Consider a steady-state slug flow (i.e., flow with uniform velocity equal to U) of an incompressible and constant-property fluid in a flat channel (see Fig. P4.36). The system is isothermal. Assume...

Consider a steady-state slug flow (i.e., flow with uniform velocity equal to U) of an incompressible and constant-property fluid in a flat channel (see Fig. P4.36). The system is isothermal. Assume that the walls of the channel contain a slightly soluble substance, so that downstream from location x = 0, a species designated by subscript 1 diffuses into the fluid. The boundary condition downstream the location where x = 0 is thus UWM (i.e., m₁
= m_1,s
at surface for x ≥ 0), whereas upstream from that location the concentration of the transferred species is uniform and equal to m_1,in
( m₁
= m_1,in
for x ≤ 0 and all y). Assume that the diffusion of the transferred species in the fluid follows Fick’s law.

(a) Derive the relevant conservation equations and simplify them for the given system.

(b) Prove that, for
 b, where b is defined in Fig. P4.36, the local and average mass transfer coefficients can be found from