Flow Rate and Mean Velocity

Volume flow rate - Discharge.



More commonly we need to know the volume flow rate - this is more commonly know as discharge. (It is also commonly, but inaccurately, simply called flow rate). The symbol normally used for discharge is Q. The discharge is the volume of fluid flowing per unit time. Multiplying this by the density of the fluid gives us the mass flow rate. Consequently, if the density of the fluid in the above example is 850then:

Mass flow rate


If we want to measure the rate at which water is flowing along a pipe. A very simple way of doing this is to catch all the water coming out of the pipe in a bucket over a fixed time period. Measuring the weight of the water in the bucket and dividing this by the time taken to collect this water gives a rate of accumulation of mass. This is know as the mass flow rate.
For example an empty bucket weighs 2.0kg. After 7 seconds of collecting water the bucket weighs 8.0kg, then:



Performing a similar calculation, if we know the mass flow is 1.7kg/s, how long will it take to fill a container with 8kg of fluid?





Discharge and mean velocity

If we know the size of a pipe, and we know the discharge, we can deduce the mean velocity




Discharge in a pipe
If the area of cross section of the pipe at point X is A, and the mean velocity here is . During a time t, a cylinder of fluid will pass point X with a volume At. The volume per unit time (the discharge) will thus be



So if the cross-section area, A, is  and the discharge, Q is , then the mean velocity, , of the fluid is



Note how carefully we have called this the mean velocity. This is because the velocity in the pipe is not constant across the cross section. Crossing the centreline of the pipe, the velocity is zero at the walls increasing to a maximum at the centre then decreasing symmetrically to the other wall. This variation across the section is known as the velocity profile or distribution. A typical one is shown in the figure below.




A typical velocity profile across a pipe
This idea, that mean velocity multiplied by the area gives the discharge, applies to all situations - not just pipe flow.



Example


Calculating Volume from Flow Rate: The Heart Pumps a Lot of Blood in A Lifetime

How many cubic meters of blood does the heart pump in a 75-year lifetime, assuming the average flow rate is
5.00 L/min?

STRATEGY

Time and flow rate Q are given, and so the volume V can be calculated from the definition of flow rate.

SOLUTION

Solving Q=V/t for volume gives

V=Qt

DISCUSSION

This amount is about 200,000 tons of blood. For comparison, this value is equivalent to about 200 times the volume of water contained in a 6-lane 50-m lap pool.

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