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### Understanding basic principles of flow calculations

Valve size often is described by the nominal size of the end connections, but a more important measure is the flow that the valve can provide.

**John Baxter and Ulrich Koch**

Swagelok CompanySwagelok Company

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**Article Summary**

And determining flow through a valve can be simple. Using the principles of flow calculations, some basic formulas, and the effects of specific gravity and temperature, flow can be estimated well enough to select a valve size easily and without complicated calculations

The principles of flow calculations are illustrated by the common orifice flow meter (see Figure 1). We need to know only the size and shape of the orifice, the diameter of the pipe and the fluid density. With that information, we can calculate the flow rate for any value of pressure drop across the orifice (the difference between inlet and outlet pressures).

For a valve, we also need to know the pressure drop and the fluid density. But in addition to the dimensions of pipe diameter and orifice size, we need to know all the valve passage dimensions and all the changes in size and direction of flow through the valve.

However, rather than doing complex calculations, we use the valve flow coefficient, which combines the effects of all the flow restrictions in the valve into a single number (see Figure 2).

Valve manufacturers determine the valve flow coefficient by testing the valve with water at several flow rates, using a standard test method2 developed by the Instrument Society of America for control valves and now used widely for all valves.

Flow tests are done in a straight piping system of the same size as the valve, so that the effects of fittings and piping size changes are not included (see Figure 3).

Since liquids are incompressible fluids, their flow rate depends only on the difference between the inlet and outlet pressures (Δp, pressure drop). The flow is the same whether the system pressure is low or high, so long as the difference between the inlet and outlet pressures is the same. Equation 1 shows the relationship.

The water flow graphs show water flow as a function of pressure drop for a range of Cv values.

Gas flow calculations are slightly more complex because gases are compressible fluids whose density changes with pressure. In addition, there are two conditions that must be considered: low pressure drop flow and high pressure drop flow.

Equation 2 applies when the low pressure drop flow outlet pressure (p2) is greater than one half of the inlet pressure (p1):

The low pressure drop air flow graphs show low pressure drop air flow for a valve with a Cv of 1.0, given as a function of inlet pressure (p1) for a range of pressure drop (Δp) values.

When outlet pressure (p2) is less than half of inlet pressure (p1), high pressure drop, any further decrease in outlet pressure does not increase the flow because the gas has reached sonic velocity at the orifice and it cannot break that “sound barrier”.

Equation 3 for high pressure drop flow is simpler because it depends only on inlet pressure and temperature, valve flow coefficient and specific gravity of the gas:

The high-pressure drop air flow graphs show high pressure drop air flow as a function of inlet pressure for a range of flow coefficients.

The flow equations include the variables liquid specific gravity (Gf) and gas specific gravity (Gg), which are the density of the fluid compared to the density of water (for liquids) or air (for gases).

However, specific gravity is not accounted for in the graphs, so a correction factor must be applied, which includes the square root of G. Taking the square root reduces the effect and brings the value much closer to that of water or air, 1.0.

For example, the specific gravity of sulphuric acid is 80% higher than that of water, yet it changes flow by just 34%. The specific gravity of ether is 26% lower than that of water, yet it changes flow by only 14%.

The effect of specific gravity on gases is similar. For exam ple, the specific gravity of hydrogen is 93% lower than that of air, but it changes flow by just 74%. Carbon dioxide has a specific gravity 53% higher than that of air, yet it changes flow by only 24%. Only gases with very low or very high specific gravity change the flow by more than 10% from that of air.

Figure 5 shows how the effect of specific gravity on gas flow is reduced by use of the square root.

Temperature usually is ignored in liquid flow calculations because its effect is too small. Temperature has a greater effect on gas flow calculations, because gas volume expands with higher temperature and contracts with lower temperature. But similar to specific gravity, temperature affects flow by only a square root factor. For systems that operate between -40°F (-40°C) and +212°F (+100°C), the correction factor is only +12 to -11%. Figure 6 shows the effect of temperature on volumetric flow over a broad range of temperatures. The plus or minus 10% range covers the usual operating temperatures of most common applications.Figure 4 shows how taking the square root of specific gravity diminishes the significance on liquid flow. Only if the specific gravity of the liquid is very low or very high will the flow change by more than 10% from that of water.

**Flow calculation principles**The principles of flow calculations are illustrated by the common orifice flow meter (see Figure 1). We need to know only the size and shape of the orifice, the diameter of the pipe and the fluid density. With that information, we can calculate the flow rate for any value of pressure drop across the orifice (the difference between inlet and outlet pressures).

For a valve, we also need to know the pressure drop and the fluid density. But in addition to the dimensions of pipe diameter and orifice size, we need to know all the valve passage dimensions and all the changes in size and direction of flow through the valve.

However, rather than doing complex calculations, we use the valve flow coefficient, which combines the effects of all the flow restrictions in the valve into a single number (see Figure 2).

Valve manufacturers determine the valve flow coefficient by testing the valve with water at several flow rates, using a standard test method2 developed by the Instrument Society of America for control valves and now used widely for all valves.

Flow tests are done in a straight piping system of the same size as the valve, so that the effects of fittings and piping size changes are not included (see Figure 3).

**Liquid flow**Since liquids are incompressible fluids, their flow rate depends only on the difference between the inlet and outlet pressures (Δp, pressure drop). The flow is the same whether the system pressure is low or high, so long as the difference between the inlet and outlet pressures is the same. Equation 1 shows the relationship.

The water flow graphs show water flow as a function of pressure drop for a range of Cv values.

**Gas flow**Gas flow calculations are slightly more complex because gases are compressible fluids whose density changes with pressure. In addition, there are two conditions that must be considered: low pressure drop flow and high pressure drop flow.

Equation 2 applies when the low pressure drop flow outlet pressure (p2) is greater than one half of the inlet pressure (p1):

The low pressure drop air flow graphs show low pressure drop air flow for a valve with a Cv of 1.0, given as a function of inlet pressure (p1) for a range of pressure drop (Δp) values.

When outlet pressure (p2) is less than half of inlet pressure (p1), high pressure drop, any further decrease in outlet pressure does not increase the flow because the gas has reached sonic velocity at the orifice and it cannot break that “sound barrier”.

Equation 3 for high pressure drop flow is simpler because it depends only on inlet pressure and temperature, valve flow coefficient and specific gravity of the gas:

The high-pressure drop air flow graphs show high pressure drop air flow as a function of inlet pressure for a range of flow coefficients.

**Effects of specific gravity**The flow equations include the variables liquid specific gravity (Gf) and gas specific gravity (Gg), which are the density of the fluid compared to the density of water (for liquids) or air (for gases).

However, specific gravity is not accounted for in the graphs, so a correction factor must be applied, which includes the square root of G. Taking the square root reduces the effect and brings the value much closer to that of water or air, 1.0.

For example, the specific gravity of sulphuric acid is 80% higher than that of water, yet it changes flow by just 34%. The specific gravity of ether is 26% lower than that of water, yet it changes flow by only 14%.

The effect of specific gravity on gases is similar. For exam ple, the specific gravity of hydrogen is 93% lower than that of air, but it changes flow by just 74%. Carbon dioxide has a specific gravity 53% higher than that of air, yet it changes flow by only 24%. Only gases with very low or very high specific gravity change the flow by more than 10% from that of air.

Figure 5 shows how the effect of specific gravity on gas flow is reduced by use of the square root.

**Effects of temperature**Temperature usually is ignored in liquid flow calculations because its effect is too small. Temperature has a greater effect on gas flow calculations, because gas volume expands with higher temperature and contracts with lower temperature. But similar to specific gravity, temperature affects flow by only a square root factor. For systems that operate between -40°F (-40°C) and +212°F (+100°C), the correction factor is only +12 to -11%. Figure 6 shows the effect of temperature on volumetric flow over a broad range of temperatures. The plus or minus 10% range covers the usual operating temperatures of most common applications.Figure 4 shows how taking the square root of specific gravity diminishes the significance on liquid flow. Only if the specific gravity of the liquid is very low or very high will the flow change by more than 10% from that of water.

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