Parts 1 and 2 of this series (*Pumps & Systems*, September and October 2015) discussed the engineering principles that dictate the operation of various elements of a piping system. Using those principles, we developed a mathematical model of an example piping system, used the model to simulate the operation of the physical system, and determined if the equipment is operating within the confines of the model. The model was compared with with the physical piping system to identify and isolate problems within the piping system to arrive at a course of corrective action. Part 3 will examine the results and model validation, exploring ways to evaluate the model.

Using the valve stem indicator on the control valve, we can determine the valve is approximately 75 percent open. Using this position, the manufacturers C_{v} table for valve position and the flow rate of 1,000 gpm through the flow meter, we can calculate the differential pressure across the control valve. If the calculated differential pressure across the control valve matches the differential pressure calculated above, we can validate the control valves inlet and outlet pressures. Using the ANSI/ISA – 75.01.01 Flow Equations for Sizing Control Valves standard, Equation 7 can determine the C_{v} value.

The control valve manufacturer provides the F_{p} value. Table 1 displays the C_{v} data for FCV-101.

_{v}and valve coefficient values as a function of the valve position. This can be used to simulate the operation of the specific control valve under various operating conditions. (Courtesy of the author)

The calculated C_{v} value falls between the 70 and 80 percent valve open position. Performing a linear interpolation between the 70 and 80 percent open results in a calculated valve position of 74 percent for FCV-101 using the valve sizing equation.

The calculated results correlates with the observed valve position. Because we have an independent method of calculation using the model with the actual observed values, we can validate the results of the control valve.

The total energy values at the inlet and outlet of the head exchanger and the inlet and outlet of the flow meter have not been validated. Because the total energy calculated by the model at PI-101 and the inlet of FCV-101 have been validated, one could assume that the total energy calculated accurately reflects the physical piping system even without the validation of installed instrumentation.

Based on the correlation between the calculated energy of the model and the observed values provided by the installed plant instrumentation, one could assume that the model accurately represents the operation of the system under this condition.

Imagine two week after the system has been validated the operating tank level in TK-101 has increased from 10 to 15 feet of level. How will that effect the system?

Again, the boundary parameters drive the system operation; in this example system, the boundary parameters consist of the supply tank, the destination tank and the flow control loop FCV-101 that regulates the flow rate to a set value. Changing any of these boundary values will change the operation of every item in the system.

With a change of operating parameters, the plant operates at 30 psig and a 20 foot level in destination tank PV-102. Figure 5 shows the values of the installed instrumentation. We will use the existing model to determine if it still accurately reflects the system operation. The flow rate through the system is still maintained at 1,000 gpm by the flow control loop. We can see that the position of FCV-101 is now 82.6 percent open. We will evaluate the existing system to see if the model accurately reflects what the system is showing.

The pump suction and discharge pressures have not changed. This is to be expected because no change occurred in the supply tank level or the flow rate through the suction pipeline, pump and discharge pipeline. As a result, the pressure values in PI-100 and PI-101 should be the same as before. Because the flow rate through the heat exchanger, flow meter and all the connecting pipelines upstream of control valve FCV-101 are the same, the inlet pressure to the control valve should be 173.12 feet of fluid.

Now we will look at the system from the control valve outlet to the destination tank PV-102. Using the Bernoulli equation, we will determine the static head at the destination tank (see Equation 8).

Because the flow rate through the pipeline connecting the control valve outlet and the destination tank has not changed and the flow rate through the system remains at 1,000 gpm, the total energy at the control valve outlet is 145.15 feet (139.68 + 5.47 ft).

The new head loss across the control valve is 27.97 feet (173.12-145.15). Converting the head loss across the control valve to differential pressure results in a value of 12.04 psi. Plugging the flow rate of 1,000 gpm and the differential pressure of 12.04 psi into the valve sizing equation, we can determine the Cv of the control valve is 290.23. Using the Cv values found in Table 1 results in a calculated valve position of 82.6 percent. These calculated results correlate with the results displayed in the operating system.

As demonstrated, the mathematical model developed using the manufacturer's supplied data and the basic engineering equations that calculate head, pressure and head loss accurately reflect the operation of fluid piping systems. This provides us with two powerful points of knowledge:

- The model accurately reflects the operation of the physical piping system.
- Any deviation between the model and the operation of the physical system can be traced to equipment not operating according to the manufacturer's supplied data or standard engineering calculations.

Next month we will use this knowledge to determine when a physical piping system is not operating according to the piping system model and how to correct the problem. Read Part 1 of this series here and Part 2 here.