Advances in Heat Transfer (eBook)

V Volume-Averaged Transport Equations: Mass

In this section we present the volume-averaged form of the continuity equations for the solid, liquid, and gas phases. The results for the solid and liquid phases are straightforward, as is the total continuity equation for the gas phase; however, the continuity equation for water vapor is complex and requires the solution of a closure problem in order to obtain the final form.

A SOLID PHASE

Since the σ-phase is a rigid, solid phase, the continuity equation is replaced by Eq. (105) and the local volume-averaged equivalent is given by

σσ=constant

Here the intrinsic average density is defined according to Eq. (46).

B LIQUID PHASE

In this case we begin with Eq. (110),

ρβ∂t+∇⋅ρβvβ=0

and form the superficial average to obtain

ρβ∂t+∇⋅ρβvβ=0

We can use the general transport theorem, in the form given by Eq. (56), to express the first term in Eq. (139) as

ρβ∂t=∂∂β∂t−1V∫Aβγtρβw⋅nβγdA−1V∫Aβσρβw⋅nβσdA

Since the speed of displacement of the β-σ interface is zero, the last term in this result is zero and Eq. (139) takes the form

ρβ∂t−1V∫Aβγtρβw⋅nβγdA+∇⋅ρβvβ=0

Integration and differentiation in the last term in Eq. (141) can be interchanged using the spacial averaging theorem given by Eq. (53). This leads to

⋅ρβvβ=∇⋅ρβvβ+1V∫Aβγtnβγ.ρβvβdA+1V∫Aβσtnβγ.ρβvβdA

Since the β-σ interface is impermeable, nβσ · vβ is zero, and we can use this condition to express Eq. (141) as

ρβ∂t⏟accumulation+∇⋅ρβvβ⏟convection+1V∫Aβγtρβvβ−w⋅nβγdA⏟massrateofevaporation=0

It is quite reasonable to neglect variations of the density, ρβ , and this allows us to use Eq. (47) to express the superficial average density as

β=εβρββ=εβρβ

Substitution of this result into Eq. (143) provides

∂tρβεβ+∇⋅ρβvβ+1V∫Aβγtρβvβ−w⋅nβγdA⏟massrateofevaporation=0

Since we are treating the liquid-phase density as a constant, we could remove ρβ from both the time and space derivatives in Eq. (145); however, it is convenient to retain the conservative form of Eq. (145).

The mass rate of evaporation represents a crucial term in any theory of drying, and we identify this term as

˙=1V∫Aβγtρβvβ−w⋅nβγdA

With this representation, our volume-averaged continuity equation for the liquid phase becomes

∂tρβεβ+∇⋅ρβvβ+m˙=0

and we are ready to move on to the gas-phase continuity equation. Before doing so, we should remind the reader that when we assume that the density can be treated as a constant, we are specifying the dependent variable in one of our governing differential equations. This means that Eq. (138) is no longer the governing differential equation for ρβ , and it becomes the constraining equation for pβ in Eq. (111). Because of this, we are forced to discard the equation of state [Eq. (39)] for the pressure, pβ, as is commonly done for the solution of incompressible flows.

C GAS PHASE: TOTAL

In this case we begin with the total continuity equation given by Eq. (114),

ργ∂t+∇⋅ργvγ=0

and repeat the procedure leading to Eq. (143). This provides us with

ργ∂t+∇⋅ργvγ+1V∫Aγβtρβvβ−w⋅nγβdA=0

however, we cannot simplify this result by neglecting variations in the density ργ. We first attack the accumulation term by expressing the superficial average density as

γ=εγργγ

and this leads to

∂tεγργγ+∇⋅ργvγ+1V∫Aγβtργvγ−w⋅nγβdA=0

We deal with the convective transport term by using the spatial decomposition represented by Eq. (57),

γ=ργγ+ρ˜γvγ=vγγ+v˜γ

and then follow the analysis of Carbonell and Whitaker [5] to obtain

∂tεγργγ+∇⋅ρ˜γv˜γ+1V∫Aγβtργvγ−w⋅nγβdA=0

Here we should note that the convective transport has been expressed in terms of the superficial average velocity according to the following analysis:

γγvγγ=1ργγvγγ=εγργγvγγ=ργγvγ

The quantity ˜γv˜γ represents the dispersive flux of mass, and while dispersion is often considered to be important in the species continuity equation, it is generally neglected in the total continuity equation, and we will follow that practice here. This allows us to express Eq. (153) as

∂tεγργγ+∇⋅ργγvγ+1V∫Aγβtργvγ−w⋅nγβdA=0

On the basis of the definition of the mass rate of evaporation given by Eq. (146) and the jump condition given by Eq. (128), we can express Eq. (155) as

∂tεγργγ+∇⋅ργγvγ−m˙=0

We are now ready to attack the gas-phase continuity equation for water.

D GAS PHASE: WATER VAPOR

The gas-phase continuity equation for water is of crucial importance, since there are many drying processes that are controlled by vapor-phase moisture transport. We begin with Eq. (113a) and form the volume average to obtain

ρAγ∂t+∇⋅ρAγvAγ=0water

We can repeat the development given by Eqs. (148) through (151) to obtain

∂tεγρAγγ+∇⋅ρAγvAγ+1V∫AγβtρAγvAγ−w⋅nγβdA=0

The jump condition given by Eq. (126) allows us to make use of the mass rate of evaporation to express this result as

∂tεγρAγγ+∇⋅ρAγvAγ−m˙=0

At this point, the manner in which we treat the continuity equation for the water vapor depends on the drying process under consideration. If the gas-phase mass average velocity is to be determined by Darcy's law, we decompose the species velocity according to

Aγ=vγ+uAγ

and then use Fick's law to express the flux of species A according...