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17 March, 04:43

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.

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  1. 17 March, 05:19
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    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
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