THERMAL PROPERTIES OF FOODS AND METHODS OF THEIR DETERMINATION Thesis for the Degree of M .. S. MICHIGAN STATE UNIVERSITY GERARD A.REIDY ‘ 1968 .Or- O t 1 3 ""3 .1'3? ARY B‘IL‘IHO“ n 5mm L- '- ‘rsi‘cy I ' “In- ' an...“ I E BINDING n IIDAS a SDIIS' NQEJEEEELJIE IIIIIIIIIIII IIII L, PLACE IN RETURN BOX to remove othIo chockwtfrom you: “record TO AVOID FINES Mano on or baton due due. DATE DUE DATE DUE DATE DUE _ 431242 ELEV/U i m (S ”3312612 I pc ; , ms 2A9 . <2 mm seezaarfla m 3 I) W 4051mm “ MSU Is An Affirmative Action/Equal Opportunity lgsthalonm LW/f» ABSTRACT THERMAL PROPERTIES OF FOODS AND METHODS OF THEIR DETERMINATION by Gerard A. Reidy Many investigators have reported values for thermal properties (a.k,c) of various foods. A number of different methods were employed in making these determinations. No- where in the literature have these values been fully com- piled. nor has a thorough analysis been made of the experi- mental methods used. This study describes the techniques most widely used in evaluating the thermal conductivity and thermal diffus- ivity of foodstuffs. values of thermal properties of all classes of foods have been compiled from the literature and are tabulated. Results obtained from various steady and un- steady state methods are discussed and sources of error associated with the experiments critically analyzed. The study has made clear that the majority of investi- gators fail to report sufficient information with their data to enable other researchers to apply the data with confidence. Information is suggested which should be re- ported by all investigators to allow a complete evaluation Gerard A. Reidy of their work. It is recommended that the methods most suitable for a particular food are x the parallel plate steady state method for liquids and dehydrated foodsgand transient methods. preferably the ”Probe“ or numerical methods, for solid foods of significant moisture content. THERMAL PROPERTIES OF FOODS AND METHODS OF THEIR DETERMINATION BY 9 “I? II ‘ Gerard ANTReidy A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Food Science 1968 ACKNOWLEDGMENTS The author extends his sincere appreciation to Professor A.L. Bippen and Dr. R.C. Nicholas for their constant guidance and advice throughout this study and to Dr. F.N. Bakker-Arkema, Agricultural Engineering Department, for his many helpful suggestions and for serving on his committee. The author also wishes to thank Dr. B.S. Schweigert, Chairman of the Department of Food Science for his interest in the program and for making the assistantship available. To Dr. T.I. Hedrick, appreciation is extended for the consideration shown to the author and his wife both prior to and after their arrival at Michigan State University. The Institute for Industrial Research and Standards (Ireland) is thanked for financial assistance which en- couraged the author to pursue his studies. Finally, the author wishes to thank his wife. Ann- Marie. for her constant encouragement and patience shown throughout his studies and her invaluable help in typing this manuscript. 11 TABLE OF CONTENTS Page IACKNONLEDGMENTS ii LIST OF TABLES vi LIST OF FIGURES vii LIST OF SYMBOLS ix Chapter I. INTRODUCTION ' 1 II. REVIEW OF METHODS FOR DETERMINING THERMAL CONDUCTIVITY AND THERMAL DIFFUSIVITY 3 A; Theory of Conduction Heat Transfer 3 B. Steady State Methods 9 (1) Parallel Plate 9 (ii) Concentric Cylinder 25 (iii) Concentric Sphere 30 0. Transient Methods 32 (1) Heating and Cooling Curves 33 (11) ”Probe” Method #1 (111) “Fitch“ Method an (iv) Zierfuss Method 46 (v) Numerical Methods ‘ 48 111 Chapter III. DISCUSSION OF RESULTS FOR.k AND R AND THE VARIOUS METHODS USED A. B. Steady State Methods 1. 2. 3. Parallel Plate Method Concentric Cylinder Method Concentric Sphere Method Transient Methods ’4. 5. 6. 7. a. 9. Heating and Cooling Curves "Probe” Method "Fitch” Method During Freeze-Drying Graphical Method Numerical Method Note on Accuracy of Temperature measure- ments IV. CONCLUSION AND RECOMMENDATIONS REFERENCES IAPFENDIX A. B. C. D. E. Table A; Thermal properties of foods Theory of "probe" method for determining thermal conductivity Finite difference methods for transient heat conduction Finite difference method used by Matthews (1966) .- Non-linear estimation procedure, Beck (1966) iv 52 52 53 66 71 73 73 92 98 107 113 ,115 118 120 126 A1 A60 A62 A65 A66 F. Theory of heating in a finite cylinder as applied by Babbitt (1945) Standard errors for thermal properties of cherry flesh. reported by Parker and Stout (1966) Solutions for various heat conduction conditions in solids (Olson and Schultz; 19n2) Page A69 A70 A71 Table 1. 7. 8. 9. LIST OF TABLES Thermal Conductivities of Some Gases, quuids and Solids Factors which Introduce Errors when Determining Thermal Conductivity by Steady State Methods Factors which Affect the Accuracy of Determin- ation of Thermal Diffusivity from Heating and Cooling Curves Factors Causing Inaccuracies in ”Probe" Method Factors Contributing to Inaccurate Determin- ation of k by ”Fitch" Methods Summary of Advantages for Steady State Methods. Summary of Disadvantages for Steady State Methods Summary of Advantages for Transient Methods Summary of Disadvantages for Transient Methods vi Page 75 93 99 122 123 124 125 Figure 1. 2. 3. h-a. n-b. 5'3- 5-b- 7. 8. 9. 10. 11. LIST OF FIGURES Typical Temperature vs. Time Curve for a Pbint in a Body Subjected to Heating, with Convection Cooling Boundary Condition . ”Single” Sample Parallel PLate Method “Twin” Sample Parallel Plate Method Parallel Pdate Method with Guard Heater at Side of Main Heater only Approximate Representation of Temperature Profile Parallel Plate Method with Guard Heaters at Side of and Overhead Main Heaters .Approximate Representation of Temperature Profile Heat Flow through Composite Sample Arrangement Test Cell - Concentric Cylinder (Woolf and Slbbltt. 195A) Thzrmal Diffusivity Apparatus (Dickerson, 19 5 Detail of Thermal Conductivity Probe Thermal Conductivity Apparatus used by Saravacos and Pilsworth (1965) Cross Section of Heat Conductivity Unit used for Determination of Thermal Conduct- ivities of Powders (tha e; 31.. 1966) layout of Thermal Conductivity Measuring Apparatus (OJha gt,a;.. 1966) vii 15 29 NO #2 63 69 7O Figure 13. 1h. 15. 16. 17. 18. 19. 20. Radial Location of Thermocouples in Annulus Vertical Section of Spherical Steady State Apparatus used by Oxley (1933) Apparatus for Heating Canned Soybean Oil Meal at Constant Surface Temperature (Housen. 1957) Concentric Tube Apparatus for Determination of Thermal Conductivity (Hougen. 1957) Graphical Presentation of the Mathematical Idne Heat Source Relationship (Hooper and Chang, 1953) Cell No.1 (Helvey, 1954) Cell No.2 (Helvey. 195A) Schematic of Apparatus for the Measurement of Thermal Conductivities of Peor Conduct- ors; (A) Fitch Thin Disk; (B) Modified Cenco-Fitch. as used by Bennett. Chance and Cubbedge (1962) viii Page 70 7h 84 85 97 100 101 105 11 o‘tfltfltdm O 0 ll LIST OF SYMBOLS cross-section area. ft2 constant = B/WPIaJl(/&9)’ in equation (23) constant. equation (F2). Appendix constant a 6T/6t. equation (27a) radius of cylinder of sphere. ft length of finite cylinder. thickness of infinite slab. width of rectangular prism. or length of rectangular parallelepiped (brick) in Olson and Schultz (l9N2) solutions only (Appendix) galvanometer constant. equation (31) constant. equation (10). hr. constant = 3H/3H+K constant. equation (F2). Appendix. 0F. half height of finite cylinder. ft specific heat at constant pressure. Btu/1b - oF. ch. But/1b3 - OF. weight x heat capacity of fluid layer weight x heat capacity of internal metal core constant. equation (F2). (F3). Appendix. oF/hr diameter of external sphere. ft diameter of internal sphere. ft natural logarithm ix Jon J10 dimensionless relationship used by Woodside (1957). to describe the edge temperature of a sample between hot and cold plates. surface area of calorimeter core. ft2 Fourier number = “t/L: shape factor between two surfaces. equation (12) the reciprocal slope of the asymptote of the heating curve a time to reduce Tm - T by 90%.hrs. guard ring width. ft 1 dimensionless number = (IIKIK?)CO 3K01 heat of sublimation of ice. Btu/lb convection heat transfer coefficient. Btu/hr-ftz-OF electrical current. amps Bessel function of the first kind. zero order Bessel function of the first kind. first order D1/D parameter depending on geometry of body. equation 22 thermal conductivity. Btu/hr-ft-OF. thermal conductivity of a particular substance at a particular reference temperature. Btu/hr-ft-OF. thickness of a slab or rectangular sample. ft characteristic dimension of a body e.g. radius of a sphere or cylinder; half thickness of plate thickness of the dried layer in a freeze-drying food sample. ft _ l (1 K)Eg 2nKF1 dT/ constant rate of temperature change. dt hb/k. Biot number ha/k, Biot number. initial moisture content. dry basis final moisture content. dry basis galvanometer deflection at time t galvanometer deflection at t = 0 chBB'AT/Dz 2 , Grashof number I“. #15. . Prandtl number th/k. Biot number eccentricity between axes of internal and outer cylinder; ft heat flow rate. Btu/hr heat flow rate per ftz. Btu/hr-ft2 heat loss due to a temperature unbalance of 1°F between hot plate and guard plate. Btu.hr component of qt occurring across the gap between the hot and guard plates thermal ”contact” resistance at cold plate. Btu/hr-ftZ-OF thermal ”contact” resistance at hot plate. Btu/hr-ftz-OF radial distance from central axis. ft inside radius of hollow sphere or cylinder. ft outside radius of hollow sphere or cylinder. ft xi #3 P3 5 H ' 5 :1 <2 d' d' ta la 0 o .1 of rate of sublimation of ice. lb/hr half width of specimen sample. ft function tabulated by Olson and Schultz (1942). 2 2 2 Z soluble solids temperature at hot surface. °F temperature at cold surface. °F temperature at outside radius. r = absolute temperatures. oR. equation (12) temperature of hot plate. oF temperature of cold plate. °F can temperature. OF initial temperature. °F ----) a temperature at center of sample. °F temperature at radial distance r. 0F temperature of heating medium. °F surface temperature of sample. oF interface temperature. °F time. hr correction time factor. hr overall heat transfer coefficient. Btu/hr-ftz—OF a function of location. equation (20) internal heat generation. Btu/ft3-hr Tm - T / Tm - To xii cumulative moisture weight loss. lbs. flesh.weight of each cherry in grams. Appendix. (G) work energy. Btu. incremental thickness of sample. ft thermal diffusivity. ftz/hr constant temperature coefficient of thermal conductivity for a particular substance. l/°F. dummy integral parameter. equation (54) factor related to Al and ml. equations (45). (49) B J (B ) roots of NB1 = 1 1 1 / Jo(Bi) factor related to o and mi. equations (45). (49) coefficient of volume expansion thickness of heater plate. ft diameter of heating wire. ft parameter used by Charm (1963). defined by equations (45). (47) viscosity. lb/ft-hr parameter used by Charm (1963). defined by equations (48). (49) positive root of JOSKIr) = 0 density. lb/ft3 xiii I. INTRODUCTION The thermal properties of foods are regarded as thermal diffusivity. thermal conductivity and specific heat. though thermal diffusivity is not an independent property (a = kéoc). Knowledge of thermal properties of food substances is essential to researchers and designers in the field of food science for a variety of purposes e.g. predicting the drying rate or temperature distribution within foods of various compositions and geometric shapes when subjected to different drying. heating and cooling conditions; or to allowoptimum design of heat transfer equipment. dehydrating and sterilizing apparatus. Such information is not always conveniently available. In many instances where thermal prOperties have been re- ported. values disagree significantly. This apparent dis- agreement could be due not only to instrumental errors but also to the methods used. many of which are not suited to particular foods. Reports often fail to report pertinent data. thus limiting the usefulness of their results since it is known that thermal properties of foods vary with temperature and pressure conditions. Complete information on the composi- tion of the sample is often lacking (e.g. moisture content. fat content. protein content etc.). Some researchers do not indicate the direction in which heat is applied during a test (e.g. whether parallel or perpendicular to muscle fibers in a meat sample). Thermal properties are dependent on these factors and unless all are recorded. future workers will not be able to use the reported values with confidence. A survey of the present position of the field is made in this study. the objectives of which are 1 (a) to discuss and critically review the methods used by previous investigators to evaluate thermal conductivity and/or thermal diffusivity of foodstuffs; (b) to offer recommendations for future thermal property determinations of foods. (c) to gather from the literature available values for thermal properties (thermal conductivity. thermal diffusivity. specific heat) of foodstuffs. II. REVIEW OF METHODS FOR DETERMINING THERMAL CONDUCTIVITY.AND THERMAL DIFFUSIVITY A. Theory of Conduction Heat Transfer Thermal conductivity is defined according to Fourier's law for heterogeneous isotropic continua which states that the heat flux qx. in the x direction is equal to the temper- ature gradient across an incremental thickneSSIAx. times the thermal conductivity. k: AT 9.21 qx = '33}. = 'kbx (1) Ax-—> O The type of continuum in which k is measured is of ut- most importance. The continuum can be either "homogeneous" or non-homogeneous” (”heterogeneous") and "isotropic" or "anisotropic". A "homogeneous" continuum is regarded as such if the thermal conductivity is the same at all points; while a continuum is said to be "isotropic“ if the thermal conductivity is the same in all directions. The terms "non- homogeneous" and "anistropic" are regarded as the opposite of the above respective definitions. For the solution of many heat conduction problems continua are considered to be homogeneous and isotropic. For such a case equation (1) can be written in the form: -k (T - T ) The vector form of the heat conduction equation is q = "k OVT (2) which. in a non-homogeneous anistropic continuum reduces to q = qx + q y + qz (3) where in cartesian co-ordinates. 6T 5T 6T qJr = -(k11'5'; + 1:123; + 1:133?) ‘OT CT CT qy = ‘(kzl'é'i' "' 1‘22??? + 1‘23???) (3a) 1 or ‘5T 6T q2 = -(k31-5-§ + k32'5y- -|- k33‘5?) (Carslaw and Jaeger. 1959) Normally the k determined is reported as if the system was isotropic or orthorhombic. Many investigators fail to mention this in their reports and in cases where a simple value for ”k” is reported for a material the result obtained may be worthless unless the qualification is made that the reported value applies to a particular axis or to an isotropic body. Due to crystalline symmetry. certain simplifications can be made in some instances if the axes are chosen in the proper crystallographic directions. Two particular systems will be mentioned: (a) the orthorhombic system. in which there are either two perpendicular diad axes or a diad axis with a plane of symmetry through it. In this system. only three k's result along mutually perpendicular axes. Equation (3a) then reduces to II I PT I qx xbx ' (3b) .0 II I W I qz I 23; 3 (b) the cubic system in which it is possible to inter- change the axes of the above system i.e. an isotropic body. As expected for this system. k: = ky = k2 = k. The value for k in any body usually depends on a number of factors. e.g. the physical structure and state. the temperature and pressure. and the chemical composition. All of these factors must be considered in the case of foodstuffs. Of the constituents in a food. moisture con- tent is probably the most important. Results indicate that thermal properties of most foods increase in value with in- creasing moisture content,ultimately approaching the value for water. Due to the large variation in values for thermal conductivity between gases. liquids and solids (see Table l) the thermal conductivity of food would be expected to vary over a wide range. Table 1. Thermal conductivity of some gases. liquids and solids. Gases k T°F Liquids ' k T°F Solids k T°F Air .014 32 Ammonia 0.29 50 Iron 39.0 64 ‘ Nitrogen .014 32 Benzene .096 50 Ice - 1.3 32 Sulfur dioxide .005 32 Water .331 50 Celluloid 0.12 86 Different samples of the same foodstuff may have varying thermal properties depending not only on the particular moisture content; but also on physical structure. e.g. com- pare freeze-dried and air dried samples. In solids where thermal conductivity is temperature dependent. an assumption of linearity is usually made i.e. k = kb (1 + BAT). where k0 is the thermal conductivity at a certain reference temperature. and B is a constant for the particular substance. (ST represents the difference between the actual and reference temperature. The equation governing unsteady state temperature dis- tribution within a body subject to internal heat generation u per unit volume is flog-€- =3 V. (koVT) +11 _ ((4’) If a body is subjected to a constant heat flux for a period of time. a heating curve approximating that in Fig.1 will result. At stage ”B” the body is said to be in a ”steady state“ condition i.e. %%' = 0. I I I <51- I HEAT FLUX REMOVED l —, =0 I/ k I l (.5 A I s | c o; heatin I steady-state I coolin 3 9—11 m 9 § L A {I I I a I I “I: I l I I I I I I T I ME,I Fig. / TYPICAL TEMPERATURE Vs. TIME CURVE FOR A POINT IN A BODY SUBJECTED TO HEATING,WITH CONVECTION COOLING BOUNDARY CONDITION. In cases where there is no internal heat generation e.g. no heat of respiration. the term u also becomes zero. The solution of the steady state problem is thus relatively simple and is dependent on the boundary conditions. For un- steady state problems. where temperature may increase or de- crease with time. the solution will depend on the initial condition. For unsteady state when dealing with homogeneous isotropic bodies. by dividing equation (4) by k'we get on the left hand side the term 1°73- : 71‘. where a is called the “thermal diffusivity' of the body. It is d which is norm- ally determined in unsteady state tests. and k is then evaluated indirectly through use of the values for the properties/o and c which can be independently determined for identical samples of the substance under the same con- ditions of temperature and pressure. It will be shown later that it is also possible to evaluate both a and k simultaneously. Note also that equation (4) assumes that no work (e.g. through deformation of the solid) is being done. i.e..AW=0. The solution to unsteady state problems may be simple or complex. however. for many practical cases an approxi- mate solution is available. The accuracy of the values of k determined by experiments under steady state and unsteady state conditions will depend on the agreement between the. experimental boundary conditions and the boundary conditions corresponding to the problem for which the solution is available. A number of steady state and unsteady state (transient) methods are discussed below. B. Steady State Methods. A steady state system implies that there is no rise in temperature in the substance with time i.e. the term %%.= 0. The steady state solution of the heat conduction equation for homogeneous isotropic systems with no internal heat generation yields the following basic equations for the three most usual configurations: -»kA(T1 - T2) Infinite plate : Qcart = ___——_IT____' (5a) for the parallel plate method; Infinite hollow 2wk(T1 - T2) ( ) cylinder. per 1 Q = ——————- 5b unit length °Y1 1“ r2/r1 41Tk(Tl - T2) = ———:-5— r1 1‘2 (5°) Hollow sphere : Qsph {r2 1 It can be seen that by measuring the heat quantity Q. the inner and outer temperatures T1 and T2 and some linear dimensions. k can be calculated. (1) Parallel Plate. Schematic drawings of two typical parallel pLate systems are shown in Figs. 2 and 3. A,E :insulation B,C,D:guard heater F :main heater stample Hzcold plate B fi'éiegg I D H D I COOLING m LIQUID our Fig. 2 "SINGLE" SAMPLE PARALLEL PLATE METHOD. — . A: cold plate 2% “W” /Mb“ U C 1 J D % szain heater .1 ........................ e/ 3% down: It... / A Etinsulation Fig. 3 "TWIN" SAMPLE PARALLEL PLATE METHOD. 11 A method frequently used is the “guarded hot plate” method standardised by the American Society for Testing Materials (A.S.T.M.. 1955). It should be noted that this method is specifically for ”dry” samples. With the excep- tion of dehydrated foodstuffs no other food substances meet this requirement. The "guarded hot plate” method is fre- quently applied because of its apparent simplicity. How- ever. there are many difficulties associated with the method. Consider the case of a heater on a sample with in- sulation on the upper face only (see Fig.4) with the heater plate as a system of uniform temperature Th and thickness 6. with a steady internal heat generation u: Ou q1 + q2 1 U = -7———————————-————— 1 h + L1/k1 + L2/1:2 (Ih - I”) k A _ _______3_ Q2 - L3 q1 = U’A (Th - I”) q2 L3/1:3 = q1/U 3.1.31 and q2 = U L (a) (b) Too Fig. 4 (O)PARALLEL PLATE METHOD WITH GUARD HEATER AT SIDE OF MAIN HEATER ONLY. (b) APPROXIMATE REPRESENTATION OF TEMPERATURE PROFILE. T TH L3 I 3—}— TH L—I— , . L3: TC Tea (0) (b) Fig. 5 (O)PARALLEL PLATE METHOD WITH GUARD HEATERS AT SIDE OF AND OVERHEAD MAIN HEATER. (D)APPROXIMATE REPRESENTATION OF TEMPERATURE PROFILE. A: main heater ("hot plate") B: sample :cold plate :guard heater (side) : insulation :guard heater TINOO 13 The term q2 is the heat loss. For this to be negligible 3 ——> 0 or L3 ——> a. both of which are physically impossible. The temperature profile will (i.e. q2 -> 0). either k appear similar to that in Fig. 4. To completely eliminate q2 it is necessary to install the second guard heater F. over the main heater A (Fig.5) which. if properly regulated. will allow all of the heat to pass through the sample. Means must also be provided to prevent heat loss by conduction or convection from the side faces of both the heater and the sample. Guard heaters (D in Fig. 5) are satisfactory for prevention of heat loss from the main heater. However. they introduce problems when used to pre- vent heat loss from the samples (see next section). The perfect guard heater for this case would be one providing a thermal gradient equal to that between the hot plate and the cold plate. Some investigators try to approach this by placing a number of guard heaters along the side surface in such a way as to provide incremental temperature decreases approaching the continuous gradient. Another method is to provide a heater with a temperature equal to the average of the hot and cold plate temperatures. Yet another method is to extend the whole sample laterally as shown in Fig. 3 and provide insulation at the outside edge. Since the area of interest is only that directly below the test heater (called the "test area“) this latter method minimizes the error due I‘ll! I'I III. All-ll l‘ IIDI '1'.Ilul' .IIVIII III!!! I, 11 ll 14 to non-linear heat flow. An analysis of the errors result- ing from some of these practices is included in the follow- ing section. Factors Which cause Inaccurate Measurements a. Measurement of Heat Input. The accurate measure- ment of the heat flow through the sample probably consti- tutes the greatest difficulty with this method and undoubt- edly gives rise to the largest errors in the determined k values. Since thermal conductivity is directly proport-I ional to the quantity of heat in the definition equation. any percentage error in the estimation of heat input will automatically mean a similar error in the calculated ther- mal conductivity. Normally the electrical input is measured and taken as the total heat throughput. However. if any of this heat escapes to the surroundings. the calculated value wiliebe too high. One way to avoid this measurement of heat input (actual) would be to use a two layered system where one of the substances would be a material of known thermal con- ductivity. Referring to Fig. 6 in the usual notation where Q is the actual heat input. k 1 k2 1 1 Thus. the necessity to measure Q is eliminated. The measurements ofo1 andex2 can also be eliminated. by using ///////////////////////////////I OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO AAAAAAAAAAAA 16 a common material in both layers i.e. k1 e k . Thus Axl _ T1 - T2 (7) 4x2 ' II.l _ T1 1 2 . The thermal conductivity measurement can now be carried out simply by having a material of known thermal conductivity and by measuring the temperatures of the upper and lower 3 faces and the inter-faces. b. Heat Loss Due to Nan-linear Heat Flow. Heat losses from the edge of the sample and the hot plate are generally rectified either by guard plates or by insulation. both methods giving varying degrees of satisfaction. Woodside (1957) made an analysis of the errors in a guarded hot plate apparatus resulting from the non-linear temperature distribution in the sample between the hot and cold plates. The assumption that a linear distribution exists is rarely true due to either (a) edge heat losses. (b) temperature difference between the guard plates (Figs.2 and 5) and the hot plate. (0) the non-homogeneity of the foodstuff or (d) uneven moisture distribution within the sample due to diffusion (see discusSion in Chapter III.) Woodside (1957) made a theoretical analysis on the basis of two assumptionssv 1. that the hot and cold plate (Fig. 5) surfaces are both isothermal. (this assumes no air gap between the guard plate and the hot or cold plate. and also 17 that perfect "balance'I exists. i.e. no temperature differential between the guard plate and hot or cold plate ) and 2. the edge of the sample is maintained at a uniform temperature higher than that of the cold plate. but lower than that of the hot plate. He used a factor ”so” to describe the edge temperature in dimensionless form. If the hot plate temperature is T1 and the cold plate temperature is zero the temperature of the edge is eo T1 . where e varies between 0 and 1. He 0 presented his result as follows: k ( 1/kexp ) a "81; (8) Cosh RSgis) 1 Cash n(gis) 1 L + L " coin (5335-?%§3--:r;) + (l-eo)£n (COEh (g?) - 1 where g = the guard ring width; 3 = the half width of the specimen. Equation (8) shows that the error in thermal conduct- ivity measurement depends on the ratios of specimen width to specimen thickness and guard ring width to specimen thickness: and also on so. which will be influenced by the temperatures of the hot and cold plates. Thus the error may be calculated once the various dimensions and the edge tem- perature are known. When eo = 0.5 (the edge temperature 18 being the average of hot and cold plate temperature) the result is k i as ( /kexp) = SinhLTTIEE I (9) L In ( Sinh %§ ) Equation (9) predicts very small errors. A graphical re- presentation of the above results shows that. as expected. the experimental k approaches the true k with decreasing thickness and increasing guard ring width. Thus it is possible to arrive at optimum dimensions for a parallel plate apparatus having selected the required degree of accuracy. It may be difficult to arrive at a value for ”so” particularly if the parallel plate test is a vertical type where four different edge temperature distributions will exist. as compared with the horizontal type where the four temperature distributions are the same. Woodside and Wilson (1957) carried out an experiment which showed that for any hot plate apparatus it was poss- ible to measure two constants which allow the error in ex- perimental thermal conductivity arising from unbalance in temperature between guard and hot plates to be easily cal- culated. In cases where the temperature of the guard plate is lower than that of the test heater plate. a heat loss will occur resulting in too hish a value for kexp, the opposite effect will result in kexp values lower than the 19 true k. Let the heat loss qt due to the above temperature effect be divided into two components. qo. the heat flow across the gap and Bk the error heat flow through the sample. both with reference to a temperature unbalance of 10F. across the guard and heater plates. Both the terms of qo and B can easily be measured for any apparatus by run- ning two tests on two specimens of known thermal conduct- ivities. and both terms are constants for any apparatus. These constants are obtained from the relationship qt = q0 + Bk (10) which is plotted for the apparatus. B will be the slope of the graph and q0 the intercept on the vertical qt axis. qt can be calculated from the simple equation A AT Ak qt "-" L (11) where Ak 2 the error in thermal conductivity (deviation from real known k). A number of specimens of known thermal conductivity were tested. Graphs of qt vs. k for the specimens were plotted. and the experimental results confirmed the theory. Woodside and Wilson (1957) show in their investigation that the errors arising in thermal conductivity measurements with the parallel plate method are proportional to the mag- nitude and direction of the temperature unbalance between the guard area and the test area. the size and design of 20 ' the hot plate. the thermal conductivity and thickness of the test specimens and the temperature differential be- tween the hot and cold plates. They reported errors as high as 6% for tests which met the ASTM requirements for temperature balance. The largest errors occured. as espected. with specimens of large thickness and low thermal, conductivities. The authors recommended that a ratio speci- fying maximum gap width to test area dimensions is more satisfactory than merely specifying gap width (gap width is the distance between the guard plate and the hot plate. < 1/8" in the.ASTM specifications). c. Heat Loss Due to Convection and Radiation. Other forms of heat ”loss” may occur in fluid foods i.e. heat loss from convection and radiation. When calculating the thermal conductivity. heat flow by conduction only is assumed. so therefore any heat transfer which takes place by other means must be regarded as a loss. It is reasonable to assume that in solid foods heat flow by convection and radiation can be neglected unless in the case of a very porous medium where internal convection may have to be considered. However. both should be considered for fluids. Tsederberg (1965) states that convection heat transfer can be neglected when the product of NGr and NPr numbers is below 1.000. Allowing for errors in the determination of NGr and NPr' Tsederberg states that NGr x NPr less than 700 or 800 guarantees negligible convection heat transfer. This 21 condition is usually satisfied by employing a horizontal parallel plate with a distance of the order of 1/8' between the hot and cold plates. Radiation heat losses can be estimated from the Stefan Boltzman law which states that the radiant heat flow q12 between two surfaces is given by .12 .1 oF12(TL’1a—Tga) (12) The factor of F12 depends on the emissivities and relative 8 areas of the surfaces. The factor a = 0.17 x 10' Btu/ 2 _ hr _ o 4. ft Woolf and Sibbit (1954) justified neglecting errors due to radiation when they used a steady state method to determine k of fluids. They carried out tests in a tarn-_ ished brass instrument and in a polished brass instrument. and both results were identical. Other investigators who demonstrated that errors due to radiation heat transfer could be neglected were Powell and Challoner (1956). Their investigation concerned the determination of thermal conductivity of seven liquids. using the parallel plate method. The initial and final emissivities of the hot and cold plates were measured in- dependently and the results indicated that their combined emissivity was of the order 0.1. Powell and Challoner. therefore. concluded that radiated heat amounted to more than 1% of the conducted heat only when testing carbon 22 tetrachloride. the liquid having the poorest thermal con- ductivity. d. fl§§in' or ”Contact” Thermal Resistance. If equation (5a) is to be true. perfect thermal contact must be maintained between the hot plate and the sample. and between the cold plate and the sample. If this condition is not met. a resistance to heat flow at these interfaces occurs. Let the contact resistances be Hop and th. Then 1 l 1 L -- = —- 4- ~— + --- (13) U1 ch th ktrue and Q :2 U1 A AT (14) For the cases of perfect thermal contact. . ‘k U = if“! (15) and Q a UAAT (16) Therefore. if perfect contact is falsely assumed (very often the case for food samples) a value for thermal con- ductivity‘will be determined which will be lower than the true value. since E'L— - E-L- : % (J'— + El“). exp true th 0P Two methods of overcoming the above problem are used. One is to lightly "wet” the surfaces of the sample which will be in contact with the hot and cold plates. However. 23 this practice may alter the moisture content of the test sample. If the moisture is not absorbed into the sample a liquid film may form which will have a higher thermal. conductivity than the sample. resulting in incorrect de- terminations of k. The alternative practice is to apply pressure to the sample to ensure perfect contact with the hot and cold plates. Foodstuffs will yield when subjected to pressure.l moisture may be expelled. and the structure of the sample may be altered. Emphasis has been given to inaccuracies in parallel plate methods. It is important that a designer know the degree of accuracy of those reported values which he has used. It seems to the writer that many investigators should replace the word ”accuracy" with ”precision". when evaluating their results. In many instances the analysis does not refer to absolute accuracy but rather to repro- ducibility of results. A description of a parallel plate method developed by Black et_g;, (1966) seems to justify the above statement. This apparatus allows the determination of thermal conduct- ivity of solid. foamed. fibrous or particulate materials in an evacuated or non-evacuated medium. while the test mat- erial can be subjected to a test pressure of 15 p.s.i. The test temperature can be varied from -320°F to 350°F. J'luill’ III! I. 24 Basically the apparatus consists of a sample chamber. containing the hot plate and heating coil. which can be evacuated. and the cold plate assembly. The cold plate assembly has a base (which contacts the sample) on top of which is a ”measuring” vessel which in turn is surrounded by a guard vessel. A gap separates the measuring vessel from the guard vessel. The gap can be evacuated. resulting in decreased heat transfer between the vessels. cryogen is placed in both vessels. When heat flows through the sample. some of the cryogen in the measuring vessel boils off and the measured rate of this boil off determines Q. Both the sample chamber (including the hot plate) and cold plate assembly are totally enclosed in a bell jar which in turn can be evacuated. The result is thermal isolation of the measuring vessel from direct contact with the ambient temperature. The reader is referred to the original paper for a complete description of the apparatus. Despite the precautions taken by Black g§_§;.. the authors indicate that an error up to 8% might be expected. This is interesting when one considers much less elaborate experimental procedures in which extremely low errors (often < 2%) are claimed. The error components are made up of each of the four factors in the equation Q.IA£ k’AAT An error of 1% is predicted in measuring Q. the boil off 25 rate of nitrogen: a maximum error of 4% is expected for the measurement of A; in measuringfiAx a variation of i .003“ is found. which for a 0.4 inch thick sample gives about 1% error: an error of 1% - 2% is given for temperature measure- ment. Furthermore. it should be noted that these errors do not comtemplate any distortion of the heat flow path which could arise from anistropic material or uneven contact be- tween sample and hot and cold plate. The accumulative effect of these errors would indicate a probable maximum error of 't 9 - 10% which as indicated above is worth noting in view of high accuracies claimed by other investigators. (ii) Concentric Cylinder When a hollow cylinder of inside radius r1 and outside radius r2 is subjected to a steady state heating condition the heat flux across the annulus can be measured by measur- ing the inside and outside temperatures T1 and T2 respect- ively. Zak (T1 - T2) Q = 73—7—- W ( 2/r1) where Q is the heat flux in a radial direction per unit length. Therefore. when a foodstuff is placed in an annular space between two concentric cylinders and is subjected to steady state heating (6.8. a hot oil bath or electric heater maintained at uniform temperature for the inside cylinder) 26 the thermal conductivity of the foodstuff can be calculated by measuring the temperatures and the heat input. An alternative procedure is to utilize materials of known thermal properties which can reduce the necessary measure- ments to that of temperature measurements only ( as des- cribed for the parallel plate method). The method known as the ”hot wire method" is included in this category and simply consists of using a wire (on the order 0.1 mm. dia- meter) as both the heater and as a resistance thermometer. The method is more applicable to fluid foods than solids. The value of the thermal conductivity measured can be applied at the average temperature and pressure test con- ditions. Factors affecting accuracy.-9The disadvantages associ- ated with the method are similar to those for the parallel plate method and can be summarized as : (a) heat losses: (b) convection and radiation heat transfer: (c) the maintenance of true concentricity of the cylinders throughout the experiment: and (d) contact resistance. (a)--It can readily be seen that the problem of heat losses will be less serious than with the parallel plate system. since the heat losses are confined to end losses. One way-to eliminate or minimize such losses is through the use of guard heaters at the ends which are maintained at 2? constant temperature. This arrangement will again intro- duce an error due to distortion of the heat flow lines since the maintenance of a constant temperature across the annulus acts contrary to the required temperature differ- ential being measured. This difficulty can be largely over- come either by insulating the foodstuff in the annulus from the effects of the guard heater (thus confining the heat from the latter to the central cylinder part only) or by employing a relatively long cylinder and confining the test area to a short central section. Heat losses through thermocouple leads will always occur. (b)--Convection heat losses are mainly confined to fluid foods and can be minimized by either employing a very small annular gap width or a low temperature differential. or carrying out the test with the apparatus in the hori- zontal rather than the vertical position. Basically this problem is similar to that of the parallel plate system. Errors due to convection and radiation losses for thin annulus techniques for liquids and gases have been reported and are approximately 1 - 2%. Grimm and Kreder (1966) give a heat loss of i 1.58% for the determination of the thermal conductivity of liquids. They also state that the losses due to convection increase as the product of the NGr and NPr numbers increases. (c)--With reference to lack of concentricity. the errors due to this are probably more serious with the ”hot- 28 wire” method than with the standard concentric cylinder method. Several measurements along the test length are essential to ensure uniformity of gap width and for fluids where the "hot-wire” method is usually employed the error will be more serious due to the small annular gap. Tsederberg (1965) recommended the use of the following equation when eccentricity exists: ./(r2-I-rl)2-n2 + J(r2-rl)2-n§ J(r2+r1)2-n2 - ../(r2--rl)2-n2 2w T1 - T2 1! = Q in (17) Centering mechanisms are available which maintain a con- stant gap between inner and outer cylinder. The possib- ility of further heat loss through such a mechanism by conduction should be recognized. even though it is small. The centering mechanism should be placed well beyond the measuring section. (d)--The comments made on contact thermal resistance for the parallel plate system. apply also to the concentric cylinder method when used to determine k for solid foods. Maximum errors of the order 1% - 2.5% are claimed for the hot wire method (Tsederberg. 1965). I. Ix w N W MISSILE New SSWSS\ 7 m .r o... r m s \ .............. Elsaluuauaafim m m g m m m m m S C W T TEST CELL 30 (iii) Concentric Sphere‘ The steady state heat transfer equation for a hollow sphere of inner radius r1 and outer radius r2 is : 4TTk(T1 - T2) r1 r2 Q = --T;;L:»rl) 1.L, (50) Equation (5c) allows calculation of the thermal conductivity of the material once temperature. heat flux and linear measurements are made. The inner surface at r1 is main- tained at a uniform temperature usually by means of an electrical heater. Theoretically this method should be the most accurate of the steady state methods since errors due to heat loss are practically eliminated. However. because of many practical difficulties. the method has not become popular. Foodstuffs in general do not conform to the ideal spherical shape. e.g. for a granular type material it is difficult to fill the hollow sphere homogeneously and obtain a uniform bulk density. Foodstuffs of approximately spherical shape have to be cut in half to allow insertion of the heater and this operation will introduce an error in heat flow measurement. The discussion above concerning the use of composite samples to eliminate measurement of Q: errors due to radia- tion. convection and eccentricity between geometric axes: and the problem of thermal contact resistance. is in general 31 applicable also to the spherical steady state method. A development of interest might be that of Grayson (1952) for determining thermal conductivity of tissue and blood - the ultimate purpose being to measure the heat transferred by flowing blood in humans or animals and hence the actual blood flow rate. He examined the validity of the method of internal calorimetry based on the mathematical equation (18) below. presented by Carslaw (1921) which states that with an electrically heated spherical source of heat in an infinite mass of material. a steady state is approached in which the relation between heat production and heat loss is represented by the equation : 2 Q=IR=4wkr(T1-T (18) 2) Grayson investigated this statement by setting up two experiments: (a) embedding a heating thermocouple in a large mass of gel (10% gel in water) and plotting the I2 vs. (Tl-T2) relationship; (b) keeping (T1 - T2) constant at 1°C. and determining 12 in a number of substances of different known thermal conductivities. His results showed that the 12 vs. (T1 - T2) relation- ship was in fact linear for small values of (T1 - T2) though in one case using water only the results indicated heat transfer by convection. He also demonstrated the existence 32 of the linear relationship between I2 and k. C. Transient Method§_ Unsteady state solutions to heat conduction equations depend on initial and boundary conditions. The equation to be solved is : clog-$- e --(k gT) + W‘k 165%) + 52(1: g—T) + u (19) As stated previously the initial condition is usually given as a particular temperature at time zero. Many boundary conditions are possible and of course in general the simpler the boundary condition. the simpler the solution. The most commonly encountered boundary conditions include; (a) known surface temperatures: (b) specified heat flux across the boundaries: (note that insulation conditions at the boundary imply no heat flux.) (c) heat transfer to the ambient by convection: (note that for the case where h -—> w. the boundary condition of type (a) is implied when now the surface temperature will be the same as the ambient temperature): (d) interface of two substances - 1.2. - in contact. when the assumption is made that no temperature differential exists at the interface. This neglects contact resist- ance o 33 Many solutions for different situations are available in standard references such as Carslaw and Jaeger (1959). However. from the experimental point of view the greatest difficulty lies in approaching the boundary conditions as- sumed. The degree to which they are attained determines the validity of the ideal solution. Since many solutions to transient problems involve a first term approximation. it is essential to check that the second term can be neglected. The main advantages of the methods are that rapid results are possible. no heat flow measurements are re- quired in many cases and a very small temperature differ- ential is acceptable. All of these factors are highly important in determining k for foods. various methods of interest are described below; specific applications to foodstuffs are discussed in Chap- ter III. (1) Heating and Cooling Curves. Many investigators have applied the solutions for heating and/or cooling of bodies of different geometries under various boundary conditions and determined a. These solutions have found widespread use in the food canning industry. A good mathematical treatment is given by Ball and Olson (1957). The basis of the method is the exponential relation- 34 ship between temperature change of the product and time. after a certain heating up period. frequently referred to as a "lag time“. The straight line which results from plotting temperature vs. time on semi-log paper. gives a slope from which a can be calculated. Tsederberg (1965) described a method called the ”Regular Regime Method”. By knowing the temperature dis- tribution. the geometry of the test body and the boundary conditions. a can be determined. This ”regular regime“ type cooling is reached after a time t determined by the Fourier number : > 0.3 - 0.5 '11 ll (ELIE. where Lo represents a linear dimension depending on the geometry of the body (e.g. radius for sphere or cylinder; half thickness for plate). T -T The dimensionless temperature differential ud = Tm-T m 0 changes exponentially. -mt ud = M U0 e (20) where M is a constant. U0 is a function of location. and m 1s 8% . the same at all points. in the body. From this equation we see that 6 u d —E— . (21) s l I and 35 The value for m is determined experimentally. As an example of application of the method Tsederberg cites the case of a composite sphere where the inner sphere has known properties and the outer has unknown properties. For the case where h -—> oo, k = %1 (mm) where M. B1 and H below are dimen- sionless numbers : 1 K08 H 1 1+1: K208 _ ' _.. _ . _ ______i___.__ “- 217K 1.1’ Bl‘3'H-t—K’ H- 31! Cl’ 1 K = 155 ratio of diameters of internal and external spheres: c1 c1 1 h 1 h o _ o co = eat capacity of interna core x weig t; l 1 l C a G c = heat capacity of fluid layer x weight; F1 = surface area of calorimeter core. The value for m0° is found by the slope of the linear part of the curve resulting from the plot of log ud vs. time. For the case of h -—> w . a = K1 m. where 2 l K = a5' for sphere: (223) a l L K = 2 40 2 2 for cylinder of (22b) _:__5. IL lenthL (a)+(L) 3 I K1 1 I la (22) = --§-—-—I2 2 or alrcctangg Ith c %)+(‘l)+(‘%) Paraeepipe" sides L1. L2. L3. Thus a is evaluated. 36 Hurwicz and Tischer (1952) also presented a theoretical analysis which could be used to determine a. They solved the case of a cylindrical container whose surface tempera- ture is kept at retort temperature during heating and at a constant cooling temperature during cooling. The initial condition is that the substance is at a uniform temperature. They presented the final solution both in the form of a general solution and a first approximation. the latter being . 2 1 w 2 a LILl ID for o t1 (23) ‘where 2L1 = the length of can and z = the vertical distance from geometric center. By measuring the temperature at various radial and longitudical positions after a certain time a value for a can be calculated. Kopelman's (1966) work was based on simulating the boundary conditions of the convective heat transfer coef- ficient approaching infinity. He set up an experimental apparatus of cylindrical form with the following two bound- ary conditions: 3? %%=0 atreorortgo (24a) g5. = %'(T - T1). r = a for t 3.0 (24b) Note : where h -—>-w . T - T1 = 0 and t = T1 therefore T a T1 at r = a where T1 is ambient temperature: and the initial condition of T = T0 at time = 0 for all r. By having a length to diameter ratio in the 3 : l to 6 : 1 range the problem could be treated as one dimensional. The general solution to this problem is : T -T 2 r 2 't = 2 (—) J (B -)exp(-B 01—) (25) To'Tl 1:1 81 J2(B )+J2(B ) ° 13 1 a2 0 i 1 i where the 81's are the roots of the transcendental equation ‘ J NBiot = 5.173%}? (26) Kopelman states that the advantages include the lack of necessity for measuring heat quantities: the location of the thermocouple nor the timing of a sudden change of tem- perature is not critical. His results for thermal pro- perties of apples. meat and apple sauce are discussed in the next chapter. Ball and Olson (1957) discussed the effect of a displacement of the thermocouple on the re- sults for the lag factor. They showed that for a 307 x 409 can. a displacement of 0.205 inches (from the geometric 38 center) towards the end coupled with a displacement of 0.138 inches from the axis. produces an error only slightly in excess of 2%. Dickerson (1965) described an apparatus that permits the ”rapid” determination of the thermal diffusivity of foods. under controlled transient conditions. However. the determination required two hours which is hardly ”rapid". The method is again based on the transient heat transfer condition whereby temperatures of the sample increase linearly with time. It allows evaluation of the thermal diffusivity over the heating range of the test. giving a number of values from a single experiment. The sample was placed in a cylinder and inserted into a hot water bath which was heated at a constant rate. The cylinder was regarded as of infinite length and temperature gradients parallel to the axis were assumed non-existent. This led to the governing equation: 2 A! .2;I 1H£1 6t = A0 = constant = 01(61‘2 + r 6r) (27a) T -_- Act a Ta fort>0 , r=a. (27b) dT (11‘ = O fort>0. r=0. which gave the temperature profile and the temperature gradient as 4 a (T - T I __g = ( 2.)? (28a) A0 a a A r dT o a; = 20. (28b) Dickerson compared this simple solution with the com- plete solution given by Carslaw and Jaeger (1959) and concluded that the simple solution is acceptable for values of at/fi: > 0.55 (note Tsederberg's (1965) recommendation that at/L2 >.0.3 - 0.5). A constant heating rate is c assumed over all the surface but this is hardly likely due to convection currents in the hot water bath which would interfere with the convection heat transfer coefficient between the cylinder surface and the fluid. With Dickerson's experiment this was not possible to check since Only one thermocouple was used to record surface temperature. The thermal diffusivity was calculated by plotting the temperatures of the outside surface of the tube and the center of the tube against time. using equation (28a) with r = a i.e. a a Aoa2/4(Ta-To). verification of the appar- atus was carried out by determining thermal properties of reagent grade chemicals for which reliable values of specific heat were available. The thermal conductivity was determined using a method described by Vos (1955) based on the condition described 4O WATER BATH — HEATER STIRRER J 1 F 1/ FOOD W I SAMPLE 11; 1- THERMOCOUPLES CHROMIUM +PLATED BRASS TUBE k j Fig. 8 THERMAL DIFFUSIVITY APPARATUS.( DICKERSON.I965) 41 later for the ”probe” method. Vos shows that the initial transient is eliminated when 4at/502 > 50. A wire diameter of 0.01 inches was used which is well within D'Eustachio and Schreiner's (1952) recommendation of 0.02 inches. An accuracy of 5% is claimed. This method is further discussed when evaluating the measurement of thermal diffusivity of Chicken a la King. The method. although very similar. seems simpler than the ”regular regime” method described by A Tsederberg (1965). (ii) ”Probe” Methodyh A transient method which could be considered for determining thermal conductivity in granular materials is the so called ”probe” method. The details of the apparatus developed by Hooper and Iepper (1950) are shown in Fig. 9. The central electrical resistance wire acts as a heat source of constant heat flux. The principle of the system is based on the solution of the cylindrical heat conduction equation for the particular case of the temperature at the source (in this case a line source) in an infinite homo- geneous body initially at uniform temperature (see Appendixx For a heat input of Q per unit length the temperature rise 4T in the time interval between t1 and t2 is given by AT .2 519;}; tin-,5?- (29) 42 SERIES THERMOCOUPLES LEADS \ . 7 COPPER POWER LEADS ALUMINUM SLEEVE / LI w THERMOCOUPLE JUNCTIONS -3/I6"ALUMINUM TUBE OVERALL LENGTH L56 FT AP” HEATER 0.98 OHMs PER FOOT /STEEL TIP / . Fig. 9 DETAIL OF THERMAL CONDUCTIVITY PROBE 43 By plotting the temperature against the natural logarithm of the time a straight portion of the curve should occur with a slope Q/4nk. A correction factor is necessary due to the fact that the probe is of finite diameter. The correction suggested by the investigators is a time to. to be subtracted from each observed time. By plotting %% vs. time. the time correction to is found where %%.= zero. (to is nearly a constant for a particular instrument). The corrected equation.will therefore be t - t 2 o k = REIT 8n ( E;_:—E; ) (30) A minimum requirement is that the probe is surrounded by at least three inches of material and that no initial temper- ature gradients are present. An alternative modification to correct for the finiteness of the probe was suggested by Hooper and Chang (1953). (see next chapter). Of-interest is the fact that the probe system has given satisfactory results in moist materials (e.g. soils) and that the test can be performed within ten minutes. The absolute accuracy is not known. though comparisons with the hot-plate method indicate it is at least equal and results have been reproducible to within 0.5%. .A temperature differential of only a few degrees F is required. Hooper and Chang (1953) reported values for k of butter. wheat and margarine using the “probe” method. 44 The ”probe” method has been further developed by D'Eustachio and Schreiner (1952) who claim to be able to detect temperature changes of 0.010F. By using a small diameter wire (0.02”) and lowering the heat capacity. end losses can be regarded as negligible. However. the authors point out the limitation of the method for measuring absolute values. since there is no simple way of evaluating the effects of non-radial heat flow. (11;) Fitch Method. In 1935 a method for determining k was developed by Fitch. This type of test (with various modifications) has found wide use especially for the rapid determination of thermal conductivity of poor conductors. The determination is simple and can be carried out in ten minutes. which is the main advantage. However. for absolute measurements the accuracy is unsatisfactory due to factors listed below. Basically the method consists of placing a sample of low thermal conductivity between two copper blocks fitted with thermocouples to indicate temperature difference. The lower block. which is embedded in insulation material was described originally by Fitch as being only 4.4 cm. in diameter and having a mass of 346 grm. The upper block serves as the bottom of a well insulated vessel of about 800 ml. capacity. with a diameter of 8.4 cm. and an unknown mass. This vessel is filled with boiling water. ice. solid 45 carbon dioxide. or any convenient material that will give the temperature at which the measurement is to be made. It is necessary to have good thermal contact between the sample and the copper blocks unless the sample is thick. For soft materials the weight of the upper vessel is sufficient to provide good contact. The faults associated with these requirements are obvious i.e. distortion and possible alter- ation of the physical structure of the original sample. The temperature difference is indicated by a low resistance galvanometer such that 1 aON = T - T (31) where a0 = galvanometer constant and N = deflection of the galvanometer. The heat change in the lower block over time At is 1 1:0 = c(mass)(AT1) = k A TL. T Lt (32) where (T - T1) is the temperature difference between hot and cold blocks. Since aON = T-T1 , -dT1 :- ao.dN. In the limit we can write kAao N'dt -c(mass)(a .dN) = -———-———-—— (33) o L from Which N k A (it (31+) N’ = c(mass)(L) Upon integration this becomes 46 in (319') = a_(ECmE—ASEE-TLT (35) . k A t ; . . gnN OIMSST‘L) = on N (36) Therefore by plotting 6n N vs. time t. the resulting straight line has a slope equal to (-k A/c(mass)(L)) which allows k to be evaluated. Kopelman (1966) rejected the use of a modification of this method. the ”Cenco - Fitch" method. on finding that a minimum error of 15% was possible due to and effects and heat losses from the ”sink” (lower copper block). His reasons were : (a) the unpredictable errors arising from the applied pressure which may cause extrusion of liquid. or change the thickness or the physical properties of the sample: (b) the unlikelihood of obtaining linear heat flow (i.e. neglecting radial flow) with a foodstuff: (c) the unavoidable and practically unmeasurable heat loss from the sink. Note that errors (b) and (c) are similar to those encount- ered with the parallel plate steady state method. (iv) Zierfuss Method Zierfuss (1963) developed an interesting method for the rapid determination of thermal conductivity of poor 47 conductors. It was based on the equation given by Carslaw and Jaeger (1959) for two semi-infinite bodiesbrought into ”ideal” thermal contact : T-Tl /5 T1 - T = k: “2 T; (37) l 2 kl/a1 + kZ/a2 where the subscripts l. 2. 1 refer to the hot body. cold body and interface respectively. He showed that marked temperature differences are limited to regions near the interface for short times and therefore justified approxi- mating small bodies with semi-infinite bodies. By differ- entiating equation (37) with respect to a2 he showed that a good thermal conductor e.g. copper. was best for the reference (hot) body. He found that only thirty seconds were required to develop a steady temperature at the inter- face by using as a hot body a U-shaped capillary containing an alloy of low melting point ( 40°C.) in the liquid state. Using this arrangement he found that a minimum height of 2.5 cm. for a sample of poor thermal conductivity was necessary. The electrical circuit was so balanced that the galvanometer reading "p” was equal to T1 - Ti/Tl - 2 (see equation 37) and therefore a function of k2 and a2. It seems the problems with the method as described are (a) the radiation losses due to the small portion of the face in contact with the hot body and (b) the theoretical assumption of linear heat flow. The accuracy of the method 48 is not known. but due to time considerations and simplicity the approach might warrant further investigation. (v) Numerical Methods Powerful methods for solving heat conduction equations have been recently developed. They are numerical methods. which did not become popular earlier due to tedious and comples calculations. The digital computer overcomes these limitations. With the use of the digital computer a large number of temperature readings and calculations can be obtained on a sample in a short time. Basically two numer- ical methods exist. the forward approximation and the back- ward approximation. the latter involving the simultaneous solution of a number of equations and called an implicit method. The ”Crank-Nicholson” approximation. which util- izes both the forward and backward methods give the great- est accuracy. These methods are outlined in the Appendix. Larson and Koyama (1965) showed a method for deter- mining a and k of a substance which was ”backed” by a material whose properties would be known. They employed a Xenon flashtube to transmit a heat pulse to the composite sample for an extremely brief period of time and utilized the digital computer to solve the equations. The thin sample and brief time of heating justified their assumption that heat losses were negligible. They rightly stated that without the digital computer their solution would be 49 practically impossible. Beck (1966) described a statistical procedure called ”non-linear estimation” which can be used to simultaneously determine several properties appearing in certain partial differential equations. k and c (and hence a) of solids were calculated simultaneously with a claimed error of less than 0.1% arising from numerical approximations. Beck further indicated certain cases for which the properties cannot be found accurately. The method assumes one dimensional heat flow by con- duction through a sample with constant c and k over the temperature range and time tested. By combining c and . c1 2 /oc (heat capacity). The governing equation for transient heat transfer becomes: 2 6 T 1 6T 1‘! _ = C (38) 6x2 6t 1 Values are selected for k and c . and by using a finite difference approximation approach. values of TJ (t) can be calculated at position x and time t. The experimental 3 temperature Te.j(t) is measured at this point. This non- linear estimation procedure then minimizes the sum of squares functions F for n thermocouples and the duration of the experiment. tm : t m i (Tj(t)-Te n F (k. cl) = z th))2 dt (39) 50 with respect to k and c1. TJ(t) is a function of k and c 1 Beck used the Taylor series approach for minimizing F and assumed that the temperature was a linear function of k and cl. or : T(k. cl) = TI 1 1 koco + Tk. Ak + To Ac (40) n: = ig-l-T; (k. c1) (41) clk o o Tcl ._.. 1'11 (k. e1) (42) Do "elk o o Ak = k-ko and Acl = cl-cg‘ (43) where ko and c: are the zeroth estimates of k and cl. At the minimum value of F it is necessary that 9'2 = '6': = O (44a.b) 6k 6cl Equation3(44a.b) are solved simultaneously to get the new first corrected estimates of k and c1. which are then used in the next iteration. This procedure may be continued until the required degree of agreement is attained e.g. so: if or cl < 0.0001. Thus with the aid of a digital computer. as small an error as is desired can be Obtained. See Appendix for further details of the method. 51 In summary it can be stated that the main advantage of non-steady state methods is that rapid results can be obtained. The greatest difficulty lies in simulating the required boundary conditions. III. DISCUSSION OF RESULTS FOR k AND a AND THE VARIOUS METHODS USED A. (Steady State Methods General The parallel plate method has found wider applications than either the concentric cylinder or concentric sphere methods. The most important criticism that can be made concern- ing steady state methods has been ignored by many investi- gators. This concerns the migration of moisture from hot surfaces towards cold surfaces due to temperature differ- ential and moisture concentration differential. Moisture diffusion will take place until the ”driving force” due to temperature differential is balanced by the force due to moisture concentration differential. Therefore. steady state methods are unacceptable when applied to food sub- stances which often have upto 80 - 90% moisture content. A sample whould be ”dry” when using steady state methods. Practically. the method should be confined to liquids and dehydrated foods. A maximum moisture content of 10% would seem generous to impose as a limit for dehydrated samples. One practice which minimizes the error caused by moisture 52 53 movement is to maintain a small temperature differential across the sample. but this will likely lead to larger errors in temperature measurement. The magnitude of the error introduced by moisture migration is not possible to measure. Suffice to state that since readings are taken only when steady state conditions have been reached (and therefore moisture migration stopped). the results apply to a sample with different physical characteristics than the original sample. The use of more than one piece to make up a sample also introduces an unmeasurable error. The homogeneity of the sample is affected and conceivably thermal resist- ances could be introduced at the interfaces between samples. A list of the most common sources of errors is tabulated in Table 2. Reference should be made to the previous chapter which deals in more detail with the errors. The following discussion reviews applications by a number of investigators of steady state methods to foods. lI Parallel Plate Methqi a. Some of the earliest work on thermal properties of meats was carried out by Awberry and Griffiths (1933). They measured the thermal conductivity and.specific heat Of beef at very low temperatures (down to -180°C.). and from these measurements calculated the thermal diffusivity. 54 TABLE 2. Factors which introduce errors when determining thermal conductivity by steady state methods. No. Description 1 Measurement of heat flow (Q). 2 Measurement of test area (A). 3 Measurement of temperature drop across sample (AT). 4 Measurement of sample thickness (L orIAx). 5 Unacceptability of steady state methods for foods of high moisture content. due to moisture migration through the sample. 6 Use of more than one ”piece“ for test specimen. 7 Non-linear heat flow (radial heat losses). 8 Temperature unbalance between guard and main heaters 9 Presence of ”skin” or ”contact” resistance at surface of sample. 10 Convection heat transfer within sample. 11 Radiation heat transfer across sample. 12 Non-homogeneity of sample. 13 Other heat losses from apparatus. The parallel plate apparatus was immersed in a medium cooled with carbon dioxide snow. No mention is made of a guard heater. Heat loss was estimated by running a test on a cork sample with known thermal conductivity. Accuracy of the electrical input to the hot plate is not discussed: this aspect is all the more important since an average temperature gradient of only 1°C. per cm. was 55 used. The average of six values for k was reported but no indication given of the range of results. Their selection of c = 0.36 is open to question. It should be also noted that 0 reported below freezing point is partly due to latent heat. therefore to apply the re- sults of a. k and c in the range -40°C. to 80°C. is incor- rect. Three results only are given for c at 4.8. 14.7 and 18.4% moisture in the 18 to 40°C. range. but the authors nevertheless fit a straight line to the data and read off values for c at 74% moisture. This practice is hardly acceptable. The results reported could also be in error due to factors 1 through 5. 7. 9 listed in Table 2. The com- position of the sample with which the thermal conductivity test was performed was not given. b. Miller and Sunderland (1963) published results for k of beef muscle obtained using the parallel plate ”twin-sample” method as recommended by the National Bureau of Standards (1955). They rightly pointed out that al- though the thermal conductivity of beef depends on temper- ature. moisture content. direction of heat transfer with respect to the fiber. and pressure for dehydrated samples. several investigators failed to specify these factors. However. pressure was employed by them to ensure good ther- mal contact between the sample and the plates. but they 56 failed to indicate the range of pressures applied. The edge heat loss from the primary heater was mini- mized by maintaining a negligible temperature difference across the gap between the primary and guard heaters. A 32 - junction thermopile made from 30 BS - gage chromel and constantan wire was used to measure the temperature difference. Good thermal contact was ensured by planing the surfaces of the frozen samples. wetting the surfaces of the hot and cold plates. and applying a compressive force by tightening four corner bolts between the top and bottom cold plates. After weighing the samples to deter- mine moisture loss during storage and testing. no measur- able loss was reported. The authors concluded that the values obtained are accurate to within i 2.75% of the actual value. and stated that this includes maximum errors of 2.25% from instrumen- tation. 0.16% due to measurement of temperature difference across the sample. and 0.34% in assuming that the hot and cold plates are isothermal surfaces. They used the analy- sis by Woodside and Wilson (1957) to measure the error caused by small temperature difference between primary and guard heaters and calculated the plate constants qo and B as 0.30 Btu/hroF. and 4.24 ft.. respectively. Miller and Sunderland supported their claim of accuracy by determining values of the thermal conductivity of known materials. i.e. a rigid cellular glass insulation and a semi rigid glass 57 fiberboard. which agreed within 0.5% and 1.35%. respect- ively. of the known values. The two samples. however. were of low moisture content and a comparison with the beef tested (69.5% moisture) was not justified. The authors failed to acknowledge the problem which exists with steady state methods when applied to samples of high moisture content. A point to be noted. as pointed out by the authors. is that freezing may alter the physical characteristics of the structure of the tissue and therefore values of thermal conductivity measured above the freezing point may differ from those of samples not previously frozen. In addition to the error of i 2.75% referred to above. inaccuracies caused by factors 5. 6 and 7 in Table 2 should have been considered. as also should have been the possible alteration of the physical structure of the sample due to the applied pressure. The authors did not state the average temperature differential across the sample. 0. The thermal conductivities of a variety of meats. fats and gelatine gels in the frozen state were determined by Lentz (1961). In contrast to recommendations by the American Society for Testing Materials (1955). Lentz used only the test sample on one side of the hot plate. while inserting insulation material of precisely known thermal conductivity on the opposite side. The entire apparatus. 58 having been placed in an enclosed container. was submerged in a bath controlled (1 0.100) at about the temperature of the sample. The cold plate temperatures were controlled to within 1 0.02°c.. and the power to the heater plate was determined with an accuracy of i 0.5%. To prevent errors from air spaces between samples and plate or between pieces of samples. the samples were wetted; while. to ensure reasonably uniform composition of the sample. temperature differences of only 2 - 3°C. were maintained between hot and cold plates. This latter precaution violates the recommendations of ASTM Standards (1955) which gives 40°F. as the minimum temperature differential. A disadvantage of using a small temperature differential is the possibility of large errors in the determined k values resulting from relatively small errors in temperature measurement. Unfortunately Lentz does not report the density of the samples nor does he include an error analysis of his re- sults. Factors effecting the accuracy include points 2 through 6 in Table 2. and also (i) i 0.5% in the measurement of Q (ii) the effect on the temperature gradient through the sample by submersion in a liquid of constant tem- perature. This should be minimal due to the exten- sion of the sample under the guard heaters. the small temperature differential across the sample. 59 and the control of the liquid temperature to 10.10C. (iii) any deviation in the thermal conductivity of the insulation from the assumed true value; (iv) the fact that during freezing of the sample the cold plate was maintained at -30°c. and the hot plate was heated ”so that a film of liquid was maintained adjacent to it”. Since freezing time for all samples was 2.4 hours. it is likely that moisture migration from hot to cold plates occured before freezing was complete. resulting in non- uniformity of the sample. Not withstanding the above mentioned factors. the research work as reported by Ientz indicates carefully performed experiments with recording Of most of the import- ant parameters. an awareness not shown by many investiga- tors. The k values determined for the various foods in the frozen state and below -10°C. seem to be as accurate as any others reported in the literature. d. Hill. Leitman and Sunderland(1967) reported values of k for various meats using a modified parallel plate method. They also presented in their paper a model which allows approximate calculations for k of meats as a function of moisture content. The only reference made by the authors concerning accuracy states that they measured k of a slab of paraffin wax and a slab of southern yellow pine and compared the 60 results with handbook values. The samples chosen are not comparable to foodstuffs at temperatures above 32°F.. and ”handbook values” as such cannot be considered to be the best reference sources for absolute values. The accuracy of the measurements depended on points 1 through 6 in Table 2. and also on the fact that both water and fat were lost in the temperature range 32-150°F. (an average of 19% was reported: two thirds of the moist- ure loss and all the fat loss occured above 80°F.). Silica gel was used "to keep the insulation dry" and the question might well be raised if the silica gel also aided in removing some of the moisture from the sample. Since it seems from this report of Hill §§_g;, that the k values determined refer to a sample which had changed considerably from its original natural state. it is difficult to recommend that the published values could be reliably applied. Therefore the more interesting aspects of the above study are the comparison of thermal conductivities when heat flows are perpendicular or paral- lel to the fiber direction. e. The parallel plate method was used by Woodams (1965) in determining the thermal conductivity of a number of emulsions (glycerol. olive oil and peanut oil in water) and mixtures (sausage). The accuracy of the experimental procedure was shown to be unreliable by comparing the results obtained for 61 water with those of Riedel (1945) whose results are regarded as the most reliable of those available in the literature. Woodams then allowed for a total of six correction factors : (1) external resistance in D.C. heating coil circuit: (2) radial heat loss from apparatus through the insulation: (3) vertical heat transfer between the hot plate and the auxiliary coil; (4) heat loss by conduction through thermocouple leads; (5) thermoelectric heating at the Chromel-Alumel junction. and (6) heat loss through lucite ring between the hot and cold plate. The summation of these heat losses was 0.9788 Btu/hr. By reducing the heat input Q this amount. Woodams obtained values in good agreement with Riedel (1945). The accuracy claimed by Woodams for these corrections is open to question. Each one of the six factors was calculated by using either average or assumed values for temperature. resistance. thermal properties of materials. etc. The statement that ”to ensure all heat produced by the heating coil passed between the two plates. the apparatus was well insulated with polystyrene. vermiculite and fiber- glass” is vague and gives no such assurance. 62 Allowing for the experimental inaccuracies. the results given for glycerol. olive oil and peanut oil are acceptable. In the case of the sausage mixture with a moisture content over 60%. the moisture migration problem makes the values presented unreliable. and also invalid- ates the formula for k suggested by the author. f. Saravacos and Pilsworth (1965) investigated the thermal conductivity of freeze-dried potato starch. gel- atine. pectin. cellulose gum and egg albumen gels using a guarded hot plate apparatus designed to allow the detere mination to be made under vacuum as well as under atmos- pheric pressure. .A schematic sketch of the apparatus in Fig.10 shows details of the experimental set-up. The average temperature differential across the sample was 10°C. The thermal conductivity was first determined at atmospheric pressure on a sample of 12 - 15% moisture content. After evacuating the apparatus by pumping con- tinually for twenty four hours. measurements under vacuum were made. By admitting dry air the pressure was gradually increased again to atmospheric pressure and a final k value determined. The dry samples (2% moisture) were allowed to absorb moisture at room temperature. until an equilibrium moisture content of 12 - 15% was attained. thus violating the ASME recommendations for ”dry” material. The error introduced by the moisture migration suggests that the error of 2% 63 McLEOD PRESSURE GAGE D'Fng'fig‘ \ COLD TRAP F MANOSTAT e MANOMETER . G) I OIL Hg VACUUM PUMP :81: OIL BATH HEAT CONTROL- MEASUREMENT TOP SHIELD~ GUARD RING / J / HOT PLAR SAMPLE COLD PLATE Fig. /0 THERMAL-CONDUCTIVITY APPARATUS, USED BY SARAVACOS AND PILSWORTH ( I965) 64 found when calibrating the instrument with standard materi- als measured by the National Bureau of Standards is opti- mistic. The latter error could have been caused by assum- ing the temperature of the oil bath to be the same as that of the cold plate. However. the results for the ”bone dry“ samples should be accurate within the 2% range referred to. The large difference between thermal conductivity for all samples at atmospheric pressure and under pressure is worthy of note. There is no reason why accurate data cannot be obtain- ed with this method when the samples used have been freeze- dried. The ability to test under controlled vacuum makes the method attractive for obtaining useful data for freeze- drying processes. g. Spells (1960-61) employed a steady state parallel plate method to determine k of rat blood. human blood,cow's milk. skimmed milk. cream. egg white. egg yolk and cod liver oil. He used a thin (3/16”) glass disk. a central conductor (copper) and a thin (1/32") liquid layer in series. A top heating block (copper) was placed over the glass disk: a cold copper block was placed below the liquid sample. The apparatus was enclosed in an insulated box. Spells could not have had significant convection heat losses due to the thinness and size (2.5m1.) of the sample. The fact that his figures agree well with those of Riedel suggest that his claim of 5% accuracy may be justified. 65 since the work of Riedel has been accepted generally as the most accurate available. According to Challoner and Powell (1956). Riedel ”appears justified in claiming to have made determinations which are accurate to within 1 u or 2%. Factors likely to have effected the accuracy of Spells work would include : (i) (11) (111) heat leakage and evaporation (also possibly result- ing in a "surface resistance” due to loss of sample) of the liquid sample where the edge was exposed to air space: assuming the temperatures of the top block. central conductor and bottom block represented the tem- perature drop across the liquid layer and glass disk. A minor error could result from this assump- tion. assuming that the heat loss from the liquid layer was directly proportional to the temperature differ- ential across the layer and calibrating the appara- tus on this assumption. using reference materials. Two possibilities for errors arise here: (a) the validity of the assumption: (b) if the assumptiOn is true. the heat loss may not be the same for the test samples as for the ref- erence materials. Results of k for fluids should be accurate. provided 66 the necessary precautions against heat loss are taken and mechanical or electrical measurements can be accurately made. 2, Concentric Cylinder Method. The concentric cylinder method has found the most widespread application for determining k of liquids and gases. One possible explanation for this may be the difficulty in getting suitably shaped samples of suffic- ient size of solid foods. Of the latter froup. powders or grains would appear most suitable. a. The method was used by Woolf and Sibbitt (1954) to determine the thermal conductivity of liquids (see Fig. 7). During the test the cell was suspended in a liquid bath of fourteen gallons capacity, where the temperature was thermostatically controlled to a constant temperature. 1 .0036°F. This eliminated heat loss to the ambient from the test apparatus. Also. only a small section of cylinder was used as the test length. thus minimizing the possibility of heat loss from the actual testing area. Woolf and Sibbitt discussed the accuracy of their measurements. Since measurements for thermal conductivity of water in a tarnished brass instrument were identical with those measured in a polished aluminum instrument. they justified making no correction for losses due to radiation. 67 The authors quoted the work of Mull and Reiher (1930) Beckman (1913) and Jakob (1946) who estimated that when the product of NGr NPr was less than 1.000. the error due to convection heat transfer is less than 2%. To remain within this limit it is necessary to use a very thin liquid layer and a small temperature difference of the order of 10F. For glycerol and olive oil the product of NGr NPr was 122 and 158 respectively. well within the stated limits. The effect of the horizontal liquid layer could be questioned from the view point of convection currents and heat loss (Fig. 7). Woolf and Sibbitt state : "For most liquids. the end losses were estimated by making runs at several temperature differences and extrapolating the thermal conductivity to the zero temperature differential. In most cases the cor-, rections were negligble. The maximum error was 1.5% corresponding to a temperature differential of about 50F. This experimental correction also accounted for the influence of convection“ Concentricity was maintained through use of three centering screws. 120o apart. minimizing errors due to eccentricity. ‘ This writer doubts the accuracy of the claim that temperature differences of 10F. were accurate to 0.5% maximum. Fig. 7 shows that the thermistors are actually located in the walls of the inner and outer cylinder. Therefore. the temperature drop measured takes place part- ially in the cylinder walls and partially across the liquid layer. For accurate determination of temperature differ- 68 once across the liquid layer therefore. the very small thickness of the brass cylinder between the thermistor wall and the liquid layer would have to be accurately determined. and the brass material may differ slightly in composition from the standard material for which the thermal conductivity is known. Woolf and Sibbitt conclude that the maximum error in their data 1s-"probab1y less than 2.5%! This figure seems optimistic in view of the doubtful accuracy of the temper- ature measurement mentioned above. Notwithstanding this latter criticism. the determin- ation of thermal conductivity of liquids using the con- centric cylinder apparatus under steady state conditions has been shown by Woolf and Sibbitt to give results within experimental accuracy. b. tha §§,gi. (1966) applied the method to non-fat dry milk powder and wheat flour (see Figs. 11. 12. 13). Since the moisture contents varied in the range of 4.2% to 8.8% it can be accepted that moisture migration should have been much less an inhibiting factor than for samples in the 60% - 80% moisture content range. Unfortunately. the authors gave no indication of the reliability of their results. The reliability depends on each of the factors in the following equation : 3n r2 1‘1 k =3 W (unit length) (5b) 69 OUTLET WATER TUBE THERMOCOUPLES THERMOMETERS 5% POWDERED MATERIAL % INSULATION ' CEMENT ASBE-SETOS CYLINDER EL CYLINDERHTO SUPPORT CABLE PER TUBE (6 DIA.,IO" LONG) PORT FOR THERMOCOUPLES coqm ‘U'OITI GUARD TUBESIG" DIA., 4" LONG) .4 i -" s \ 4| ‘ A." I [ . ‘I. i: I ' . .3 . I. I is. I ‘ l v . I I I . ‘ J I. . . . . . . . "‘ ' VIE-".1 . n . .‘. ._ .._........._._._.....‘.'....._......._...‘ .‘.'.’,'.' .3... _...'.. ..‘............._.‘ .._.. ..*?_'.I . , 0'". G v - u o o o n n o u o u a I "' _ . {" .v. .-‘_- t ‘ a, i ' a . J a 1"; I. -_.'.,4. . , ‘ ..~ “r‘,.. - .-_ . - “—‘.v\,£‘..~. I 9 u - . o . - . ‘ . 5* - . . ' ' ... o“. J. "'4'. a. R ‘9 ‘_ .3 .‘f, . A_'\,. .Q .I '3 :..o’ a . .P. 'l ‘ .‘s‘ .'- _‘~ \ I; "o ‘. I - ‘. ’0 ‘ \ 54 ' - _ _ . r . ‘ . f: .0.‘ ‘5”. e ' :‘fl Q 'i" . ‘-._ 313-: O - .I INLET Fig. // CROSS SECTION OF HEAT CONDUCTIVITY UNIT USED FOR DETERMINATION OF THERMAL CONDUCTIVITIES OF POWDERS. (tha et al, l966) 70 WATER IN ' l ;: PLOATII :1 HEAT I. \ I III; CONSTANT HEAD CONDUCTIVITY n 24" "I: WATER DEVICE ”N'T——’ I: I: 5: II 'I ' H \::l A ._I Lil/IL L” FUNNEL I. \ WATER / W "W W W “1 VT" Fig. /2 LAYOUT OF THERMAL CONDUCTIVITY MEASURING APPARATUS. (Olha et al, I966) Fig. /3 RADIAL LOCATION OF THERMOCOUPLES IN ANNULUS 71 The temperature measurements were probably very accurate. but there is no indication given that the linear measurements were equally so. Since Q was calculated from the temperature r1se 1n the flowing water. 1t 1s doubtful if the error introduced by this measurement could be neglected. Also. since the values were determined for the mid-section only. there is no guarantee that the thermal conductivity of the powder at the top is similar to that at the bottom. where settling probably had occurred (therefore. change is density) after eight hours. the length of time allowed before steady state conditions were assumed to exist. The average bulk density only was given. Since a higher bulk density would mean that less free air space was present. one would expect that higher thermal conductivite 183 would result due to the low k values associated with gases. Therefore. since possible errors due to factors 1 through 5 and also 9.in Table 2. could have been present but are not acknowledged. it is not possible to evaluate the accuracy of the data. An excellent review of liquid thermal conductivity research is given by Filippov (1968). .A maximum error of about 4% is reported for the concentric cylinder method. 3, Concentric Sphere Method. The use of the concentric sphere method has been con- fined to grains. With the exception that heat losses 72 should be minimized. it is subject to all the errors of steady state methods. Table 2. It also has the problem of finding suitably sized samples. Oxley (1933) used the method to determine k of a number of grains Fig.14. The apparatus was placed in a thermostatically con- trolled air chamber in which the temperature was controlled to within 1 0.l°c.. and in wh1ch the air was ”stirred vigorously by a small high-speed fan”. It was claimed that heat flaw could be measured with an accuracy of 1%. How- ever. the accuracy of the temperature measurement must be questioned. since the inner sphere temperature was measured by a mercury-in-glass thermometer and the outer sphere temperature was taken as that of the air temperature. The average temperature during the tests was about 30°C, (869E) with a maximum temperature differential between the inner and outer spheres of about 49°F. The bulk density was not given. For practical pur- poses the thermal conductivity of grains in bulk conditions are most useful. Therefore. this thermal conductivity will depend on a number of factors in addition to the thermal conductivity of the grain itself i.e. (1) size of grain. since the larger the grain the less total surface-to- surface contact (therefore thermal resistance) and the larger free air spaces: (11) temperature of the air: (iii) air convection currents: (iv) moisture flow within the grains and from the grains (v) moisture flow through the 73 air spaces from hot to the cold surface: and (vi) bulk density of the grain (therefore depth of grain). Oxley was aware of these difficulties and states that ”no absolute value for the thermal conductivity can be obtained and it is necessary to choose a method of determination which approximates as closely as possible the conditions for which calculations involving thermal conductivity are to be made”. .Acknowledging the relatively low moisture contents in the grains these results are probably acceptable as practical guidelines for grains in bulk. Bulk density values. unfortunately not given. would have made the results more useful. The results can be used as guide- lines only. however. because of possible errors from points 1 through 5, 9 and 10 1n Table 2. B. Transient Methods g, Heating and Cooling Curves. The general method of determining a from heating and cooling data has already been outlined in Chapter 11. A number of solutions for different geometries under various boundary conditions are listed in the Appendix. From this list it is obvious that a wide choice of conditions is available. to the investigator. If theoretical conditions can be simulated in practice. it is usually possible to evaluate an average value of a: sometimes k can also be MERCURY IN GLASS THERMOMETER FILLING PORT .. . ., SPACE OCCUPIED - -° 4 BY GRAIN HEATER -- INNER ALUMINUM SPHERE (DIAM.=IO cm) EBONITE ROD. = 1' _, OUTER UPPER . 1 ' ' ' ’ , SPHERE(D|AM.=30cm) STAND~~ Fig. /4 VERTICAL SECTION OF SPHERICAL STEADY STATE APPARATUS USED BY OXLEY (I933). 75 evaluated directly. Table 3 lists some factors which limit the accuracy and usefulness of the method TABLE 3. Factors which affect the accuracy of determin- ation of thermal diffusivity from heating and cooling curves. No. Description 1 Measurement of temperature. 2 Measurement of time. 3 Location of thermocouples in sample. 4 Convection heat transfer. 5 Arrangement of sample in container 6 Fitting of straight line to heating and cooling data. 7 USB of average values only for a and k. 8 Initial temperature distribution in sample. 9 Physical alteration of sample during test. 10 Use of approximations for infinite series in theoretical solution. 11 Simulating theoretical boundary conditions in practice. 12 Moisture diffusion within sample during test. 13 Invalidity of the implied assumption that the sample is homogeneous and isotropic. 14 Use of several small pieces to make up the test sample. 76 a. Charm (1963) presented a method for evaluating heat transfer coefficients in freezers and thermal con- ductivity of frozen foods. which was based on heating curves and relationships between a. f (the reciprocal slope of the asymptote of the heating curve). and other variables as presented by Hall and Olson (1957) and Olson and Jackson (1942). The method is relatively simple and the value for thermal conductivity of frozen codfish has a claimed accu- racy of less than 10% which may be justified. The procedure consists of plotting the temperature - time relationship for a cylindrical frozen sample immersed in brine and solving by trial and error the following equation: 2 0 2 2 2 13%}- : A1I/‘o (”5) The term 11 is evaluated from the relationships hb NBiot = '1? = ml (”6) B 1 A1 = if (“7) The term/Ab is evaluated from the relationships ha 1 _k- .-..- ml (48) B1 _ .1 llo - a (’4'9) 77 Charm presents the graphical relationships between :111 and 81. and mi and Bi. which will allow h and k to be evalua- ted. To calculate h it is necessary to run a test using a sample with known properties (i.e. ice) and the value for h so obtained is then used when evaluating k for the unknown sample. This method as described by Charm would appear to have the advantage of being relatively easy both in experi- mental procedure and mathematical calculations. The error analysis for k evaluated under these test conditions is also simple. Since the sample is in the frozen state there should be no problem with moisture migration from hot to cold locations. It should also be noted that the thermal diffusivity and thermal conductivity are simultaneously determined when using the trial and error solution. However. the accuracy of the method would appear subject to factors 5.6.7.10.ll.l3 and possibly 9 in Table 3. Unfortunately. Charm does not give details of the com- position of the sample. The author stated that ”the results are sufficiently accurate for most purposes" and this is probably true for design problems which include a safety factor. b. The apparatus used by Dickerson (1965) to measure thermal diffusivity of chicken "a la king” has previously 78 been mentioned. He does not give details of the composi- tion of the sample. The method is relatively simple and has the advantage of being suitable for determining the apparent a for prod- ucts such as semi-fluid foods. powdered products. soups and solid foods from.which cylindrical samples can be taken. Except possibly for the latter foods. the term "apparent'I would seem appropriate. since Dickerson does not allow for heat transferred by convection. However. for practical applications an ”apparent” 8 might be more useful than true values for a. especially since the cylin- drical test apparatus may be physically similar to a con- tainer in which the foods may be heated or cooled. The time for testing is much shorter than for steady state systems and moisture migration therefore is less of a problem. As mentioned earlier it is unlikely that con- stant rate heating was present over all the surface. due to convection currents in the hot water bath. The small temperature difference across the wall of the diffusivity tube affects the temperature measurement. though probably by a very small amount. The alignment of the center thermocouple also affects the results. though Dickerson points out that an off-center deflection of 0.21” causes an error of only 4%. To ensure a linear rate of temper- ature rise. a low initial (as well as uniform) temperature of the sample is required. 79 Other factors which affect the accuracy of the result obtained by this method are points 5. 6. 7 and 10,Table 3. Dickerson's method appears of limited value for solid foods since it is unlikely that many samples meeting the require- ments (2.15” diameter. 9” length) are available. a. Hurwicz and Tischer (1952) used a method previous- ly described for determining thermal diffusivity of beef over a range of temperatures. The accuracy of the method depends on maintaining the surface temperature of a cylinder at constant retort temperature during heating and at con- stant cooling temperature when cooling. The first approxi- mation of the solution for such conditions is given by equation (23). The thermal diffusivity is evaluated from the slope of the heating and cooling curves. The various constants are already known. Thermal diffusivity at twelve locations was determined and the average computed. Since results for ninety minutes heating differed the least. these values were regarded by this writer as the most reliable. For the thirty minute tests. values differed by over 50% (Tfi=225°F) and 47% (Tm a 250°F): while for the ninety minute tests. corresponding differences were 15% and 33% respectively. From the complete results presented by Hurwicz and Tischer we can note that for all lacations the apparent a is greater (in some cases substantially) at the higher temperature and therefore it seems clear that use of an 80 average value for a is incorrect unless a safety factor is included. Thermal diffusivity is obviously dependent on temperature. Hurwicz and Tischer discussed the errors involved. including the deviation of experimental data from theor- etical. and errors due to temperature. length and time re- cording instrumentation. They show in a thorough analysis that the errors in evaluating thermal diffusivity was of the order 1 10%. When one also considers the range of results as given above. a safety factor of the order 1.25 would not seem unreasonable if one decides to use the average ”apparent” a. d. Hurwicz and Tischer (1956) also used the above method to evaluate the thermal diffusivity of round of beef in cans. They presented complete results for 25 locations at temperatures ranging from 225°F. to 315°F. The conclusion drawn by these investigators that a does not depend on location is not substantiated by the results as presented. The average values reported should be used with caution in view of the variations in thermal diffusivity at various temperatures : 81 Temperature OF. 225 243 261 279 297 315 Average a. ftz/hr .0143 .0171 .0181 .0180 .0178 .0167 a min. ftz/hr .0111 ..0131 ..0104 .0141 .0135 .0104 s max. ftz/hr .0176 .0196 .0206 .0234 .0286 .0434 max -am n 00 a min, 60% 50% 100% 70% 110% 250% Such a wide variation in results suggests that the non-homogeneity of the beef sample was most likely the dominant factor. Since twenty five locations were used the likelihood of finding spots differing widely in com- position (e.g. moisture content. fat. protein or muscle fiber) was increased. The results also confirm that a varies with temper- ature. The affect of high temperatures (above 212°F) on the moisture content or moisture distribution is not taken into account: indeed. the authors fail to report the composition of the product either before or after the experiment. Meat shrinks on heating in the cylinderical can,therefore. one assumes that changes in the physical properties of the meat with temperature are coupled with extrusion and probable evaporation of some of the moisture. In view of the stated unreliability of the a values it appears that the cooling and heating curves presented by Hurwicz and Tischer are of more practical interest than are the average thermal diffusivities given. Contributing 82 factors to the inaccuracy of the results include all those listed in Table 3 with the possible exception of points 4 and 1h. . e. It is not possible to extimate the accuracy of the values for cherry flesh as reported by Parker and Stout (1966). The thermal diffusivity was determined by approx- imating an infinite cylinder (insulation conditions at each end of a cylinder filled with cherry flesh) and placing the cylinder. which is suddenly lowered in temperature. in a cold bath. The cylinder surface is maintained at uniform temperature T8. The solution for this physical situation is given by Schneider (1955) : Th0 " T8 2 ‘2 exp (.11: a t/az) 0(9) (50) To - T8 Dal Mn J1 Mn - -9 C(0) e a function of the Fourier Modulus. which is tabula- ted by Schneider (1955). They also determined the specific heat and speciric gravity which thus allowed k to be calculated. The authors give no indication of how closely their experi- mental conditions approach theoretical conditions and the description of the procedure does not allow such an evalu- ation to be made from the literature. The standard errors given by the authors (see Append1X9 are an indication of the precision of the results but reveal nothing concerning the absolute accuracy of the 83 measurements. or the degree to which the ideal theoretical equations were approached. f. The thermal conductivity and thermal diffusivity of soybean oil meal was determined by Hougen (1957). 4A steady state concentric cylinder method was employed to evaluate k; trial and error solutions for fitting heating and cooling curves for cylindrical containers were used to evaluate a. The latter method is relatively simple. Olson and Schultz (19h2) presented solutions for predicting temper- atures at points in bodies of a variety of shapes (see Appendix). The solution for a finite cylinder is given by Tm - T ‘ t t Tm ‘ To a CO (£50 80 (9—50 (51) a a’ values for Co and SO are calculated approximately (since they represent an infinite series): 1m - T is measured T - T m o experimentally The correct solution will be given by choos- ing the value of a which will give the same Tm ' T value T - T m o as that which is experimentally measured. The experimental set-up showing the container in the autoclave is shown in. Fig. 15. The apparatus used to determine k is shown in Fig. 16. Hougen used a reference material in one annular space and soybean oil meal in the other. reversing the position of the materials for alternate runs. The results were averaged. 84 To Brown self - balancing electronic potentiometer Pressure Gauge Pressure Control Therm findun N V’ ouple ion 33l Fig. /5 APPARATUS FOR HEATING CANNED SOYBEAN OIL MEAL AT CONSTANT SURFACE TEMPERATURE (HOUGEN,|957) 85 CONDENSER CONCENTRIC COPPER CYLINDERS THERMOCOUPLE JU NCTI ON S CON DENSATE RETURN BOILER w J HEATER Fig. /6 CONCENTRIC TUBE APPARATUS FOR DETERMINATION OF THERMAL CONDUCTIVITY(HOUGEN, I957) 86 Excellent agreement was obtained between calculated and experimental values for c. Due to the large temperature differentials used (168°F. maximum to 31°F. minimum) small errors in temperature measurement are not critical. A steady state method is not ideally suited for determining k of samples of 131 moisture content. The reported value for thermal conductivity also applies at one point only. and the effect of differences in k with changing bulk density is not investigated. Temperature at three loc- ations only were taken into account (1. 2. 3). It seems to this writer that a simpler method for this type of apparatus would be to also measure the temperature at point I . 2 . and choose fg_ 2 £1_. This arrangement allows the r r 1 2 _ thermal conductivity to be calculated from the relationship k1 T2 - T _=T-T R2 1 2 where either k1 or k2 apply to the reference sample. This arrangement also allows a larger temperature differential T2 - T3 than used by Hougen. Hougen gives an excellent description of the sample used. It is unfortunate that Hougen did not experimentally derive c and compare the evaluated results with those calculated. The accuracy of a determined will depend on how closely the infinite series was approximated. 87 g. The thermal diffusivity of a navy bean powder ” mixture was determined by Evans and Board (1954) by plot- ting the heating and cooling curves for a container of the mixture when placed in a retort at 240°F.. and a water bath at 104°P. They assumed that the temperature of the side walls and the bottom of the container reached the medium temperature immediately but that a thermal resist- ance existed at the top due to head space effect. The solution for this situation is the product of the solutions for an infinite slab and an infinite cylinder. subject to the above boundary conditions and the initial condition that at time zero. the product was at a uniform temper- ature. By plotting the heating and cooling curves (on semi-logarithm scales) a straight line should result of slope 8 2 2 -c [Tl/a2 + all/Ba (52) From equation (52) a can be evaluated. The value of Y1 a 2.40h8; a and L between the intercepts of £n(Tm-Tc) 1 and £n(Tm-Tt) on the £n(Tm-T) axis ( =‘2n553313I17;7)' The results of Evans and Board clearly indicate the inaccuracy of the method (as was also evident from the data of Hurwicz and Tischer. 1956). especially at higher temperatures. The authors suggests that the ”abnormal” values for a of 0.0075 and 0.006 may be ”due probably to 88 movement of the thermocouples during processing”. Commenting on the unexpectedly high values for a at the higher temperature they offer as the most likely explanation that evaporation took place in the headspace. particularly near the hot wall of the can which is approxi- mately at retort temperature. This. if true. would in- crease the value of a determined. Excepting factors 5. 12. 13 and In all the factors in Table 3 would affect the results. It is very likely that the difference between the practical situation and the ideal conditions for which the solution is available (constant medium and surface temperatures. constant thermal conductivity. constant specific heat and constant density negligible heat resistance at side walls and bottom etc.). are significant enough to render the method inaccurate. h. ‘Hany investigators assume that variations of ther- mal properties of foodstuffs with temperature are small enough to be neglected. Evans (1958) examined the validity of this assumption. He used an.agar gel mixture (95$ water 5% agar) and a beanabentonite mixture of 75%'water content. Evans noted that the thermal diffusivity of water varied by nearly 20% between 68°F. and 248°F and correctly reas- oned that a similar variation could be expected in food- stuffs which may contain 20 - 70% water. The above author's presentation consisted of two parts; 89 (i) theoretical calculations using the Crank-Nicholson approximation to show heating and cooling curves for a substance of varying thermal properties; and (11) experimental verification of the theoretical results. using agar gel and bean-bentonite mixtures referred to above. The main purpose of the investigation was to demon- strate the effect on heating and cooling curves by a sub- stance whose thermal properties were temperature dependent. However. since the purpose of this dissertation concerns the actual thermal properties fo foods. the most important aspect shown by Evans results is that thermal properties do in fact vary with temperature and as expected. more so for those foods with high moisture content. It is not possible to state how reliable the results for the agar gel and bean-bentonite mixtures are except to note that in the case of the latter. values for a between samples when (i) heating. vary by over 13%. and when (ii) cooling. by over 4% and also that a evaluated (iii) under heating conditions may vary by as much as 19% from the a evaluated under cooling conditions. Similar variations for the agar gel are (i) 5% (ii) 3% and (iii) 18% respect- ively. It is clear that average values reported should be used with caution. since Evans shows that the thermal properties do vary with temperature. Numerical methods are 90 the only satisfactory way to deal with problems where thermal properties vary with temperature or from one location to another. 1. The experimental set-up used by Kopelman (1966) has been referred to in the previous chapter. He reported values of k and a for apple flesh. meat and apple sauce. determined under what appears to be carefully controlled conditions. Of interest is the fact proved by Kopelman that k in fibrous systems is greater parallel to fibers than across them. Kopelman calculated that for NCr x Npra 600 max.. a diameter of 5.2 mm. was tolerable for the air cell sizes. Internal convection heat transfer could there- fore be neglected. He further showed that internal heat generation by biological material (e.g. respiration) was negligible. The convective heat transfer coefficient was deter- mined using aluminum and copper cylinders. The assumption that the same coefficient held true for the containers of food could have been in error. Another source of possible error could have been the assumption that specific heat of the samples was the summation of the weight fraction times the specific heats of the various non-gaseous components. Other factors affecting the accuracy of the determination include factors 1. 5. 7. l3 and 1h in Table 3. J. The thermal conductivity and diffusivity of No.1 seed Marquis wheat was determined by Babbitt (l9h4). He 91 used a 1 ft. diameter x 2 ft. high cylinder of grain. with a heating wire stretched along the axis of the cylinder. Three sets of thermocouples were used along three diameters of the cylinder. each set measuring the temperature at twelve locations. Both k and a were evaluated from know- ledge of this temperature distribution. The theory is based on the solution of the equation describing the case of a right circular cylinder with a central line source of heat in which the initial temperature is constant and uniform and equal to that of the surroundings. and the final equilibrium temperature distribution is independent of time and dependent only on the radius (see Appendix). By knowing the temperature of the wheat at a partic- ular radius and at a time sufficiently large to Justify using the first approximation of the solution. a was determined as 0.0045 ftz/hr. The plot of final temperature vs. log r agreed with the theory. and a straight line re- sulted. From this. k was found to be 0.087 Btu/ft-hr-OF. It is interesting to note that this was in perfect agree- ment with Oxleys (1933) result for No. 1 Manitoba wheat. although the latter wheat had a moisture content of 11.7%. compared to the No. 1 seed Marquis wheat moisture content of 9.2%. The bulk density was given as “9.9 lbs/ft3. The comments made on thermal conductivity of grain when re- viewing Oxleys paper apply also in this case. 92 A point to be noted with reference to Babbitts results is that temperatures in the lower half of the cylinder were about 0.5°F. lower than temperatures at corresponding points in the upper half. Babbitt suggests this is due to convection. This writer suggests that it is also due to the slightly higher bulk density in the lower section. With a higher bulk density (therefore less free air spaces) the thermal conductivity would be expected to be higher. This in fact would be the case as indicated by the equation for calculating k. 5. ”Egobe“ Method {A general description of the "probe” method has been given previously. .Although the method is relatively simple and eliminates many of the problems associated with steady state methods and heating/cooling curve methods it sur- prisingly has found only limited applications to food- stuffs. Contributing factors to inaccuracies are listed in Table #. page 93. a. An investigation of all three thermal properties. a. k and c. was carried out by Kazarian (1962) on samples of soft white wheat and yellow dent corn over a wide moist- ure content range. The determination of k was made using an apparatus similar to the ”probe” method developed by Hooper and Lepper (1950) and D'Eustachio and Schreiner 93 TABLE h. Factors causing inaccuracies in ”probe” method. No. Description 1 Measurement of Q. But/hr. 2 Measurement of AT. oF. 3 Measurement of time. 4 Use of approximations for infinite series solution. 5 Use of a finite heat source instead of theoretical “line source”. 6 Non-homogeneity of the probe. Invalidity of the implied assumption that the sample is homogeneous and isotropic. 8 Difference in composition of probe material and sample. 9 Precision of location of temperature measuring point. 10 Non-radial heat flow. (1952). The determination of c was-made by employing transient heat flow conditions in a slab of the material. by lowering the face temperature suddenly to zero and keeping it constant at that temperature. Of interest is the fact that Kazarian was able to use the same sample over the moisture content range by starting at the lowest moisture and conditioning the sample thereafter to the re- quired moisture level. The heating time for the corn was chosen as 16 m1n.; 10 min. heating time was suitable for wheat. Cooling data for the determination of a was recorded over periods of 9h 35 - #0 min. The accuracy of k evaluated by the ”probe” method would be affected by the factors in Table A. However. Kazarian shows that the use of the approximate solution introduces a negligible error. Further he examined the possibilitylof non-radial flow during thermal diffus- ivity tests by noting that the temperatures at the end locations fell on the cooling curve plotted for the center temperature. Kazarian used a correction factor as advised by Hooper and Lepper (1950) (which he found to be 6 sec.) to compensate for the use of a finite source. The ”probe“ method. as already suggested. is probably well suited to granular materials; this test would seem to substantiate that view point. The largest source of error can be limited to temperature measurement. a problem which presents itself for all methods. That the latter Problem may have occured with Kazarian's experiment is suggested by a comparison of the calculated and evaluated values for thermal diffusivity. which differed by 11.6% and 16.1% for wheat and corn re- spectively. Allowing for the fact that an error was as- sociated with the determination of c. and that Kazarian used actual density values in his calculation. while a was evaluated under bulk density conditions. an accuracy at least within 10% could be assumed for the k and a values reported. 95 b. Hooper and Chang (1953) used the probe method developed earlier by Hooper and Lepper (1950). to deter- mine the thermal conductivity of a number of materials in- cluding wheat. butter and margarine. The theory of the method is given in the Appendix. In the previous work Hooper and Lepper (1950) used a time correction factor to compensate for the finiteness of the line heat source which was approximately constant for each instrument. However. in this later work they suggest an alternative method to correct for the above factor. The governing equation for an infinite line heat source of constant strength and zero diameter placed in an infinite. homogeneous sample initially at a uniform tem- perature throughout. is AT e 2%; 157155;) (53) where r is the radial distance from the line source at which the temperature rise T is measured after time t. and co -3?) d8 1‘ e 2 at o where 80 is an integral parameter. A correction factor is necessary not only due to the finiteness of the line source. but also due to points 6. 8 and 9 in Table A. 96 An examination of equation (53) shows that it is poss- ible to draw the three series of curves shown in Fig. 17 (B. c. D). By measuring T1. T2 etc. after times t1. t2. etc. it is possible to plot graph (A). Fig. 17. When a value of Q/k is selected. a series of points D will be located on graph (D). If the resultant curve conforms to the contours in (D) the correct value of Q/k has been sel- ected. By measuring Q. k can be calculated. Should the Q/k value chosen be incorrect the resultant graph on (D) will be either too steep or too shallow. The correct value of Q/k is determined by trial and error. although in most situations the approximate k value can be estimated and this should allow the correct Q/k to be chosen on the third or fourth attempt. The method has the limitations of (i) being subject to human error in the reading of the graphs (B. C. D) in 1' ”2 etc. on (D): (11) lacking a means of evaluating the accu- Fig. 17. and the plotting of the graphs (A). and D racy of the results; (iii) applying only to those foods that can be arranged to approximate an ”infinite” mass e.g. meats. butter. granular or powdered materials; (iv) not applying to liquid foods due to convection currents and (v) other factors listed in Table 4. However. by using the above approach. the largest errors are likely to be associ- ated with measurements of Q and AT. The advantages of the method include (i) being 97 (B) (A) LL _Q_ It 1:. k a ‘2 ‘/\ / '2 tt 4 it Q 7 . Al ‘ § If In R / 1(r/2f—dr ) TIME,M|N. l ) i (C) (D) 8 E N N \ 0.0 I(r/2/d"t ) TIME,M|N. Fig. /7 GRAPHICAL PRESENTATION OF THE MATHEMATICAL LINE HEAT SOURCE RELATIONSHIP.(HOOPER AND CHANG, I953) 98 applicable to a much wider range of materials than food- stuffs; (ii) allowing measurements to be made in short time e.g. 5 minutes being typical; and (iii) the important fact that. as pointed out by Hooper and Chang: "If the method is not applicable under certain test conditions. or if observations have been incorrectly made. a constant r/di will not be obtainable. Thus. improper application is automatically avoided”. The above authors found good agreement between results from this method and the parallel plate method when applied to dry materials. The results reported for wheat are also in good agreement with those reported by Kazarian (1962). and about 15$ lower than results of Babbitt (1945) and Oxley (1998). However. these comparisons are only of academic interest. since different types of wheat are being compared. and the density of the sample used by Hooper and Chang was much lower than that used by Babbitt. Therefore a lower k value would be expected. 6I “Fitch” Method. The ”Fitch” or modified ”Fitch" methods have been previously described. Limiting factors have also been discussed previously. These are summarized in Table 5 on the following page. a. The experimental apparatus used by Helvey (1954) to determine the thermal conductivity of ”finely crystal- lized'l honey at various temperatures is shown in Figs. 18 99 TABLE 5. Factors contributing to inaccurate determination of k by ”Fitch” methods. No. Description Measurement of Q. Btu/hr. Measurement of A. ftz. Measurement of temperature. Measurement of L or Ax. ft. Measurement of time. Convection and radiation heat transfer. \‘I O\U\ PK») N 5—! Applied pressure on the sample which may cause extrusion of liquid. change in thickness or alter- ation of physical properties of the sample. 8 Non-linear heat flow through the sample. 9 Heat losses from the sink. 10 Non-homogeneity and anisotropy of the sample. and 19. Both are of the modified Fitch methods type and were used to cross-check results. In cell No. 1 heat was supplied by water or steam to the copper container. and the heat transferred through the sample to the copper block in the water bath. In cell No. 2 heat was supplied by a 100 ohm resistance coil to the upper chamber Uc which was maintained at 2°C higher than the lower water chamber Lo. The sample was placed in C. having a cross-section of 10 cm2 and a height of 1 cm. This container C is contained in a plastic disk J; it has IOO COPPER CONTAINER MANUAL STIRRER-\j7 ,// :4T ].-:_«—STEAM OR WATER COPPER CONSTANTAN OCOUPLES _,.< CELL —+H-< _i I: COPPER ' BLOCK :3 «FINSULATION WATER BATH L J L J 0:13P Fig. /8 CELL No.| (HELVEY, I954) IOI Pm MECHANICAL @Q AGITATOR O I .m 77 Var. C 1 . :0 rt— ~L—Ifl L oA I ”H SoA-B FLASK T T RESISTANCE j E H HEATER I ’ Mfl i NHHHH. Uc I! L n KM>THERMOCOUPLES . ““7 was” 3253/ es; / i SAMPLE . /W PITT COPPERr i BLOCK WITH 1 FINS I I h J LC—r-t'“ E’ - MANUAL AGITATOR i :3. Fig. /9 CELL No. 2 (HELVEY, |954) 102 a copper top and copper bottom. The whole equipment is placed in a vacuum bottle. Both cells are of simple construction. but at best only approximate measurements could be expected. More elaborate equipment to accurately determine the heat input and prevent heat losses might make the method feasible for fluid foods such as honey. However. the main objections to the method described by Helvey are x (1) measurement of heat input. As for the Fitch method there is no way to account for the heat losses from the top and bottom water chambers. Not necessarily true is the assumption that the temperature in the chambers is the same as that in the copper blocks. and that the temperature of the copper blocks is the same as the end of the sample in contact with it. Where a small temperature difference was maintained in Cell No. 2 the latter error would be more serious. (11) convection currents,especially in cell No. 1. though Helvey takes the precaution of heating vertically downwards. He does not give dimensions for cell No.1. If the method was applied to solid foods. it would have all the objections of the Fitch method. In any event the parallel plate procedure can be quite acceptable for fluids. b. The Fitch method was used by Bennett. Chace and 103 Cubbedge (1962) to estimate the thermal conductivity of Marsh grapefruit and valencia orange rind. The authors conclude that the apparatus as described by them is satis- factory for determination of k of homogeneous samples of fruits and vegetables. While one could agree with this statement when applied to cases where approximate values will suffice. their results confirm further the limitations of the method. A thorough study was carried out to determine those test conditions which would give the most reliable results. By comparing the results evaluated for k of a sample of silastic silicone rubber with the assumed ”true” value as measureiby the Bureau of Standards. they decided that the least error would be given for a test period of 20 mintues. a sample thickness of 0.25” and plate and plug temperatures of 130°F. and 78°F. respectively. For the rubber sample. their results show that the error due to these conditions are 9%. 1.95% and 3.55% respectively. An examination of the reported k values for rind of grapefruit and orange will reveal that in the case of Marsh grapefruit a 13% increase in sample thickness was associ- ated.with a 12% increase in the k evaluated; in six tests 'with Valencia orange the results showed random variation. the extreme case being when a decrease in sample thickness of about 30% resulted in an increase in k evaluated of 14%. .Also note that pressure on the samples was only 0.017 p.s.i. 10h while in further work (described later) a pressure of 1.0 p.s.i. was used. so a contact resistance was probably present in this earlier work. The method is simple and minimum equipment is required. However. for accurate work the results are too unreliable. due to the factors tabulated in Table 5. the most important of which would appear to be unmeasureable heat losses from sample and plug as indicated by the work of Bennett .e_t, 9‘1. By taking accumulative instrument errors into consider- ation. the possible error of 15% suggested by Kopelman is probably slightly conservative. c. The above study was further developed and reported by Bennett. Chance and Cubbedge (196A). The original apparatus was modified to allow the use of smaller sample sizes. The authors also calculated an overall ”apparent” k from the combination of the thermal conductivities of the juice vesicle and the rind. and the results presented should be a useful contribution but the assumptions made (necessary for application of the calculation equations) should be kept in mind I (l) assumed homogeneity of fruit rind and juice vesicles; (2) assumption that the fruit was of spherical shape with a spherical void in the center. The comments made above relating to the earlier work apply also to this case. and it is worth noting the vari- ation in the results for both orange and grapefruit. The I05 “Mow—T“ (A) INSULATION I 3 h if." \l 3.1-! INSULAI ION '7‘. '.Q. I .' ‘ . 'v“. 0‘ ‘ 1 . '- {I ';.' AA... AA A AA >‘L’\q . f . A“: v 4. 4 0' . '1. .0 I ‘5 ‘ 57:": .3: ...t .7), 5‘35: .... v..,.'.\ , . ‘. .' 'u...‘ HEAT SOURCE Q 5, ' : ‘1' o J! - . c-e‘“ i I ’h“ "’JO”. V r ~ I '75....1 ' ' 'V ...;I_\‘ PM"? L»‘\'.‘ l 9" AMPLE I'd... . a. _ mtg-:2- 1::.--" ' ...-" . A-.l' PM ‘ ' .- .. l.. " Eififii-S _ w ‘1 . ° 5 .. ‘T‘V ‘ n g Q t t. . v A.-.“ .. ... «Aux; um -.I . ;. § - . ......... ’.‘,'.-- "- ... 1‘.‘1 u‘ 7 ul“. l‘.“"‘..' .A“..'.'. \‘ . ' .' 3‘, ‘. .--.-~--‘II.'.'."_~: :fah.‘ .'_' '2; - n ,L ‘ ‘ I ..u‘. u... .15.... b‘ .g. . . ."/ O n . ' ' 'vp‘...- I " o. . - .4.-"J 9. I - A “ "k F— o . 1". fl ‘..' ‘9‘ w—vr‘ j- (1 .. ~ . ' 1' ' ' r '0' ._ o l"“ ‘0‘!" '. -,.~ ADJUSTABLE I’M/a it (B) PLUG Fig. 20 SCHEMATIC OF APPARATUS FOR THE MEASUREMENT OF THERMAL CONDUCTIVITIES OF POOR CONDUCTORS; (A),FITCH THINDISC; (B). MODIFIED CENCO-FITCH, AS USED BY BENNETT,CHACE AND CUBBEDGE (I962) “106 extreme variations are of the following order 8 16% for orange juice vesicle. 23% for orange rind; 22% for grapefruit juice vesicle: and 20% for grapefruit rind. d. Walters and May (1963) investigated the thermal conductivity of the muscle and skin of broilers and hens using a modified Cenco-Fitch apparatus. The above authors presented only their average results but stated that no significant effect on k values was produced by variations in percent moisture. percent fat or of carcasses from which samples originated. This conclusion is surprising since it contradicts the logical assumption that k in- creases with larger moisture content. an assumption that_ has been proved true by experiment (Evans. 1958). Simil- arly it would be expected that k should depend on the quantity of other components i.e. fat. protein. in a food- stuff. An examination of the report shows that in the case of muscle the range of moisture content and fat content 'was only of the order of 6% i.e. 69.1 to 79.9%'and 0 to 5.9%. respectively. This narrow range would not justify the above conclusion. However. the corresponding ranges for skin samples were nearly 30%. so in this instance their conclusion cannot be explained. The samples were under test for only 10 to 20 minutes 107 and therefore the problem of moisture migration associated with the parallel plate method should be largely overcome. On the other hand the errors due to the heat losses from the sink and possible changes in physical characteristics of the sample for applied pressures may be of the order of 15% (Kopelman. 1966). Walters and May compared their apparatus with the parallel plate method and decided to introduce a correction factor of 1.11. The apparatus used indicates that the reported results were evaluated under careful conditions. The hot water bath could be controlled to within a claimed 1 0.l°F. and the potentiometer used for measuring plate temperature differences had 0.01 millivolt scale divisions. In cases where k values of broiler or hen muscle (of similar composition to that tested by Walters and May) are required for the 40°— 80°F. range. it seems safe to use these reported values with an accuracy of 10 to 15%. 2. During Freeze-DryiggI Of particular importance for design purposes of freeze- drying processes is information about the thermal proper- ties of the product when subjected to freeze-drying con- ditions. “Effective“ values (e.g. including the influence of latent heat. convection etc.) could conceivably be as useful as absolute values to the designer. provided test conditions are in good agreement with practical conditions. 108 a. The thermal conductivity of salmon. haddock and perch during freeze-drying was investigated by Lusk. Karel and Goldblith (1964). They placed frozen blocks of the samples between two heating platens and this assembly was mounted on a Mettler balance and placed in the freeze- drier. The loss in moisture. and the temperature at the center and surface of the sample was recorded at regular intervals. These investigators assumed that all the heat transferred across the dried layer was equal to the heat used for ice sublimation i.e. l k A (Ts '- To) AH r = t (55) Ld By substituting for La from the equation it can be seen that by plotting the graph of rt vs. (TS - Tb/WL. it will be possible to evaluate k from the slope of the graph AZ/OD (mL .- mf) k (TS . T3“) rt 2 AH le (57) This writer feels the following comments should be made concerning the results presented by Lusk gt 9;. (1) It will be noted that the values for k increase as thickness of sample increases. This clearly indic- ates that leakage losses occur. and the assumption (ii) (iii) (iv) (v) 109 of negligible end effects is not true. Therefore it would appear that if any of the reported values are to be used. the most accurate would be that for the sample of least thickness. The accuracy of the assumption that all the heat is transferred for heat of sublimation. should be clari- fied. The writers claim that the sensible heat was less than 1% of the heat for sublimation but do not present calculations in their publication to support this. After initial drying this may be true. but consideration of the test figures for the first three hours would indicate the sensible heat to be of the order 4% of the heat of sublimation (755 moisture. temperature rise of approximately 100°F in dried layer. average specific heat value of 0.5). Internal convection heat transfer is also neglected. An error of about 4% is given by the authors because of instrumentation inaccuracies. While the species of fish was stated. more complete details of composition of sample (e.g. density etc.) should be reported to allow more accurate applica- tions of the reported values. The indication that thermal conductivities during freeze-drying are essentially the same as the con- ductivities of dry materials would appear to ignore the fact that thermal conductivity of ice is consider- 110 ably higher than food solids (Lentz. 1961). b. In a paper which discusses the considerations involved in the design of a pilot scale freeze—drier using dielectric heating. Harper. Chichester and Roberts (1962) present the thermal conductivities of ”the beef sample in the presence of air plotted as a function of pressure”. However. they make no mention of how they calculated k. the measurements or assumptions made or theoretical equations employed. Therefore it is not poSsible to com- ment on the reliability of the results given. c. Further investigations of the thermal conduct- ivity of beef during freeze-drying were carried out by Massey and Sunderland (1967). The experiment was performed at chamber pressures varying between 0.2 and 3.0 Torr under quasi-steady-state conditions (assuming steady state over short time intervals) and was similar to that performed by Lusk gt_§l. (1964) for freeze-dried fish. An important consideration of MaSsey and Sunderland was the convective heat transferred by the vapor while moving from the frozen interface through to the dry surface layer. a point not noted by other investigators and which involved an error contribution of 2.5 to 3.5%} as judged from the findings of the above authors. Their findings also indicated increasing thermal conductivity with in- creasing pressure. probably due to increase of k of gases with increasing pressure. The different values of k when 111 using the heat of sublimation for ice or for beef juice should be noted. though for freeze-drying calculations either value can be used provided consistent values of k and latent heat are used. As stated. the method is similar to that of Lusk gt a1. (1964) and is based on the three equations below : war -T) Ts o .. __g=__.1-rtr on (58) T i wL e 2 A L/OD (mL - mi.) (60) From equations (58). (59) and (60) k can be found a T "L rt(AH -I- f s c dT) T ’2 1""— Massey and Sunderland do not try to estimate the accuracy of their results. but an analysis of their report would suggest that errors may have been possible due to some of the considerations stated below. (1) How close did the actual freeze-drying approximate the one dimensional model? Although the authors refer to removing some samples before drying was com- pleted to observe this factor. they fail to state how 112 closely the model was approximated. (ii) How true was the slope of the linear graph? Many points recorded did not fall near the straight line. (iii) The authors admit that quasi-steady-state conditions did not exist at the start and finish of drying: this is evident from the graphs presented. (iv) An error was likely present because of using the average values of A and /OD (average of beginning and final values) i.e. the assumption of constant drying. is not true. The error in/Db'would be quite small but probably not so in the case of A. (v) Any deviation from the assumption of uniform Q to each surface would invalidate. equations (58). (59) and (60) because 3 (a) the temperature at the center of the sample would not equal the temperature at the interface; (b) L value for left and right hand sides would not be the same; and (0) heat transfer would take place in frozen region due to temperature differential (a. above) (vi) A small error might be introduced by the assumption that the temperature of the vapor is the same as the temperature of the product at the same position. (vii) Since the authors point out the difference in results obtained on a sample 1.5 inches thick. this indicates that some end-effect was probably present for the 113 0.5 inch samples. This fact suggests that the one- dimensional equations are not completely valid and that ”rt” and “Q” in equation (58) are not numeri- cally equal to ”rt” and ”Q” in equation (59). It is not possible to estimate the extent to which the above points (i) to (vii) may have contributed to errors in evaluating the values of k reported. The advantages of the method are pointed out by the authors: ”thermal conductivity is measured during the dry- ing process and the actual gas vapors are present in the void spaces of the meat”. For design purposes for freeze-drying operations the use of the values reported would seem acceptable. 8I Graphical Method. Keller (1956) explained a graphical method for evalu- ating k in frozen concentrated orange juice. The method was a modification of the Schmidt (1924a'b.1942) plot. which is based on the choice of (AI)2/a At = 2. for one directional heat flow. By dividing the width of the sample into a number (n) of increments of thickness Ax (rectang- ular p1ate) or.Ar (cylinder) and choosing.At such that the above equation is satisfied. it is possible to approximate graphically the temperature distribution within the sample. Keller measured the temperatures and consequently derived 114 the thermal properties from the plot. Since his work covered the frozen region. the values he reported were ”effective” i.e. included for the latent heat of freezing. The method. as conceded by Keller in his conclusion. is only an approximation. the accuracy depending on-a number of factors besides instrumental accuracy 3 (i) (ii) (iii) (iv) (v) no guarantee of the validity of the assumption that radial heat flow only exists; a stainless steel cylinder was used to estimate the convective heat transfer coefficient h from the cool- ing medium to the sample. Surface effect of the tin containing the orange juice need not be identical to that of the stainless steel cylinder. therefore an error could be introduced also with this assumption. The effect of a thermal resistance on the inside surface of the tin container should also be consid- ered. Note that an error of x% in h value also means a similar error of x% in k; the larger the radius. the larger the number of in- crements that are necessary, (accuracy depends on number of increments chosen; the larger the number. the greater the accuracy obtained); being subject to the physical limitations of accur- ate linear measurement; using effective values of c from work by Riedel (1949) which included some inaccuracies; and 115 (vi) the fact stated by Keller: ”One complication which could cause the results to be erroneous is radial stratification of the ice and liquid portion of the frozen concentrate in the cylinder. This would cause the ice to become more concentrated towards the surface. and the center to be practically free of ice". Although "effective” values only are reported. and these are only approximations. the results should be use- ful for practical applications which after all is the ultimate objective. The conclusion given by Keller is representative of the work: "The graphical method is useful to predict approxi- mate temperature distributions during unsteady heat flow through cylindrical containers of frozen concentrated fruit juices. Useful values for the density. specific heat and thermal conductivity of three concentrations (20°. 40° and 600 Brix) of frozen citrus concentrates were deter- mined. Until.more specific information is available. these results should provide reasonable estimates for other fruit and vegetable juices”. 2. Numerical Methods. Matthews (1966) employed a transient method to deter- d of raw and heated potatoes (Excel variety). using the 116 finite difference method of solution (see Appendix). This method seems to the writer as the most suitable method. for foods in the solid state. It has several advantages 3 (i) the conditions under which the tests are carried out are similar to practical conditions; (ii) the length of time for a test can be as short as two minutes; for Matthews' tests. the time of a test ranged from 4 to 15 minutes. but the maximum time over which data was collected was 100 seconds; (iii) due to the short length of test time. moisture migra- tion is less likely to be a problem; also. the assumption of no heat losses from boundary surfaces could introduce only a minimal error. (iv) errors can be confined to instrumental measurements; sophisticated temperature measuring instruments could minimize these errors. Matthews used a hot plate which was placed in contact with a 0.953 inch diameter x l 1/32 inch high cylindrical sample of potato. surrounded by polyurethane insulation. The temperatures at the interface between the hot plate and sample. at a point a distance L from the interface. and at the bottom of the sample were recorded every 10 seconds. Measurements indicated that the boundary conditions were true for 2% minutes. Therefore the readings up to 100 seconds only were used. Thermal diffusivity was evaluated using a computer 117 program drawn up in accordance with the description out- lined in the Appendix. The accuracy of the method depends on instrumental accuracy and the choice of the incremental widths used. Matthews suggests that for engineering purposes. accuracy to i 10% only should be assumed. This is probably highly conservative. A development of the above method has been referred to in Chapter II Beck (1966) described a numerical method which allowed the simultaneous determination of k and c. The main objection to the method is the assumption of constant thermal properties. but this is true of all methods. However. this can be overcome by taking a large number of readings at different average temperatures. This method seems applicable to fluid samples which meet the requirement that NCr NPr‘< 800 (as for steady state methods) for dehydrated and natural foods from which reasonably homogeneous samples can be taken. However. where thermal properties in a radial direction are required. the apparatus as described by Matthews would not be suitable. Modific- ations would be required to ensure heat flow in a radial direction. In this respect. there is no reason why the "probe’I method cannot be successfully applied. 118 Note on Accuracy of Tempgrature Measurement Thermocouples are normally used to measure temper- ature in thermal conductivity determinations. Various types. with different wire diameters. are available. Errors can be serious or small. The reader is referred to publicadsions such as Jones (1965). Holman (1966). Cook and Rabinowicz (1963). Doebelin (1966). Schneider (1955) and other tests for details on accuracies and sour- ces of errors with thermocouples. Iron/constantan is widely used. and may have an accuracy of t 0.5°F. in the 32 - 250°F. range if cali- brated. The limit of error in this range for a thermo- couple of premium grade wire as supplied by the manu- facturer is normally 3 20F. Copper/constantan is usually more accurate. often having a guaranteed accuracy of i 3/4OF. with premium grade wire (within the 32 - 25°F. range). Chromel/alumel is also used and has accuracies similar to those for iron/constantan. Time constants for thermocouples are dependent on wire diameter - the larger the diameter. the longer the response time. For precise work. very fine wire sizes are available (0.0005 to 0.015 inches in diameter). Holman (1966) states that the time constant of an iron/constant couple of 0.0005 inch diameter for a step change from 200 to 100°F. in still water is about 0.001 seconds 119 However. typical thermocouples have time constants of a few seconds (1 to 4). Cook and Rabinowicz (1963) list the sources of error in thermocouples as (a) voltaic effects. (b) parasitic e.m.f.'s. at extraneous junctions. (c) corrosion (e.g. oxidation) problems. (d) radiation effects. (e) conduction errors and temperature/time response. IV. CONCLUSION AND RECOMMENDATIONS Methods which have been used to determine the thermal conductivity of foods have been discussed in detail. The most important features of the various methods are tab- ulated in Tables 6. 7. 8 and 9. Reference to these tables will guide an investigator in selecting the method which is most suitable for a particular food product. A concluding comment seems in order; every method1 is an application of a mathematical solution which is only true for a homogeneous sample. The unlikelihood of this condition being true has been stressed. However. this fact should not discourage investigators since this as- sumption is necessary to make acquisition of useful data feasible. It is necessary that complete data be reported if results of thermal properties of foods are to be of value. Suggested information to be reported includes: 1The only possible exception is a numerical method. but even for this case. the resulting equations may become too complex and unwieldy. without making the assumption of homogeneity. 120 a. b. c. d. f. S. h. 121 direction of heat flow with respect to arrangement of sample; density of sample; moisture content of food; content of other components of food (e.g. fat. protein. salt etc.); estimate of absolute accuracy of results; complete description of experimental set-up and appar- atus used (e.g. diameter of thermocouple wires. heat input and it's control. method and precision of linear measurements. etc.). average temperature and pressure to which reported values apply; and such information as may be necessary to completely describe a sample. e.g. type of meat. viscosity of a juice. species of a fruit or vegetable. etc. 122 TABLE 6. Summary of advantages for steady state methods. Applicable to methods Simple mathematical solution. all Suitable for liquids and gases. all Suitable for dehydrated foods in powdered. granular or solid form. all Errors can be minimized through use of reference material in two-layered systems. all Magnitude of error due to non linear heat flow can be evaluated. 1 Small test sample is acceptable. 1. 2 Suitable for dehydrated food samples under vacuum. all Key to methods: 1. parallel plate; 2. concentric cylinder; 3. concentric sphere. 123 TABLE 7. Summary of disadvantages for steady state methods Applicable to methods Error due to non-linear heat flow. all Non-homogeneity and anistropy of most solid food samples. all Unmeasurable error due to contact resist- ance. all Cannot be correctly applied to foods of significant moisture content. all Error due to eccentricity. 2. 3 Difficult to acquire correct geometrically shaped sample in one piece. 2. 3 Heat loss from test apparatus. 1. 2 Length of time required (several hours). all Key to methods: 1. parallel plate; 2. concentric cylinder; 3. concentric sphere. 124 TABLE 8. Summary of advantages for transient methods. Applicable to methods Test conditions approximate processing conditions. all Short time required (minutes). 5. 9 Suitable for foods of high moisture content. all No heat flow measurements necessary. 4. 8. 9 Location of thermocouple or temperature measuring point not critical. 4. 5 Suitable for frozen foods. 5. 5: 8. 9 Suitable for foods ”in situ“. 5 Small errors. after correction. 5. 9 Simplicity. 4. 5. 6 Key to methods: 4. heating and cooling curves; 5. ”probe” method; 6. ”Fitch" method; 7. during freeze-drying; 8. graphical method; 9. numerical method. 125 TABLE 9. Summary of disadvantages for transient methods. Applicable to methods Non-homogeneity and anisotropy of solid food samples. all Alteration of physical characteristics of sample. 4. 6 Immeasurable heat losses. 6 Difficult to maintain theoretically ideal conditions. 4. 7. 8 Complex mathematical solutions. 4. 9 Men-linear or non-radial heat flow. all Inconvenient sample size. 4. 5 Assumption that thermal properties are independent of temperature. all can be tak- en into 809 001111.13 by 90 Key to methods: 4. heating and cooling curves; 5. ”probe” method; 6. ”Fitch” method; 7. during freeze-drying; 8. graphical method; 9. numerical method. 1. 7. 8. 9. 10. 11. REFERENCES Andersen. S. A. 1959. “Automatic Refrigeration”., Maclaren and Sons Itd.. for Danfoss. Nordborg. Denmark. ASME. 1959. "Thermodynamic and Transport Properties of Gases. Liquids. and Solids”. McGraw-Hill Book Co.. New York. ASHE. 1949. ”The Refrigerating Data Book". The Am. Soc. of Refrigerating Engineers. West 40th St.. New York. ASTM. 1955. 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Unbalance methods in guarded hot plate measurement. Symposium on thermal conductivity measurements and applications of thermal insulations. Special Technical publication No.217. p.32. Am. Soc.-for Testing Materials. 1916 Race St.. Phi lade 1phia . Pa. Woolf. J. F. and W. L. Sibbitt. 1954. Thermal cofiductivity of liquids. Ind. Eng. Chem. 46: 19 7. Woolrich. W. R. 1933. Latent heat of foodstuffs. . University of Tenn. Eng. Exp. Sta. Bu11.. No.17. Zeirfuss. H. 1963. Apparatus for the rapid determine ation of thermal conductivity of poor conductors. J. Sci. Instr. 40:69. Zhadan. V. Z. 1939. Heat capacities of food products. Konservnaya i Plodoovoshch Prom. 1:27. Zhadan. V. Z. 1940. Specific heat of foodstuffs in relation to temperature. Kalte-Ind. 18(4):32. APPENDIX A1 l? TABLE A. - Thermal properties of foods at b. ca d. A2 a,b.0.do Key for pages A3 - A59 gathered from the literature; converted to Btu-ft-hr-oF-lb units: average values given in some instances: the following symbols are used in Table A: e = thermal diffusivity. ftZ/hr. k = thermal conductivity. Btu/ft-hr-OF. c = specific heat (at constant pressure). BtU/lb-OFQ f.= density. lbs/ft3 ma%é % moisture content. wet basis. ToFa temperature. 0F. M = method number (see below). Rf.= reference number. 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N. o .8 Qx o x voom {A27 o¢.H we mNN.o 0N.H 0NH «0m.o 0m.H mo mHm.o ummm mq.H 0NH NmN.o mq.H we mNN.o mm.H 0NH 0Nm.o mm.H we NmN.o mm.H 0NH mom.o mm.H we mNm.o mHaa< 000 um xmvcH m>Huomuwmm mmUHsh ooH om ou mN NN.o o.oN oN H.N o.a oH H.H H.H o NN.o H.N oH- H0.o Nw.o oN- Nm.o 0H.o on- mq.o CwNOdeQ o\o OHVI O¢.O Ewwho wUH omN- NmN.o «NH- mmN.o NHH- wom.o OH- mm¢.o Nm mN.H NN Hm.H «H mm.H m mm.H q- mm.H mH- H.H mUH HQH o x .A28 o.Hm mmN.o m.Hq ooN.o m.mN cam.o 0 cu umHsoncmauma 0.NH ~00.0 mumHHH .umm NN.N w.HH mNo.o 00 H vamuw EDHEmua Nme cmmH 0.0 0.HN 00.0 0mN- HHN.o «NH- qu.o NHH- omN.o mN oq- mmm.o cw~oum m mcHNmmum m>on< oN 0“ 00 0N.o 0“ wo.o HamH maHNmmum sonm H.om Nq.o m maHNmmum m>on< H.cm Na.o HnmquoH m 00 00.0 mumaaHM mm NHN ou Nm 00.0 mamauHx mq.H 0NH o.Nm qu.o mq.H we o.Nm mNN.o mm.H 0NH 0.00 HNm.o mm.H we o.wo NwN.o mm.H 0NH N.mw on.o mm.H we N.mw mNm.o mamuo «H.H 0NH o.am me.o Hq.H we o.¢m NmN.o mm N oq.H 0NH q.oo wom.o ummm .wuaou\mmUHah .mm 2 mucmEEoo may N.o.E o 3 000m £539 0 00.0 oH- mm.o oN- mm.o mHHHom m338 “mums wa.N om- om.o HoH .HHH NHN.o OH- H.No NH.o smwkmco; .mconz mm NHN ou Nm mH OH H om.o ou NH.o Nm m .mcHa m u ummu Ho nuwamH 0H 0H OH H.NH mmH.o mcHHmemz om m N.mH NoH.o 23H?H .ouHmz mm NHN cu Nm m.mH 0“ m.NH mH.o oH HH.o Hcoumomz mm m.NH meo.o HHo Hmg cosmH NN NHN oH Nm Nm mm.o mHmmH H.NHH HHN.o m.mHH NHN.o H.Nm HmN.o 0.0m HNN.o o.NH oHN.o m.HN mN0.o 0 cu m.mH 000.0 HHHHSHAH mHmHHH .HHH N0.o H.oH ONN.o NH H .mwmuw asHamHa .me aHmH o.0 o.HN mmN.o o.NHH HNN.o 0.mHH HNN.o NH H H.Na N.HN HNN.o cmuopm .choo\nEMH .mm 2 mquEEoo wok N.u.E n\ o x b voom .A30 00 N mHH Scum N 00 m0H H OH ¢0H umm Nm.¢ mwmum>< 0.NNH N.0mH n.00H 0.0m MNH 00 Hm 0NH 00 0NH NNH 00 N.qm NHN ou Nm 00 00H 0» mm 0¢H Ou «OH «OH Cu on 00H ou Nm «0H ou Nm mm Cu Nm 00 Cu 0m 0N 0H 00 00 m.N0 0.N0 N0.0 00m.0 cmmamwcoo mHosa no :mmum xHHz .vuaoo \3mvkmcoz NmaonE .mM 2 muamEEoo hoH $0008 000m IMBl 00 00.0 00 00.0 cm Hm.0 mm Nm Ha.0 0NH 500.0 NNH 00m.0 00 wNm.0 m0 N umw NH.0v N.Hm HHm.0 m0 NHN ou Nm Hm 00.0 Eme HHHm.N 0NH 00m.0 HHHm.N 00 mNN.o HNHHmHuHaH Hum.H 0NH NNm.0 umm Nm.Nv 00 N HH0.H 00 NON.0 wmumuucmocoo H.mNH HN.0 HsmMH HN.0 0.m0H 0N.0 0.0m on 0H.0 N.MNH 0m.0 0.0HH 0m.0 N.moH 0N.0 N.mn N0 NN.0 m.mNH mm.0 0.0mH Hm.0 0.HOH Nm.0 N N m.mN om Hm.o nmmcmucoo .vucou\xHHz .mm 2 mucmaaoo moa N.u;= «K o x voom A32 moH H qud 5253 HH NHN.0 mHomzz m.OOH HN0.0 m.HNH Omm.O m.NO 0.0 HO0.0 O.NOH HO0.0 0.0NH HNN.O m.mOH O.m NON.O 0.0mH OO0.0 mCOHUHE mufiHZ m.NMH oomoo A.E.Q...H.ZV HO N HOuOEHHO ammz H.NOH N.H 0.0m NHN.O Hmwsoa HHHz OOH H NO mm.O OHH O0.0 HOH m0.0 Om H0.0 OH HON NO0.0 OH Om.O NO H0.0 ONH N0.0 OO OO Hm.O OHH N0.0 NNH O0.0 Om HOH O0.0 eme .Ouaoo\xHHz .mm 2 mquEEOU .moH. o\o.U.E Qx U & HvOOh 4A33 0mm 000.0 ONO HN0.0 HON OO0.0 OHN OO0.0 NHN OH0.0 ONH OO0.0 OHH OH0.0 HOH HH0.0 ON OO OOH.O HOHOOO mHHO OO O NO OH OO N.NH OHN0.0 OOHHOOO OHHsz O.HH NOO.O HH :eOOm: O.NH OOH.O xH OOO N.HH NO0.0 O.HH OHH.O dd :EOOM: m.NH 940.0 HH Cdm 0.00 NOOO.O N.HN HOO0.0 O O N.O NO0.0 OHOO O O OH O ON.O OH NN.O OOHHO .OHOz OO NHN OH NO OO .OO.O OOHHOz OO NHN OH NO OO O0.0 OOHHO 00 NHN ou Nm 00 H00 mEoouHHmsz .mm 2 3:95:00 mom. N.u.E Q\ o x a 000m A31- Nmm HN000 ONO ONO.O HON HH0.0 OHN HHO.O NHN OOO.O ON ONH N0.0 OOH OHO.O . HOOOOO.O + ONH.O n O OHH OO0.0 NO OHOOHH NNH NO0.0 OH- NHO.O OO- ONO.O HOOHOO.O + OOH.O n O ON- NHO.O HOOHOOOOOHONOO NO OHHOO HO- HON.O OOOOOOHHOO NNH OOH.O HOH OOH.O HHOOOO.O + NOH O n O OO HOH.O NO OHHOHH OO ONH.O ON- OO0.0 HO- OHO.O HNNHOO.O + NHH.O n O NHH- HO0.0 NO OHHOO OOH- OON.O OOOOOOHHOO OHO OO0.0 OOO ONO.O HOH OO0.0 ONH OO0.0 ON - NOO OOO.O HOHOOO .OHOOONOHHO .Hm H.H mucmasoo “HOH. N.o.E Q\ o x 6 000m -A35 O0 00 00H.0 NHN 000.0 00 00 N00.0 NHN H00.0 0NH 000.0 0HH N H0 N00.0 HNH N00.0 00H 000.0 MHH H NOH 000.0 m>HHO me 0m0.0 000 0H0.0 ¢0H Hmm.0 0NH 000.0 Nam 000.0 0mm Nam.0 0Nm 0mm.0 HmN Nmm.0 qu mHm.0 NHN mHm.0 0NH 00H.0 0HH NOH.0 HOH 0NH.0 ON OO OOH.O OOOOOHH 0H0 mq0.0 qu wm0.0 AvmumamwouUN£v 0N 0NH 000.0 0mmmc0uuou .OHOOONOHHO .mm 2 muCUEEOU «HOH o\O.U.E Q\ U x UOOh -A36 ONH HOH.O OO OOH.O OO HNH.O ON OH HHH.O OHHHHOO OOH OO0.0 ..GOHmHDEm Hun—G3 Gun ._....HO mOH “00.0 OHH H HOOOOO OOHOHOOOO OHOOHO: ON NO0.0 NO OO ONOO O HOOOOO O.NHOOO.O + HHOH.O n O ONH ONHO.O OHOOHH HO ONO0.0 «0.0 u N mvHom %uumm mmum H.HOH u OOHH-é. OOHHOOHOHOOOOO HO- OOHH.O 0.NN u wsHm> :mwocmNUOLH mm: mH0m.0 HO u OOHO> OOHOOH HHH- ONHO.O OOHOOHOO.O + NHO0.0 n O OOH- OHON.O HOOHOOOOOHONOOOO NHH OHHOO HON- OOHN.O HOOOOH HNH NHOO.O OOH OOO0.0 OoHNHHOO.O + OHNH.O n O OOH OOO0.0 OHOOHH NHH NON0.0 O0.0 u N OOHOO OHHOO OOHO OH- ONN0.0 0.H0 n msHm> awwocwzuoHnH mm- mmNm.0 O.NO u OOHO> OOHOOH OOH- NNNN.O OOHOOHOO.O + OHHH.O n O NON- OOON.O HOOHOOOOOHONOO NHH OHHOO ONN- OOOH.O HOOOOO .VHGOUNOHHO .mm 2 muamEEou mom. N.o.8 Q\ o x a coon A37 ONm mmm.o HwN m~m.O wHN mHm.o NHN Nom.o 05H NwH.O me mmH.o mm MOH.O ON Hm mMH.O wGDH Nm N.om mNOH.O UGOEHm umm3m me Hem-O 00m mHo-O HOH NH©.O wNH wmm.o Nam Hmm.o 0mm Hem-O ONm wdm.o HwN Hmm.O wHN NNm.O NHN wom.o 05H mmH.O OHH mnH-O HOH 00H.O we me.O 0N Hm 0HH.0 ammnhom No N.mm HHOH.O mEMmmm me mum-O Nmm mHm.O ON mom HwH.O wHkumm .quOUNOHHo HO OH OHOOOEOO ..H OH H.. O .5 K O OH HOOOHH .A38 0H: m0 0.H 00.H NH0.0 ammouw m0 xHHm O0N 0N: m0 0.0 00.0 No.0 vmumuucmuaou moHsn mmawuo NH 0.00 N.H0 NOHN.0 wmcHQEoo mH 00 cu nH 0NH.0 NH .H.O.O H 0.00 OOOH.O OOHO NH .H.O.O 0.0 0.00 OHON.O OHOHOO> OOHOO AmHoamHm>v mmwcmuo 00H 00 Cu 0N Hm.0 m.mH 0N HH.N 0.0 0H ~0.H N.m 0 0N.0 0m.H 0H: N0.0 Nm.0 0N: mm.0 00.0 om: mH.0 00H amNOHHCS N 0H: n.00 HH.0 mmmsmuo HOOHOHOOOOOO 00H omH OH 00 m.m mH.0 mmmem mm NHN Cu Nm 00 OH 00 mm on 00.0 OCOHGO Nmm 000.0 0N 0mm 000.0 many .0ucoU\mHHo .mm mucmfiaou may N.o.E Q\ o x a 000m 1¥39 HNH ONH O OOHNHOO.O + HOO.O n O OO NOO.O HHO NO.NH OO OHH.O OOHOO OHHOH OOHH NNH.O OO OHH.O HHH OOOHHHOOHO HH O.H OOO.O OOHHOO HOOOOO OOH OO OH ON HO.O 0.0H ON HO.H O.N OH NO.H OO.O O NN.O OO.H OH- NO.O HO.O ON- O0.0 OO.O OO- OH.O OOH OOOOHHOO N OH- 0.00 HH.O OOOOOOO OOHOOOHH 3OHOO O.ON O.ON OH.O O OOHOOOHH O>OOH O.ON O.ON HO.O OOHOOHOO OO NHN OH NO OO OH OO NO.O OH ON.O OOHOHOO O OO H.H ON.O OOO O OH- ON O.H OO.O HOO.O OO O OHHO oOO ON- ON H.H OO.O ON ON OOO.O OH NN O.N OO.O NOO.O O O.ON O.H NO.O OOO.O OH- ON H.H OH.O OOO O OO O xHOO oOH ON- ON O.H NN.O HO.O ON O.HO H.H ON.O OOO.O OH O.OO O.H OO.O NOO.O .OHOOO OO O O OO H.H OO.H HNO.O NOOHOO OOOOHO .mm 2 3:95:00 H.HOH N.o.HHH O\ u x a 000m 00H HOH A140 mm HOH mm HHH OOH OO OH HO ON.O N.O ON OO.H 0.0 OH O0.0 OO.H O OO.O ON.O OH- OH.O OO.O ON- HH.O HH.O OO- OO.O OOOOHHOO N OH- O.HN NO.O OO OH OO HN.O ON OO.H OH OO.O O OO.O OH- OO.O OOHHOO OHOOHOO HOHOa NON.O ON- HH.O .HOH NO.H OO- OH OH- N.HO OH.O NHN OH NO OO.O OH OO.O ON HN.O OH ON.H O OO.O OH- OO.O ON- HO.O OOHHOO OHOOHOO HOHOa NH.NH OO- OH.O .HOH NHO.O OH- H.ON HH.O NHN OH NO OO.O OH NO.O OOH OOH.O HHH OOH.O vwzmuxumHm mmwm 050m mmm AuumHuumnv mummm NmHumn Hummm .vucoo\umuu5n usamwm .mm muGMEEOU .mQH. o\O.U.E Q\ 000m mmoous Nm .HOO NO ..OO as OO.O HOHO.O OO.O OHHOOHOO ONN0.0 Hm H mumnHm cu HOOHHHO-HmnH 0 00H :NNQ: 00.0 00N0.0 HOO NO ONOO.O 00.0 zuHmouom NNN0.0 HO H OHOOHO OH HOHHOHOO O OOH :OHO: OH.N ONN0.0 HOO NOH HOHO.O NO.O OHHOOHOO HONO O HOO H0 H mumnHm cu HmHHmuma 0 00H mH cu HH HH.H HNN0.0 vapwumemHm cHuowm 00H 00 cu 0N Nw.0 0.HH 0N Nm.H 0N.0 0H Hm.0 1O m0.N 0 0N.0 mm mN.H 0H: 00.0 mH.0 0N- Nm.0 mH.0 om: 0H.0 OOH OOOOHHOO N OH- OO OH.O HOOHHOOOO OOOHO wcHNmmuw szmm NH.0 mm wcHNomum m>on< 00 Hw.0 ammoum 00 NHN 0H Nm HH HH.0 AuHmv vaHQ mm Cu N0 0H.0 ummma mammHu up kumumamm H on 0N HN.O NmumNmH CH vmxuma 0 cu 0H NN.0 NOH H Nvmemom vcm meHmnm m.m- ou 0H HH 0N.0 Umhm-xumHm .vucou\mmmm .mm 2 mucmaaoo mOH N.O.E n\ o x 000m m0H H umm Nmm 00 cu mHu 0.0 NH.0 umm uoHumuxm Nm 0NN.0 mN mHH.0 HH HNm.0 NN H H: 0.0N mHN.0 m.HHH mwN.0 m.00H 00N.0 N.00 NON.0 0.mH HON.0 0.0m 0mN.0 0.mN 0mN.0 umm NN.0 m.HN mHN.0 ?H Nwwwuw EsHEmna Nme NammH 0.NH 000.0 ”M 0H H «mumnHm ou umHsoHvawauma O N.0 0.0m me.0 0.0mH NHm.0 N.NHH 000.0 0.0m mmN.0 0.0H 00N.0 0.NH NON.0 umm N0.N H.NH 0mN.0 vamuw EDHEmua Nme NammH 0.mH mmm.0 OH H .OHOOHH OH HOHHOOOO O N.O H.ON OON.O HHOO NOH H OOHOOO OHHOOHH .OOHOH O- OH N OO HH.O NOH H HHmEm H Cu 0 0m NH.0 00 0N ou 0N H0.0 masHm HO O H.H N.O... N O O OO.H Az-3 AGHE-moNHv wmummm 00H N.m0 mm 3mm 00H N.m0 chuum Hmoxm mm NHN oH Nm 0N H0.0 vmxoma NHomooH «mmcHuum NOH OO OOOOHHNO .OOOOOz NH OO HN.O NOH OOHOOO NHHOOHH .OOHOOz O HO OO.O OOOHOHOO mm NHN Ou NM 00 H0.0 050m Caduom mm 00N.0 mN NN0.0 chuw m 00N.0 mHaEmm Cu umHsochmaumm O H: mNN.0 NN .HOH NH.O OH- NN OON.O mm 0NN.0 mN HNN.0 HH MN0.0 0 000.0 chuw mHaEmm Cu HmHHmuma O H- 000.0 NN .HOO NH.O OH- NN OH0.0 OOH OOOH mm NHN ou Nm Om N0.0 Nm 00H.0 mN HmH.0 HH nHH.0 NN H- H.m 00H.0 umm .vHGOONHHom .mm 3:08:50 ..H O.H. N. o .E Q\ o 3 000m .wm ES no.0 :NAHQ: mmO0.0 ONHOH ONNO.O HOO OOHHO-OOOOHH HO OO.O NHHOOHOO OOH OH OH HH O.O NON0.0 .HOHOHO OHOHOO OOH O.N OH.O HOOOHOO HOOzO OOH OO OH OO O.HO N0.0 NOOHO HOOHH OOH HOH OH HO O.O OH.O OOHHHO OOH HOH OH HO H.O HH.O OOOHO O0.0H HH0.0 OO.OH OHH.O OO.O OOH.O HO OH.N OOO O mm OH.H NOO.O OH.N OO0.0 ON.O NHO.O OOHOO =ONO OOH HOH OH HO OH.O HHO.O OOHOHONOOO OO NHN OH NO OO NO.O OOHHOO AmmsHm> wmuumHmmv HOHe-OoOOOO OOHOOO HOH 0.00 OOO0.0 HOH HOH 0.00 NOOO.O HOHO-O.NOHO OOHOOO ONH O.OO HOO0.0 OOH ONH O.OO NOOO.O HOHa-HoNOHO OOHOOO NOH N.OO OOOO.O NN OOH NOH N.OO OHO0.0 OHOHHO HOOxO .wucOQNOmoumuom .mm muamEEoo .mmH N.O.E n\\ o x a 000m A145 0.0H 00m.0 O.HH HHH.O HOOHHOHHO HH =eOOO: H.NH OHH.O OOOOHOHO OHHOO OO NHN OH NO 0.0H OH 0.0H HH.O OH NH.O OOHO OOHOOOHO 3OHOO N.OO HH.O H mcHummum m>on< N.00 Hw.0 xumHn «mmHuumnmmmm OO NHN.OH NO. O.HN NH.O . OOHOHOO waHNmmum onmm 0H.0 O OOHOOOHH OOOHH ‘ _ 0.00 OO.O OOHOHOOO OOH H 0.00 OON.O HO>HH OOH H 0.00 OON.O NOOOHN HHOHOO NOH H OH OH ON OO ON.O HOHHOO OH ON ONO.O OOOOOO mmusm chuw OHOOOa OH HOHOOHOOOOHOO O“ OO OH OH 0.00 OH O.NO N.HO HHO.O OHHO OOO OHH O OHO OHHOOe OH OO OH OH ON OH OO N.OO OON.O OHOOOa OOH :Hmuw OHOOOE OH HOHOOHOOOOHOO 0* OO OH OH 0.00 OH O.NO N.HO OHO.O OHHO HOHHOHO OHH O OHO OOOO OO OO OH OH O.HN OH H.OO N.OO OON.O OHOOOE HOHHOHO NHUHaom .mm 2 mquEEou MOB N.u.E. n\ o x a 000m 1U+6 0.NH N.0H N.0N mm.m0 00N.0 N.0H m.MN 0.0N Nm.m0 NNN.0 m.NN m.HH 00N.0 OHH O.NN 0.0N NON.O OHOHxHe OOOOOOO chuoum N umw N 0NH- HNH.0 00- 00H.0 0m 0H- HOH.0 mm NHN ou Nm Nm.0 cu NN.0 uHmm mm 00N.0 MN 000.0 HH HH0.0 Am3m£ms masocmnuooaoV 0 000.0 mHomse Cu umHDUHwammumm o H- 000.0 NN .HOH NH.O OH- ON ONN.O MN 0N0.0 HH 000.0 AumHmm oEHmm mammov 0 000.0 chuw ou umHDUchmauma 0 H- HHN.0 NN .HOH NO.NH OH- NO ON.O OOeHOO 0.0 0mm.0 O.HH OHH.O AOOHHOHHO HH :Eoom: 0.NH mHH.0 umHHmnm N.0H 0NN.0 0.0H NOH.O NOOHHOHHO ONH :EOOM: O . N H md . O fiwflom .vucoo\mon . Hm muamfiaou ."H O.H. N. u .E Q\ 0 OH voom O OO.O OH- HO.O ON- OH.O OOHHOO OHOOHOO HOHOs NOO.O OO- OH.O HOH w. .HOH NHO.O OH- 0.0N H.O OHHHHO HHOO OOH OO OH HO OO.O O.NH ON OO.H 0.0 OH N0.0 H.H O O0.0 O.N OH- OO.O OO.H ON- OO.O N.O OO- NH.O OOH OOOOHHOO N OH- O.ON OH.O OeHHHO OO OH NO HH.O Or ON OH O NH.O mm NN H HOH NOO H- OH OH- O.H OH.O HOOOOHH HOOO OOH OO OH HO OO.O H.O ON OH.H O.N OH O.O OO.H O H0.0 ON.O OH- HO.O CwNOHwfifl o\o ONI OOH.O OO- OH.O OOH OH- H.ON NH.O HOOHH OOO O.OH O.HN H.ON OO.HO OHN.O N.HH N.HN H.ON NO.HO NNN.O O.OH H.HN N.ON HO.HO HNN.O H.OH O.HN O.ON NN.HO OON.O N.NH H.OH N.ON OO.NO NHN.O .OHOOO mHH H 0NH 0.NH 0.0N N500 HHN..0 \muzuxHE mwmmnmm . mm 2 3:95:00 ...H O.H. N . u .E Q\ 0 OH 6 000m £U+8. maumm N.Nm ou H.0H Hm.0 NOH H OOOHOOO NNO OH OOOHO OOtz O.O OH H.O OO.O N.OO OH N.ON O0.0 H.OH OH N.ON ON0.0 O.H OH O.ON NH0.0 O- OH O.NH HO0.0 NOH H OOHOOO NHHHOHH .OOOHO OOtz O.N- OH OH OO OH0.0 OOHHHOHOOHHO O O.OO HO.O HOHOHs O OO OO.O HOaaOO OO H OO OH NO ON.O HOOOOO Avmumuvxsmvv OOH HOH- O.O OH.O HOOH OO NHN OH NO OO OH OO H0.0 OH 0.0 HOOOHOO HOOHOHO .H HOOa OH OOOO HHH3 NON OH OO N0.0\N.0 u uamuaoo HHo Ax HomV AxumHum> cHoocHHv OO N NOO.HO n OHOHOHO HOHOO OO OH OH N.OH O.OO OHN.O HO.O OHOO.O HOOE HHO OOOONOO OO NHN OH NO NO O0.0 HOHHOO ON OH OO O0.0 ON HO.H HOH OH 0.0N N0.0 OEHHHO HHOO .0ucoo\aEHu:m .mm 2. muamaaou MOB N.o.E. 0\. u x a voom Afl+9 0m 0mm.0 OO NNO.O OO OHO.O OO N NO OO OO0.0 ONH NOO.O OOH OO0.0 OHH ON0.0 NNH ON0.0 HOH NO0.0 OO OO0.0 OO OH0.0 OO OO0.0 OO N HOHOH u NOOH .OHOz NO OOH ON0.0 OOOHHOHOO OOOHOOO OOH OO OH OO OO.O O.NH ON NO.H 0.0 OH N0.0 H.H O O0.0 O.N OH- O0.0 OO.H ON- OO.O N.O OO- H0.0 OOH OOOOHHOO N OH- O.OO OO.O OH OO NO.O .mEm 0.m\0H uLmea mwmum>m NOH H OHO .EHHH .OOOHO HHOEO H.O- OH O OO H0.0 NOH H sHHH .OOOHO OOtz H.N- OH O OH HO.O 005mm50mcs Numk NOH H ...me ©H\NH «mmNHm ww-HNH O.NI Ou N.N O.HH Hm.O .HuuGOU\mm..HHHOnH3mH-Hm .mm 2. mucmaaoo mOH N.o.E n\ u x a 000m A50 0NH H0m.0 00H 00m.0 0HH 00N.0 NNH 00N.0 HOH HON.0 0w NNN.0 00 NNN.0 00 HON.0 Ow Nm 00 00N.0 0NH mNm.0 00H HNm.0 0HH 0Hm.0 NNH 0Hm.0 HOH H0m.0 00 NON.0 00 00N.0 00 NON.0 O0 Nm 0N HNN.0 0NH NHm.0 00H NHm.0 0HH 0mm.0 NNH 0mm.0 HOH HNm.0 00 0Hm.0 00 00m.0 om 00m.0 O0 Nm 00 NON.0 0NH 000.0 00H m0m.0 NNH 0mm.0 .vuaoo Ow HOH 00 mHm.0 \mcoHusHOO mmouosm .mm mucmaaou NOH N.O.E x 0000 A51 mm NHN ou Nm m.0 umwsm 3mm 0H.0 whammmum ustHm umwa: «Hawam wmumvsoa mHmch H00.0 umwsm wmuovsba hHmaHm 0m0.0 mchuw 0mNHm mch HH0.0 maHmuw vaHm HmEhoz 0H0.0 whammmua ustHm nmvcs chHmuw m>HuwG HmEuoz H00.0 mchuw m>Humc HmEuoz 0H0.0 OO N OOHOHO OOHOOO HOH OH OOH OO0.0 HOOOO 0NH m0N.0 00H 00N.0 0HH 00N.0 NNH 00N.0 HOH NHN.0 00 0HN.0 00 HmN.0 00 NNN.0 00 N Nm 0H HNN.0 0NH HON.0 00H 00N.0 0HH 0NN.0 NNH 0NN.0 HOH HON.0 00 00N.0 00 NmN.0 om 0HN.0 .0uaoo 00 N Nm om 0NN.0 \maoHusHom mmouosm .mM 2 mummEEoo NOH N.O.E 0\ u x 0000 A52 no.0 0m- 00H cmuoumcs N 0H- NOH 00H 00 cu N0.0 m.m 0N N0.0 m0.H 0H 00.0 00.0 0 N0.0 HH.0 0H: 0H.0 HH.0 0N: mH.0 cmnouwcs N 0m: NH.0 00H 0H: HH.0 Nm mcoHusHOO Hmwnm 0NH 00m.0 NNH 00N.0 00 NNN.0 00 N coHumHucmucou N0.0m N.Hm 00N.0 0NH 0Nm.0 NNH HHm.0 00 NON.0 00 N coHumuucwocoo Nom N.Hm 0NN.0 0NH 0H0.0 NNH 0mm.0 m0 00m.0 00 N coHumHucmucou NON N.Hm HON.0 0NH 0mm.0 NNH mHm.0 OO NNO.O HOOOOOHOO 00 N coHHmuuawucoo NH.0H N.Hm NHm.0 mcoHusHom Hmwsm .mm 2 WUCQEOU “HO.H. a\O.U.E vH A53 N.NH 0H Hm.H 00.0 0 N0.0 mm.m 0H- mN.0 MN.H 0N: H0.0 HN.0 om: 00.0 00H swuoumca N 0H- 0H.0 N0m 00H 00 on on 00.0 H.HH 0N N0.H mH.N 0H 00.0 mw.m 0 NN.0 mm.H 0H- H0.0 0N.0 0N: 00.0 HH.0 0m: 00.0 00H ammoumcs N 0H: 0H.0 NON 00H 00 cu 0m N0.0 00.0 0N 0H.H m0.H 0H 00.0 H.N 0 0N.0 00.H 0H: 00.0 NH.0 0N: N0.0 HH.0 0m: 0H.0 00H cmmoumcs N 0H- 0H.0 NmH 00H 00 cu Hm H0.0 00.0 0N 0m.H 0.N 0H 0N.0 mm.H 0 00.0 N0.0 0H: 00.0 00H NN.0 0N: H0.0 NOH .0ucoo\maoHu:H0m Hmwsm .mm mucmEEoo hoe N.0H: 0\ o a 000m .ASH- 0N NNo.0 om 000.0 NN 0m mm N00.0 0N 000.0 cm 000.0 NN NHHOOOO =HOOHOOO<= OO OO NHO.O 00 H0.0 ouownoa waHNmmum 30Hmm HN NNH.0 OO OOHHOOHO O>OO< HN NON.O wcHNmmum sonm H0 Nam.0 mm wcHNwwum m>on< H0 NHN.0 :Hoo ummBm N.0 00m.0 O.HH NN0.0 HHOOHOOO OHN :EOOM: m ...VH NHH.O NAUHOMQ QENBW 00H 00 Cu 0N 0N.0 N.0H 0N N.m N.NN 0H N0.H 0.HH 0 No.H 00.0 0H: m0.0 00.N 0N: 00.0 0N.0 0m: H0.0 00H :mwouma: N 0H: 0H.0 N00 00H 00 Cu 0N 00.0 00H 0.HN 0N N0.N Nom .choo\mcoHusH0m ummnm .wm mucmEEoo mOH N.O.E n\ o x 6 000m A55 NM 00N.0 MN NON.0 umm NH.M HH NON.0 Nchuw Cu ustoHUamauma o H: 0M0.0 NN NmHomafi me MH- HN HN0.0 NM 00N.0 MN HH0.0 HH NN0.0 Hmm NH.N O NON.0 NchHw cu uwH50chmmHma o H- mMN.0 NN NmHomnE ummmum MH- HN NON.0 NM NOM.0 MN 00N.0 umm NH.N HH NH0.0 NchHw ou HmHHmuma 0 m M00.0 NN Numme ummmum MH- HN 000.0 mmxuna 00H 00 cu HM 00.0 M.N 0N mH.H N.M 0H 0N.0 H0.H 0 N0.0 H0.0 0H: N0.0 0N.0 0N: NH.0 ammoumc: N 0M: MH.0 00H 0H: 0.H0 0H.0 mmOumEOH 0N N00.0 00 NN0.0 NN 0M 0H 000.0 .0uaoo\ooumHOH .wm mucmEEou NOH N.o.E u x 000m 1&56 Om NHN oH NM M0 NN.O 0.0MH HON.0 H.00H HON.0 O.HN OON.O N.NH 00N.0 H.0H 00N.0 0.MN NON.0 aHmHm ou HmHHmuma 0 0.0N MNN.0 .HOO NN.O 0.0H HH0.0 0H H wamuw EDHEOHQ Nme cme N.0 MN 0M0.0 H.HHH MON.0 0.0NH NON.0 N.OoH 0NN.0 om OON.0 0.NH MNN.0 chHw cu umHsochmauma 0 M.HN 00N.0 .HOH NH.N O.OH OON.O OH H .OOOHO esHsOHO .OOH OOOH O.HH ON OON.O HOOO 00 mo.H om 00.H 0H 00.H OM 00.H 0M O0.H 0N ON.H 0H H0.0 0 H0.0 0H- 00.0 0N- 0H.0 OOHHOO ansHoO Hmums NOO.H 0M- HH.0 HOH .HOH NNN O OH- H.HO NH.O OOHOHOH .IIIIIII .mM 2 mucmafioo may N.0H: 0\ o x 000m A57 0.0N 0N0.0 0.0N 0H0.0 OHN :EOOM: ©.mm Nwm..O wwmumm N0 00 Cu MN NN0.0 0N Hamucoo kuma u 3 3H.0 + 0HN.0 ummLS 00H 00 Cu 00 NOM.0 00H 00H ou MN HH0.0 umm 00H N Cu MH HN.0 :me NN umm N0.0 H0 0N MH.0 NN umm N0.0 0H HN.0 mHm£3 wHN 0.00 HHO.H N000.0 NHN N.00 000.H NOM.0 0000.0 0NH 0.00 N00.H 00M.0 H000.0 0HH H.H0 000.0 0NM.0 N000.0 HOH 0.H0 000.0 NOM.0 0000.0 00 M.N0 000.0 0H0.0 0000.0 NM H.N0 000.H 0NM.0 N000.0 kum3 0w NHN Cu NM 0N H0.0 cemHam> 0w NHN Cu NM 00 HN.0 0w NHN cu NM NN N0.0 umHuau .choo\Hmm> .Hm mucmaBou mOH N.u.E n\ o x 6 000m .A58 Acumzupozv 00 M N0 OH 00 0.NH 0000.0 H .02 mnouHcmz HthoE NQEmvv 00 M M0 on 00 0.0H 0000.0 H .02 mnouHamz 00 M N0 OH 00 N.HH N00.0 H .oz mnoHHcmz 0N.H 0HM.0 00.H 0HM.0 0.H MMM.0 H.0H NOM.0 N.MH 00M.0 M.0H 0HH.0 HN :53? O. NH NHH . O 3822 OO O NO OH OO O.NH HO0.0 HOHHOOHH HH.0 NOM.0 N0.0 00M.0 0H.H NOM.0 0H.H 000.0 H.N HHM.0 0.N 0HM.0 N.H NNM.0 H.0 HMM.0 0.0 000.0 N.0H 00H.0 N.0H 0NH.0 0.0H 0NH.0 HN M.NN HOH.0 mwmumm .vuaoo\umm£3 .00 z mucmesoo may N.o.E Hu\ 0 x a 000m 1¥59 ONH OO0.0 HO OO NHO.O ONH ONO O NNH OH0.0 OO ONO.O OO O u HOH N OO OH0.0 NOnz O.OOH OO0.0 O.OHH OOO.O H0 00: 0.0 00N.0 3on H.O-HOE O.OH NNO.O OONO.O OHOOO.O N.NH OOO.O OON0.0 OHOOO.O O.OH ONH.O NHNO.O HOOOO.O O.OH ON0.0 OON0.0 NHOO0.0 HO NNH OH OH N.OH NHO.O ONOO.O OOOOO.O HHOO N O.O OO NO.O NOO.O OHOO.O H .Oz OHOOHOz .0u:00\000£3 . 00 00:08:60 .0 O.H. N. 0 .E Q\ 0 x 0 000.0 A60 g; Theorx of "probe" method for determining thermal conductivitz. (a). Equation for temperature T after time t at a point a distance r from instantaneous point source of heat Q (Btu) at t H o in a solid initially at To : 2 Q - /h t T-To_-. 3581. <1 (B1) QPc(nat) r2 = (x-x')2 + (y-Y')2 +(Z-Z')2 (B2) (b). Equation for temperature T after time t at a point a distance r from instantaneous line source of heat strength Q Btu/ft (at t = 0) in a body initially at To : w 2 Q -r /4ct , T - To = - 3/2 ii e dz §oc(flat) 2 [00%flat e-I‘ /’+at (BB) ri = (x-x')2 + (y-y')2 (B4) Note: /000 = k (BE) (c). Equation for temperature T1 after time t at a location a distance r1 from a continuous line source of heat strength Q5 Btu/hr-ft in an infinite solid initially at To 3 A61 2 Q. t1 -r1/l+c(t1-t') o e T1 ’ To = EFE’ £ t1 - t' dt' (B6)- Q' °° -u _JL e du T1 - To = 4—1l’K 2 I u (B7) rl/hat1 QO 2 = m E1 ('1'1/1441151) (B8) where the exponential integral w -u e du -E1(-x) H J‘ —-r— (139) x For sufficiently small X, B1 (-x) H 0.5722 +-£n x (B10) . Q3 2 1'? co 1 - T0 = -m (005722 + n Kart—J: ) (B11) (d). Similarly T2 at time t2 will be given by Q' E ri T2 - To = - Efifi' (0.5722 + n 55%;) (B12) (e). If AT = '1‘2 - T1 = temperature rise at location r1 between times t2 and t1 : A62 Q' t2 AT = "0k £11 (31') Q' t (BIB) 2 k =3 TTZT an (F5) 12; Finite difference methods for transient heat conduction. (a). Forward difference approximation 3 n-é n n+§ qn-i "'"> <"" (111+;- <-—-‘Ax -—> <--.Ax -—H> Consider the node n above. The energy balance equation for this node gives a heat in - heat out = increase in internal energy- i.e. qn-é A ' qn+% .A = qint. energy (Cl) _ 3 6T (A-AX) qint. energy ” /° 3% - T: = f0, T A AX (CZ) t t Cindi A 35% (Tn_1- Tn) (03) A63 k A t qni-QA = Ax ”n+1" Tn) (CM Substituting (C2), (C3), (CHI) into (Cl) we get : - t t t Tt+l Tn-l + (M-Z) Tn + T 1 (C5) n = M where M = sz/aAt. For stability it is required that M Z 2. (b). Backward difference approximation: Consider node n above again. The backward difference approximation can be written as follows : Tt+1 _ Tt e n 41 Clint. energy = lo. t AA]: (02) k-A t+1 t+l qn-é A ..- Tx (Tn _1-nT ) (co) kA t+l_ Tt+1 t (n+2) Tt+1- (T§+1+T:'1) Tn O _____1L____1___.i_ (08) The main advantage of this approach is that equation (08) is stable for any positive value of M. However, more A6h calculations are needed than for the forward difference method. since it is necessary to solve as many simultan- eous equations as there are nodes. The forward and backward difference methods have approximately the same accuracy. (c). Crank-Nicholson procedure consists of adding the backward and forward difference approximation : - t+1_ PGJIT— QEiLT-J 2 Hail—Ts.— .- [ix t k(Tn-.1" -T; ) Tn ) +k (Tt “*1 ( C9) (1- ) Q AAX 7=é Note that when -?H 1. (C9) (08) H Backward difference approximation 3 7H 0, (C9) (CS) H Forward difference approximation; Equation (C9) can be redwritten as : tHl t+1 t+1 - t t Tn_1 - (24.210113n + Tn+1- -[rn_1 ..(2H2M)T;:L + T ml] (010) The method involves the solving of a set of simul- taneous equations of the form (C9),(ClO). In general, the Crank-Nicholson procedure gives considerably more accurate results than either the forward or backward difference A65 approximations. 2; Finite difference method used by Matthews (1266), This approach is essentially the same as that of Beck's (1966) non-linear estimation procedure. The method is outlined below a 2 (a) determine the magnitude of Ax from M H 3%; 3 (b) at the first and last node a programmed time step (c) (d) (e) (f) is used to calculate temperatures with respect to time 3 Tt, thl _______ Tt+1 where (t+1) - (t) H programmed time step. (t+1) - (t) = time interval of data. Assume an a. and calculate temperatures at each node from the equation - +1 . Tt+l __ Till-1 + T311 + T: (”'1’ n ‘ n+1 The method of least squares is used to minimize with respect to c the difference between the cal- culated temperature and measured temperature. The change in a is calculated, added to the originally assumed a and a new value for a is obtained. The new value for a is used in an iteration pro- cedure to determine a more accurate value for c. A66 Assumptions Made 3 (1) Initial and boundary conditions of the body were: T (x,0) 0 T (1.13) 2: T1 (ii) One-dimensional heat flow. (iii) Constant a with respect to time and temperature. (iv) The last node was at x a ”a §L_ Non-linear estimation procedure. Beck (1266) (a). Let c1 H /0c (El) (b); For one-dimensional heat conduction.with constant k s 2 ké—g- H cl-g—E- (E2) 6x (c). To determine k and c1 we minimize the sum of - squares function F with respect to k and c1 for n thermo- couples and the time of the experiment, tm : 1 n tn 2 F(k.c ) H 2 I (TJ(t) - Te'3(t)) dt (E3) 0 321 where TJ(t) is the calculated. Te,3(t) the experimental temperature. (d). Beck used the Taylor series iterative procedure which assumes at each step that the temperature is a linear function of k and c1 x T(k.c1) H T lk 010 + TkAk + Tcl Ac1 0 where 6T 1 T :3 '6'1'!‘ (2k,0 ) 01 k 0' o T = $21 (kficl) 1 6c co.ko AK :3 k - k0 A01 = 01 - 00 A67 (E#) (E5) (E6) The properties kO and c: are the zeroth estimates of k and 1 co . (e). (f). 9: 6k When F is a minimum 3 g: 601 F with respect to k gives a or» Wu! H 2 GP 6k Similarly 9: 6c where NT llbdb J 1 -Ak + T 7”- T + T 3 1 k,J 1 o koco 1 2(NTfiAk + (Tk,Tcl)Ac + 1k) 1 2(NTOIAc + (Tk,Tcl) k + Io) n tm 2 I (T )2 3H1 o k’3 dt (Sol - T ) T O .3 63 kOJ (E7avb) Substituting (Eh) into (E3) and differentiating dt (E8) (E9) (E10) (E11) A68 n m 2 NT 1 H z I (T 1 ) dt (E12) 0 3:1 0 0 .J < > 3 1mm T M (313) T ’T = . t k c1 3H1 o k'3 c1 J 11 tm 1k H 3:1 g Tk.3 (TJ - Te,3) dt (Elu) a: in I H T 01 3:1 0 01.3 (TJ - Te'J) dt (E15) (8). Equations (E7a5b) are to be solved simultaneously to give the corrections Ak, Ac1 3 . ’ l H - (E16) T T NT Ac I ( k! 01) 01 NT (h). Improved values of k and c1 are then given by 1 1 1 (E17) H ko +Ak c = 00 +150 kl A69 2; Theor: of heati in a fin te c l nder as a 1 ed b Babbitt . The temperature at any point in a cylinder with a line sorce of heat along it's a318, initially at uniform temperature. with a final temperature distribution inde- pendent of the time and dependent only on the radius (i.e. finite convection at surface) is given by x - 2 - t T H 2 .An Jo (mnr)(l - e Eng ) (F1) n21 I .- mn s are roots of the equation Jo(mna) H 0 An is a constant given by AnH BZmn31(22mna)+C[anl(:na)Zna-;2J () F2 %7’J1 (Inn 9.)2 B2 and C2 are constants found from the final temperature distribution 8 Tr(s.s.) H B2 + C2 log r (F3) Also. Tr(s.s.) is given by 2 I R 12 Tr(8.S.) = We“ -§"—£n 1' (Fit) Tr(s.s.) steady state temperature at r. A70 Therefore k can be calculated. knowing either B2 or CZ. no The first part of the expression for T (i.e. 2 A J (m r)) n=1 n o n is simply the equilibrium temperature (at t - 'w). The second part can be approximated by choosing t sufficiently large that only the first term gives a good approximation. Measuring the temperature at a known radius and selected time t allows a to be calculated. QL’ Standard errors for thermal ro rties of cherr flesh. refiorted bi Parker and Stout (:535). (1) O H 0.900 + 0.0051(61) + 0.020 “e i 0.060 (2) c H (5.320 .. 0.0159 sl)10"3 2': 0.1116 x 10‘”3 (3) k H -0.275 - 0.0009 s1 + 0.280,“ + 0.327 c 1‘ 0.002 s f' H grams/c.c. S H % soluble solids. W H flesh weight of each cherry (grams). A71 20mm: 0 Ham? .00:0H0 :00300n N030Hz .90300 8090 N0 .0000 .00000 0:» 3090 000809 p0Hm owHOH 0 0o :0H00m .UHHom owH0H 0 00 00000 HOHO OHHsHasH 19:0 0:0 aoa0 90H00H0:00900 03» 0o :oHpoom .900:HH00 H0 00:0pmHn isepsa so 0000 0:0 900: :onom 00H:H0:Hnampa0:a pdDN 0000900 aon0 .0HHom owa0H .oHHom H O.H .. 0 0 00:0pmHQ 0 00 0000 0:0Ha 0 H00: :on0m 00H:H0:H H300 0 . 0 .900080H0 0.0H 00 Hmwvm N 0mwvo .Hop:00 0000H:w0a 00 90090 0800 0:0 0 p 0HH005000 0o 0H 300:0H 000:3 H00:HH00 900:HH00 00H:Hm pdnm .o:0 aoa0 .00:0 0:» 00 0:0 .s00:HHho H Hm N Am 000 0 .mHN0 :0 H00: H00:HH00 0:0H 0 0o :OHmom 00H:H0:HIHa0m 0 .00:0 0000H00:H ans 900:HH00 H.mm 00 00H:H0 so .00:0 :009 aos0 000: 0HN0 :0 (on 900:HH00 0:0H 0 00 :onmm H00:HH00 00H:H0:H Np :0>H0 000090 :H p:Hom 0:0H0>H:00 H00Hmasm 0:02 prssom 0:0 :omHo mUHHom :H 0:0H0H0:00 :0H0000:00 p00: 0:0Hn0b 900 0:0H00Hom A72 Hwy; 0 HOO.O; a HMO-.3 TOOPNOHE .le .O IINr.E a HBO-H»: O HO! 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L .339 -1139 _ ~R§e 0(6) = Ale +A2e +430 ..... 2 An = Bn J1 {Rn} P(e) = probability integral 9 . 2 = "L I 8-x dX «Fr 0 e -.= -—d-— x = integration variable. HICHIGQN STQTE UNIV. LIBRARIES 293100214968