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ABSTRACT

INFLUENCE OF FOOD PRODUCT PROPERTIES
ON THE FREEZING TIME
By

Rong-Ching Hsieh

The complicated factors associated with prediction of the freezing time of a food

' product are due to the continuous changing of food product properties and the lack of

accuracy in measurement of the freezing characteristics of the product. Although Plank’s
equation has been widely used to predict the freezing time of a food product, the
assumptions used in formulating the equation are too risky to be accepted for a transient
heat transfer problem. Several attempts have been made to overcome these difﬁculties

by “utilizing a finite difference model and digital computers. For the most successful
simulation model as developed by Gorby (1974), thermal property data are required only
for unfrozen products. In addition, the temperature function curves for product properties
and the freezing history curves can be generated besides the predicted freezing time.

The objective of this investigation was to utilize the computer simulation techniques
to predict the freezing times and temperature history curves for different food products.
Furthermore, the influences of some product properties of an actual food product on the
freezing time can be explained by investigating the independent effect of the product

properties.

The computer simulation program utilized in this investigation requires minimum

input information. The input information consists of the properties of a food product
for temperatures above the initial freezing point, the initial product temperature and
freezing conditions. The criterion for the freezing time was established by freezing the
product to the instant when the mass average enthalpy of the product is equivalent to
the value at —25°C which is the assumed storage temperature. The freezing time is a
function of freezing conditions which include freezing medium temperature, surface heat
transfer coefficient and product diameter. Further reduction of the freezing medium
temperature below —200°C has only a small effect on reducing the freezing time. The
same result can be predicted with further increase of the surface heat transfer coefficient
above 200 W/mOK. The predicted freezing time is directly proportional to the product
diameter.

Under given freezing conditions, the freezing time is a function of initial freezing point,
initial water content, product density and thermal conductivity of a food product.
Influences of these parameters on the freezing time were discussed by using hypothetical
combinations of product properties. A food product with a lower initial freezing point, a
higher initial water content and a higher initial product density will have a longer freezing
time. The deviations in the predicted freezing time among actual food products and
hypothetical test products were due to the combined and complex influences of the product
properties. It has been found that the prediction of freezing time is most sensitive to the
accuracy of the measurement of the initial product density and the initial freezing point if
the freezing point of a food product is above —O.5°C. The inﬂuence of the accuracy of
initial thermal conductivity data on freezing time is not important in the range of 0.45 to
0.55 W/moK investigated. The combined effect of the inaccuracy in measuring these

product properties will be signiﬁcant even if the influence of an individual product property

is small in predicting the actual freezing time and other freezing characteristics. However,

the influences of product properties on the freezing time for actual food products

are consistent with the trend predicted by the independent inﬂuence analysis.

Approved ﬂfﬁﬁw‘

Major Professor

Chairman

Department of Agricultural
Engineering

INFLUENCE OF FOOD PRODUCT PROPERTIES

ON THE FREEZING TIME

By

Rong«Ching Hsieh

A THESIS

Submitted to
Michigan State University
in partial fulﬁllment of the requirements

for the degree of

MASTER OF SCIENCE

Department of Food Science and Human Nutrition

1976

To My Parents

ACKNOWLEDGMENTS

The author would like to express his deep appreciation to Professor Dennis R. Heldman,
Chairman of the Agricultural Engineering Department, for his suggestion of the topic and
his guidance and support throughout the course of this thesis.

Dr. Richard C. Nicholas, professor of Department of Food Science and Human Nutrition,
and Dr. James V. Beck, professor of Department of Mechanical Engineering are also
acknowledged for their assistance.

Many thanks to Mr. David P. Gorby, process engineer of the Pillsbury Company in
Minnesota, for his brilliant advices and suggestions.

The author would also like to express his gratefulness for his wife, Sue-whei, who did

all the typing work.

iii

TABLE OF CONTENTS

LIST OF TABLES ........................................................................................... Pige
LIST oF FIGURES ......................................................................................... vi
NOMENCLATURE ......................................................................................... viii
I INTRODUCTION ..................................................................... 1
II LITERATURE REVIEW ........................................................... 3
III THEORETICAL CONSIDERATIONS ...................................... 6
3—1 Unfrozen Water Content ............................................. 6
3—2 Thermal Conductivity .................................................. 9
3—3 Product Density ........................................................... 10
3—4 Enthalpy ...................................................................... 1 1
3—5 Apparent Speciﬁc Heat ................................................ 12
3—6 Heat Transfer ............................................................... 1 2
3—7 Criterion for Freezing Time ......................................... 14
IV EXPERIMENTAL CONSIDERATIONS .................................... 17
V RESULTS and DISCUSSIONS ................................................. 20
VI CONCLUSIONS ......................................................................... 47
VII SUGGESTIONS FOR FURTHER STUDY ................................ 49
BIBLIOGRAPHY ........................................................................................... 50
APPENDIX A ............................................................................................... A1
APPENDIX B ............................................................................................... Bl
APPENDIX C ............................................................................................... C1

LIST OF TABLES

Table Page
1 Maximum allowable deviations in measuring product properties

from the actual values for the state variations in the prediction

of freezing time ................................................................................... 34
A—l Values of physical properties of several food products utilized

in the simulation ................................................................................. A1
A—2 Predicted freezing times for different food products with different

freezing conditions and product sizes ................................................. A2
A—3 Predicted freezing times of several food products with the product

diameter of 1cm ................................................................................. A3
A—4 Predicted freezing times of several food products with the product

of 2cm ............................................................................................... A3
A—5 Predicted freezing times for different food products of diameter

2 centimeter at different surface heat transfer coefﬁcients with

initial product temperature 10°C and freezing medium temperature

-35 °c .............................................................................................. A4

LIST OF FIGURES

Figure Page
1 Types of temperature history profile as a function of freezing
medium temperature ...................................................................... 15
2 Predicted unfrozen water content for different food products ....... 22
3 Predicted thermal conductivity for different food products ........... 23
4 Predicted product density for diffenent food products .................. 24
5 Predicted enthalpy for different food products ............................. 25
6 Predicted apparent speciﬁc heat of different food products ........... 26

7 Predicted influence of initial freezing point on the freezing time... 28

8 Influence of initial water content on the freezing time .................. 29
9 Influence of unfreezable water content on the freezing time ......... 30
10 Influence of initial product density on the freedng time ............... 31
1 1 Influence of initial thermal conductivity on the freezing time ...... 32

12 Predicted freezing time versus freezing medium temperature

for various food products of different product diameters ............. 36
13 Predicted freezing time versus surface heat transfer coefﬁcient

for various food products with different product diameters ......... 37
14 Predicted freezing time versus product diameter for various

food products at different surface heat transfer coefficients ........ 38

vi

Figure
15

16

17

18

19

20

Page
Predicted temperature history for different food products ................ 39
Predicted temperature history of strawberries freezing at
different values of surface heat transfer coefficient ........................... 40

Predicted freezing time versus initial water content for different

food products at different values of surface heat transfer coefﬁcient. 42
Predicted freezing time versus initial freezing point for different

food products at different values of surface heat transfer

coefﬁcient .......................................................................................... 43
Predicted freezing time versus intial product density for different

food prodcuts at different values of surface heat transfer

coefﬁcient .......................................................................................... 44
Predicted freezing time versus initial thermal conductivity

for different food products at different values of surface heat

transfer coefﬁcient ............................................................................. 45
Schematic representation of the spherical nodal heat transfer

system of a ﬁnite difference method ................................................. B1

vii

pa

CPi

dm

1K
IPD
IWC

ICP

NOMENCLATURE

Conversion to degrees absolute

Constant used in equation (9)

Volume fraction of the discontinuous phase used in equation (9)
Apparent speciﬁc heat of food products, J/goK (CPA)*

Speciﬁc heat of ice, J/g O1((CPI)

Speciﬁc heat of water, J/g C’K (CPW)

Differential mass, g

Mass average enthalpy of food products, J/g-product
Molar Gibbs free energy of solvent, J/g—mole

Heat content of the solvent, J/g-mole

Total heat content of food products, J/g-product
Averaged surface heat transfer coefﬁcient, W/mZOK (H)
Initial thermal conductivity of food products, W/moK
Initial product density, g/cm3

Initial water content of food products, g-water/g-product
Initial speciﬁc heat of food products, J/g° K

Thermal conductivity of food products, W/m °K (K)
Thermal conductivity of continuous phase, WIm° K (KC)
Thermal conductivity of the discontinuous phase, W/m°K(KD)
Latent heat of fusion of pure solvent , J/g-solvent (HS)

Cubic root of the volume fraction of solids or discontinuous phase

viii

AG

Molecular weight of pure solvent, g/g-mole
Effective molecular weight of solute, g/g-mole
Mass of pure solvent, g
Mass of solute, g
Gas constant, 8.314 J/g-moleOK
Constant used in equation (B—13)
Absolute temperature,° K
Initial freezing point of food products,°C or °K
Freezing point of pure solvent, °K (TO)
Cryoscopie temperature,°C or°K
Temperature of node n at time t, 0C
Volume, cm3
Speciﬁc volume, cm3/g (SV)
Total water content, g-water/g-product
Mole fraction of solvent in solution
Thermal diffusivity, m2/ hr
Density, g/cm'3
Density of ice, g/cm3 (DI)
Density of water, g/cm3 (DW)
Density of product solids, g/cm3 (DS)
Time, minute or hour (TIME)

Computing time increment, hr (DTHETA)

' All symbols in the parenthesis are and in the computer program.

ix

I. INTRODUCTION

The determination of the unsteady state heat conduction temperature distribution
of frozen food products differs from that associated with ordinary cooling. Food products
contain heat sources (latent heat evolution) during ice formation and the food product
properties vary with temperature. Cryoscopie temperatures of the product are discontinuous
and nondifferential.

The most critical factor limiting accuracy in predicting the freezing time is the lack of
accurate method to describe the freezing characteristics of food products. Product properties
data for food products are either inaccurate and/or difﬁcult to be measured experimentally.
To solve the complicated features of the continuous changing of product properties of
frozen food products, numerical techniques and digital computers were utilized by Lescano
and Heldman (1973), Heldman and Gorby (1974—B) and Gorby (1974) in recent years.
These approaches provide an accurate and efﬁcient way of predicting the freezing time,
the refrigeration requirements and the product properties below the initial freezing point.
This prediction model can be based on the available product properties data above the
initial freezing point and the assumption of an ideal solution being present in food product
system. In addition, simple mathematical caculations of thermal properties based on phase
theories of physical chemistry and some empirical equations, such as Kopelman equation,
Dickerson’s equation to evaluate the speciﬁc heat of ice, the freezing point depression
equation, and the basic equations to describe the transient heat conduction problems, are

assumed to be reasonably correct.
The major difﬁculty involved in analyzing the influences of product properties of

food products is the combined effect of these properties. The investigation of the

1

independent influence of several product properties on freezing time is becoming essential
in development of accurate methods.

The objectives of this investigation were:

1. To utilize the computer techniques which predict the product properties at below
the initial freezing point and to apply the assembled prediction models developed
by Gorby (1974) in predicting freezing time and temperature history for isotropic
products.

2. To analyze the computer output data and compare freezing times, temperature
history curves and product properties during freezing processes for several different
food products.

3. To investigate the independent effect of different product properties on freezing
time.

4. To explain the differences between freezing time and temperature history curves
for different food products.

After the relationships between product properties and freezing time have been

established, the freezing time of other food products with similar shapes and freezing
conditions can be predicted. Furthermore, the relative significance of the accuracy in

exPerimental determination of product properties will be more evident.

II. LITERATURE REVIEW

It has been pointed out by Barlett (1944), Ede (1949), Riedel (1949-A, 1949-B,
1956, 1957), Long (1955) and Hill (1966) that the water in a food product crystallizes
over a range of temperatures. This fact was interpreted as the continuous depression of

freezing point and variation in ice-water-solids composition by Staph and Woolrich

(1951). Detailed explanations of the properties of water and ice during food freezing was
presented by Fennama and Powrie (1964), Luyet (1969), Dickerson (1969) and Bakal
and Hayakawa (1972).

The unfreezable water content may have a signiﬁcant inﬂuence on refrigeration
requirement for freezing. Energy can be saved by just reducing the food product to the
temperature at which all the freezable water has been frozen. Duckworth (1971) proposed
the techniques of differential thermal analysis (D.T.A.) to measure the unfreezable water
content in various food products by utilizing powdered C02 or liquid nitrogen as a coolant.

Heldman (1974-B) predicted the relationship between unfrozen water fraction and
temperature during food freezing by utilizing freezing point depression. It was established
that accurate prediction of the relationship of unfrozen water percentage and temperature
for products with considerable unfreezable water requires knowledge of the percent
unfreezable water or improved knowledge of freezing characteristics of the product.

Several attempts had been made by Dickerson (1969) and others (Smith (1952), Long
(1955), Lentz (1961), Hill (1967), etc.) to measure thermal properties of food products
experimentally. The results were tabulated by Woodams and Nowrey (1968), Reidy(1968),

Dickerson (1969) and Morley (1972). Long (1955) predicted the thermal conductivity

of frozen codﬁsh using a model of sodium chloride solution to simulate the cod muscle
as a mixture of protein with a certain concentration of salt solution.

The prediction of refrigeration requirements during freezing was demonstrated by
Heldman (1966) and veriﬁed experimentally for ice cream by Heldman and Hedrick (1970)
using calorimetric determinations. The most widely used and straight forward empirical
equation to predict the freezing rate in food products is the Plank (1913) equation. This
equation was modiﬁed by Nagaoka et al.(1955) to take into account the initial water content
and the sensible heat removed. Eddie and Pearson (1958) also modiﬁed Plank’s equation
for inﬁnite slab products and pointed out that experimental measurements of thermal
properties of food products are frequently not possible or may be inaccuate due to the
inherent instrument errors. Carslaw and Jaeger (1959) presented the Neumann Solution
to the transient heat conduction problem. This solution was applied to the food freezing
problem by Charm and Slavin (1962). A detailed analysis of several expressions for
predicting freezing rates conducted by Bakal and Hayakawa (1972) indicates that all approaches
have limitations of not taking into account the variable products properties during freezing.

To overcome the above limitations for analytical and empirical solutions, the numerical
approach to solving the transient heat conduction by computer simulation with changing
product properties was developed by Albasiny (1956), Hohner and Heldman(1970), Charm
(1971), Cordell and Webb (1972), Bonacina and Comina (1973), Fleming (1973), Cullwick
and Earle (1973), Lescano and Heldman (1973), Comina and Bonacina (1974), Heldman
and Gorby (1974-B), Gorby (1974), and Tarnawski (1976). Several different computer
simulations were compared by Heldman (1974—A).

An analysis of the output information for sweet cherries has been thoroughly discussed

by Gorby (1974). The predicted values of unfrozen water content, thermal conductivity,
product density, enthalpy and apparent speciﬁc heat for sweet cherries has been presented.
In addition, the influence of surface heat transfer coefficient, product size and initial
product temperature on freezingtime was investigated. It has been shown that the initial
product temperature (under normal handling conditions) has a little effect on the freezing
time compared to the other factors. Heldman and Gorby (1975) demonstrated that low
freezing medium temperature (—195°C or —320°F) in an individual-quick-freezing (IQF)
process resulted in the heat content of the product being reduced to the desired level.

Therefore, the refrigerant can be used more efﬁciently based on the heat content level.

III. THEORETICAL CONSIDERATIONS

Freezing time of food products change not only with the freezing conditions but
also with the food product properties. The freezing conditions imposed on the food
products during freezing are the freezing medium temperature, the surface heat transfer
coefficient between the surface of food product and the freezing medium, the initial
product temperature and the radius or other geometric factors of the product.

Changes in product properties during freezing are due to the continuous depression of
the freezing point and therefore the continuous changing of the unfrozen water content.

All the product properties are assumed to be constant above the initial freezing point.

3—1 Unfrozen water content

From the phase equilibrium described in the physical chemistry, the relationship
between the mole fraction of solvent in a solution and the freezing point of the solution
can be obtained by introducing the chemical potential of a pure solvent in the solid state
and the chemical potential of the pure solvent in the liquid state. Under equilibrium
conditions:

EA (Solid) — 6A (Liquid)
RT

 

where G A = molar Gibbs Free energy
R = gas constant (8.314/g—moleoK)
T = absolute temperature (OK)

XA = mole fraction of solvent in the solution

From the Gibbs-Helmholtz equation:

T T2

 

(2)

and differentiation of equation (1) with respect to T, the following equation is obtained:

EA (liquid) — EA (solid) LMA dlnXA
= = (3)
RT2 RT2 dT

 

 

Integration of equation (3) with. respect to T from the freezing point of pure solvent

to any temperature below the freezing point results in the following equation:

 

LM 1 1
A
——[— — —-]=1nXA (4)
R To T
where:
mA M
XA = I A = mole fraction of solvent (5)
mA/MA + mB/MB
and:
L = latent heat of fusion of pure solvent (J/g-solvent)

To = freezing point of pure solvent (OK)

T = freezing point of the solution (OK)
m A = mass of pure solvent (g)

mB = mass of solute (g)

M A molecular weight of pure solvent (g/ mole)
MB = effective molecular weight of solute (g/ mole)
T — To = freezing point depression (OK)

In the case of food freezing, it is assumed that the liquid water is the solvent and ice

together with food solids (including soluble solids) are the solute.

The unfrozen water content at a given temperature below the initial freezing can be

obtained from the mole fraction of the unfrozen water. From equations (4) and (5):
= LM 1 1
R To T

and: MBXAMA
mA =

 

MB(1 — XA) (7)

where m A is the effective mass of water and MB is the apparent molecular weight of product
solids. MB in equation (7) can be determined by utilizing equation (6) with initial freezing
point of the food product as T and solving equation (7) for MB using the initial water content
of the food product.

A small amount of water may remain unfrozen even at very low temperature in some
food products. This portion of water is considered “inactive” in the dynamic freezing point
depression. The above method was thoroughly discussed by Lescano (1973) and Heldman
(1974-B).

The modiﬁed (or effective) water content used in equation (7) to determine the apparent
molecular weight of product solids, is:

MA = (Initial water content) — (Unfreezable water content)
Hence, the total water content in the system at any temperature below the initial freezing
point will be:

WC = M A + (Unfreezable water content) (8)
The unfrozen water content (WC) obtained from the above equation can be utilized in
predicting the thermal properties of food products at temperature below the initial freezing
point by determining the water, ice and solid fractions at any freezing instant. This method
was proposed by Heldman (1974-B) and was shown to give good agreement with experimental

data.

3—2 Thermal Conductivity

Long (1955) and Lentz (1961) have described the dependence of thermal conductivity
of frozen food on the extent of product freezing The results indicated that the thermal
conductivities of a frozen food could be predicted by using an appropriate mathematical
model and the frozen water fraction in the product as a function of temperature,

The mathematical model used by Long (1955), Lentz (1961) and Lescano (1973) was
the Maxwell (1904) equation as ﬁrst adapted to food products by Eucken (1940). The

Maxwell equation is:

1—(1— k /k )b
k=kc[ ad “ ] (9)
1+(a—1)b

 

where:
kc = thermal conductivity of the continuous phase

kd = thermal conductivity of the discontinuous phase

 

 

a = 3kC
2kc + kd
Vd . . .
b = = volume fraction of the discontInuous phase
Vc + Vd

A set of mathematical expressions was proposed by Kopelman (1967) to predict the
thermal conductivity in a food system for both isotropic and anisotropic systems. In
an anisotropic system with the thermal conductivity parallel to the fiber, the values are
approximately 15 to 20% higher than that perpendicular to the ﬁbers were proposed.

In this investigation, the Kopelman (1967) isotropic model was utilized. This model
has been used successfully to predict thermal conductivities of frozen sweet cherries by

Gorby (1974). For a two-component system:

1—Q 7
k = k (10)
C[1_Q(1-M)J

10

where:

k
=2__‘i
QM(1k

c
M3 = volume fraction of solids or discontinuous phase

Since the frozen food product consists of water, ice and product solids, a modification of
the Kopelman model has been introduced by Heldman and Gorby (1974-A). It was found
that the best agreement with experimental data is obtained by first considering two of the
three product phases and reducing the system to a binary solution followed by reducing
the system to a another binary system for a second use of the same expression. Water was
considered a continuous phase for the water-ice system during the ﬁrst use of the Kopelman
equation, then the water-ice mixture was treated as a continuous phase and the product
solids was taken as discontinuous phase for the second use of the equation.

The value of the thermal conductivity of product solids was found by using the Kopelman

model and the experimental (or initial) thermal conductivity of the unfrozen product.

3—3 Product Density

Constant densities for water, ice and solids below the initial freezing point have been
assumed in predicting the product density. In an ideal case, the product density change is
only due to the proportion changes of the mixture.

The speciﬁc volume (or volume per unit mass ) was found by adding the volume per
unit mass of product fractions. Thus:

V = 1/Y = mi/Pi + mw/Pw + ms/Ps (11)

with the density of product solids ( PS) found by utilizing equation (11) with the initial

product density and solving for the density.

11

3—4 Enthalpy
The total heat content of a frozen food product as proposed by Lescano and Heldman
(1973) was expressed by considering zero enthalpy at —40°C.
HT = mSCps (T + 40) + me + mWCPW (T + 40) + miCpi(T + 40) _
mw (at 40%) L (12)
At —40° C, where enthalpy is zero, the total water content (mw) is equal to the water
content at -40°C. This portion of the unfrozen water at —40°C was considered unfreezable
water content and used as an additional input parameter in the prediction of unfrozen
water contents for cod ﬁsh and lean beef by Heldman (1974-B).

The solids content (ms) can be obtained from the initial water content. The unfrozen
water content (mw) can be predicted by equations (6), (7), and (8). The unfreezable water
content (mW at —-40°C) is known from experimental data. The speciﬁc heat of solids (CPS)
is determined from the initial product speciﬁc heat and the initial water content by the

following equation:

ICP = IWC ' pr + ins ' CPS (13)

In general, ms, C C and L are not functions of temperature, whilem w’ mi and Cpi

ps’ pw
are predicted. The specific heat of ice (Cpi) as a function of temperature was obtained by

Dickerson (1969):

Cpi=A+B'T (14)
where:

A = 1.9507941

B = 0.00206153

T = degree K.

12

Heldman and Gorby (1974-B) predicted the refrigeration requirements using equation (12).

3—5 Apparent Speciﬁc Heat
The apparent specific heat of a food product is deﬁned as the differential change in
enthalpy if the enthalpy-temperature function is continuous:
cpa = AH/AT (15)
ifAT is chosen as 0.028% (0.0S°F) and AH can be obtained from equation (12), the

apparent speciﬁc heat can be predicted.

3—6 Heat Transfer

Several attempts has been made to solve the heat transfer problem and predict the
freezing time of food products. The exact solution from Plank’s (1913) equation is not
applicable due to the assumptions required when applied to food products.

Some modiﬁcations of Plank’s equation have been proposed. Yet, the major problems
imposed by an unsteady state heat transfer have not been solved. The techniques of a
ﬁnite difference model to solve the transient heat transfer problem has been proved
accurate and effective.

For the transient heat conduction problem of a spherical product with temperature
dependent thermal property values, the appropriate differential equations of energy

balance in terms of ﬁnite difference model can be generated from the following equation:

' __ 16
qnm _ qnout _ qisltored ( )

where qn is the heat transfer rate at node n. The expanded differential forms of the above
equation for the center, interior and surface nodes are presented in Appendix B.
There are several ways to solve the energy balance differential equation by the finite

difference methods. The most commonly used methods are the forward difference

13
(explicit) and the backward difference (implicit) for the time derivative. The simplest
ﬁnite difference representation of equation (16) utilizes the central difference formula
for the spatial derivative and a forward difference for the time derivative. This method
is presented in Appendix B.

The following assumptions have been made in formulating the finite difference model:

(1) Energy transport occurs only in the radial direction.

(2) The initial product composition and temperature are uniform and the food products
are homogeneous and isotropic.

(3) Mass transfer within the product or on the products surface is neglected.

(4) The specific heats of solids and water, density of solids, water and ice, and the latent
heat of fusion for water are constant.

(5) All product properties are constant above the initial freezing point.

(6) At a discontinuous point of product properties there exists a separation limit phase,
at which temperature is Tkr (cryoscopic temperature). This divides the examined
food product into unfrozen and frozen parts.

(7) The freezing medium temperature and surface heat transfer coefﬁcient are assumed
to be constant throughout the freezing processes.

The above assumptions have been proposed by Gorby (1974) and Tarnawski (1976).
The solution of equation (16) can be put in various forms for different approaches. A set
of solutions for the food product was derived by Gorby (1974) for the surface, interior and
center nodes from both explicit and implicit methods. The implicit method was used in
this investigation.

The implicit method, or backward difference method, has the advantage that any time
increment can be used (Kreith, 1973). The explicit method, or forward difference method,

has the problem of stability. Choosing a computing time increment which is small enough

14

to stabilize the calculation for a given value of shell thickness in the computer is essential

upon using the explicit method. The method is described in Appendix B.

3—7 Criterion for Freezing Time

The freezing time in a quick freezing process is small compared to conventional freezing,
Because of the fast freezing rate, the temperature distribution within the product at the end
of freezing may span a wide temperature range. Therefore, the commonly used method
which applied thermal arrest time or center temperature criterion can not be utilized for the
quick freezing processes. Two types of temperature history diagrams shown in Figure 1
were found by Gorby (1974). Comparison of these two ﬁgures shows that the lower freezing
medium temperature [case (a)]has a more dramatic inﬂuence since a large temperature
gradient exists between the freezing medium and the product surface. Beyond the end
point of the freezing process (tF) the product is exposed to environmental parameters with
a surface heat transfer coefﬁcient of 1 BTU/hrft2°F (5.7 W/m2 °K) and a storage temperature
of —20°F (—29 0C) until the product temperature is uniform at point tE. The time period
between tF and tE is the equilibration time. Shorter freezing time and longer equilibrating
time were predicted by using lower freezing medium temperature.

It has been proposed by Gorby (1974) that the total heat content be a criterion for
food freezing processes. The freezing process stops at the instant when the total enthalpy
from equation (12) reaches the equivalent product enthalpy at the desired storage
temperature.

The mass average enthalpy (EMA, J/g-product) is defined mathematically as:

EMA = LI Hdm (17)
m m

The above equation can be approximated by the same divisions in the finite difference

    
  
  
 
   
  

 

 

    
 

 

 

 

  

 

 

Center
Tkr tF = freezing time
:5: (tE -— tF) = equilibration
g time
o.
E
E‘ O
-20 F I
O
(—29 C) I
I
I
Surface :
I
I
I
l
0 tr; Time IE
(81)
\ Center I
T \
g kr _. l
:9. I
8. I
a I
Q
I"
Surface \
0
-—20 F —- '7
(—29°C) I j
I
I I
I I
I l
I I
I I
l I
0 Time tF tE
(b)

Figure 1. Types of temperature history profile as a function of freezing medium
temperature when (a) the freezing medium temperature is very low and
(b) the freezing medium temperature is higher.

16

model as:
I M
EMA = —— Z H(I)V(l) (18)
V I=I

where M is the number of nodes and V is the total volume of the individual product.
The uneven temperature distribution at the end of freezing process will redistribute
heat within the product until it reaches equilibrium. This provides the lowest possible

residence time in the freezing medium and also the lowest refrigerant requirement.

IV. EXPERIMENTAL CONSIDERATIONS

Asparagus, carrots, cherries, peas, plums and strawberries were chosen to compare

the freezing time. The reason for selection of these products were as follow:

1.

They are commercially available in frozen form and the quick freezing process is
the best and most efficient way to preserve these products and maintain product

quality.

. The product properties of these food products cover the range typical of most fruit

and vegetable products encountered in food freezing preservation.

. These products are isotropic or can be described by isotropic conditions.

. These products are of comparable size or can be cut into comparable sizes for use

in comparing the effects of product properties on the freezing time.

A computer program based on the mathematical model presented in the preceding

chapter was developed by Gorby (1974, see Appendix E). The program can generate the

predicted product properties as a function of temperature and the predicted temperature

history under an input set of initial conditions.

The following input information is necessary to generate the program:

1.

Properties of food product

a) Initial product temperature (TI), degree C.
b) Initial freezing point (TF), degree C.

c) Initial water content (IWC), decimal.

d) Unfreezable water content (UFWC), g-watcr/g-product.

17

18

0) Initial product density (IPD), g/cm3
f) Initial specific heat (ICP), J/g°x

g) Initial thermal conductivity (IK), W/mOK

2. Freezing medium temperature (TINF), degree K

3. Convective heat transfer coefﬁcient (H), W/mzoK

4. Computing time increment (DTHETA), hour.

5. Print time increment (TPRINT), minute.

6. Product diameter (DIA), cm

7. Equivalent enthalpy temperature for the program to stop (STEMP), degree C

8. Number of shells into which product will be divided (LAYERS), integer
A STEMP value of —25°C was utilized and 10 LAYERS was assumed throughout the
investigation unless otherwise speciﬁed.

Additional input data were required:

1. Conversion to degrees absolute (A) = 273.15

2. Gas constant (R) = 8.314J/g-moleoK

3. Heat of fusion for water (HS) =334.72 J/g

4. Density ofwater (DW) = 1.00 g/cm3

5. Density of ice (D1) = 0.917 g/cm3 (at 0 °C)

6. Freezing point of pure water (TO) = 0°C

7. Thermal conductivity of ice (KI) = 2.21 W/moK (at 0 °C)

8. Thermal conductivity of water (KW) = 0.57 W/mOK

9. Speciﬁc heat of water (CPW) = 4.2 j/gOK

The following parameters were chosen to evaluate:

1. Product diameters of 1cm for asparagus and peas, 2cm for carrots, cherries, plums,

and strawberries.

19

2. Initial product temperature: 10°C

3. Surface heat transfer coefﬁcients: 25, 70, 170, and 340 W/mzoK

4. Freezing medium temperatures: —35, —75, —130, and -——19S°C

Times for the products to reach an equivalent heat content at —25°C were predicted.
Table A-1 shows the property data for these products. These data were obtained or
estimated by Riedel (1951), Reidy (1968), Dickerson (1969) or Bedford (1976).

A set of test runs was conducted using carrots to generate the relationships between
the freezing time and each separate product property by ﬁxing all the other parameters
except the one which was investigated. The variations of these parameters were in the
range of the property values of these products.

A diameter of 2cm was assumed for asparagus and peas in order to compare the
freezing rates of the same product size with the rest of the products.

The output information includes:

1 . Predicted apparent speciﬁc heat, thermal conductivity, product density, unfrozen

water content and enthalpy of food products as a function of temperature.

2. Predicted temperature history for the center and surface nodes.

3. Predicted freezing time for the whole product to reach the equivalent enthalpy

temperature of —25 0C.

V. RESULTS AND DISCUSSIONS

Four input values for the freezing medium temperature (-35°C, —75°C, —130 oC, and
-—195°C) and surface heat transfer coefﬁcient (25, 70, 170 and 340 W/mzoK) were
selected in order to generate reasonable curves for their influences on the freezing time.

It is safe to assume that all the individual products or cut pieces are homogeneous. The
pits in cherries and plums have thermal properties similar to those of the fruit but have
slightly lower water contents. Since the freezing time is very short, the effect of the cylindrical
or cubical shapes of carrots and asparagus pieces was assumed to be very small since the food
product shapes investigated are spherical.

The temperature influence on product properties was investigated in order to demonstrate
the influence of these properties on the freezing time. Figure 2 illustrates the predicted
values of unfrozen water content as a function of temperature for the various products.

The unfrozen water contents remains constant until the initial freezing point is reached and
is reduced rapidly due to the crystallization of water in the food product while the product
temperature remains constant. The freezing point of these food porducts decreases due

to the higher ratio of solids to water content. The unfrozen water fraction remains almost
constant at the temperatures below —40°C.

Figure 3 illustrates the predicted thermal conductivity changes during the freezing
process. The signiﬁcant increase at temperatures below the initial freezing point is related
to the presence of ice which has a higher (about 4 or 5 times higher) thermal conductivity
than that of water. The values of thermal conductivity in the frozen region are functions

0
of unfrozen water content and temperature. At temperatures below about —50 C the

20

21

increase in the values of thermal conductivity is very small and values remain fairly constant
at very low temperature. It is also shown that the food product with a higher initial thermal
conductivity does not necessarily have a higher thermal conductivity in the frozen region
because other factors such as the product density and water content of the food products
influence the thermal conductivity at temperatures below the initial freezing point. This

is demonstrated by Kopelman’s equation (Equation 10) where M3, volume fraction of
solids or discontinuous phase, is a function of the product density and unfrozen water
content at that instant. In addition, the temperature functions of product density and
unfrozen water content are unique for each food product and their combined influences

on the evaluation of thermal conductivity below the initial freezing point are complicated.

The temperature influence on the product density is presented in Figure 4. The shape
of the curves is similar to those for unfrozen water content (Figure 2) since the determination
of product density in the frozen region is directly related to the amount of unfrozen water
present in the food product (Equation 12). By comparing the values of product properties
at initial temperature and at temperatures far below the initial freezing point, it is evident
that the value of thermal conductivity increases almost 4 times the initial value, whereas
the product density decreases about 10%.

The enthalpy values for the food products at temperatures below the initial freezing
point are presented in Figure 5. Removal of latent heat contributed most to the reduction
in heat content at temperatures just below the initial freezing point where the temperature
decrease is very small. The large amount of enthalpy reduction in that region results in
very high values of apparent speciﬁc heat as illustrated in Figure 6. Curves in Figure 6 are
the temperature derivatives of the corresponding curves in Figure 5.

In order to investigate the independent influence of product properties on the freezing

time, a series of tests were conducted using carrots as the test product. The results are

22

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l

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30 " I ['3

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I”

'III

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lj‘l

1 Asparagus l II”

25 - 2 Carrots ‘ “I

3 Cherries "ll

4 Peas ll:

5 Plums '1'.

6 Strawberries I III

'III
20 I-
15 I.
10 I"
5 P

0 I I I I I 4
—40 —30 —20 —10 0 10
Temperature (° C)

Figure 6. Predicted apparent speciﬁc heat of different food products.

27

presented in Figures 7, 8, 9, 10 and 11. These test runs were set up in the range of the
product properties typical for the products. The results shown in Figure 7 illustrate that

a food product with a higher initial freezing point has a shorter freezing time. It is fairly
safe to assume that the temperature function curves of unfrozen water content and enthalpy
for the products with different initial freezing points will have similar shape of curves.
Besides, these curves do not cross each other. The only difference between these curves is
the position of the discontinuous point and the whole curve have been shifted. The above
reasoning is based on the fact that all the parameters except the initial freezing point have
been kept constant. Therefore, the product with a higher initial freezing point will possess
lower unfrozen water content and enthalpy values than the other product with a lower
initial freezing point at the same freezing temperature, and a shorter freezing time for the
product with a higher initial freezing point is expected.

Figure 8 shows that a food product with a higher initial water content has a longer
freezing time. It is apparent that the product with a higher initial water content has a
higher total heat content due to the extra contribution of the latent heat from higher
initial water content.

The influence of the unfreezable water content on the freezing time is signiﬁcant for
ﬁbrous products. A higher value of unfreezable water content results in less freezable
water while keeping the total water content constant. Less freezable water present in a
food product will require shorter time to freeze. This is consistent with the results
presented in Figure 9.

Figure 10 predicts the influence of initial product density on the freezong time. From
the heat transfer equation presented in Appendix B, it is evident that a product with a
higher density will have a slower heat transfer rate. Thus, a longer freezing time can be

expected for the product with a higher density.

28

 
   
 
 
   

IWC = 0.88
UFWC = 0

IPD: 1.06 g/cm3
IK =0.50 W/m°x
TINF = —75 “c

h = 70 W/m2 °K
Diameter = 1cm

5.8 "

5.4 f-

 

 

3?
.53 5.2 +-
E,
Q)
..E.
I?"
be
.5
N
8 5.0 I-
I}:
4.8 — ‘?
1.74 minutes at —0.1 0C
4.7 ! l I I

 

-5 -—2 51 0
Initial Freezing point, TF, °c

Figure 7. Predicted influence of initial freezing point on the freezing time.

Freezing Time (minute)

5.4

5.3

5"
N

U1
.—I

5.0

4.9

29

 

 

 

l
_ TF = 1.11°C
UFWC = 0
IPD = 1.06 g/em3
IK = 0.50 W/m°K
TINF = —75 'c
h = 70 W/m2°K
Diameter = 1cm
P
p
I I I L
0.70 0.75 0.80 0.85 0.90

Initial Water Content, IWC, g-water/g-product

Figure 8. Influence of initial water content on the freezing time.

Freezing Time (minute)

5.5

5.4

S"
I.»

01
N

5.1

5.0

30

TF = —1.11°C
IWC = 0.88

IPD = 1.06 g/cm3
1x = 0.50 W/m °x
TINF = —75 °c

h = 70 W/m2°K
Diameter = 1cm

 

 

 

1
0.0? 0.06 0.09
Unfreezable Water Content, UFWC; g-water/g-product

Figure 9. Influence of unfreezable water content on the freezing time.

Freezing Time (minute)

31

 

 

 

6.0 l—
TF = —1.11°C
IWC = 0.88
UFWC = 0
5.8 .. IK = 0.50 wIm°x 0
TINF = —75 °C
11 = 70 W/m2°K
Diameter = 1cm
5.6 -
5.4 )-
5.2 -
5.0 —
4.8 -
4.6 I l I I I
0.98 1.00 1.02 1.04 1.06 1.08

Initial Product Density, IPD, g/cm3

Figure 10. Influence of initial product density on the freezing time.

Freezing Time (minute)

32

 

 

 

 

6-0 " TF = —1.11°c

IWC = 0.88

UFWC = o

IPD= 1.06

TINF = —75 °C
5.9 .. h = 70 W/m2 '1(

Diameter = 1cm

0 O
5 8 -
A 49—0-—
9 G)
0
5.7 P
5.6 I-
5,5 It
I I
0.45 0.50 0.55

Initial Thermal Conductivity, IK, W/m2°K

Figure 11. Inﬂuence of initial thermal conductivity on the freezing time.

33

The influence of the initial thermal conductivity shown in Figure 11 demonstrates
that the values of the initial thermal conductivity have almost no influence on the predicted
freezing time. The values of freezing time in Figure 11 were generated by dividing the
product into 20 layers and utilizing a smaller computing time increment to get more
accurate results. Similar shapes of curve were obtained by dividing the product into 10
layers which had been used throughout this investigation. Better results were obtained by
dividing the food product into more layers and utilizing a smaller computing time increment,
and accordingly, a longer computing time is required. The results for the freezing times of
different initial thermal conductivities generated by dividing the product into 10 layers had
even larger deviations and were unable to identify the influence. Therefore, a 20-1ayer
division was utilized and the results appeared to be close to a horizontal straight line which
means the initial thermal conductivity has little effect in predicting the freezing time.

The results in Table 1, which were generated from Figures 7 to 11, illustrate the
maximum allowable deviations from the actual property values when measuring product
properties with 1 to 4% variations in the prediction of freezing times. The results are
based on arbitrarily chosen conditions: (a) initial product temperature of 10°C, (b) freezing
medium temperature of —75°C, (c) surface heat transfer coefﬁcient of 70 W/m2°K and
(d) product diameter of 2cm. Values of allowable deviations for product properties in
freezing time prediction presented in Table 1 were calculated by dividing the freezing time
variations by the average slope of curves in Figure 7 to 11 and multiplying the stated
ranges of product properties. The product with a higher initial freezing point requires a
more accurate measurement. A 0.1g-water/g-product inaccuracy in measuring the initial
water content or unfreezable water content will cause 3% of deviation in predicting the
freezing time.

The tolerance of deviations for the initial product density values appears to be the

34

 

 

 

 

 

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35
smallest. A 0.036 g/cm3 deviation in measuring the initial product density will cause
4% of variation in predicting the freezing time. The slope of the curve in Figure 11 is
zero which shows that the inﬂuence of the inaccuracy in measuring the thermal
conductivities is very small in the range of 0.45 to 0.55 W/moK.

Comparing the temperature history of freezing with various values of initial thermal
c orductivity indicates that at any freezing instant, the product with a higher initial thermal
conductivity has a higher surface temperature and a lower center temperature. In addition,
the mass average enthalpy values at any instant are almost the same. The consistency of
the values of mass average enthalpy explains the close freezing time. The mass average
enthalpy is the criterion to terminate the freezing process. The freezing process will be
stopped once the product reaches the mass average enthalpy of that equivalent to the
temperature of -25°C.

Figures 12 and 13 illustrate the influence of freezing conditions on the freezing ti me
for the various products. At freezing medium temperatures below -200°C, reduction in
the freezing medium temperature has a small effect on the freezing time. However, at
freezing medium temperatures above —100°C, the appropriate selection of coolant with
a lower boiling temperature will decrease the freezing time significantly. The influence of
the surface heat transfer coefficient on the freezing time is very large at the values of
surface heat transfer coefficient lower than 100 W/m2°K. Above the value of 200 W/mzoK
for the surface heat transfer coefficient, an increase in surface heat transfer coefficient has
a very small influence on the freezing time. The results in Figure 14 demonstrate the
increase in freezing time for product with larger diameters. The freezing time is almost
doubled with a two-fold increase in the product diameter. Although the total heat content
to be removed is proportional to the mass (or approximately the volume) of the product,

the exposed surface area of the product to the freezing medium is also increased. The net

36

       

 

 

 

I
I
36 .- I
I
I
I
32 h- I
I
I
TI 10°C I
28 - =
h = 25 W/m2°K :
I
I
A 24 ‘- |
3
:s
.s I
E I
«E; 20 r- I
i: I
.§° I
I
5; 16 .. I
I
I
12 — . I
Diameter = 2cm I
3 I
I
4 ‘ :TINF=-25°C
Diameter = 1cm I
I
I
0 I l I l L
-200 —150 —100 —50 O

Freezing Medium Temperature (°C)

Figure 12. Predicted freezing time versus freezing medium temperature for
various food products of different product diameters.

37

      

 

    

 

 

36 -
|
32 - I,
I.
‘ O
28 I- TI'= 10 C a
TINF = —35 C
1 Asparagus
A 24 — 2 Carrots
35’ 3 Cherries
.S 4 Peas
:8: 20 - 5 Plums
.5 3
[— 2
.316-
g 5
I-h
12 '-
1
8 _ Diameter = 2cm
4 —
Diameter = 1cm
OF I I I I 1 I I
50 100 150 200 250 300 350

h, Surface Heat Transfer Coefﬁcient (W/m2° K)

Figure 13. Predicted freezing time versus surface heat transfer coefﬁcient for
various food products with different product diameters.

38

3
36 #- T1 = 10°C 2
TINF = —35 °C 1
h = 25 W/m2°K
5
1 Asparagus
32 .. 2 Carrots 4

Freezing Time (minute)
N N
C 4s
T I

H
0"
I

3 Cherries
4 Peas
5 Plums
28 p
32
15 h = 70 W/m2°K
12 - 4
8 ..
3.
41:5 h = 170 W/m2°K
0 i 1 ' I

1.0 1 .5
Product Diameter (cm)

 

 

N
C

Figure 14. Predicted freezing time versus product diameter for various food
products at different surface heat transfer coefﬁcients.

39

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effect shows that the freezing time is approximately proportional to the diameter of
the product.

Two plots of freezing history were shown in Figures 15 and 16. Figure 15 compares
the freezing history of asparagus, cherries and peas which have different initial water
contents (asparagus = 0.926, cherries = 0.867 and peas = 0.758). The center temperature
of asparagus stays in the horizontal region for a longer period compared to the other two
products, and a longer freezing time was expected for asparagus. Figure 16 shows the
inﬂuence of surface heat transfer coefficient on the temperature history of strawberries.
At a lower value of the surface heat transfer coefficient, the product center temperature
stayed at the cryoscopic temperature for a longer period. The time that the center
temperature stayed at initial freezing temperature covers almost one half of i the total
freezing time. The horizontal portion of the curve was reduced as the temperature history
curve was moving toward the center of the product and eventually approached a continuous
curve when the surface heat transfer coefficient was very large. The influence of the latent
heat removal on the surface temperature was not very clearly shown on the temperature
history curve, yet the convex shape of surface temperature history curves was recognized
as the retardation of the latent heat removal during water crystallization. This was
illustrated by equation (4) such that at very low temperatures, there is still some water
unfrozen. The removal of latent heat of unfrozen water retards the temperature drop.

The inﬂuences of several product properties on the freezing time for the actual food
products is demonstrated in Figures 17, 18, 19 and 20. The combined effect of these
properties on the freezing time is complicated. However, the trend of curves shown in
Figures 17 through 20 were found to be consistent except for some cases where abnormally
high values of the parameters were used. The unexpected longer freezing times of cherries

and carrots shown in Figure 17 was due to the higher product density. A considerably

Freezing Time (minute)

42

 

 

 

 

 

 

30 r- T1 = 10°C a
TINF = -—35 C
Diameter = 2cm
1 Asparagus
2 Carrots
25 r— 3 Cherries
4 Peas
5 Plums
6 Strawberries
20 "'
15 _
3 2 6 1
5 4W A
4? h = 70
10 -
4.42 W
5 I I I I
0.75 0.80 0.85 0.90 0.95

Initial Water Content, IWC, g-water/g-product

Figure 17. Predicted freezing time versus initial water content fondifferent
food products at different values of surface heat transfer
coefficient.

Freezing Time (minute)

43

 

30r-

T1 = 10°C
TINF = 45%
Diameter = 2cm

1 Asparagus
2 Carrots

3 Cherries

4 Peas

5 Plums

6 Strawberries

N
UI
I

N
c
j

 

 

 

15 I-
3 2 6
5 1
4
h = 70
10 r-
3 =
4 D 2 h 170
a e C) 6 I] 1
5 L | I I
—2.5 -—2.0 -1.5 —1.0 —0.5

Initial Freezing Point (°C)

Figure 18. Predicted freezing time versus initial freezing point for different

food products at different values of surface heat transfer
coefficient.

Freezing Time (minute)

44

 

 

 

 

 

35 .—
30 I.
TI= 10°C 0
TINF = —35 C
Diameter = 2cm
25 " 1 Asparagus
2 Carrots
3 Cherries
4 Peas
S Plums
20 _ 6 Strawberries
h = 70
15 F 3
M
1
4
r
10 ..
h = 170
4 W
5 I I I L
1.00 1 .02 1 .04 1 .06 1 .08

Initial Product Density, IPD, g/cm3

Figure 19. Predicted freezing time versus initial product density for different
food products at different values of surface heat transfer coefﬁcient.

 

45

 

 

 

 

35 -
30 -
T1 = 10°C 0
f; TINF = —35 C
‘2 Diameter = 2cm
3 _ 1 Asparagus
E 25 2 Carrots
'[u 3 Cherries
a“ 4 Peas
3 S Plums
E 6 Strawberries
20 I-
15 -
10 -
4 a 5 h = 1709 6
E—’ a D
5 1 I .L I
0.46 0.48 0.50 0.52 0.54

Initial Thermal Conductivity (W/mzoK)

Figure 20. Predicted freezing time versus initial thermal conductivity for

different food products at different values of surface heat
transfer coefficient.

46

shorter freezing time for peas compared to that of plums is because of the lower product
density for peas. Low initial water contents of plums and peas caused the shorter freezing
times in Figures 18 and 19 in which the other points were consistent with the previous
predictions. The small influence of initial thermal conductivity was demonstrated in
Figure 20. The shorter freezing times of peas and plums were caused by the lower initial

water content.

V. CONCLUSIONS

. A computer prediction model which requires only the unfrozen product property
data can be used to compare the freezing times of different food products.

. A food product possesses a specific cryoscopic temperature where all the temperature
function curves of product properties are discontinuous and nondifferential.

. More accurate results can be obtained in predicting the freezing time by dividing the
food product into more layers in the ﬁnite difference formulation. Accordingly, a
longer computing time is required.

. At freezing medium temperatures below ~2000C, reduction in the freezing medium
temperature has a small effect on reducing the freezing time. Furthermore, above

- the value of 200 W/m2°K for the surface heat transfer coefﬁcient, an increase in
surface heat transfer coefﬁcient has a very small influence on the freezing time.

. The freezing time of a food product is a function of the initial freezing point, water
content, product density and thermal conductivity. In addition, the independent
influences of the above product properties on the freezing time can be predicted
through the usage of hypothetical combinations of product properties.

. A food product with a lower initial freezing point, a higher initial water content and
a higher initial product density has a longer freezing time.

. The freezing time is most sensitive to the accuracy of the measurement of the initial
product density and the initial freezing point while the initial freezing point of the

food product falls above —0.5 6'C. Little influence of the accuracy of initial thermal

47

48
conductivity data is observed for fruit and vegetable products.

8. The functional curves of the influences of product properties on the freezing time for
actual food products were found to be consistent with the trend predicted by the
usage of independent influence analysis. Unexpected longer or shorter freezing times
can be explained by the abnormally high or low values of the product properties except
the one under investigation.

9. The combined inaccuracy of measuring the product property values may have a very

signiﬁcant effect on prediction of the exact freezing time.

VI. SUGGESTIONS FOR FURTHER STUDY

. To experimentally verify or modify the model used in this paper to predict thermal
properties and freezing time.

. To utilize the anisotropic model for thermal conductivity in a simulation for ﬁbrous
products.

. To modify the model to take into account the changes in product density (about 10%)
and product diameter which occur during freezing.

. To modify the model to predict thermal properties and freezing time for a slab, a

cube or a cylindrical product.

. To utilize the present model or modiﬁed model to investigate the inﬂuence of product
properties on freezing rate for other food products to check the consistency.

. To utilize this model to predict the product properties below the initial freezing point
for food products of which the thermal properties above the freezing point have been

experimentally measured.

49

BIBLIOGRAPHY

Albasiny, E. L. 1956. The solution of non-linear heat conduction problems on the pilot
ACE. Proceedings of A. I'. E. E. (103B). Supplement Nos. 1—3: 158.

Bakal, A. and K. Hayakawa 1973. Heat transfer during freezing and thawing of foods.
Advances in Food Research. _2_0: 217—256.

Bartlett, L. H. 1944. A thermodynamic examination of the “latent heat” of foods.

Refrig. Eng. ﬂ: 377—380.

Bedford, C. L. 1976. Personal communication, April 20.

Bonacina, C. and G. Comina 1973. On a numerical method for the solution of the
unsteady state heat conduction equation with temperature dependent parameters.
Proceedings of the XIII International Con gress of Refrigeration. AVI Publishing Co.,
Inc. Vol 11: 329—336.

Carslaw, H. S. and J. C. Jaeger 1959. Conduction 9! PM 12 Sikhs. Claredon Press,
Oxford, England.

Charm, S. and J. Slavin 1962. A method for calculating freezing time of rectangular

package of food. Int. Inst. Ref. Comm., 2, 3, 4. 6A and 9:567;

Charm, S. E. 1971. The Fundamentals 5! Food Engineering. 2nd Ed. The AVI Publishing

 

Company, Inc. Westport, Conn.
Comina, G. and C. Bonacina 1974. Application of ( omputer codes to phase-change problems

in food engineering. In Current Studies 51} t_hg Thermqphysical Properties of Eoodstuffs.

 

Published by International Institute of Refrigeration. pp. 15—27.
Cordell, J. M. and D. C. Webb 1972. The freezing of ice cream. International Symposium

on Heat and Mass Transfer Problems in Food Engineering. Paper D3. Wageningen, The

50

51
Netherlands. Oct. 24—27.

Cullwick, T. D. C. and R. L. Earle 1973. Prediction of freezing time s of meat in plate
freezers. Proceedings of the XIII International Congress of Refregeration. AVI
Publishing Company, Inc. Vol II: 397—401 .

Dickerson, R. W., Jr. 1969. Thermal properties of food. In the Freezmg Preservation o_f
M. 4th Ed. Vol II. AVI Publishing Company, Inc. Vol II: 397—401.

Duckworth, R. B. 1971. Differential analysis of frozen food systems.I. The determination
of unfreezable water. J. Food Technol. 6: 317—327.

Eddie, G. C. and S. F. Pearson 1958. The freezing time of ﬁsh in airblast freezers. Journal
of Refrigeration.l: 124.

Ede, A. J. 1949. The calculation of the freezing and thawing of food stuffs. Modern
Refrig. 2: 52.

Fennama, O. and W. D. Powrie 1954. Fundamentals of low-temperature food preservation
Adv. Food Research 1_3_: 219—347.

Fleming, A. K. 1973. The numerical calculation of freezing processes. Proceedings of
the XIII International Congress of Refrigeration. AVI Publishing Company, Inc.

V0. 11: 303—31 1 .

Gorby, D. P. 1974. Computer simulation of food freezing process. Special problem
report. Agricultural Engineering Department. Michigan State University. East Lansing,
Michigan.

Heldman, D. R. 1966. Predicting refrigeration requirements for freezing ice cream. Quart.
Bull. Mich. Agri. Exp. Stat.f_I_9_ (2): 144—154.

Heldman, D. R. 1974-A. Computer simulation of food freezing processes. Paper presented

at the IV International Congress of Food Science and Technol; gy. Madrid, Spain.

Sept. 22—27.

52

Heldman, D. R. 1974 -B. Predicting the relationship between unfrozen water fraction
and temperature during freezing using freezing point depression. ASAE Trans.
11 (1 ): 63—66.

Heldman, D. R. and D. P. Gorby 1974-A. Prediction of thermal conductivity in frozen
foods. ASAE Paper No. 74—6017. American Society of Agricultural Engineers.
St. Joseph, Mich.

Heldman, D. R. and D. P. Gorby 1974-B. Computer simulation of ice cream freezing.

In Current Studies pp the Thermal Physical Properties _o_f Food Stuffs published by

 

International Institute of Refrigeration, Paris, France. pp. 29—36.

Heldman, D. R. and D. P. Gorby 1975. Computer Simulation of Individual-Quick-
Freezing of Foods. ASAE Paper No. 75—6016. American Society of Agricultural
Engineers. St. Joseph, Mich.

Heldman, D. R. and T. I. Hedrick 1970. Refrigeration requirements for ice cream freezing.
J. Dairy Sci._5_3(9): 1194—1200.

Hill, J. E. 1966. The thermal conductivity of beef. M. S. Thesis. Georgia Institute of
Technology. Atlanta, GA.

Hill, J. E., J. D. Litman and J. E. Sunderland 1967. Thermal conductivity of various
meats. Food Technol.ﬂ: 1143.

Hohner,G. A. and D. R. Heldman 1970. Computer simulation of freezing rates in foods.
Paper presented at the 30th Annual Meeting of Food Technologists. San Francisco,
California, May 24—27.

Kopelman, I. J. 1966. Transient Heat Transfer and Thermal Properties in Food Systems.
Ph.D. Thesis. Food Science Department, Michigan State University, E. Lanlsing, Mich.

Kreith, F. 1973. Principles pf Heat Transfer. 3rd Ed. Intext Educational Publishers,

New York and London.

53

Lentz, D. P. 1961. Thermal conductivity of meats, fats, gelatin gels, and ice. Food
Technol._1§:243—247.

Lescano, D. E. and D. R. Heldman 1973. Freezing rates in codﬁsh muscle. ASAE Paper
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Long, R. A. K. 1955. Some thermodynamic properties of ﬁsh and their effect on the
rate of freezing. Journal of Science in Food and Agriculture _6_ : 621.

Luyet, B. 1969. Basic physical phenomena in the freezing and thawing of animal and
plant tissue. In Freezing Presservagigp 9f E99515. 4th Ed. Vol II. The AVI Publishing
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Maxwell, J. C. 1904. _A_ m o_r_i Electricity Ed Magnetism. 3rd Ed. Clarendon Press,
Oxford, England.

Morley, M. H. 1972. Thermal Properties of Meat: Tabulated Data. Meat Research
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Nagaoka, J., S. Takagi and S. Hotani 1955. Experiments on the freezing of ﬁsh in an
airblast freezer. Proc. 9th Int. Congr. Refrig., Paris_2_: 4.

Plank, R. Z. 1913. Zeitschrift fur die gesarnte Kalte-Industrie 295 109. Cited by A. J.
Ede 1949. The calculation of the freezing and thawing of food stuffs. Mod. Refrig.
_5_2_: 52.

Reidy, G. A. 1968. Thermal Properties of Foods and Methods of Their Determination.
M. S. Thesis. Michigan State University, E. Lansing, Mich.

Riedel, L. 1949-A. Measurements of the thermal conductivity of sugar solutions,fruit
juices, and milk. Chemic.-Ingenier-Technik.2_1: 340—341 (German).

Riedel, L. 1949-B. Refractive index and freezing temperatures of fruit juices as a
function of concentration. Lebensmittel_8__9_: 289—297 (German).

Riedel, L. 1956. Calorimetric investigation on the freezing of ﬁsh. Kalte technik_8(12):

54

374—377 (German).

Riedel, L. 1957. Calorimetric investigations of the meat freezing process. Kaltetechnik
2(2): 38—40 (German).

Staph, H. E. and 5W. R. Woolrich 1951. Specific and latent heats of foods in the freezing
zone. Refrig. Eng.5_9:1086—1089, 1114.

Tarnawski, W. 1976. Mathematic model of frozen consumption products. International
Journal of Heat and Mass Transfer 1_9_: 15—20.

Woodams, E. E. and J. E. Nowrey 1968. Literature values of thermal conductivities of

foods. Food Technol._22(4): 150.

APPENDIX A

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APPENDIX B

EXPLICIT NUMERICAL SOLUTIONS

AND STABILITY

The analysis of a spherical transient heat conduction problem can be simplified
by cosidering heat transport being only in the radial direction. In the case of
spherical food freezing, the direction of positive heat flow is defined to be pointed

radially outward. Then, the energy equilibrium equation at any node n of the sphere

 

 

 

 

is:
' t d
quin _ qnout; = :0“? (B1)
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’ T! 1' I ' .' }
Eo—N I I I . A
l I I I
I I I I I
/ I, I, I
'13-: / a) I I I
I IﬁT lg ”.... I, / ll
/ /
I‘ /)§ "1 I, I /
__,,.,r/ 199/ / / / A.
‘ .——//// 8M4 / //
...-.— // éMp/M‘N
/’ / [/7I—1+I=Tao
/ .
ll /’// ///// A
______”"’ / /
3, /
/ /
. /
b__ _'__,—v //’
l———--"”’/

 

Figure B—l Schematic representation of the spherical nodal heat transfer system
of a ﬁnite difference method.

B1

B2

For the general interior nodes (except the center node), the above equation can be

 

 

 

 

 

 

 

 

written as:
k + k A
n1 n) “1(Tn—1—TnI—
2 A r
k + k A dT
n n—I n2 = __
( 2 ) Ar (Tn ’ Tn+1 ) ﬁn Cnvn (16 (B2)
where: Tn = nodal tempetature
kn = thermal conductivity of the product at node n and temperature Tn
Ar = thickness of the shell
Anl = area of shell closer to the center
An2 = area of shell closer to the surface
Vn = volume of the shell n
Pn = density of the product at temperature Tn
Cn = apparent specific heat of the product at temperature Tn
6 = time
Since 3'6— = 11 A9 :1 , equation (B2) can be solved by the explicit method as follows
6+1 e e
Tn A6 Tn =(En—1 + kn )Anl Tn-l — Tug) _
2 ﬂ. Cn Vn A r
k + k +1) A 0 _ e
( n I1 I12 Tn Tn+1 ) (B3)
2 ﬁeryn A 1»
Let W1 = (kn—1 +kn )Anl ’wz = (kn +kn+1 lAnZ

 

 

2.I'JnCnVnA r 2 PnCnVnAr

Equation (B3) may be simplified as:
T 2+1 = WIA6(Tn?_—1 _ Tr? ) " W2A6( Tn6 — Tnfl ) + Tne

= ( 1 — wlAe— wzoe) Tn +W106Tn._1 +W245Tn+1 (B4)

B3

To stabilize the above equation, the coefficient of T: should not be negative, thus

1
AMW “35’

Since kn—l < kn< kn+1 because of Tn—l >Tn >Tn+1 and the thermal conductivity of
food products increases with decreasing temperature (refer to Figure 3). Besides,
Anl < An2
VII / A112 9:: Ar
In order to minimizeAe, equation (BS) can be expressed as:
Pncn Ar2

Ae< __-——— , n=2,3,....,M (B6)
2kn+1

For the center node:

in_0

q _

out

k+k
<1 =(1 2)

A12 (T1 —T2)
2 Ar

 

qstored. P1C1V1—_ dT

dt
9
dT T 9+1 — '1‘
Taking Y2= L1 + k2) A12,a n—-d = 1 1

 

, equation (B1) for the center node

can be solved by the explicit method as:
9+1 9 6
T1 = ( 1 — Yer) T1 + YZAOTZ (B7)

To stabilize the above calculation, it requires:
1 2 C V Ar
A6 5 __ P1 1 1

 

Since

Ar/Z

V1/A12

B4

Thus
P1C1 “2 .,
A8 .<:——— (B8)
4 2k2
Equations (B6) and (B8) can be combined as:
pncn Arz
A6 < , n = 1,2,3, ....,n (B9)

 

~

2kn+1
for the center and interior nodes.

For the surface node M,
9M. =(M—1 M ) ‘2‘“ (TM—l—TM)
. r I
qMout = h AM2 (TM— TM)
qﬁored = PMCMVM "ii—F;—

where
h = surface heat transfer coefficient
Toa= freezing medium temperature

+1
Let X1=(kM—1+kM)AM1 ’ x = hAM2 and dT = TM ‘TM

2 PMCMVMA r PMCMVM de A 6

equation (B1) for the surface node can be solved by the explicit method as

 

 

 

Tlgﬂ = (1 .. wlae _ w2A9)T§+wlAeT£_1 +W2A9Toa (B10)

To stabilize the above equation, it requires that

1
A6 _<_ —-———- (B11)
W1 +WZ
Since
kM — 1 < kM

AMl < AM2

B5

Equation (B11) can be written as

C A 2
A6 g—ZM M r (1312)
2( kM-i-hAr)

in order to minimize the right hand term of equation (B11).

To obtain the minimum value ofAe, it is necessary to consider the temperature
function of f , C and k, and take the minimum value of P and C, the maximum value
ofk. From ﬁgures 3, 4 and 6,

k a: 4 k.
fmin it 0.9 P .
Cmin c: C o
where k0 , f, , C. are the product property values above the initial freezing point.
Comparing equations (B9) and (B12), A6 in equation (B12) is smaller than that in
equation (B9) because of hAr is positive and kMZ kn+1’ CM< Cn’ whereas the influence
of density is very small.
This concludes our search for the minimum value of the computing time increment
(AG) expressed as the initial property value to stabilize the explicit finite difference

solution throughout the freezing processes, and

1 ﬂCozsr2
A6 5. —S-[2 (ko+hAr)] (B13)

 

where S 2 5.

APPENDIX C

UNITS CONVERSION

Temperature C = ~34 F+32 )
K = C + 273.15

=-3-(F +459.67)

K=%R
3.98
Length 1m = Wft
Mass lKg = 2.2046lbm
lb
Density 1Kg/m3 = 1 10—3 g/cm3 = 0.0624 .55"—
Speciﬁc Heat 1 J/SOK = 0.239 BTU/lbmoF
Thermal Conductivity 1 W/moK = 0.578 BTU/hrft c‘F

Surface Heat Transfer Coefﬁcient 1 W/mzoK = 0.176 BTU/hrft2°F
Energy 1 J = 9.48 ><1o-4 BTU

Heat Transfer Rate 1 W = 3.414 BTU/hr

C1

HICHIGQN STATE UNIV. LIBRRRIE

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