By Jeff Sines, Senior Product Engineer, Engineered Software, Inc.

Fluid flow through a piping system that consists of components such as valves, fittings, heat exchangers, nozzles, filters, and pipelines will result in a loss of energy due to the friction between the fluid and internal surfaces, changes in the direction of the flow path, obstructions in the flow path, and changes in the cross-section and shape of the flow path.

The energy loss, or head loss, will be seen as a pressure drop across the piping system and each component in the system. There are different ways to characterize the impact of these factors with regard to the flow rate through the component and the resulting pressure drop.

One way is to characterize the amount of ** resistance** that is offered by the system or components. The resistance can be given a numerical value as a resistance coefficient (K) or by the pipe length (or equivalent length), which can be used in the 2

\[H_{L} = f \frac{L}{D}\frac{v^2}{2g} \text{ or, } H_{L} = K\frac{v^2}{2g}\] |

Where: H_{L} = head loss (ft)

f = Darcy friction factor (unitless)

L_{eqv} = pipe length (ft)

D = pipe internal diameter (ft)

v = fluid velocity (ft/sec)

g = gravitational constant (ft/sec^{2})

K= resistance coefficient (unitless)

The pressure drop resulting from the head loss is given by:

\[dP = \frac{\rho H_L}{144}\] |

Where: dP = pressure drop (psi)

rho = fluid density (lb/ft^{3})

The resistance of a piping system is constant when all the valves (including control valves) remain in one position. When the position of a valve is changed, the resistance of the valve and therefore the entire system is changed. The change in resistance results in a change of flow rate and pressure drop in the system. Conversely, if the resistance remains constant and the differential pressure changes, the effect on the flow rate can be calculated.

Another way to characterize the performance of the system or a component is by its ** capacity**. The capacity can be expressed with a numerical value as a flow coefficient (C

\[Q = C_v\sqrt{\frac{dP}{SG}} \text{ therefore, } C_v = \frac{Q}{\sqrt{\frac{dP}{SG}}}\] |

Where: C_{v} = flow coefficient (unitless)

Q = flow rate (gpm)

SG = fluid specific gravity

The flow coefficient is commonly used by control valve manufacturers to characterize the performance of their valves with a change in valve position. Nozzle and sprinkler manufacturers also use the flow coefficient, except that they generally refer to it as a discharge coefficient. For a nozzle or sprinkler, when the inlet pressure changes, the flow rate through the nozzle will change proportionally, as given by the C_{V} equation above and holding C_{V} constant.

Just as with the system resistance, the system capacity remains constant as long as the positions of the valves in the system are constant. A constant flow coefficient allows for calculating the flow rate when the differential pressure changes.

This concept can be used to simplify piping systems for analysis using Engineered Software's PIPE-FLO or Flow of Fluids programs. If the pressure (or differential pressure) and flow rate at a point in the system is known and the system resistance downstream from this point is constant, the downstream system can be simplified and represented with a flow coefficient, C_{V}.

Consider the following simple system of water at 60 °F being fed from a 10 psig pressure source. The upper branch goes to a 5 psig boundary pressure and the lower branch goes to a boundary flow rate of 50 gpm. The inlet and both outlet boundary conditions are set by the user. PIPE-FLO calculates the flow rate in the upper branch (246.5 gpm) and the pressure at the outlet of the lower branch (7.847 psig).

To determine what happens to the flow rate and pressure in the upper branch when the flow rate in the lower branch changes, assuming the configuration of the upper branch remains the same (i.e. valve positions in the upper branch are not changed), the system should be modified to obtain an equivalent system, as shown below. A short length of pipe (0.0001 ft) is added to the upper branch at the outlet with a boundary pressure of 0 psig and a fixed C_{V} fitting installed in the pipe. The C_{V} is calculated using the C_{V} formula above, using SG=1.0 (for water at 60 °F), the calculated flow rate, and a differential pressure of 5 psid.

\[C_v = \frac{Q}{\sqrt{\frac{dP}{SG}}} = \frac{246.5}{\sqrt{5-0}} = 110.238\] |

The calculated model confirms that the two systems are equivalent since the calculated flow rate in the upper branch is 246.5 gpm and the calculated pressure at the node is 5 psig.

Now the model can be evaluated for a change to the flow rate in the lower branch. When the flow rate is increased to 200 gpm, the pressure at the common junction decreased from 7.923 psig to 6.049 psig. This results in a lower differential pressure in the upper branch and therefore a lower flow rate in the upper branch. The flow rate in the upper branch goes from 246.5 to 214.8 gpm, and the pressure at the outlet goes from 5 psig to 3.797 psig.

This method of simplifying a system can be used when the flow rate and pressure are known at a given point in the system and it is known that the system downstream of that point remains constant with regard to valve positions. It can also be used when the differential pressure is known for a given flow rate and the components within the boundaries that define the differential pressure remain constant.

This method can also be used for evaluating the performance of a closed loop system. Consider the following system with three heat exchangers, each with a flow rate of about 200 gpm. The flow control valves are manually operated valves and have Cv data entered for the range of valve positions (each valve has different Cv values so they are not identical). The three heat exchangers have identical head loss curves. This is a common configuration for a plant with heat exchangers located throughout the facility.

The top and middle branches from the inlet pipe of the heat exchanger to the outlet pipe of the control valve can be simplified using fixed C_{V} fittings by calculating the values based on the above formula:

\[\text{Top branch } C_v\text{:}\] \[C_v = \frac{Q}{\sqrt{dP}} = \frac{199.7 \text{ gpm}}{\sqrt{47.78-9.984 \text{ psid}}}=32.4829\] |

\[\text{Middle branch } C_v\text{:}\] \[C_v = \frac{Q}{\sqrt{dP}} = \frac{199.2 \text{ gpm}}{\sqrt{50.09-7.673 \text{ psid}}}=30.5858\] |

These C_{V} values are entered as fixed C_{V} fittings in a very short length pipe (0.0001 ft) to ensure all the pressure drop is across the fitting and not the pipe itself. The system below shows the simplified system with the C_{V} values installed and the model calculated. Note that the flow rates in the top and middle branches are the same as the original system, as well as the inlet and outlet pressures of the branches. This confirms that the calculated C_{V} values are equivalent to the original branches consisting of the pipes, heat exchanger, and control valve.

Now a flow rate change in the bottom branch can be evaluated to see the impact on the flow rates and pressure drops in the top and middle branches. This is similar to an operator in one part of the plant making an adjustment to the flow rate through his heat exchanger, and operators of the heat exchangers in other parts of the plant seeing the impact on their flow rates.

First, evaluate the flow rate change in the original system so it can be used as a comparison to the simplified model with the fixed C_{V} fittings. The system below shows the bottom flow rate adjusted to 50 gpm (from the original 200 gpm). A lower flow rate in the bottom branch causes the inlet pressure to rise (also, this pressure increases due to increased pump total head at a lower pump flow rate) and the outlet pressure to drop. The inlet pressures to the middle and top branches increase in response and the outlet pressures decrease slightly. The resulting increase in differential pressure causes the flow rates to increase in the top and middle branches.

The flow rate in the top branch has increased from 199.7 gpm to 208.9 gpm, with an inlet pressure of 51.23 psig and an outlet pressure of 9.786 psig. The flow rate in the middle branch has increased from 199.2 gpm to 208.3 gpm due to the inlet pressure increase to 53.74 psig and the outlet pressure decrease to 7.269 psig.

The same impact can be seen in the model using fixed C_{V} fittings to simplify the top and middle branches. The calculated results are shown below. The top branch flow rate is 209 gpm in the simplified model, and the middle branch flow rate is 208.5 gpm. Both flow rates are comparable to the original model with the reduced flow rate in the bottom branch. The inlet and outlet pressures are also very close to the original model. __Conclusion__

The flow coefficient C_{V} represents an important relationship between the flow rate and resulting differential pressure caused by the head loss across a pipeline, component, valve, fitting, or a portion of a system consisting of these devices. By calculating a C_{V} from a given flow rate and pressure drop, a large system can be simplified to a single pipeline with a fixed C_{V} fitting installed. This method can only be used if the resistance of the system being simplified remains constant with no changes in valve position.