Reference no: EM131218178

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The Bernoulli equation, shown in Equation 1, relates pressure, velocity, and gravitational potential energy for incompressible fluid systems at steady-state.

u^2/2+ gh+p/ρ=constant (1)

where u is the average fluid velocity, g is the acceleration due to gravity, h is the height, p is the fluid pressure, and ρ is the fluid density. Using the assumptions that the fluid frictional losses and change in potential energy are negligible, a mechanical energy balance equation for a fluid circuit with a centrifugal pump can be derived from the above Bernoulli equation. Equation 2 relates the mechanical energy state at the fluid circuit outlet to that of the inlet, and the shaft work done by the pump per unit mass, -W ?_s:

(u_2^2- u_1^2)/2+ (p_2- p_1)/ρ+ W ?_s=0 (2)

The average fluid velocities can be represented in terms of volumetric flow rate, q, using the following relationships:

u_1= 4q/(πD_1^2 ) and u_2= 4q/(πD_2^2 ) (3)

where D_1 is the inlet diameter (2 inches) and D_2 is the outlet diameter (1.5 inches). The pressure drop across the pump, p_2- p_1, can be represented by the sum of the discharge pressure and the suction pressure, p_d+ p_s. To express Equation 2 in units of power to obtain the work done on the fluid per unit time, -W ?_s, it must be multiplied by the constant mass flow rate, ρq:

-W ?_s= (8ρq^3)/π^2 (1/(D_2^4 )- 1/(D_1^4 ))+ q(p_d+ p_s ) (4)

The electrical power supplied to the motor of the pump, P_e, depends on the current and voltage:

P_e=IV(5)

The rotational power delivered from the motor to the impeller, P_u, is calculated as follows:

P_u=T_m ω(6)

where the motor torque is defined as T_m and the angular frequency of the motor as ω. The motor frequency, in rotations per minute, is related to the angular motor frequency by Equation 7.

ω= (2πf_m)/(60 sec/min)(7)

The efficiency of the pump motor can be determined using the previous power calculations, shown below:

η_m= P_u/P_e (8)

The efficiency of the impeller can be calculated as follows:

η_i= (-W ?_s)/P_u (9)

Finally, the overall efficiency of the pump can be calculated by multiplying Equation 8 by Equation 9:

η= (-W ?_s)/P_e (10)