For fully developed laminar flow through a parallel-plate

Chapter 8, Problem 8.5

(choose chapter or problem)

For fully developed laminar flow through a parallel-plate channel, the x-momentum equation has the formThe purpose of this problem is to develop expressionsfor the velocity distribution and pressure gradient anal-ogous to those for the circular tube in Section 8.1.(a) Show that the velocity profile, u(y), is parabolicand of the formwhere is the mean velocityum??a212??dpdx? nd?dp/dx??p/L, where ?pis the pressure dropacross the channel of length L.(b) Write an expression defining the friction factor, f,using the hydraulic diameter Dhas the characteris-tic length. What is the hydraulic diameter for theparallel-plate channel?(c) The friction factor is estimated from the expression, where Cdepends upon the flow crosssection, as shown in Table 8.1. What is the coeffi-cient Cfor the parallel-plate channel?(d) Airflow in a parallel-plate channel with a separationof 5 mm and a length of 200 mm experiences a pres-sure drop of ?p?3.75 N/m2. Calculate the meanvelocity and the Reynolds number for air at atmos-pheric pressure and 300 K. Is the assumption of fullydeveloped flow reasonable for this application? Ifnot, what is the effect on the estimate for um?