Chapter 16 Simultaneous heat and mass transport (Material presented in this chapter are based on those in Chapters 20, �Diffusion mass transfer in fluid systems� second edition by EL Cussler) Processes that involve simultaneous heat and mass transfer are plenty and commonly found in nature. Cooling towers is one such example. In this chapter, development of the heat and mass transfer relations for cooling towers will be discussed.
16.1
Design of cooling towers
Cooling towers are a very economical way to cool large quantities of water. The tower is packed with inert materials (Fig. 16.1). Hot water is sprayed into the top of the tower. Sprayed hot water trickles through the inert material. While flowing through the inert material, hot water evaporates. Air enters the bottom of the tower and rise through the packing. Depending on the size of the tower, forced convection may be introduced by pumping air or the tower may experience only free or natural convection. Typical question in cooling towers is to estimate the height of the tower given the amount of water to be cooled. A steady state differential mass balance for the conservation of water 197
198 CHAPTER 16. SIMULTANEOUS HEAT AND MASS TRANSPORT
Figure 16.1: Design of cooling towers vapor is given by GyH2 0 |z − GyH2 0 |z+dz + (∆za)k(CH2 0,i − CH2 O ) = 0 d(GyH2 0 ) ⇒− + ka(CH2 0,i − CH2 O ) = 0 (16.1) dz where, y is the mole fraction, a is the surface area per unit volume, G is the molar flux of wet air. As CyH2 0 = CH2 0 where C is the total concentration and if G ≈ na ir = molar flux of dry air, which is valid under dilute limit, then Eq. (16.1) can be written as −nair
dyH2 0 + kaC(yH2 0,i − yH2 0 ) = 0 dz
(16.2)
Next a steady state differential energy balance for the wet air is given by −nair C˜p,air
dTair + ha(Ti − Tair ) = 0 dz
(16.3)
and for water, the energy balance, that is, energy lost from water is gained by the air, is ˜ dTH2 0 dH −nH2 0 C˜p,H2 0 − na ir =0 (16.4) dz dz
16.1. DESIGN OF COOLING TOWERS
199
As kH2 0 kair and Ti ≈ TH2 0 , multiplying Eq. (16.2) with ∆Hvap and adding to Eq. (16.3) gives − nair ∆Hvap
dyH2 0 + kaC∆Hvap (yH2 0,i − yH2 0 )+ dz dTair ˜ −nair Cp,air + ha(Ti − Tair ) = 0 dz i d h˜ ⇒ nair Cp,air Tair + ∆Hvap yH2 0 = dz kaC∆Hvap (yH2 0,i − yH2 0 ) + ha(TH2 0 − Tair ) (16.5)
From Chilton-Colburn analogy, ν 23 h k ν 23 = v D ρCˆp,air v α where Cˆp,air is defined based on the mass density. For gases, h = ρCˆp,air k = C C˜p,air k
(16.6) α D
≈ 1. Thus, (16.7)
where, C˜p,air is defined in terms of the concentration. Using Eq. (16.7), Eq. (16.5) can be re-written as nair
˜ dH ˜ i − H) ˜ = kaC(H dz
(16.8)
˜ = C˜p,air Tair + ∆H ˜ vap yH2 0 , which signifies the enthalpy of wet air where H ˜ i = C˜p,air TH2 0 + ∆H ˜ vap yH2 0,i , which signifies the per mole of dry air and H enthaly of wet air/per mole of dry air at the interface. Expression in Eq. (16.8) can be intergrated to obtain the height of the tower: H Zl Z˜ out ˜ nair dH l = dz = (16.9) ˜i − H ˜ kaC H 0
˜ in H