Abstract: A temperature history detected by measuring sensor, along with other relevant system’s parameters have been used to predict the surface film conductance through transient temperature measurements in fish flesh samples during their cooling in a chilled air duct at a constant temperature of 1°C. The Inverse Heat Conduction Problem (IHCP) solution was performed by using the sequential function specification method to estimate heat flux, which was then utilized to solve the direct problem for the temperature distribution at any position including at the sensor position on the fish sample using Crank-Nicolson implicit finite difference scheme. The predicted and measured temperature distribution profiles were compared numerically, yielding good agreement indicating the accuracy of the present approach in calculating surface film conductance.
INTRODUCTION
In many food processing applications, including cooling and freezing, transient heat transfer occurs between the cooling medium and the solid item. In such circumstances, surface film conductance values are predicted usually by the appropriate Nusselt-Reynold’s-Prandtle correlation (Becker and Fricke, 2004; Abbas et al., 2006).
Based on experimental and theoretical studies, researchers (Hafiz and Ansari, 2000; Ansari et al., 2003, 2004) reported that, these values of surface film conductance are found to give poor results and the actual values are higher than those predicted from the Nusselt -Reynold’s-Prandtle correlation.
A critical look to those literatures indicates that still there is a need to develop a new approach (Fricke and Becker, 2002), which yields an accurate surface film conductance during precooling process, which may be of great importance for designers of cold storage; refrigerators and heat transfer equipments in food industry.
MATERIALS AND METHODS
The present research was performed by the means of air blast cooling duct at the Food Engineering Lab of Faculty of Food Science and Technology in University Putra Malaysia in 2006. As experimental and theoretical investigations were carried out on a slab shaped samples of freshwater Malaysian Patin fish.
The work was started initially with mass density and water mass fraction (W)
measurements of the fish samples whereas the thermal conductivity and specific
heat were determined (Abbas et al., 2006) as follows:
| Fig. 1: | Schematic diagram of air blast cooling duct |
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(1) |
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The air-blast cooling duct, shown in Fig. 1, was designed and fabricated for the measurement of surface film conductance of fish sample, which requires temperature–time records inside fish flesh during its transient cooling. This plant as well as the experimental procedure had been reported and detailed earlier in literature in a similar work (Abbas et al., 2006). The temperature of the circulating air inside the test duct was maintained constant at 1°C and the velocity of the air was kept constant throughout the experiments at 6 m sec-1.
The test container of rectangular shape which is shown in Fig. 2, was designed to allow symmetrical one-dimensional heat transfer to take place within the samples and to avoid any moisture evaporation to the air stream. The characteristic length, zo, of the fish sample is half the thickness of test container (1.2 in. or 1.27 cm). A copper-constantan thermocouple bead (sensor) was installed inside the fish flesh, at the depth of zo/5 from the sample surface. The thermocouple was connected to a data logger to obtain the temperature measurements at a specified equal time interval, which was maintained at 1 min while time of the experiment was 30 min (Seven-eights cooling time).
Initially, the refrigeration system of the chilling duct was run until a constant temperature of 1°C was achieved. Then the fish package was suspended in the test section of the air duct such that the conducting surfaces were parallel to the direction of flow of chilled air stream and the data logger was activated to record time temperature data to be used for the estimation of the surface film conductance.
Inverse Heat Conduction Problem Formulation
Estimation of temperature at any position and time from transient heat conduction
differential equation with prescribed boundary and initial conditions is known
as the direct method. While determination of the boundary conditions, initial
condition or thermal properties from transient temperature measurements is known
as an Inverse Heat Conduction Problem (IHCP).
| Fig. 2: | Test container details |
There are several pioneer algorithms proposed for the solution of the IHCP including, the exact matching algorithm (Stolz, 1960) the function specification algorithm (Beck, 1962, 1968, 1970) and the regularization algorithm (Tikhonov and Arsenin, 1977). In this research the function specification algorithm will be used to solve for the surface heat flux.
For isotropic slab shaped fish samples, with constant thermal properties, initially
at uniform temperature and exposed suddenly to symmetric cooling on both sides,
the governing heat conduction equation, center boundary condition and surface
boundary condition are given by the following system of Eq. 3-6.
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with initial and boundary conditions:
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Estimation of the surface heat flux qm can be obtained from minimization
of the following sum of squares function:
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Where, m is index for discrete time. The estimation of qm involves
temperature measurements at time tm, tm+i…..tm+r-i,
the heat flux at t<tm-i is assumed to be known. The heat flux
for time tm to tm+r-i can assume different functional
form such as constant, linear, cubic, parabolic or other form. Based on a temporary
assumption of constant heat flux for time tm to tm+r-i
(Beck et al., 1996), minimized Eq. 7 with respect to
qm and used Taylor series expansion and developed the following algorithm
for calculating heat flux.
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Where Xm+i-1 is the sensitivity coefficient defined by
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From the governing heat conduction equation and the prescribed boundary conditions
the following are the equations for the sensitivity coefficients.
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With initial and boundary conditions:
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the following equation was used to estimate the surface film conductance at
discrete time step:
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Computer Program
A FORTRAN computer program was written to solve numerically Eq.
3-6 and Eq. 10-13
based on Crank-Nicholson implicit finite difference discretization. Equation
8 is also incorporated into the program for calculating the heat flux sequentially
in time. The program can handle different values of r. However, r = 3 is found
to be adequate based on a preliminary runs at different r-values. The numbers
of nodes used were 50 and the time increment is 1 sec.
RESULTS AND DISCUSSION
Table 1 shows the measured and calculated thermo physical
properties of the fish sample.
| Table 1: | Thermophysical properties of a slab shaped fish sample |
| Fig. 3: | Experimental and calculated temperature history at the sensor
position |
The above information along with initial and known boundary condition were
used as an input to the developed computer program so as to calculate the heat
flux in sequential manner at each time (m). Once the heat flux at the heat transfer
surface side is known the problem became a direct problem, the same computer
program then calculates the temperature at any position including at the sensor
location. The accuracy of the calculated heat flux was checked by comparing
the calculated temperature at the sensor position with the measured value, using
the Root Mean Squares of the error (RMS) defined by:
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The above equation resulted in a value of RMS of 0.2°C, for the range of precooling Fo>0.2 (i.e., >4 min) until reaching the seven-eights cooling time. The incorporated error is within the allowable error encountered during temperature measurements by thermocouples indicating the accuracy of the calculated heat flux. Figure 3 shows comparison between the calculated temperature from the estimated heat flux and the measured temperature at the sensor position.
Ansari et al. (2003, 2004) developed plots similar to that shown in Fig. 3 and they reported that, the better coincidence between the calculated and the measured temperature histories, the more accurate approach is. Based on above it is clear that the excellent agreement between these two temperature histories in the present study, indicating the reliability, accuracy and superiority of the author approach among the existing ones. Beck et al. (1996) investigated different types of function form for heat flux including cubic, parabolic, linear and constant for the function specification method and found that the constant heat flux form resulted in excellent and efficient estimation of the heat flux compared to the experimental heat flux data and the other functional form.
Figure 4 shows that the calculated surface film conductance values which decrease gradually with time in a non-linear fashion, which was fitted to several nonlinear model using least square method, the following model yielded the best fit with the minimum standard error of 2.16 and maximum R2 = 0.99
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| Fig. 4: | Surface film conductance versus time curve |
It is notable to mention that t in the above equation must be expressed in minutes and it’s valid for the period of the present study. The value of surface film conductance determined by the proposed method revealed good agreement between the measured results and the predicted ones.
CONCLUSIONS
With known thermo-physical properties, transient temperature-measurement records at a location of 0.254 cm from the tested sample surface, the boundary heat flux has been estimated through the IHCP technique. The predicted variable values of the heat flux along with the others system parameters were used to find out sequentially the surface film conductance at each time step during the experimental period by applying the finite different technique. A correlation model for surface versus time was developed. This relationship can be used along with the finite difference technique to predict temperature variation at any location in the food flesh.
Nomenclature| c | specific heat of fish (J kg-1. K) |
| h | surface film conductance (W m-2. K) |
| k | thermal conductivity (W m-1. K) |
| N | Number of measurements |
| q | heat flux at the surface boundary, (w m-2) |
| r | number of future temperature measurements |
| T | temperature (°C) |
| t | time (s) |
| W | water content, % (on wet mass basis) |
| Y | measured temperature (°C) |
| z | distance from the centre (m) |
| zo | half thickness of the sample (m) |
Subscripts and Superscripts
| c | mcooling medium |
| m | discrete time index |
| o | initial |