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J 4 4 . .4 4 .. ... .. o .4. 4 l l .4 . .. 4 44' . 4 .......o.oo...¢ .J.4.,.. a“.!44r.4.\¢.. 434 4....49~...4444.4..4.o¢4.. .4 4.4.4.4.. ‘ra T4..&J¢Lr¢\4.‘.\..-L§ LIBRARY *5 Michigan 53%“: UnichS‘itY {an ABSTRACT MOISTURE ABSORPTION BY FREEZE - DRIED MEAT CUBES Freeze-dried beef cubes were rehydrated isothermally. A mathematical model recently employed by Young (1968) was used in conjunction with a non-linear estimator technique to compute the diffusion coefficient as a function of moisture content. The three-dimensional diffusion equa- tion was solved numerically by an alternating direction explicit procedure. Approved , hajogrofessb /E_5: 57. Department Chairman MOISTURE ADSORPTION BY FREEZE - DRIED MEAT CUBES Gonzalo Roa A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER.OF SCIENCE Department of Agricultural Engineering 1969 ACKNOWLEDGMENTS Thanks to the members of the Drying Research Group of the Agricultural Engineering Department, especially to Dr. Fred W. Bakker-Arkema, David R. Thompson and John R. Rosenau. The author also wants to express his acknowledgments to the Rookefeller Foundation who sponsored his program. Special recognition to Dr. Carl W. Hall, Chairman of the Agricultural Engineering Department for his collaboration to the author's program and his contributions to the development of the new Agricultural Engineering Programs in Latin America. This thesis is dedicated to my parents, Pedro and Zoila. ii NOTE This thesis has been written in the form of a technical paper. It has been presented in its present form (minus the Appendix) as ASAE Paper No. 69-893 at the 1969 Winter Meeting of the American Society of Agricultural Engineers, Chicago, Illinois, December 9-12, 1969 by Gonzalo Roa. F. W. BakkeroArkema Major Professor iii TABLE OF CONTENTS ACKNOWLEDGMENTS. NOTE . . . . . . . . . . . . . . . . . LIST OF TABLES . . . . . . . . . . . . . . . . . . . . LIST OF FIGURES. LIST OF SYMBOLS. . . . . . . . . . . . Chapter I. INTRODUCTION. II. EXPERIMENTAL PROCEDURES . A. Preparation of Samples . . . . . . B. Conditioning of Samples. III. ANALYSIS. . . . . . . . . . . . . . . . . . . A. Mass Transfer Equation . . . . . . . . . . B. NUmerical Solution . . . . . . . C. Estimation of Parameters . IV. RESULTS . . . . . . . . . . . . . . . . . . . . . 1. Experimental . . . . . . . . . . . . . . . . . A. Moisture Adsorption Curves and Temperature History of the Cube . . . . . . . . . . . B. External Resistance. . . . . . . . . . . . C. Isotherm . . . . . . . . . . . . . . . . . D. Input Parameters . . . . 2. Numerical. . . . . . . . . . . . . . . . . . . A. External Resistance. . . . . . . . . . B. Diffusion Coefficients . . . . . . . . . . V. SUMMARY. . . . . . . . . . . . . . . . . . . . . . BIBLIOGRAPHY . . . . . . . . . . . . . . . . . APPENDIX Digital Computer Program iv page ii iii vi vii UN GUI-l5 41‘ 10 10 ll 12 12 13 l4 l8 Table LIST OF TABLES Numerical Results with Optimized Variable Diffusion Coefficient ft"2 = 2 - __ Deff 10.961 (M) 0.16621 0!) + 0.002011 hr .. Numerical Results with Optimized Constant Diffusion Coefficient ft? 1.9535 hr . LIST OF FIGURES Apparatus for equilibration of freeze-dried beef cubes. Experimentally determined moisture adsorption curves for freeze-dried beef cubes. Experimentally determined temperature curves during the adsorption process indicated in Figure 2. Equilibrium moisture content versus relative humidity (Isotherm) for freeze-dried beef cubes. Adsorption curves for points located in the main dia- gonal of the cube. Moisture distribution within the cube at different times. Optimized polynomial for the diffusion coefficient as function of moisture content. Comparison between the numerical solution (with optimized diffusion coefficient polynomial) and experimental points. vi 3.11. . x,y,z At AX:AY9A3 LIST OF SYMBOLS diffusion polynomial coefficients. isotherm polynomial coefficients. Biot number 1b , m vapor water concentration,'EE3 ft2 diffusion coefficient, h;— lbm density, 2:3 porosity, decimal grid spacing, ft . . ft convective mass transfer coeffic1ent, h; space sub-indices Colburn correlation factor characteristic length of the cube, ft _ . lbm m01sture content, dry basis, ft: Molecular weight of water “ 18 lb water rzpor partial pressure, TEE lb water vapor partial pressure at saturation, Egg relative humidity, decimal lpg'ft . Q ~ L ' universal gas constant, 1,545 lbmfmole -'R time, hours temperature, ”R space coordinates time step, hr grid spacing, ft‘ I . INTRODUCTION It is well'known that dehydrated food products exhibit an optimum moisture content for storage (Rockland, 1969). At moisture levels below the optimum, autoxidation type deterioration reactions decrease the product stability. At higher levels, microbiological spoilage may occur. During a study of the hygroscopic and textural behavior of freeze-dried beef cubes dehydrated to 0.5 percent moisture content (Heldman and Bakker-Arkema, 1969) it became necessary to find out how long the cubes could be subjected to a certain temperature-humidity environment before they had reached this optimum (or any other) moisture level. Although much research has been conducted on the desorption rates of moisture by foodstuffs (van Arsdel, 1963) relatively little has been published on the rates of adsorption of water by foods. No information at all could be found concerning the diffusion coefficients for water adsorption by freeze- dried beef at very low moisture contents. A number of new simulation models have recently been proposed for moisture sorption and desorption in capillary porous systems (King, 1968; Young, 1968; Harmathy, 1969). For each one of the models, it is assumed that the mass transfer within the medium takes place in the gaseous phase. Young's model was chosen in this study for analyzing the rate of water adsorption within freeze-dried meat cubes, mainly because it does not require knowledge of as many basic product properties as the King and Harmathy simulation models. ' This study was partially supported by the U. S. Army Laboratories, Natick, Massachusetts. Contract No. DAAGl7-67-C-Ol65. Young's model for simultaneous heat and mass transfer in porous, hygroscopic solids (Young, 1968) is a modification of a model first pro- posed by Henry (1939) for moisture flow in a textile package. Henry's analytical solution of the model assumed a linear dependence between the equilibrium moisture content and the vapor concentration and temperature of the environment. Also, the moisture diffusion coefficient and the thermal diffusivity were assumed to be constant during the mass transfer process. The agreement between Henry's experimental and theoretical data was only fair probably because of the simplifying assumptions which were necessary to make an analytical solution possible. Young's numerical solution of Henry's model did not require that the moisture diffusion or theimal diffusivity coefficients remain constant. Unfortunately, however, Young's model cannot be properly evaluated since no experimental data was presented by the author. Also, Young's solution is one-dimensional and was solved with forward and central difference methods rather than with the more stable and faster alternate direction type procedures (Allada and Quon, 1966). II. EXPERIMENTAL PROCEDURES A. Preparation of Samples Commercial low fat beef was cooked at an oven temperature of 325° F until the center of the roast attained a temperature of 160° F. The cooked roast was wrapped in aluminum foil and frozen to -20° F within 48 hours. After freezing the beef roast was cut into approximately 2% inch cubes and the samples were dried to about one percent moisture content in a freeze-drier with "plate" temperature of 145° F and an absolute pressure of 0.5 mm of Hg. Conditioning of Samples The freeze-dried beef cubes were conditioned at 100° F to differ- ent moisture contents at four relative humidities (18.4, 40.0, 60.0, and 80.0 percent). Nine cubes were arranged in a wire tray for each conditioning process. Orientation of the meat fibers was randomized. The tray was placed in an insulated chamber. Conditioned air was blown through the chamber by a conditioning unit capable of controlling the air dry-bulb temperature to within t 3/4 °F and the relative humidity to within i 1/2 °F. The air velocity was controlled by a variable speed fan; the stream was directed perpendicularly to the plane of the trays. A laminar-flow element was used to measure the free stream velocity (between 6 and 21 ft/sec.). The air and cube temperatures were measured by 20 gage copper-constantan thermocouples. No tempera» ture gradients within the cubes could be measured during any of the experiments. An electronic hygrometer was used for measuring the relative humidity of the air. A schematic of the experimental set-up is given in Figure l. The moisture adsorption history was determined by weighing the wire trays at ten minute intervals. This procedure was continued until a constant weight was achieved.. The initial and final moisture contents were determined by measuring the weight of the final products after they had been placed in a forced convection air oven at 170 'F for eighteen ,2. hours. An additional experiment was conducted to investigate the differ- ences in the convective mass transfer coefficient resulting from differ- ent air flow patterns on the three different cube faces. All but one of the faces of each cube were made impermeable by wrapping the cube with aluminum foil. The cubes were positioned in three trays, each tray containing nine cubes. The position of the permeable face with respect to the air flow direction was different for each tray; the top faces were permeable for tray number one, the side faces for tray number two, and the bottom faces for the third tray. III. ANALYSIS Mass Transfer Equation Henry's (1939) isothermal vapor diffusion equation in fibrous material is of the following form (Young, 1968): .9. 92.5.. 19.9.. fig :99. _ 9! f [ ax (Deff ax ay (Deff by) 32 ( eff ea) 3 f at + (1 f) ds at (1) The left-hand side of the equation represents the moisture flow rate to the differential volume; the right—hand side represents the moisture sink composed of porous spaces and solid fibers. The following assumptions were made in deriving the equation: 1) the material is isotropic; 2) mass transfer in meat cubes is isothermal at low relative humidities (See Figure 2); 3) porosity is a35umed not to change during the process. (This is approximately correct for freeze-dried products.); and 4) there is instant local equilibrium between the condensed water and the vapor phase. The last assumption permits the use of the experimental isotherm curve (Figure 3) for expressing the local moisture content as a func- tion of the surrounding water vapor concentration. Five data points were fitted to a polynomial of the form. M = a 03 + B 02 + B c + B 1 2 3 4 Equation (1) can then be rewritten as: a. .2229. 92:2. 529,... -99- 5x (Deff 5x + By (Deff ay) + 52 (Deff 62) (K1 c + K2 c + K3) 5: (2) Wherf‘ (3.0) (B1) (l—f) (d5) Klr f (2.0) (32) (l-f) (d8) K' = 2 f K. = f + (B3) (l-f) :1S 3 f Since the convective external resistance to mass transfer is negligible (see next section) fixed concentrations on the cube faces were used as the boundary conditions. It was assumed that the initial moisture concentration within the cubes was uniform. Numerical Solution Equation (2) is a non-linear partial differential equation for which no analytical solution is available. A numerical method developed by Allada and Quon (1966) was selected to solve the problem. The tech- nique called an alternating direction explicit procedure (ADEP) is stable and accurate. Equation (2) was solved for each grid point assuming 1) D is a constant, and eff 2) Deff is a function of the local moisture content. In the second case, assuming equal grid spacing (5x = Ay = A2 = h), the computational formulas are: (n) 3 Y1 1 (n) (n) (n) (n-l) Ci.j.k Y1 + 1 [he (Ci-1.1.k + Ci.j-1,k + Ci,j,k-1 + Ci+1,j.k (n-l) (n- 1) - (n- 1) Ci,j+1.k +C C.j.k+1) + (i 3) Ci k] (3) and Y . C(n+1) g. 2 1 __ (n+1) (n+1) (n+1) (n) 1,3,k gg-qrf [he + c + c + c (C1+1.j.k 1.3+1,k i.j.k+1 1-1.j.k + (n) c (n) )+ C__ _ 3) c (n) J . . + . . laJ'lak 1:3:k 11",j k (4) where: Y = l 1 (n- 1) (n 1) 3 K1 Ci,j,k +K2 Ci,j,k +K3 +— h2 Y = 1 2 (n) (n) 3 (K1 Ci,j,k K2 C1 ,,j k +K3 + h? and (3) (B ) (1-f) (as) K = ‘ 1 (f) (At) (D ff ) 1.j k (2) (B2) (l-f) (d ) K a: S 2 (f) (At) (Deff > i.j.k f + (B3 ) (l-f) (d ) K - s 3 ' (f) (At) (0 ff ) i.j.k . .. -1 _, \ ' l I ' ‘, .f l “u - - 1?: .. .... ... . . f. ...... - _.__ -'- D (M )9+A (M )+A A . . . . e£fi,j,k l 1,3,k 2 1,3,k 3 (n-l) = (n-l) 3 +. (n-l) 2 + (n-1)_ + Mi,j,k Bl (€1,333 B2 “1,3319 B3 (€1,313 B4 (n) = (n) 3 + (n) 2 + (n) + “1,3,1. Bl (Cum) B2 “1.1.19 B3 “1.3.19 B4 (5) Sub-indices refer to the three coordinates and super-indices to two different time levels. At time level 5; (even time step) the computation starts in one corner and continues through all the grid points until the corner diagonally opposite the starting point has been reached (Equation 3). Equation (4) applies for time step ggj;l (odd time step). The starting point is the final point of the previous time step. The truncation errors are minimized in the ADEP Procedure. The accuracy of the complete program was determined on the simple case: 5x2 eye 322 at with boundary conditions C = 0 at x = y 3 x = 0 and initial condition: C = 1.0 at t = 0 where: Ax = Ay = A2 = h = .16 ft At = .005 hr JAE... = 0.2 (AX)2 Results were compared with tabulated values from Schneider (1955). Differences were noted only in the third decimal point. Only one-eighth of the cube was considered in order to save computer time. Therefore, three insulated faces had to be considered as boundary conditions. A minimum number of (6 x 6 x 6) grid points was required to obtain results with a maximum error of less than one percent. The computational speed of the model was 3,300 node evaluations per second on a CDC 3600 computer and 5,300 on a CDC 6500. Estimation of Parameters Solution of the system will reproduce the experimental data, if the assumptions in deriving the partial differential equation are justified and if the correct values of the parameters are chosen. For evaluating hD and De a Non-Linear Least Squares (GAUSHAUS) ff estimator model which uses iterative linear approximations was adopted (Meeter, 1965). The main computer program reads in the experimental data and initializes the parameters. The Allada-Quon model was written as a subroutine and generated the function values (moisture content at each elapsed time.) GAUSHAUS was then used to minimize the square differences between the computed and experimental values. It was necessary to fit the experimental data with polynomials of sixth degree I. because of the time step requirements of the numerical method, special attention was given to the initial parameter estimates. In face, the linear approximation technique did not give the true results of bad guesses are made. For the case where Deff was considered constant, a set of several values of different orders of magnitude was considered as the initial estimates. In this way, local minima of the sums of squares of the differences between the numerical solution and the experimental data were discarded. The global minima was selected as the one which presented the minimum sum of squares. This constant value was used as a guide to estimate the coefficients of the diffusion polynomial. Similar procedures of elimination were used in this case. IV. RESULTS 1. Experimental Moistpre Adsorption Curvgs and Temperature History of_the Cube Figures (2) and (3) present the average moisture content and the product temperature of meat cubes adsorbing moisture at four different relative humidities at an air temperature of 100° F. Moisture equilibrium is reached rapidly compared to other biological products, due to the high void fraction of the freeze-dried beef. Equilibrium times are larger for high relative humidities because the amount of water transferred is larger. The increase in temperature due to the heat of condensation is noticeable only for the 40, 60 and 80 percent samples. Adsorption at 18.4 percent relative humidity can be considered as an isothermal process. B. - 10 - External Resistance The experiments with impermeable faces simulate: 1. Ewo dimensional stagnation flow on the bottom face (when this face is the only permeable one). 2. air flow over a plate for the lateral faces; and 3. turbulent flow on the top face. In general, for each of these air patterns, different convective mass transfer coefficients define different external resistances. The experimental results showed that the rates of moisture adsorp- tion were approximately the same for all three cases. Equal or negligible convective resistances are deduced for each case. Cubes subjected to different airflows, presented slightly different rates of adsorption, but the differences were attributed to heterogeneity of the product and not to differences in external resistances. Isotherm Equilibrium moisture contents at different relative humidities were fitted to a third degree polynomial: = 3+ 2+ + M BlC BZC BBC B4 ~1.9211 x 10’2 on 11 3.2166 x 10‘1 or: II -4.0934 x 10'3 w ll 4.5648 x 10‘5 bi fl - 11 - A linear least-square computer program was used for the fit. The sum of squared differences was 2.38 x 10-6. The isotherm is shown in Figure 4. Ngoddy (1969) in his study of a general theory of moisture adsorp- tion in biological products presented data for isotherms at the same temperature for freeze-dried meat powder. Good agreement exists between the two isotherms although Ngoddy's moisture values are slightly lower. The differences are the result of different structures of the meat. Input Parameters The product constants assigned to the computational formulas (3) and (4) were obtained from a laboratory analysis: Fiber density and porosity were determined by decomposing randomly selected samples of lean beef. The known specific gravities of the components (water, protein, fat and ash) were assigned to the partial weights in order to obtain the partial volumes. Results of this analysis were: porosity (f) = 0.63 ’ = .29. den31ty (ds) 84.0 ft3 Bulk density of the product was also obtained by direct weighing of known volumes of meat product. The resulting bulk density was 1b lb PZs.‘ This value correspondS to a fiber density Of 67°12 PEG (assuming a porosity of 0.63). The average value taken for the 25.0 computations was d8 = 75.56 lb/fta. .1 ...... - 12 - The vapor concentrations of the boundaries were calculated from the known relative humidity and temperature of the air. It was assumed that water vapor behaves as an ideal gas. Thus: (pw) (MW) C ' (Roi'(T> .but, pw = (pw S) (R.H.) (p ) (M ) “’8 w (R.H.) so, C = (R0) (To) Replacing values c = 2.8448 x 10‘3 (R.H.) Initial vapor concentrations were calculated from the known initial moisture contents and the isotherm polynomial. 2. Numerical External Resistance The Allada-Quon model for moisture adsorption was initially written to take into account different convective resistances over each face. The equivalent convective mass transfer coefficients (hD) along with the average diffusion coefficient were estimated by the GAUSHAUS subroutine. Two different sources were used in making the first estimate of hD: 1. by using the Reynolds analogh and the Colburn correlation factor (jD) (Rohsenaw and Choi, 1961); and 2. from tabulated values from (Barker, 1965); Sets of different orders of magnitude (10.2 - 10—6) were given to the first estimate of the diffusion coefficient. The result gave a Biot number hD L D eff B. = 1 > 400 where L is the characteristic length of the cube (L = .0416 ft). Thus, the external resistance could be considered negligible and a fixed concentration at the walls was used as a boundary condition. Diffusion Coefficients The following results refer to the adsorption process carried out at 100° F and 18.4 percent relative humidity. The diffusion coeffi- cient polynomial (5) was optimized by the non—linear GAUSHAUS esti- mator to obtain the minimum sum of squares. The final optimized polynomial was: Deff = 10.961 (M)2 - 0.16621 On)-+ 0.002011 The results of the simulation are shown in Figures 5, 6, 7 and 8. Numerical values are also shown in Table 1. Figure 5 shows the simulated adsorption moisture content for points located on the main diagonal. Figure 6 illustrates the distribution of the moisture for each of the points on the main diagonal at different elapsed times. Figure 7 presents the optimized diffusion coefficient. Fish (1958) published similar curves for potatoes. King (1968) explains that the increasing value of the coefficient is a logical behavior since - 14 - in his adsorption model, it is proportional to the inverse of the isotherm slope (Figure 4). At very low moisture contents the isOv thermal slope decreases when the moisture content increases; this causes the diffusion coefficient to increase. In Figure 8 and Table 1. the experimental points and the simulated adsorption curve are compared. The sum of squares is 0.1371 and the confidence limits are shown in the table. value was optimized using the same method. ff , . -3 ft2 The numerical value was 1.95 x 10 h;— A constant De ; the sum of squares was 0.138. The results are presented in Table 2. The constant De value gives approximately the same results as ff these obtained by the polynomial---compare the differences between computed and experimental values in Tables 1 and 2. In general, for larger ranges of relative humidity, a constant value for D is not expected to fit the data as well as a polynomial. During the optimization process, five interactions and twenty-seven evaluations of the numerical programs were necessary. The execution time on the computer (CDC 3600) was 5% minutes. V. SUMMARY A stable, fast and accurate nwmerical technique was employed for solving a non-linear diffusion equation which describes the moisture adsorption in a cube of freeze-dried beef. A non-linear leastosquare estimator of parameters was used to determine the mass convective and diffusion coefficients. The diffusion coefficient was assumed to be a quadratic function of the moisture content. The method of solution is - 15 - general. The use of the multi-dimensional method makes it useful in solution of problems of irregular shapes. The method can be employed to solve systems of simultaneous heat and mass transfer without the computational difficulties of implicit techniques. 4. D4" 9'. . Ilnl 9.04 _ l6 - TABLE 1 NMMERICAL RESULTS WITH OPTIMIZED VARIABLE DIFFUSION COEFFICIENT 3 ft2 8 2- - Deff 10.961 an) 0.16621 (MJC) + 0.002011 hr mm . Numerical Average Difference Between Approximate Confidence Time Moisture Content Experimental and Intervals for each Numerical Values Nemerical Value hr %, d.b Z, d.b Z, d.b O .684 .110 .684 .684 .1 1.515 - .247 1.600 1.429 .2 1.811 - .173 1.878 1.742 .3 2.004 - .086 2.056 1.951 .4 2.150 — .007 2.196 2.104 .5 2.271 .041 2.313 2.228 .6 2.375 .065 2.419 2.331 .7 2.469 .073 2.514 2.423 .8 2.552 .067 2.599 2.506 .9 2.627 .055 2.671 2.584 1.0 2.694 .041 2.731 2.658 1.1 2.753 .027 2.784 2.723 1.2 2.806 .016 2.833 2.778 1.3 2.851 .008 2.878 2.824 1.4 2.891 .003 2.919 2.863 1.5 2.926 .002 2.954 2.898 1.6 2.957 .003 2.985 2.929 1.7 2.984 .006 3.011 2.956 1.8 3.008 .010 3.035 2.980 1.9 3.029 .013 3.056 3.001 2.0 3.048 .017 3.076 3.020 2.1 3.064 .019 3.093 3.040 2.2 3.079 .019 3.108 3.050 2.3 3.092 .016 3.121 3.063 2.4 3.104 .013 3.134 3.074 2.5 3.114 .007 3.145 3.084 2.6 3.124 .000 3.155 3.093 2.7 3.133 - .006 3.164 3.102 2.8 3.141 - .013 3.172 3.109 2.9 3.147 ~ .020 3.180 3.116 3.0 3.154 - .025 3.185 3.122 3.1 3.159 - .027 3.191 3.128 3.2 3.164 - .027 3.196 3.133 3.3 3.169 - .025 3.200 3.138 3.4 3.173 - .020 3.204 3.142 3.5 3.177 - .015 3.207 3.147 3.6 3.181 - .011 3.210 3.151 Sum Squares: 0.137 . .-- -‘.- v“... ...---- [Ill lu[l|linl‘ (I [III -17- TABLE 2 WE E *— NUMERICAL RESULTS WITH OPTIMIZED CONSTANT DIFFUSION COEFFICIENT = 1.9535 x 10‘3 35—52- hr. Numerical Average Difference Between Approximate Confidence Time Moisture Content Experimental and Intervals for each Numerical Values NUmerical Value hr 2, d.b. %, d.b. Z, d.b.‘ .0 .684 .110 .684 - .684 .1 1.419 - .151 1.426 - 1.413 .2 1.783 - .145 1.794 - 1.768 .3 1.961 - .038 1.981 - 1.942 .4 2.107 .036 2.131 - 2.083 .5 2.233 .079 2.259 - 2.206 .6 2.345 .096 2.374 - 2.315 .7 2.447 .095 2.478 - 2.415 .8 2.539 .081 2.572 - 2.506 .9 2.622 .061 2.656 - 2.588 1.0 2.697 .038 2.731 - 2.663 1.1 2.764 .017 2.797 - 2.731 1.2 2.823 - .002 2.855 - 2.791 1.3 2.876 - .017 2.906 - 2.845 1.4 2.922 - .027 2.951 - 2.893 1.5 2.962 - .034 2.989 - 2.935 1.6 2.997 - .036 3.022 - 2.971 1.7 3.027 - .037 3.050 - 3.004 1.8 3.053 — .035 3.074 - 3.032 1.9 3.076 - .033 3.095 - 3.056 2.0 3.095 - .031 3.113 - 3.078 '2.1 3.112 - .029 3.128 - 3.096 2.2 3.126 - .029 3.141 - 3.112 2.3 3.139 - .030 3.152 - 3.126 2.4 3.150 - .033 3.162 - 3.138 2.5 3.159 - .037 3.170 - 3.149 2.6 3.167 - .042 3.177 - 3.158 2.7 3.174 - .048 3.183 - 3.166 2.8 3.180 - .053 3.188 - 3.173 2.9 3.185 - .058 3.192 - 3.179 3.0 3.190 - .061 3.196 - 3.184 3.1 3.194 - .062 3.199 - 3.188 3.2 3.197 - .060 3.202 - 3.192 3.3 3.200 - .056 3.204 - 3.196 3.4 3.202 - .049 3.206 - 3.198 3.5 3.204 - .042 3.207 - 3.201 3.6 3.206 - .036 3.209 - 3.203 Sum Squares: 0.138 BIBLIOGRAPHY Allada, S. R. and D. Quon (1966) A stable,_explicit numerical solution 2; the conduction equation for multi;ggmeg§ional nonhomogeneous media. Heat Transfer, Los Angeles, Chemical Engineering Progress Symposium Series, 26, 64, 151. Barker, J. J. (1965) Heat transfer in pacggg beds. Industrial Engineer- ing Chemistry. 57,34, P.4. Fish, B. P. (1958) Diffusion and thermodynamics of water in_potato starch gel. In Fundamental Aspects of the Dehydration of Foodstuffs. Society of Chemical Industry, 143. Harmathy, T. Z. (1967) Simultaneous moisture and heat transfer in porous systems with particular reference to dgying. Industrial Engineering Chemistry. Fundamentals, 8, 2, 92. Heldman, D. R. and F.1W. BakkermArkema (1969) Investigatigg:of the energegics of water binding in dehydrated foods at very71ow moisture levels in rglgtion to quglity pgggmeters. Departments of Agricul- tural Engineering and Food Science, Michigan State University, East Lansing (Unpublished mimeo). Henry, P. S. H. (1939) Diffusion in absorbing media. The Royal Society of London. 171A, 215. King, C. J. (1968) Rates of moisture sorption and desorption in porous, dried foodstuffs. Food Technology, 22, 4, 165. Meeter, D. A. (December, 1965) Non-Linear Least-Squares (GAUSHAUS). Michigan State University, Computer Laboratory No. 0000087(Mimeo). Kgoddy, P. 0. (1969) A Generalized Theory of Sorption Phenomena in Biological Materials. .Michigan State University Agricultural Engineering Department. (Unpublished Ph.D Thesis) pp. 104. Rockland, L. B. (1969) Water activity and storage aggbility. Food Technology, 23, 10, 11. Rohsenaw, W. M. and H. Choi (1961) Heat, Masgg and Momentum Transfeg. Prentice-Hall, Englewood Cliffs, New Jersey, pp. 416. Schneider, P. J. (1955) Conduction Heat Transfer. Addison-Wesley Publishing Co., Inc. Cambridge, Mass:1 pp. 378. Van Arsdel, W. B. (1963) Food_Dehydration. Principles. Vol. 1. The AVI Publishing Co., Westport, Connecticut, pp. 66. Young, J. H. (1968) Simultaneous heat and mass transfer in a porous, hygroscopic solid. ASAE paper no. 68 353, presented at Utah State University, July, 1968. 7. A/P COND/T/ON/NG (AM/NCO) UN/T. 2. EQUILIBRA T/ON CHAMBER 3. LAM/NAP Alf? FLOW METER \\\\\\\\\ @ Figure 1. Apparatus for equilihrntinn vi- francs-dried beef ('Hl'kfi. — 0/0’ 0.8. CONTENT MOISTURE A—-————— Q «0‘ r- '\ EXPERIMENTAL A 60% RH. O E h— / <:) 6’9. 56 C) o S __ E] 40. /o C) 76.4 ‘7. Q Q' _' o G JV} 0- N o a air at .' 700 ”F O - 0 Q5 A O O Q T'— KO A0 W I . n O on \f " , o -/H @- 7 ¢"M'§'. Q .¢ (\i C") l L i J | 0.0 70 2.0 3.0 4.0 5.0 TIME - HOURS Figure 2. Experimenrnllv determined Muistdrc adsznwvtiun claw CS hi"€l: Cllhgis_ F01? frnwnzy-(iritcl CUBE TEMPERATURE— °F 96 700 702 704 706 96 94 EXRER/MEN TAL ——_‘ (30.0 °/o :9. H. “.__ 60.0 O/a A) H. ———-—‘ 40,0 °/o R, H, Jr\\ ————-‘ 78.4 .O/o RH. I I \\ \ air at: 700 °F \\ \ \ \\\ \\ \ \ \ '\.\‘\ y\“\~ ~\‘y““~ "if T‘“ “ \¥ f l l I L J l .0 .250 .500 .750 7.00 7.25 7 50 TIME — HOURS Figure 3. Experimentnlix d [LYHlHCd tompcratlxu lervcs chlrizn, tht a1bs.nqwti(ut pr wqus Wldileltvd in FWHI'L‘ 2. °/., DB. MO/STURE CONTENT —— Q»— S gh— Li? Q. S..— Q'— 03 QB to Q_ (*3 Q Q I i 1 L l 0.0 20. 40. 60. 00 700. RELAT/VE HUM/O/TY — °/o [quililn'imtl [n.‘lSLL'L‘L “01%an W {5.15 relathw Till.ni«i"Lf (..‘H‘Hu'l'r-L) 1’“1‘ Il‘rw:’.{"--(‘li‘i€d in. 2' thin. r.— 3.5 ,QB. °/o N: 6, surface: NUMERICAL N, NODES FROM CENTER 70 fl '3)?“ s URFA C E K 9 \S’ - 2 6 -—-—AVERAGE MOISTURE CONTENT E S U 7 3:4 f/ air at : 700 :7: m N 5 2 0 Ln. 1 L 1 l L l 0.2 0.5 1.4 2.0 2.6 3.2 3.6 FigLnx‘ 5. Adsorption curves TIME -- HOURS £01: pCUIIts lflifitCCI ll] thcgrwa U1 Liiagu)nal. of LhLa uxfl».. v MOISTURE CONTENT -—°/a,D.B. 3. 5' 3.0 2.5 2.0 7.0 7.5 0.5 0.0 NUMER/CA L M» 3.2 H, 2.5H". 2. 0 HR. 7. 4 HR. air 32‘ : 70007-— 76 °/. 7?. H. 02 HR. I l l l l ___ 2 3 4 5 5 NODES —FROM CENTER 70 SURFACE Figtm b. D-Iuistule distrilmLion within the tube at d' {fervent Limes. X70 2 DIFFUSS/ON COEFFICIENT —FT/HR Q b—- N 2 ’gffAflMC) +A2x7MC)+ A3 9 _ A, = 709670000 .0 A 2 507552700 A3 = 0.0020770 Q _ ,. 03 Q _ \r' Q r— 7», (III at: 7000 F 2* MA%RH- Q .. l I l l J l 0.0 7.0 7.5 2.0 2.5 3.0 3.5 MOISTURE CONTENT - °/o,D.B. Figure 7. Optimized polynomial for the diffiasion coefficient as function of moisture gontent. °/o . D. B. 2.5 MOISTURE CON TENT 3.5 3.0 /(r*’*’ T T— r' I/ y 7 ——"NUMERICAL SOLUTION 0‘ EXPERIMENTAL DA'TA _ AIR AT: 7170 ”F and 7’5 o/Io RH 1 1 1 1 1 0.0 7.0 2.0 3.0 4.0 50 TIME '“ HOURS Figure 8. Comparison between the numerical so ution (with optimized diffLsinn cucifinicnt p01}nhwnial) aiui cxpcxflmunital [N31HES. IFIc_m1uZC >420 ozmaun_uzcv ZC_»Cua_C LC aubcs u muo~u Jd—ECZ>JCQ ...ucr: -3..:0u232 hr; )0. 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