Chapter 8, Problem 8.5P

For fully developed laminar flow through a parallel-plate channel, the x-momentum equation has the form

   $\mu \left(\frac{d^{2}u}{d{y}^2}\right)=\frac{dp}{dx}=\text{constant}$

The purpose of this problem is (o develop expressionsfor the velocity distribution and pressure gradient analogous to those for the circular tube in Section 8. I.

(a) Show that the velocity profile, $u(y)$ , is parabolicand of the form

   $u(y)=\frac{3}{2}{u}_m\left[1-\frac{y^2}{{\left(a/2\right)}^2}\right]$

where $u_m$ is the mean velocity

   $u_m=-\frac{a^2}{12\mu }\left(\frac{dp}{dx}\right)$

and $-dp/dx=\Delta p/L$ , where $\Delta p$ is the pressure drop across the channel of length L.

(b) Write an expression defining the friction factor, f, using the hydraulic diameter $D_h$ as the characteristic length. What is the hydraulic diameter for the parallel—plate channel?

(c) The friction factor is estimated from the expression $f=C/{\mathrm{Re}}_{Dh}$ , where C depends upon the flow cross section, as shown in Table 8.1. What is the coefficient 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 $\Delta p=3.75{\text{N/m}}^2$ . Calculate the mean velocity and the Reynolds number for air al 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 $u_m$ ?