air flux (G) across a cross-sectional area normal to the flow of Ab

Fig. 18.6. Illustration of the various ways of expressing evaporative loss of water and evaporative heat removal (convective heat removal is not considered here). (a) In the case in which the solid and gas phases are treated as separate phases; (b) In the case in which the solid and gas phases are treated as a single pseudo-homogeneous phase air flux (G) across a cross-sectional area normal to the flow of Ab

Fig. 18.6. Illustration of the various ways of expressing evaporative loss of water and evaporative heat removal (convective heat removal is not considered here). (a) In the case in which the solid and gas phases are treated as separate phases; (b) In the case in which the solid and gas phases are treated as a single pseudo-homogeneous phase

We can therefore write the local rate of evaporationRevap (kg-H2O h-1) as:

As in the case of convective heat removal, it is common to combine the mass transfer coefficient and the area to obtain an overall transfer coefficient ("kA").

The local rate of heat removal from the solid phase by evaporation (Qevap, J h-1) is given by:

where A. is the enthalpy of vaporization of water (J kg-H2O-1).

Note that an isotherm can be used to order to write the driving force in terms of the water content. In this case the driving force for evaporation is the difference between the water content that the solid actually has (W, kg-water kg-dry solid-1) and the water content that it would have it were in equilibrium with the gas phase (Wsat, kg-water kg-dry solid-1). This affects the units used in the mass transfer coefficient. Chapter 22 will describe how the equation is written in this case.

At times the solids and air are assumed to be in moisture equilibrium, or, in other words, the air phase is assumed to be saturated with water at the temperature of the solids (this is the assumption of a pseudo-homogeneous bed). This has the consequence that the humidity at a particular position can be expressed as a function of the temperature at that position. In this case, it is not necessary to write equations describing solids-to-air water transfer and evaporative heat transfer, as these are subsumed within the terms that describe the water and heat removal associated with the flow of gas through the bed, as explained in the next section.

18.5.2 Water Removal Due to Air Flow Through the Bed

The flow of moist air through the bed typically leads to the removal of water from the bed (see Fig. 4.3). The overall rate of water removal (kg-water h-1) from the bed is:

where Hinht and Houtiet are the humidities (kg-water kg-dry-air-1) at the air inlet and outlet, respectively. G (kg-dry-air m-2 h-1) and Ab (m2) are as described in Sect. 18.4.3.

For an overall energy balance on a bioreactor, the rate of heat removal due to evaporation will then be:

However, as before, for beds that are not well mixed, it is often of more interest to write balances that allow the calculation of the temperature and humidity as functions of the position within the bed. In this case, a balance equation is written over a thin layer of the bed. The approach is different depending on whether the solids and gas phases are treated as different phases or are lumped together and treated as a pseudo-homogenous phase.

18.5.2.1 Solids and Gas Treated as Separate Phases

If the solids and gas are treated as separate phases, then the convective flow term within the mass balance equation for water will appear as:

dz since the humidity difference (kg-water kg-dry-air-1) between the inlet and outlet of this thin layer is simply the humidity gradient (dH/dz, kg-water kg-dry-air-1 m-1) multiplied by the thickness of the thin layer (Az, m).

Typically during the manipulations of the water balance equation, it will be divided through by the volume of the thin layer, such that the term will appear without containing Ab and Az. Note also that if the humidity at the outlet of the thin layer is higher than the humidity at the inlet of the thin layer, then the flow of air will be reducing the humidity of the thin layer, and therefore this term will be preceded by a negative sign if it appears on the right hand side of an equation such as Eq. (18.3). In fact, it is often put on the left hand side of the balance equation.

Note that evaporation removes energy from the solids and not from the air phase. Energy removal from the solids phase, which does not flow, has already been taken into account by Eq. (18.16). Therefore the energy balance on the air phase will not contain a term of the form of Eq. (18.19) multiplied by the enthalpy of evaporation.

18.5.2.2 Solids and Gas Treated as a Pseudo-Homogeneous Phase

When the assumption is made that the air is always saturated at the temperature of the solids (i.e., the assumption of a pseudo-homogeneous bed), the rate of evaporation (Revap, kg-H2O h-1) is still written in the form of Eq. (18.19). In this case the rate of evaporative heat removal is given by

This is not inconsistent with Sect. 18.5.2.1, since Eq. (18.16) is not used when the assumption of a pseudo-homogeneous bed is made. Further, even though evaporation removes the energy from the solids and not the gas, this makes no difference since the solids and gas are assumed to equilibrate immediately to the same temperature. The Antoine equation can be used to calculate the saturation humidity (Hsat) as a function of temperature, so it is useful to apply the chain rule of differentiation to cause the term "dHsat/dT' to appear explicitly in the equation:

dH sat dH sat dT

dz dT dz

An equation relating dHsaJdT to the temperature is developed in Sect. 19.4.1.

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