35 points

Consider a flow loop constructed using cylindrical tubes of circular cross-section as shown in the

figure below. Flow is being driven using a pump that drives liquid of density p through a Reducer

section. The pump inflow tube has a cross-sectional diameter Do; and the reducer geometry

decreases the diameter to Do, where is a constant number 1.0. Two pressure ports are

included in the loop to measure pressure before and after the reducer segment, and an inverted

U-tube manometer with manometer liquid of density Pm is connected to measure the differential

pressure. Both the operating liquid in the loop, and the manometer liquid can be assumed to be

incompressible. All viscous effects can be assumed to be negligible.

Problem 2

FLOW RATE

Ih

ID,

Do

Part a: Assume that in this problem, only the pump flow rate Q going in to the reducer is known./nIt can be shown that the flow rate Q for flow with uniform velocity U flowing through a plane

surface of area A can be written as: Q=UA. Use this expression to find the flow velocity of the

operating fluid at the reducer inlet.

Part b: Using an elementary version of mass balance for incompressible fluids, obtain the flow

velocity at the reducer outlet.

Part c: In terms of the reducer parameter as described in the problem statement above, quantify

the change in: (1) flow velocity; and (2) flow rate across the reducer.

Part d: Obtain an estimate for the differential manometer reading (h as shown in the figure), in

terms of all provided problem variables. In order to accomplish this, select a reference height along

the manometer tubes, and equate the pressure on left and right tubes of the manometer at that

chosen reference. Hint: Note that this manometer measures the pressure P as it appears in the

Bernoulli's equation, and it is safe to assume that pressure does not vary across the cross-section

of the tubes.

Part e: Is the pressure higher or lower when measured from the smaller tube? Is the sketch above

correctly representing the pressure differential?