Lesson 6 - Principles of Fluid Mechanics 3

The means by which we quantify pressure loss through various fittings involves the use of experimentally determined values, termed ‘loss coefficients’, which are applied to a kinetic energy term for the flow.

The approach we shall use to determine the magnitude of the pressure drop through a fitting is one involving the concept of the loss coefficient k. We use a series of experimentally derived multiplying factors (k) which represent the number of velocity heads which are lost through the fitting. So firstly, let us deal with what a velocity head is. To do that, we will revise Bernoulli’s equation which we considered in a previous lesson.

Revisiting Bernoulli’s equation

Bernoulli’s equation proposes that the three forms of energy – gravitational potential, kinetic and pressure energies – would always be conserved so that they would sum to a constant value:

The equations for quantifying these three forms of energy are given in the following expression:

Where ρ is the fluid density, h is the vertical height of the fluid, u is the fluid velocity and P is the fluid pressure.

(Strictly speaking, this equation assumes a fixed temperature, and no gain or loss of heat or work).

We have previously applied this in a number of brewing situations. But, now let us pay particular attention to the kinetic energy term, which is given by:

Remember that u is the average fluid velocity (m/s).

Let us remind ourselves of two things we previously considered:

1. We discussed the kinetic energy of a moving fluid to relate to the energy a fluid possesses when the fluid is moving with a velocity u. The higher the fluid velocity, the greater the quantity of kinetic energy.

2. The concept of ‘head’ can be used to relate a pressure to the equivalent value at the base of a column of fluid of height h (m) with density ρ. So, a pressure P would be equivalent to a head of P/ρg.

Now that we are armed with some Bernoulli’s equation concepts fresh in our minds, what is a velocity head? Well simply put, a velocity head is the kinetic energy of a fluid, expressed as a head. (You can think of it as a ‘kinetic energy head’ if you wish). This takes the form of:

We should note that in the equation above, we cancelled out the common ρ on the top and bottom of the equation.

Let us look at a worked example using the concept of velocity head.

Example:

What would be the pressure equivalent of one velocity head for water flowing with an average velocity of 1 m/s (density = 1,000 kg/m³).

As velocity u increases, and hence the kinetic energy of the flow increases, then the velocity head would also increase.

Let us now see how we use the concept of velocity head in determining the pressure (or head) loss in a flow due to a fitting.