For the operating point of the pump, a system characteristic between the head required ‘H’ and the discharge to be maintained ‘Q’ is generally expressed as

Option 2 : Parabolic equation

__Concept:__

Before manufacturing the large-sized pumps, their models which are in complete similarity with the actual pumps (also called prototypes) are made. Tests are conducted on the models and the performance of the prototype is predicted. The complete similarity between the model and actual will exist if the following condition is satisfied.

I)\(\left ( \frac{\sqrt{H}}{DN} \right )_m=\left ( \frac{\sqrt{H}}{DN} \right )_p\;\;\;\;(1)\)

II) \(\left ( \frac{Q}{D^3N} \right )_m=\left ( \frac{Q}{D^3N} \right )_p\;\;\;\;\;(2)\)

III) \(\left ( \frac{P}{D^5{N^3}} \right )_m=\left ( \frac{P}{D^5{N^3}} \right )_p\;\;\;\;\;(3)\)

From (1)

\(\sqrt{H}∝\;N\)

From (2)

Q ∝ N

Combining (1) and (2)

\(\sqrt{H}∝\;Q\)

∴ H ∝ Q^{2}

∴ the head 'H' varies with the square of discharge 'Q'

The relation between head 'H' and discharge 'Q' can be better understood with the Operating characteristic curve as shown in the figure which gives the relation between the manometric head (H), power (P) and efficiency (η) with respect to the discharge when the speed (N) is kept constant.

Option 4 : 384 hp

**Concept:**

The relationship between power and diameter of a centrifugal pump is governed by the following equation:-

P ∝ D^{5} N^{3}

Where

**Calculation:**

In the given question it is said that the only diameter is changed and other parameters are constant then

P ∝ D^{5}

\(\frac{{{P_1}}}{{D_1^5}} = \frac{{{P_2}}}{{D_2^5}}\)

\(\Rightarrow \frac{{12}}{{{{127}^5}}} = \frac{{{P_2}}}{{{{254}^5}}}\)

\(\Rightarrow {P_2} = {\left( {\frac{{254}}{{127}}} \right)^5} \times 12 = {2^5} \times 12\;hp = 384\;hp\)

Option 2 : free vortex flow

**Explanation:**

**Centrifugal pump works on the principle of force vortex**. Where external torque is provided to the impeller by the means of the prime mover.

In a **pump**, there are **two important parts**, first is the **impeller** which **creates velocity through rotation**. And the second is the **casing** which **converts this velocity into pressure by the change in the area**.

When fluid is inside the impeller, then the speed of the fluid experience a rotational motion because of the torque provided by the prime mover, and flow is characterized as a** forced vortex flow**. When the fluid comes out from the rotating impeller at that time also **fluid has a vortex motion because of the inertia of the fluid. But since the external torque is absent outside the casing therefore it becomes free vortex flow.**

__Additional Information__

There are two types of casing generally available for the centrifugal pump.

**1.Volute casing**:

In volute casing area gradually increase as you can see in the above figure. Because of the gradual increase in the area the velocity decreases and pressure increases.

**2.Diffuser casing:**

In diffuser casing impeller periphery is designed in such a way that its area gradually increases which promotes the rise in the pressure at the expense of the velocity.

Option 3 : N^{3}

** Explanation**:

**Power of the centrifugal pump is given by,**

\(P = \frac{{\rho Qg{H_m}}}{{1000}}\)

i.e. P ∝ Q × Hm

From modal law of the pump, \(\frac{{Q}}{{ND^3}}=C\)

i.e. P ∝ D3 × N × Hm

From modal law of the pump, \(\frac{{H}}{{N^2D^2}}= C\)

∴ P ∝ D3 × N × D2 × N 2

P ∝ D^{5 }× N^{3}

**∴ P ∝ N ^{3} **

Option 2 : Net Positive Suction Head

**Explanation:**

**Net Positive Suction Head:**

The Net positive suction head (NPSH) is defined as the** absolute pressure head at the inlet to the pump**, **minus the vapour pressure head (in absolute units) plus the velocity head.**

\({\rm{NPSH}} = \frac{{{P_1}}}{{\rho g}} - \frac{{{P_v}}}{{\rho g}} + \frac{{{v_s}^2}}{{2g}}\)

Where, P1 = Absolute pressure at the inlet of the pump, Pv = Vapour pressure at the inlet, vs = velocity of the fluid in the suction pipe

**Stagnation head:** The stagnation head is the sum of the static head and velocity head.

\(H_{stag} = \frac{{{P_1}}}{{\rho g}} + \frac{{{v_s}^2}}{{2g}}\)

**Hydraulic head:** The elevation of a water body above a particular datum level is called a hydraulic head.

- The hydraulic head represents the potential energy stored in the fluid.
- It is also called the
**datum head.**

H_{Hydraulic }= Z (height of the free surface of the water from datum)

**Total energy head:** Total head is the sum of **pressure head, kinetic head, and potential head.**

\({\rm{Total\;head}} = \frac{p}{{\rho g}} + \frac{{{v^2}}}{{2g}} + z\)

A centrifugal pump driven by a directly coupled 3 kW motor of 1450 rpm speed, is proposed to be connected to a motor of 2900 rpm speed. The power of the motor should be

Option 4 : 24 kW

__Concept:__

**Velocity in pump:**

**\(V = \frac{{\pi \times D \times N}}{{60}} = \sqrt {2gh} \)**

From above, \(h = \frac{{{\pi ^2} \times {D^2} \times {N^2}}}{{3600 \times 2g}}\)

**Discharge of a pump:**

Q = A × V

\(Q = \frac{{{\pi ^2} \times {D^3} \times N}}{{240}}\)

**Power of a pump:**

P = ρ × g × Q × h

\(P = \rho \times g \times \frac{{{\pi ^2} \times {D^3} \times N}}{{240}} \times \frac{{{\pi ^2} \times {D^2} \times {N^2}}}{{3600 \times 2g}}\)

P = K × N^{3}

\(\frac{{{P_1}}}{{{P_2}}} = \frac{{N_1^3}}{{N_2^3}}\)

__Calculation:__

__Given:__

P_{1} = 3 kW, N_{1} = 1450 rpm, and N_{2} = 2900 rpm.

From, \(\frac{{{P_1}}}{{{P_2}}} = \frac{{N_1^3}}{{N_2^3}}\)

\(\frac{3}{{{P_2}}} = {\left( {\frac{{1450}}{{2900}}} \right)^3}\)

P_{2 }= 24 kW.

Option 4 : Open impeller

**Explanation:**

Closed impellers (Two-sides shrouded):

- In the closed or shrouded impellers, the vanes are covered with shrouds (side plates) on both sides
- The back shroud is mounted into the shaft and the front shroud is coupled by the vanes
- This ensures full capacity operation with high efficiency for a prolonged running period
- This type of impeller is meant to pump
**only clear water, hot water and acids**

Semi-open impeller (One-side shrouded):

- It has a plate (shroud) only on the backside
- The design is adapted to industrial pump problems which require a rugged pump to handle
**liquids containing fibrous material**such as paper pulp, sugar molasses and sewage water etc.

Open impeller:

- In open impeller, no shroud or plate is provided on either side i.e. the vanes are open on both sides
- Such pumps are used where the pump has a very rough duty to perform i.e. to handle
**abrasive liquids such as a mixture of water, sand, pebbles and clay, wherein the solid contents**may be as high as 25%.

Thus, centrifugal pumps dealing with **water containing slurry and sewage**** have **an open impeller.

If the characteristics of a pump are as shown in figure. What is represented by abscissa?

Option 3 : Discharge

__Explanation:__

Characteristic curves are necessary to predict the behaviour and performance of the pump when the pump is working under different flow rate, head and speed.

If the speed is kept constant, the variation of manometric head, power and efficiency with respect to discharge gives the operating characteristic curves of a pump.

The abscissa represents the discharge of the pump.

Option 3 : inward radial flow reaction turbine

**Explanation:**

- The centrifugal pump acts as a reverse of an inward radial flow reaction turbine.
- This means that the flow in centrifugal pumps is in the radial outward direction.

- The centrifugal pump works on the principle of forced vortex flow which means that when a certain mass of liquid is rotated by an external torque, the rise in pressure head of the rotating liquid takes place.
- In a centrifugal pump casing, the flow of water leaving the impeller is a free vortex.

Option 2 : It is used for large discharge through smaller heads

**Explanation:**

Centrifugal or radial pumps are simple and versatile constructions with a wide range of impeller design. The motion of water in it is from the centre towards the periphery. The motion is caused by the centrifugal force created in the pump as a result of the revolving motion of the working wheel. It is used for large discharge through smaller heads.

Water enters at the centre of the impeller. The water passes between the vanes and is pushed radically towards the casing and then onwards through the discharge nozzle. The vanes create an increase in both water velocity and pressure.

The housing that encloses the rotating element and seals the pressurized liquid inside commonly called as Casing. The casing is of spiral shape and it terminates in a delivery pipe.Option 3 : \(N_{s}=\frac{N\sqrt{Q}}{H_{m}^{3/4}}\)

__Explanation:__

**Specific speed:**

- It is defined as the
**speed of a geometrically similar pump that would deliver one cubic meter of liquid per second against the head of one meter.** - It is used to compare the performances of 2 different pumps.
- Its dimension is M
^{0}L^{3/4}T^{-3/2}and given by the formula and is given by

**\(N_{s}=\frac{N\sqrt{Q}}{H_{m}^{3/4}}\)**

Where N_{S} = Specific speed, Q = Discharge, H = Head under which the pump is working, N = Speed at the pump is working.

__Additional Information__

There is also one dimensionless specific speed, which is called **shape numbe**r, and given by

**\(N_{s}=\frac{ω \sqrt{Q}}{gH_{m}^{3/4}}\)**

Where ω = Angular speed of the pump, Q = Discharge, H = Head, g = Acceleration due to gravity

Option 3 : Directly proportional to cube of the speed of its impeller

__Explanation__:

Power of the centrifugal pump is given by,

\(P = \frac{{\rho Qg{H_m}}}{{1000}}\)

i.e. P ∝ Q × Hm

From model law of the pump:

\(\frac{{Q}}{{ND^3}}=C\)

∴ Q ∝ D3 × N

i.e. P ∝ D3 × N × Hm

From the model law of the pump:

\(\frac{{H_m}}{{N^2D^2}}= C\)

∴ H_{m} ∝ N2 × D2

i.e. P ∝ D3 × N × D2 × N2

P ∝ D5 × N3

P ∝ N^{3}

i.e. **the power of a centrifugal pump is directly proportional to the cube of the impeller speed.**

Option 2 : N

**Explanation:**

Centrifugal Pumps:

- Centrifugal or radial pumps are simple and versatile constructions with a wide range of impeller design.
- The motion of water in it is from the centre towards the periphery.
- The motion is caused by the
**centrifugal force created in the pump as a result of the revolving motion of the working wheel.** - Water enters at the centre of the impeller.

**The discharge of a centrifugal pump is given by:**

Q = Area × Velocity of flow

Q = (π × D × B × V_{f})

where Q = Discharge, V_{f} = Flow velocity, D = Diameter of the impeller, B = Width of the impeller

As we know B ∝ D

\(V_f \propto u={πDN\over 60 }\)

Vf ∝ DN

where N = Speed in rpm

**∴ Q ∝ D3 × N**

**Q ∝ N**

Option 2 : 10π m / s

__Concept:__

The tangential velocity of impeller at inlet is given as:

U = πDN/60

Where

D is inlet diameter of impeller

N is speed in rpm

__Calculation: __

**Given:**

D = 50 cm; N = 1200 rpm

\(U = \frac{{\pi\; \times \;0.5\; \times \;1200}}{{60}}\)

**∴**** U = 10π m/s**

__Note:__

1. In above, equation if it is asked to find tangential velocity at outlet then we D as outlet diameter.

2. If both inlet and outlet diameters are given and it is asked to find out the tangential velocity of impeller 9 nothing specified inlet or outlet), then we should consider the average of inlet and outlet diameters.

Option 3 : D^{3}

__Explanation__:-

With the use of the **Buckingham Pi theorem**,

For Discharge in Pumps;

The dimensionless analysis of the centrifugal pumps proves that the discharge is written as

- Two geometrically similar pumps said to be homogeneous when

**Q ∝ D3N**

\(\frac{Q}{{{D^3}N}} = Constant\)

where Q = Flow rate, D = Diameter of the impeller, N = Rotational Speed.

The Flow rate is given by,

\(Q = C_d\sqrt {2gH}\)

where C_{d} = coefficient of discharge, A = Reference area, and H = Head.

**For the pumps with the same speed (N), discharge is proportional to D ^{3}. i.e Q ∝ D3**

__Important Points__

Some relations for Pumps is as follows,

- For Head H ∝ D
^{2}N^{2} - For Power P ∝ D
^{5}N^{3} - For Specific Speed = \({N_S} = \frac{{N\sqrt Q }}{{H_m^{3/4}}}\)

Head coefficient is given by,

\(\frac{{H}}{{{D^2}N^2}} = C\)

A centrifugal pump gives maximum efficiency when its blades are

Option 2 : Bent backward

__Explanation:__

Backward Curved Blades: vane exit angle is less than 90°

Forward Curved Blades: vane exit angle is more than 90°

Radial Blades: Vane exit angle is 90°

The blades of the compressor or either forward curved or backward curved or radial. Backward curved blades were used in the older compressors, whereas the modern centrifugal compressors use mostly radial blades.

Pumps are not usually designed with forward curved vanes since such pumps tend to suffer unstable flow conditions.

Backward curved blades are slightly better in efficiency and are stable over a wide range of flow. While forward-curved blades are used for higher pressure ratio.

Forward Curved Vanes |
Small Volume |
High-Pressure ratio |
High speed, High noise, Low Efficiency |

Backward curved Vanes |
Large Volume and size |
Low to High-Pressure Ratio |
High Efficiency, Low Noise |

Radial Vanes |
Medium Volume and Size |
Medium to High-Pressure ratio |
Good Efficiency |

Option 3 : Manometric head

__Explanation:__

**Centrifugal Pumps:**

A centrifugal pump is a machine which converts the **kinetic energy **of the water into** pressure energy** before the water leaves its casting. The flow of water leaving the impeller is** free vortex **. The impeller of a centrifugal pump may have volute casing, vortex casing and volute casing with guide blades.

- The static head is the actual difference in the elevations.
- The kinetic and velocity heads are the same which exits because of the kinetic energy of the water in the pump because of the flow.
- The total manometric head is the difference in pressure (in metres) between the pump’s inlet and outlet points. This value is always higher than the actual difference in elevation between these two points; when pumping is going on, the pump needs also to overcome friction losses occurring as the water flows through the intake and outlet pipes.
- Thus, the pump will only deliver the water if the pressure rise in the
**impeller is equal to or more than the**manometric head.

**Different heads in centrifugal pump:**

- Suction Head (hs): It is the vertical height of the centre line of the centrifugal pump above the water surface in the tank or pump from which water is to be lifted.
- Delivery head (hd): The vertical distance between the centre line of the pump and the water surface in the tank to which water is delivered is known as delivery head.
- Static Head (H): The sum of the suction head and the delivery head is known as the static head. H = hs + hd
- Manometric Head (Hm): The manometric head is defined as the head against which a centrifugal pump has to work. It is given by the following expressions:

- Hm = Head imparted by the impeller to the water – Loss of head in the pump
- Hm = Total head at the outlet of the pump – Total head at the inlet of the pump
- Hm = Suction Head (hs) + Delivery Head (hd) + Friction head loss in the suction pipe (hfs) + Friction head loss in the Delivery pipe (hfd) + Velocity head of water in the delivery pipe (V2/2g)

Option 1 : Radially

Centrifugal or radial pumps are simple and versatile constructions with a wide range of impeller design. The motion of water in it is from the centre towards the periphery. The motion is caused by the centrifugal force created in the pump as a result of the revolving motion of the working wheel.

- Water enters at the centre of the impeller. The water passes between the vanes and is pushed radically towards the casing and then onwards through the discharge nozzle. The vanes create an increase in both water velocity and pressure.
- The housing that encloses the rotating element and seals the pressurized liquid inside commonly called as Casing. The casing is of spiral shape and it terminates in a delivery pipe.
**Centrifugal pump acts as a reverse of an inward radial flow reaction turbine. This means that the flow in centrifugal pumps is in the radial outward directions. The flow enters the chamber along the axis of the chamber and is discharged radially.**

The casing of a centrifugal pump is designed to minimise

Option 4 : Loss of kinetic energy

__Explanation:__

**The casing of the centrifugal pump is designed to minimise the Loss of kinetic energy.**- The volute (spiral geometry) of a centrifugal pump is the casing that receives the fluid being pumped by the impeller, maintaining the velocity of the fluid through to the diffuser.
- As liquid exits, the impeller has high kinetic energy, and the volute directs this flow through to the discharge.
- The concept that pressure decreases with an increase in the applied area is valid only for static conditions. But in this case, the total energy of the unit volume of fluid should remain constant for the flow to be continuous.
- Now, from Bernoulli's equation, the total energy is mainly divided into kinetic energy, potential energy, and pressure energy.
- Therefore when there occurs a drop in the kinetic energy as in the case of the fluid coming out from an eye of an impeller; it is recovered in the form of a rise in the pressure of the fluid.
- Velocity will not remain constant, instead, it will decrease due to a sudden increase in the area.
- It does not eliminate the loss of head due to change in the velocity instead it converts the head into a pressure head.

The internal and external diameter of the impeller of a centrifugal pump are 200 mm and 400 mm respectively. The pump is running at 1200 rpm. Find the velocity of flow if the vane angle at inlet 20°.

Given tan 20° = 0.363

Option 3 : 4.57 m/s

**Concept:**

Tangential velocity at inlet \({u_1} = \frac{{\pi {D_1}N}}{{60}}\)

\(\left( {\tan \theta } \right) = \frac{{{V_{f1}}}}{{{u_1}}}\)

**Calculation:**

**Given:**

N = 1200 rpm, Inner Diameter = 200 mm = 0.2 m, Outer Diameter = 250 mm

**Now,**

Tangential velocity at inlet \({u_1} = \frac{{\pi {D_1}N}}{{60}}\)

\(= \frac{{\pi × 0.2 × 1200}}{{60}} = 12.56\;{m}/{s}\)

Now,

∴ Vane angle at inlet \(\left( {\tan \theta } \right) = \frac{{{V_{f1}}}}{{{u_1}}} = \frac{{V_{f1}}}{{10.47}}\)

∴ tan 20° × 12.56 = V_{f1}

∴ Vf1= 4.57 m/s