the surface of the sample and the furnace completely enclosed the sample as it was heating. A thermocouple was used, therefore, for this portion of the study. The surface temperatures recorded are tabulated in Table 2 and plotted in the graphs that follow. Figure 15 shows the results obtained for the relationship between the internode of the stems of the corn plants and the surface temperature obtained when exposed to 2000° F for two seconds with the electric furnace. Correlation was poor between the surface temperature and the stem internode above the surface of the ground. Likewise, the correlations were poor between the surface temperature and stem diameter and between the surface temperature and the density of the stem samples as shown in Figures 16 and 17. The maximum 49 Table 2 SURFACE TEMPERATURES OBTAINED WITH STEM SEGMENTS OF CORN PLANTS EXPOSED TO 2000° F FOR.TWO SECONDS Treatment Internode Stem Diameter Stem Density Maximum Surface No. (In) (lb/fta) Temperature (°F) 1 1 0.50 57.0 315 2 l 1.00 58.2 315 3 l 1.00 64.5 322 4 l 1.10 58.2 233 5 l 0.84 63.6 520 6 l 0.68 62.5 288 7 l 0.64 62.6 554 8 1 0.72 57.3 274 9 l 0.75 63.0 254 10 3 0.44 50.6 431 11 3 0.85 38.4 541 12 3 1.00 31.8 409 13 3 0.75 37.2 346 14 3 0.60 47.1 470 15 3 0.58 47.2 299 16 3 0.60 28.9 306 17 3 0.68 41.2 256 18 5 0.40 50.0 415 19 5 0.79 31.1 294 20 5 0.90 21.2 330 21 5 0.65 26.6 234 22 5 0.52 31.6 375 23 5 0.50 38.6 253 24 5 0.55 22.9 284 25 5 0.55 24.3 494 26 7 0.28 48.9 348 27 7 0.71 40.2 294 28 7 0.57 28.8 425 29 7 0.70 20.8 447 30 7 0.55 28.0 312 31 7 0.50 28.5 307 32 7 0.40 48.3 257 33 7 0.50 23.5 287 50 Figure 15 THE SURFACE TEMPERATURE OBTAINED ON SEGMENTS 0F CORN STEMS EXPOSED To 2000° F FOR Two SECONDS AS AFFECTED BY THE HEIGHT ON THE STALK AT WHICH THE SAMPLE Is TAKEN o o o 500 . o o o o o A 0 an. 400 ‘ ° 0" Average 0 E ° .3 F: 00 8 ° 8 333 ‘ o 0 o o E ° ° ° 9' O O o o W o O 3% a 200 4 :> 0) E F! 2&1004 Cor. Coef. = 0.075 0 . . V I U I I V o 1 3 5 7 INTERNODE ON CORN STEM ABOVE THE GROUND SURFACE Q Aej 0.0 0.1;. 51 Figure 16 THE SURFACE TEMPERATURE OBTAINED ON SEGMENTS OF CORN STEMS EXPOSED TO 2000° FOR TWO SECONDS AS AFFECTED BY THE DIAMETER OF THE STEM Cor. Coef. = 0.099 026 0:8 1'.O 1:2 DIAMETER 0F STEM (in) 52 Figure 17 THE SURFACE TEMPERATURE OBTAINED ON SEGMENTS OF CORN STEMS EXPOSED TO 2000° F FOR TWO SECONDS AS AFFECTED BY THE DENSITY OF THE STEM O O O O 0 0° 0 O O O O 0 0° 0 0 00 o o 0 o o O O O O 0 Cor. Coef. = 0.025 20 30 110 S DENSITY 0F STEN (lbs/ 0 60 ft3) 53 surface temperature recorded when exposing a corn stem segment to 2000° F for two seconds was 554° F and the minimum was 233° F. The inconsistency of data obtained in these experiments can be related in part to the non-uniformity of the characteristics of the corn stem segments and partly to the faulty location of the thermocouple. For example, attempts were made to measure the densities and moisture contents of segments of the stems. Though all of the plants were approx- imately the same age, it was found that the densities for all nodes ranged from 20.0 to 72.5 pounds per cubic foot. The overall variation of moisture contents on a dry basis for all internodes was 248 to 772 percent; however, the greatest variations for an internode was for the seventh internode, for which the moisture content varied from 248 to 772 percent. On a wet basis, however, the overall variation in moisture contents was from 71.3 to 88.5 percent. The relationship between density and the internode of the corn stem above the ground is shown in Figure 18. Obviously, the density of the stem does not vary linearly from the ground to the top of the corn plant. The relationship between moisture content and the internode of the corn stem above the ground is shown in Figure 19. It is difficult to describe accurately the physical characteristics of the stem of a corn plant. These characteristics are important, however, when attempting to relate reactions of prediction models with reactions of the plant. Therefore, predictions of temperature distributions within the stem of a plant cannot be very reliable unless more accurate measurements of the physical characteristics are made. 54 Figure 18 THE RELATIONSHIP BETWEEN THE DENSITY OF A CORN STEM AND THE HEIGHT ON THE STALK AT WHICH THE SAMPLE IS TAKEN 601 50* a? .p 64 \ 3 saw 31 H H H 30" o ' I I II f 0 1 3 5 7 IL'T‘I'PY'ZRTIODE ON CORN STEM ABOUT} THE GROUND SURFACE 55 Figure 19 THE RELATIONSHIP BETWEEN THE MOISTURE CONTENT OF A CORN STEM AND THE HEIGHT ON THE STALK AT WHICH THE SAMPLE IS TAKEN 6004 fl ’2 D. 'o 500. ‘25 E: B Z O H O m E: E4 a O 21 14004 0 ' V l I 1— 0 1 3 S 7 INTERNODE ON CORN STEM AROVE THE GROUND SURFACE 56 The loss of moisture from the sample during heating was always less than 0.5 percent (w.b.); however, the total loss for the heating and cooling cycle was approximately 1.0 percent. Although the moisture loss was relatively small, all occurred from a very thin layer at the surface of the sample and from the ends. As the heat was applied to the surface for a short period of time, the evaporative cooling effect of the moisture could be significant. DISCUSSION OF PARAMETRIC EFFECTS The influence of any one parameter on the temperature distri- bution in a body cannot be entirely separated from the remaining parameters. It was difficult, if not impossible, to allow changes in one parameter in real cases without causing a change in one or more of the other parameters. For each parameter Study, the temperature distributions obtained were compared with the temperatures obtained with the assumed standard sample illustrated in Figure 20. A discussion follows for each parameter and its influence on the temperature and, in some cases, its influence on one or more of the other parameters. 1) hH (BTU/Hr Ft? F) -- overall heat transfer coefficient for the surface during heating. The temperature at the surface of the model is directly proportional at any given time to the magnitude of the surface coefficient for heat transfer as shown in Figure 21. This effect is further explained under a following section discussing the influences of the diameter of the body. Additionally, the influence of the surface coefficient is TEMPERATURE (°F) 57 Figure 20 TEMPERATURE DISTRIBUTION HISTORY FOR THE STANDARD SAMPLE USED IN THIS STUDY 320 _ CASE 1: A0 = h x 10-5 hr Ar = 00002 ft P8 = 00013.0 ft 280... r0 = 0.035 :1: h}; = no BTU r 1%? a ho - 2 BTU hr rtZF k1 = 0.2 BTU/hr rt F k2 = 0.3 BTU/hr rt? I cp1= 0.91 BTU/lb F P = o lb/ft 92 =10 lb /ft3 tfx=1£31°F (Heating) §f1=gg F(Cooling) = 200: 160 ‘ 120‘ \ \ «1'0"» O O 80 Low ‘ \\ h l L l J J 6 8 10 [-1-— Ar fig DEPTH FROM SURFACE x 103 (ft) 17 58 Figure 21 TEMPERATURE AT THE SURFACE AS AFFECTED BY HEAT TRANSFER COEFFICIENT FOR HEATING 560 (CYLINDER) 480 o. - zox Io'5m r, . 0.040 n 400 ‘3 .~° 3 320 I 2. E a 240 .0 p. rC (CYLINDER) 91 = 20 x 10-5 hr k2 0.1 214.0. 0.2 2001 0.3 €+ E: E 1.0 E ii 9'120 80 . V V V I o 6 16 2h. 32 up 0 x 105(hr) 62 The thermal conductivity of the radial segment smaller than rc had no effect on the surface temperature for the heating time considered. At a position of 0.004 ft from the surface of the cylinder, higher thermal conductivity values in the segment greater than rc resulted in more rapid temperature changes. The effect was not as pronounced for changes in thermal conductivity for the segment less than rc. These effects are illustrated in Figures 24 and 25. 4) cp (BTU/Lb F) -- specific heat of the model. The specific heat was assumed to be constant and uniform in radial segments of the cylindrical body. One stepwise change in specific heat was assumed to occur at the critical radius, re. The changes considered in the specific heat had no appreciable effect on the temperature distribution of the body. A very slight effect was found for changes in the specific heat at r'> rc. This effect as shown in Figure 26 for the surface of the model was inversely related to the magnitude of the specific heat. As shown in Figure 27, the effects of changes in specific heat at a depth of 0.004 ft were very slight for r > rC and there were no effects from changes at r S re. 5) p (Lb/Ft?) -- density of model. The density was assumed to be constant and uniform in radial segments in the cylindrical model. One stepwise change in density was assumed at the critical radius, rc. 63 Figure 24 TEMPERATURE AT A DEPTH 0F 0.004 FT AS AFFECTED BY THE THERMAL CONDUCTIVITY AT r < rC (CYLINDER) 12C>1 k1 01.: 20 x 10"5 hr 1‘8 = 00014.0 ft 100 0.6 1&0 - 0.3 0.2 A __ 0.1 a. o g} 100 «4 E: a: p/ m ‘3: A. m 5" 90A 80'! L 0 u u v v . . 0 32 6h 96 128 160 o x 105 (hr) 64 Figure 25 TEMPERATURE AT A DEPTH OF 0.004 FT AS AFFECT!” BY THE THERMAL CONDUCTIVITY AT r > rc (CYLINDER) ' 1201 01 = 20 x 10"5 hr r8 = 0.01m :1: k2 no - 0.3 0.2 3': ' 0-6 Ema « 1.0 ;§ 0.1 [I] A. E 90A 801 o , . . . . 6 32 614. 96 128 160 o x 105 (hr) 65 Figure 26 SURFACE TEMPERATURE AS AFFECTED BY SPECIFIC HEAT FOR r > re (CYLINDER) ' 280' 01 = 20 x 10"5 hr r8 = 000,40 ft 2&0- 20 0.87 01 0.91 1.00 160- 120 80 o 53 12> 21; 3'2 EC 0 z 105 (hr) 66 Figure 27 TEMPERATURE AT A DEPTH OF 0.004 FT AS AFFECTED BY SPECIFIC HEAT FOR r >rC (CYLINDER) 120. 01 = 20 x 10'5 hr rs = 0.0u0 rt °p2 110* 0.87 0.91 A 1.00 fig ‘1. 100- E3 53 .¢ [1: CI] 04 g 90" 80‘ 01 t ' I I r o 32 6E 96 126 160 9 = 103(hr) 67 For the segment less than re, a change in the density had no noticeable effect on the temperature distribution. A change in density for the segment greater than re, however, had a very pronounced effect on the surface temperature, which resulted in an important effect at a depth of 0.004 ft from the surface. See Figure 28 for the effects at the surface and Figure 29 for the resulting effects at a depth of 0.004 ft. A 50% increase in the density resulted in approximately a 20% lower peak temperature at the surface. If sudden drying occurs in the surface layer resulting in a decreased density, a rapid increase in surface temperature can result. Under this circumstance, however, the thermal conductivity factor probably decreases at a rate comparable to the rate of decrease in density. 6) r8 (Ft) -- radius of the model. The radius of the model had a greater potential effect on the temperature at the critical point, rc, than any of the other parameters included in this study with the exception of the temperature of the surrounding fluid (heat source). Although this effect appeared relatively insignificant on the surface as indicated in Figure 30, the internal effect was extremely pronounced as shown in Figure 31. The temperature at the 0.004 ft depth in Figure 31 rose to 125° F in a period of 0.0016 hour (5.76 seconds) for the 0.01 foot radius and eventually peaked at 130° F in 0.004 hour (14.4 seconds). A peak temperature of 108.9° F at a similar depth in the cylinder of 0.02 ft radius was reached in a much shorter period, 0.002 hour. As the radius was increased, the time required to reach the peak temperature 68 Figure 28 SURFACE TEMPERATURE AS AFFECTED BY DENSITY FOR I > r (CYLINDER) 320. 01 = 20 x 10"5 hr 1‘8 =3 000 0 ft 280- 2110‘ {:2 no ET ‘3. . 60 £3200 65 E? E... 180- 80 0 U U I Y T I U 0 8 16 24 32 no G x 105 (hr) 69 Figure 29 TEMPERATURE AT A DEPTH 0F 0.004 FT AS AFFECTED BY DENSITY AT r > r (CYLINDER) 2&0. 200. 0 = 20 x 10"5 hr 1'8 8 000,40 ft i=7 9.. 160‘ E E3 8“. ‘° 2 5?. 120- “0 60 65 80' {5 (Fr . 4r . r .7 0 32 6h 96 128 160 O x 105 (hr) TEMPERATURE (°F) 70 Figure 30 SURFACE TEMPERATURE OF CYLINDER AS AFFECTED BY DIAMETER 3201 Cylinder ......... Semi-infinite wall 2801 211.0- / \\ rs a” ‘\ \ .02 \\ . CO3 \\\ . 001,4, \\ . 005 “~. .OCHB 160-1 ‘\\ o .32 \Wall 120.. 80 o T t T f v 7 0 8 16 2h. 32 14.0 C x 105 (hr) TEMPERATURE (°F) 71 Figure 31 TEMPERATURE AT A DEPTH OF 0.002 {IT . AS AFFECTED BY CYLINDER DIAMETER" I's 0 .01 ft 120-: Cylinder ---- Semi-inf inite Wall 110- 0 .02 0 .01; 100;. =0 '08 E0 .32 90T ,. __ ___ _____________ Wall 801 ,7 ’ ’ ’ ’ ’ C U V I U j 0 32 61;. 96 128 160 72 continued to decrease. The effects resulting from changes in the size of the cylinder can be caused by several factors. The surface area per unit volume per unit of length increases as the cylinder diameter decreases, thereby allowing more heat per unit of volume. A point at a given depth in a small cylinder is close to a greater surface area than at a similar depth in a large cylinder. Therefore, the temperature of a point at the considered depth in a small cylinder is influenced more readily by a rise in the surface temperature than at a comparable depth in a large cylinder. 7) tf1 (°F) -- temperature of fluid surrounding the model. The surface temperature obtained was directly proportional to the temperature of the flame used as a heat source as illustrated in Figure 32. The temperature at some fixed point in the flame was normally not constant; however, for this study, the flame temperature at the point considered was assumed to be constant. A heat source with a temperature varying from atmospheric up to 2500° F for an average temperature of 1641° F during the heating period gave a surface temperature of the model quite different from that obtained with a constant heat source at a temperature of 1641° F. See Figure 33, which indicates a higher surface temperature for the constant temper- ature°source. Only one fluid temperature, 77° F, was used for the cooling period. 3204 2804 2401 200‘ TEMPERATURE (°F) 14 0‘ E? 1204 73 Figure 32 (CYLINDER) SURFACE TEMPERATURE AS AFFECTED BY SURROUNDING FLUID TEMPERATURE 20x16'5nr 0.0Uo ft md 2L 9 x 105 (hr) 32 TEMPERATURE (°F) 74 Figure 33 SURFACE TEMPERATURE WITH CONSTANT XE VARYING HEAT SOURCE (CYLINDER) Constant 16h1°F -__- Varyi Average 380‘ 16uI§E5 6 = 16 x 10' hr r5 = OQOLLO ft 2&0. 200‘ 1601 120- 80+ r o- (I) 16 r 2h 32 AC 9 x 109 (hr) v I v I ‘V— 75 DISCUSSION OF ERRORS The accuracy of the predicted temperatures by the numerical analysis procedures used is almost entirely dependent upon the accuracy of measuring or assuming the values of the contributing parameters. These parameters for biological materials vary widely due to non- homogeneity of the samples. The errors of measurement of some of the characteristics of the stems of corn plants include the following estimates: 1) Density - i 1% - by water displacement and weighing 2) Specific heat - i 2% by Kopelman (1966). 3) Thermal diffusivity - i 10% - Matthews (1966). 4) Thermal conductivity - i 12% - lecp) 5) Diameter - i 1% by measurement. 6) Moisture content - i 1% (w.b.) by oven. To evaluate the relative effects of each of the above indicated errors, each parameter must be considered for its relative contribution to the temperature distribution. For example, the error of measuring the thermal conductivity is obviously greater than that for measuring the diameter of the sample. However, the effects of changes in the diameter are greater than a comparable percentage change in the thermal conductivity. Kardas (1964) evaluated errors obtained when using finite- difference solutions to the heat flow equation. His analysis was for a slab with errors indicated at the surfaces and center of the slab. The procedure involves the use of two figures reported by Kardas and factors such as Biot No. (ha/k), Fourier No. (QT/32) and r (n2 A0) in the 76 expression, a n? = k (l i 6r) - E where: e discretization error E error parameter from graph a § distance, rS for cylinder n = number of increments These errors are less than 12%. Likewise, Freed ££_§l. (1961) reported that errors resulting from using finite-difference methods are less than i2%. Combining all of the above indicated sources of errors in the numerically approximated solution, the predicted temperatures will have an estimated error up to i12%- An additional source of error could be the evaporative cooling effect of the moisture losses from the surface of the plant stem. Measurements in this study indicated that approximately one-half gram of the total weight of each 100 gram sample was lost during the heating period. If all of this moisture was lost from the surface of the stem, not including the ends, the heat energy required to evaporate the moisture can be significant. 77 SUMMARY The use of finite-difference techniques for describing the temperature distribution in an agricultural specimen was demonstrated. The accuracy of the calculated temperatures depended on the accuracy with which the properties of the specimen were described. The application of an earlier developed technique for measuring the thermal diffusivity of the specimen was demonstrated. This technique was found to be readily usable with some slight modification of the thermal diffusivity apparatus. Of the parameters discussed in relation to their effects on the temperature distribution of the selected model, the diameter of the cylinder is most critical. The diameter of the plant stem is a reliable criterion for recommending intensity and time of exposure to flame for weed control in crops. The moisture content of a plant stem probably has considerable influence on the internal temperatures resulting from flaming. Although the moisture losses from the samples during heating was small (less than 1% by weight) as found in this study, significant evaporative cooling probably resulted. A uniform temperature of the flame had a greater effect on the temperature of the body than an asymptotically increasing and decreasing temperature. For more weed kill, therefore, exposure to a constant energy source for a given period of time would be more effective than a higher energy for a shorter period of time. 78 SUGGESTIONS FOR FURTHER WORK In order that processes involving the application of heat to biological materials can be better understood and recommendations be made for improving these processes, there is a real need to continue to measure and aCCumulate data on the physical properties of these materials. It is necessary to determine these properties as they are related to temperature and mositure content. Evaluation of the effects of moisture losses from the surface of a biological material as related to the surface temperature is an important area of study. Techniques and equipment for measuring the surface temperatures and moisture losses from the surfaces of biological materials are needed. It is difficult to attach sensing devices such as thermocouples or thermistors to maintain intimate contact with the surface of a biological material without altering the characteristics of the surface. The suggested procedures for predicting temperatures and determining heat damage should be tested on some fruits and vegetables, such as apples, pears, squash, potatoes and sugar beets to obtain optimum rates of applying heat if used for curing and storing them. The effects of coupled heat and mass transfer (moisture diffusion and evaporation) were not shown. More information relative to the coupling phenomena was needed to show these effects. The heat equations and the diffusion equations could be developed though the coupling equations were not available. The heat equations would show the temperature history when the body is considered as a semi-infinitely thick wall. The 79 resulting temperatures would be lower than those obtained for an infinitely long cylinder with similar heat exposure. Sample out- puts for the temperature distributions obtained for a semi-infinite wall as well as for a long cylinder are shown in Appendix C. Using prediction equations with properties of the body to be considered and data on the heat tolerance of the plant tissue as demonstrated, the relative damage to a plant with applied heat can be predicted. These prediction techniques may be applicable to other processes such as curing and storing sweet potatoes, burning the leaves from sugar cane and foliage from potatoes, rapid drying of biological materials, and applying heat for defoliating cotton when physical properties of the plants and plant products are known. 80 APPENDIX A List of computer symbols used: CPl CP2 DR DX DT HH HC PKKl PKKZ RC RO RHOl RHOZ TAH TAC 'TI cp (BTU/Lb F) -- Specific heat (r'< re). 1 cp (BTU/Lb F) -- Specific heat (r 2 re). 2 Ar (Ft) -- Radius increment. X (Ft) -- Depth increment. A6 (Hr) -- Time increment. hH (BTU/Hr Ft? F) -- Overall heat transfer coefficient (0 0 ) 1 NUmber of internodes ( distance increments ). N + l -- Total number of nodes. k (BTU/Hr Ft F) -- Thermal conductivity (r < re). 1 k (BTU/Hr Ft F) -- Thermal conductivity (r 2 re). 2 r (Ft) -- Radius rC (Ft) -- Radius at which properties change. rS (Ft) —- Outside radius. p (Lb/Fta) -- Density (r < re). 1 p (Lb/Fta) -- Density (r 2 re). 2 t (°F) -- Temperature. tf1(°F) -- Temperature of surrounding fluid (6 s U ). 1 tf1(°F) -- Temperature of surrounding fluid (9 > 0 ). 1 9 (Hr) -- Accumulated time. ti (°F) -- Initial temperature (9 = 0) 81 X -- X (Ft) -- Depth in wall. XC -- Xc (Ft) -- Depth at which properties change. XL -- XL (Ft) -- Total depth considered. YYY -- 0 (Hr) -- Total time of run. Other computer symbols were used for calculational purposes only and were defined in comment statments in the computer programs included under Appendix B. 82 APPENDIX B Sample Computer Programs used: 1) PROGRAM TEMPCYL -- to calculate the temperature distribution in a long cylinder. 2) PROGRAM TEMPWAL -- to calculate the temperature distribution in a semi—infinite wall. 31 32 33 34 35 36 40 41 42 43 44 45 47 52 55 60 65 065?) “CDC30(DC1 83 PRCCRAM TLMPCYL FIND TEMPERATURE DISTRIBUTION IN AN INFINITE CYLINCER AF TER A SHORT EXPOSURE TU FLAME UNIFORMLY CVLR.ITS SLRFACE. DIPENSIDN A(30C193135019C1300190(3C011TIBJD)9R139319 1RK133019CCI3CJ11DD(3C01FXIBOC)9UI3U51 CDPMDN A98.CngT9CCyDDgTT,DT,TA,DR,RD,RHD1cRHCZpPCClv IPCCZIPKKlvpKKZICPIQCPZ’HQHHpHC9RC’N’TAH’TACgTH’DTH RC=UUTSIDE RADIUS, RH01=DENSITY AT R.LT.RC9RFC2=DEASITY AT R.GT.RC9 CP1=$PECIF1C,HEAT,AT R.LT.R£J-C£2?$PEQLEIC FEAT AT R.GT.RC: PKK1=THERMAL CONDUCTIVITY AT R.LT.RCP PKK2=THERMAL CCNDUCTIVITY AT R.GT.RC:T(AN1 IS THE TEMPER- ATCRE AT THE SLRFACE. Til) IS THE TEPPERATURE AT THE CENTER. ICC FORMATi1X9F8.591X910F8.31 103 READ 1049CASEFCT1YYYFRHD1CBHDZJRCCCP1FCPZFPKKlsPKKZITHs 1TIvN 104 FCRMATI F4.39F8.59F6.3,2F7.2g5F6.3g F7.51F7.29141 IF‘CASE.EQ.P.J)GD TD 1003 AN=N+1 READ 1059 DTHCTANCUTAHFTACFHHpHC,RC 195 FCRMA"IXOF70503£Z&212F79ZJF613) YYY=9.JDbO RC=RU-3.005 PRINT 205 2C5 FCRMATTIHll/1 PRINT 196 1J6 FCRMATIISX,42HCA$E. DT m YYY RHnl_ RHQZ -CPl. CPZ, 13X911HPKK1 PKKZ /1 PRINT 1379CA$EvDT9YYYgRHDI9RHDZ:CP19CP2,PKK1.PKKZ 137 FCRMATI15X.F4.C1F8.5,F7.492F6.294Fb.3//1 PRINT 108 105 FCRMATTIQX.41H R0 RC HH HC TAP IAC . 11X.9HTI N .llw ' PRINT 109,RC9RC.FH,HC.TAH,TAC9TI9N 159 FCRMAT119X9ZF6.312F7.293F7.1914//1 TT IS TIMEaTTK) 15 TEMP AT NUDE K, PKKK=K, PCCC=RHCFCP J=C 1T3000 AA=N DR=ROIAN CC 200 KV=19NN J; TIKM)=TI BECINNINC OF PROGRAM CALCULATIONS. 211 CALL CDETS 212 CALL TRIDI TT=TT+DT CC 40 II’IQAN L=hN=II+1 4i U(II)=TIL) IF(N.CE.2?1CD TD 45 IF‘N.EQ.15)CD TD 54 . IF(N.EQ.101CD TD 43 PRINT 4101TT,(L(K11K=1:NN1 41C FCRMATI10X9F8.596F8.3/1 66 67 74 101 192 103 110 115 116 123 124 131 132 13? 14v 141 144 147 152 155 16d 161 162 163 164 165 166 451 44 45 111 112 455 831 802 901 14.23“; 84 CC T0 455 PRINT 1109TT91DIK19K=1971 PRINT 4519(LTK19K=89AN1 FCRMATT18X94F8.3/1_H GC T0 453 PRINT 11*31TT91UTK10K‘1971 PRINT 1119(LTK19K38914) GC T0 459 PRINT 11DvTToTLTK19K=1971 FURNATIlDXcF8.5aTF8.3I PRINT 111v (UTK19K38'141 FCRMAT118X97F8.3T PRINT 1129 (UTK19K3150211 FCRMATT18X97F8.3/T J’J‘I IFIJ.EQ.1lICU T0 801 IFTJ.EQ.251GU T0 801 IFTJoEQ.39)GD T0 801 IFTJ.EQ.53)CD T0 801 IFTJ.EQ.67)GO T0 801 BC T0 991 PRINT 802 FCRMATIIHIT IFITT‘YYY) 211D211!1C3 YYY 15 MAX TIME FOR CALCULATIONS. CDATINUE STCP EAC Uv‘ ~Joxnb 12 16 17 20 21 23 24 25 26 27 32 33 34 35 36 37 43 41 42 43 44 4s 46 47 50 51 52 53 54 55 56 57 60 63 64 65 66 67 7) 71 535 222 49 22 It. 85 SUERDUTINE CCETS TFIS PROGRAF GENERATES THE COEFFICIENTS. CIPENSIDN,AI3GCJ¢BI3COI.C133010013LDJ41135CJ.RJ3ub19- 1RKI3391'CCT3801900(3C019XI3001vUI3LJ) CDPMDN Ag8.C.D.T.CC.CD.TTgDTgTAgDRgRflgRHfllthCZ.PCCla 1PCC29PKK19PKK2.CP19CP29H.HH.HC.RC.hyTAHyTAC.TH.CTH AA=N AA=N+1 CRR=DR*DR .. IFTTT.GT.O.CCZ4)DT=0.CCO4 CCT=DRRIDT PCC1=RH01*CP1 PCC2=RH025CF2 PECT1=PCC1'DDT/Z.J PEETZSPCCZ'CDT/2.J 31173000 RI1)=D.3 BT1)=-(PKK1+PDCT11 C(1)= PKKI C(1)= (PKK1-PCCT11*TI11-PKK1'T121 IFITT.GT.3.CCJ15)GU T0 49 TA=TAH P=FH 50 T0 225 TA3TAC P=fC RDA=RD-DR RCE=RD+DR RCAA=2.0*RO+CR PKRA=PKK2*RDA PKR8=PKKZ‘RC8 AIAN)=PKK2*PDA . , BIAN)=‘TAINA1+P'DR*RCB+Z.U*RC*PDDT2) CIANI=3.0 DTAN)=-A(NN)*T(N1+(ATAN)+H*DRORDB-2.3*RDlPDDT21'TTAK1- 1 2.0'H'DR9RC85TA DC 2 K32'N KL=K+1 KJ=K-1 PKJ’K-l PKL=K+1 PK=K PN=N RTK)=DR*PKJ . 1F(R(K).GE.RC1 GO TO 3 PKKK=PKK1 PCCTSPCDTI GC T0 4 PKKK=PKK2 RCCT=PDDT2 RKIK13PKKKGTR1K1‘0.5!DRI KK=KL ‘ RKIKK) IN SUERCUTINE CDETS EQUALS CCNDUCTIVITY'RADIUS AT 86 c PCSITIDN ONE-HALF NUDE BEYOND (+7 NUDE K. thx) 15 AT c CKE-HALF NOCE LESS (-) THAN NUDE K. 72 RKTKK1=PKKKC(RTK1+O.5'DR) 73 A(K)=RK(K)-FKKK§DR[2.0 ,. ‘ . 74 B(K)=-(RK(K)+RK(KK)+4.6.pooT-(R4K)+DRI2.O)i 75 C(K)=RK(KK1*FKKK!DR/2.d 76 o¢x7=~c¢x)*7(KK)+(RK(K1+RK7KK7-4.o.ponr.(a(x)+0R/2.:J)4 1 TlK)-A(K)*T(KJ) 77 2 ccnrxwus 131 RETURN 102 ENC N's «Jaynb 10 12 13 15 16 17 20 21 23 24 h) 87 SUE—REG T 1 NE. FIT [EDI DIFENSIDN A1305138130019C130513DI3U019T130019RI3001: 1RK130013CC13001900130019X130019UT3DU) CCHMDN A.8.CaC.I.CC3CD.ITLDI.IA:DR.RD.RHQ1.RhcagPCC13mm_ IPCCZ.PKKl.PKK2.CP1.CP2.H.HH.HC.RCghpTAHoTAC.THgoTH THIS SUBROUTINE SDLVES A TRIDIAGDNAL SET OF EQUATICAS. CC 15 C*o DC 15 0* AN=N+1 CCTI)=C(1)/E(1) CD11)=D(1)/&(11 DC 1 K=29NN KJ=K~1 KL=K+1 CCTK)=C(K1/(BIKI-AIK)*CC(KJ11 CDIK)=(D(K)-A(K)*DD(KJ)1/181K)-A(K1*CC(KJ)) ITBNIFB01NN) DC 2 KK=24NN KKK=NN-KK+1 KKKK=KKK+1 T(KKK)=DD(KKK)-CC(KKK)*T(KKKK) RETURN END” 2- -22 . . ~-. .-.-_.1.. . _.-, . F.-. .. ~ -m-1—._. ._-., -—- m_-.-r —. .-._ -mg, . - w.. —- 7 r. 12 13 14 15 16 17 2d 21 22 23 24 25 26 27 31 32 33 34 35 37 40 41 42 45 53 6} 61 62 67 74 nonnnn 88 PRCGRAN TFNPhAL FIAD TEMPERATURE DISTRIBUTION IN AN INFINITE CYLINCER A1 TER A SHORT EXPOSURE TO FLAME UNIFORMLY OVER ITS SLRFACE. CIVENSION AI3OCIcBI3CQ)9C1380110139313TI30013X133C)o, 1XK131019CC130019CDT3C019YI3091 COPMON A,8.CyDuT5CC9CDoTTgDTgTAyDXoXLaRHOI.RFO29PCC19 IPCCZ.PKK19PKK2pCPlyCP2.HQHH9HC.XCyNyTAHyTACoThgnTng RC=OUTSIDE RADIUS: RFOI=DENSITY AT R.LT.RC.RFC2=CEASITY AT R.GT.RC. CP1=SPECIFIC HEAT AT R.LT.RC’ CPZ=SPECIFIC HEAT AT R.CT.RC32PKK13THERMAL CCNDUCTIMITX-AT RLLI‘RQJ. _ PKKZ3THERMAL CCNDUCTIVITY AT R.GT.RC9TTNN1 IS THE TEMPER- ATLRE AT THE SLRFACE. . _ T11) 15 THE TEVPERATURE AT THE CENTER. FCRNATTIX:FE.531X110F8.31 READ 174’CASEvCTvYYYrRHOIvRHOZQXLICP19CP21PKK1yPKKZ0TH, ITIvN _-TM. . w-. - . .~.- .3 . 154 FCRMATI F4.UyF9.59F6.312F7.295F6.3. F7.5.F7.29141 IFICASE.EQ.C.CTGO TO 1000 AN=N+1 READ 165, DTFpTANCgTAH.TAC.HF.HC.XC CE FCRMATT1XpF7.593F7.292F7.2.F6.31 YYY=Q . 33.16 PRINT 2‘5 2:5 FCRMATTIHI/IT PRINT 136 186 FORMAT115X942HCASE DT YYY RHOI RHCZ CPI CPZ: 13X911HPKK1 PKKZ /) PRINT 1C79CASEaDTaYYYaRHOIaRHOZ9CP11CRZ¢PKK1:PKKZ 137 FCRMATT15X,F4.C9F8.59F7.492F6.294F6.3//) {‘3 D “Jr; 1 ..- 1 1 PRINT 138 188 FCRMAT119X941H XL XC HH HC TAH TAC 9 11X99HTI N I) PRINT 1899XL’XC9FH9HCpTAHgTAC9TI9N 1’39 FORMAT11’9X9ZF6..312E7.2:3F7.1.21.4/l) ._ ,. . , TT IS TIMEpTIK) I5 TEMP AT NCDE K9 PKKK=K9 PCCC=RHClCP J‘L TTgLoO AA=N CX=XLIAN CC 230 KM=13AN 20C TTKM1=TI BEGINNING OF PROGRAM CALCULATICNS. 211 CALL CDETS 212 CALL TRIDI TT=TT+DT IFIN.GE.ZU)GD TD 45 IFTN.EQ.15)CO TO 44 IFTN.EU.1C)CU TD 43 PRINT 41“.TT.1TTK).K=19NN) 41C FCRMAT113X9F8.596F8.3/1 GC T0 454 43 PRINT 110.TT.(T(K1.K=1.71 PRINT 45151TIK19K=81AN1 451 FORMAT118X34F8.3/1 75 76 143 115 111 116 117 124 125 132 133 134 137 142 145 153 153 154 155 156 157 160 161 44 11; 111 112 456 0mm "3 C; t.) HR).— 19 \. VU. 89 ,2.__-—-—-‘—--‘— ..-, CC TO 458 PRINT 1179TT91T1K’9K‘197) PRINT 1119(TIK19K=89141 GC TO 450 PRINT 1199TT91T1K19K51971- . 22,2 FCRMAT113X9F8.597F8.31 PRINT 1119 (TTK19K389141 FCRMATI18X97F8.31 PRINT 1129 (T1K19K3159211 FCRMAT118X97F8.3/) J=J+1 IFIJ.EQ.11)CO TC 801 IFIJ.EQ.ZSIGO TO 801 IFTJ.EQ.39)CO TO 891 IFIJ.EQ.53TCO TO 801 IFIJ.EQ.67ICO TO 801 CC TO 901 PRINT 832 FCRMATT1HIT IFITT-YYYI 21192119103 YYY 13 MAX TIME FOR CALCULATIONS. CONTINUE STCP EAC 67 7c 71 C 49 222 10 90 SUEROUTINE CUETS TFIS PROGRAN GENERATES THE COEFFICIENTS. DIPENSIDN-AI3Q£1¢BI$UDIILJQOOIIDLBLQIJTIJDDIIXIBuCIQ. 1XK130019CC13C019CDT3CO).Y13001 COAMON A989C903T9CC9809TT9DT.TAgDXgXLgRH019RFC2.PCCla 1PCC2.PKKI.PKKZ9CP19CP2.H.HH.HC.XC.N,TAH.TAC.TH,DTH,X Ah=N AA=N+1 Cxx=vaDx _ IFITT.GT.O.LC335)DT=L.56004 IFKTT.GT.0.:524)CT=J.CC64 CCI=DXXIDT PCC1=RF01*CP1 PCC2=RH02*CF2 PECT1=PCC1*CDT PCETZ=PCC2§£CT IFTTT.GT.O.CCCIS)GO ID 49 TA=TAH F=+H cc T0 222 TA=TAC A A 3. (1.0+FCDTZIPKK2) .\ CD A H “V.” rap-0| 0(1) XD=2.C*PDDT1/PKK1 AIBN)=1.O 8(AN)=-(1.V+XDI C(hN)=a.‘ C(hN)=-T(N)+I1.5-XD)*T(AN) CC 2 K=29N KL=K+1 KJ=K-1 PKJ=K-1 PKL=K*1 PK=K PA=N X1K1=DX§PKJ IF1X1K1.LT.XC) CC T0 3 PKKK=PKK1 PECT=PCDT1 GC T0 4 PKKK=PKK2 PECT=PCDT2 XK(K1=PKKK*(X1K)-3.5*DX) KK=KL XKIKK)=PKKK*1X(K1+O.5*DX) PCSITION ONE-HALF NUDE BEYOND 1+) NUDE K. ONE-HALF NDCE LEssnIfT THAN NUDE K. XCF=2.C*PDDTOXIK) A1K1=XKIK1 BTK)=-(XK(K)+XK(KK)+XDP) RKIK) 1.J-PEDT2/PKK2)‘T(11tT121-2.C*H{DK!(TArT(111/PKKZ IS AT L 73 74 76 77 a C 91 C(K)=XK1KK1 CT“1=~r~ Ts NODE 1 <+—08-€> 1 TEMP (°F) 320 280 2‘10 200 160 120 . 80 .._..._.. ...-.--. .-———. -.- CASE NO. 2: 108 p, = 60 1.13/1-‘7a .\ .\‘\‘. \‘\.. \‘\‘ x at 21 20 19 18 17 16 11001: 1(—- 011 ——-> 1 109 320 280 CASE 110. 3: p,= 65.0 LB/FT" 2110 200 \ I60 11\ TEMP I (°F) 80 21 20 I9 18 I7 I6 NODE I(— DR -—) I TEMP (°F) 320 280 2110 200 I60 I20 80 110 CASE NO. £1: 92 = ‘10 LB/FT" \\ 1 ‘ ‘ x. s" \\ \ ' \ \ \ x ‘ ' \\ \ \\ \7 ‘\‘ \\ ‘ ‘.‘.,‘ N '\.~§;\\;\-;;~—?-. T.%—— N \ 21 20 19 13 '7 '6 11001: 1<—-011'—>1 TEMP (°F) 320 280 290 200 I60 I20 . 20 CASE NO. 5: MODE I9 111 .-—.—--—v. -...—._--_. --- 18 17 1(——011——>1 TEMP (°F) 320 280 2L10 200 160 120 80 112 CASE NO. 35: Ta = 0.1 FT .1 x\ \ .\\ N» \\ \\\ \. 21 20 19 18 17 11001: 1(— DR ——)1 '16 TEMP (°F) 320 280 290 200 I60 120 80 113 CASE 110. 6: rs = 0.08 FT 1 -. \ ___........_._ ~ \.:—~..:_:‘ :77. . m... .... 21 20 19 18 17 MODE Ié——DR—)I 114 320 28° CASE NO. 7: rs = 0.05 FT 2110 200 N.“ 160 TEMP (°F) 120 80 \\\ ‘\.‘ ~‘ ::‘~- Wu" ‘-—\__‘___ - H U a 21 20 m 19 18 17 _" ”16“" NODE I (-DR —) I TEMP (°F) 320 280 240 200 I60 120 80 _"’ 20 CASE NO. 8: I9 NODE 115 1' S = 0.03 FT 18 ‘ 17 19011—911 116 320 280 CASE NO. 9: rs = 0.02 FT 240 200 160 \ a. \ .\\ TEMP (°F) 21 20 19 18 I7 MODE 1(3—-08-—%>1 TEMP (°F) 320 280 290 200 . I60 120 80 117 CASE NO. 10: rS = .01 FT 21 20 19 18 17 16 MODE 1é—0R—-§1 TEMP (°F) 320 280 2110 200 160 120 . 80 .fl "’ CASE N0. 11: k: 118 = 0.1 BTU/HR FT F 21 20 MODE I9 18 I7 neon—>1 16’ 119 320 280 CASE NO. 12: k1 =1 0.3 BTU/HR FT F 2110 200 160 TEMP 1 (°F) / 120 c o.‘ "." , ‘-- .. _‘ , -' ‘~ 2 —_~-“_ MODE 1 1 TEMP (°F) 320 280 290 200 160 120 ‘ CASE NO. 13: 120 k, = 0.6 BTU/HR FT F 21 20 MODE 19 18 17 11 TEMP (°F) 320 280 2110 200 I60 120 80 CASE NO. IA: 121 = 1.0 BTU/HR FT F 21 20"""-'_m”197"WWW—18'“' 17 "16‘ 11001: 1<—011 ——> 1 122 320 280 I CASE NO- '5: ka = 0.1 BTU/HR FT F 240 200 160 I\\ 120 TEMP (°F) 80 2' 20 I9 18 17 15 NODE 1<——08—91 Iilil-lllvlll TEMP (°F) 320 280 290 200 160 120 80 21 CASE NO. NODE 16: I9 123 kg = 0.2 BTU/HR FT F 18 17 16—08—91 TEMP (°F) 320 280 290 200 I60 120 80 CASE NO. 17: 124 kg = 0.6 BTU/HR FT F 21 20 MODE 19 I8 17 I 6-- DR«-—>I TEMP (°F) 320 280 290 200 \ I60 120 CASE MO. 18: 80 k, = 125 1.0 BTU/HR FT F 21 20 MODE 19 18 17 l€—'DR -91 126 320 CASE NO. 19: cpl = 0.87 BTU/LB F 280 2‘10 200 160 TEMP \ (°F) 120 . 80’ -.- . ....-- ;__-.-—. .....2 20 19 18 17 1’6 NODE 1 (-——DR -—9 1 127 320 280 CASE no. 20, c ‘01 ° 1.00 BTU/LB F 280 .- o. 'd'—— 200 160 \\ .... ‘ \ (°F) 120 . \\ \ 80 “. 21 20 '9 18 17 MODE |<%——0R 1 TEMP (°F) 128 320 280 CASE NO. 21: sz = 0.91 BTU/LB F 21.. I 1 1 1 200 1 '1 160 120 . \ ‘\ :5 ‘\1 say. 80 ...- . >‘frrra\\\ \— rm... 0 21 20 I9 18 I7 MODE l<$—-DR ——€>I .161 129 320 230 CASE M0. 22: sz = 1.00 BTU/L8 F 2110 200 160 TEMP (°F) 120 . \ \‘\‘ 80 \\\\’\. “‘9‘ ~ ‘2 2 ‘2 777777 L I“ " v v > . ..- .1..._...... _ LM‘ ”W‘TTM“ W”: .....b .. 0 E , -.- -, _-.._ 21 20 19 18 17 16 MODE lé—DR-él TEMP (°F) 320 280 2110 200 I60 120 80 130 CASE NO. 23: hH. = 2.0 BTU/HR FT2 F 21 20 I9 18 I7 MODE I (—-*DR ~—>I I6 - #0." TEMP (°F) 320 280 290 200 I60 120 131 CASE NO. 21.: hH = 11.0 BTU/HR FT” F 20 MODE 19 18 I7 FG-rDR'—4> I 16 TEMP (° F) 320 280 2‘10 200 160 120 80 132 CASE 110. 28: 11C = 1.0 BTU/HR FT‘a F "I 20 19 18 17 I 16 NODE 1<-—011-—-) 1 133 320 280 CASE NO. 29: hC = 5.0 BTU/HR F73 F 2‘10 200 1 160 '20 ‘ \ 80 TEMP (°F) -.._.. *”_—-_ .— ‘_—___———.——._.c -- --..» 21m 20 19 18 17 ’16 1100: 14—011 ——>1 134 320 2‘10 200 I60 TEMP (°F) 120 1, 21 20 19 18 17 MODE I<--DR --)I 135 320 280 CASE NO. 33: tf1 = 1000.0 F 2110 200 160 15131 .201 \\\\\\\ \ 21 20 19 18 17 MODE lé—DR -—7 1 TEMP 1°11 320 280 280 ZOO 160 120 80 CASE NO. 39: 136 t.. f1 - 2000.0 F 21 20 19 MODE 18 ‘ 17 19011—91 16 137 LIST OF SELECTED REFERENCES Bakker-Arkema, F.W. and Bickert, W.G. (1967). Computer Simulation of the Cooling of a Deep Bed of Cherry Pits. 1. Analysis, Quarterly Bulletin, Michigan Agricultural Experiment Station, Michigan State University, East Lansing, Michigan, Vol. 50, No. 2, pp. 204-211. Bakker-Arkema, F.w.; Bickert, W.G.; and Patterson, R.J. (1967). Simultaneous Heat and Mass Transfer During the Cooling of a Deep Bed of Biological Products Under Varying Inlet Air Conditions. Journal of Agricultural Engineering Research, Vol. 12, No. 4, pp. 297-307. Beck, J.V. (1962). Calculation Surface Heat Flux from an Internal Temperature History. ASME Paper No. 62-HT-46, presented at the ASME-AICHE Heat Transfer Conference, Houston, Texas. Beck, J.V. (1963). Calculation of Thermal Diffusivity from Temp- erature Measurements. Journal of Heat Transfer, Transactions of the ASME, May, 1963, pp. 181-182. Beck, J.V. (1964). The Optimum Analytical Design of Transient Experiments for Simultaneous Determinations of Thermal Con- ductivity and Specific Heat. Ph.D.Thesis, Mechanical Engin- eering Department, Michigan State University, East Lansing, Michigan. Bennett, A.H.; Chace, w.G., Jr.; and R.E. Cubbedge. (1962) Estimating Thermal Conductivity of Fruit and Vegetable Components -- The Fitch Method. ASHRAE Journal, Vol. 4, No. 9, pp. 80-85. Carslaw, H.S. and Jaeger, J.C. (1965). Conduction of Heat in Solids. Oxford University Press, London, Reprinted by D. R. Hillman and Sons, Ltd. Crank, J. and Nicholson, P. (1947). A Practical Method for Numerical Evaluation of Solutions of Partial Differential Equations of the Heat Conduction Type. Proceedings The Cambridge Philo- sophical Society, Vol. 43, pp. 50-67. Dusinberre, G.M. (1961). 999$.IIEPEESIL931C9192199§-PY-2191E9 Differences: Internétional Textbook Co., Scranton, Pa., pp. 64-71. Eucken, A. (1940). Allgemeine Gesetzmassigkeiten fur das Warme- leitvermogen verschiedener Stoffarten und Aggregatzustande, Forschungsgebiete Ingenieur, Ausgabe A., Vol. 11, No. 1, pp. 6-20. 138 Finch, D.I. (1962). General Principles of Thermoelectric Thermometry. Research and Development Department, Leeds and Northrup Co., published by Reinhold Publishing Corporation, pp. 34-43. Fitch, A.L. (1935). A New Thermal Conductivity Apparatus. American Physics Teacher, Vol. 3, p. 135. Freed, N.H. and Rallis, C.J. (1961). Truncation Error Estimates for Numerical and Analog Solutions of the Heat Conduction Equation. Journal of Heat Transfer, Vol. 83, pp. 383-384. Gurney, H.P. and Lurie, J. (1923). Charts for Estimating Temperature Distributions in Heating or Cooling Solid Shapes. Industrial and Engineering Chemistry, Vol. 15, pp. 1170-1172. Hartree, D.R. (1958). Numerical Analysis Oxford at the Clarendon Press. Kardas, A. (1964). Errors in a Finite-Difference Solution of the Heat Flow Equation. Journal of Heat Transfer, Vol. 86, pp. 561- 562. Kethley, T.W.; Cown, W.B.; and Bellinger, F. (1950). An Estimate of the Thermal Conductivities of Fruits and Vegatables. Refrig- erating Engineering, Vol. 58, pp. 49-50. Kopelman, I.J. (1966). Transient Heat Transfer and Thermal Properties in Food Systems. Ph. D. Thesis, Food Science Department, Michigan State University, East Lansing, Michigan. Kreith, F. (1958). Principles of Heat Transfer. International Text- Book Co., Scranton, Pa. Lalor, W.F., and Buchele, W.F. (1966). The Development and Preliminary Testing of an Air-curtain Flame Weeder. Report to Iowa LP-Gas Association, Des Moines, Iowa. Lentz, C.P. (1961). Thermal Conductivity of Meats, Fats, Gelatin Gels and Ice. Food Technology, Vol. 15, pp. 243-247. Lentz, C.P. (1964). The Role of Engineers in Food Research. Food Technology, Vol. 18, No. 14, p. 104. Luikov, A.V. (1964). Heat and Mass Transfer in Capillary-Porous Bodies. Advances in Heat Transfer, Academic Press, Vol. 1, pp. 123-184. 139 Matthews, F.V., Jr. (1966). Thermal Diffusivity by Finite Differences and Correlation with Physical Properties of Heat Treated Potatoes. Ph. D. Thesis, Agricultural Engineering Department, Michigan State University, East Lansing, Michigan. Maxwell, J.C. (1904). A Treatise of Electricity and Magnetism. Dover Publications, Inc., New York, New York, Vols. 1 and 2. (1954) Mironov, V.F. (1962). Inzh. Fiz. Zh., Vol. 5, p. 10. Moore, R.P. (1960). Tetrazolium Testing Techniques. Proceedings of 38th Annual Meeting, Society of Commercial Seed Technologists, pp. 45-51. Parker, R.E. and Stout, B.A. (1967). Thermal Properties of Tart Cherries. Transactions of the ASAE, Vol. 10, No. 4, pp. 489-491, 496. Patin, T.R. (1967). Internal Temperature in Cotton Stalks Induced by Flame Cultivation and the Resulting Cell Damage. M.S. Thesis, Agricultural Engineering Department, Louisiana State University, Baton Rouge, Louisiana. Perumpral, J. (1965). Determination of Temperative Patterns of Flame Cultivator Burners. M.S. Thesis, Agricultural Engineering Department, Purdue University, Lafayette, Indiana. Reidy, G.A. (1968) Thermal Properties of Foods and Methods of Their Determination. M.S. Thesis, Food Science Department, Michigan State University, East Lansing, Michigan. Richtmeyer, R.D. (1957). Difference Methods for Initial—Value Problems. Interscience Tracts in Pure and Applied Mathematics, No. 4. Riedel, L. (1951). The Refrigerating Effect Required to Freeze Fruits and Vegetables. Refrigerating Engineering, Vol. 59, pp. 670- 673. Schneider, P.J. (1957). Conduction Heat Transfer. Addison-Wesley, Inc., Reading, Massachusetts. Smilie, J.L.; Thomas, C.H.; and Standifer, L.C., Jr. (1964). Farm With Flame. Louisiana Agricultural Extension Publication 1364, Louisiana State University, Baton Rouge, Louisiana. Smith, G.D. (1965). NUmerical Solution of Partial Differential Equations. Oxford University Press, London. 140 Watson, H. (1961). A Study of Flame Patterns Obtained from a Flame Cultivator Burner as Affected by Fuel Pressure. Special Problem Report, Agricultural Engineering Department, Louisiana State University, Baton Rouge, La. Weber, W.J., Jr. and Rumer, R. R., Jr. (1965). Intraparticle Trans- port of Sulfonated Alkylbenzenes in a Porous Solid: Diffusion with Nonlinear Adsorption. Water Resources Research, Vol. 1, No. 3, pp. 361-373. Woodams, E.E. and Nowrey, J.E. (1968). Literature Values of Thermal Conductivities of Foods. Food Technology, Vol. 22, pp. 494- 502. , National Bureau of Standards, Thermal Conductivity of Sample of Silastic Silicone Rubber. Submitted by the U.S.D.A., Test Report, Test Folder G28359. 1302 11111111111 .11 I 93 03 11111111111111