A centrifugal pump impeller has an outside diameter of 290mm and the with at the outlet is 110mm. The vanes are set back at an angle of 35 degrees to the tangent to the impeller at the vane tip. The diameter of the suction and delivery pipes is 290mm. Gauges at points on the delivery and suction pipes close to the pump and Ravi each 1.5m above the sump level showed heads of 18.6m above and 3.4m below atmosphere respectively while the pump was delivering 180 l/s at a speed of 1200 rev/min. It requires 68kW to drive the pump. Assume the water enters the impeller radially and without shock. Calculate the overall efficiency, manometric efficiency Abe the pressure head loss in the suction pipe
To calculate the overall efficiency of the centrifugal pump, we need to consider the manometric efficiency and mechanical efficiency.
1. Manometric Efficiency:
The manometric efficiency (n_m) of a pump is the ratio of the actual head developed by the pump (h_m) to the theoretically ideal head (h_i), expressed as a decimal.
h_i can be calculated using the following formula:
h_i = (P_d - P_s) / (ρ*g)
where:
P_d = Delivery gauge reading + atmospheric pressure (converted to meters of head)
P_s = Suction gauge reading - atmospheric pressure (converted to meters of head)
ρ = Density of water (1000 kg/m^3)
g = Acceleration due to gravity (9.81 m/s^2)
2. Mechanical Efficiency:
The mechanical efficiency (n_mech) of a pump is the ratio of the power output at the impeller (P_out) to the power input to the pump shaft (P_in), expressed as a decimal.
P_in can be calculated by the following formula:
P_in = Q * h_m / ρ * n_p
where:
Q = Flow rate (180 l/s) - convert to m^3/s (1 m^3/s = 1000 l/s)
h_m = Head developed by the pump (in meters)
ρ = Density of water (1000 kg/m^3)
n_p = Pump efficiency - assume 0.75 (75%)
P_out is given in the problem as 68kW.
3. Total Efficiency:
The overall efficiency (n_overall) of the pump is the product of the manometric efficiency and the mechanical efficiency.
n_overall = n_m * n_mech
Now, let's calculate the required values step by step:
1. Calculate h_i:
Convert atmospheric pressure (in meters of head):
P_atm = 101325 Pa = 10.33 m (approximately)
P_d = 18.6 + 10.33 = 28.93 m
P_s = 3.4 - 10.33 = -6.93 m (negative since below atmospheric)
h_i = (28.93 - (-6.93)) / (1000 * 9.81) = 3.365 m
2. Calculate P_in:
Q = 180 / 1000 = 0.18 m^3/s
h_m = P_d - P_s = 28.93 - (-6.93) = 35.86 m (round to two decimal places)
P_in = 0.18 * 35.86 / (1000 * 0.75) = 0.0861 kW
3. Calculate n_overall:
P_out = 68 kW
n_mech = P_out / P_in = 68 / 0.0861 = 790.93
n_overall = n_m * n_mech = 0.759 * 0.79093 = 0.599 (approximately)
Therefore, the overall efficiency of the centrifugal pump is approximately 0.599 or 59.9%.