Page 232 - Modelling in Transport Phenomena A Conceptual Approach

212 CHAPTER 7. UNSTEADY-STATE MACROSCOPIC BALANCES

t
Cooling
air + Solid product
(-1 L)
c

Figure 7.3 Schematic diagram of a spray cooling tower.

It should be remembered that mathematical modeling is a highly interactive
process. It is customary to build the initial model as simple as possible by making
assumptions. Experience gained in working through this simplified model gives a
feeling and confidence for the problem. The process is repeated several times, each
time relaxing one of the assumptions and thus making the model more realistic. In
the design procedure presented below, the following assumptions are made:
1. The particle falls at a constant terminal velocity.

2. Energy losses from the tower are negligible.
3. Particles do not shrink or expand during solidification, i.e., solid and melt
densities are almost the same.
4. The temperature of the melt particle is uniform at any instant, i.e., Bi << 1.
5. The physical properties are independent of temperature.
6. Solid particles at the bottom of the tower are at a temperature T,, the solid-
ification temperature.

7.6.1 Determination of Tower Diameter

The mass flow rate of air can be calculated from the energy balance around the
tower:
Rate of energy Rate of energy lost
( gained by air ) = ( by the melt particles ) (7.61)
or,
ha(ep,,) [(~a),,t - (T)J = hrn { ePrn [(Trn)i,, - GI + A} (7.6-2)

where the subscripts a and m stand for the air and the melt particle, respectively,
and is the latent heat of fusion per unit mass.