refrigertion load cal methods in food processing

F O R U M
22
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
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, 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 from the inside 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.
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
THE OFFICIAL JOURNAL OF AIRAH - July 2002

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 (temperature and heat load
falls exponentially with time). 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 engineers 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 a 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) + c2 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) Baehr (1953), Rutov (1958) and Fikiin and
Fikiina (1971). 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 Pflug, 1965. 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 have 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] Earle and Fleming, 1967; Bailey and Cox, 1976
or a beef carcass can be described as a collection of slabs and
cylinders [8] Davey and Pham, 1997. 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 Smith et al., 1968; Cleland and Earle, 1982; Lin et
al., 1993, 1996a, 1996b. The most advanced of these methods
is that of Lin et al. [9-11] 1996a, 1996b. Crocombe et al,
1999. 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 time 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
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23

F O R U M
24 THE OFFICIAL JOURNAL OF AIRAH - July 2002
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]):
(7)
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], (1992a) Cleland et al. [17,18], McNabb et al. [19,20])
Cleland et al., 1987b,c; Hossain et al., 1992a,b.
Freezing heat load
A simple method for calculating freezing heat loads was
proposed by Lovatt et al. [21,22]. 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 the peak initial heat
load depends on what happens in the thinnest parts of the
product while freezing time depends on what happens in the
thickest parts).
Calculating Product Heat Load in an Industrial Cooler or
Freezer
The methods described in this section 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 seconds or less. The finite difference equations can be
set up and solved by any mathematically competent
engineering graduate. However, this method is practical only
for objects that have a reasonably regular shape, such as a
rectangular box or a finite cylinder.
R H1 H2 Bi
EFreezeh T1 T2 2
+ 1+
tf =
+
–

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,
distorted 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 1, using the standard k-
model and the RNG (ReNormalisation Group Theory) k- model
respectively. Both of these are empirical models with several
empirical coefficients. It can be seen that different heat
transfer coefficients are predicted by the two models.
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. Such a calculation gave good results for beef
chilling (Pham and Nguyen [23])
F O R U M
25
3
3

F O R U M
Figure 1. Heat transfur coefficient predicted using the
standard k- model (dotted curves) and the 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 can be due to the following sources:
• errors inherent in 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.
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
2). 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 0.01ºC temperature
rise in the air, which can be easily measured with a digital
voltmeter accurate to 1µV. There are some errors (normally
less than 5%) introduced by heat loss through the tunnel
walls and other unknown causes. The tunnel was calibrated
with an electric heater to correct for these errors.
A typical heat load curve for beef chilling (Davey and Pham
[8]) is shown in Figure 3. 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).
7. CONCLUSIONS AND FUTURE RESEARCH NEEDS
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.
3 3
+
–
+
–

Future research needs include
• more research on physical properties of foods
• better turbulence models for convection cooling, although
this may be some time in coming. While waiting for this,
• more heat transfer convection data for a variety of
situations, including:
• packages with air voids when there may be convection and
radiation within voids, and partial ventilation through
packages.
• heat transfer in porous foods where there are interactions
between heart and nass transfer
F O R U M
27
Table 2. Comparison of FD and FE model against
experimental data for 55 tests.
Method Mean SD Range Correl.
(%) (%) (%)
Finite Differences 0.1 7.6 -11 to 12 –
Cleland & Earle (1984b) -0.2 6.3 -10 to 10 0.728
Hung & Thompson (1983)a -1.7 8.7 -8 to 21 0.373
Eqn A2 of Cleland et al (1987a) -0.9 6.7 -13 to 9 0.723
de Michelis & Cavclo (1983) -1.7 11.4 -17 to 18 0.858
Pham (1984) 0.4 7.1 -11 to 9 0.787
Pham (1986a) 1.1 7.3 -11 to 12 0.772
Range = range enclosing 90% of the percentage differences.
Correl. = corrrelation coefficient with finite differences.
Table 1. Comparison of simplified
freezing time prediction methods with
experiments and finite difference
calculations.
Figure 2. Wind tunnel cum flow calorimeter
designed by Davey and Pham [25]
Figure 3. Comparison of heat load prediction by FE, FD
and experimental results for a beef chilling experiment.
FD Model FE Model
Representation of beef side 7 simple- 13 cross-
shaped regions sections
Average % error in heat
removed during first 2 hours -12.6 % -5.6 %
Average % error in weight loss
after 20 hours of chilling -1.25% 2.32 %
Time to simulate 20 hour process
on Pentium 166 Mhz Computer < 1 min 4-5 hours

F O R U M
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refrigertion load cal methods in food processing

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