Question

I need a partial differential EQUATION to govern the pressure change for a steady state fluid inside a horizontal pipe.

please, make some clarafication if possible.

thanks.

Answer 1

For a horizontal pipe(dz/dx = 0).

Hence, it follows from p = g (H - z) ,where p is the pressure intensity, that p/t = g (H/t) and p/x = g (H/x), where z = elevation of the pipe centerline above the specified datum and piezometric head, H, above a specified datum and the discharge, Q, flow velocity V . Now, Q = V A

By substituting these relationships into

------------------------(i)

-------------------------(ii)

we obtain

-----------------(iii)

------------------------(iv)

Now, Steady-state equations corresponding to Eqs. (iii) and (iv) may be obtained by substituting H/t = 0 and

Q/t = 0. Hence, it follows from Eq. (iii) that Q/x = 0; i.e., Q is constant along the pipe length.

Substituting Q/t = 0 into Eq. (iv), simplifying the resulting equation, and writing it in a finite-difference form, we obtain

---------------------------------------(v)

where H = head loss in pipe length x for a flow of Q.

Note that this equation is the same as the Darcy-Weisbach friction equation