Can someone pleasee help me with this one? Thanks! In a diaphragm valve, a membr

Question

Can someone pleasee help me with this one? Thanks!

In a diaphragm valve, a membrane is partially lowered to impose a pressure drop, and thus control flow rale. Consider a 2D model of such a valve, with valve length L, and flow rate per unit thickness Q'. The membrane height at any point along the length of the valve is: H(x) = H0 - 4 delta x/L[1-x/L] The velocity profile may be assumed parabolic at any x position along the valve (for some alpha): V(x,y) = alpha y/h[1 - y/H] Assuming that all flow resistance is due to viscous shear on the lower wall and membrane, derive an expression for the pressure drop across this valve. You may assume that the membrane is parallel to the x-axis when calculating local shear (i.e. the membrane is only deformed shallowly). There is no need to evaluate this nasty integral.

Explanation / Answer

First, obtain the value of alpha in the velocity profile from,

integral 0 to H vdy= Q (total volume flow rate)

Now substitute this velocity profile in the first navier-stokes eqn,

u(du/dx) + v(du/dy)= (1/p)(dp/dx) +(u/p)(d^2 u/dx^2 +d^2 u/dy^2)

Here v=0 and u= v(x,y) (given velocity profile) (u=viscosity, p=density)

Solve in terms of dp/dx ,

Let,

Say it comes out as,

dp/dx= f(x) , f(x) is something that u will solve..

Now

delta p= integral from 0 to L f(x) dx

Substitute H as H(x) from the given profile....

u will easily get the answer..i did it and i got it....rate well and cmment if u have any probs, i will rply asap...

Related Questions

Navigate

• Browse (All)
• Browse C
• Subjects

---