## Flow rate.

**Introduction: 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 known as the mass flow rate.

**Volume flow rate Discharge:** More commonly we need to know the volume flow rate - this is more commonly known 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 850 kg m^{3}then:

An important aside about units should be made here: As has already been stressed, we must always use a consistent set of units when applying values to equations. It would make sense therefore to always quote the values in this consistent set. This set of units will be the SI units. Unfortunately, and this is the case above, these actual practical values are very small or very large (0.001008m^{3 }/s is very small). These numbers are difficult to imagine physically. In these cases it is useful to use derived units, and in the case above the useful derived unit is the liter. (1 liter = 1.0 × 10 -3m^{3}). So the solution becomes 1.008l / s. It is far easier to imagine 1 liter than 1.0 × 10-^{3}m^{3}. Units must always be checked, and converted if necessary to a consistent set before using in an equation.

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

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

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 center line of the pipe, the velocity is zero at the walls increasing to a maximum at the center 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.