## II. FUNDAMENTALS OF OPERATION

### D. Horsepower Calculations

(see Notes 3, 4, 5, and 6 at the end of this section)

1. The required horsepower (hp) for driving a single acting reciprocating pump is calculated by the following equation:

hp = Pd Q (100) - Pi Q (Em-5) 1714 Em 1714 (100) Where: D = pump displacement, U.S. gallons per minute (GPM) Q = pump capacity, (GPM). Where Q = D x Ev Pd = liquid pressure at pump discharge, lbs/in2 gauge (PSIG). Pi = liquid pressure at pump inlet, lbs/in2 gauge (PSIG). If Pi < 50 PSIG, then Pi Q (Em - 5) = 0 1714 x (100) Em = mechanical efficiency of pump, percent (for the above Pd and Pi < 50 PSIG): a. 90% for pumps without built-in reducer (power input from crankshaft). b. 85% for pumps with built-in or bolted-on gear reduction (power input from pinion shaft). c. Reduce Em by the mechanical efficiency losses of any intermediate speed reduction between drive and pump.G eneral Em values:: V-belt drive 5% HTD drive 5% Parallel shaft gear reducer 5% Etc... Em - 5 = efficiency of power recovery due to liquid pressure at pump inlet. Ev = volumetric efficiency, where Ev = Q

Example 4: Find the required horsepower (hp) and speed (rpm) for driving a single acting 2-3/4" x 5" triplex plunger pump under the following operating conditions:

**Pump displacement** = D = 138.9 GPM Pressure at pump inlet = Pi = 200 psig Pressure at pump discharge = Pd = 2020 psig Mechanical efficiency of pump = Em = 75% Volumetric efficiency of pump = Ev = 80% Q = D x Ev = 138.9 x 0.8 = 111.12 GPM hp = (Q x Pd) (100) - (Q x Pi ) ( Em - 5) 1714 x (Em) 1714 x (100) hp = (111.12 x 2020) (100) - (111.12 x 200) (75-5) 1714 x (75) 1714 x (100) hp = 165.53 gpr = 0.129 gal./plunger x 3 plungers/rev. gpr = 0.387 gal./rev. (see Table 1) n = GPM = 138.9 = 358.9 rpm gpr 0.387 2. The required horsepower (hp) for driving a double acting duplex piston pump is calculated by the following equation: hp = (Q) (Pd - Pi) (100) 1714 (Em) Where, D = pump displacement, gallons per minute (GPM) Q = pump capacity, GPM where (Q = D / Ev ) Pd = liquid pressure at pump discharge, lbf/in2 gauge (psig). Pi = liquid pressure at pump inlet, lbf/in2 gauge (psig). Em = pump mechanical efficiency, percent (for above Pd and Pi). a. 90% for pumps without built-in reducer (with power input from crankshaft) b. 85% for pumps with built-in reducer (with power input from pinion shaft) 3. Reduce Em by the mechanical efficiency losses of any intermediate speed reduction between driver and pumpG eneral Em values: V-Belt drive 5% HTD drive 5% Parallel shaft gear reducer 5% Etc... EV = pump volumetric efficiency, % = Q (100) D

**Example 5: **Find the required horsepower (hp) and pump speed (n) for driving a double acting 5" x 10" duplex piston pump with the following conditions.

Pump capacity = Q = 281.7 GPM Liquid pressure at pump inlet = Pi = 50 psig Liquid pressure at pump outlet = Pd = 330 psig Mechanical efficiency of pump = Em = 90% Volumetric efficiency of pump = EV = 85% Piston Rod diameter = d = 1- 1/2" hp = Q (Pd - Pi) 100 = 281.7 (330-50) 100 1714 (Em) 1714 (90) hp = 51.13 gpr = [0.850 x 4] - [ π (1.5)2 ] x 10 x 2 x 1 = 3.24 4 231 D = ( Q x 100 ) / EV = (281.7 x 100) / 85 = 331.41 GPM n = D = 331.4 = 102.3 rpm gpr 3.24 3. Quick calculation of horsepower requirement of a reciprocating pump. Where the above formulas are very accurate, they are somewhat detailed. A quick and easy method of calculating horsepower which is almost as accurate as the other methods, is as follows:

A flow of one barrel per hour (1 BPH) of any specific gravity liquid against one pound per square inch gauge pressure (1 psig) requires 0.00040833 correction factor for theoretical horsepower. If the pump mechanism is approximately 90% mechanically efficient, the conversion factor can be corrected, for convenience, to 0.00045 to reflect this efficiency loss.

Therefore, hp = 0.00045 x Pd x BPH capacity BPH capacity = D Ev = (BPH displacement x EV) 100 100 Where, Pd = liquid pressure at pump discharge, psig EV = pump volumetric efficiency, %

Example 6: Find the required horsepower (hp) to displace 100 gallons per minute (GPM) against a Pd = 1000 psig at a Ev = 95%.

hp = 0.00045 x 1000 x 100 x 95 = 61.1

0.7 x 100

**Note 3: **One barrel is 42 U.S. gallons. Gallons per minute (GPM) divided by 0.7 equals barrels per hour (BPH). Displacement is the theoretical volume swept by the plunger or piston on the discharge stroke during any selected time period.

*Under the conditions and definitions given above, the pump speed must be a higher rpm in order to displace enough volume to deliver the required amount of liquid.*

Note 4: Mechanical efficiency (E_{m}) expressed as a percentage of total horsepower requirement can be used only if the pump is to be applied at or near its maximum designed rating. The horsepower required to operate a large pump at a small horsepower will usually be a considerably higher percentage of the total horsepower for the application.

*We suggest that if the hydraulic horsepower - calculated without correction for mechanical efficiency - is less than 50% of the maximum design rating for the pump, you contact WGI for our recommendation for the driver horsepower.*

**Note 5: **If the horsepower requirement is less than 15, as calculated by any of the above methods, we recommend that the motor driver be one size larger than the calculated requirement. This is because the horsepower required by a speed reduction device (belt, gear reducer, etc.) is relatively fixed and cannot be factored into the equation as a percentage of the smaller horsepower requirements.

Note 6: These computations are intended as a guide to standardization and may be modified if efficiency ratings for the pump application are lower than E_{v} = 95% and/or E_{m} = 90%.