# Determination of heat transfer coefficients of foods.

Abstract

Cooling and freezing of food by forced convection is one of the most significant applications of industrial refrigeration. In order for cooling and freezing operations to be cost-effective, it is necessary to optimally design the refrigeration equipment. This requires estimation of the cooling and freezing times of foods and the corresponding refrigeration loads. These estimates, in turn, depend upon the surface heat transfer coefficient for the cooling or freezing operation.

This paper describes a study which was initiated to resolve deficiencies in heat transfer coefficient data for food cooling and freezing processes by forced convection. Members of the food refrigeration industry were contacted to collect cooling and freezing curves as well as surface heat transfer data. A unique iterative algorithm was developed to estimate the surface heat transfer coefficients of foods based upon their cooling and freezing curves. Making use of this algorithm, heat transfer coefficients for various food items were calculated from the cooling and freezing curves collected during the industrial survey. Nusselt, Prandtl and Reynolds numbers were determined for the various food items, based upon their flowfield parameters, product dimensions and calculated heat transfer coefficients. Heat transfer coefficient correlations for various foods items were then developed, based upon these dimensionless parameters. These correlations are important in the design and operation of food cooling and freezing facilities and will be of immediate usefulness to engineers involved in the design and operation of such systems.

Keywords: Heat transfer coefficient; food; refrigeration.

1. Introduction

Preservation of food is one of the most significant applications of refrigeration. Cooling and freezing of food effectively reduce the activity of microorganisms and enzymes, thus retarding deterioration. Furthermore, crystallization of water reduces the amount of liquid water in food items and inhibits microbial growth (Heldman, 1975).

Optimally designed refrigeration equipment is required to maximize the efficiency of food cooling and freezing operations. In addition, it is necessary that the refrigeration equipment fits 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 forced convective cooling or freezing operation.

a. Heat Transfer during Cooling and Freezing of Foods

In many food-processing applications, including cooling and freezing, forced convective heat transfer occurs between a fluid medium and the solid food item (Dincer, 1993). Knowledge of the surface heat transfer coefficient is required to accurately design equipment in which forced convection heat transfer is used to process foods. Newton's law of cooling defines the surface heat transfer coefficient, h, as follows:

q = hA ([t.sub.s] - [t.sub.m]) (1)

where q is the heat transfer rate, t is the surface temperature of the food, [t.sub.m] is the surrounding fluid temperature and A is the surface area of the food through which the heat transfer occurs.

During a convective heat transfer process, energy is theoretically transferred by convection alone. However, in practice, conduction, radiation and mass transfer may also occur at the same time. In blast cooling/freezing operations, the conductive heat transfer component is very small and can be neglected. The significance of the radiation heat transfer and evaporative cooling due to mass transfer must be considered on an individual basis and their effects upon the surface heat transfer coefficient must be properly incorporated. Researchers often define an "effective" heat transfer coefficient, which includes the effects of convection and radiation heat transfer, as well as the energy transfer due to evaporation of moisture from the surface of the food item (Lind, 1988). However, this paper focuses upon those cooling/freezing operations which are dominated by forced convective heat transfer and the resultant heat transfer coefficients do not account for conduction, radiation or latent heat transfer.

A detailed literature survey and discussion regarding heat transfer coefficients for foods is given by Arce and Sweat (1980). Since then, additional studies have been performed to measure or estimate the surface heat transfer coefficient during cooling, freezing or heating of food items (Alhamdan et al., 1990; Ansari, 1987; Chen et al., 1997; Daudin and Swain, 1990; Dincer, 1991, 1993, 1994a, 1994b, 1994c, 1995a, 1995b, 1995c, 1995d, 1996, 1997; Dincer et al., 1992; Dincer and Genceli, 1994, 1995a, 1995b; Dincer and Dost, 1996; Flores and Mascheroni, 1988; Frederick and Comunian, 1994; Khairullah and Singh, 1991; Kondjoyan and Daudin, 1997; Mankad et al., 1997; Stewart et al., 1990; Vazquez and Calvelo, 1980, 1983; Verboven et al., 1997; Zuritz et al., 1990). However, collectively, these studies present surface heat transfer coefficient data and correlations for only a very limited number of food items and process conditions. Therefore, the objective of this study was to determine the surface heat transfer coefficients for a wide variety of foods during forced convective cooling and freezing processes.

b. Determination of Heat Transfer Coefficients

Techniques used to determine heat transfer coefficients generally fall into three categories:

1. Steady-state temperature measurement methods

2. Transient temperature measurement methods

3. Surface heat flux measurement methods

Of these three techniques, the most popular method is the transient temperature measurement technique, in which the heat transfer coefficient is determined by measuring product temperature with respect to time during a cooling or freezing process.

All cooling processes exhibit similar behavior. After an initial "lag", the temperature at the thermal center of the food item decreases exponentially (Cleland, 1990). A cooling curve, shown in Figure 1, depicting this behavior can be obtained by plotting, on semi logarithmic axes, the fractional unaccomplished temperature difference versus time. The fractional unaccomplished temperature difference, Y, is defined as follows:

where [t.sub.m] is the cooling medium temperature, t is the product temperature and [t.sub.i] is the initial temperature of the product.

[FIGURE 1 OMITTED]

Y = [t.sub.m] - t / [t.sub.m] - [t.sub.i] = t - [t.sub.m] / [t.sub.i] - [t.sub.m] (2)

This semi logarithmic temperature history curve consists of one initial curvilinear portion, followed by one or more linear portions. Simple empirical formulae, which model this cooling behavior, have been proposed for estimating the cooling time of foods and beverages. These models incorporate two factors, f and j, which represent the slope and intercept, respectively, of the temperature history curve.

As shown in Figure 1, the j factor is a measure of the "lag" between the onset of cooling and the exponential decrease in the temperature of the food. The f factor represents the time required to obtain a 90% reduction in the non-dimensional temperature difference. Graphically, the f factor corresponds to the time required for the linear portion of the temperature history curve to pass through one log cycle. The f factor is a function of the Biot number while the j factor is a function of the Biot number and the location within the food item.

The general form of the cooling time model is: where [theta] is the cooling time. In addition, the slope of the linear portion of the cooling curve, C, can be written in terms of the f factor:

Y = [t.sub.m] - t/[t.sub.m] - [t.sub.i] = j exp (-2.303[theta]/f) (3)

For simple geometrical shapes, such as infinite slabs, infinite circular cylinders and spheres, infinite series solutions for cooling or freezing time may be derived from the one-dimensional transient heat equation (Carslaw and Jaeger, 1980). After the initial "lag" period has passed, the second and higher terms of the infinite series solution are assumed to be negligible (Dincer and Dost, 1996).

C = 2.303/f (4)

The iterative technique developed in this paper for determining the heat transfer coefficients of irregularly shaped food items is based on the first-term solution for the dimensionless center temperature of a sphere, given as:

Y = 2Bi sin [[mu].sub.1]/[[mu].sub.1] - sin [[mu].sub.1] cos [[mu].sub.1] exp (-[[mu].sup.2.sub.1]Fo) (5)

where Bi is the Biot number:

Bi = hd/k

in which d is the shortest dimension of the food item and k is the thermal conductivity of the food item, and Fo is the Fourier number:

Fo = [alpha][theta]/[d.sup.2] (7)

where a is the thermal diffusivity of the food item. For a sphere, the parameter, [[mu].sub.1], is specified by the following characteristic equation:

By comparing Equations (3), (4), and (5), it can be seen that:

cot [[mu].sub.1 = 1 - Bi/[[mu].sub.1] (8)

-C[theta] = -[[mu].sup.2.sub.1]Fo (9)

Since the Fourier number, Fo, of a cooling process can be readily determined, and, provided that the value of C can be determined from a cooling curve, the value of [[mu].sub.1] can be obtained by rearranging Equation (9):

[[mu].sub.1] = [square root of (C[theta]/Fo)] (10)

Then, the Biot number, Bi, can be obtained from Equation (8) and the surface heat transfer coefficient, h, may be obtained through algebraic manipulation of the definition of the Biot number, Equation (6).

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. This iterative technique utilizes a shape factor, called the "equivalent heat transfer dimensionality," to extend the analytical method to irregularly shaped food items (Cleland and Earle, 1982; Lin et al., 1993, 1996a, 1996b). 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 of heat conduction in a sphere, Equation (5), as follows:

Y = 2Bi sin [[mu].sub.1]/[[mu].sub.1] - sin [[mu].sub.1] cos [[mu].sub.1]exp(-[[mu].sup.2.sub.1] Fo E/3) (11)

resulting in the following modification to Equation (10):

[[mu].sub.1] = [square root of (C[theta]/Fo 3/E)] (12)

Lin et al. (1993, 1996a, 1996b) present equations for determining the equivalent heat transfer dimensionality, E, as a function of Biot number and shape of the food item (short cylinder, squat cylinder, rectangular rod, ellipsoid, etc.). Values of E range from 1.0 to 3.0, with E = 3.0 being the equivalent heat transfer dimensionality of a sphere. Thus for a sphere, Equations (11) and (12) reduce to Equations (5) and (10).

To determine the heat transfer coefficient of irregularly shaped food items, a value of [[mu].sub.1] is obtained via Equation (12) by assuming a value for the equivalent heat transfer dimensionality, E. Then, the Biot number can be calculated from Equation (8). From the Biot number, the equivalent heat transfer dimensionality can be obtained using the equations of Lin et al. (1993, 1996a, 1996b). The value of [[mu].sub.1] is then recalculated via Equation (12), 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, Equation (6).

c. Cooling and Freezing Curve Database

Members of the food refrigeration industry were contacted to collect cooling and freezing curves as well as 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 and freezing curves for various food items were collected from the industrial survey. These cooling and freezing curves were determined through the use of thermocouples imbedded within the food items. Cooling and freezing curves were collected for food items in the following categories:

1. Diary: cheese, cream and ice cream

2. Prepared Foods: entrees, croquettes, pasties, pizza, sandwiches, soup

3. Poultry: whole and portions of chicken and turkey

4. Fruit/Vegetable

5. Fish/Seafood: various fish fillets, fish sticks, shrimp, scallops, squid

6. Bakery Products: raw dough, bread, rolls, cakes, pies and pastries

7. Meat: beef, lamb, pork and sausage

8. Beverages

9. Miscellaneous

The collected cooling and freezing curves were digitized and a database was developed which contains the digitized time-temperature data obtained from these curves. The temperatures were non-dimensionalized and the natural logarithms of these non-dimensional temperatures were taken. The slopes of the linear portion(s) of the logarithmic temperature versus time data were determined using the linear least-squares-fit technique.

d. Calculated Heat Transfer Coefficients

The slopes of the logarithmic cooling curves, in conjunction with the iterative algorithm described previously, were used to determine the heat transfer coefficients for the food items. A sample of the calculated heat transfer coefficients for pizza is presented in Table 1. Table 1 shows the calculated heat transfer coefficient along with food dimensions, mass, air temperature, air velocity with direction and packing description.

e. Heat Transfer Coefficient Correlations

For those food items having a significant amount of heat transfer coefficient data, non-dimensional analyses were performed to obtain simple heat transfer coefficient correlations which can be used to predict the heat transfer coefficients of those food items. The Nusselt number, Nu, a non-dimensional heat transfer coefficient, is defined as follows:

Nu = hd/[k.sub.m] (13)

where [k.sub.m] is the thermal conductivity of the cooling medium.

Physical reasoning indicates a dependence of the heat transfer process on the flow field, and hence on the Reynolds number. The Reynolds number, Re, is defined as follows:

Re = [[rho].sub.m]Ud/[[mu].sub.m] (14)

where U is the free stream velocity of the cooling medium, [[rho.sub.m] is the density of the cooling medium and [[mu].sub.m] is the dynamic viscosity of the cooling medium.

The relative rates of diffusion of heat and momentum are related by the Prandtl number, and the Prandtl number is expected to be a significant parameter in the final solution. The Prandtl number, Pr, is defined as follows:

Pr = [[mu].sub.m][c.sub.m]/[k.sub.m] (15)

where cm is the specific heat capacity of the cooling medium.

Previous experimental and analytical work has shown that an exponential function is perhaps the simplest type of relation to use (Holman, 1990):

Nu = c x [Re.sup.m] x [Pr.sup.n] (16)

where c, m and n are constants to be determined from experimental data.

Using the Data Analysis ToolPak available in Microsoft Excel, a non-linear fit was performed on Nusselt, Reynolds and Prandtl number data to obtain heat transfer coefficient correlations in the form given by Equation (16). The resulting Nusselt-Reynolds-Prandtl correlations for cake, chicken breast and pizza are summarized in Table 2 and plotted in Figures 2 through 4. Table 2 also gives the level of significance, F-statistic, and the coefficient of determination, [r.sup.2], for the correlations. Generally, a significance level less than 0.05 indicates that the correlation represents the data significantly better than the mean. The coefficient of determination, [r.sup.2], indicates the proportion of variation in log(Nu x [Pr.sup.-0.3]) explained by the variation in log(Re).

[FIGURES 2-4 OMITTED]

2. Conclusion

This paper described a study which was initiated to resolve deficiencies in heat transfer coefficient data for food cooling and freezing processes by forced convection. An algorithm was developed to estimate the surface heat transfer coefficients of foods based upon their cooling and freezing curves. Making use of this algorithm, heat transfer coefficients for various food items were calculated from the cooling and freezing curves collected during the industrial survey. In addition, non-dimensional analysis was performed on the calculated heat transfer coefficient data to obtain Nusselt-Reynolds-Prandtl correlations.

The data and correlations resulting from this project will be used by designers of cooling and freezing systems for foods and beverages. 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.

Nomenclature

A surface area

Bi Biot number

C cooling coefficient

[c.sub.m] specific heat of the cooling medium

d smallest dimension of the food item

E equivalent heat transfer dimensionality

f cooling time parameter

Fo Fourier number

h heat transfer coefficient

j cooling time parameter

k thermal conductivity

[k.sub.m] thermal conductivity of the cooling medium

Nu Nusselt number

Pr Prandtl number

q heat transfer rate

Re Reynolds number

t product temperature

[t.sub.i] initial product temperature

[t.sub.m] cooling medium temperature

[t.sub.s] surface temperature of the food

U free stream velocity of the cooling medium

Y fractional unaccomplished temperature difference

[alpha] thermal diffusivity of the food item

[theta] cooling time

[[mu].sub.1] characteristic parameter

[[mu].sub.m] dynamic viscosity of the cooling medium

[[rho].sub.m] density of the cooling medium

References

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Dincer, I. 1995c. Development of New Correlations for Forced Convection Heat Transfer During Cooling of Products. International Journal of Energy Research 19(9): 791-801.

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Lin, Z., A.C. Cleland, D.J. Cleland, and G.F. Serrallach. 1996b. A Simple Method for Prediction of Chilling Times: Extension to Three-Dimensional Irregular Shapes. International Journal of Refrigeration 19(2): 107-114.

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Verboven, P., B.M. Nicolai, N. Scheerlink, and J. De Baerdemaeker. 1997. The Local Surface Heat Transfer Coefficient in Thermal Food Process Calculations: A CFD Approach. Journal of Food Engineering 33(1): 15-35.

Zuritz, C.A., S.C. McCoy, and S.K. Sastry. 1990. Convective Heat Transfer Coefficients for Irregular Particles Immersed in Non-Newtonian Fluid During Tube Flow. Journal of Food Engineering 11 (2): 159-174

Brian A. Fricke, Ph.D., Deep Bandyopadhyay, Arun K. Ranjan, Mark F. McClernon, Ph.D., P.E., Bryan R. Becket, Ph.D., P.E.

Civil and Mechanical Engineering Division University of Missouri--Kansas City 5100 Rockhill Road

Kansas City, MO 64110-2499 U.S.A.

Cooling and freezing of food by forced convection is one of the most significant applications of industrial refrigeration. In order for cooling and freezing operations to be cost-effective, it is necessary to optimally design the refrigeration equipment. This requires estimation of the cooling and freezing times of foods and the corresponding refrigeration loads. These estimates, in turn, depend upon the surface heat transfer coefficient for the cooling or freezing operation.

This paper describes a study which was initiated to resolve deficiencies in heat transfer coefficient data for food cooling and freezing processes by forced convection. Members of the food refrigeration industry were contacted to collect cooling and freezing curves as well as surface heat transfer data. A unique iterative algorithm was developed to estimate the surface heat transfer coefficients of foods based upon their cooling and freezing curves. Making use of this algorithm, heat transfer coefficients for various food items were calculated from the cooling and freezing curves collected during the industrial survey. Nusselt, Prandtl and Reynolds numbers were determined for the various food items, based upon their flowfield parameters, product dimensions and calculated heat transfer coefficients. Heat transfer coefficient correlations for various foods items were then developed, based upon these dimensionless parameters. These correlations are important in the design and operation of food cooling and freezing facilities and will be of immediate usefulness to engineers involved in the design and operation of such systems.

Keywords: Heat transfer coefficient; food; refrigeration.

1. Introduction

Preservation of food is one of the most significant applications of refrigeration. Cooling and freezing of food effectively reduce the activity of microorganisms and enzymes, thus retarding deterioration. Furthermore, crystallization of water reduces the amount of liquid water in food items and inhibits microbial growth (Heldman, 1975).

Optimally designed refrigeration equipment is required to maximize the efficiency of food cooling and freezing operations. In addition, it is necessary that the refrigeration equipment fits 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 forced convective cooling or freezing operation.

a. Heat Transfer during Cooling and Freezing of Foods

In many food-processing applications, including cooling and freezing, forced convective heat transfer occurs between a fluid medium and the solid food item (Dincer, 1993). Knowledge of the surface heat transfer coefficient is required to accurately design equipment in which forced convection heat transfer is used to process foods. Newton's law of cooling defines the surface heat transfer coefficient, h, as follows:

q = hA ([t.sub.s] - [t.sub.m]) (1)

where q is the heat transfer rate, t is the surface temperature of the food, [t.sub.m] is the surrounding fluid temperature and A is the surface area of the food through which the heat transfer occurs.

During a convective heat transfer process, energy is theoretically transferred by convection alone. However, in practice, conduction, radiation and mass transfer may also occur at the same time. In blast cooling/freezing operations, the conductive heat transfer component is very small and can be neglected. The significance of the radiation heat transfer and evaporative cooling due to mass transfer must be considered on an individual basis and their effects upon the surface heat transfer coefficient must be properly incorporated. Researchers often define an "effective" heat transfer coefficient, which includes the effects of convection and radiation heat transfer, as well as the energy transfer due to evaporation of moisture from the surface of the food item (Lind, 1988). However, this paper focuses upon those cooling/freezing operations which are dominated by forced convective heat transfer and the resultant heat transfer coefficients do not account for conduction, radiation or latent heat transfer.

A detailed literature survey and discussion regarding heat transfer coefficients for foods is given by Arce and Sweat (1980). Since then, additional studies have been performed to measure or estimate the surface heat transfer coefficient during cooling, freezing or heating of food items (Alhamdan et al., 1990; Ansari, 1987; Chen et al., 1997; Daudin and Swain, 1990; Dincer, 1991, 1993, 1994a, 1994b, 1994c, 1995a, 1995b, 1995c, 1995d, 1996, 1997; Dincer et al., 1992; Dincer and Genceli, 1994, 1995a, 1995b; Dincer and Dost, 1996; Flores and Mascheroni, 1988; Frederick and Comunian, 1994; Khairullah and Singh, 1991; Kondjoyan and Daudin, 1997; Mankad et al., 1997; Stewart et al., 1990; Vazquez and Calvelo, 1980, 1983; Verboven et al., 1997; Zuritz et al., 1990). However, collectively, these studies present surface heat transfer coefficient data and correlations for only a very limited number of food items and process conditions. Therefore, the objective of this study was to determine the surface heat transfer coefficients for a wide variety of foods during forced convective cooling and freezing processes.

b. Determination of Heat Transfer Coefficients

Techniques used to determine heat transfer coefficients generally fall into three categories:

1. Steady-state temperature measurement methods

2. Transient temperature measurement methods

3. Surface heat flux measurement methods

Of these three techniques, the most popular method is the transient temperature measurement technique, in which the heat transfer coefficient is determined by measuring product temperature with respect to time during a cooling or freezing process.

All cooling processes exhibit similar behavior. After an initial "lag", the temperature at the thermal center of the food item decreases exponentially (Cleland, 1990). A cooling curve, shown in Figure 1, depicting this behavior can be obtained by plotting, on semi logarithmic axes, the fractional unaccomplished temperature difference versus time. The fractional unaccomplished temperature difference, Y, is defined as follows:

where [t.sub.m] is the cooling medium temperature, t is the product temperature and [t.sub.i] is the initial temperature of the product.

[FIGURE 1 OMITTED]

Y = [t.sub.m] - t / [t.sub.m] - [t.sub.i] = t - [t.sub.m] / [t.sub.i] - [t.sub.m] (2)

This semi logarithmic temperature history curve consists of one initial curvilinear portion, followed by one or more linear portions. Simple empirical formulae, which model this cooling behavior, have been proposed for estimating the cooling time of foods and beverages. These models incorporate two factors, f and j, which represent the slope and intercept, respectively, of the temperature history curve.

As shown in Figure 1, the j factor is a measure of the "lag" between the onset of cooling and the exponential decrease in the temperature of the food. The f factor represents the time required to obtain a 90% reduction in the non-dimensional temperature difference. Graphically, the f factor corresponds to the time required for the linear portion of the temperature history curve to pass through one log cycle. The f factor is a function of the Biot number while the j factor is a function of the Biot number and the location within the food item.

The general form of the cooling time model is: where [theta] is the cooling time. In addition, the slope of the linear portion of the cooling curve, C, can be written in terms of the f factor:

Y = [t.sub.m] - t/[t.sub.m] - [t.sub.i] = j exp (-2.303[theta]/f) (3)

For simple geometrical shapes, such as infinite slabs, infinite circular cylinders and spheres, infinite series solutions for cooling or freezing time may be derived from the one-dimensional transient heat equation (Carslaw and Jaeger, 1980). After the initial "lag" period has passed, the second and higher terms of the infinite series solution are assumed to be negligible (Dincer and Dost, 1996).

C = 2.303/f (4)

The iterative technique developed in this paper for determining the heat transfer coefficients of irregularly shaped food items is based on the first-term solution for the dimensionless center temperature of a sphere, given as:

Y = 2Bi sin [[mu].sub.1]/[[mu].sub.1] - sin [[mu].sub.1] cos [[mu].sub.1] exp (-[[mu].sup.2.sub.1]Fo) (5)

where Bi is the Biot number:

Bi = hd/k

in which d is the shortest dimension of the food item and k is the thermal conductivity of the food item, and Fo is the Fourier number:

Fo = [alpha][theta]/[d.sup.2] (7)

where a is the thermal diffusivity of the food item. For a sphere, the parameter, [[mu].sub.1], is specified by the following characteristic equation:

By comparing Equations (3), (4), and (5), it can be seen that:

cot [[mu].sub.1 = 1 - Bi/[[mu].sub.1] (8)

-C[theta] = -[[mu].sup.2.sub.1]Fo (9)

Since the Fourier number, Fo, of a cooling process can be readily determined, and, provided that the value of C can be determined from a cooling curve, the value of [[mu].sub.1] can be obtained by rearranging Equation (9):

[[mu].sub.1] = [square root of (C[theta]/Fo)] (10)

Then, the Biot number, Bi, can be obtained from Equation (8) and the surface heat transfer coefficient, h, may be obtained through algebraic manipulation of the definition of the Biot number, Equation (6).

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. This iterative technique utilizes a shape factor, called the "equivalent heat transfer dimensionality," to extend the analytical method to irregularly shaped food items (Cleland and Earle, 1982; Lin et al., 1993, 1996a, 1996b). 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 of heat conduction in a sphere, Equation (5), as follows:

Y = 2Bi sin [[mu].sub.1]/[[mu].sub.1] - sin [[mu].sub.1] cos [[mu].sub.1]exp(-[[mu].sup.2.sub.1] Fo E/3) (11)

resulting in the following modification to Equation (10):

[[mu].sub.1] = [square root of (C[theta]/Fo 3/E)] (12)

Lin et al. (1993, 1996a, 1996b) present equations for determining the equivalent heat transfer dimensionality, E, as a function of Biot number and shape of the food item (short cylinder, squat cylinder, rectangular rod, ellipsoid, etc.). Values of E range from 1.0 to 3.0, with E = 3.0 being the equivalent heat transfer dimensionality of a sphere. Thus for a sphere, Equations (11) and (12) reduce to Equations (5) and (10).

To determine the heat transfer coefficient of irregularly shaped food items, a value of [[mu].sub.1] is obtained via Equation (12) by assuming a value for the equivalent heat transfer dimensionality, E. Then, the Biot number can be calculated from Equation (8). From the Biot number, the equivalent heat transfer dimensionality can be obtained using the equations of Lin et al. (1993, 1996a, 1996b). The value of [[mu].sub.1] is then recalculated via Equation (12), 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, Equation (6).

c. Cooling and Freezing Curve Database

Members of the food refrigeration industry were contacted to collect cooling and freezing curves as well as 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 and freezing curves for various food items were collected from the industrial survey. These cooling and freezing curves were determined through the use of thermocouples imbedded within the food items. Cooling and freezing curves were collected for food items in the following categories:

1. Diary: cheese, cream and ice cream

2. Prepared Foods: entrees, croquettes, pasties, pizza, sandwiches, soup

3. Poultry: whole and portions of chicken and turkey

4. Fruit/Vegetable

5. Fish/Seafood: various fish fillets, fish sticks, shrimp, scallops, squid

6. Bakery Products: raw dough, bread, rolls, cakes, pies and pastries

7. Meat: beef, lamb, pork and sausage

8. Beverages

9. Miscellaneous

The collected cooling and freezing curves were digitized and a database was developed which contains the digitized time-temperature data obtained from these curves. The temperatures were non-dimensionalized and the natural logarithms of these non-dimensional temperatures were taken. The slopes of the linear portion(s) of the logarithmic temperature versus time data were determined using the linear least-squares-fit technique.

d. Calculated Heat Transfer Coefficients

The slopes of the logarithmic cooling curves, in conjunction with the iterative algorithm described previously, were used to determine the heat transfer coefficients for the food items. A sample of the calculated heat transfer coefficients for pizza is presented in Table 1. Table 1 shows the calculated heat transfer coefficient along with food dimensions, mass, air temperature, air velocity with direction and packing description.

e. Heat Transfer Coefficient Correlations

For those food items having a significant amount of heat transfer coefficient data, non-dimensional analyses were performed to obtain simple heat transfer coefficient correlations which can be used to predict the heat transfer coefficients of those food items. The Nusselt number, Nu, a non-dimensional heat transfer coefficient, is defined as follows:

Nu = hd/[k.sub.m] (13)

where [k.sub.m] is the thermal conductivity of the cooling medium.

Physical reasoning indicates a dependence of the heat transfer process on the flow field, and hence on the Reynolds number. The Reynolds number, Re, is defined as follows:

Re = [[rho].sub.m]Ud/[[mu].sub.m] (14)

where U is the free stream velocity of the cooling medium, [[rho.sub.m] is the density of the cooling medium and [[mu].sub.m] is the dynamic viscosity of the cooling medium.

The relative rates of diffusion of heat and momentum are related by the Prandtl number, and the Prandtl number is expected to be a significant parameter in the final solution. The Prandtl number, Pr, is defined as follows:

Pr = [[mu].sub.m][c.sub.m]/[k.sub.m] (15)

where cm is the specific heat capacity of the cooling medium.

Previous experimental and analytical work has shown that an exponential function is perhaps the simplest type of relation to use (Holman, 1990):

Nu = c x [Re.sup.m] x [Pr.sup.n] (16)

where c, m and n are constants to be determined from experimental data.

Using the Data Analysis ToolPak available in Microsoft Excel, a non-linear fit was performed on Nusselt, Reynolds and Prandtl number data to obtain heat transfer coefficient correlations in the form given by Equation (16). The resulting Nusselt-Reynolds-Prandtl correlations for cake, chicken breast and pizza are summarized in Table 2 and plotted in Figures 2 through 4. Table 2 also gives the level of significance, F-statistic, and the coefficient of determination, [r.sup.2], for the correlations. Generally, a significance level less than 0.05 indicates that the correlation represents the data significantly better than the mean. The coefficient of determination, [r.sup.2], indicates the proportion of variation in log(Nu x [Pr.sup.-0.3]) explained by the variation in log(Re).

[FIGURES 2-4 OMITTED]

2. Conclusion

This paper described a study which was initiated to resolve deficiencies in heat transfer coefficient data for food cooling and freezing processes by forced convection. An algorithm was developed to estimate the surface heat transfer coefficients of foods based upon their cooling and freezing curves. Making use of this algorithm, heat transfer coefficients for various food items were calculated from the cooling and freezing curves collected during the industrial survey. In addition, non-dimensional analysis was performed on the calculated heat transfer coefficient data to obtain Nusselt-Reynolds-Prandtl correlations.

The data and correlations resulting from this project will be used by designers of cooling and freezing systems for foods and beverages. 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.

Nomenclature

A surface area

Bi Biot number

C cooling coefficient

[c.sub.m] specific heat of the cooling medium

d smallest dimension of the food item

E equivalent heat transfer dimensionality

f cooling time parameter

Fo Fourier number

h heat transfer coefficient

j cooling time parameter

k thermal conductivity

[k.sub.m] thermal conductivity of the cooling medium

Nu Nusselt number

Pr Prandtl number

q heat transfer rate

Re Reynolds number

t product temperature

[t.sub.i] initial product temperature

[t.sub.m] cooling medium temperature

[t.sub.s] surface temperature of the food

U free stream velocity of the cooling medium

Y fractional unaccomplished temperature difference

[alpha] thermal diffusivity of the food item

[theta] cooling time

[[mu].sub.1] characteristic parameter

[[mu].sub.m] dynamic viscosity of the cooling medium

[[rho].sub.m] density of the cooling medium

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Brian A. Fricke, Ph.D., Deep Bandyopadhyay, Arun K. Ranjan, Mark F. McClernon, Ph.D., P.E., Bryan R. Becket, Ph.D., P.E.

Civil and Mechanical Engineering Division University of Missouri--Kansas City 5100 Rockhill Road

Kansas City, MO 64110-2499 U.S.A.

Table 1. Calculated Heat Transfer Coefficients for Pizza. Heat Transfer Length, Description Coefficient, m W/([m.sup.2] x K) Pizza 12.8 0.24 Pizza, Box of 6 pieces, 4 2.9 0.178 oz./piece Pizza with Topping, 12 inch Diameter, On 12.7 Cardboard Pizza, Canadian Bacon, 12 11.5 inch Diameter Pizza, Ham and Pineapple, 7.6 16 inch Diameter Pizza, Sausage and Cheese, 12.1 Round, No Packaging Pizza, Sausage and Cheese, Rectangular, Sliced, No 13.7 0.406 Packaging Pizza, Sausage and Cheese, Rectangular, Sliced, No 17.4 0.406 Packaging Pizza, Sausage Special Deluxe, Round with 5.3 Cardboard Backing and Clear Shrink Wrap Pizza, Sausage Special Deluxe, Round with no 16.3 Backing, Unwrapped Sheet Pizza 9.1 Pizza, Vegetable, 12 inch 14.5 Diameter Pizza Crust with Uncooked 17.3 Ingredients on Top Diameter, Height, Description m or m Width, m Pizza 0.022 Pizza, Box of 6 pieces, 4 0.203 0.019 oz./piece Pizza with Topping, 12 inch Diameter, On 0.413 0.013 Cardboard Pizza, Canadian Bacon, 12 0.305 0.02 inch Diameter Pizza, Ham and Pineapple, 0.406 0.02 16 inch Diameter Pizza, Sausage and Cheese, 0.178 0.029 Round, No Packaging Pizza, Sausage and Cheese, Rectangular, Sliced, No 0.305 0.032 Packaging Pizza, Sausage and Cheese, Rectangular, Sliced, No 0.305 0.032 Packaging Pizza, Sausage Special Deluxe, Round with 0.305 0.022 Cardboard Backing and Clear Shrink Wrap Pizza, Sausage Special Deluxe, Round with no 0.318 0.019 Backing, Unwrapped Sheet Pizza 0.394 0.013 Pizza, Vegetable, 12 inch 0.305 0.02 Diameter Pizza Crust with Uncooked 0.152 0.013 Ingredients on Top Mass, Air Description gm Temperature, [degrees]C Pizza 392 -30 Pizza, Box of 6 pieces, 4 680 -34.4 oz./piece Pizza with Topping, 12 inch Diameter, On 1077 -34.4 Cardboard Pizza, Canadian Bacon, 12 1077 -34.4 inch Diameter Pizza, Ham and Pineapple, 1497 -34.4 16 inch Diameter Pizza, Sausage and Cheese, 211 -26 Round, No Packaging Pizza, Sausage and Cheese, Rectangular, Sliced, No 1549 -28.9 Packaging Pizza, Sausage and Cheese, Rectangular, Sliced, No 1548 -34.4 Packaging Pizza, Sausage Special Deluxe, Round with 504 -27.5 Cardboard Backing and Clear Shrink Wrap Pizza, Sausage Special Deluxe, Round with no 753 -28.9 Backing, Unwrapped Sheet Pizza 680 -34.4 Pizza, Vegetable, 12 inch 624 -34.4 Diameter Pizza Crust with Uncooked 170 -34.4 Ingredients on Top Air Air Flow Description Velocity, Direction m/s Pizza 3 along height Pizza, Box of 6 pieces, 4 3 along oz./piece width Pizza with Topping, 12 inch Diameter, On 3 Cardboard Pizza, Canadian Bacon, 12 3 inch Diameter Pizza, Ham and Pineapple, 3 16 inch Diameter Pizza, Sausage and Cheese, 3.8 along Round, No Packaging diameter Pizza, Sausage and Cheese, along Rectangular, Sliced, No 3.8 width Packaging Pizza, Sausage and Cheese, along Rectangular, Sliced, No 3.8 width Packaging Pizza, Sausage Special Deluxe, Round with 3.3 along Cardboard Backing and diameter Clear Shrink Wrap Pizza, Sausage Special along Deluxe, Round with no 3.3 diameter Backing, Unwrapped Sheet Pizza 3 Pizza, Vegetable, 12 inch 3 Diameter Pizza Crust with Uncooked 3 along Ingredients on Top diameter Table 2. Nusselt-Reynolds-Prandtl Correlations for Selected Food Items. Number Food Type Reynolds Number of Data Range Points Cake 4000<Re<80000 29 Chicken Breast 1000<Re<11000 22 Pizza 3000<Re<12000 12 Level of Coefficient of Food Type Significance Determination, (F-statistic) [r.sup.2] Cake 5.34E-12 0.833 Chicken Breast 0.00115 0.418 Pizza 0.00814 0.520 Food Type Nu-Re-Pr Correlation Cake Nu = 0.00156 x [Re.sup.0.960] x [Pr.sup.O.3] Chicken Breast Nu = 0.0378 x [Re.sup.0.837] x [Pr.sup.0.3] Pizza Nu = 0.00517 x [Re.sup.0.891] x [Pr.sup.0.3]

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Author: | Becker, Bryan R. |
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Publication: | Transactions of the Missouri Academy of Science |

Geographic Code: | 1U4MO |

Date: | Jan 1, 2005 |

Words: | 4552 |

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