VWE {:9 g]

{53;}: - ,
."""“ _b ‘ ‘VJJ.
‘.

ABSTRACT

HEAT AND MASS TRANSFER IN ONIONS

BY

John R. Rosenau

The response of the product to its environment
must be known if the processing of a biological product
is to be Optimized. The overall purpose of this study
was to model the heat and moisture transfer response of
onion bulbs to any given boundary condition.

The "Alternating Direction Explicit Procedure"
for the solution of the transfer equations was adapted
to a special finite difference grid system. This grid
system was constructed orthogonal to the principle trans-
fer directions within the onion bulb. The model was
designed to be easily adapted to any axially symmetric
body.

The model parameters required to model heat and
moisture transfer in onions were obtained.

The heat transfer portion of the model simulated
the actual process very well as indicated by small RMS
and maximum differences between predicted and measured

center temperatures. The model showed that for heat

John R. Rosenau

transfer considerations, the bulb may be modeled as a
sphere with thermal conductivity equal to the radial
thermal conductivity of the bulb.

The mass transfer portion of the model was in only
fair agreement with experimental weight loss measurements
conducted as a test of the model. The differences are
attributed mainly to the effect of respiration and the
effect of the cracking and loosening of the outer scales

during the drying process.

Approved V4 ,
Major Professor

Approved

 

epartment Chairman

HEAT AND MASS TRANSFER IN ONIONS

By
Y\
(M?

John R? Rosenau

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Agricultural Engineering

1970

027/01

ACKNOWLEDGMENTS

The interest, counsel, and time generously extended
by the author's major professor Dr. F. W. Bakker-Arkema
(Agricultural Engineering) is gratefully acknowledged.

The author appreciates the help and guidance offered by
Dr. J. V. Beck (Mechanical Engineering), Dr. W. G. Bickert
(Agricultural Engineering), Dr. A. M. Dhanak (Mechanical
Engineering), and Dr. R. Hamelink (Mathematics).

The author thanks Dr. G. E. Merva (Agricultural
Engineering) and Dr. B. F. Cargill (Agricultural Engineer-
ing) for their interest and help in defining the research
problem.

The financial assistance given the author by
Michigan State University, the National Science Foundation,
and the Agricultural Engineering Department is acknowledged.

Special thanks are extended by the author to his
wife, Marion, for her help and encouragement during this

study.

ii

TABLE OF CONTENTS

Page
LIST OF TABLES . . . . . . . . . . . . . v
LIST OF FIGURES. . . . . . . . . . . . . vi

SYMBOLS . . . . . . . . . . . . . . . Vii

Chapter

I. INTRODUCTION . . . . . . . . . . . 1
II. LITERATURE REVIEW. . . . . . . . . . 5
2.1 Introduction. . . . . . . . . . 5

2.2 Specific Heat . . . . . . . . . 7

2.3 Density . . . . . . . . . . . 10

2.4 Thermal Conductivity . . . . . . . 11

2.5 Moisture Diffusion. . . . . . . . 13

2.6 Mathematical Modeling. . . . . . . 25

III. OBJECTIVES . . . . . . . . . . . . 33
IV. MATHEMATICAL MODEL . . . . . . . . . 34
4.1 Introduction. . . . . . . . . . 34

4.2 Finite Difference Grid . . . . . . 35

4.3 Solution of Node Equations . . . . . 46

V. DENSITY . . . . . . . . . . . . . 55
VI. HEAT TRANSFER . . . . . . . . . . . 57
6.1 Introduction. . . . . . . . . . 57

6.2 Axial Conductivity. . . . . . . . 60

6.3 Radial Conductivity . . . . . . . 63

6.4 Heat Transfer Simulation. . . . . . 65

iii

Chapter Page

VII. MOISTURE TRANSFER . . . . . . . . . 67
7.1 Introduction . . . . . . . . . 67

7.2 Determination of Moisture Diffusivity. 68

7.3 Mass Transfer Model Testing . . . . 76

VI I I . CONCLUSIONS . . . . . . . . . . . 7 9
Suggestions for Further Study . . . . . 80
BIBLIOGRAPHY . . . . . . . . . . . . . 81
APPENDIX . . . . . . . . . . . . . . 85
1. Tables . . . . . . . . . . . 85

2. HTRAN. . . . . . . . . . . . 91

3. MTRAN. . . . . . . . . . . . 103

4. HMTRAM . . . . . . . . . . . 116

iv

LIST OF TABLES

Table Page

1. Experimental Values and Results of the Onion
Density Determination Data . . . . . . 85

2. The Fourier Series Approximating the Onion
Shape I O O O O O O O O O O O O 86

3. Results of the Axial Thermal Conductivity
Experiments . . . . . . . . . . . 87

4. Results of the Radial Thermal Conductivity
Experiments . . . . . . . . . . . 88

5. Results of the Radial Moisture Diffusivity
Experiments . . . . . . . . . . . 89

6. Data Points for the Onion Moisture Isotherm
at 86 F (Saravacos, 1960) . . . . . . 90

Figure

2.6.1.

LIST OF FIGURES

A Portion of a Two Dimensional Finite

Difference Grid in Cartesian Coordinates

Relationship Between PERIM (z), DPERIM (z),

and z 0 O O O O I O O O O O 0

Node Points for a Two-Inch Diameter
Sphere . . . . . . . . . . .

Node Points for a Two-Inch Diameter Onion
A General Surface Node. . . . . . .
A General Central Axis Node . . . . .
An End Node . . . . . . . . . .

Test Case Model Error Using the ADEP on a
Two-Inch Diameter Sphere . . . . .

Temperature Sensitivity Coefficients at
the Center of a Two-Inch Diameter Onion

Cross Sectional View of the Sample Holder
Used in the Axial Conductivity Tests .

Heat Transfer to Onion Bulbs Subjected
to a Step Change in Ambient Temperature

Cross Sectional View of the Denny Osmometer.

Onion Moisture Isotherm . . . . . .

Predicted Moisture Diffusivity Dir. . .

Moisture Loss from Onion Bulbs at 70 F and

31% Relative Humidity . . . . . .

vi

Page

27

36

39
40
42
44

44

50

59

61

66
69
71

74

77

0 0 OJ

C

Gfap

Gfrp

ml

Le

Nu

Pr

Re

Sc

Sh

SYMBOLS

Permeability (lbmft/lbfhr)
Concentration (moles/ft3)

Heat Capacity (Btu/F)

Diffusion Coefficient (ftz/hr)
Internal Energy (Btu)
Activation Energy (1bfft/mole)
Axial Geometric Factor (ft)
Radial Geometric Factor (ft)
Enthalpy (Btu)

Molar Enthalpy (Btu/mole)

Molar Mass Flux (moles/ftz)
Lewis Number (dimensionless)
Molecular Weight of Water
Nusselt Number (dimensionless)
Pressure (lbf/ftz)

Prandtl Number (dimensionless)
Universal Gas Constant (1bfft/mole°R)
Reynolds Number (dimensionless)
Schmidt Number (dimensionless)

Sherwood Number (dimensionless)

vii

:3‘ 5‘ Q: OI

w u.

:3

'0

.0

Temperature (°R)
Volume (ft3)
Partial Molar Volume (ft3/mole)

Volume of the Node i,j (ft3)

Mole Fraction (dimensionless)

Specific Heat (Btu/lbmF)

Molar Heat Capacity (Btu/mole F)

Diameter (ft)

Convective Heat Transfer Coefficient (Btu/hr ftzF)
Convective Mass Transfer Coefficient (ft/hr)
Colburn j-factor (dimensionless)
Conductivity (But/hr ftF)

Moisture Content (d.b.) (dimensionless)
Moles (moles)

Partial Pressure (lbf/ftz)

Heat (Btu)

Heat flux (Btu/hrz)

Radius (ft)

Time (hr)

Distance (ft)

Distance (ft)

Distance (ft)

Velocity (ft/hr)

Water Potential (atm)

Mobility (moles ft/hr lbf)

viii

7 Specific Gravity (dimensionless)
Y Activity Coefficient (dimensionless)
U Chemical Potential (Btu/mole)

0 Density (1bm/ft3)

pdm Density of the Solids (lbm/ft3)
w Mass Fraction (dimensionless)
Subscripts

I End Node Index

J Surface Node Index

a Axial Direction

a Ambient

c Based on Concentration

dm Dry Matter

m Based on Moisture Content

r Radial Direction

i Node Index

1 ith Species

j Node Index

w Water

3 Solids

5 Surface

wv Water Vapor

x Direction

y Direction

2 Direction

ix

Based on Chemical Potential

Fixed Coordinate System

Coordinate System Moving with the "Medium"
Time Index

Coordinate System Moving with the "Solids"
Coordinate System Moving with the "Volume“
Saturated

Pure

I . INTRODUCTION

The remarkable efficiency of the agricultural
production system in the United States has been achieved
through advances in three basic areas: (i) the development
of improved growing practices and genetic varieties, (ii)
the mechanization of crop and animal production, and (iii)
the evolution of new processing techniques to handle the
resulting production increases. The development of these
processing methods is extremely important as the benefits
of increased production are wasted if the system for
handling, processing, and distributing this production
is inadequate.

Inherent to the methods incorporated for the pro-
cessing of agricultural products is the response of the
products to their total processing environment. If the
effect of all of the mechanical, chemical, thermal, and
biological forces acting on a product can be predicted,
the deve10pment of Optimal processing systems for that
product is made possible.

This research project examines a part of the total

problem outlined above--the development of a mathematical

model simulating the response of yellow globe onions (var.
Abbott and Cobb 192) to a convective heat and moisture
transfer boundary condition.

There are two main reasons for choosing this thesis
topic. First, the onion production industry in Michigan is
in the process of rapid change. The harvesting and material
handling operations have been totally mechanized as have the
packaging and shipping operations. However, problems still
remain. The first occurs just after harvest when the onions
are "cured." Normally, this is done by leaving the covered
pallets of freshly harvested and topped onions in the field.
During the curing period, the onion necks dry, helping to
prevent disease organisms from entering the bulb through
the place where-the green top was attached. In addition,
the skins develop the traditional golden color, while the
outermost skin drys, cracks, and falls off ("shucks"), N
taking with it any attached field dirt. With warm dry
weather, curing takes about a week; with cool rainy weather,
up to three weeks.

The author has found that the curing process can
be shortened to about three days by passing air at 100 F
and 90% relative humidity over the bulbs. More work is
needed, however, to Optimize this artificial curing
process.

The processing step following curing is storage.

While storage conditions of 32 F and 75% relative humidity

have been recommended (Franklin et_§l., 1966), more work
needs to be done to identify the various effects of
respiration, moisture transfer, and spoilage on the weight
loss of marketable onions in storage.

Another problem area in the processing sequence
occurs when the bulbs are brought in from storage. If
run through the packaging line at storage temperatures
(about 35 F), water condenses on the bulbs. In the some-
what dusty atmosphere of the packing room, this moisture
picks up dirt causing an inferior product. The onions
thus need to be warmed before processing.

The Optimal design of curing, storage, and heating
equipment depends on the prediction of the response of the
Onion bulb to its environment. Thus, a mathematical model
of the bulb is needed.

When the environment is significantly affected by
interaction with the product (such as in the case of a
deep bin), the model of the bulb may be included within
larger models (Bakker-Arkema gt 31., 1969), to predict
the response of the entire process. One problem associ-
ated with such nested models, however, is the large amount
of computer storage and time required for their solution.

The second reason for the research was the hope
that studying the heat and moisture transfer processes in
onion bulbs would lead to a better understanding of these

processes in other high moisture products. The onion has

a relatively large size and a structure which allows
sections to be removed with a minimum of cell damage.
Thus, experiments may be performed on sections of the
product to determine its transfer properties without

changing these properties through the sectioning process.

II . LITERATURE REVIEW

2.1 Introduction

 

The investigation of the processes of heat and
moisture transfer can be divided into two parts--the trans-
fer processes occurring within the product, and the inter-
action of the product surface with the environment. Con-
cerning moisture migration, Van Arsdel (1963) writes:

The drying of a moist substance always involves the
movement of a quantity of water away from a dry sur-
face. The separation is usually regarded for purposes
of analysis as the result of two successive phenomena:
(1) migration of water within the moist body to its
surface; and (2) conveyance of the vaporized water
away from the body . . . the factors that determine
the rate of movement of water within the body can be
regarded as independent of the external conditions.

A useful analysis of the process can be made on the
basis of this simplified picture, even though in some
cases it may become evident that vaporization is in
fact occurring in an ill-defined zone within the moist
body instead of only at its geometric surface.

The surface-environment interaction can be described
by two convective transfer coefficients. The first, h, is

defined by the equation

4" = h (TS - Ta) (2.1-1)

and the second, hD’ by the equation

J8 = h (c — c ). (2.1—2)

In general, these coefficients are determined by the nature
of the air flow pattern around the product. When radiation
heat transfer is significant, h can be modified to include
its effects.

Much research has been performed investigating
these coefficients. Since onions are usually processed in
deep bins and pallet boxes, the work of Barker (1965) in
reviewing the subject of heat transfer in packed beds
should be mentioned. By reference to Barker's article,

a "Colburn j-factor" can be obtained for any given Reynolds
number and product shape. The heat transfer coefficient h

is then obtained by the relation

hd 1/3

 

 

k = Nu = j Re Pr (2.1-3)
air
and the mass transfer coefficient hD by
h d
DD = Sh = j Re Sol/3 (2.1-4)
air

where the Reynolds number Re is given by the onion bulb
diameter times the mass flow rate of air per square foot
of bed area divided by the absolute viscosity. The
Prandtl number Pr is given by the kinematic viscosity

divided by the thermal diffusivity ka. , and the Schmidt

1r

number So by the kinematic viscosity divided by the mutual
diffusivity of water in air D . .
air
While the above formulas can only be considered
approximate, they do characterize the interaction between

the product and its environment. The transport processes

within the product are outlined in the following sections.

2.2 Specific Heat

 

If heat is added to a simple closed constant pres-
sure system in which no work other than that used to change
the system volume is performed, the change in internal

energy of that system is given by

AB = q - PAV. (2.2—1)

The quantity H is defined by the equation

H = E + PV. (2.2-2)

Thus, in this process,

AH = q. (2.2-3)

During the heat addition process the temperature
of the system rises. This then is the basis for the

definition Of heat capacity, namely,

C = (——d . (2.2-4)

If the system is homogeneous, the specific heat is

given by
32
and the molar heat capacity by
_ C
c : _r12° (2.2.6)

When the system can be considered a homogeneous
mixture, the enthalpy Of the system is the sum of the

partial enthalpies, i.e.,
H = Z n. H. (2.2-7)
i

where ni is the number of moles of species i present and

Hi, the partial molal enthalpy, is defined by the equation

_ 3H _

Likewise,

c = z n. 6.. (2.2—9)
i

In general, 5: is not a constant for a particular
species but varies with changes in temperature as well as
in the relative mole fractions making up the mixture.

Nevertheless, it is customary as a first (and usually quite

accurate) approximation to consider the 53's for a food
system as being constant.

Siebel (1892) considered food as composed of
solids and water for which he used the respective specific
heats of 0.2 and 1.0 Btu/1bmF.

Charm (1963) used this approach with the equation

C = 0.5 w + 0.3 w + 1.0 w (2.2-10)
S m

f

wherein wf, ms, and ww refer to the mass fractions of fat,
solids, and moisture. If this formula is applied to
onions of 90% moisture content (w.b.), 9.8% solids non-
fat, and 0.2% fat, c is calculated as 0.93 Btu/lbmF. If
the formula were applied to 80% moisture content onions
(19.6% solids nonfat and 0.4% fat), c would be 0.86
Btu/1bmF. Ordinanz (1946) gives identical values of c

for onions at 80 and 90% moisture content (w.b.).

Reidel (1951) separated the solids portion of
fruits and vegetables into XO "soluble" and Xu "insoluble"
fractions by examining the refractive index of the juice.
He tested onions of 85.5% moisture content (w.b.) and used
weight fractions of l3.% (of the total weight) as soluble

solids, and 1.5% as insoluble solids in the equation

c = (l - Xu)(1 — 0.57 X0) + 0.29 Xu. (2.2-11)

In this case c is 0.916 Btu/lbmF. For 90% moisture con-

tent onions (w.b.), assuming the soluble and insoluble

10

fractions are in the same proportion to each other, c
would be 0.942 Btu/lbmF.

In light of the above, it was concluded that
further work on determining the specific heat of onions
is not needed at the present time. Charm's equation
(eqn. 2.2-10) was used to determine c in the calculations

involving specific heat in this research.

2.3 Density

 

The importance of a product's specific heat has
been described in the previous section. Density merits
consideration in that it, when multiplied by the specific
heat, expresses the system's heat capacity on a per unit
volume basis.

It is often easier to determine a food product's
specific gravity y (by measuring its buoyancy in water)
than its density. The density is then determined by
multiplying the specific gravity by the density Of water.
While the density of water is slightly dependent upon its

3 (at 70 F)

temperature, it does not vary from 62.27 lbmft
by more than 0.4% in the range of 40 to 100 F (Holman,
1963). Thus, in this temperature range the following

equation is sufficiently accurate.

0 = 62.27 Y (2.3-l)

11

The author was unable to find literature values
for the density of onion flesh. It thus became necessary

to determine this property eXperimentally.

2.4 Thermal Conductivity
In the one dimensional case, Fourier's equation of

heat conduction

defines a material's thermal conductivity k. Consideration
of the law of the conservation of energy in conjunction
with the above leads to the following familiar differential
equation for the temperature history of a body in which

there is no heat generation:

8T _ 3 8T _
no 3-5 — 32 (k 82). (2.4 2)
This simple definition of k, however, must be
expanded when the multi-dimensional case is studied since
thermal conductivity is a tensor and not a scalar property.

Thus, in three dimensions (Arpaci, 1966),

.. 22
61x kll k12 k13 ax
" _ _ a—'I‘
qy — k21 k22 k23 ' 3y
- .. 3_T_ _
qz k31 k32 k33 32 . (2.4 3)

 

 

 

 

 

 

12

If a material is orthorombic, and if its principle
directions are coincident with the x, y, and 2 directions,

the following equations are equivalent to equation (2.4-3):

.u _ 32 -

. 8T ‘

II = k _ 204-5

qy y 3y . ( )
and

_ 22 ..

qz - k2 32 - (2.4 6)

The law of conservation of energy can be applied
to these equations, along with pertinent initial and bound-
ary conditions, to generate a model for the temperature-
time history of a product. (See Section 2.7 of this chap-
ter and all of Chapter IV.) Since this history is dependent
upon the thermal conductivity of the product, these values
must be determined and included in the mathematical model.
Van Arsdel (1963) gives a general description of
the values of thermal conductivity which might be expected
in biological products:
In fresh fruits and vegetables, whose moisture content
is very high, the conductivity is not far from that of
pure water. As drying takes place, however, con-
ductivity falls. If shrinkage is complete, so that
the dry product is free from internal voids, the de-
crease in conductivity is only minor, but if the body
becomes highly porous as it dries the low conductivity

of the air in the open spaces reduces the overall con-
ductivity markedly.

13

Since the onion is a high moisture content vege-
table, one would expect its thermal conductivity to be
close to that of pure water which is 0.349 Btu per hr ft F
at 70 F (Holman, 1963). Since the thermal conductivity of
water varies from the value at 70 F by a maximum of 4.9%,
in the temperature range of 40-100 F, one would also ex-
pect the thermal conductivity of onion flesh to be quite
constant within this range.

Reidy (1968) has conducted an extensive review of
experimental techniques used to determine thermal proper-
ties. He noted that steady state techniques could not be
expected to give accurate results with high moisture bio-
logical products due to the problem of moisture migration
within the product during the test. He concluded that
numerical methods should be used with transient type
experiments involving these materials.

As the author has been unable to find values for
thermal conductivity of onion flesh in the literature, one
of the objectives of this research (see Chapter III and
Chapter VI) was to determine values for the thermal con-

ductivity of onion flesh.

2.5 Moisture Diffusion

 

GOrling (1958) outlined the process of moisture
movement within a biological product as a combination of
five different mechanisms: (i) liquid movement caused by

capillary forces, (ii) diffusion of liquids caused by

14

concentration differences, (iii) surface diffusion, (iv)
water vapor diffusion, and (v) vapor flow caused by differ-
ences in total pressure.

Before proceeding with the subject of diffusion,
the distinction between the movement of moisture via dif-
fusion and via bulk flow should be mentioned. Bulk flow
is flow caused by the action of a pressure gradient or
gravity on water existing within a relatively large channel
such as a pipe. Poiseuille flow and Knudsen flow are
familiar examples of bulk flow. While such flows exist
in the xylem vessels of the larger plants during the grow-
ing process, the movement of water within the flesh of a
biological product can be considered a diffusion process,
i.e., the movement is caused by a chemical potential
gradient (as discussed later in this section) and not by
a total pressure gradient or gravity directly (Saravacos,
1962).

The traditional literature on the subject of dry-
ing used the theory and mathematics associated with heat

transfer. Thus, the diffusion equation

3C

J=-D‘ w

w 0 -§;' (2.5-l)

was written in obvious analogy to the heat conduction
equation and all of the mass transfer mechanisms were
grouped together under one overall diffusion coefficient

D5 and one driving force sew/32.

15

Babbit (1940) pointed out that the true driving
force for moisture movement is not the concentration
gradient. He showed this by noting that water can be
made to diffuse against a concentration gradient. He con-
cluded that the driving force for diffusion was the
gradient of the equilibrium water vapor pressure.

Kramer (1969) showed that the true driving force
for diffusion of water through a medium is the gradient
of the chemical potential of water Buw/Bz. Kramer, how-
ever, followed the soil physics literature convention and
used the chemical potential in another form. By dividing
the chemical potential by the molar volume of pure liquid
water, the chemical potential is transformed into WW,

the "water potential,‘ which is dimensionally equivalent
to pressure. Kramer mentioned that the transformation is
employed simply for convenience.

By using the chemical potential gradient instead
of the concentration gradient, a diffusion equation
analogous to equation (2.5-l) can be developed. If JS

is the molar flux of water with respect to a reference

frame moving at the same rate as the "medium," then

C 0 3p
m _ w w w _
Jw - 32 (2.5 2)

where 0w may be thought of as a mobility.* The solids

 

*Hartley and Crank (1949) presented a similar
development but with emphasis on liquid diffusion. Their
expression 1/0AnN is the same as 0w.

16

may also move with respect to this reference frame and
the flux relative to the solids, J3, is the flux needed
to determine drying rates.

ms

If v is the velocity of the medium with respect

to the solids,

s _ m ms _
Jw - Jw + v Cw (2.5 3)
and
J8 = o = J‘“ + vms c . (2.5-4)
S S S

. (2.5-5)

The rate at which the solids move with respect to

the medium defines another mobility 03 by the equation

m 3“5
JS — ' CS {25 w. (2.5'6)
From the Gibbs-Duhem equation (Moore, 1962),
C
dp = lap . (2.5—7)
S C w
5
Therefore,
311
Jm= c {2 ——w (2.5-8)

U)

and

then

n

W

 

2
3p C 8n
— _ __E. w __E. -
— Cw Qw 82 C s 32 ' (2'5 9)
Cw auw

= - (Cs 9w + Cw as) C; 32 " (2.5-10)

defined by

Cw Buw
= — ' -

RT D 32 , (2.5 11)
_. Bil”. _
— Cs (CS Qw + CW 08). (2.5 12)

Since
0 pw

- “w - RTlnaw — RT1n(waw) — RTln 5:. (2.5-13)

the equation for the molar flux of water may be stated in

a number of equivalent ways.

The most traditional involves

 

the concentration of water. Thus,
an alna Blna alnx
w _ w = w w _
‘52— ‘ RT 32 RT 231an 32 ' (2'5 14)
But,
Cw

Expanding de yields

BXW BXw
de = 36— de + EC- dC
W S
or
CS CW
dXW = '7 de " —'2' dCs
C C

S

18

From the Gibbs-Duhem relations,

 

 

 

 

 

dC V
__§ = _ .2
de V
5
Therefore,
Csvs + vaw l
de = 2_ dC = 2_ de
C V V
s s
and
alnx 8C
w = 1 w
32 C C V. 8w
w s
yielding,
J5 = - (CS 9w + Cw QS)RT Elnaw BCW
w

CCV
SS

31an 32 '

(2.5-16)

(2.5-17)

(2.5-18)

(2.5-l9)

(2.5-20)

(2.5-21)

This development shows that Dé as used in equation

(2.5-1) is given by

19

 

 

RT Elnaw
D8 = (Cs Qw + Cw as) alnX
C C V w
s s
Blna
= D11 i alnx‘”. (2.5—22)
CVs w

At this point the distinction between D; and DC should be
explained. Dé as defined in equation (2.5-l) relates the
molar flux with respect to the solid to the concentration
gradient. DC as used in many texts on mass transfer
(especially Bird, Steward, and Lightfoot, 1960) is the
mutual diffusion coefficient which relates the molar flux
with respect to the volume fixed reference plane to the

concentration gradient. Thus,

The relation of Dé to DC can now be determined. Since

v _ mv _
Jw — J3 + v Cw’ (2.5 24)
vmv = - v'm = — V' Jm - V Jm , (2.5-25)
w w s s
and
JV = Jm - c (\7 Jm + \7 Jm) , (2.5-26)
w w w w w s s

substituting for J? and J: into equation (2.5-26) yields

20

V—_ — _. —
Jw — (CS Qw + CW 98) VS Cw . (2.5 27)

Substituting for Buw/Bz gives finally

BE Blnaw BCW
C :31an 82

 

J = - (C Q + C Q )
S W W S

w (2.5-28)

which is identical with the result given by Hartley and

Crank (1949). Thus,

 

D!
59. = , (2.5-29)
C

When shrinkage upon drying is negligible, the partial
molar volume of water V; is zero and the term 1/C8Vé is

unity. In this case, Dé and DC are identical.

Fish (1958) found that in potato starch gel, Dé
varied with temperature and the variation could be de-

scribed by Arrhenius' equation as

-E
l _ I __a_ ..
DC — DO exp ( RT) . (2.5 30)

The constant D6 varied little with moisture content but
the activation energy Ea increased with decreasing moisture
content.

Jason (1958) obtained good agreement between the
experimental and theoretical results in the drying of fish

muscle by using two values for Dé--the first during the

21

initial falling rate drying period and the second during
the later stages of drying. He postulated that the trans-
ition from the first to the second was associated with

the uncovering of the unimolecular water layer normally
bound to the portein molecules.

Jason (like Fish) noted an Arrhenius type of
relationship between the diffusion coefficient and temper-
ature. His results agree with those of Fish in that D5
was quite constant but Ea increased with decreasing
moisture content.

Wang (1958) used a different equation to model
moisture diffusion. While he solved a problem in simul-
taneous heat and mass transfer, the equations he used

degenerate for a one dimensional isothermal case to

J = W (2.5-31)

 

where pw is the equilibrium vapor pressure. Since

 

 

3p 3lnp 3p
__W==RT W = 53 W , (2.5-32)
32 32 p 32
w
equation (2.5-10) can be rewritten as
C 3p
8:- 1&4: -
Jw (Cs 9w + CW 98) C p 32 . (2.5 33)

22

Thus,

MWRT cw D' M.w cw
= (CS 9w + CW 525) C— = —“——— . (2.5-34)
W S pWV

B

 

Since Cw includes all of the moisture existing per
unit volume in the material, the gas law does not provide
a simple relationship between pw and Cw' One cannot say
a priori that B' is a constant.

Young (1968) used a related approach to model
simultaneous heat and mass transfer in spherical, porous,
hygroscopic solids. He assumed that the gradient of the
density of water vapor in the pore spaces pwv is the driv-
ing force for mass transfer, thus making a distinction
between pgm, the overall moisture density and pwv' the
density of the water vapor in the void fractions of the
material. Two further assumptions were that the overall
moisture content (d.b.) was related to the vapor concen-

tration in the void fraction and the temperature by

m = a + B pwv - y T (2.5-35)

where a, 8, and y are constants and that the diffusion
coefficient associated with the vapor concentration

gradient driving force could be modeled as

I _ —
Dwv — D1 + Dzm + D3T (2.5 36)

and D are constants.

D2' 3

where D1,

23

It can be shown that D&v is related to D; by the equation

D' m p
D¢V = _£LEV_JBE . (2.5-37)
WV

While Young did not attempt to verify his model experi-
mentally, he did solve the model numerically for many
different input conditions. One important result was

the illustration of the fact that the heat and mass trans-
fer equations may be solved separately whenever temper-
ature equilibrium is reached much faster than moisture
equilibrium. Young developed a modified Lewis number as

a criterion for this condition. (See Chapter VIII.)

In spite of the fact that Dév is not a constant,
extremely good fits between experimental data and the
corresponding mathematical models have been obtained in
some cases by using constant values of D&V., For example,
Roa and Bakker-Arkema (1969) found that in freeze-dried
meat cubes, a model using a constant Dév and one using a
second order polynomial D¢v fitted the adsorption data
almost equally well.

Another possibility for the driving force should

be mentioned--the moisture content (d.b.), m. Since

3uw 3lnaw

 

32 3m. 35 ' (2°5’38)

24

equation (2.5—10) becomes

CWRT 3lnaw

C 3m
3

J

 

8-- 2E -
w - (CS 0w + CW 98) 32 . (2.5 39)

The mass flux of water with respect to the solids J: is

given by
= M J . (2.5-40)
w

Therefore,

C 31na

 

 

.s _ _ w w 3m _
3w — M.W RT (CS 0w + CW 05) Csm 31nm 32 (2.5 41)
defining Dfi in the equation
.5 _ . 3m
3W " - pdm Dm E (2.5-’42)
as
31na
. _ 52_ W _
D — C5 (C5 0w + CW 98) 31nm . (2.5 43)
It follows that
" 31naw
D' = D’ C V = D = D' . (2.5-44)
s s c u 31nm

Fish (1958) found that in potato starch gel the
term 3lnaw/3lnm varied from approximately 1.50 at moisture
contents between 0.02 and 0.18 (d.b.) to approximately

0.02 near saturation. He also found that, at 35 C, Di was

25

7

constant (2.4 x 10— cmz/sec) for samples over 0.30 mois-

ture content (d.b.). For samples below this moisture con-

10 cmZ/sec. at 0.8%

tent, DA decreased rapidly (l x 10-
moisture content).

The mass of water included in an incremental
volume of material dV is pdmmdv. The law of the conser-
vation of mass when combined with equation (2.5-42) yields
in the one dimensional case

_ . 3m)

3
—3_z(m32 . (2.5 45)

OJIO)
H'B

Thus, the use of the moisture content as the driving force
for moisture transfer leads to an equation that is rela-
tively simple to solve in that the driving force for mois-
ture transfer and the measure of water existing within the
material are the same. For this reason, the moisture con-
tent was used as the driving force in the mathematical

model as described in Chapters IV and VI.

2.6 Mathematical Modeling

 

As shown in Sections 2.4 and 2.5, heat and moisture
transfer within a material can be described by parabolic
partial differential equations such as (2.4-2) and (2.5-45).
This section outlines in general terms how these equations
can be put into finite difference form, and describes a
particular method--the "alternating direction explicit

procedure" for solving the finite difference equations.

26

For the purpose of illustration, constant property,
isotropic, two dimensional heat conduction shall be con-
sidered. The modifications needed to extend this to the
heat and moisture transfer problem under consideration in
this research are straightforward and are outlined in
Chapter IV. The partial differential equation for the

constant property, isotropic, two dimensional case is

2 2
pc 22-: k (§—%-+ g—g) . (2.6-l)

3x 3y
In order to develop the finite difference approxi-

mations to this equation, a rectangular grid is super-
imposed on the material as shown in Figure 2.6.1. To

make the notation easier, the node points are referred

to by subscripts i, j, and n, with i and j referring to
the increment in the x and y directions, and n to the
increment in time.

By the Taylor series expansion theorem,

 

 

2 T. . - 2 T. . + T. .
2.2 = 1-1.3 1'1 1+1'3 + o (sz) (2 6-2)
2 2 . .
3x Ax
Similarly,
2 T. . - 2 T. . + T. .
2.2 = lLlfl 1'1 1L3+1 + o (Ayz) . (2.6-3)

27

 

i-l, 1+: i, j+l i+l,j+l

 

 

i-l,j i,j i+l,j

 

 

 

 

 

JUJ-I i+l,j-l

 

 

Figure 2.6.1. A Portion Of a Two Dimensional Finite
Difference Grid in Cartesian Coordi-
nates.

28

Combining (2.6-2), (2.6-3), and (2.6-1),

 

 

dT. . T. . - T. . T. . - T. .
_1_'_l = i 1-113 1'1 + 1+1d 1:3
dt pc AX2 AX2

2

T. . — T. . T. . - T. .
+ 1.3-1 1,1.+ 1.1+1 1.]
Ay Ayz

+ 0(AX2) + 0(Ay2) . (2.6-4)

Equation (2.6-4) is the basic equation upon which the
various finite difference schemes are built. Given the
initial temperatures at every node point, the numerical
methods attempt to approximate the term (dTi,j/dt)ave in

the equation

(n+1) (n) d Ti '
To I = To 0 + ——’_1 At (2.6-5)
i,j i,j dt ave

in order to Obtain the temperature-time history of the
material. The evaluation of the right hand side of
equation (2.6-4) by the inclusion of the node temperatures
at different times gives rise to the various numerical
methods for solution of the conduction problem.

The "forward difference explicit method" uses
only node temperatures at time n in order to calculate

the right hand side of equation (2.6-4). Thus,

29

 

 

 

 

 

T(n+1) _ T(n) T(n) T(n) T(n) T(n)
TiI j 1Ij = j; (i-le1Ij + i+le1Ii
At pc sz Ax2
T_(n) T(n) T(n) T(n)
+1Ij--11Ij + iIi+l Tin
Ay2 Ay2
(2.6—6)

and rearranging,

 

 

 

 

 

Tm) T(n) T(n) _ T(n)
T(n+1) = T(n) + Atk i-le1,j + i+1,j i,j
Ti,j i,j pc Ax2 sz
T(n) T(n) T(n) Tm)
+1Ij-11Ij +irj+1ltj . (2.6-7)
Ay2 Ay2

Thus, the method is "explicit, meaning that the calcu-

(n+1)

lation of the temperature Ti' uses only one equation

I
and not a set of simultaneous equations. Unfortunately,
the method is unstable if either Of the ratios kAt/pch2
or kAt/pcAy2 becomes larger than about 1/4. This requires
that the time step At be kept so small that the method is
impractical.

The "backward difference method" uses the node
temperatures as evaluated at time n+1 for evaluation of
the right hand side of equation (2.6-4). Since this yields
a system Of simultaneous algebraic equations giving the

node temperatures at the time n+1, the method is an

"implicit" one. While the method is stable for any At,

30

the requirement of solving the simultaneous equations
renders the method impractical.

The "Crank-Nicolson" method uses the average of
the node temperatures at times n and n+1 for the right
hand side of equation (2.6-4). The method is stable but
has the same limitation as the backward difference method
in that the requirement of solving the system of simul-
taneous algebraic equations created by using the node
temperatures at time n+1 makes the method impractical.

A fourth method is known as the "alternating
direction implicit procedure" (ADIP) or the "Peaceman-
Rachford" method. This method uses node temperatures at
time n in the first two terms within the brackets on the
right hand side of equation (2.6-4) and temperatures at
time n+1 in the second two terms. In calculating the
temperatures at time n+2, temperatures at n+2 in the first
two terms within the brackets and at n+1 in the second two
are used. The systems of resulting simultaneous algebraic
equations differ from those Obtained in the backward
difference or the Crank-Nicolson methods in that the un-
known node temperatures appearing in any equation are all
from the same nodal row or column. This makes the solution
much easier since it is easier to solve I algebraic simul-
taneous equations J times than to solve I times J simul-
taneous equations once. This method is stable and
practical but the "alternating direction explicit pro-

cedure" (ADEP), as described next, is faster and easier

31

to program and was therefore used in this research (Allada
and Quon, 1967).

The alternating direction explicit procedure
(ADEP), like the ADIP, is a two pass system. On the for-
ward pass, the first and third terms within the brackets
on the right hand side of equation (2.6-4) are evaluated
at time n+1, and the second and fourth terms are evalu-
ated at time n. On the return pass the first and third
terms of the right hand side of equation (2.6-4) are
evaluated at time n+1 and the second and fourth at time
n+2. The method thus progresses through two time steps
during the full forward and return passes.

Barakat and Clark (1966) described a variation of
the ADEP in which two separate solutions are generated.
In one, the first and third terms within the brackets on
the right hand side of equation (2.6-4) are evaluated at
time n+1 while the second and fourth are evaluated at
time n. In the other solution, the second and fourth
terms are evaluated at time n+1 and the first and third
at time n. The final solution is obtained by averaging
the two solutions wherever desired. Thus, for a given
At, this variation requires twice as much computer time
and storage as the first ADEP method. Barakat and Clark,
however, argued that the averaging ADEP is better than

the non-averaging. They pointed out that the truncation

At 32T

error in the first pass includes the terms - A; 3E3; (n+1)

32

A; _32T
Ay 3t3y (n+1).

convergence, the ratios At/Ax and At/Ay must go to zero.

and - This would indicate that, to Obtain
The truncation errors on the return pass, however, have
the opposite sign of the ones on the forward pass. This
indicated to Barakat and Clark that these errors cancel

in the averaging ADEP, relaxing the requirement that the
ratios At/Ax and At/Ay go to zero. Barakat and Clark com-
mented that this condition needs further examination. The

tOpic is pursued further in Chapter IV.

III. OBJECTIVES

The objectives selected for this research are as

follows:
1.

2.

To determine the density of onion bulbs.

To determine the thermal conductivity of

onion flesh in the axial and radial directions.
To determine the moisture permeability of
onion skins.

To develop a mathematical model that simulates
the response of an onion bulb to its environ-

ment.

33

IV. MATHEMATICAL MODEL

4.1 Introduction

 

The partial differential equations governing the
heat and moisture transfer in an orthorombic material were
developed in Sections 2.4 and 2.6. It was also mentioned
that, by consideration of initial and boundary conditions
and of the laws of the conservation of mass and energy, a
mathematical model suitable for computer solution could be
developed to simulate the response of a biological product
to its environment. Examination of the onion bulb, how-
ever, reveals that there are two considerations that should
be mentioned before undertaking the development of such a
model.

First, examination reveals that the bulb can be
considered axially symmetric. This is the basis for the
decision to use the cylindrical coordinate system. A
reduction from a three dimensional problem to a two
dimensional one is easily accomplished in this case by
including only radial and axial dimensions.

The second consideration is that the principal

directions of the flesh (i.e., "axia1"--coplanar with the

34

35

central axis and tangential to the onion rings, and
"radial"--perpendicu1ar to the rings) are dependent upon
position. A grid system that is orthogonal to these
principal directions is desirable because the node equations
describing heat and mass transfer will then be simplified.
Under such a system, transfer from one node to another is
independent of the potential gradient existing at right
angles to a line connecting the two nodal points. The
following section describes the generation of such an
orthogonal grid system.

The grid system is the basis for three computer
programs. The first of these, HTRAN, was written to simu-
late the temperature response of a biological product
under the assumption of no mass transfer. The second
program, HMTRAN, simulates both the heat and moisture
responses using the moisture content (d.b.) as the mois-
ture transfer driving force. The third program, MTRAN,
is identical with the second except that only moisture

transfer is considered.

4.2 Finite Difference Grid

 

In the development of the finite grid system, two
computer subroutines are required. The first, PERIM (z),
associates with any point (2,r) the length of the radius
passing through (2,r) to the surface. The second DPERIM
(z), associates with (2,r) the slope of the tangent line to

the surface with respect to the axis. (See Figure 4.2.1.)

36

DPERIM (z)- fan 0

0

    

-PERIM(:H

 

 

 

:fi-r

Figure 4.2.1. Relationship Between PERIM
(z), DPERIM (z), and z.

37

Expressions for PERIM (z) and DPERIM (2) were
determined for onion bulbs by fitting a ten term Fourier
sine series to the onion shape. This was done by taking
35mm slides of ten individual onions. From the projected
images, the length to maximum diameter ratio was found to
be 3/2. The diameters corresponding to various points
along the central axis of the bulbs were recorded. The
ratios of the diameters to the maximum diameter of each
bulb gave about 200 points outlining the shape of an
average onion with unit diameter. The ten term series
was fitted to these points by the least squares technique
as performed by GAUSHAUS--a computer program supplied by
the Michigan State University Computer Laboratory. Table 2
of the Appendix gives the resulting series. DEPERIM (z)
is obtained by term by term differentiation of PERIM (z)
with respect to 2.

To develop the grid proper, node points were placed
on the central axis. Since the base and the tip of the
onion were thought to play a relatively important role in
the mass transfer process, the node points were placed
progressively closer together near the base and near the
tip. When, for the purposes of model testing, the grid
was developed for a Sphere, eleven node points were placed
on the central axis at points corresponding to 0.16, 1/16,
2/16, 4/16, 6/16, 8/16, 10/16, 12/16, 14/16, 15/16, and
16/16 Of the diameter. When developed on the onion bulb,

an additional point was placed at the length ratio of 9/16

38

so that the total grid would follow the bulb shape more
closely (see Figures 4.2.2 and 4.2.3).

From each of the central axis node points, a line
was projected outward along the principal material direc-
tions. This was done in the following manner: at each
central axis node point, an incremental distance was
formed as one hundredth of the distance between the two
adjacent axial node points. Using this incremental dis-
tance, a grid line was projected outward from the node
point adjusting itself to remain along the radial princi-
pal direction, which was always assumed at right angles
to the axial principal direction. The axial principal
direction was determined by assuming the tangent of the
angle between the principal direction and the central
axis is given by DPERIM (z) multiplied by the ratio r/PERIM
(2). When the grid line so generated reached the surface
of the bulb, it was sectioned to give nodes at points
corresponding to 0/32, 8/32, 16/32, 20/32, 24/32, 26/32,
28/32, 30/32, 31/32, and 32/32 of the line length. The
total grid is shown in Figures 4.2.2 and 4.2.3 for the
two-inch sphere and the two-inch diameter onion bulb,
respectively. When the grid was developed for HMTRAN
and MTRAN, two additional rows of points were added near
the surface at the ratios (as described above) of 61/64
and 63/64. This was done because HMTRAN and MTRAN had to
be designed for cases in which the modeling time would be

much less than the time constant for mass transfer.

39

.mumcmm Hmpofimfia cocHIOBE a How mucflom mpoz .N.N.v musmflm

3: 1.5sz
38 9.6 «.6 ad

 

 

 

.Im) q 4 . . . . u m4 . m
.. ... m m m .H .. f.
.. .. m x. .5 mod
I 3.0
. 1 mod
.. u I mod
I o 8

 

(‘41)) SfllOVH

.COHcO HODOEMHQ SOCHIOBB o How mucflom mooz

3: 1:0sz
36 cud

.m.m.w musmflm

 

0.0 N_.0 00.0 V0.0 «0.0 0
_ L a a _ 1 q _ .2. _ m4 _ q a M. A u :1 0

r. .. . .. m m n ..
o . coo an n n no on. on .L o a
... n m ... «00 V
O o ooh an m o m
4 . m n . l . n
.. .u H. .. coo s
.... 1 mod mm

1 00.0

10.0

 

41

After the grid was developed, the nodal volumes
were determined. The volume of a general interior node
(i,j) was calculated by the following equation (see

Figure 4.2.4):

 

 

 

Vol = firiLi (Bl + Bz)ri+1.j + (A1 + A3”in
1I3 8 ri+l,j + ri,j
+ (B5 + B6)ri-l,j + (Al + A3)ri,{]
r. . + r. .
1-1,] 1’]

+
H

[(87 + B8)ri,j+1 + (A2 + A4)riLj

ri,j+1 i,j

(B + B )r. ._
.+ 3 4 1,3 l. (4.2-1)

4.
P
N
+
3’
a;
H
p.
L____

. . + . .
rl'J-l 1’]

The volume of a surface node was calculated in the same

manner. Referring to Figure 4.2.5,

_ "ri,J B2ri+lLJ + A3ri,J BSri-1,J + A3ri,J
Vol. 4 +

. . . + .
r1+1,J + r1,J r1-1,J r1,J

+
r +r

BBri,J-l + A2ri,J B4ri,J+l + A4ri,J
i,J-1 i,J ri,J+1 + ri,J

(4.2-2)

The volume of a general central axis node is given by (see

Figure 4.2.6)

_ fl 2 2 _
Voli’l — —§§-[A2(3Al + Bl) + A4(3Al + B6) 1 . (4.2 3)

 

42

 

 

 

Ba
Figure 4.2.4. A General Interior Node.

 

 

) "
I I ...—.1- .1
B

5

 

Figure 4.2.5. A General Surface Node.

43

The volumes of the two end nodes are simply (see Figure

4.2.7)
3
Vol = Vol = -— A2 . (4.2-4)

The transfer of heat or moisture from one nodal
volume to another is directly proportional to what
Dusinberre (1961) calls the "geometric factor." This
geometric factor is defined as the effective tranSport
area divided by the distance between the two node points.

Referring again to Figure 4.2.4 for a general
interior node, the geometric factor between node (i,j)

and node (i+1,j) is given by

(B + B )r. . + (A + A )r. .
Gfapi . = 4n 1 2 l+lLJ 1 3 1L3 (4.2-5)
I

j 6A2 + B3 + B8

The geometric factor between node (i,j) and (i,j+l) is

given similarly as

6A + B + B °

Gfrpi . = 4n
'3 1 1 6

The geometric factor between two surface nodes

(i,J) and (i+l,J) as shown in Figure 4.2.5, is

Gfa = 2“ Bz ri+1,J + A3 ri,J
pi,J 3A2 + B3

 

(4.2-7)

The geometric factor between a surface node and the

ambient surroundings is somewhat different from the above

44

 

 

 

B. .
l+l,l H-l,2
A2 ' 90
A; L
l,l (,2
A4 3 B7
I-I,l i-I,ZJ
94 I“

Figure 4.2.6. A General Central Axis Node.

 

 

 

 

Figure 4.2.7. An End Node.

45

in that transfer between the two is proportional to a
heat or mass transfer coefficient, instead of a con-
ductivity or a mass diffusivity. Referring to Figure

4.2.5,

Gfrpi = W
I

J ri,J (A2 + A4) . (4.2-8)

With reference to Figure 4.2.6, the geometric
factor between a central axis node and its axial neighbor
is given by

2
1r (Bl + Al)

Gfapi’l = 4 (3A2 + BB) (4.2-9)

 

The geometric factor between a central axis node and its

radial neighbor is given by

(4.2-10)

Gfrpi’l = A2 + A4 + B7 + 138 .

The end nodes present a somewhat different case
in that heat and moisture transfer takes place between
them and all of the adjacent nodes as well as with the
ambient. Thus, referring to Figure 4.2.7, the geometric
factor between the end node (1,1) and the node (2,j) is

given by

2’1 (4.2-11)

 

64m

Gfapl'j =

46

The geometric factor between (1,1) and the surface node

(2,J) is

(4.2-12)

wIN
O

Gfapl J =

The geometric factor between (1,1) and (2,1) is given by

2
2 Bl
Gfapl’l - '3' T . (4.2‘13)

2

The geometric factor between (1,1) and the ambient is
Gfrpl,J = __4_ . (4.2-14)

After having determined the geometric factors, the
transfer parameters are assigned to each node. In HTRAN
these consist of ka, kr’ c, p, as well as h for the surface
and end nodes. In addition to the heat transfer parameters,
the initial values for D$a, Dfir,

in HMTRAN. In MTRAN, only D' , D'
ma mr

and hD have to be assigned

, and hD are assigned.
This completes the development of the grid system.

The next section describes the solution of the system of

resulting node equations.

4.3 Solution of Node Equations

 

The Alternating Direction Explicit Procedure
(ADEP) for the solution of the conduction equation was
reviewed in Section 2.7. The present section describes

how this method was utilized to solve the node equations

47

resulting from the finite difference grid outlined in the
previous section. The case of pure heat conduction is
developed first, after which the modifications necessary
to include moisture transfer are described.

As was mentioned in Section 2.7, the ADEP is a
two pass method. However, when attempting to model a
homogeneous, isotropic sphere for the purposes of model
testing, it was found that, for the present grid system,
the two pass system was not truly symmetrical. Instead,

a four pass system was used which proved satisfactory.

The first pass starts at the bottom center of the product

and progresses row by row (moving outward upon each) until
it reaches the top. If this progression is designated as

northeast, the other three passes may be simply described

by the directions southwest, southeast, and northwest.

The following paragraphs outline the equations used
for the first pass. The equations used on the other passes
are all similar.

For an interior node, as shown in Figure 4.2.4,
the equations are the same as described in Section 2.7.

Thus, for the first pass,

(n) (n+1)
Vol T ./At + kaGfapi—l,jTi-l,j

. .C. . .. .
T(n+1) = ‘31.] 1I3 1I15 1IJ

 

1,] pi,j Ci’j Voli’j/At + kaGfapi-l,j + krGfrpi,j-l
(n+1) [ (n) _ (n>J
+ krGfrpi,j-1T + kaGfapi,j Ti+1gi Ti,j

 

(n) _ (n)]
+ krGfrpi.j [ 1.1+1 Tin .

(4.3-l)

48

Heat transfer between the product surface and the
ambient is proportional to the surface heat transfer coef-
ficient. Thus, the above equations are modified for a
surface node, as shown in Figure 4.2.5, to yield for the

first pass,

 

 

T(n) (n+1)
T(n+1) = pLJ cLJ V011, J Ti, J/At + k aGfapi- -L JTi- 1, J
1,J pi,J ci,J Vo J/At + k aiGfap -l J + k rGfrpi j- -l
(n+1) [ (n) T(n)
krGfrp1,J-l T ,J-l + kaGfapi,JTi+l,J Ti, J
m]
+ h Gfrp ,J [Ta T1,J . (4.3-2)

 

Since a central axis node, as shown in Figure

4.2.6, has only three faces, the first pass equation is

T(n) (n+1)
T(n+1) _ 01,1 1,1 V011,1T1/At + k aGfapi- 1, lTi-l, 1

T1,1 pi l c. VO l/At + k aGfapi_ 1 1

(n) T(n)
+ k aGfapL 1 [TTi+l,l T1,1)

+ krGfrpi'l [T in) - T(n)] .

The end nodes are unique in that each has more than
four adjacent nodes as shown in Figure 4.2.7. The first

pass equation for the node (1,1) is

49

 

J
(n)
p c Vol T /At + Z k Gfap .
T(n+l)= 1,1 1,1 1,1 1,1 i?1 a 1,3
1,1 01,1 Cl,l Voll,1/At + hl Gfrpl’J

(n) _ (n)
T2,j T1,j + h1 GfrpLJ Ta . (4.3-4)

 

The heat transfer model was tested by comparing
the analytical and model temperature histories for the
center of a two-inch diameter sphere initially at uniform
temperature and subjected to a step change in ambient
temperature. The surface heat transfer coefficient was
set at 500 Btu/hr ftZF, the initial temperature was uni-
form at 40 F, the ambient temperature was 100 F, the
density was 58.8 lbm/ft3, the specific heat was 0.93 Btu/
lbmF, and the axial and radial thermal conductivities were
0.3 Btu/hr ft F. The results, as shown in Figure 4.3.1,
indicate that the model agrees reasonably well with the
analytical solution which was calculated by the following
equation (Grigull, 1964):

B 2 sin Vn - vn cos v --v2 kt
T = 100 - 60 Z v - sin v cos v ech———7§—).
m=l n n n pc r

 

(4.3—5)

The eigenvalues vn in the above equation are the solutions

to the following transcendental equation:

50

 

0.3 0.4 0.5 0.6 0.7
TIME Um)

0.2

0.l

 

_ _

 

2.0 "

_ _

o o o. o. o.
In In 9. av
NEH—RGNQENP ...zm_m2< 02¢ JdEz. NI... zumxrrum mozmmmuua
NIP no m¢<hzmomma < m< ommmmmaxm mommm ...—moo: mm<o ...mm._.

Figure 4.3.1.

Test Case Model Error Using the ADEP on

a Two-Inch Diameter Sphere.

51
v cos v = (1 - 25) sin v (4 3-6
h n k n ° '

Only eight terms of the infinite series were used in deter-
mining the analytical solutions. The analytical results
for times less than 0.04 hours were discarded due to the
nonconvergence with only eight terms.

The author tried various modifications of the pro-
gram in an attempt to minimize the error shown in Figure
4.3.1. As shown in the Appendix, HTRAN was written so
that only a few statements would have to be changed to
convert the ADEP to that of Barakat and Clark (see Section
2.6). When this was tried, however, it was found that the
error was not affected. Since the Barakat-Clark scheme
requires in this case four times more storage and four
times more computing time, the Barakat-Clark modification
was abandoned.

The second modification tried was to further sub-
divide the region by adding more nodes in the radial
direction. This modification reduced the positive peak
shown in Figure 4.3.1, but did not affect the negative
one.

The third modification tried was to vary the axial
spacing of the node points to one in which the incremental
distances in 2 were more uniform. This did not signifi-

cantly affect the error.

52

The fourth modification tried was to further sub-
divide the region by adding more nodes in the axial
direction. This increased slightly the magnitude of
both the positive and negative peaks shown in Figure 4.3.1.

The fifth modification tried was to reduce the
time increment by a factor of 100. This did not affect
the magnitude of the error.

The sixth modification tried was to replace the
geometric factors based on the average area between the
nodes by factors taking into account the fact that the
area between nodes is not constant. For example, if this
variation is assumed linear, the axial geometric factor

between two general interior nodes becomes (see Figure

 

 

4.4.2)
B+B..-A+A..
Gfa = fl( 1 2)r1+l,3 ( l 3)r1,3 (4 3_7)
pi,j (B + B )r. . '
A 1n 1 2 71+1L3
2 7A1 + A3)ri,j

If the variation is assumed quadratic, the radial geo-
metric factor between two general interior nodes becomes

(see Figure 4.2.4)

 

 

 

 

Gfrp. . = w/ri’j(A2 + A4F[(B7 + B8)ri,j+l - (A2 + A4)ripfl
1,3
A tan-1 /(B7 + BBri,j+l _ 1
1 (A2 + A4)ri j
I

(4.3-8)

53

When these equations were used in the model, the positive
peak shown in Figure 4.3.1 remained the same but the magni-
tude of the negative peak was increased.

Barakat and Clark (1966) gave the following
equation for the truncation error for a uniform grid in

a cartesian coordinate system:

- At 32T

i,j,n+l AX atax i,j,n+l

M

II
m'o)
('1' H

 

 

 

 

+ . . . . (4.3-9)

 

The failure of the modifications to improve the
model error appears to contradict equation (4.3-9). It
would thus seem that a different expression for the trun-
cation error would be needed for this grid system. The
author, however, was unable to derive such an equation

due to the complexities of the grid system.

As shown by the listings of HTRAN, HMTRAN, and
MTRAN in the Appendix, the modifications necessary to model
mass transfer are minor: first, since the diffusion coef-
ficient Dfi is a function of the moisture content, it was

re-evaluated for each node at the start of each group of

54

four passes; and second, to prevent instability at the
boundary, the mass flux at the surface was always evalu-
ated at the time n+1 for each pass. This was done by
approximating the boundary layer vapor concentration at

time n+1 by

 

(n+1) (n)
m. . C
céggl) = 1'3(n) WVS . (4.3-10)
m. .
1,]

(n)

was evaluated
wvs

The boundary layer vapor concentration C

as a function of min; by means of the straight lines
I

approximation shown in Figure 7.2.7.

V. DENSITY

A very simple experiment was performed to determine
the density of onion flesh. Eleven bulbs were weighed
individually in air. A metal sinker was then sewed to
each and each weighed in water. The temperature of the
bulbs and of the water was 70 F. The moisture content of
the bulbs was 9.00 (d.b.).

Table l of the Appendix shows the results. The
statistical data shown was calculated with the aid of
BASTAT, a basic statistical computer routine supplied by
the Michigan State University Agricultural Experiment
Station. As shown in the table, the average specific
gravity was 0.944. This corresponds to a density of 58.8
lb/ft3. The moisture content of the bulbs was 9.00 (d.b.).

The assumption that shrinkage upon drying is negligible

yields
and
3
de = 5.88 lbm/ft . (5-2)

55

56

Equations (5-1) and (5-2) were used in all calculations

involving the density in this research.

VI . HEAT TRANSFER

6.1 Introduction

 

The thermal conductivity of wood is considerably
greater parallel to the grain (0.23 Btu/hr ft F) than
perpendicular to the grain (0.12 Btu/hr ft F) (Rohsenow
and Choi, 1961). It was felt that this anisotropy might
also be present in onion flesh. Thus, experiments were
performed to determine the conductivity in both the "axial"
and "radial" directions (see Section 4.1).

The mathematical model for heat and mass transfer
has been described in Chapter IV. In most biological
products, the time scale for heat transfer phenomena is
much smaller than that for moisture transfer phenomena.
This allows separation of the two for modeling purposes
(Young, 1968).

When conducting an experiment to determine a
property such as thermal conductivity, the experiment
should be planned so that the measured quantity, such as
temperature, is highly dependent upon the desired property.

The "sensitivity coefficients,‘ defined in this case by
the ratio of the change in temperature at a particular

location and time to an assumed change in thermal

57

58

conductivity, give an indication of the accuracy obtainable
in a particular experiment. More than one property may be
obtained from a single experiment if the respective sensi-
tivity coefficients are not closely correlated to each
other and are of sufficient magnitude. The sensitivity
coefficients AT/Akr and AT/Aka for the center (the point

on the central axis at which the onion diameter is a maxi-
mum) of a two-inch diameter onion are shown in Figure 6.1.1.
To determine these sensitivity coefficients, both kr and

ka were assumed to be 0.3 Btu/hr ft F while the initial
temperature was taken as 40 F with the boundary condition
consisting of a step change in ambient temperature to 100 F
with a heat transfer coefficient of 500 Btu/hr ft F. The
changes in the center temperature versus time due to small
(1%) changes in the assumed values of thermal conductivity
were obtained by solving the model three times--once at the
base values, once with an increased kr, and once with an
increased ka. The sensitivity coefficients were calcu-
lated by dividing the resulting changes in center tempera-
ture by the causal change in thermal conductivity. As the
figure shows, the center temperature is much more sensi-
tive to kr than to ka' Thus, placing an onion initially

at 40 F in a constant temperature bath at 100 F could be
expected to give kr (as described in Section 6.3) but not
both kr and ka. An alternative method (described in the

next section) was used to find ka.

59

 

 

 

 

SENSITIVITY COEFFICIENTS (Pam: I'm/em.)

l l2 t
96 -
80 -
64 —
RADIAL
48 -
32 F
IG -
l AXIAL
o J L J i
0 0.!25 0.250 0.500 0.750 |.0
TIME Um)
Figure 6.1.1. Temperature Sensitivity Coefficients

at the Center of a Two-Inch Diameter
Onion.

60

6.2 Axial Conductivity

 

As outlined in the previous section, the tempera-
ture history of an onion, when subjected to a step change
in ambient temperature with a uniform heat transfer coef-
ficient existing over its surface, is not highly dependent
upon the axial conductivity of the onion flesh ka. How-
ever, since the axial conductivity is a parameter in the
model, and since it was desired to be able to model the
case in which the surface heat transfer coefficient is a
function of 2 (see Figure 4.2.1), determination of ka was
included among the objectives of the research.

In order to determine the axial conductivity, the
sample holder shown in Figure 6.2.1 was constructed. The
holder consists of two metal cups lined with insulation.

A heat source consisting of a copper block one-inch in
diameter by 0.66 inches long was placed in a cavity in the
top cut. Using a cork borer, a one-inch diameter by one-
inch long sample was cut from the center of an onion and
placed in the bottom cup. After cooling the bottom cup

and specimen to 40 F and heating the top cup and block

to 100 F, the cups were brought together and the tempera-
ture histories of the block and sample were taken by means
of three thermocouples attached to the block and five
thermocouples imbedded in the sample. An existing system
developed by Dr. J. V. Beck (Mechanical Engineering Depart-
ment) was used to collect and record the data. The thermo-

couple voltages were amplified and sent to an IBM 1800

61

 

COPPER HEAT
SOURCE

   
  
  

THERMOOOUPLE

SAMPLE

 

INSULATION

CASE

Figure 6.2.1. Cross Sectional View of the Sample
Holder Used in the Axial Conductivity
Tests.

62

hybrid computer which converted the voltages to tempera-
tures, and punched the temperature-time histories onto
cards. Each experimental run lasted 7.5 minutes during
which 150 readings were taken at each thermocouple.

Two existing nonlinear estimation computer pro-
grams developed by Beck (1964, 1966, 1968) were used to
analyze the data. The first used the temperature history
of the copper block to estimate the heat flux from the
block to the sample. The heat flux was assumed constant
during each of the time intervals between the thermocouple
readings. These discrete heat fluxes were adjusted by the
program so that the sum of the square differences between
the experimental block temperatures and the temperatures
as generated by a one dimensional Crank-Nicolson finite
difference model of the block, which included the dis-
cretized heat fluxes, was a minimum.

The second program used the temperature history
of the sample and the heat fluxes determined by the first
program to find the thermal conductivity of the sample.
As in the first program, the sum of the squared differ-
ences between the experimental temperatures and those
generated by a one dimensional Crank-Nicolson model of
the sample was minimized--in this case by adjusting the
thermal conductivity.

The results of five experimental runs are shown

in Table 3 of the Appendix. The second column in the

63

individual results section of the table shows the root mean
square (RMS) difference between the calculated and the
experimental sample temperatures for each run. From this
column, it can be seen that the numerical model fits some
runs better than others. It is believed that the higher
RMS values were caused by inaccuracies in the measurement
of the thermocouple position within the sample. Thus, it
would seem justifiable to weight the results of each run

by the inverse of the RMS value to obtain a better estimate
of the axial conductivity. This weighted average is given
in the results section of Table 3 as 0.30 Btu/hr ft F and

was used in the calculations involving ka in this research.

6.3 Radial Conductivity

 

As shown in Section 6.1, the temperature history
at the center of an onion subjected to a step change in
ambient temperature is highly dependent upon the radial
thermal conductivity kr. This is the basis for the experi-
ments performed to determine kr'

Thermocouples were placed at the center (the point
on the central axis corresponding to the maximum diameter)
of each of five onions. After cooling to a uniform temper-
ature of 40 F, the onions were placed in an agitated water
bath at 100 F. The thermocouple voltages were amplified
and sent to the IBM 1800 hybrid computer which punched the
temperature-time histories onto cards. Readings were taken

every ten seconds over an interval of twenty-five minutes.

64

The convective heat transfer coefficient h was found to
be 500 Btu/hr ft2 F from the temperature history of a
copper block placed in the bath. While the geometry of
the onion is different from that of the block, it was felt
that this manner of determining h was sufficiently accur-
ate since the Biot number of a two-inch diameter sphere
with a thermal conductivity of 0.3 Btu/hr ft F and a
surface heat transfer coefficient of 500 Btu/hr ft F is
139. This would indicate that the surface heat transfer
coefficient is sufficiently large so that the temperature
response of the onion is governed by the transfer proper-
ties of the onion and not by the surface heat transfer
coefficient.

To determine kr' a non-linear estimation program,
GAUSHAUS, as supplied by the Michigan State University
Computer Laboratory, was used in conjunction with the heat
transfer model described in Chapter IV to find a kr that
minimized the sum of the squared differences between the
experimental and calculated temperatures for the center
node of the onion. The results are shown in Table 4 of
the Appendix. As in the case of axial conductivity, the
weighted (by l/RMS temperature difference between the
calculated and experimental temperatures) mean is included

(0.217 Btu/hr ft F).

65

6.4 Heat Transfer Simulation

 

The heat transfer model was solved for a set of
convective heat transfer coefficients ranging from 0.5-
500 Btu/hr ft2 F and a set of bulb diameters ranging from
1-3 inches. The results are shown in Figure 6.4.1. The
ratio of the heat transferred to that transferred at
infinite time is plotted versus the dimensionless grouping
th/pckr at various values of hd/kr.

From this chart, it is possible to determine the
time necessary to raise the average bulb temperature to
any desired level if the initial temperature, the bulb
diameter, the ambient temperature, and the convective
heat transfer coefficient are known.

If Figure 6.4.1 is compared to the corresponding
figure for spheres (Holman, 1963), it is seen that the two
agree within the accuracy that these charts may be read.
Thus, for process evaluation, such figures may be substi-

tuted for Figure 6.4.1.

en:

.musumummfima ucmflnfi< CH
emcmcu meum m on pmpomnbsm madam cease on Hemmcmue ummm

.80Q\~N£
no. No. 6. _ ..o. N.o.

 

66

-

q A _ _

.H.e.e messes

n..o.

N
0

Q’
0'
EN”. SLINIANI 1V OBHHBJSNVUJ.

‘9.
CD
lVI-IJ. 0.1. OBUUBJSNVUJ. .LVBH :IO OllVH

“I
o

 

0..

VII. MOISTURE TRANSFER

7.1 Introduction

In Section 2.5 it was shown that D; and D5 are

related by the equation

Blna
w

 

Dm = D . (7.1-l)

I
u 31nm

This means that the two equations modeling one dimensional

moisture transfer

 

s Cwa 3“W
0 — ' -
Jw — RT Du 82 (7'1 2)
and
3'5 = - o 0' -——3m (7 1—3)
w dm m 32 °

are equivalent if D; and DA satisfy equation (7.1-1).
Section 7.2 of this chapter describes the eXperi-

ments conducted to determine D; and D$ in onions. Section

7.3 describes the results of including D$ in the mathe-

matical model outlined in Chapter IV.

67

68

7.2 Determination of Moisture
Diffusivity

 

A modified Denny osmometer as described by Dumbroff
and Webb (1968) (Figure 7.2.1) was constructed to determine
the moisture diffusivity of onion flesh. The osmometer
consists basically of a distilled water chamber connected
to a sample holder. The rate of water loss from the
chamber can be monitored by observing the movement of the
water meniscus in a capillary tube attached to the chamber.
The unit was originally designed to be used with another
chamber on the other side of the sample in which was placed
a stirred sugar solution. Since sugar solutions become
quite viscous at water activities below 0.95 and saturated
near 0.9, this chamber was removed for water activities
below 0.95. The entire osmometer was placed in an air
stream with closely controlled temperature and relative
humidity produced by an "Aminco-Aire"* unit. Close control
of temperature was necessary since the position of the
meniscus within the capillary tube is very temperature
sensitive.

It was desired to model Db over the respective
ranges of water activity and temperature of 0.4 to 1.0
and 40 to 100 F. Inspection of equations (2.5-4), (2.5-8)
and (2.5-12) shows that if the velocity of the medium with

respect to the solids vmS is small, as is small, and

 

*A device producing a stream of constant tempera-
ture and humidity air manufactured by the American Instru-
ment Co., Inc., Silver Spring, Maryland.

69

.ueumEoEmo macmo ecu mo BmH> Hmcofluomm mmouo .H.~.n whomflm

no.0 u to
CO... Gm>ozmm

     

      

   

20.5.5»
m<oam

rah;
Duds—EMS

 
   

      

    

 

 

MJQZ<m

 

 

   
 

mummfib mun... «5.3.... mun: >¢<Jn=a<0

   

70

D; = R T Q . (7.2-l)

This is the rationale behind attempting to approximate

l
Dur by

Q
D' = RT [01 m 2

“r + 03(T — 530)] . (7.2-2)

Table 5 of the Appendix summarizes the results of

Q and Q of

the experiments conducted to determine 91, 2 3

equation (7.2-2). By equation (7.1-2),

 

 

an
s _ _ pdm m . w _
JW'- RT D 82 ' (7'2 3)
and
02

l p m RT[Q m + Q (T-530)]da
jsAL — f d“ l 3 w (7.2-4)
w — a

aw w

where 5; is the water activity of the fluid to the right of
the sample (see Figure 7.2.1). Evaluation of this integral
requires that m be expressed as an explicit function of aw.
Saravacos (1960) determined the moisture isotherm
for onion flesh at 80 F. His data are summarized in
Table 6 of the Appendix and in Figure 7.2.2. In order to
express m explicitly in terms of aw, the isotherm was
approximated as shown in Figure 7.2.2 by three connecting
straight line segments passing through the points (0,0),
(0.022, 0.22), (0.530, 0.863), and (9.00, 1.0). The first

and last of these are not from Saravacos' data-~the first

MOISTURE CONTENT (d. b.)

71

lot

L____L__.=ij vpi V 1L" E 1 1 L
0 0.|25 0.250 0.500 0.750 LO
WATER ACTIVITY

 

0

Figure 7.2.2. Onion Moisture Isotherm.

72

was included so that the straight lines approximation to
the isotherm would pass through the origin; the second,
so that the approximation would pass through the initial
onion bulb moisture content at saturation. Thus, the

simplified relationship between m and aw is given by

m = .l a for 0 < a < 0.22 , (7-2-5)
w — w —

m = 0.15181026 + .79004666 aw for 0.22

and
m = -52.824817 + 61.8248175 aw for 0.863

< a < 1.0 . (7.2-7)

The non-linear estimation routine, GAUSHAUS, was
used again (see Section 6.3) to vary the parameters 01, 02,
and 93 so that the sum of the squared differences between
the experimental mass fluxes and those predicted by
equation (7.2-4) was a minimum. The integration of
equation (7.2-4) was performed numerically by a Romberg

integration routine. The resulting values are shown in

the following equation:

18 m 6.023

+ 1.779 X 10-l4(T-530)).
ur

D' = RT[4.5535 x 10'

(7.2-8)

73

Inspection of equation (7.2-9), which is shown in
graphical form in Figure 7.2.3, reveals that negative
values are predicted for DLr when both the temperature
and moisture content are low. This was prevented when
using equation (7.2-8) in the mass transfer model by
setting Dfi equal to zero whenever this occurred.

In Section 7.1, it was shown that D$ is obtained
by multiplying DL by Blnaw/Blnm. Differentiation of

equations (7.2-5), (7.2-6), and (7.2-7) yields

 

Elnaw
31nm = l for O i m i 0.022 , (7.2-9)
Blna

w m

 

31m = m + 0.15181026 for 0.022 < m 3 0.0.530, (7.2—10)

 

and

alna
____E.= m
31nm m + 52.824817

 

for 0.530 < m i 1.0 . (7.2-ll)

In Section 6.1, it was shown that the temperature
history of the onion bulb is much more dependent upon kr
than ka. Likewise, the mass transfer within the product
is much more dependent upon D$r than upon Dfia. With this
in mind, it was decided to model Dfia by simply assuming
it to be a constant multiple of Dfir. To determine the

ratio between D$ and Dfir, five experimental runs with

a

0.22-inch-thick slices cut from the center portion of the

bulb were performed under the same conditions as run 6

74

.an >HH>HmDMMHQ ensemfloz pmuoflpmum .m.m.> wusmflm

 

A... .3 hzmkzoo map—.902
m t n N _ o

 

 

 

1
‘9 ‘3 C“ C)

I l l I J J I 1 JV
‘f 0| C! a) “I Q’ GI C, I!

 

_1_
‘9
0'

(“I *3») ,0I I: do All/\Isnddlo

75

of Table 5 of the Appendix. The average mass flux was

2 2

1.80 x 10’ lbm/ft hr. Comparing this result with

3

4.35 x 10' 1bm/ft2hr, the mass flux obtained from

equation (7.2-4) under the same conditions yields

D' = 4.14 x 102 0' . (7.2-12)
ua ur

Before concluding this section, the validity of a
tacit assumption made earlier should be established. The
above method rests upon the assumption that the resistance
to mass transfer through the specimen lies wholly within
the sample proper and the surface resistance to the right
of the sample is negligible. (See Figure 7.2.1.) (The
surface resistance to the left of the sample is obviously
zero since the fluid is pure water.) Reynolds analogy
states that the convective heat transfer coefficient and

the convective mass transfer coefficient are related by

 

hD = C 2/3 (7.2-13)

where pai is the air density, c . the specific heat, and

r air

Le the ratio of the thermal diffusivity of air to the

mutual diffusion coefficient of the air-water vapor mix-

ture. If h is assumed to be 5 Btu/hr ft2 F; pair’

0.075 lbm/ft3; c , 0.24 Btu/lbm F and Le, 0.845, then

air

hD is 312 ft/hr. Since

76

h M

.s _ D w _ _
Jw - RT ( wvs Pwva) ’ (7'2 l4)

 

the maximum water vapor pressure difference from the sur-
face to the ambient (using the mass flux of 1.4 x 10-3
1bm/ft2hr from run 4 of Table 5 of the Appendix) is
1.425 x 10-3 psi. Since the minimum water pressure drop
from the pure water chamber to the ambient is 0.036 psi

(run 5), the drop in pressure from the surface to the

ambient is negligible.

7.3 Mass Transfer Model Testing

 

To obtain data with which to test the mathematical
model outlined in the previous sections and Chapter IV,
five two-inch diameter onion bulbs were placed in the air
stream of an “Aminco-Aire" unit for twenty-four days. The
temperature was maintained at 70 F and the relative humidity
at 31%. One of the onions sprouted after a week; the
average weight loss from the other four versus time is
shown in Figure 7.3.1.

Predicted moisture losses for the same conditions
were obtained by solution of the mathematical model and
are included in Figure 7.3.1. The convective mass transfer
coefficient hD was set to 300 ft/hr. The initial moisture
content was assumed to be 9.00.

As Figure 7.3.1 shows, there is only fair agreement
between the predicted and experimental values. The differ-

ence between the two is believed due to three factors:

WEIGHT L083 1: I03 (lb...)

2<,+ 77

'9

I7-
I6-
l5”
l4-
l3-

 

 

 

 

l0 *-
9 '" ‘91
_. 0“
8 9‘“ c
Q(’
7 " (j' o
6 )-
5 I- ‘ PR ' ' A
4
3
. 2
I
11 i J 1 L
200 300 400 500 600
TIME (ha)

Figure 7.3.1. Moisture Loss From Onion Bulbs at
70 F and 31% Relative Humidity.

i)

ii)

iii)

78

The predicted values are based on the assumption
that the initial moisture content distribution
is uniform. This accounts for the predicted
moisture loss values being greater than the
experimental values at small time values.
Respiration losses are not accounted for in
the predicted weight losses.

As the onion dries, the outer scales crack

and loosen uncovering additional surface area
of higher moisture content. This accounts

for the experimental moisture losses being

higher at the longer times.

VIII . CONCLUSIONS

The first objective for this research as given in
Chapter III was to determine the density of onion bulbs.
The final value obtained was 58.8 1bm/ft3.

The second objective listed was to obtain values
of thermal conductivity in the axial and radial directions.
The final values obtained were 0.30 Btu/hr ft F and 0.217
Btu/hr ft F for the axial and radial thermal conductivity
respectively.

The third objective was to obtain values for the
moisture diffusivity. The predicted values for the radial
diffusivity DLr and the axial diffusivity Dfia are given by

the equations

0hr = RT[4.553 x 10’18 m6°023 + 1.779 x 10'14(T-530)J
(8-1)
and
0' = 414 0' . (8—2)
113 Llr

The final objective was to develop a mathematical

model to simulate the reSponse of the onion bulb to its

79

80

environment. To achieve this objective, the Alternating
Direction Explicit Procedure (ADEP) for solving the trans-
fer equation in cartesian coordinates was adapted to a non-
uniform grid system based on the onion shape.

The heat transfer portion of the model simulated
the actual process very well as indicated by the small
RMS and maximum differences between the predicted and
experimental center temperatures observed during the
tests for radial thermal conductivity (see Table 4).

The mass transfer portion of the model was in only
fair agreement with the experimental weight losses observed

as a test of the model.

Suggestions for Further Study
The author concludes that further work is needed:
1) to determine the magnitude of the weight losses
due to respiration compared to those due to
moisture loss,
ii) to measure the effect of the cracking and
loosening of the outer layers of the bulb, and
iii) to include this last effect in the modeling

of the mass transfer process.

BIBLIOGRAPHY

BIBLIOGRAPHY

Allada, S. R. and Quon, D. (1966) A stable, explicit
and numerical solution of the conduction equation
for multi-dimensional non-homogeneous media.
Chemical Engineering Progress Symposium Series.
Heat Transfer. Los Angeles. 26:151.

 

 

Arpaci, V. S. (1966) Conduction Heat Transfer. Addison-
Wesley Publishing Company, Reading, Massachusetts.

 

Babbitt, J. D. (1940) Observation on the permeability of
hygroscopic materials to water vapor. Canadian J.
of Research. 18(A):105.

 

 

Barakat, H. Z. and Clark, J. A. (1966) On the solution
of the diffusion equations by numerical methods.
Trans. ASME, J. Heat Transfery Series C. 88:421.

Bakker-Arkema, F. W., Rosenau, J. R., and Clifford, W. H.
(1969) Measurements of grain surface area and its
effect on the heat and mass transfer rates in fixed
and moving beds of biological products. ASAE Paper
No. 69-356.

Barker, J. J. (1965) Heat transfer in packed bed.
J. Ind. Engr. Chem. 57:43.

 

Beck, J. V. (1964) The Optimum Analytical Design of
Transient Experiments for Simultaneous Determi-
nations of Thermal Conductivity and Specific Heat.
Unpublished thesis, Michigan State University, East
Lansing, Michigan.

Beck, J. V. (1966) Analytical Determination of Optimum,
Transient Experiments for Measurements of Thermal
Properties. Proc. Third International Heat Transfer
Conference. IV:74.

Beck, J. V. (1968) Surface heat flux determination using
an integral method. Nuclear Engr. and Design.
7:170.

 

81

83

Jason, A. C. (1958) A study of evaporation and diffusion
processes in the drying of fish muscle. Fundamental
Aspects of the Dehydration of Foodstuffs. Society of
Chemical Industry, London.

 

 

Jones, H. A., and Mann, L. K. (1963) Onions and Their
Allies. Interscience, New York.

 

Jost, W. (1960) Diffusion in Solids, Liquids, Gases.
Academic Press, Inc., New York.

Katchalsky, A. (1961) Membrane permeability and the
thermodynamics of irreversible processes. Membrane
Transport and Metabolism. Proceedings of a Sym-
posium held in Prague, August 22-27, 1960.

 

Kedem, O. (1961) Criteria of active transport. Membrane
Transport and Metabolism. Proceedings of a Sym-
posium held in Prague, August 22-27, 1960.

 

Kozlowski, T. T. (ed.) (1968) Water Definits and Plant
Growth. Vol. I. Development, Control and Measure-
ment. Academic Press, New York.

 

Kramer, P. J. (1969) Plant and Soil Water Relationships;
A Modern Synthesis. McGraw-Hill Book Co., New York.

 

 

Kuprianoff, J. (1958) Bound water in foods. Fundamental
Aspects of Dehydration of Foodstuffs. Society of
Chemical Industry, London.

 

 

Lewis, G. N. and Randall, M. Revised by Pitzer, K. S.
and Brewer, L. (1961) Thermodynamics. 2nd Edition.
McGraw-Hill Book Co., New York.

 

Lykov, A. W. (1955) Experimentalle und Theoretische
Grundlagen der Trocknung. V. E. B. Verlag Technik,
Berlin.

 

 

Lykov, A. W. and Mikhaylow, Y. A. (1961) Theory of
Energy and Mass Transfer. Prentice—Hall, Inc.,
Englewood Cliffs, New Jersey.

 

 

Meeter, D. A. (1965) Non-Linear Least-Squares (GAUSHAUS).
Michigan State University Computer Laboratory Pro-
gram Documentation No. 87.

 

Moore, W. J. (1962) Physical Chemistry. Prentice—Hall,
Inc., Englewood Cliffs, New Jersey.

 

84

Ngoddy, P. O. (1969) A Generalized Theory of Sorption
Phenomena in Biological Materials. Unpublished
thesis, Michigan State University, East Lansing,
Michigan.

Ozisik, M. N. (1968) Boundary Value Problems of Heat
Conduction. International Textbook Co., Scranton,
Penn.

 

Reidy, G. A. (1968) Thermal Properties of Foods and
Methods of Their Determination. Unpublished thesis,
Michigan State University, East Lansing, Michigan.

Riedel, L. (1951) The refrigerating effect required to
freeze fruits and vegetables. Refrigerating
Engineering. 59:670.

 

 

Roa, G. and Bakker—Arkema, F. W. (1969) Moisture ad-
sorption by freeze—dried meat cubes. ASAE Paper
No. 69-893.

Rohsenow, W. M. and Choi, H. Y. (1961) Heat, Mass, and
Momentum Transfer. Prentice-Hall, Inc., Englewood
Cliffs, New Jersey.

 

Saravacos, G. D. (1962) A study of the mechanism of
fruit and vegetable dehydration. Food Technology.
16:78.

Saravacos, G. D. (1960) Studies of the Mechanism of
Fruit and Vegetable Dehydration. Unpublished thesis,
Massachusetts Institute of Technology, Cambridge,
Massachusetts.

Siebel, E. (1892) Specific heats of various products.
Ice and Refrigeration. 2:256-7.

 

Van Arsdel, W. B. and Copley, M. J. (1963) Food Dehye
dration. Vol. 1. Principles. AVI Publishing
Company, Inc., Westport, Connecticut.

 

Wang, J. K. (1958) Theory of Drying. Unpublished thesis.
Michigan State University, East Lansing, Michigan.

Wright, R. C., Lauritzen, J. I., and Whiteman, T. M.
(1935). Influence of Storage Temperature and
Humidity on Keeping Qualities ogOnions and Onion
Sets. Technical Bulletin No. 475. United States
Department of Agriculture.

Young, J. H. (1969) Simultaneous heat and mass transfer
in a porous, hygroscopic solid. Transactions of the
ASAE. 12(5):720.

APPENDIX

85

 

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86

 

 

 

 

TABLE 2.--The Fourier series approximating the onion shape.
d 10 . i n z
PERIM (a) = 2 131 Ai Sin (1.5 d)
1 Ai
1 3 0.8322400
2 0.3689900
3 0.0462150
4 0.0074205
5 0.0822030
6 -0.0114820
7 0.0194130
8 0.0074970
9 0.0145270

10 -0.0046454

 

87

TABLE 3.--Results of the axial thermal conductivity experi-
ments.

 

RMS Difference Between
Run ka(Btu/hr ft F) Calculated and Experimental
Temperatures (F)

 

Individual Experiments

 

 

 

1 0.32518 _ 1.38

2 0.251 1.44

3 0.322 0.56

4 0.324 1.36

5 0.241 2.55

Results

Average Axial Conductivity 0.293 Btu/hr ft F
Standard Deviation 0.042 Btu/hr ft F
Standard Error of the Mean 0.019 Btu/hr ft F

Weighted (by l/RMS) Average
Axial Conductivity 0.30 Btu/hr ft F

 

 

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mouuo>< xmzm\a use evacuees

 

 

88

 

 

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90

86 F (Saravacos,

1960).

TABLE 6.--Data points for the onion moisture isotherm at

 

Water Activity

Moisture Condent (d.b.)

 

0.112
0.220
0.328
0.436
0.539
0.633
0.756
0.863

0.907

0.012
0.022
0.077
0.148
0.190
0.305
0.330
0.530

0.925

 

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MICHIGAN STATE UNIV. LIBRARIES
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