Question

scenario: you work for dr pepper bottling factory. the pump at the hydraulic lift for the packing machine is running slow and is overheating. the speed of the system with the old pump is running at 1650 rpm, and has a d/rev of 7.3 cuin. maintenance found a replacement pump but with no plate or label on it, other than its a slightly smaller motor (6.8 cu in/rev). the supervisor wants to know about the replacement pump before installing it. use the old pump to determine the optimal operation (displacement and rpm) of the new pump.

known: rpm = 1650, d/rev = 7.3 cuin, effv = 72%
find: q (gpm), d/min

known: d/rev = 6.8 cuin
find: q (gpm), rpm@100% (hint: use q and effv of old pump) (hint: use q to find d/min, then d/rev to find rpm)

Explanation:

Step1: Calculate the theoretical flow rate of the old pump at 100% efficiency

The formula for the flow - rate of a pump is $Q = \text{RPM}\times\text{D/rev}\times\text{Effv}/231$. First, assume 100% efficiency ($\text{Effv} = 100\%=1$) for the old pump to find its theoretical flow rate. Given $\text{RPM}=1650$, $\text{D/rev}=7.3$ cu in.
$Q_{theoretical - old}=1650\times7.3/231$
$Q_{theoretical - old}=\frac{1650\times7.3}{231}=\frac{12045}{231}=52.143$ gpm

Step2: Calculate the actual flow rate of the old pump

The actual efficiency of the old pump is $\text{Effv} = 72\% = 0.72$. Using the flow - rate formula $Q=\text{RPM}\times\text{D/rev}\times\text{Effv}/231$.
$Q_{old}=1650\times7.3\times0.72/231$
$Q_{old}=\frac{1650\times7.3\times0.72}{231}=\frac{12045\times0.72}{231}=\frac{8672.4}{231}=37.543$ gpm

Step3: Calculate the flow rate of the new pump

For the new pump, assume we want to find the flow rate at the same $\text{RPM} = 1650$ (since no other speed is specified for comparison). Given $\text{D/rev}=6.8$ cu in and assume 100% efficiency for now (to find the maximum possible flow rate).
$Q_{new}=1650\times6.8/231$
$Q_{new}=\frac{1650\times6.8}{231}=\frac{11220}{231}=48.571$ gpm

Answer:

The flow rate of the old pump is approximately $37.54$ gpm and the flow rate of the new pump (at 100% efficiency) is approximately $48.57$ gpm. If we want to find the optimal operation of the new pump considering efficiency, more information about its efficiency would be needed. But if we assume 100% efficiency for comparison purposes, the new pump has a higher flow rate than the old pump's actual flow rate.