Page 177 - Industrial Ventilation Design Guidebook

4.3 HEAT AND MASS TRANSFER 1 3 9

for the mixture of the material and the gas, and therefore the pressure must be
calculated for the surface area of the total material <j>p. When <f> is held con-
stant, independent of x, Eq. (4,302) is obtained.
In a steadv-state case. Ea. (4.302) is simolified to

and with Eq. (4.301),

with v — r\/d = rf(p/p. If the flow velocity is zero, Eq. (4.303) can be inter-
preted as saying that the resistance force is linearly proportional to the veloc-
ity difference between the gas and the material and also linearly proportional
to the dynamic viscosity of the gas.
Equation (4.303) is valid but it is lacking something. The resistance force
fnx that applies to the component A has to be found, and not that for the
whole mixture. The force applying to the whole mixture f mx is the sum of the
partial forces f^ x and f^ x :

Assuming that the force is divided along the ratio of the mass flows, Eq.
(4.303) gives

Summing (4.305a) and (4.305b), f mx is obtained for Eq. (4.303). This is due to
the fact that p Av Ax + p Bv Bx = pu x.
We now consider the resistance force f mx caused by the diffusion. This
force resists the diffusion flow in a porous material together with f^. Writing
the linear momentum equation for component A in accordance with Eq. (4.302),

This gives a model for f^ x , Eq. (4.305b), but not a model for force /j^. While
force f mx gives the flow force caused by the material, it is normal to represent this
fact so that fj x gives the pure diffusion resistance force that is not caused by the
material. This requires treating /j^ independently from the material or porosity.
For {£ = 1 or k = °o, where f^= 0, Eq. (4.306) gives

In a steady-state case the results are

In a steady-state case at constant pressure (p = p A +p B — constant), Pick's
law (Eq. (4.273)) is valid: