For fully developed laminar flow through a parallelplate

Chapter , Problem 8.5

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For fully developed laminar flow through a parallelplate channel, the x-momentum equation has the form The purpose of this problem is to develop expressions for the velocity distribution and pressure gradient analogous to those for the circular tube in Section 8.1. (a) Show that the velocity profile, u(y), is parabolic and of the form where is the mean velocity um a2 12 dp dx um u(y) 3 2 um 1 y2 (a/2) 2 d and dp/dx p/L, where p is the pressure drop across the channel of length L. (b) Write an expression defining the friction factor, f, using the hydraulic diameter Dh as the characteristic length. What is the hydraulic diameter for the parallel-plate channel? (c) The friction factor is estimated from the expression , where C depends upon the flow cross section, as shown in Table 8.1. What is the coeffi- cient C for the parallel-plate channel? (d) Airflow in a parallel-plate channel with a separation of 5 mm and a length of 200 mm experiences a pressure drop of p 3.75 N/m2 . Calculate the mean velocity and the Reynolds number for air at atmospheric pressure and 300 K. Is the assumption of fully developed flow reasonable for this application? If not, what is the effect on the estimate for um?