To measure the mass flow rate of a fluid in a laminar flow through a circular pipe, a hot wire type velocity meter is placed in the center of the pipe. Assuming that the measuring station is far from the entrance of the pipe, the velocity distribution is parabolic where Umax is the centerline velocity (r = 0), r is the radial distance from the pipe centerline, D is the pipe diameter.

Question

(a) Derive an expression for the average fluid velocity at the cross-section in terms of

Umax and D.

(b) Obtain an expression for the mass flow rate.

(c) If the fluid is mercury at 30°C, D = 10 cm, and the measured value of Umax is 0.2

cm/s, calculate the mass flow rate from the measurement.

GIVEN

• Fully developed flow of mercury through a circular pipe

• Parabolic velocity distribution: u(r)/Umax = 1 – (2r/D)2

• Mercury temperature (T) = 30°C

• Pipe diameter (D) = 10 cm = 0.1 m

• Measured center velocity (Umax) = 0.2 cm/s = 0.002 m/s

FIND (a) An expression for the average fluid velocity (u )

(b) An expression for the mass flow rate (m )

c) The value of the mass flow rate (m )

SKETCH

Answer 1

(a) The average fluid velocity is calculated as follows





(b) The mass flow rate is given by



(c) Inserting the values of these quantities into this expression