The velocity profile in fully developed laminar flow in a circular pipe of inner radius R 5 2 cm, in m/s, is given by u(r) 5 4(1 2 r2/R2). D

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The velocity profile in fully developed laminar flow in a circular pipe of inner radius R 5 2 cm, in m/s, is given by u(r) 5 4(1 2 r2/R2). Determine the average and maximum velocities in the pipe and the volume flow rate.

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Thu Thảo 5 months 2021-08-10T20:11:46+00:00 1 Answers 192 views 0

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    2021-08-10T20:13:40+00:00

    The question is not clear and the complete clear question is;

    The velocity profile in fully developed laminar flow In a circular pipe of inner radius R = 2 cm, in m/s, is given By u(r) = 4(1 – r²/R²). Determine the average and maximum Velocities in the pipe and the volume flow rate.

    Answer:

    A) V_max = 4 m/s

    B) V_avg = 2 m/s

    C) Flow rate = 0.00251 m³/s

    Explanation:

    A) We are given that;

    u(r) = 4(1 – (r²/R²))

    To obtain the maximum velocity, let’s apply the maximum condition for a single-variable continual real valued problem to obtain;

    (d/dr)(u(r)) = 0

    Thus,

    (d/dr)•4(1 – (r²/R²)) = 0

    4(d/dr)(1 – (r²/R²)) = 0

    If we differentiate, we have;

    4(0 – (2r/R²)) = 0

    -8r/R² = 0

    Thus, r = 0 and with that, the maximum velocity is at the centre of the pipe.

    Thus, for maximum velocity, let’s put 0 for r in the U(r) function.

    Thus,

    V_max = 4(1 – 0²/R²) = 4 – 0 = 4 m/s

    B) Average velocity is given by;

    V_avg = V_max/2

    V_avg = 4/2 = 2 m/s

    C) the flow can be calculated from;

    Flow rate ΔV = A•V_avg

    A is area = πr²

    From question, r = 2cm = 0.02m

    A = π x 0.02²

    Hence,

    ΔV = π x 0.02² x 2 = 0.00251 m³/s

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