Influence of Nusselt Number on Weight Loss During Chilling Process

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Procedia Engineering 32 (2012) 90 – 96

I-SEEC2011 Influence of Nusselt number on weight loss during chilling process

R. Mekprayoon and C. Tangduangdee

Department of Food Engineering, Faculty of Engineering, King Mongkut’s University of Technology Thonburi, Bangkok, 10140, Thailand

Elsevier use only: Received 30 September 2011; Revised 10 November 2011; Accepted 25 November 2011.

Abstract

The coupling of the heat and mass transfer occurs on foods during chilling process. The temperature of foods is reduced while weight changes with time. Chilling time and weight loss involve the operating cost and are affected by the design of refrigeration system. The aim of this work was to determine the correlations of Nu-Re-Pr and Sh-Re-Sc used to predict chilling time and weight loss, respectively for infinite cylindrical shape of food products. The correlations were obtained by estimating coefficients of heat and mass transfer based on analytical solution method and regression analysis of related dimensionless numbers. The experiments were carried out at different air velocities (0.75 to 4.3 m/s) and air temperatures (-10 to 3 ÛC) on different sizes of infinite cylindrical plasters (3.8 and 5.5 cm in diameters). The correlations were validated through experiments by measuring chilling time and weight loss of Vietnamese sausages compared with predicted values.

© 2010 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of I-SEEC2011 Open access under CC BY-NC-ND license. Keywords: Analytical solution; Chilling process; Nusselt number; Sherwood number; Weight loss

1. Introduction

Chilling process involves heat transfer between a food product and cooling medium to reduce the product temperature to a desired level. It is widely used to inhibit pathogenic bacteria, to prevent product spoilage or to improve product quality, especially for meat product and carcasses. During the chilling process, it does not involve only heat transfer but also mass transfer by water evaporation at the product surface that causes weight loss. It was reported that during conventional chilling weight loss was about 2% for pork carcasses and about 6% for cooked sausages, resulting in economical loss.

* Corresponding author. Tel.: +662-470-9342; fax: +662-470-9240. E-mail address: [email protected].

Nomenclature Į thermal diffusivity (m2/s)

Bi Biot number for heat transfer (=heffLc/k), Biot number for mass transfer (=hmLc/Dsa), C, m, n constant D mass diffusivity of water vapor in air or mass diffusivity of water in sample (m2/s) 2 2 Fo Fourier number for heat transfer (=at/Lc ), Fourier number for mass transfer (=Dsat/Lc ), h heat transfer coefficient (W/m2 ÛC) or mass transfer coefficient (kg/m2 s)

J0, J1 zeroth- order and first-order Bessel function of the first kind, respectively k thermal conductivity (W/m ÛC)

Lc characteristic length (m)

Nu Nusselt number (=hLc/ka)

Pr Prandtl number (=Cpȝa/ka) r radius coordinate

Re Reynolds number (=ȡvLc/ȝa)

Sc Schmidt number (=ȝa/ȡDa)

Sh Sherwood number (=hmLc/Da) t time (s) ĳ temperature ( ÛC ) or moisture concentration (Kg water/Kg dry matter) v velocity (m/s) Y dimensionless mass or temperature ȡ density (kg/m3)

ȝ viscosity (kg/m s) or eigen value Subscripts a air e equilibrium eff effective value i initial m mass s surface sa sample 92 R. Mekprayoon and C. Tangduangdee / Procedia Engineering 32 (2012) 90 – 96

The major factor affecting the weight loss is cooling rate, which depends on temperature and velocity of a medium. The study of correlation between cooling rate and weight loss caused by moisture transfer is important for designing refrigeration system to minimize weight loss with an appropriate cooling rate. Many attempts to estimate the cooling process parameters i.e. heat and mass transfer coefficient (heff, hm) have been proposed by many researchers. Dincer [1], Becker and Fricke [2] estimated the heff values based on analytical solutions of refrigerating processes then developed Nu-Re-Pr correlations for many types of foods. Juditpasert et al. [3] simulated food model using finite difference method to determined heff values and to create the Nu-Re-Pr correlations. These studies focused on heat transfer only, but not on mass transfer. Kondjoyan and Daudin [4] estimated the heff and hm values using the psychometric method. The results demonstrated less than 3% of experimental error. Stewart et al. [5] estimated the heff values of apples and potatoes during cooling by measuring temperature different at the center and the surface of fruits. The estimated heff values were very close to that published in ASHARE (1985). The method is simple, but time consuming since the h-value was determined at the steady-state period. Tocci, and Mascheroni [6] and Erdogdu [7] determined heat transfer coefficient during freezing process by comparing the measured thermal histories with theoretical curves calculated numerically with different Bi numbers. The Bi number giving the best fit curve for the experimental data was then used to calculate the corresponding h value whilst mass transfer coefficient was estimated from analogies such as Ranz-Marshall and Chilton-Colburn. However, this method is valid for low mass flux, smooth and constants surface temperature [7]. The aim of this work was to study effects of Nusselt number of heat and mass transfer on the chilling time and the weight loss and to establish the correlations of process variables (air temperature, velocity and size of a sample) in terms of dimensionless groups. The correlations would be useful for simulation of heat and mass transport phenomena during the cooling process.

2. Materials and methods

2.1 Mathematical model

Assumptions of mathematical modeling are: (1) one-dimensional heat and mass transfer, (2) uniform initial temperature and moisture within a sample, (3) homogenous and isotropic properties, (4) constant thermo-physical properties, and (5) negligible heat generation and radiation. The governing equation for heat or mass transfer of infinite cylindrical shape is: 2 w ĳ 1 wĳ 1 wĳ Eq. (1) 2 r r t wr w : w Where ĳ is temperature or mass concentration, ȍ is thermal or mass diffusivity (m2/s), and r is radius coordinate. The governing equation is subjected to initial and boundary conditions as follows: wĳ ĳ r,0 ĳȠ Eq. (2) 0 Eq. (3) wr or wĳ wĳ k heff ĳs ĳf Eq. (4) Da hm ĳs ĳo Eq. (5) wr sr wr sr

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2.2 Estimation heat and mass transfer coefficients based on analytical solutions

The first term of infinite series solutions is enough to obtain the correct solution when the Fourier number is greater than 0.2 [8]. The fractional differences of temperature and moisture at the center of infinite cylindrical bodies can be calculated from eq. 6 [8] and eq. 7 [9], respectively. 2 ĳĳ a 2 J1 ȝ1 2 ĳĳ e 4Bi § 2 · Y exp ȝ F Eq. (6) Ym exp¨ ȝ F0 ¸ Eq. (7) 2 2 1 0 ĳ ĳ 2 § 2 2 · © 1 m ¹ ĳi ĳa ȝ1 Jo ȝ J ȝ i e ȝ ¨ȝ Bi ¸ 1 1 1 1 © 1 ¹ The dimensionless terms of temperature or mass are plotted against time according to eq. 8 and eq. 9 to determine slopes and intercepts of Y-axis.

Y Ct)jexp( Eq. (8) Ym jmexp( Cmt) Eq. (9) where j or jm are intercepts of Y-axis (lag factor), C or Cm are slope. Comparing between eq. 6 and 8 and eq. 7 and 9, Bi numbers are estimated from eq. 10 and eq. 11. Consequently, h and hm are obtained via Bi numbers of heat and mass transfer, respectively [2].

J1 ȝ1 J1 ȝ1 Bi ȝ1 Eq. (10) Bim ȝ1 Eq. (11) Jo ȝ1 Jo ȝ1 2.3 Nu-Re-Pr and Sh-Re-Sc correlations

Once the heat and mass transfer coefficients were attained, Nu- and Sh numbers of various experimental conditions were estimated and correlated with Re- and Pr numbers as shown in eqs. 12 and 13. The constant parameters (C and m) were determined by plotting in log-log scale and obtained by regression analysis, except for n, which is fixed at 0.33 for the chilling process [8].

hL c m n hmLc m n Nu CRe Pr Eq. (12) NuSh m CRe Sc Eq. (13) k a Da 2.4 Description of experiments

The experiments were carried out in an air-blast chiller by varying air velocities (0.75, 1.9, 3.1, and 4.3 m/s), air temperatures (-10, -3, and 3qC) and sizes of infinite cylindrical plasters (3.8 and 5.5 cm in diameter) used as a food model in order to achieve homogeneous thermo-physical properties. The plasters were prepared by mixing 60% of water with 40% of gypsum powder. The mixture was then poured in an infinite cylinder mold (3.8u19 cm and 5.5u27.5 cm) and leaved in the ambient temperature for setting. A couple of plasters with the same size were soaked in water for 1.5 hours [10], and then heated up until their temperature reached 73qC to imitate cooked meat products. Thermocouples (T-type) were inserted at the center and the near surface of a sample for measuring temperatures whilst the other was used for determining weight loss. The samples were hanged in the refrigerated chamber and then temperatures were recorded via data logger every minute, and the sample weight was collected manually every 5 minutes. The records were terminated when the core temperature reached 4qC. The observed temperature and mass loss versus time were used to establish the correlation of dimensionless parameters, which was further used to validate with a real food sample, Vietnamese sausage (Moo-yaw in Thai, 4u11.5 cm) by varying air velocities (0.75, 1.9, 3.1, and 4.3 m/s) and air temperatures (-10, -3, and 3qC). Thermal conductivity, thermal and mass diffusivity for samples obtained from literatures are shown in Table 1. 94 R. Mekprayoon and C. Tangduangdee / Procedia Engineering 32 (2012) 90 – 96

Table 1 Thermo-Physical properties of samples

kind of sample Parameters Sources

Wet plaster k (W/mÛC) 0.91 Carson et al., 2006 -7 Į (m/s2) 2.56 x 10 Carson et al., 2006 2 4.73 x 10-7 Voumbo et al., 2010 Dsa (m /s) Vietnamese sausage k (W/mÛC) 0.407 Singh & Heldman, 1993 2 -7 (Moo-yaw) Į (m/s ) 1.41 x 10 Markowski et al,2004 2 -7 Dsa (m /s) 1.31 x 10 Dincer & Yildiz, 1995

3. Results and discussion

3.1 Studying relations of heat and mass transfer 3.1.1 Influences of Nusselt number on chilling time

From the experiments, it was obvious that higher Nu number provokes a reduction of chilling time. In other words, the Nu value is inversely proportional to the chilling time as shown in Fig. 1.

Fig. 1. Effect of Nusselt number (Nu) on chilling time Fig. 2. Comparing influence of Nusselt number (Nu) on chilling time at different sizes In general, the Nu number is directly proportional to air temperature, air velocity and sample size. However, the effect of the sample size on the chilling time was independently influenced by the Nu number. As shown in Fig. 2, at the same Nu value the chilling time of the larger size (5.5 cm in diameter) was longer than that of the smaller one. Thermal resistance due to conduction (Lc/k) dominates that of convection (1/h). Therefore, increasing in size brings about an increase in Bi number or thermal resistance, in consequence, the chilling time increases.

3.1.2 Influences of Nusselt number on weight loss

Fig.3. shows the relationship between the Sh number or Nu number of mass transfer (Num) and the percentage of weight loss. The result indicated that an increase in Sh number did not enhance the percentage of weight loss as expected, although the increasing Sh number is accomplished by an increase in the Re number as shown in Fig.4. In this experiment, the core temperature was used as the criterion to cease the cooling process; therefore, the short period of cooling time enhanced by high Sh number resulted in low moisture loss. This implied that mass transfer of the sample was controlled by the internal mass transfer resistance (Lc/Dsa). In other words, the percentage of weight loss was mainly affected by air temperature and size of the sample rather than the air velocity. This result agreed with the work of Tocci & Mascheroni [6].

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Fig. 3. Effect of Sherwood number (Sh) or Nusselt Fig. 4. Effect of Reynolds number (Re) on Sherwood number of mass transfer (Num) on weight loss number (Sh)

3.1.3 New correlations of Nu-Re-Pr and Sh-Re-Sc

Heat and mass transfer simultaneously occurs during the chilling process. Process parameters were grouped in dimensionless terms of Nu and Sh numbers and used to estimate the heat and mass transfer coefficients, respectively. The dimensionless groups were plotted in logarithmic scale as presented in Fig. 5. and new correlations were proposed in eq. 14 and eq. 15. These correlations were valid in the designed range of 32 Nu 111, 0.015 Sh 0.102 and 925 Re 7815.

(a) (b)

Fig. 5. Regression analysis of (a) Nu-Re-Pr correlation and (b) Sh-Re-Sc correlation during cooling of an infinite cylindrical plasters

0.4296 0.33 0.3654 0.33 Nu 2.35Re Pr Eq. (14) Sh 0.00245Re Sc Eq. (15)

3.2 Validation of Nu-Re-Pr and Sh-Re-Sc correlations

The correlations of Nu-Re-Pr and Sh-Re-Sc presented in the previous section were used to calculate for heat and mass transfer coefficients for chilling Vietnamese sausages and then the predicted chilling time and weight loss were computed. The predicted results were validated through observed data from experiments. The goodness of fit was then analyzed using R2 values, which showed high values of 0.9887 and 0.9807 for heat and mass transfer, respectively, as shown in Fig.6. The experiments were conducted to varies cooling conditions of air temperatures (-10 to 3ÛC) and air velocities (0.75 to 4.15 m/s).

(a) (b)

Fig. 6. Validation of (a) chilling time and (b) percentage of weight loss obtained from prediction using the new correlations compared with the observed data of Vietnamese sausage cooling 96 R. Mekprayoon and C. Tangduangdee / Procedia Engineering 32 (2012) 90 – 96

4. Conclusions

Chilling time and weight loss were studied at different Nusselt number and Sherwood number. The process variations used in this study were air temperature, air velocity, and size of samples. During cooling process, an increase in the Nu number resulted in a reduction of chilling time. It is true only in the case of an increase in air velocity, but not the cases of air temperature and size increasing. Although a larger size of the sample and higher air temperature causes higher Nu number, the chilling time increases due to high internal thermal resistance and low temperature driving force. In addition, since the final temperature was fixed at 4qC. The higher Re number promotes higher Sh number and higher Nu number, which enhanced heat transfer rate, resulting in a reduction of chilling time, in turn lowers the weight loss. Generally speaking, weight loss during chilling process at a certain core temperature was controlled by heat transfer rate. The new correlations of Nu-Re-Pr and Sh-Re-Sc were proposed using an infinite cylindrical food model (plaster) and employed to estimate heat and mass transfer coefficients in order to predict the temperature and moisture content of Vietnamese sausages during cooling. The chilling time and the weight loss were then calculated and validated through the experiments. A good agreement between the observed data and the predicted results was obtained with high R2 of 0.9887 and 0.9807 for heat and mass transfer, respectively.

Acknowledgements

The authors are very grateful to office of the National Science and Technology Development Agency (NSTDA) of Thailand for financial support in this research.

References

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