MA THS CHECKING

4.4. 1 CHECKS AGAINST P REVIOUSLY VALIDATED SOLUTIONS

After a model has been implemented in a numerical solution, checks must be made to ensure that what h as been implemented numerically is what was actually formulated. This i s done by comparison against an existing sol ution which h as already been

validated. Due to the complexi ty of the transport model formulated here, a d irect comparison like thi s could not be made. Tf the model was simplified by t urning off moisture or h eat transport, then individual parts ofthe model could be checked separately.

Moi sture transport was switched off by making the binary moisture ditTusivity of water vapour in air equal to zero. As di ffusi on i s the only method by whi ch moisture i s transported in the mode l , only h eat transfer by conduction wi l l occur. Th i s reduced the transport model to the simple one di mensional heat transfer by conduction in an i n fi ni te

slab case. I f the surface h eat t ransfer cocificients were set very high then the

temperature h istory of the slab could be predicted using the Fourier solution for heat

conductIOn in an i n fi n i te s l ab . The pred icti ons by this method were com pared to the transport model predictions for the case of a lOOmm l actose slab of initially uni form

temperature (20°C), heated on both sides by a step change on the surface to 40°C. During a simulation of over eight hours, the maximum di fference in any nodes between the two models was O. 06°C. This shows the heat conduction part of the transport model

is a reasonable representation of t he actual syste m .

A s i m i l ar comparison was performed against a previously val i dated fi nite di fference model for heat conduction i n an infinite slab with convection occurring at the slab boundaries, (RADS, Corne l i us 1 99 1 ). The same situation described above was

simulated with both surface heat transfer co-efficients set to 1 0W/m2K. The maximum difference observed d uring an eigh t hour simul ation was O.045°C at the centre of the slab, and O. 1 4°C at the slab surfaces. This showed the term describing convection at the l actose slab surfaces, h ad been correctl y impl emented in the transport model.

however, this was difficult. With an analytical sol ution, the moisture concentration

changes by ddfusion only. With the transport model, the water vapour concentration

changes also because of the equilibrium rel ationshi p between the gas phase and sol id phase viithi n a particular node. Therefore, thi s condensation/evaporation process was disabled before a comparison with the analytical sol ution was made.

To disabl e the evaporation/condensati on process required reformulation and could

not be done by simply changing system input data. To reformulated the model for the purpose of checking the models mathematical correctness defeats the whole purpose of

the check. For this reason no compari son against existing models was carried out for the moisturc transport portion of thc transport model, and this was compared d i rectly against experi mental resul ts.

4.4.2 N U M E RI CAL ERROR C H ECKING

The fundamenta l principl e which f<mns the finite difference n umerical scheme, used

i n t h i s work 10 solve the coupled heat and moisture transport equation, is the

discretisation of both time and space, by cutting the continua of time and space i nto a series of time-steps and nodes, over which the properties o C the material arc averaged.

As the ti me-step approaches zero and the number of n odes approaches i n fi ni ty, the real conti nua are more closely m ode l l ed. This has the effect of rapidly increasing simulation time and i ntroduces rounding errors which accumulate in the calcul ated results. Some trade otT is therefore required so n umerical errors are at acceptabl e l evels and

simulation times are also sensible.

A series of simulations were run using typical values for system inp uts. The time­

step i n these runs was changed from 1 second to 1 0 seconds and the number of nodes in

a total slab thickness of 1 00 mm was changed i n the range 5 to 20 nodes. The temperature and relative humidity predictions for nodes at 0, 20, 40, 60, 80, and 1 00 m m were compared. For these positions the maximum temperature and relative humidity difference compared with the run with 20 nodes and a 1 second time step

were compared for each time-step. From these results it was shown that the error in using only 5 nodes lead to differences i n predictions of up to 20% RH than that

predicted using 20 nodes. The effect of time-step had less influence on predictions. This analysis showed that 20 nodes and a

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accurate predi ctions. The simulation time for these settings is of the order of 2 minutes per hour of real time simulated. This was acceptable for the investigation of caking carried out in further work.

4.4.3 M ATHS C H EC KING S U MMARY

The model describing the transport of heat and moisture through a slab of bulk lactose developed in this work was show'll to give accurate predictions when compared against existing solutions. This indicated that the implementation of the formulated model was performed correctly.

Checks on the level of numerical errors accumulating i n the solution of the model indicated that a minimum of twenty nodes and time steps of less than five seconds are required to obtain accurate predictions .