t . S + 1 (4.7)
Equation (4.7) is of the typical fonn for a first order system. The first order time constant (1:) for the system is given by Eq (4. 8).
Ps
(4.8)
If standard lactose properties are used and a surface mass transfer coefficient
(kg)
of 1 .8 x 1 0-6 s/m (from Sherwood number(
Nsh)
of 2 for negligible flow past a particle ofaverage size, McCabe and Smith 1 984) the first order time constant ('r) is of the order of less than a second. This shows that the sorption process occurs very quickly and the assumption that local moisture equilibrium between the solid and the gas phases is valid.
4 . 2 . 3 . 3 N EGLIGIBLE CONVECTION
Natural convection can be neglected if the Rayleigh number is less than 4n:2 (Bear and Bachmat, 1 99 1 ).
Pa.g.a.K.L.!J. T
A. 1l (4.9)
If convection is unimportant i n the worst case situation that occurs in i ndustry, then convection will not play an active role i n the transport of moisture and heat in the simplified one dimensional case. The worst case scenario consists of lactose powder, recently packaged into a I m' cubic bulk bag placed in a store room.
The properties for l actose i n thi s situation are: Ct 1 /298 K ( liT)
j1 1 . 8 x 1 0.5 Pa.s
LlT
,-
20cC ( i m mediately after bagging) r I m (dimension of a bulk bag) K 1 . 98 x 1 0'7 m2 (Hodges 1 992)This gave N/?a = 0.05, which is m uch less than the critical Rayleigh number (4rrC) and therefore natural convect ion is not sign i ficant in the transport of heat and moisture i n bulk l actose under the types o f conditions under consideration.
4 . 2 . 3 . 4 N EGLIGIBLE H EAT AND MOISTU R E TRANSPO RT D U E To C HANGES IN AIR D E NS ITY
Upon heating a node of fixed volume, the air density will decrease and air will flow out of that fixed volume carrying heat and moisture with it. Because of this, a heat and moisture transport term due to air expansion and contraction with changing temperature should be incl uded i n the model. It was assumed that the amount of heat and moisture transported in this way was negligible. Having made this assumption, rather than making a further assumption that air density was constant i n the temperature range of
\,
i nterest ( 1 5 - 40°C), i t was desirable to al low it to change with temperature. As a result, i f a i r density i s all owed to change and no effort m ade to account for the air that expands over the boundaries of the n ode of fixed vol ume, then moisture and dry air balances over the whole system wil l resul t in a net l oss of air and moisture. Some heat wil l be associaled with the air l ost from the system and therefore a total heat balance over the system wil l al so show a discrepancy. Quantification of the amount of moisture and heat l ost from the system was required to show how significant this effect was.
The dri v i ng force for moisture diffusion in the porous system was expressed as a water vapour concent ration (kg water vapour! m; dry air) and absolute humidity was
back calculated from water vapour concentration. Thi s means the only way the total amount or moisture in a node of fixed volume coul d change was by the processes
modell ed (diffusion and condensation) For this reason no moist ure was overlooked by the model and the tota l moisture in the system i s accounted for. The balance equation J()f moisture in the system was \vritten i n temlS of a moisture concentration (kg water vapour/ m3 air) and only dry air was l ost.
The effect of thermal expansion at heated parts of a porous slab and contraction at parts of the slab undergoing coo l ing, wil l promote extra moisture movement in the direction of the temperature gradient (hot to cold).
Quantification of the sign i fi cance of heat l osses and gains in dry air to the system by thermal expansion and contraction was most easily achieved by checki ng the eiTeel of the assumpt ion that they are negl igible, after a working simulation was compl eted. Once the model was form ulated, and numericall y solved, a running total o f the dry air lost, along with the heat lost with this air, was made using a simple dry air m ass balance at each time step. Thi s heat loss term was then be compared with the total heat entering over the boundaries. The results of this test are discussed in Section 4 . 5 . 2 .
4 . 2 . 3 . 5 MODE OF MOISTU R E MOVEMENT:
Other modes by which moisture transfer in the lactose bed can occur are surface diffusion, capillary action and ditIusion through the lactose particles themsel ves. The structure of the l actose crystal is very ordered and contains few spaces in the crystalline matri x where free moisture can be present. Because of this, diffusion through the particle is not significant.
Because the free moisture in the lactose system exists on the surface of the crystal particles, a thin fi lm of saturated lactose syrup i s present. This fi lm is likely to be concentrated at contact points between adjacent particles due to surface tension forces. It can be seen from the moisture sorption isotherm that the amount of free moisture present in atmospheres of l ess than
95%
relative humidity is very small, consisting of only a few mono-layers thick. It was concl uded that flow of moisture by surface diffusion would be insignificant.4 . 2 . 3 . 6 OTHE R ASSUMPTIONS MADE
All other assumptions made are reasonable assumptions necessary to enable a simple model formulation.
4.2.4 MATHEMATICAL FORMULATION