N umerical solutions also introduce rounding error which keeps on increasing as the simulation time i ncreases. Therefore a study to determine the appropriate space and time steps that could give both stabi lity and accuracy was performed.
A series of simulations were run using typical i nput data for a block of butter by choosing a grid of
I Ox I Ox I 0 nodes in all the directions. This grid has a maximum time step of 1 750 seconds using equation 5 - 1 1 . Simulations with time step of 875 seconds and 1 750 seconds were compared for thawing a typical butter block. The results had a difference of less than 0.026°C from each other over the entire 70 hour period of simulation for the centre temperature of the block. It took 37seconds on a 1 .67GHz computer to complete the simulation w ith the time step of 1 750 seconds and halving the time step doubled the simulation time.
Another simulation was run by increasing the nodes to 20x20x20 in each direction. This gives a maximum time step of 486 seconds using the stability criteria. It took 947 seconds to complete the same simulation and the results showed less than 0.07°C difference in the temperature of the centre of the block compared with the 1 0x I Ox I 0 block. These results i ndicated that the numerical errors were not significant if an appropriate time step is selected using the stability criteria.
Equation (5- 1 1 ) is a function of density, thermal conductivity and specific heat capacity. Density i s assumed t o be constant but thermal conducti vity and specific heat capacity are functions of
temperature. Therefore whi le using temperature dependent properties minimum values of specific heat capacity and maximum values of thermal conductivity should be used to find the appropriate time step.
5.5.2 Checks Against Previously Validated Solutions
The numerical solution was checked against two existing and validated solutions. A comparison was made against a previously validated finite difference model RADS developed by Cleland and Cleland ( 1 99 1 ) for heat conduction in a solid with convective boundary conditions and uniform i nitial conditions. The comparison was made for a I m x I m x I m cuboid with constant thermal properties
(p
= 923 kg m·3, A. = 0. 1 9Wm·I K I , cl' = 3.6 kJ kg· 1 °C l ) being heated from 4°C to 1 2°C. A maximum difference of less than O. 0 I °C was found for the centre of the block for a 250 hour simulation with I Ox I Ox I 0 nodes in each direction.The other check was made against the analytical solution (Carslaw and Jaeger, 1 959). For heat conduction i n a block with uniform initial conditions (Ti = 4°C, Ta= 1 2°C) and convective boundary conditions (h = 1 1 .4Wm-2K 1 ). When the analytical solution was compared with the model
predictions the maximum temperature difference for the centre of the block temperature was found to be less than 0.03°C.
5.6 Evaluation of Mathematical Model
In this section a sensiti vity analysis was done on all the i nput data to the model and then the model was validated against the experimental data.
5.6.1 Sensitivity Analysis
Once the model was rigorously checked for numerical errors and validated against already existing models, a sensitivity analysis was performed to see how sensitive the model predictions were to each of the i nput parameters data including density, thermal conductivity, enthalpy, heat transfer coefficient, dimensions, i nitial freezing point, ambient temperature and initial butter temperature.
The best estimates of the above input data was taken and applied to a 25kg block of butter being heated up from an initial temperature of -9.5°C to an ambient temperature of 11°C. Simulations were run for both completely frozen and unfrozen unsaIted butter.
Table 5 . 1 gives the upper and lower bounds for the predictions for defined % errors in each corresponding input. The percentage error was different for different inputs and was selected from the measured data to reflect the uncertainty in each value.
The density of butter does not change significantly within the temperature range of interest. Sensitivity analysis shows a negligible change in the predictions with I % density change. So usi ng a constant value for modeling heat transfer in butter is appropriate. Uncertainty in thermal
conductivity of the butter was estimated to vary by about 1 0% below Tif and about 5% above Tif'
Sensitivity analysis on the thermal conductivity shows that it has negligible effect on predicted temperature in the freezing range but gave some differences in the final thawing time. Thus using an appropriate value for thermal conductivity for the thawing and freezing range is important. Uncertainty of ± I 0% in the enthalpy of the butter had large effects on the thawing time predictions as shown in Table 5. 1 . Lower specific heat capacity predicts shorter thawing time and higher
specific heat capacity predicts longer thawing times. Heat transfer coefficients are a function of the air flows and the packaging material. The overall resistance is domi nated by the packaging and the air gaps. The effect of the resistance due to convection and radiation is not significant as shown in the Table 5. 1 . Dimensions have significant effect on the thawing time predictions. So accurate determination of the dimensions is very important. Ambient temperature also affects the thawing time significantly. In storage rooms the ambient temperature changes with time due to the incoming and outgoing of the load traffic and imprecise refrigerati on system controls. If the measured
ambient temperature has large changes, it is important to use time - variable ambient temperature for the model input. The initial temperature of the product is also necessary, although it has negligible effect on the final thawing time predicted.
Table 5. 1: Sensitivity analysis of the thawing model to changes in input variables
Input property data Prediction band (----+ x %, -- - x %)
With Freezing Without Freezing
Densi ty " " p = 970 ± 1 % , 0
j
-s -, .1°0 20 '" .1°0 io '" .. 50 " Tom.I"'ou�1 Tome (Hours) Thermal conductivity 10 "�
=0.29 ± L O% ... .. ... � Au = 0.22 ± 5 %i
0 -5 -, .1°0 " °0 io '" .. 5' " 20 40 60 80 TIme (Hours)Enthalpy /specific heat " "
/, ... / .. " '-'--- H= Eq (5.4) ± 1 0% ... ..-.. - .// . cl' =3.2 ± I O% .,., -, -, -1°0 20 .0 " .0 .1°0 io '" 5' " T"".{Hours)
Heat Transfer coefficient
" "
h" = 4.3 ± 30%
-, -,
.100 .1°0 '" '0 5' "
Dimensions ,. V = ±5% .1°0 T ... 'I-iourl) Ambient Temperature ,. /-/"'--- Tu = ±2°C • -1°0 .0- T_(Hours) Initial Temperature ,. T; = ±2°C
5.6.2 Model Predictions - Thawing
,.
.1°0 -,0- 20 10 .&Q T_(Hours)
,.
In chapter 4, the experimental data collected for thawing and freezing of butter pats and blocks was summarized. Using the guidelines developed for the number of nodes and the time step in section 5 .5 and the thermal properties data determined in chapter 3, the model predictions were compared against the measured data for thawing butter pats and blocks. Two modeling approaches were taken to model the thawing behavior of butter.
The first approach (Equilibri um Properties -Model - I ) assumed equilibrium thawing behavior where the latent heat is absorbed over a range of temperatures up to the initial freezing point. This data is gi ven in Table 3. 1 3. Thus in this case the temperature of the butter wi 11 not rise above T if until all the latent heat is absorbed at that position.
The second approach (Sensible Heat Only - Model -
2)
was taken by using a temperatu re dependent enthalpy relationship measured in chapter 3 where the butter was frozen only to-2oDe
before thawing and no significant water freezing was observed. This was done to approximate the thawing of butter that had not fully frozen prior to thawing.5.6.2.1 Predicting Thawing of Pats
5. 6.2. 1.1 Salted Butter Pat (B18, SPH - 1)