Question

The wall shear stress in a fully developed flow portion of a 12-in.-diameter pipe carrying water is 2.00 lb/ft^2. Determine the pressure gradient, ∂p/∂x, where x is in the flow direction, if the pipe is: a. Horizontal b. Vertical with flow up. c. Vertical with flow down.

Answer 1

Answer:

a) -8 lb / ft^3

b) -70.4 lb / ft^3

c) 54.4 lb / ft^3

Explanation:

Given:

- Diameter of pipe D = 12 in

- Shear stress t = 2.0 lb/ft^2

- y = 62.4 lb / ft^3

Find pressure gradient dP / dx when:

a) x is in horizontal flow direction

b) Vertical flow up

c) vertical flow down

Solution:

- dP / dx as function of shear stress and radial distance r:

                      (dP - y*L*sin(Q))/ L = 2*t / r

                      dP / L - y*sin(Q) = 2*t / r

Where            dP / L = - dP/dx,

                      dP / dx = -2*t / r - y*sin(Q)

Where            r = D /2 ,

                      dP / dx = -4*t / D - y*sin(Q)

a) Horizontal Pipe Q = 0

Hence,           dP / dx = -4*2 / 1 - 62.4*sin(0)

                      dP / dx = -8 + 0

                      dP/dx   = -8 lb / ft^3

b) Vertical pipe flow up Q = pi/2

Hence,           dP / dx = -4*2 / 1 - 62.4*sin(pi/2)

                      dP / dx = 8 - 62.4

                      dP/dx   = -70.4 lb / ft^3

c) Vertical flow down Q = -pi/2

Hence,           dP / dx = -4*2 / 1 - 62.4*sin(-pi/2)

                      dP / dx = -8 + 62.4

                      dP/dx   = 54.4 lb / ft^3