# Calculating the total system head in USCS units

USCS stands for “United States Customary System Units” as opposed to the SI (Le Syst`eme International d`Units) or metric units that have been adopted by the International standards Organization (ISO).

It turn out that “head” is a very convenient term in the pumping business. Capacity is measured in gallons per minute, and each gallon of liquid has weight, so we can easily calculate the pounds per minute being pumped. Head or height is measured in feet, so if we multiply these two together we get foot- pounds per minute which converts directly to work at the rate of 33,000 foot pounds per minute equals one horsepower.

Pressure is not as convenient a term because the amount of pressure that the pump will deliver depends upon the weight (specific gravity) of the liquid being pumped and the specific gravity changes with temperature, type of fluid, and fluid concentration.

If you will refer to FIG 1, you should get a clear picture of what is meant by static head. Note that we always measure from the center line of the pump to the highest liquid level

To calculate head accurately we must calculate the total head on both the suction and discharge sides of the pump. In addition to the static head we will learn that there is a head caused by resistance in the piping, fittings and valves called friction head, and a head caused by any pressure that might be acting on the liquid in the tanks including atmospheric pressure, called ” surface pressure head”.

Once we know these heads, we will then subtract the suction head from the discharge head and the amount remaining will be the amount of head that the pump must be able to generate at the rated flow. Here is how it looks in a formula:

System head = total discharge head – total suction head

H = hd – hs

The total discharge head is made from three separate heads:

hd = hsd + hpd + hfd

- hd = total discharge head
- hsd = discharge static head
- hpd = discharge surface pressure head
- hfd = discharge friction head

The total suction head also consists of three separate heads

hs = hss + hps – hfs

- hs = total suction head
- hss = suction static head
- hps = suction surface pressure head
- hfs = suction friction head

As we make these calculations, you must sure that all calculations are made in either “feet of liquid gauge” or “feet of liquid absolute”. In case you have forgotten “absolute means that you have added atmospheric pressure (head) to the gauge reading.

Now we will make some actual calculations:

Figure #2 demonstrates that the discharge head is still measured to the liquid level, but you will note that it is below the maximum height of the piping.

Although the pump must deliver enough head to get up to this maximum piping height, it will not have to continue to deliver this head when the pump is running because of the “siphon effect”. There is of course a maximum siphon effect. It is derived from: 14.7 psi (atmospheric pressure) x 2.31 feet / psi = 33.4 feet maximum siphon effect.

We will begin with the total suction head calculation

- The suction head is negative because the liquid level in the suction tank is below the centerline of the pump:

hss = – 6 feet

- The suction tank is open, so the suction surface pressure equals atmospheric pressure :

hps = 0 feet gauge

- You will not have to calculate the suction friction head, I will tell you it is:

hfs = 4 feet at rated flow

- The total suction head is a gauge value because atmosphere was given as 0,

hs = hss + hps – hfs = -6 + 0 – 4 = -10 feet of liquid gauge at rated flow

The total discharge head calculation

- The static discharge head is:

hsd = 125 feet

- The discharge tank is also open to atmospheric pressure, thus:

hpd = 0 feet, gauge

- I will give you the discharge friction head as:

hfd = 25 feet at rated flow

- The total discharge head is:

hd = hsd + hpd + hfd = 125 + 0 + 25 = 150 feet of liquid gauge at rated flow

The total system head calculation:

H = hd – hs = 150 – (-10)= 160 feet of liquid at rated flow

Note: did you notice that when we subtracted a minus number (-10) from a positive number (150) we ended up with a positive 160 because whenever you subtract minus numbers it is the same as adding them? If you have trouble with this concept you can learn more about it from a mathematics book.

Our next example involves a few more calculations, but you should be able to handle them. In this example we are going to learn how to handle a vacuum application. Pipe friction numbers are taken from the Hydraulic Institute Engineering Data Book. You can get a copy of this publication from your library if you want to see the actual charts. I have some of this information in the chart section of this web site.

Specifications:

- Transferring 1000 gpm. weak acid from the vacuum receiver to the storage tank
- Specific Gravity – 0.98
- Viscosity – equal to water
- Piping – All 6″ Schedule 40 steel pipe
- Discharge piping rises 40 feet vertically above the pump centerline and then runs 400 feet horizontally. There is one 90° flanged elbow in this line
- Suction piping has a square edge inlet, four feet of pipe, one gate valve, and one 90° flanged elbow all of which are 6″ in diameter.
- The minimum level in the vacuum receiver is 5 feet above the pump centerline.
- The pressure on top of the liquid in the vacuum receiver is 20 inches of mercury, vacuum.

To calculate suction surface pressure use one of the following formulas:

- inches of mercury x 1.133/ specific gravity = feet of liquid
- pounds per square inch x 2.31/specific gravity = feet of liquid
- Millimeters of mercury / (22.4 x specific gravity) = feet of liquid

Now that you have all of the necessary information we will begin by dividing the system into two different sections, using the pump as the dividing line.

Total suction head calculation

- The suction side of the system shows a minimum static head of 5 feet above suction centerline. Therefore, the static suction head is:

hss = 5 feet

- Using the first conversion formula, the suction surface pressure is:

hps = -20 Hg x 1.133/ 0.98 = -23.12 feet gauge

- The suction friction head, hfs, equals the sum of all the friction losses in the suction line. Friction loss in 6″ pipe at 1000 gpm from table 15 of the Hydraulic Institute Engineering Data Book, is 6.17 feet per 100 feet of pipe.

in 4 feet of pipe friction loss = 4/100 x 6.17 = 0.3 feet

Friction loss coefficients (K factors) for the inlet, elbow and valve can be added together and multiplied by the velocity head:

FITTING | K | FROM TABLE |

6″ Square edge inlet | 0.50 | 32 (a) |

6″ 90 flanged elbow | 0.29 | 32 (a) |

6″ Gate valve | 0.11 | 32 (b) |

Total coefficient, K = 0.90

Total friction loss on the suction side is:

hfs = 0.3 + 1.7 = 2.0 feet at 1000 gpm.

- The total suction head then becomes:

hs = hss + hps – hfs = 5 + (-23.12) – 2.0 = -20.12 feet, gauge at 1000 gpm.

Total discharge head calculation

- Static discharge head = hsd = 40 feet
- Discharge surface pressure = hpd = 0 feet gauge
- Discharge friction head = hfd = sum of the following losses :

Friction loss in 6″ pipe at 1000 gpm. from table 15, is 6.17 feet per hundred feet of pipe.

In 440 feet of pipe the friction loss = 440/100 x 6.17 = 27.2 feet

Friction loss in 6″ elbow:

from table 32 (a), K = 0,29

from table 15, V^{2}/2g = 1.92 at 1000 gpm.

Friction loss = K V^{2}/2g = 0.29 x 1.92 = 0.6 feet

The friction loss in the sudden enlargement at the end of the discharge line is called the exit loss. In systems of this type where the area of the discharge tank is very large in comparison to the area of the discharge pipe, the loss equals V^{2}/2g, as shown in table 32 (b).

Friction loss at exit = V^{2}/2g = 1.9 feet

The discharge friction head is the sum of the above losses, that is:

hfd = 27.2 + 0.6 + 1.9 = 29.7 feet at 1000 gpm.

- The total discharge head then becomes:

hd = hsd + hpd + hfd = 40 + 0 + 29.7 = 69.7 feet, gauge at 1000 gpm.

- Total system head calculation:

H = hd – hs = 69.7 – (-20.2) = 89.9 feet at 1000 gpm.

Our next example will be the same as the one we just finished except. that there is an additional 10 feet of pipe and another 90° flanged elbow in the vertical leg. The total suction head will be the same as in the previous example. Take a look at figure # 4

Nothing has changed on the suction side of the pump so the total suction head will remain the same:

hs = -20.12 feet, gauge at 100 gpm.

Total discharge head calculation

- The static discharge head “hsd” will change from 40 feet to 30 feet, since the highest liquid surface in the discharge is now only 30 feet above the pump centerline.(This value is based on the assumption that the vertical leg in the discharge tank is full of liquid and that as this liquid falls it will tend to pull the liquid up and over the loop in the pipe line. This arrangement is called a siphon leg).
- The discharge surface pressure is unchanged:

hpd = 0 feet

- The friction loss in the discharge pipe will be increased by the additional 10 feet of pipe and the additional elbow.

In 10 feet of pipe the friction loss = 10/100 x 6.17 = 0.6 feet

The friction loss in the additional elbow = 0.6 feet

The friction head will then increase as follows:

hfd = 29.7 + 0.6 + 0.6 = 30.9 feet at 1000 gpm.

The total discharge head becomes:

hd = hsd + hpd + hfd

= 30 + 0 + 30.9

= 60.9 feet, gauge at 1000 gpm.

- Total system head calculation

H = hd – hs = 60.9 – (-20.12) = 81 feet at 1000 gpm.

For our last example we will look at gauges. Take a look at FIG 5:

Specifications:

- Capacity – 300 gpm.
- Specific gravity – 1.3
- Viscosity – Similar to water
- Piping – 3 inch suction, 2 inch discharge
- Atmospheric pressure – 14.7 psi.

Divide the heads into two sections again:

The discharge gauge head corrected to the centerline of the pump, in feet of liquid absolute is found by adding the atmospheric pressure to the gauge reading to get absolute pressure, and then converting to absolute head:

hgd = (130 + 14.7) x 2.31 / (1.3 Specific Gravity) + 4 = 261.1 feet, absolute

Note the 4 foot head correction to the pump centerline.

The discharge velocity head at 300 gpm. is found in table 9 of the Hydraulic Institute Engineering Data Book

hvd = 12.8 feet at 300 gpm.

The suction gauge reading is in absolute terms so it needs only to be converted to feet of liquid, absolute.

hgs = 40 x 2.3 / 1.3 +2 = 73.08 feet absolute

Note the 2 foot head correction to the pump centerline.

The suction velocity head at 300 gpm. is found in table 11 of the Pipe Friction Manual:

hvs = 2.63 feet at 300 gpm.

The total system head developed by the pump =:

H = (hgd + hvd ) – ( hgs + hvs ) = (261.1 + 12.8) – (73.08 + 2.6)= 198.22 feet absolute at 300 gpm.