Coeficiente Convectivo Articulo

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Heat transfer coefficients for forced-air cooling andfreezing of selected foods

Bryan R. Becker*, Brian A. FrickeMechanical Engineering, University of Missouri-Kansas City, 5100 Rockhill R oad, Kansas City, MO 64110-2499, USA

Received 1 April 2003; received in revised form 13 February 2004; accepted 19 February 2004

Abstract

To maximize the efficiency of cooling and freezing operations for foods, it is necessary to optimally design the refrigeration

equipment to fit the specific requirements of the particular cooling or freezing application. The design of food refrigeration

equipment requires estimation of the cooling and freezing times of foods, as well as the corresponding refrigeration loads. The

accuracy of these estimates, in turn, depends upon accurate estimates of the surface heat transfer coefficient for the cooling or

freezing operation. This project reviewed heat transfer data for the cooling and/or freezing of foods. A total of 777 cooling

curves for 295 food items were obtained from an industrial survey and a unique iterative algorithm, utilizing the concept of

equivalent heat transfer dimensionality, was developed to obtain heat transfer coefficients from these cooling curves. NineNusseltReynoldsPrandtl correlations were developed from a selection of the 777 heat transfer coefficients resulting from this

algorithm, as well as 144 heat transfer coefficients for 13 food items, collected from the literature. The data and correlations

resulting from this project will be used by designers of cooling and freezing systems for foods. This information will make

possible a more accurate determination of cooling and freezing times and corresponding refrigeration loads. Such information is

important in the design and operation of cooling and freezing facilities and will be of immediate usefulness to engineers

involved in the design and operation of such systems.

q 2004 Elsevier Ltd and IIR. All rights reserved.

Keywords:Cooling; Freezing; Food; Calculation; Cooling time; Freezing time; Heat transfer coefficient

Coefficients de transfert de chaleur de certains produits

alimentaires lors du refroidissement aair forceMots-cles:Refroidissement; Congelation; Produit alimentaire; Calcul; Temps de re frigeration; Temps de congelation; Coefficient de transfert

de chaleur

1. Introduction

In many food processing applications, including blast

cooling and freezing, transient convective heat transfer

occurs between a fluid medium and the solid food item [1].

Knowledge of the surface heat transfer coefficient is

required to design the equipment wherein convection heat

transfer is used to process foods. Newtons law of cooling

defines the surface heat transfer coefficient, h, as follows:

q hAts 2 tm 1

The surface heat transfer coefficient, h, is a lo ca l

phenomenon which depends upon the velocity of the

surrounding fluid, product geometry, orientation, surface

roughness and packaging, as well as other factors.Researchers have noted that the most significant factor

International Journal of Refrigeration 27 (2004) 540551

www.elsevier.com/locate/ijrefrig

0140-7007/$35.00 q 2004 Elsevier Ltd and IIR. All rights reserved.

doi:10.1016/j.ijrefrig.2004.02.006

* Corresponding author. Tel.: 1-816-235-1255; fax: 1-816-

235-1260.

E-mail address:[email protected] (B.R. Becker).
http://www.elsevier.com/locate/ijrefrighttp://www.elsevier.com/locate/ijrefrig


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influencing the surface heat transfer coefficient is the

velocity of the fluid flowing past the product. Thus, it is

common to report the results of experimentally determined

surface heat transfer coefficients in terms of Nusselt

ReynoldsPrandtl correlations. These correlations give the

heat transfer coefficient as a function of product shape and

size as well as fluid velocity. Since the heat transfercoefficient is not constant over the surface of a body, the

value of the heat transfer coefficient reported by these

Nusselt Reynolds Prandtl correlations is actually the area

averaged value of the local heat transfer coefficient.

A small number of studies have been performed to

measure or estimate the surface heat transfer coefficient

during cooling, freezing or heating of food items [1 34]. In

addition, a detailed literature survey has been compiled by

Arce and Sweat [35]. These studies present surface heat

transfer coefficient data and correlations for only a very

limited number of food items and process conditions. Hence,

the objective of this study was to determine the surface heat

transfer coefficients for a wide variety of foods during blast

cooling and freezing processes.

2. Review of existing techniques to determine the surface

heat transfer coefficients of foods

Techniques used to determine heat transfer coefficients

generally fall into three categories: steady-state temperature

measurement methods, transient temperature measurement

methods and surface heat flux measurement methods. Of

these three techniques, the most popular methods are the

transient temperature measurement techniques.

Transient methods for determining the surface heat

transfer coefficient involve the measurement of producttemperature with respect to time during cooling or freezing

processes. Two cases must be considered when performing

transient tests to determine the surface heat transfer

coefficient: low Biot number (Bi # 0.1) and large Biot

number (Bi . 0.1). The Biot number, Bi, is the ratio of

external heat transfer resistance to internal heat transfer

resistance and is defined as follows:

BihZ

k 2

A low Biot number indicates that the internal resistance to

heat transfer is negligible, and thus, the temperature within

the object is uniform at any given instant in time. A large

Biot number indicates that the internal resistance to heattransfer is not negligible, and thus, a temperature gradient

may exist within the object.

In typical blast cooling or freezing operations for foods,

the Biot number is large, ranging from 0.2 to 20[36]. Thus,

the internal resistance to heat transfer is generally not

negligible during food cooling and freezing and a

temperature gradient will exist within the food item.

One method for obtaining the surface heat transfer

coefficient of a food product with an internal temperature

gradient involves the use of cooling curves. For simple, one-

dimensional food geometries such as infinite slabs, infinite

circular cylinders or spheres, there exist empirical and

analytical solutions to the one-dimensional transient heat

equation. The slope of the cooling curve may be used inconjunction with these solutions to obtain the Biot number

for the cooling process. The heat transfer coefficient may

then be determined from the Biot number.

All cooling processes exhibit similar behavior. After an

initial lag, the temperature at the thermal center of the food

item decreases exponentially [36]. As shown in Fig. 1, a

cooling curve depicting this behavior can be obtained by

plotting, on semilogarithmic axes, the fractional unaccom-

plished temperature difference versus time. The fractional

unaccomplished temperature difference, Y, is defined as

follows:

Y tm 2 t

tm 2 ti

t2 tm

ti 2 tm3

The lag between the onset of cooling and the exponential

decrease in the temperature of the food item is measured

with thej factor, as shown inFig. 1.

FromFig. 1,it can be seen that the linear portion of the

cooling curve can be described as follows:

Y j exp2Cu 4

whereCis the cooling coefficient, which is minus the slope

of the linear portion of the cooling curve.

For simple geometrical shapes, such as infinite slabs,

infinite circular cylinders and spheres, analytical

expressions for cooling or freezing time may be derived.

To derive these expressions, the following assumptions are

made: (1) the thermophysical properties of the food item andthe cooling medium are constant, (2) the internal heat

Fig. 1. Typical cooling curve.

B.R. Becker, B.A. Fricke / International Journal of Refrigeration 27 (2004) 540551542


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generation and moisture loss from the food item are

neglected, (3) the food item is homogeneous and isotropic,

(4) the initial temperature distribution within the food item

is uniform, (5) heat conduction occurs only in one

dimension, and (6) convective heat transfer occurs between

the surface of the food item and the cooling medium. With

these assumptions, the one-dimensional transient heatequation may be written as follows for infinite slabs, infinite

circular cylinders and spheres:

1

za

z

z

a f

z

1

a

f

u

5

The initial and boundary conditions are as follows:

fz; 0 ti 2 tm 6

zf0; u 0 7

2k

zfZ; u

hfZ; u 0 8

In order to non-dimensionalize the solutions of Eq. (5), twodimensionless parameters are introduced, namely, the Biot

number, defined in Eq. (2), and the Fourier number, defined

as follows:

Foau

Z2 9

3. Iterative technique to determine heat transfer

coefficients of irregularly shaped food items

The technique developed in this paper to determine heat

transfer coefficients from experimental cooling curves is

based upon the infinite series solution of Eq. (5), given byCarslaw and Jaeger [37], for the dimensionless center

temperature of a sphere:

YX1n1

AnBn 10

After the initial lag period has passed, in which case Fo $

0:2;thesecond andhigher terms of Eq.(10) are assumed to be

negligible[21].Thus, Eq. (10) can be simplified as follows:

YA1B1 11

whereA1andB1are given as follows:

A1 2Bisinm1

m1 2 sinm1cos m112

B1 exp2m21Fo 13

and m1is a parameter specified by a characteristic equation:

cotm1 12Bi

m114

The parameter, m1;may also be determined from the cooling

coefficient,C, defined in Eq. (4). By comparing Eqs. (4), (11)and (13), it can be seen that:

2Cu 2m21Fo 15

Since the Fourier number, Fo, of a cooling process can be

readily determined, and, provided that the value ofCcan be

determined from a cooling curve, the value of m1 can

be obtained by rearranging Eq. (15):

m1

ffiffiffiffiffiCu

Fo

r 16

Then, the Biot number,Bi, can be obtained from Eq. (14) and

the surface heat transfer coefficient, h, may be obtained

through algebraic manipulation of the definition of the Biotnumber, Eq. (2).

Table 1

Geometric parameters and equations from Lin et al. [41]

Shape p1 p2 p3 Eo

Infinite slab b1 b2 1 0 0 0 1

Infinite rectangular rod (b1 $ 1: b2 1) 0.75 0 21 Eo 1 1

b1

1

b2

Brick (b1 $ 1: b2 $ b1) 0.75 0.75 21 Eo 1 1

b1

1

b2Infinite cylinder (b1 1: b2 1) 1.01 0 0 2

Infinite ellipse (b1 . 1: b2 1) 1.01 0 1 Eo 1

1

b1

1

b1 2 12b1 2

2

Squat cylinder (b1 b2; b1 $ 1) 1.01 0.75 21 Eo 1

1

b1

1

b2

Short cylinder (b1 1: b2 $ 1) 1.01 0.75 21 Eo 1 1

b1

1

b2Sphereb1 b2 1 1.01 1.24 0 3

Ellipsoid (b1 $ 1: b2 $ b1) 1.01 1.24 1 Eo 3b1 b2 b

211 b2 b

221 b1

2b1b21 b1 b22

b1 2 b220:4

15

B.R. Becker, B.A. Fricke / International Journal of Refrigeration 27 (2004) 540551 543


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Since the analytical method described thus far is only

applicable to spherical food items, an iterative technique

was developed to handle irregular shaped food items. Thisiterative technique utilizes a shape factor, called the

equivalent heat transfer dimensionality, to extend the

analytical method to irregularly shaped food items[3841].

This equivalent heat transfer dimensionality, E, compares

the total heat transfer to the heat transfer through the shortest

dimension.

The equivalent heat transfer dimensionality,E, is used to

modify the analytical solution, Eq. (13), as follows:

B1 exp 2m21Fo

E

3

17

resulting in the following modifications to Eqs. (15)

and (16):

2Cu 2m21Fo

E

3 18

m1

ffiffiffiffiffiffiffiffiffiCu

Fo

3

E

r 19

Lin et al. [3941] give the equivalent heat transfer

dimensionality,E, as a function of Biot number:

EBi4=3 1:85

Bi4=3

E1

1:85

Eo

20

Eo and E1are the equivalent heat transfer dimensionalities

for the limiting cases ofBi 0 and Bi !1, respectively.

For both two-dimensional and three-dimensional food

items, the general form for the equivalent heat transfer

dimensionality at Bi !1,E1, is given as:

E1 0:75p1fb1 p2fb2 21

where

fb 1

b2 0:01p3exp b2

b2

6

" # 22

The geometric parameters,p1,p2and p3, are given inTable

1for various geometries. The definition of the equivalentheat transfer dimensionality for Bi 0, Eo;is also given in

Table 1for various food geometries.

To determine the heat transfer coefficient of irregularly

shaped food items, the value ofm1 is obtained via Eq. (19)

and then the Biot number can be calculated from Eq. (14).

From the Biot number, the equivalent heat transfer

dimensionality can be obtained by using Eqs. (20)(22).

The value ofm1 is then recalculated via Eq. (19), using the

updated value of equivalent heat transfer dimensionality.

This process is repeated until the value of the Biot number

converges. Finally, the heat transfer coefficient, h, may be

determined through algebraic manipulation of the definition

of the Biot number, Eq. (2).

Table 2

NusseltReynoldsPrandtl correlations for selected food items

Food type Reynolds number

range

Number of

data points

Level of

significance

(F-statistic)

Coefficient of

determination, r2NuRe Prcorrelation

Beef patties (Unpackaged) 2000 , Re , 7500 7 0.182 0.324 Nu 1.37Re 0.282Pr0.3

Cake (packaged and unpackaged) 4000, Re , 80 000 29 5.34 10212 0.833 Nu 0.00156Re 0.960Pr0.3

Cheese (packaged and unpackaged) 6000 , Re , 30 000 7 0.196 0.307 Nu 0.0987Re 0.560Pr0.3

Chicken breas t (unpackaged) 1000 , Re , 11 000 22 0.00115 0.418 Nu 0.0378Re 0.837Pr0.3

Fish fillets (packaged and unpackaged) 1000, Re , 25 000 28 1.27 1026 0.601 Nu 0.0154Re 0.818Pr0.3

Fried potato patties (unpackaged) 1000, Re , 6000 8 0.0143 0.660 Nu 0.00313Re 1.06Pr0.3

Pizza (packaged and unpackaged) 3000, Re , 12 000 12 0.00814 0.520 Nu 0.00517Re 0.981Pr0.3

Sausage (unpackaged) 4500,

Re,

25 000 14 0.314 0.0918 Nu 7.14Re

0.170

Pr

0.3

Trayed entrees (packaged) 5000 , Re , 20 000 42 0.213 0.0386 Nu 1.31Re 0.280Pr0.3

Fig. 2. Cooling curve for catfish.



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4. Cooling curves

Members of the food refrigeration industry werecontacted

to collect cooling curves and surface heat transfer data for

various food items. These contacts included food refrigeration

equipment manufacturers, designers of food refrigeration

plants, and food processors. An effort was made to collect

information on as many food items as possible.

A total of 777 cooling curves for various food items were

collected from the following sources: (1) Advanced Food

Processing Equipment, Inc.; (2) Freezing Systems, Inc.; (3)

Frigoscandia Equipment, AB; and, (4) Technicold Services,

Fig. 3. NusseltReynoldsPrandtl correlation for unpackaged beef patties.

Fig. 4. NusseltReynoldsPrandtl correlation for packaged and unpackaged cake. If present, packaging consists of either an aluminum tray, or

an aluminum foil cover and a paper tray.



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Inc. These cooling were determined through the use ofthermocouples imbedded within the food items.

A typical cooling curve is shown in Fig. 2. These

collected cooling curves were digitized and a database was

developed which contains the digitized time-temperature

data obtained from these curves. The temperatures were

non-dimensionalized according to Eq. (3) and the natural

logarithm of these non-dimensional temperatures weretaken. The slopes of the linear portion(s) of the logarithmic

temperature versus time data were determined using the

linear least-squares-fit technique. These slopes were then

used in conjunction with the techniques described in Section

3 to determine the heat transfer coefficients for the food

items.

Fig. 5. NusseltReynoldsPrandtl correlation for packaged and unpackaged cheese. If present, packaging consists of either a pouch, a paper

tray with plastic lid, or a plastic box with lid.

Fig. 6. NusseltReynoldsPrandtl correlation for unpackaged chicken breast.



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5. Calculated heat transfer coefficients

Using the iterative algorithm, 777 heat transfer

coefficients for foods were calculated from the database

of 777 cooling curves and tabulated with a description of

their packaging, dimensions, and weight, as well as the

air temperature and air velocity used to cool or freeze thefood items.

5.1. Effects of packaging

Packaging affects the heat transfer coefficients of food

Fig. 7. NusseltReynoldsPrandtl correlation for packaged and unpackaged fish fillets. If present, packaging consists of either a hard plastic

tray or a plastic pouch.

Fig. 8. NusseltReynoldsPrandtl correlation for unpackaged fried potato patties.



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items in several ways. It insulates the food item by

presenting a barrier to the transfer of energy from the

food, thus lowering the heat transfer coefficient. Packagingmay also create air-filled voids around the food item which

further insulates the food and lowers the heat transfer

coefficient.

Furthermore, the algorithm developed in Section 3

makes use of a density for the food item which includes

the packaging. In the algorithm, this density is calculated

from the outside dimensions of the package around the fooditem and the mass of the food item plus the packaging. Thus,

the density used to calculate the heat transfer coefficient is

affected by the packaging, resulting in a heat transfer

coefficient for the food item within its packaging.

Fig. 9. NusseltReynoldsPrandtl correlation for packaged and unpackaged pizza. If present, packaging consists of either a cardboard backing

or a cardboard backing and shrink wrap.

Fig. 10. NusseltReynoldsPrandtl correlation for unpackaged sausage.



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5.2. NusseltReynoldsPrandtl correlations

Non-dimensional analyses were performed to develop

Nusselt Reynolds Prandtl correlations for selected food

items. The Nusselt number is a dimensionless heat transfer

coefficient defined as:

Nuhd

km23

Physical reasoning indicates a dependence of the heat

transfer process on the flow field, and hence on the Reynolds

number,Re:

RermUd

mm

24

The relative rates of diffusion of heat and momentum are

related by the Prandtl number, Pr, and hence, the Prandtl

number is expected to be a significant parameter in the

determination of heat transfer coefficients:

Prmmcm

km25

An exponential function is commonly used to relate the

Nusselt number, Nu, to the Reynolds and Prandtl numbers:

Nu CRem

Prn

26

where C, m and n are constants determined from

experimental data.

To obtain Nusselt Reynolds Prandtl correlations forfoods, in the form given by Eq. (26), the Reynolds, Prandtl

and Nusselt numbers were determined for each coolingcurve using Eqs. (24)(26) in conjunction with the reported

air temperature and commodity size. Based on information

from the heat transfer literature[42], the exponent,n, in Eq.

(26), was set at 0.3. Using the Data Analysis TookPak

available in the Microsoft Excel software package [43],

regression analysis was performed on the collective

logNuPr20:3 vs. log(Re) data for a particular food item.

This regression analysis yielded the constant C and the

exponentm in Eq. (26).

The resulting Nusselt Reynolds Prandtl correlations

are summarized inTable 2and plotted inFigs. 3 11.Table

2 also gives the level of significance, F-statistic, and the

coefficient of determination, r2, for the correlations.

Generally, a significance level less than 0.05 indicates thatthe correlation represents the data significantly better than

the mean. The coefficient of determination,r2, indicates the

proportion of variation in log(NuPr20.3) explained by the

variation in log(Re).

6. Conclusions

This study was initiated to resolve deficiencies in heat

transfer coefficient data for food cooling and/or freezing

processes. Members of the food refrigeration industry were

contacted to collect cooling curves and surface heat transfer

data. In addition, a literature search was performed to collectcooling curves as well as surface heat transfer data for

Fig. 11. NusseltReynoldsPrandtl correlation for packaged trayed entrees. Packaging consists of either an aluminum tray, a plastic tray, an

aluminum tray with paper lid, a plastic tray with film lid, a paper tray with paper lid, or a paper tray with plastic lid.



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various food items. Techniques to determine surface heat

transfer coefficients from cooling curves were also collected

and reviewed.

A unique iterative algorithm was developed to estimate

the surface heat transfer coefficients of irregularly shaped

food items based upon their cooling curves. This algorithm

utilizes the concept of equivalent heat transfer dimension-

ality to extend to irregularly shaped food items existing

techniques for the calculation of the surface heat transfer

coefficient, previously applicable to only regularly shaped

food items.

Making use of this algorithm, 777 heat transfer

coefficients for 295 different food items were calculatedfrom the cooling curves collected during the industrial

survey. An additional 144 surface heat transfer coefficients

were collected from the literature for 13 different food

items. Nine Nusselt Reynolds Prandtl correlations were

developed from a selection of this data. As shown by their

level of significance and coefficient of determination,

reported in Table 2, these nine correlations satisfactorily

represent the experimental data. The NusseltReynolds

Prandtl correlations given for cake, chicken breast, fish

fillets, fried potato patties and pizza were found to be more

representative of the data than the correlations given for beef

patties, cheese, sausage and trayed entrees.

The data and correlations resulting from this project will

be used by designers of cooling and freezing systems for

foods. This information will make possible a more accurate

determination of cooling and freezing times and correspond-

ing refrigeration loads. Such information is important in the

design and operation of cooling and freezing facilities and

will be of immediate usefulness to engineers involved in the

design and operation of such systems.

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