Question

The ratio of the theoretical power input (P$_1$/P$_2$) of two centrifugal fans having the same specific speed and operating at same speed is given by

• A. (D$_1$/D$_2$)$^2$
• B. (D$_1$/D$_2$)$^3$
• C. (D$_1$/D$_2$)$^4$
• D. (D$_1$/D$_2$)$^5$

Hint:

Fan laws are essential for designing ventilation systems in mines, ensuring efficient airflow and energy use.

Answer 1

The Correct Option is B

For centrifugal fans, the power input (P) is related to the fan laws, which describe how performance scales with changes in diameter (D), speed (N), and other parameters. When two centrifugal fans have the same specific speed and operate at the same speed (N$_1$ = N$_2$), the power scales with the cube of the diameter ratio. The fan power equation is P $\propto$ $\rho$Qv, where Q (flow rate) $\propto$ D$^3$ and v (velocity pressure) $\propto$ D$^2$, but since speed is constant, the overall power scales as P $\propto$ D$^3$ when considering the air volume and pressure relationships. Thus, the ratio of power inputs P$_1$/P$_2$ = (D$_1$/D$_2$)$^3$. Options (D$_1$/D$_2$)$^2$, (D$_1$/D$_2$)$^4$, and (D$_1$/D$_2$)$^5$ do not align with this relationship. Thus, the correct answer is (D$_1$/D$_2$)$^3$.