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GONZALO RCA

1969

Thesis for the Degree of M. S. .

MICHIGAN STATE UNIVERSITY

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MOISTURE ABSORPTION

BY FREEZE - DRIED MEAT CUBES

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LIBRARY *5

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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-I5

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

ft2diffusion coefficient, h;—

lbm

density, 2:3

porosity, decimal

grid spacing, ft

. . ftconvective mass transfer coeffic1ent, R;

space sub-indices

Colburn correlation factor

characteristic length of the cube, ft

_. lbm

m01sture content, dry b8618, If:

Molecular weight of water “ 18

lb

water vzpor partial pressure, IE5

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, autoxidstion 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.. 19.5.. 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) tNere 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 + B1 2 3 4

Equation (1) can then be rewritten as:

e. .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)

Klf- f

(2.0) (32) (l-f) (d8)

K' =2 f

K. = f + (B3) (1-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)

Cum Y1 + 1 [175 (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 +CC.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 YZ—IIT [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:k11::jk (4)

where:

Y = l1 (n- 1) (n 1) 3

K1 Ci,j,k +K2 Ci,j,k +K3 +—h2

Y = 12 (n) (n) 3

(K1 Ci,j,k K2 C1 ,,jk +K3 +h?

and

(3) (B) (1-f) (as)K = ‘

1 (f) (At) (Dff )

1.jk

(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) (0ff )

i.j.k

.

.. -, _,\

' l

I ' ‘,

.f

l “u

- - I? 1

.. .... ._.

.

. f.

...... - _.__ -'-

D (M )9+A (M )+AA . . . .e£fi,j,k 1 1,3,k 2 1,3,k 3

(n-l) = (n-l) 3 +. (n-l) 2 + (n-l)_ +

Mi,j,k Bl (€1,333 B2 “1,3319 B3 (€1,313 B4

(n) = (n) 3 + (n) 2 + (n) +

“1,3,1. Bl “11.1.19 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

_P-AL = 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

Mgistpre 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:

l. 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 10’2an ll

3.2166 x 10‘1on H

-4.0934 x 10'3w

ll

4.5648 x 10‘5bi 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.denSLty (d8) 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

lb lb

EZa.‘ This value correspondS to a fiber density Of 67°12 EEG

(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'kT>

.but, pw = (pw S) (R.H.)

(p ) (M )

“’8 w (R.H.)

so, C =

(80) (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. =i

> 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 l. 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 ft2The 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 polynomia1---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.

’4"‘v.

.Ilnh

9‘4

_ l6 -

TABLE 1

NMMERICAL RESULTS WITH OPTIMIZED

VARIABLE DIFFUSION COEFFICIENT 3

ft?8 2- -

Deff 10.961 08) 0.16621 (MJC) + 0.002011 hr

mm

. Numerical Average Difference Between Kpproximate Confidence

Time Moisture Content Experimental and Intervals for each

Numerical Values Numerical 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

. .-o -‘.-v“... N....

[l

lu[l|linl‘

(IIII

-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,_exp1icit numerical solution

2; the conduction equation for multi;ggmeg§ional nonhomggeneous

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 P.8W. BakkermArkema (1969) Investigatigg:of the

energegics of water binding in dehydrated foods at very710w 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. 17LA, 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).

Nguddy, P. 0. (1969) A Generalized Theory of Sorption Phenomena in

Biological Materials. .Michigan State University Agricultural

Engineering Department. (Unpublished Ph.D Thesis) pp. 104.

neckland, L. B. (1969) Water activity and storage sigbility. Food

Technology, 23, 10, ll.

Rohsenaw, W. M. and H. Choi (1961) Heat, Masgg and Momentum Transfer.

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. 4/9 COND/T/ON/NG (AM/NCO) UN/T.

2. EQUILIBRA T/ON CHAMBER

3. LAM/NAP AIR FLOW METER

\

\\\\\\\\

@

Figure 1. Apparatus for equilibration

of i‘I'emct-(li‘ied beef L'ulus.

—0/0’

0.8.

CONTENT

MOISTURE

A—-—————

Q

MS I-

'\ EXPERIMENTAL

A 60% RH.

OE 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 I'—

‘0 11° W

I .

n

O on

\T‘

" , o -/H G

7 ¢"M'§'.

Q ,¢

(\i

C")

l L i I |

0.0 70 2.0 3.0 4.0 5.0

TIME - HOURS

Figure 2. Experimentallydetermined muisturo

adsznwvtiun claw CS

hi‘Cl: Clihtas_

F01? frnwnzy-(lrircl

CUBE

TEMPERATURE—

°F

96

700

702

704

706

96

94

EXPERIMENTAL

——_‘ 00.0 °/o :9. H.

“.__ 60.0 O/a A) H.

———-—‘ 40,0 °/o R. H.

Jr\\ ————-‘ 70.4 .O/o RH.

I

I \\ \ air at: 700 °F

\\ \ \

\\\ \\ \\ \

'\.\‘\ \\“\~ ~\‘~““~

"If T‘“ “ \¥

I

l l I L J l

.0 .250 .500 .750 7.00 7.25 7 50

TIME — HOURS

Figure 3. Experimentalix d [LYHlHCd tomperntlie

tlirvcs chlrizn, tht aihstnqitiiut pr wqus Wldllzltvd

in FWHI‘L‘ 2.

°/.,DB.

MOISTURE

CONTENT

——

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S

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'53.

Q.S..—

Q—

03

me

to

9..(*3

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0.0 20. 40. 60. 00 700.

RELATIVE HUMIDITY — °/o

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N: 6, surface:

NUMERICAL

N, NODES FROM CENTER 70

a 35’“ 5URFA CE

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2 e -—-—AVERAGE MOISTURE CONTENT

E

SU I

3:4 I, air at : 700 :F

U, L

5

2 0

Ln.

1 l l l i l

0.2 0,6 1.4 2.0 2.6 3.2 3.6

Figinx‘ 5. Adsorption curves

TIME -- HOURS

£01: ptdzits lfliiitCCI 111 thLJIJa hi tiiagu)nal. of thta clnit.

v

MOISTURE

CONTENT

-—°/a,D.B.

3.5

3.0

2.5

2.0

7.0

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0.5

0.0

NUMERICA L

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I l l I l ___

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NODES —FROM CENTER 70 SURFACE

Figlme 6. Hoistuie distribution within the

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2

DIFFUSS/ON

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Q I—-

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2* 04%RH-

Q L

l I l l l l

0.0 7.0 7.5 2.0 2.5 3.0 3.5

MOISTURE CONTENT - °/o,O.B.

Figure 7. Optimized polynomial for the

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of moisture content.

°/o

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2.5

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1 1 1 1 1

0.0 7.0 2.0 3.0 4.0 50

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