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Refrigeration can be costly in terms of equipment and energy, and if not done correctly will fail to
achieve its objectives and lower the quality and safety of the product. To ensure that refrigeration is
effective, we need to be able to calculate processing times, product temperatures, heat loads and water
diffusion into and out of the product.
The calculation of food cooling and freezing may be complicated by phase change. The product
undergoing cooling often has very complex shape and composition. Heat transfer may be coupled with
moisture transfer and the equations governing the two processes should be solved simultaneously. The
heat transfer coefficient is often difficult to determine for the infinite variety of real-life situations such
as packaged products, cryogenic cooling, highly turbulent flow, swirling and non-parallel flow, etc. and
within the same chiller the heat transfer coefficient will vary greatly from place to place. The heat
transfer coefficient may also vary along the surface of a product due to the complex turbulent flow
pattern and the development of the flow boundary layer, a problem still largely unsolved even by the
most advanced computational fluid dynamics packages. Food products often have inconsistent
compositions, shapes and sizes, which lead to variable thermal behaviour and quality after processing,
so that it is difficult to quantify accurately the effects of different processing practices.
This paper will review methods for modelling foods with complex shapes, from highly simplified
models to detailed numerical models including CFD. Typical results on the accuracy of various
methods will be reported.

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Content Preview

"Food for Thought - Cool" AIRAH Conference, Darling Harbour, Sydney, Australia, 24 May, 2002

**CALCULATION OF PROCESSING TIME AND HEAT LOAD DURING**

**FOOD REFRIGERATION**

Q. Tuan Pham

School of Chemical Engineering and Industrial Chemistry

University of New South Wales, Sydney 2052, Australia

email: tuan.pham@unsw.edu.au

**ABSTRACT**

Refrigeration can be costly in terms of equipment and energy, and if not done correctly will fail to

achieve its objectives and lower the quality and safety of the product. To ensure that refrigeration is

effective, we need to be able to calculate processing times, product temperatures, heat loads and water

diffusion into and out of the product.

The calculation of food cooling and freezing may be complicated by phase change. The product

undergoing cooling often has very complex shape and composition. Heat transfer may be coupled with

moisture transfer and the equations governing the two processes should be solved simultaneously. The

heat transfer coefficient is often difficult to determine for the infinite variety of real-life situations such

as packaged products, cryogenic cooling, highly turbulent flow, swirling and non-parallel flow, etc. and

within the same chiller the heat transfer coefficient will vary greatly from place to place. The heat

transfer coefficient may also vary along the surface of a product due to the complex turbulent flow

pattern and the development of the flow boundary layer, a problem still largely unsolved even by the

most advanced computational fluid dynamics packages. Food products often have inconsistent

compositions, shapes and sizes, which lead to variable thermal behaviour and quality after processing,

so that it is difficult to quantify accurately the effects of different processing practices.

This paper will review methods for modelling foods with complex shapes, from highly simplified

models to detailed numerical models including CFD. Typical results on the accuracy of various

methods will be reported.

**Keywords:**

Cooling time, freezing time, product heat load, modelling

**1. INTRODUCTION**

From the refrigeration engineer's point of view, the most important parameters determining the design

of cooling and freezing processes and equipment are the processing time and the heat load. The former

determines the size of the processing equipment (chiller, freezer) since holding capacity is the product

of throughput and residence time, while the latter determines the capacity of the refrigerating

equipment (compressors, condensers) and the energy consumption.

Twenty years ago, the refrigeration engineer would estimate processing time and product heat load

from experience or, if he was really "nerdy", by using some simple empirical equation or chart. There

is a lot of uncertainty in using such methods but as products and processing methods tended to change

only slowly, design engineers had time to experiment and learn from their experience.

In the last twenty years, things have changed greatly. New markets and products are being developed

more and more quickly, while computers have also progressed at an increasing rate, doubling in speed

and capacity every three years. Commercial calculation packages such as finite element methods and

computational fluid dynamics have appeared in the market, giving the designer new tools that are very

powerful but also at times dangerous to use. This paper will attempt to give a review of the range of

computational methods that are now available.

**2. THEORY**

During the chilling and freezing of solids, heat must be conducted through layers of material before

reaching the surface, then cross over to the surrounding fluid either directly or through layers of

packaging. Processing time and heat load depend on the properties of the product (thermal

conductivity, density and specific heat), the surface heat transfer coefficient and the thermal resistance

of any packaging, and also on the size and shape of the product. The latter is often a major obstacle to

the accurate prediction of processing parameters, due to the complex shape of many products of

interest.

The processing time is usually defined as the time for the thermal centre of the product to reach a

certain temperature. In a complex geometry, it depends on what happens in the*thickest* part of the

object, for example the rump of a carcass. The heat load varies greatly during processing, being highest

at the start, and this*peak* heat load, which may determine refrigeration requirements, often depends on

what happens in the*thinnest* parts (biggest surface area per unit volume). Only in objects of simple,

regular shapes (spheres, long cylinders, large slabs) will both be determined by the same dimension.

When the internal resistance to heat transfer is negligible compared to the surface resistance, the whole

object will be at uniform temperature and the calculation of processing time and heat load becomes

trivial. In the more general case the Biot number,*Bi*, measures the ratio of internal to external

resistance:

*Bi = hR / k*

(1)

where*h* is the surface heat transfer coefficient (including the effect of any packaging, radiation and

evaporative cooling),*R* the *smallest* half-dimension and *k* the thermal conductivity of the product. The

Biot number is of fundamental importance to any study of heat transfer to solid objects. Internal

resistance can be neglected if*Bi* << 1 (uniform internal temperature). Another limiting case is when Bi

tends to infinity, for which analytical solutions have been found for many cases. Many simplified

methods use interpolation to predict cooling times and heat loads for intermediate cases.

It is vital for practicing engineer to be aware of the Biot number of their thermal processes. If it is much

larger than 1 (i.e. if internal resistance is controlling heat transfer), then there is little use trying to

improve heat transfer say by reducing packaging or increasing air velocity, and more advantage will be

gained by trying to reduce the dimensions of the product (if possible); the inverse applies when*Bi* is

much smaller than 1.

**3. SIMPLIFIED CALCULATION METHODS**

**Cooling time**

When a solid object is cooled from a uniform initial temperature*Ti* in an medium at constant

temperature*Ta *, the temperature *T *at any location in the solid as well as the mean temperature obey the

following equation, which is a sum of exponential decay terms (Carslaw and Jaeger [1]):

(*T - Ta* )/(*Ti - Ta* )* = c1 *exp(-*b1t*) + *c*2* *exp(-*b2t*) + ...

(2)

where the coefficients*bi* and *ci* depend on the Biot number and the location in the product. The series

solution is difficult to compute but graphs of the solutions are provided in many heat transfer textbooks

for some simple shapes (e.g. Holman [2] p.148). However, the situation is greatly simplified if the

temperature fall is more than about 30% of the maximum that can be achieved (i.e. (*T - Ta* )/(*Ti - Ta* ) ?

0.7). In that case, all the terms in the series solution become negligible except for the first, which has

the smallest decay rate. Thus, after the initial period the residual temperature difference and heat load

decay exponentially. The centre temperature*Tc* and mean temperature *Tm* are described during the

exponential decay phase by

(*Tc - Ta* )/(*Ti - Ta* ) = *jc *exp*(*-2.303 *t*/*f*)

(3)

(*Tm - Ta* )/(*Ti - Ta* ) = *jm *exp*(*-2.303 *t*/*f*)

(4)

from which cooling time can be calculated.*f* is the time for the residual temperature difference or heat

load to decrease by a factor of 1/10 during the decay period, while*j* is called the "lag factor" (it

measures how much the actual decay curve lags behind a true exponential decay curve). While the*j*-

factor depends on the position in the product,*f* remains the same for all positions.

Equations are available for calculating*j* and *f *for basic shapes (slab, cylinder and sphere). Pflug [3]

plotted these solutions on a graph. Ramaswamy*et al*. [4] and Lacroix and Castaigne [5] gave

approximate solutions that can be computed with a calculator.

For complex or irregular shapes, there has been two main lines of approach to the prediction of cooling

time. The first is to approximate the object by an idealised (simple) shape with an equivalent thickness

or diameter. Thus, a lamb leg or beef rump can be approximated by an equivalent infinite cylinder [6,7]

or a beef carcass can be described as a collection of slabs and cylinders [8]. As long as the real shape does

not deviate too far from the ideal, the equivalent dimension can be calculated from, say, the volume/area

ratio.

The second approach attempts to derive general expressions for the*j* and *f* factors of complex shapes

from their various dimensions. The most advanced of these methods is that of Lin et al. [9-11]. The

formulas are long and involved but simple to apply on a spreadsheet.

**Heat load during cooling**

For simple shapes, analytical and graphical solutions can be found in most heat transfer textbooks. The

*f-j* method should normally be avoided since it does not apply in the initial cooling phase, which is

when heat load is highest. However, if the average heat load*Qav *between times 0 and time *t* is needed,

it can be calculated from

*Qav *= *mc *(*Ti - Tm*) / *t*

(5)

where*Tm* can be found by the *f-j *method, as long as the exponential period has been reached. For

example, the average heat load*Q0-0.7 *between time 0 and time *f*. log10 (*jm/*0.7), at which the residual

mean temperature difference has fallen to 70% of its initial value, is

*Q0-0.7 *= 0.3 *mc *(*Ti - Ta*) / [*f*. log10 (*jm/*0.7)]

(6)

**Freezing time**

The freezing of a food can be divided into three distinct periods:

1. A precooling period, where cooling takes place without phase change.

2. A phase change period, which starts when the surface reaches the freezing point*Tf* and ice forms at

the surface. The freezing front gradually advances into the product, until it reaches the thermal

centre and the whole food can be considered frozen.

3. A postcooling period, during which the whole food continues to cool below*Tf*.

Plank [12,13] presented an analytical solution for the phase change period only. For many years,

Plank's equation was the only one available and it is still taught in many textbooks. However, it can

lead to large errors (typically 30%). There is no analytical solution for food freezing that includes all

three periods. Furthermore food freezing is more complicated than the freezing of pure water, because

the water in the food does not freeze instantaneously but does so over a range of temperature. To

predict freezing time for a simple shape, many empirical equations have been presented in the last

twenty five years. A fairly simple equation which is nevertheless quite accurate (±10%) is that by the

author (Pham [14]):

*R*

? ?*H*

*H*

*Bi*

1

? 2 ?

?

*t*

(7)

*f *=

+

1 +

?

?

?

?

?

*E*

*h*

*Freeze*

? ?*T*

*T*

1

? 2 ?

?

?

?

2 ?

where*Bi* = *hR*/*kf *, *EFreeze* is the equivalent heat transfer dimensionality (EHTD) shape factor for

freezing (1 for slab, 2 for infinite cylinders, 3 for spheres), ?*H1* and ?*T1* are the volumetric enthalpy

change and temperature difference respectively for the precooling period, and ?*H2* and ?*T2* those for

the combined freezing-postcooling period.

As in cooling, the freezing of an irregular shape can be approximated by that of an equivalent regular

shape; for example, an ellipsoid can be approximated by an equivalent sphere. The technique has also

been frequently used on specific products such as lamb legs.

Another approach for complex shapes is to first calculate the freezing time for a slab then divide it by

the shape factor*EFreeze*, which is available for many shapes (Hossain et al. [15, 16], Cleland et al.

[17,18], McNabb et al. 19,20]).

**Freezing heat load**

A simple method for calculating freezing heat loads was proposed by Lovatt et al. [21,22] (Figure1). It

is based on how the freezing front moves towards the thermal centre and involves an empirical shape

factor*N*, which may be different from the shape factor *E* for freezing time (because heat load depends

what happens in the thinnest parts of the product while freezing time depends on what happens in the

thickest parts).

Figure 1. Lovatt et al's method for freezing

heat load calculation.

**Calculating Product Heat Load in an Industrial Cooler or Freezer**

The methods described in this chapter enable the calculation of heat load from a single item of product.

However, in industry normally many items will be processed at one time, and it is important to take

into account the operational characteristics of the process. For a*strict batch* operation, when the cooler

is filled with product before refrigeration is turned on, the refrigeration must be able to cope with the

peak heat load under the starting condition. For a*continuous operation*, the heat load is constant and

can be calculated from the mean temperatures of the entering and leaving products and the throughput.

Many operations in the food industry operate in an intermediate mode; for example, a chiller or freezer

may be loaded/unloaded continuously during daytime and then continue to cool batchwise during the

night. At any time*t*, the product in the cooler will have residence times ranging from 0 to *tmax* and

hence varying heat loads. In such situations, the total heat load at time*t* is made up of the heat load

from product loaded at all times between*t - tmax *and *t*. The time from *t - tmax *to *t* can be divided into a

number of intervals, the heat load from product loaded in each interval calculated and added together.

**4. NUMERICAL (DISCRETIZATION) METHODS**

In numerical discretization methods (finite differences, finite elements, finite volume), the product is

divided into small control volumes or elements. The heat conduction equation is written for each of

these control volumes/elements, giving a set of hundreds, thousands or sometimes millions of

equations. These equations are then solved together to calculate the change with time of the

temperature in each location.

**Finite differences (FD)** was the earliest such method. The product is represented by a regular grid and

the equations for heat flow between the "nodes" (grid points) are written out and solved by computers.

Nowadays, the solution of a finite difference problem on a computer is very fast, of the order of a few

second or less. The finite difference equations can be set up and solved by any mathematically

competent engineering graduate (sad to say, not all engineering graduates are mathematically

competent). However, this method is practical only for objects that have a reasonably regular shape,

such as a rectangular box or a finite cylinder.

**Finite Elements (FE) **is preferred to finite differences for modelling objects with complex, irregular

shapes. A grid is still used but it can be irregular, consisting of triangles, distorded rectangles or

volumes of various shapes. The grid can be in two or three dimensions. Finite elements models take

more time to solve than finite difference models due to the larger bandwith of the matrix equations that

need to be solved. On a modern PC, a two-dimensional FE model may take several minutes to solve

while a three-dimensional FE model may take an hour or more. Also, it is more difficult to set up the

model, usually requiring special graphical software. However this is not due to the nature of the model

but to the complex shape that such models are applied to. FD models are easier to set up but that is only

because the real shape is usually approximated by a more regular shape.

The theory behind finite elements is mathematically abstract, and nowadays the method of**finite**

volumes (FV), which combines the flexibility of finite elements with the conceptual simplicity of finite

differences, has become widely popular. The object is divided into small control volumes of arbitrary

shape (as in finite elements), and the equations of conservation for heat, mass and momentum are

written for each control volumes. Computational effort is similar for FV and FE. FV is widely used in

computational fluid dynamics (CFD) models.

**5. CFD METHOD**

One of the big uncertainties in calculating process time and heat load is the heat transfer coefficient at

the surface of the product. Often the air is highly turbulent, the shape is complex giving rise to

boundary layer variation and eddies, which make the heat transfer coefficient vary from place to place

and become highly unpredictable. In other cases such as cartoned product, the presence of air gaps with

irregular shapes containing natural convection cells make the effective heat transfer coefficient difficult

to predict. In these cases the technologist must often carry out a large number of experiments, or make

gross approximations based on experience.

Computational Fluid Dynamics offer the promise of eliminating these uncertainties, by calculating the

flow pattern and heat transfer at the product surface from first principles, using the basic equations of

heat, mass and momentum transfer. This has become possible owing to the huge computing power of

modern computers.

For the case of*laminar flow*, CFD has indeed lived up to its promise. However, most industrial

situations involve turbulent flow, for which CFD is still unable to entirely deliver the goods. This is

because turbulent flow involves stochastic variations in the flow and temperature pattern, which must

be averaged out, and during this averaging process, empirical equations and coefficients must be

introduced and the equations lose their fundamentality. They must incorporate*turbulence models*

which are still conceptually suspect and involve a large number of empirical coefficients. The empirical

coefficients are specific to each flow situation and if they are used beyond these (i.e. in almost every

new situation practice), accuracy is no longer guaranteed.

To illustrate these points, Pham and Nguyen's [23] CFD simulation results for heat transfer coefficients

during beef chilling is presented in Figure 2, using the standard*k-*? model and the RNG

(ReNormalisation Group Theory)*k-*? model respectively. Both of these are empirical models with

several empirical coefficients.

Another obstacle against the widespread use of CFD models is the large amount of developmental time

and computation effort involved. It took about three months of an experienced user's time to build the

finite volume of a beef side model, and simulating a 20-hour run took a week on a 300-MHz personal

computer, with frequent intervention from the user to optimise relaxation factors etc. (and that's

ignoring mass transfer).

Nevertheless, computer power continues to increase quickly, and one can expect that more fundamental

turbulence models (such as the large eddy simulation model) will eventually become practicable for the

food industry and yield reliable results.

**Pseudo-steady state CFD simulation**

In view of the large computing time required to simulate a transient problem, a good compromise is to

use CFD to calculate the heat transfer coefficient for a steady state situation, which is much faster and

takes minutes instead of days. Since the htc varies only slowly with time, it can be assumed to remain

the same for a significant fraction of the process time. Thus, changes in temperature can be calculated

using a conduction model (FD, FE or FV) for the inside of the product only, which is again much faster

than a full transient CFD calculation. Figure 3 shows such a calculation for beef chilling (Pham and

Nguyen [23])

15

45

40

12

35

30

9

25

20

6

15

10

3

Temperature, deg.C

5

Heat transfer coefficienth (W/m2/K)

0

0

0

5

10

15

20

0.5

1

1.5

2

2.5

Time in chiller, h

Inlet velocity (m/s)

Figure 3. Leg centre and surface temperatures

Figure 2. Heat transfer coefficient predicted using

of beef side. ?? calculated by CFD using the

the standard*k*-? model (dotted curves) and the

RNG turbulence model, ?• measured.

RNG*k*-? model.

**6. ACCURACY OF COOLING AND FREEZING PREDICTION METHODS**

In view of the large range of calculation methods available it would be useful to compare them against

each other and against empirical data, where available. This will help the refrigeration engineer and

food engineer decide which to use should the need arise.

**Errors in predicting cooling time**

Prediction errors are due to the following sources:

• errors inherent the calculation method (for example, by approximating the cooling curve by the*f-j*

exponential)

• errors in the product's thermal properties (thermal diffusivity)

• errors in the heat transfer coefficient – effect of turbulence, radiation, evaporation, wrapping etc.

• errors in predicting the effect of complex shape

The first source of errors can be eliminated by using a strict analytical or numerical method, however

the other sources of errors cannot be entirely eliminated. Surprisingly little data is available on the

extent of these errors in practice. However, we can expect that they will be of the same order as those

incurred during freezing calculations, for which data*are* available, as will be seen below.

**Errors in predicting freezing time**

Cleland [24] did some careful analysis of the most well known freezing time prediction methods (Table

1). It can be seen that the most accurate methods, such as that of Pham presented above, can predict

freezing time to about ±10%. There is also quite good agreement between simplified formulas and

numerical methods such as finite differences.

Table 1. Comparison of simplified

freezing time prediction methods

with experiments and finite

difference calculations.

It must be noted that these error values apply to experiments carried out in the laboratory under

carefully monitored conditions, with well known test materials, so that the errors due to heat transfer

coefficients, product shape and material properties are minimal. In industry, larger errors can be

expected, although with some expertise, errors of perhaps no more than ± 15% can be achieved.

**Errors in predicting cooling and freezing heat load**

The author and his associates at MIRINZ, Massey University (NZ) and the University of New South

Wales seem to have been the only group to have gathered reasonably accurate heat load data during

cooling and freezing. These data have been obtained using a flow calorimeter technique [25] (Figure

4). The hot product, for example a carton of meat or a beef side, is put in a wind tunnel. A thermopile

(a sensitive differential temperature sensor) is made by having an array of thermocouples distributed

over the inlet flow area and another over the outlet flow area. The thermocouple pairs are connected in

series, thus multiplying the thermoelectric signal. For example, with 20 thermocouple pairs, a voltage

signal of about 8 µV is obtained for every 0.010C temperature rise in the air, which can be easily

measured with a digital voltmeter accurate to 1µV. There is some errors introduced by heat loss

through the tunnel walls, but as long as conditions are stable these effects can be removed by a simple

baseline correction.

A typical heat load curve for beef chilling (Davey and Pham [8]) is shown in Figure 5. Davey and

Pham [26] did a comparison of predicted and measured heat load (averaged over the first two hours of

chilling) for 55 beef chilling tests (Table 2). It can be seen that the predictions for both a finite

differences model and a 2-D finite element models are quite acceptable, although there is a definite

improvement with the FE model (5.6% average error in heat load for FE vs 12.6% for FD).

900

800

Experimental

700

FE Model

Davey and Pham (1997)

600

500

400

300

**Beef Side Heat Load (W) **200

100

0

0

5

10

15

20

25

**Time in Chiller (hr)**

Figure 5. Comparison of heat load prediction by FE,

Figure 4. Wind tunnel cum flow calorimeter

FD and experimental results for a beef chilling

designed by Davey and Pham [25]

experiment.

Table 1. Comparison of FD and FE model against experimental data for 55 tests.

FD Model

FE Model

Representation of beef side

7 simple-shaped regions

13 cross-sections

Average % error in heat removed during

-12.6 %

-5.6 %

first 2 hours

Average % error in weight loss after 20

-1.25%*

2.32 %*

hours of chilling

Time to simulate 20 hour process on

< 1 min

4-5 hours

Pentium 166 Mhz Computer

**7. CONCLUSIONS**

A wide range of methods for predicting cooling/freezing times and heat loads, from simple

approximate equations to complex CFD models. Although sophisticated, theory-based methods are

becoming more practical with the fast progress in computer hardware and software, simplified methods

can still be useful in many cases. CFD is still basically a research tool in view of the major effort

involved in setting up the simulation, large computing time and continuing difficulty in dealing with

complex turbulent flow situations. For the majority of practical cases, a good practical compromise is

the use of finite differences or finite elements code for conduction inside the product, combined with

heat transfer coefficients obtained from empirical equations or CFD simulations.

**REFERENCES**

[1] Carslaw, H.S., Jaeger, J.C. Conduction of Heat in Solids, 2nd edn; Oxford, Clarendon Press, 1959.

[2] Holman, J.P., Heat Transfer, 7th edn; New York, MacGraw-Hill, 1992.

[3] Pflug, I.J., Blaisdell, J.L., Kopelman, I.J. Developing temperature-time curves for objects that can

be approximated by a sphere, infinite plate or infinite cylinder. ASHRAE Transactions; 1965; 71; 238-

48.

[4] Ramaswamy, H.S., Lo, K.V., Tung, M.A. Simplified equations for transient temperatures in

conductive foods with convective heat transfer at surface. Journal of Food Science; 1982; 47; 2042-

2047.

[5] Lacroix, C., Castaigne, F. Simple method for freezing time calculations for infinite flat slabs,

infinite cylinders and spheres. Canadian Institute of Food Science and Technology Journal; 1987;

20(4); 252-259.

[6] Earle R.L., Fleming, A.K. Cooling and freezing of lamb and mutton carcasses. Food Technology;

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[9] Lin, Z., Cleland, A.C., Sellarach, G.F., Cleland, D.J. A simple method for prediction of chilling

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[19] McNabb, A, Wake, G.C., Hossain, Md.M., Lambourne, R.D. Transition times

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[20] McNabb, A, Wake, G.C., Hossain, Md.M., Lambourne, R.D. Transition times

between steady states for heat conduction, Part II: Approximate solutions and examples. Massey

University, Occasional Publications in Mathematics & Statistics No.21. Massey University; 1990.

[21] Lovatt, S.J., Pham, Q.T., Loeffen, M.P.F., Cleland, A.C. A new method of predicting the time-

variability of product heat load during food cooling. Part 1: Theoretical considerations. Journal or Food

Engineering; 1993; 18; 13-36.

[22] Lovatt, S.J., Pham, Q.T., Loeffen, M.P.F., Cleland, A.C. A new method of predicting the time-

variability of product heat load during food cooling. Part 2: Experimental testing. Journal or Food

Engineering; 1993; 18; 37-62.

[23] Pham, Q.T. and Nguyen, A.V. Mean Convective Heat Transfer Coefficient of Beef Carcases

Computed by a Computational Fluid Dynamic. FOODSIM 2000: International Conference on

Simulation in Food and BioIndustries, Nantes, France, June 26-27, 2000.

[24] Cleland, A.C. Food Refrigeration Processes Analysis, Design and Simulation. London; Elsevier;

1990.

[25] Davey, L.M. and Pham, Q.T., A technique for measuring product heat load during beef chilling.

Published in Heat and Mass Transfer Australasia 1996: proc. 6th Australasian Heat and Mass Transfer

Conference, New York, Begell House, pp. 453-460, 1998.

[26] Davey, L.M. and Pham, Q.T., A comparison of three models for predicting heat and weight loss

from beef sides during chilling. 20th Int. Congress of Refrigeration, Sydney, 19-24 Sep 1999.

Q. Tuan Pham

School of Chemical Engineering and Industrial Chemistry

University of New South Wales, Sydney 2052, Australia

email: tuan.pham@unsw.edu.au

Refrigeration can be costly in terms of equipment and energy, and if not done correctly will fail to

achieve its objectives and lower the quality and safety of the product. To ensure that refrigeration is

effective, we need to be able to calculate processing times, product temperatures, heat loads and water

diffusion into and out of the product.

The calculation of food cooling and freezing may be complicated by phase change. The product

undergoing cooling often has very complex shape and composition. Heat transfer may be coupled with

moisture transfer and the equations governing the two processes should be solved simultaneously. The

heat transfer coefficient is often difficult to determine for the infinite variety of real-life situations such

as packaged products, cryogenic cooling, highly turbulent flow, swirling and non-parallel flow, etc. and

within the same chiller the heat transfer coefficient will vary greatly from place to place. The heat

transfer coefficient may also vary along the surface of a product due to the complex turbulent flow

pattern and the development of the flow boundary layer, a problem still largely unsolved even by the

most advanced computational fluid dynamics packages. Food products often have inconsistent

compositions, shapes and sizes, which lead to variable thermal behaviour and quality after processing,

so that it is difficult to quantify accurately the effects of different processing practices.

This paper will review methods for modelling foods with complex shapes, from highly simplified

models to detailed numerical models including CFD. Typical results on the accuracy of various

methods will be reported.

Cooling time, freezing time, product heat load, modelling

From the refrigeration engineer's point of view, the most important parameters determining the design

of cooling and freezing processes and equipment are the processing time and the heat load. The former

determines the size of the processing equipment (chiller, freezer) since holding capacity is the product

of throughput and residence time, while the latter determines the capacity of the refrigerating

equipment (compressors, condensers) and the energy consumption.

Twenty years ago, the refrigeration engineer would estimate processing time and product heat load

from experience or, if he was really "nerdy", by using some simple empirical equation or chart. There

is a lot of uncertainty in using such methods but as products and processing methods tended to change

only slowly, design engineers had time to experiment and learn from their experience.

In the last twenty years, things have changed greatly. New markets and products are being developed

more and more quickly, while computers have also progressed at an increasing rate, doubling in speed

and capacity every three years. Commercial calculation packages such as finite element methods and

computational fluid dynamics have appeared in the market, giving the designer new tools that are very

powerful but also at times dangerous to use. This paper will attempt to give a review of the range of

computational methods that are now available.

During the chilling and freezing of solids, heat must be conducted through layers of material before

reaching the surface, then cross over to the surrounding fluid either directly or through layers of

packaging. Processing time and heat load depend on the properties of the product (thermal

conductivity, density and specific heat), the surface heat transfer coefficient and the thermal resistance

of any packaging, and also on the size and shape of the product. The latter is often a major obstacle to

the accurate prediction of processing parameters, due to the complex shape of many products of

interest.

The processing time is usually defined as the time for the thermal centre of the product to reach a

certain temperature. In a complex geometry, it depends on what happens in the

object, for example the rump of a carcass. The heat load varies greatly during processing, being highest

at the start, and this

what happens in the

regular shapes (spheres, long cylinders, large slabs) will both be determined by the same dimension.

When the internal resistance to heat transfer is negligible compared to the surface resistance, the whole

object will be at uniform temperature and the calculation of processing time and heat load becomes

trivial. In the more general case the Biot number,

resistance:

(1)

where

evaporative cooling),

Biot number is of fundamental importance to any study of heat transfer to solid objects. Internal

resistance can be neglected if

tends to infinity, for which analytical solutions have been found for many cases. Many simplified

methods use interpolation to predict cooling times and heat loads for intermediate cases.

It is vital for practicing engineer to be aware of the Biot number of their thermal processes. If it is much

larger than 1 (i.e. if internal resistance is controlling heat transfer), then there is little use trying to

improve heat transfer say by reducing packaging or increasing air velocity, and more advantage will be

gained by trying to reduce the dimensions of the product (if possible); the inverse applies when

much smaller than 1.

When a solid object is cooled from a uniform initial temperature

temperature

following equation, which is a sum of exponential decay terms (Carslaw and Jaeger [1]):

(

(2)

where the coefficients

solution is difficult to compute but graphs of the solutions are provided in many heat transfer textbooks

for some simple shapes (e.g. Holman [2] p.148). However, the situation is greatly simplified if the

temperature fall is more than about 30% of the maximum that can be achieved (i.e. (

0.7). In that case, all the terms in the series solution become negligible except for the first, which has

the smallest decay rate. Thus, after the initial period the residual temperature difference and heat load

decay exponentially. The centre temperature

exponential decay phase by

(

(3)

(

(4)

from which cooling time can be calculated.

load to decrease by a factor of 1/10 during the decay period, while

measures how much the actual decay curve lags behind a true exponential decay curve). While the

factor depends on the position in the product,

Equations are available for calculating

plotted these solutions on a graph. Ramaswamy

approximate solutions that can be computed with a calculator.

For complex or irregular shapes, there has been two main lines of approach to the prediction of cooling

time. The first is to approximate the object by an idealised (simple) shape with an equivalent thickness

or diameter. Thus, a lamb leg or beef rump can be approximated by an equivalent infinite cylinder [6,7]

or a beef carcass can be described as a collection of slabs and cylinders [8]. As long as the real shape does

not deviate too far from the ideal, the equivalent dimension can be calculated from, say, the volume/area

ratio.

The second approach attempts to derive general expressions for the

from their various dimensions. The most advanced of these methods is that of Lin et al. [9-11]. The

formulas are long and involved but simple to apply on a spreadsheet.

For simple shapes, analytical and graphical solutions can be found in most heat transfer textbooks. The

when heat load is highest. However, if the average heat load

it can be calculated from

(5)

where

example, the average heat load

mean temperature difference has fallen to 70% of its initial value, is

(6)

The freezing of a food can be divided into three distinct periods:

1. A precooling period, where cooling takes place without phase change.

2. A phase change period, which starts when the surface reaches the freezing point

the surface. The freezing front gradually advances into the product, until it reaches the thermal

centre and the whole food can be considered frozen.

3. A postcooling period, during which the whole food continues to cool below

Plank [12,13] presented an analytical solution for the phase change period only. For many years,

Plank's equation was the only one available and it is still taught in many textbooks. However, it can

lead to large errors (typically 30%). There is no analytical solution for food freezing that includes all

three periods. Furthermore food freezing is more complicated than the freezing of pure water, because

the water in the food does not freeze instantaneously but does so over a range of temperature. To

predict freezing time for a simple shape, many empirical equations have been presented in the last

twenty five years. A fairly simple equation which is nevertheless quite accurate (±10%) is that by the

author (Pham [14]):

? ?

1

? 2 ?

?

(7)

+

1 +

?

?

?

?

?

? ?

1

? 2 ?

?

?

?

2 ?

where

freezing (1 for slab, 2 for infinite cylinders, 3 for spheres), ?

change and temperature difference respectively for the precooling period, and ?

the combined freezing-postcooling period.

As in cooling, the freezing of an irregular shape can be approximated by that of an equivalent regular

shape; for example, an ellipsoid can be approximated by an equivalent sphere. The technique has also

been frequently used on specific products such as lamb legs.

Another approach for complex shapes is to first calculate the freezing time for a slab then divide it by

the shape factor

[17,18], McNabb et al. 19,20]).

A simple method for calculating freezing heat loads was proposed by Lovatt et al. [21,22] (Figure1). It

is based on how the freezing front moves towards the thermal centre and involves an empirical shape

factor

what happens in the thinnest parts of the product while freezing time depends on what happens in the

thickest parts).

Figure 1. Lovatt et al's method for freezing

heat load calculation.

The methods described in this chapter enable the calculation of heat load from a single item of product.

However, in industry normally many items will be processed at one time, and it is important to take

into account the operational characteristics of the process. For a

is filled with product before refrigeration is turned on, the refrigeration must be able to cope with the

peak heat load under the starting condition. For a

can be calculated from the mean temperatures of the entering and leaving products and the throughput.

Many operations in the food industry operate in an intermediate mode; for example, a chiller or freezer

may be loaded/unloaded continuously during daytime and then continue to cool batchwise during the

night. At any time

hence varying heat loads. In such situations, the total heat load at time

from product loaded at all times between

number of intervals, the heat load from product loaded in each interval calculated and added together.

In numerical discretization methods (finite differences, finite elements, finite volume), the product is

divided into small control volumes or elements. The heat conduction equation is written for each of

these control volumes/elements, giving a set of hundreds, thousands or sometimes millions of

equations. These equations are then solved together to calculate the change with time of the

temperature in each location.

the equations for heat flow between the "nodes" (grid points) are written out and solved by computers.

Nowadays, the solution of a finite difference problem on a computer is very fast, of the order of a few

second or less. The finite difference equations can be set up and solved by any mathematically

competent engineering graduate (sad to say, not all engineering graduates are mathematically

competent). However, this method is practical only for objects that have a reasonably regular shape,

such as a rectangular box or a finite cylinder.

shapes. A grid is still used but it can be irregular, consisting of triangles, distorded rectangles or

volumes of various shapes. The grid can be in two or three dimensions. Finite elements models take

more time to solve than finite difference models due to the larger bandwith of the matrix equations that

need to be solved. On a modern PC, a two-dimensional FE model may take several minutes to solve

while a three-dimensional FE model may take an hour or more. Also, it is more difficult to set up the

model, usually requiring special graphical software. However this is not due to the nature of the model

but to the complex shape that such models are applied to. FD models are easier to set up but that is only

because the real shape is usually approximated by a more regular shape.

The theory behind finite elements is mathematically abstract, and nowadays the method of

volumes (FV)

differences, has become widely popular. The object is divided into small control volumes of arbitrary

shape (as in finite elements), and the equations of conservation for heat, mass and momentum are

written for each control volumes. Computational effort is similar for FV and FE. FV is widely used in

computational fluid dynamics (CFD) models.

One of the big uncertainties in calculating process time and heat load is the heat transfer coefficient at

the surface of the product. Often the air is highly turbulent, the shape is complex giving rise to

boundary layer variation and eddies, which make the heat transfer coefficient vary from place to place

and become highly unpredictable. In other cases such as cartoned product, the presence of air gaps with

irregular shapes containing natural convection cells make the effective heat transfer coefficient difficult

to predict. In these cases the technologist must often carry out a large number of experiments, or make

gross approximations based on experience.

Computational Fluid Dynamics offer the promise of eliminating these uncertainties, by calculating the

flow pattern and heat transfer at the product surface from first principles, using the basic equations of

heat, mass and momentum transfer. This has become possible owing to the huge computing power of

modern computers.

For the case of

situations involve turbulent flow, for which CFD is still unable to entirely deliver the goods. This is

because turbulent flow involves stochastic variations in the flow and temperature pattern, which must

be averaged out, and during this averaging process, empirical equations and coefficients must be

introduced and the equations lose their fundamentality. They must incorporate

coefficients are specific to each flow situation and if they are used beyond these (i.e. in almost every

new situation practice), accuracy is no longer guaranteed.

To illustrate these points, Pham and Nguyen's [23] CFD simulation results for heat transfer coefficients

during beef chilling is presented in Figure 2, using the standard

(ReNormalisation Group Theory)

several empirical coefficients.

Another obstacle against the widespread use of CFD models is the large amount of developmental time

and computation effort involved. It took about three months of an experienced user's time to build the

finite volume of a beef side model, and simulating a 20-hour run took a week on a 300-MHz personal

computer, with frequent intervention from the user to optimise relaxation factors etc. (and that's

ignoring mass transfer).

Nevertheless, computer power continues to increase quickly, and one can expect that more fundamental

turbulence models (such as the large eddy simulation model) will eventually become practicable for the

food industry and yield reliable results.

In view of the large computing time required to simulate a transient problem, a good compromise is to

use CFD to calculate the heat transfer coefficient for a steady state situation, which is much faster and

takes minutes instead of days. Since the htc varies only slowly with time, it can be assumed to remain

the same for a significant fraction of the process time. Thus, changes in temperature can be calculated

using a conduction model (FD, FE or FV) for the inside of the product only, which is again much faster

than a full transient CFD calculation. Figure 3 shows such a calculation for beef chilling (Pham and

Nguyen [23])

15

45

40

12

35

30

9

25

20

6

15

10

3

Temperature, deg.C

5

Heat transfer coefficienth (W/m2/K)

0

0

0

5

10

15

20

0.5

1

1.5

2

2.5

Time in chiller, h

Inlet velocity (m/s)

Figure 3. Leg centre and surface temperatures

Figure 2. Heat transfer coefficient predicted using

of beef side. ?? calculated by CFD using the

the standard

RNG turbulence model, ?• measured.

RNG

In view of the large range of calculation methods available it would be useful to compare them against

each other and against empirical data, where available. This will help the refrigeration engineer and

food engineer decide which to use should the need arise.

Prediction errors are due to the following sources:

• errors inherent the calculation method (for example, by approximating the cooling curve by the

exponential)

• errors in the product's thermal properties (thermal diffusivity)

• errors in the heat transfer coefficient – effect of turbulence, radiation, evaporation, wrapping etc.

• errors in predicting the effect of complex shape

The first source of errors can be eliminated by using a strict analytical or numerical method, however

the other sources of errors cannot be entirely eliminated. Surprisingly little data is available on the

extent of these errors in practice. However, we can expect that they will be of the same order as those

incurred during freezing calculations, for which data

Cleland [24] did some careful analysis of the most well known freezing time prediction methods (Table

1). It can be seen that the most accurate methods, such as that of Pham presented above, can predict

freezing time to about ±10%. There is also quite good agreement between simplified formulas and

numerical methods such as finite differences.

Table 1. Comparison of simplified

freezing time prediction methods

with experiments and finite

difference calculations.

It must be noted that these error values apply to experiments carried out in the laboratory under

carefully monitored conditions, with well known test materials, so that the errors due to heat transfer

coefficients, product shape and material properties are minimal. In industry, larger errors can be

expected, although with some expertise, errors of perhaps no more than ± 15% can be achieved.

The author and his associates at MIRINZ, Massey University (NZ) and the University of New South

Wales seem to have been the only group to have gathered reasonably accurate heat load data during

cooling and freezing. These data have been obtained using a flow calorimeter technique [25] (Figure

4). The hot product, for example a carton of meat or a beef side, is put in a wind tunnel. A thermopile

(a sensitive differential temperature sensor) is made by having an array of thermocouples distributed

over the inlet flow area and another over the outlet flow area. The thermocouple pairs are connected in

series, thus multiplying the thermoelectric signal. For example, with 20 thermocouple pairs, a voltage

signal of about 8 µV is obtained for every 0.010C temperature rise in the air, which can be easily

measured with a digital voltmeter accurate to 1µV. There is some errors introduced by heat loss

through the tunnel walls, but as long as conditions are stable these effects can be removed by a simple

baseline correction.

A typical heat load curve for beef chilling (Davey and Pham [8]) is shown in Figure 5. Davey and

Pham [26] did a comparison of predicted and measured heat load (averaged over the first two hours of

chilling) for 55 beef chilling tests (Table 2). It can be seen that the predictions for both a finite

differences model and a 2-D finite element models are quite acceptable, although there is a definite

improvement with the FE model (5.6% average error in heat load for FE vs 12.6% for FD).

900

800

Experimental

700

FE Model

Davey and Pham (1997)

600

500

400

300

100

0

0

5

10

15

20

25

Figure 5. Comparison of heat load prediction by FE,

Figure 4. Wind tunnel cum flow calorimeter

FD and experimental results for a beef chilling

designed by Davey and Pham [25]

experiment.

Table 1. Comparison of FD and FE model against experimental data for 55 tests.

FD Model

FE Model

Representation of beef side

7 simple-shaped regions

13 cross-sections

Average % error in heat removed during

-12.6 %

-5.6 %

first 2 hours

Average % error in weight loss after 20

-1.25%*

2.32 %*

hours of chilling

Time to simulate 20 hour process on

< 1 min

4-5 hours

Pentium 166 Mhz Computer

A wide range of methods for predicting cooling/freezing times and heat loads, from simple

approximate equations to complex CFD models. Although sophisticated, theory-based methods are

becoming more practical with the fast progress in computer hardware and software, simplified methods

can still be useful in many cases. CFD is still basically a research tool in view of the major effort

involved in setting up the simulation, large computing time and continuing difficulty in dealing with

complex turbulent flow situations. For the majority of practical cases, a good practical compromise is

the use of finite differences or finite elements code for conduction inside the product, combined with

heat transfer coefficients obtained from empirical equations or CFD simulations.

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[2] Holman, J.P., Heat Transfer, 7th edn; New York, MacGraw-Hill, 1992.

[3] Pflug, I.J., Blaisdell, J.L., Kopelman, I.J. Developing temperature-time curves for objects that can

be approximated by a sphere, infinite plate or infinite cylinder. ASHRAE Transactions; 1965; 71; 238-

48.

[4] Ramaswamy, H.S., Lo, K.V., Tung, M.A. Simplified equations for transient temperatures in

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2047.

[5] Lacroix, C., Castaigne, F. Simple method for freezing time calculations for infinite flat slabs,

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[7] Bailey C., Cox, R.P. The chilling of beef carcasses. Proceeding of the Institute of Refrigeration;

1976; 72; 76-90.

[8] Davey, L.M., Pham, Q.T. Predicting the dynamic product heat load and weight loss during beef

chilling using a multi-region finite difference approach. International Journal of Refrigeration; 1997;

20; 470-482.

[9] Lin, Z., Cleland, A.C., Sellarach, G.F., Cleland, D.J. A simple method for prediction of chilling

times for objects of two-dimensional irregular shape. International Journal of Refrigeration; 1996; 19;

95-106.

[10] Lin, Z., Cleland, A.C., Sellarach, G.F., Cleland, D.J. A simple method for prediction of chilling

times: Extension to three-dimensional irregular shapes. International Journal of Refrigeration; 1996;

19; 107-114. Erratum, International Journal of Refrigeration; 2000; 23; 168.

[11] Lin, Z., Cleland, A.C., Sellarach, G.F., Cleland, D.J. Prediction of chilling times for objects of

regular multi-dimensional shapes using a general geometric factor. In Refrigeration Science and

Technology 1993-3. International Institute of Refrigeration, Paris; 1993; 259-267.

[12] Plank, R. Die Gefrierdauer von Eisblocken. Zeitschrift fur die gesamte Kalte Industrie; 1913;

20(6); 109-114.

[13] Plank, R. Beitrage zur Berechnung und Bewertung der Gefriergeschwindigkeit von Lebensmitteln.

Zeitschrift fur die gesamte Kalte Industrie; 1941; 3(10); 1-16.

[14] Pham, Q.T. Simplified equation for predicting the freezing time of foodstuffs. Journal of Food

Technology; 1986; 21; 209-219.

[15] Hossain, Md.M., Cleland, D.J., Cleland, A.C. Prediction of freezing and thawing times for foods

of regular multidimensional shape by using an analytically derived geometric factor. International

Journal of Refrigeration; 1992; 15(4); 227-234.

[16] Hossain, M.M., Cleland, D.J., Cleland, A.C. Prediction of freezing and thawing times for foods of

two-dimensional irregular shape by using a semi-analytical geometric factor. International Journal of

Refrigeration; 1992; 15(4); 235-240.

[17] Cleland D.J., Cleland A.C., Earle R.L. Prediction of freezing and thawing times for

multidimensional shapes by simple formulae. Part : Regular shapes. International Journal of

Refrigeration; 1987; 10; 156-164.

[18] Cleland D.J., Cleland A.C., Earle R.L. Prediction of freezing & thawing times for

multidimensional shapes by simple formulae. Part 2: Irregular shapes. International Journal of

Refrigeration; 1987; 10; 234-240.

[19] McNabb, A, Wake, G.C., Hossain, Md.M., Lambourne, R.D. Transition times

between steady states for heat conduction, Part I: General theory and some exact results. Occasional

Publications in Mathematics & Statistics No.20; Massey University; 1990.

[20] McNabb, A, Wake, G.C., Hossain, Md.M., Lambourne, R.D. Transition times

between steady states for heat conduction, Part II: Approximate solutions and examples. Massey

University, Occasional Publications in Mathematics & Statistics No.21. Massey University; 1990.

[21] Lovatt, S.J., Pham, Q.T., Loeffen, M.P.F., Cleland, A.C. A new method of predicting the time-

variability of product heat load during food cooling. Part 1: Theoretical considerations. Journal or Food

Engineering; 1993; 18; 13-36.

[22] Lovatt, S.J., Pham, Q.T., Loeffen, M.P.F., Cleland, A.C. A new method of predicting the time-

variability of product heat load during food cooling. Part 2: Experimental testing. Journal or Food

Engineering; 1993; 18; 37-62.

[23] Pham, Q.T. and Nguyen, A.V. Mean Convective Heat Transfer Coefficient of Beef Carcases

Computed by a Computational Fluid Dynamic. FOODSIM 2000: International Conference on

Simulation in Food and BioIndustries, Nantes, France, June 26-27, 2000.

[24] Cleland, A.C. Food Refrigeration Processes Analysis, Design and Simulation. London; Elsevier;

1990.

[25] Davey, L.M. and Pham, Q.T., A technique for measuring product heat load during beef chilling.

Published in Heat and Mass Transfer Australasia 1996: proc. 6th Australasian Heat and Mass Transfer

Conference, New York, Begell House, pp. 453-460, 1998.

[26] Davey, L.M. and Pham, Q.T., A comparison of three models for predicting heat and weight loss

from beef sides during chilling. 20th Int. Congress of Refrigeration, Sydney, 19-24 Sep 1999.

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