Journal of Food Engineering
Research Note The Accuracy of Prediction of the Freezing Point of Meat from General Models Md. Shtiur Department
of Food Science and Technology, University of New South Wales, PO BOX 1, Kensington, NSW 2033, Australia
(Received 18 June 199 1; revised version received 10 August 1992; accepted 19 August 1992)
ABSTRACT General (empirical, theoretical and semi-empirical) models for the pre.diction of the freezing point of meat were tested using 101 published data points. Errors of 46% or more were obtained. The error can be reduced to 24% by considering only beef. The high error indicated that generalized equations should include more measured variables such as composition or should include more specific information such as meat source and location of the meat in the animal.
NOTATION alPa 4,&4
Model parameters of eqn ( 1) Model parameters of eqn (2) Bound water (kg water/kg solids) Model parameters of eqn (3) Model parameters of eqn (4) Degree of freedom (n-p ) Molecular weight ratio (M,/M,) Empirical parameter in model 6 Molecular weight Mean percent deviation:
Publishers Ltd, England. Printed in Great Britain
Md. Shafiur Rahman
Mean square error: i: (&pi-
n P r2 T X
Number of data points Number of parameters determined by regression Regression coefficient Freezing point (“C) Mass fraction of component phases
Molar freezing point constant of water (1860 kg K/kg mole) Freezing point depression of foods ( T, - Tf)
Subscripts f,expi Experimental freezing point of meat ith data Freezing point of meat from the model for ith data f,modeli f Food materials Solute S W Water
INTRODUCTION The initial or equilibrium freezing point is one of the most important properties of food, required for prediction of thermophysical properties because of the discontinuity exhibited at that point. In a freezing process, it is necessary to predict the freezing time to ensure the quality of the product and the efficiency of the equipment. All freezing time prediction models need the initial freezing point of the foods. Freezing point is also used as the limit of chilling injury of a fresh product when stored at low temperature. Accurate freezing point data can also be used to calculate or determine the other important properties such as effective molecular weight (Chen, 1986), water activity (Fontan & Chirife, 198 1; Lerici et al., 1983; Chen, 1987a), bound, free and frozen water (Heldman, 1974; Sakai & Hosokawa, 1984; Chen, 1985; Pham, 1987; Sheardet al., 1990) and enthalpy below freezing (Chang & Tao, 1981). Hence both freezing point data and prediction models are valuable in food physics. The freezing point prediction models can be divided into three groups: empirical curve fitting, theoretical models, and semi-empirical models. A range of published freezing point prediction models are summarized in Table 1.
(Semi-empirical) Model 6
A=(d, + d,X,)
(Xw- B-K Mw- Bx,) + E-J=& 1
(6) Chen (19876)
(4) Sanz et al. (1989)
A = C,X, + C,X; + C,Xj
TABLE 1 for the Prediction of the Freezing Point of Meat
(Empirical) Model 1
Md. Shafur Rahman
Chang and Tao (1981) divided the food materials into three main groups - meat or fish, juice, fruit or vegetables - and using data for 23 foods tabulated by Dickerson (1968) used linear and quadratic equations to correlate freezing point depression and moisture content (eqns (1) and (2)). By introducing the Margule equation for activation coefficient and using a power series expansion to represent the water activity as a function of the mole fraction of the solute, a third-degree polynomial equation was proposed by Chen and Nagy (1987) (eqn (3)). Chen (1986, 1987a) and Chen et al. ( 1990) also used eqn (3) for correlating the freezing point depression. Sanz et al. (1989) developed an empirical correlation to predict the freezing point of meat products such as function of moisture content (eqn (4)). Most of the theoretical freezing point depression equations have been derived on the basis of the Clausius-Clapeyron equation and Raoult’s law which follows from the first and second laws of thermodynamics. The Clausius-Clapeyron equation is limited to ideal solutions (i.e. very dilute solutions). A theoretical model (eqn (S), Schwartzberg, 1976) was developed as an improvement on the Clausius-Clapeyron equation by introducing parameters for non-ideal behaviour. The assumptions were that (1) the freezing point of the food is nearly the same as the freezing point of water, and (2) some water is bound to the solute and unavailable for freezing. Chen ( 1986) also used eqn (5) for the prediction of freezing point. Chen and Nagy ( 1987) proposed eqn (5) for ideal solutions assuming negligible bound water (B = 0). Blond (1985) showed that at very low concentration the deviation from Raoult’s law was small, but at higher concentration greater deviation was observed for macromolecules. According to the concept of solvation introduced by Jones (1917), some water in the system combines with solute and forms a third substance. The amount of combined water for any given substance is a function of both the concentration of solution and its temperature (Chen & Nagy, 1987). Chen (1987b) proposed eqn (6) assuming B= 0 for non-ideal solutions and used an empirical constant, tf’, which was a linear coefficient of concentration (model 6). According to Chen (1987a) ‘f’ can be either positive or negative depending on the food systems. The theoretical models mentioned above were used only for liquid foods where all solids are soluble. For solid and semi-solid foods the situation is different due to the presence of insoluble solids in the system which interact with the water and soluble solids. Heldman (1974), Schwartzberg (1976), and Murakami and Okos (1989) assumed all solids are solutes but in fact, the effects of soluble and insoluble solids are not the same.
Prediction of thefreezingpoint of meat
Mellor ( 1983) used a different approach to predict the freezing point. He assumed that at - 40°C there was no free water present in the systems, i.e. enthalpy became zero at - 40°C as adopted by Riedel (1956). The mean percent deviations from the enthalpy balance model were less that 7% for sucrose, apple, grapes, beans, peas, cod, beef and egg (white and whole), but for carrot and milk the error was about 26%. The practical limitation of this model was the need for enthalpy and bound water data. Succar and Hayakawa (1990) used enthalpy correlation above freezing and below freezing to predict the freezing point. They equated both correlations at the freezing point. The main disadvantages of this method were the iteration procedure and the need for enthalpy data which are not readily available. To date only empiricial correlations have been applied to meat and no attempt has been made to use theoretically based models. The objectives of this study were to test general freezing point prediction models (empirical, theoretical and semi-empirical) for meat by considering 101 published data points (full compilation is available from the author). The precision and accuracy of these models are estimated to find their applicability. The improvement in accuracy was also tested when freezing point prediction models for beef only were developed.
RESULTS AND DISCUSSION The SAS ( 1985) GLM (linear and quadratic) and NLIN (non-linear) procedures were used to estimate the parameters of the models in Table 1. The parameters of the empiricial linear and quadratic models were estimated by simple regression, while those of the other empirical and non-linear models were estimated by considering the lowest mean square error by the SAS (1985) iteration procedure. In the case of theoretical models, the parameters p and M,,, were fixed at 1860 and 18 as used by Chen (1986, 1987a, 1988) and Chen and Nagy (1987). Then both B and E were allowed to vary to find the optimum values by the SAS (1985) NLIN procedure. For the semi-empirical model both B and E were fixed at values for the theoretical model (eqn (5)) and the empirical concentration-dependent parameter was estimated. Chen (1987b), who proposed model 6, tested his model for solutions of known solute and assumed that B= 0. Hence f could be estimated without regression because E was known for a solution of known solute. For food materials the parameters B, E, and fare unknown.
Md. Shafiur Rahman
In the literature different criteria are used to predict the degree of fit of the models. The regression coefficient (r*) is most widely used, but r2 alone is not a good criterion. Mean square error (MSE) and mean percent deviation (MPD) are also commonly used to find the goodness of fit. The advantage of mean square error is that is also considers the number of parameters in the models and gives the absolute error or residual. In process design MPD is commonly used to accept or reject the model by comparing with the standard error at 5,10 or 15%. The parameters and accuracy of the models are given in Table 2. All models gave low r2 values. Empirical model 3 was the best when considering the MSE, but model 1 (linear) was the best considering the MPD. This anomalous result indicated the scattering of the data points and made it difficult to draw conclusions. Chang and Tao (1981) first developed the general linear model to predict the freezing point of meat considering six data points, but no criteria for the goodness of the models were reported. Sanz et al. (1989) found d,= O-069 and d2= - O-439 for meat products when fitting literature data to model 4. The above
TABLE 2 and Accuracy of the Models when All 10 1 Data Points for Beef, Lamb, Pork, Poultry, Venison and Fish Were Used
Model (Empirical) Model 1 Model 2
Model 4 (Theoretical) Model 5 (Semi-empirical) Model 6
‘Values in parentheses
B= 0.192 E= 0.021
B= 0.192 E=0+021 f= - 0.077
a, = 5.76 a2 = - 5.98 A, = 19.29 A,= -51.53 A, = 36.54 C1 = 18.94 c,= -77.57 C, = 97.26 d, = 0.084 d, = - 0.548
are standard deviations.
Prediction of thefreezingpoint of meat
values were nearly the same as the values found here. But theoretical correlations can have advantages over empiricial correlations because ( 1) they can be extrapolated beyond the experimental range, and (2) the parameters can be related to the physics of the materials and can be measured experimentally. For example, the bound water (II) and molecular weight ration (E) in the theoretical model can be measured experimentally and compared to the values from regression analysis of the freezing point data. Chen and Nagy (1987) found root mean square errors of 0.5% and 2% for empirical and theoretical based models when testing freezing point data of salts, acids, bases, glycerol, alcohols, and sugars, but the above authors recommended the theoretical based model over the empirical model because the parameters in the theoretical models had physical significance. The parameters and accuracy of the freezing point prediction models of beef are given in Table 3. Both r*, MSE and MPD indicated that the prediction models improved significantly when considering only one meat species. The error (MPD) can be reduced to 24% considering beef alone while no less than 46% was found when all types were considered.
TABLE 3 Parameters and Accuracy of the Models when Only 34 Data Points for Beef Were Used Model (Empirical) Model 1 Model 2
Model 4 (Theoretical) Model 5 (Semi-empirical) Model 6
‘Values in parentheses
B= 0.185 E= 0.023
E= 0.023 f= O-0023
a, = 10.67 a*= - 13.18 A, = 30.97 A, = - 88.13 A, = 64.49 C, = 16.94 c, = - 74.04 c, = 101.74 d, = 0.072 d, = - 0.488
are standard deviations.
Md. Shajiur Rahman
The theoretical and semi-empirical models gave the best predictions when comparing r*, MSE and MPD. Pham (1987) determined bound water from enthalpy and freezing point data. His values varied from 0.318 to 0.143 kg water/kg solids for beef, lamb and fish. The values obtained for meat and beef here are also in this range. This indicated that without enthalpy data it was possible to determine B using only freezing point data from the theoretical model. Duckworth (1971) measured bound water of meat and fish within the range 0.24-0.27 kg water/kg solids by using different thermal analysis. This was slightly higher than the values found here. Schwartzberg (1976) found E- 0.027 for lean beef using one freezing data point and bound water from Duckworth ( 1971). Here, the value of E was O-023 which is similar to the above value. The 24% error found is still not acceptable for engineering design because usually errors of less than 10% are recommended for design and process calculations. Hence, the model should be more specific, i.e. relate to meat source, age, sex and location of meat in the animals, or should include more measurable variables such as composition. The factors affecting the freezing point in foods, such as molecular weight, bound water, ionic association or dissociation, are not easily measurable and varied with the measurement technique. Hence, it is better to develop models for specific meats. Future data should be more specific, mentioning species name, location of meat in the animal and proximate composition, or at the very least the fat and ash content. Recently, glass transition temperature (end point of freezing) measurement and prediction have been emphasized, due to its application in quality improvement and because chemical and microbial degradation stops below the glass transition temperature where no mobile or free water is present in the medium. It would be useful to find the relation between glass transition temperature and freezing point because the measurement of the former requires a sophisticated instrument. Water activity and freezing point can be presented at the same time to check the validity of the Clausius-Clapeyron equation in food systems. The assumptions in the theoretical equation were that (1) the volume of solid was negligible compared to the volume of liquid (the assumption in the Clausius-Clapeyron equation), and (2) the freezing point of the food was nearly the same as the freezing point of pure water. Although the theoretical model is used widely, the assumption in the equation is not practical in the whole freezing process of food materials. When B and E are introduced in the theoretical equation, the error in the equation is reduced, but in that case bound water and molecular weight ratio
Predictionof the freezing point of meat
may deviate from the exact values. Hence, improvement cal models avoiding the assumptions is also necessary.
in the theoreti-
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