M15.4 Pumping and Piping Systems

Consider the pumping and piping system shown in Figure M15-6. The pump is taking suction on the surge tank and pumping a liquid through a chemical process system (perhaps a heat exchanger and a reactor).

A typical pump head curve is shown in Figure M15-7. The pump head is a direct measure of the pressure increase between the suction and discharge of the pump that can be obtained as a function of the flow rate through the pump.

where h_p is the pump head in feet. The process will operate at the intersection of the pump head and system head curves, as shown in the following discussion.

Assume that the pressures P_a and P₄ are known. P_a is the ambient (or atmospheric) pressure and P₄ is the pressure at some known point downstream. If the height of liquid in the tank is known, then P₀ can be calculated for any given flow rate as

Equation M15.19

where h is the height of fluid between the tank level and the suction side of the pump and DP_sp is the pressure loss due to friction in the suction piping. The pressure drop is generally proportional to the square of the volumetric flow rate.

The pump discharge pressure, P₁, can be determined either using P₀ and the pump head curve as a function of flow rate,

Equation M15.20

or using the pressure drop through the discharge piping system and the final pressure, P₄.

Using the second method, we can basically determine the system head as a combination of the pressure drop through the chemical process (DP_cp) and across the orifice plate (DP_op).

Equation M15.21

The pump discharge pressure can be found as

Equation M15.22

Obviously, the problem solution is where P₁ from Equation (M15.20) is equal to P₁ from Equation (M15.22).

Numerical Example

Consider the pump head curve shown in Figure M15-7. Assume that the maximum flow rate is 240 gpm. The pump head at this flow rate is 76 feet of water. Now, let's make some assumptions about the other pressures in the system. First, we know that P₁ calculated from Equation (M15.20) is equal to P₁ from Equation (M15.22), or

Equation M15.23

For this particular example, assume that P₄ = P₀, so

Equation M15.24

At the maximum flow rate of 240 gpm, assume that the system head is

so the valve pressure drop must be

We can easily determine the system head at any other flow rate (F₂) as

Equation M15.25

That is, .

The system head is compared with the pump head curve in Figure M15-8. Notice that in practice, the pressure drop can be predicted by knowing the pipe diameter, the number of elbows and fitting, the pipe roughness, and so forth, and by calculating an overall friction factor. The pressure drop is a function of the square of the flow rate, so we are assuming that the pressure drop has been calculated or measured at a particular flow rate and we simply use the ratio of the square of the flow rates to determine the pressure drop at any other flow rate. The difference between the pump head and the system head is the pressure drop across the valve.

The relationship for flow through a valve is

Equation M15.26

Now assuming that the valve is wide open at the maximum flow rate of 240 gpm (x = f(x) = 1) and the fluid is water, we can calculate the valve coefficient as

Notice that the unit of pressure drop that we are using is feet of water.

Then, for a given flow rate and valve pressure drop, we can calculate the fraction that the valve must be open. That is,

And for a linear valve, the fraction that the valve is open is

and for an equal-percentage valve, the fraction that the value is open is

In each case, DP_v = DP_p - DP_sys. These relationships are shown in Figure M15-9.

Notice that the linear valve has a more linear characteristic at low flow rates, while the equal-percentage valve has a more linear characteristic over a wider range of flow rates, and in the higher flow rates in particular. Notice also that the gain of the linear valve is high at low valve opening and low at high valve openings. The equal-percentage valve has a gain that is roughly the same at high and low valve openings and has a maximum gain that is roughly twice the minimum gain. The linear valve has a maximum gain that is roughly 10 times the minimum gain. If a control system using the linear valve was tuned at a high flow rate (valve gain low, controller gain high), the control system might go unstable when operated at low flow rates. This problem is less likely to occur with the equal-percentage valve.