Mechanical Engineering - Hydraulic Machines - Discussion

The relationship can be explained by the Affinity laws as described below:

Law 1. With impeller diameter (D) held constant:

Law 1a. Flow is proportional to shaft speed:
(Q1/Q2) = (N1/N2).

Law 1b. Pressure or Head is proportional to the square of shaft speed:
(H1/H2) = (N1/N2)^2.

Law 1c. Power is proportional to the cube of shaft speed:
(P1/P2) = (N1/N2)^3.

Law 2. With shaft speed (N) held constant:

Law 2a. Flow is proportional to impeller diameter:
(Q1/Q2) = (D1/D2).

Law 2b. Pressure or Head is proportional to square of impeller diameter:
(H1/H2) = (D1/D2)^2.

Law 2c. Power is proportional to the cube of impeller diameter:
(P1/P2) = (D1/D2)^3.

where
Q is the volumetric flow rate (e.g. CFM, GPM or L/s),
D is the impeller diameter (e.g. in or mm),
N is the shaft rotational speed (e.g. rpm),
H is the pressure or head developed by the fan/pump (e.g. psi or Pascal), and
P is the shaft power (e.g. W).

Note that there are two sets of affinity laws:.

Affinity laws for a specific centrifugal pump - to approximate head, capacity and power curves for different motor speeds and /or different diameter of impellers.

Affinity laws for a family of geometrically similar centrifugal pumps - to approximate head, capacity and power curves for different motor speeds and /or different diameter of impellers.

Power = density * acceleration due to gravity * discharge * head loss.

Where Q = A * V = area * velocity.

Area= (pi/4) *d^2.

Velocity= (pi * d * N/60).

Head loss= (friction * length * velocity^2)/(2 * g * d).

= (f * length * (pi * d * N/60) ^2)/(2 * g * d).

Assume f, length and N are constant.

Head loss proportional to diameter.

Power = constant * d^4.

According to affinity law, discharge is proportional to rpm, head is proportional to the square of the head, &power is proportional to the cube of rpm. On the other hand, discharge is proportional to shaft diameter, head is proportional to the square of diameter & power is proportional to the cube of shaft diameter.

Correct Ans is C.

Law 2c Power is proportional to the cube of impeller diameter (assuming constant shaft speed):

{\displaystyle {P_{1} \over P_{2}}={\left({D_{1} \over D_{2}}\right)^{3}}} {P_{1} \over P_{2}}={\left({D_{1} \over D_{2}}\right)^{3}}.

There are so much of answers if we look into this from different laws and formulas. What answer should we choose in the exams?

Is someone there to give the most appropriate answer?

d^4 will be the correct answer. Since the question is asked as directly proportional the tangent flow of an impulse turbine always takes 4th power to diameter.

According to the affinity laws horsepower is directly proportional to the fifth power of impeller diameter or cube of rpm, so this answer is not right.

Affinity laws applied to axial and radial flows pumps and turbines. Its a tangent flow ; impulse turbine so I think 4th power is correct.

Yes this answer is right but its against affinity laws, also discharge varies to square of this which is also against affinity law.