# Q. What is static head and how is the static head calculated for a system that contains a fluid at different temperatures?

A. The static head of a system is that portion of the pump’s Total Head that must be added to the fluid before a single drop of fluid will move in the system. It is the amount of energy, or head, which is used to overcome the pressure and elevation differences in the piping system. It can be calculated using the following equations:

\[ \text{Static Head} = Δ \text{ Elevation Head} + Δ \text{ Pressure Head}\] \[Δ \text{ Elevation Head} = \bigg(\begin{matrix}\text{Elevation of Discharge}\\\text{Tank's Liquid Surface}\end{matrix}\bigg)-\bigg(\begin{matrix} \text{Elevation of Supply}\\\text{Tank's Liquid Surface}\end{matrix}\bigg)\] \[Δ\text{ Pressure Head} = \bigg[\bigg(\begin{matrix}\text{Discharge Tank's}\\\text{Surface Pressure}\end{matrix}\bigg) - \bigg(\begin{matrix}\text{Discharge Tank's}\\\text{Surface Pressure}\end{matrix}\bigg)\bigg]\frac{144}{\rho}\] |

where:

- static, elevation, and pressure head are in units of feet of fluid

- pressure in units of lb/in
^{2}(psi) - density (ρ) in lb/ft
^{3}of the fluid in the pump

If the system contains fluids at different temperatures the fluid density will vary. So which fluid density should be used in the equation: the density of the hotter fluid, the colder fluid, or an average density? How much difference does it make? Does it affect the differential pressure head or differential elevation head component of static head?

At first glance , based on the equations, it appears that the fluid density will affect the differential pressure head component. But the pressure difference between the two tanks has to be overcome regardless of the densities of the fluid between the two tanks. And since static head is a portion of the Total Head at the pump, the density of the fluid in the pump is used in the pressure head component of the static head calculation.

It is actually the differential elevation head component that is affected by the fluid density. A column of fluid of a given height will equate to a pressure at the bottom of the column, which will be different compared to the same column of fluid with a different density. It is the differential elevation head component that must be adjusted to ensure the terms of elevation head are related to the fluid at the pump.

Consider the following open systems in which water is pumped from a supply tank (bottom elevation = 0 ft, liquid level = 10 ft, and surface pressure = 0 psig) to an elevated product tank (bottom elevation = 150 ft, liquid level = 10 ft, and surface pressure = 10 psig).

The first system is pumping water at 60°F and the second system has a heat exchanger at the 25 ft elevation that heats the water up to 200°F.

The calculation for static head in the first system is:

\[ \text{Static Head}=(160-10)ft+\bigg[(10-0)\frac{lb}{in^2}\bigg( \frac{144in^2/ft^2}{62.37lb/ft^3}\bigg)\bigg]= 173.1ft\] |

This value of static head can be confirmed using PIPE-FLO by selecting Graph Resistance Curve from a pump's context menu, and looking at the Head value where the blue line intersects the Y axis.

The Y intercept is this screen shot is 173 ft. It is circled in red and shown at the bottom of the graph by placing the curser on the line.

In the second system, the water from the liquid surface in the supply tank to the inlet of the heat exchanger (at 25 ft elevation) has a density of 62.37 lb/ft

^{3}, but the water from the heat exchanger to the liquid surface in the product tank has a density of 60.11 lb/ft^{3}. To calculate the static head requires adjusting the elevation head between the heat exchanger and product tank by the ratio of the fluid densities to put it in terms of head at the pump:

\[ \text{Static Head}=\bigg[(160-25)ft\bigg(\frac{60.11lb/ft^3}{62.37lb/ft^3}\bigg)+(25-10)ft\bigg]+\bigg[(10-0)\frac{lb}{in^2}\bigg( \frac{144in^2/ft^2}{62.37lb/ft^3}\bigg)\bigg]= 168.2ft\] |

This value of static head can also be confirmed using PIPE-FLO:

The Y intercept in this screen shot is 168 ft. It is circled in red and shown at the bottom of the graph by placing the curser on the line.

The difference in the calculated static head between the two systems may or may not be significant enough to justify calculating the density correction in the system, depending on the degree of accuracy required in the calculations and the magnitude of the difference between the fluid densities.

For reference to older versions of PIPE-FLO see attachments below.

Static Head in Systems with Varying Fluid Temperatures old.pdf

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## Jeff Sines

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