1、4707 (RP-1123) Calculation of Food Freezing Times and Heat Transfer Coefficients Brian A. Fricke, Ph.D. Member ASHRAE ABSTRACT The freezing of food is one of the most significant appli- cations of refigeration. In order for freezing operations to be cost-efective, it is necessary to optimally design
2、 the refriger- ation equipment. This requires estimation of the freezing times offoods and the corresponding refrigeration loads. These esti- mates, in turn, depend upon the surface heat transfer coe cient for the freezing operation. Therefore, ASHRAE research project I 123-RP was initiated to deter
3、mine surface heat trans- fer coeflcients for a wide variety of food items. This paper describes the procedures used in ASHRAE research project 1123-RP to resolve dejciencies in heat trans- fer coeflcient data for foodfreezingprocesses. Members of the food refrigeration industry were contacted to col
4、lect cooling curves andsurface heat transfer data. A unique iterative algo- rithm was developed to estimate the surface heat transfer coe$ jcients of foods based upon their cooling curves. Making use of this algorithm, heat transfer coeficients for various food items were calculated from the cooling
5、 curves collected during the industrial survq. The accuracy of the calculated heat transfer coeficients was found to be within *30%. These heat transfer coeflcients will be tabulated in the ASHRAE Hand- book-Re frigeration. INTRODUCTION Preservation of food is one of the most significant appli- cati
6、ons of refrigeration. It is known that the freezing of food effectively reduces the activity of microorganisms and enzymes, thus retarding deterioration. In addition, crystalliza- tion of water reduces the amount of liquid water in food items and inhibits microbial growth (Heldman 1975). Bryan R. Be
7、cker, Ph.D., P.E. Fellow ASHRAE In order for food freezing operations to be cost-effective, it is necessary to optimally design the refrigeration equipment to fit the specific requirements of the particular freezing appli- cation. The design of such refrigeration equipment requires estimation of the
8、 freezing times of foods, as well as the corre- sponding refrigeration loads. Numerous methods for predicting food freezing times have been proposed. The designer is thus faced with the chal- lenge of selecting an appropriate estimation method from the plethora of available methods. Therefore, this
9、paper reviews basic freezing time estimation methods that are applicable to regularly shaped food items. In addition, knowledge of the surface heat transfer coef- ficient is required in order to utilize these freezing time esti- mation methods. A small number of studies have been performed to measur
10、e or estimate the surface heat transfer coefficient during cooling, freezing, or heating of food items for only a very limited number of food items and process conditions. Thus, there was clearly a need to expand upon the previous work by developing a comprehensive database of heat transfer coeffici
11、ents for a wide range of food items and process conditions. Hence, ASHRAE research project 1 123- Rp was initiated to estimate the surface heat transfer coeffi- cients of foods based upon the foods temperature history. An algorithm was developed to calculate heat transfer coefficients for various fo
12、od items based upon cooling curves collected during an industrial survey (Advanced Food Processing Equipment, Inc.; Freezing Systems, Inc.; Frigoscandia Equip- ment, AB; Technicold Services, Inc.). These cooling curves were generated from measured product center temperature versus time data. Brian A
13、. Fricke is an assistant professor of mechanical engineering and Bryan R. Becker is a professor of mechanical engineering and head of the Civil and Mechanical Engineering Division at the University of Missouri, Kansas City, Mo. 02004 ASHRAE. 145 THERMODYNAMICS OF THE FREEZING PROCESS The freezing of
14、 food is a complex process. Prior to freez- ing, sensible heat must be removed from the food to decrease its temperature from the initial temperature to the initial freez- ing point of the food. This initial freezing point is somewhat lower than the freezing point of pure water due to dissolved subs
15、tances, such as acids, salts, and sugars, in the moisture within the food. At the initial freezing point, a portion of the water within the food crystallizes and the remaining solution becomes more concentrated. Thus, the freezing point of the unfrozen portion of the food is further reduced. As the
16、temper- ature continues to decrease, the formation of ice crystals increases the concentration of the solutes in solution and depresses the freezing point further. Thus, it is evident that during the freezing process, the ice and unfrozen fractions in the frozen food depend upon temperature. Since t
17、he thermo- physical properties of ice and the unfrozen solution are quite different, the corresponding properties of the frozen food are temperature dependent. Therefore, due to these complexities, it is not possible to derive exact analytical solutions for the freezing times of foods. Numerical est
18、imates of food freezing times can be obtained using appropriate finite element or finite difference computer programs. However, the effort required to perform this task makes it impractical for the design engineer. In addi- tion, two-dimensional and three-dimensional simulations require time-consumi
19、ng data preparation and significant computing time. Hence, the majority of the research effort to date has been in the development of semi-analytical/empirical food freezing time prediction methods that make use of simplifying assumptions. These semi-analyticaliempirical freezing time prediction met
20、hods fail into two main categories. Methods in the first category, discussed in this paper, are applicable to food items that have the following regular shapes: 1. Infinite slabs 2. Infinite circular cylinders 3. Spheres Methods in the second category are applicable to food items that have irregular
21、 shapes. These methods require a two- step procedure in which the freezing time is first estimated by using one of the methods applicable to regularly shaped food items, which is then modified by using a shape factor such as the “equivalent heat transfer dimensionality” discussed below. Thus, freezi
22、ng time estimation for both regularly and irregu- larly shaped food items requires the use of the methods described in this paper. FREEZING TIME ESTIMATION METHODS . In the following discussion, the elementary method of estimating freezing time developed by Plank is discussed first, followed by a di
23、scussion of those methods, which are based upon modifications of Planks equation. The discussion then focuses upon those methods in which the freezing time is calculated as the sum of the precooling, phase change, and subcooling times. Planks Equation Although it was not developed specifically for e
24、stimating the freezing times of foods, the most widely known freezing time estimation method used for foods is Planks (1 9 13, 194 1) equation. In this method, it is assumed that only convective heat transfer occurs between the food item and the surround- ing cooling medium. In addition, it is assum
25、ed that the temper- ature of the food item is its initial freezing temperature and that this temperature is constant throughout the freezing process. Furthermore, a constant thermal conductivity for the frozen region is assumed. Planks freezing time estimation method is given as follows: L PD RD = A
26、nsari 1987; Chen et al. 1997; Clary et al. 1968; Cleland and Earle 1976; Daudin and Swain 1990; Dincer 1991, 1993, 1994a, 1994b, 1994c, 1995a, 1995b, 199.5 1995d, 1996, 1997; Dincer et al. 1992; Dincer and Genceli 1994, 1995a, 1995b; Dincer and Dost 1996; Flores and Mascheroni 1988; Frederick and Co
27、munian 1994; Khair- ullah and Singh 1991; Kondjoyan and Daudin 1997; Kopel- manetal. 1966;Mankadetal. 1997; Smithetal. 1971; 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
28、data and correlations for only a very limited number of food items and process conditions. occurs 148 ASHRAE Transactions: Research Thus, there is clearly a need to expand upon the previous work by developing a comprehensive database of heat transfer coefficients for a wide range of food items and p
29、rocess condi- tions. Hence, the objective of this study was to determine the surface heat transfer coefficients for a wide variety of foods during cooling and freezing processes. REVIEW OF EXISTING TECHNIQUES TO DETERMINE THE SURFACE HEAT TRANSFER COEFFICIENTS OF FOOD Techniques used to determine he
30、at 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 tran- sient temperature measurement technique
31、s. Transient methods for determining the surface heat trans- fer coefficient involve the measurement of product tempera- ture with respect to time during cooling processes. Two cases must be considered when performing transient tests to deter- mine the surface heat transfer coefficient: low Biot num
32、ber (Bi I O. 1) and large Biot number (Bi O. 1). The Biot number, Bi, defined in Equation 2, is the ratio of external heat transfer resistance to internal heat transfer resistance. A low Biot number indicates that the intemal resistance to heat transfer is negligible and, thus, the temperature withi
33、n the object is assumed to be uniform at any given instant in time. A large Biot number indicates that the internal resistance to heat trans- fer is not negligible and, thus, a temperature gradient may exist within the object. Low Biot Number Consider a food item with high thermal conductivity (negl
34、igible internal resistance to heat transfer) that is subjected to airflow at a constant, colder temperature. Then, during the time interval d, mcdt = h(t - t,)Ad , (9) where t = uniform temperature of the food item, rn = mass of the food item, and c = specific heat capacity of the food item. By inte
35、grating over the time interval AB, the transient method for determining the heat transfer coefficient may be obtained. where t, and t2 = the temperatures of the food item at the beginning and end of the time interval AB, respectively Large Biot Number One method for obtaining the surface heat transf
36、er coef- ficient of a food product with an internal temperature gradient involves the use of cooling curves. For simple, one-dimen- sional food geometries, such as infinite slabs, infinite circular cylinders, or spheres, there exist empirical and analytical solu- tions to the one-dimensional transie
37、nt heat equation. The slope of the cooling curve may be used in conjunction 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 t
38、emperature at the thermal center of the food item decreases exponentially (Cleland 1990). As shown in Figure 1, a cooling curve depicting this behavior can be obtained by plotting, on semilogarithmic axes, the fractional unaccomplished temperature difference versus time. The frac- tional unaccomplis
39、hed temperature difference, is defined as follows: t -t t-t, y= Freezing Systems, Inc.; Frigoscandia Equip- ment, AB; Technicold Services, Inc.). A typical cooling curve is shown in Figure 3. These collected cooling curves were digitized and a database was developed that contains the digi- tized tim
40、e-temperature data obtained from these curves. The temperatures were nondimensionalized according to Equation 11, and the natural logarithms of these nondimen- sional temperatures were taken. The slopes of the linear portion(s) of the logarithmic temperature versus time data were determined using th
41、e linear least-squares-fit technique. These slopes were then used in conjunction with the tech- niques described in the previous section to determine the heat transfer coefficients for the food items. Figure 3 Cooling curve for pizza. CALCULATED HEAT TRANSFER COEFFICIENTS To illustrate the data that
42、 were obtained from the iterative algorithm, a small sample of these calculated heat transfer coefficients (for pizzas) are given in Tables 3a (SI units) and 3b (I-P units). These tables list the heat transfer coefficients for the pizzas along with their dimensions and weight, as well as the air tem
43、perature and air velocity used to cool the food items. Nusselt-Reynolds-Prandtl Correlations Nondimensional analyses were performed to obtain simple heat transfer coefficient correlations that can be used to predict the heat transfer coefficients of food items. A least- squares-fit was performed on
44、Nusselt number, Prandtl number, and Reynolds number data to obtain a nonlinear Nusselt-Reynolds-Prandtl correlation. For this correlation, the Nusselt number, Nu, is defined as (40) where h = heat transfer coefficient, d = smallest dimension of the food item, and k, = thermal conductivity of the coo
45、ling medium. The Reynolds number, Re, is defined as 152 ASHRAE Transactions: Research Table 3a. Calculated Heat Transfer Coefficients for Pizza (SI Units) Heat Transfer Air Coefficient Diameter or Height Weight Temperature Air Velocity Airflow Description olr.m-.K-) Length (m) Width (m) (m) (gm) (OC
46、) (mas-) Direction Pizza Pizza with topping, 0.305 m diameter, on cardboard Pizza, canadian bacon, 0.305 m diameter Pizza, ham and pineapple, 0.406 m diameter Pizza, sausage and cheese, round, no packaging Pizza, sausage and cheese, rectangular, sliced, no packaging Pizza, sausage and cheese, rectan
47、gular, sliced, no packaging Pizza, sausage special deluxe, round with cardboard backing and clear shrink wrap Pizza, sausage special deluxe, round with no backing, unwrapped Sheet pizza Pizza, vegetable, 0.305 m diameter Pizza crust with uncooked ingredients on top 12.8 12.7 11.5 7.55 12.1 13.7 17.4
48、 5.27 16.3 9.12 14.5 17.3 0.24 0.41 3 0.022 0.013 0.02 0.02 0.029 0.032 0.032 0.022 0.019 0.013 0.02 0.013 392 -30 1077 -34.4 3 along height 3 0.305 1077 -34.4 3 0.406 1497 -34.4 21 1 -26 3 0.178 3.8 along width 0.406 0.305 1549 -28.9 3.8 along width 0.406 0.305 1548 -34.4 3.8 along width 504 -27.5
49、3.3 along width 0.305 0.318 753 -28.9 3.3 along width 0.394 0.305 O. 152 680 -34.4 624 -34.4 170 -34.4 3 3 3 along length Table 3b. Calculated Heat Transfer Coefficients for Pizza (i-P Units) Heat Transfer Air Coefficient Length Diameter or Height Weight Temperature Air Velocity Airflow Description (Btu.h-.ft-.OF) (in.) Width (in.) (in.) (Ib) (“F) (ftmin-) Direction Pizza Pizza with topping, 12 inch diameter, on cardboard Pizza, canadian bacon, 12 inch diameter Pizza, ham and pineapple, 16 inch diameter Pizza, sausage and cheese, round, no packaging Pizza, sau
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