Centrifugal Pump Power Calculation Using Affinity Laws

This problem involves calculating the change in brake power of a centrifugal pump when the impeller diameter is altered. We can use the pump affinity laws, specifically the relationship between brake power and impeller diameter.

Pump Affinity Law for Power

For a centrifugal pump operating at a constant speed, the brake power ($P$) required is directly proportional to the cube of the impeller diameter ($D$). This can be expressed mathematically as:

$ P \propto D^3 $

Therefore, the ratio of brake power for two different impeller diameters ($D_1$ and $D_2$) is:

$ \frac{P_2}{P_1} = \left(\frac{D_2}{D_1}\right)^3 $

Calculating New Brake Power

We are given:

• Initial brake power, $P_1 = 1.86$ kW
• Initial impeller diameter, $D_1 = 200$ mm
• New impeller diameter, $D_2 = 180$ mm

We need to find the new brake power, $P_2$. Rearranging the formula:

$ P_2 = P_1 \times \left(\frac{D_2}{D_1}\right)^3 $

Substitute the given values:

$ P_2 = 1.86 \text{ kW} \times \left(\frac{180 \text{ mm}}{200 \text{ mm}}\right)^3 $

First, calculate the ratio of the diameters:

$ \frac{180}{200} = 0.9 $

Now, cube this ratio:

$ (0.9)^3 = 0.729 $

Finally, calculate $P_2$:

$ P_2 = 1.86 \text{ kW} \times 0.729 $

$ P_2 \approx 1.35594 \text{ kW} $

The calculated brake power is approximately 1.356 kW. This value falls within the range of 1.35 kW to 1.36 kW.