HEAT TRANSFER COEFFICIENT IN FOOD PROCESSING: COMPILATION OF LITERATURE DATA

ABSTRACT

Heat transfer coefficient data in food processing were retrieved from literature and classified per process and material. Most of the data were available in the form of empirical equations using dimensionless numbers. All available empirical equations were transformed in the form of Heat Transfer Factor vs. Reynolds Number (j_H=aReⁿ). In the case when more than one equation reported for the same process and material, a new similar equation was fitted to consolidate the existing literature equations. It is expected that the resulting equations are more representative and predict more accurately the heat transfer coefficients. Average equations for each process are also proposed.

INTRODUCTION

The interface heat transfer coefficient is important in the design of food processes and processing equipment, and in the control of food packaging and storage. Heat transfer coefficients are essential in thermal processing, drying, cooling/freezing and storage operations. The surface heat transfer coefficient (h) is defined in the Newton's law of cooling:

where Q is the heat flow rate (W), A is the surface area (m²), and ΔT is the temperature gradient (°C or K). Empirical equations involving dimensionless numbers are available in classical Chemical Engineering literature, most of which are summarized by Rahman[1] in a comprehensive review, in which Food Engineering literature is also included. The above review is further updated by an exhaustive literature search made in international Food Engineering and Food Science journals during the recent years by Zogzas et al.[2]

Most of the data were available in the form of empirical equations using dimensionless numbers. According to the most common equation the heat transfer factor (j_H) was a function of the Reynolds number as follows:

where, j_H is defined as:

and the dimensionless numbers Re, St and Pr and Pr as follow:

where, u, the average (bulk) fluid velocity (m/s); d the process characteristic length that is: the particle diameter for process involving particles, the tube diameter for flow through tubes, the silo diameter for storage,[27] the can diameter for retort process.[39], [41] Following fluid properties were used: ρ, density of air (kg/m³); η, dynamic viscosity of fluid (kg/m s); h, heat transfer coefficient (W/m² K); C_p, the specific heat of the fluid (J/kg K); k, the thermal conductivity of the fluid (W/m K). All fluid properties refer at bulk temperature. The main advantage of the Eq. 2 is based on the Chilton-Colburn analogy for heat, mass and momentum transfer as follows:

where j_H is the Heat Transfer Factor, j_D is the Mass Transfer Factor and f is the Fanning Friction Factor. The scope of the present paper is to analyze the above data using the same empirical Eq. 2, to classify these per process and material and to propose generalized equations for each process based on the data of all materials.

DATA

The data analyzed in the present paper are mainly come from the following journals:

	Drying Technology, 1983–1999
	Journal of Food Science, 1981–1999
	International Journal for Food Science and Technology, 1988–1999
	Journal of Food Engineering, 1983–1999
	Transactions of the ASAE, 1975–1999
	International Journal of Food Properties, 1998–2000

A total number of 54 papers were retrieved from the above journals Zogzas et al.[2] The data refer to 7 different processes (Table 1) and includes about 40 food materials (Table 2).

Table 1. Number of Available Equations for Each Food Process

Process	No. of Equations
1 Banking
F Forced convection	1
2 Blanching
Steam	1
3 Cooling
Forced convection	9
4 Drying
Convective	16
Fluidised Bed	1
Rotary	4
5 Freezing
Forced convection	6
6 Storage
Forced convection	4
7 Sterilization
Aseptic	9
>Retort	3
Total no. of equations	54

Table 2. Number of Available Equations for Each Food Material

Material	No. of Equations
Apples	1
Apricots	1
Barley	2
Beef	1
Cakes	1
Calcium alginate gel	1
Canola seeds	1
Carrot	1
Corn	2
Corn starch	1
Figs	1
Fish	1
Grapes	3
Green beans	1
Hamburger	2
Maize	1
Malt	1
Meat carcass	1
Model food	4
Newtonian Liquids	1
Non-food material	3
Particulate liquid foods	3
Peaches	1
Potatoes	2
Raspberries	1
Rice	1
Soya	2
Soybean	1
Strawberries	1
Sugar	1
Wheat	3
Spherical particles	1
Tomatoes	1
Corn cream	1
Rapeseed	1
Meatball	1
Total no. of equations	54

REGRESSION ANALYSIS

In order to homogenize and compare the literature data by Eq. 2 was selected and all the available equations were transformed into Eq. 2. Since analytical transformation using mathematical operations does not exist a numerical transformation was used. The main steps of the numerical transformation are summarized as follow:

1.	Generate a grid of N3 values for the factors characteristic dimension of system geometry d, fluid velocity u and temperature T.
2.	Calculate the fluid properties at the grid points (e.g., density, viscosity, thermal conductivity and heat capacity etc.) and the corresponding values of Reynolds and Prandtl numbers.
3.	Calculate the heat transfer coefficient using the -equation available in the literature e.g., if the Nu number is available versus Re, h=St ρuCp if the ST number is available versus Re and so on.
4.	Calculate the corresponding jH factor using Eq. 4.
5.	Fit Eq. 2 to the available values of jH, versus Re.

RESULTS AND DISCUSSION

The results are classified per process and material and presented in Table 3. All the equations are presented in Fig. 1 to determine the range of variation of the j_H and Re. The range of variation per process is also sketched in Fig. 2. The above results are presented analytically for each process in Figs 3-9. The effect of food material is obvious in these diagrams. The results of fitting the equation to all data for each process is summarized in Table 4 and in Fig. 10.

Figure 1. Heat transfer factor (j_H) vs. Reynolds Number (Re) for all the examined processes and materials.

Figure 2. Ranges of variation of the heat transfer factor (j_H) vs. Reynolds Number (Re) for all the examined processes.

Figure 3. Heat transfer factor (j_H) vs. Reynolds Number (Re) for cooling process and various materials.

Figure 4. Heat transfer factor (j_H) vs. Reynolds Number (Re) for convective drying process and various materials.

Figure 5. Heat transfer factor (j_H) vs. Reynolds Number (Re) for rotary drying process and various materials.

Figure 6. Heat transfer factor (j_H) vs. Reynolds Number (Re) for freezing process and various materials.

Figure 7. Heat transfer factor (j_H) vs. Reynolds Number (Re) for storage process and various materials.

Figure 8. Heat transfer factor (j_H) vs. Reynolds Number (Re) for sterilization aseptic process and various materials.

Figure 9. Heat transfer factor (j_H) vs. Reynolds Number (Re) for sterilization retort process and various materials.

Figure 10. Estimated equations of heat transfer factor (j_H) vs. Reynolds Number (Re) for all the examined processes.

Table 3. Parameters of the Equation j_H=aReⁿ for Each Process and Each Material

Process/Product/Reference	a	n	min Re	max Re
Baking
Cakes
Baik et al., [3]	0.801	−0.390	40	3000
Blanching
Green beans
Zhang, et al., [4]	0.00850	−0.443	150	1500
Cooling
Apples
Fikiin et al., [5]	0.304	−0.286	4000	48,000
Apricots
Fikiin et al., [5]	0.114	−0.440	2000	25,000
Figs
Dincer, [6]	8.39	−0.492	3500	9000
Grapes
Fikiin et al., [5]	0.472	−0.516	1300	17,000
Model food
Alvarez et al., [7]	2.93	−0.569	2000	12,000
Peaches
Fikiin et al., [5]	0.186	−0.500	3700	43,000
Raspberries
Fikiin et al., [5]	0.0293	−0.320	1300	16,000
Strawberries
Fikiin et al., [5]	0.136	−0.440	1900	25,000
Tomatoes
Dincer, [8]	0.267	−0.550	1000	24,000
Drying
Convective
Barley
Sokhansanj, [9]	3.26	−0.650	20	1000
Canola Seeds
Lang et al., [10]	0.458	−0.241	30	50
Carrot
Mulet et al., [11]	0.692	−0.486	500	5000
Corn
Fortes et al., [12]	1.06	−0.566	400	1100
Torrez et al., [13]	4.12	−0.650	20	1000
Grapes
Ghiaus et al., [14]	0.665	−0.500	8	50
Vagenas et al., [15]	0.741	−0.430	1000	3000
Maize
Mourad et al., [16]	11.9	−0.901	150	1500
Malt
Lopez et al., [17]	0.196	−0.185	60	80
Placebo
Taranto et al., [18]	2.48	−0.523	200	1500
Potatoes
Wang et al., [19]	0.224	−0.200	2000	11,000
Poultry Excreta
Dixon et al., [20]	0.204	−0.600	1500	12,000
Rice
Torrez et al., [13]	4.12	−0.650	20	1000
Soybean
Taranto et al., [18]	2.48	−0.523	200	>1500
Wheat
Lang et al., [10]	149	−0.340	50	100
Sokhansanj, [9]	3.26	−0.650	20	1000
Fluidised Bed
Corn Starch
Shu-De et al., [43]	0.101	−0.355	3200	13,000
Rotary
Fish>
Shene et al., [21]	0.00160	−0.258	80	300
Soya
Alvarez et al., [22]	0.00960	−0.587	10	100
Shene et al., [21]	0.00030	−0.258	20	80
Sugar
Wang et al., [23]	0.805	−0.528	1500	17,000
Freezing
Beef
Heldman, [24]	0.650	−0.418	80	25,000
Calcium alginate gel
Shene et al., [21]	48.6	−0.535	300	600
Hamburger
Flores et al., [25]	8.87	−0.672	7500	150,000
Tocci et al., [26]	4.67	−0.645	9000	>73,000
Meat carcass
Mallikarjunan et al., [42]	0.228	−0.269	1800	20,000
Meatball
Tocci et al., [26]	0.536	−0.485	3400	28,000
Storage
Barley
Alagusundaram et al., [27]	0.257	−0.366	350,000	2,500,000
Potatoes
Xu et al., [28]	0.658	−0.425	70	90
Wheat
Chang et al., [29]	0.0136	−0.196	1500	10,000
Rapeseed
Alagusundaram et al., [27]	0.257	−0.367	350,000	2,500,000
Aseptic
Model food
Balasubramaniam et al., [30]	0.500	−0.507	5000	20,000
Sastry et al., [31]	0.448	−0.519	2400	45,000
Zuritz et al., [32]	3.42	−0.687	2000	>11,000
Non-food material
Kramers, [33]	0.748	−0.512	3000	85,000
Ranz et al., [34]	0.662	−0.508	3000	85,000
Whitaker, [35]	0.517	−0.441	3000	85,000
Particulate liquid foods
Mankad et al., [36]	0.225	−0.400	140	1500
Sannervik et al., [37]	0.0493	−0.199	1800	5200
Spherical particles
Astrom et al., [38]	2.26	−0.474	4300	13,000
Retort
Newtonian liquids
Anantheswaran et al., [39]	2.74	−0.562	11,000	400,000
Particulate liquid foods
Sablani et al., [40]	0.564	−0.403	30	1600
Corn cream
Zaman et al., [41]	0.108	−0.343	130,000	1,100,000

Table 4. Parameters of the Equation j_H=aReⁿ for Each Process

Process	a	n	min Re	max Re
Baking	0.801	−0.390	40	3000
Blanching	0.00850	−0.443	150	1500
Cooling	0.1426	−0.4552	1000	48,000
Drying/Convective	1.0400	−0.4547	8	11,000
Drying/Fluidised bed	0.1005	−0.3545	3200	13,000
Drying/Rotary	0.001	−0.161	>10	300
Freezing	1.00	−0.4856	80	150,000
Storage	0.2588	−0.3872	70	2,500,000
Sterilization/Aseptic	0.0350	−0.4501	140	45,000
Sterilization/Retort	1.0345	−0.4989	30	110,000

The total error between the actual and calculated values for the equations j_H=aReⁿ, is the sum of the following errors:

	The error of the initial equation (e.g., Nu=aRem Prn or h=St ρuCp which have been taken from the literature,
	The numerical error due to the transformation from the initial equation to jH=aRen.
	The variation among the different literature sources

Unfortunately there is not enough information available in the literature to analyze the above errors. The statistical analysis of variance methods could not be applied. Instead an order of magnitude of the total error for each process could be obtained using the results of various products. For example in cooling the heat transfer coefficient at Re=9000 is 0.0022 for apple, 0.0021 for apricots, 0.0951 for figs, 0.0043 for grapes, 0.0165 for model foods, 0.0020 for peaches, 0.0016 for raspberries, 0.0025 for strawberries and 0.0018 for tomatoes, pears and cucumbers. The above values give an average value of 0.0023 and a standard deviation of 30%, which could be considered as the estimation error using Eq. 4.

CONCLUSION

Heat transfer coefficient values for process design can be obtained easily from the proposed equations and graphs. The range of variation of this uncertain coefficient can also be obtained in order to carry out valuable process sensitivity analysis. Estimation can be done, using the proposed equations when experimental data is missing.

Acknowledgments