FLUID MECHANICS : Hydraulic machines Notes

By Deepanshu Rastogi|Updated : April 1st, 2021



Impulse and Reaction Principles

Turbo machines are classified as impulse and reaction machines depending on the relative proportions of the static and dynamic heads involved in the energy transfer. To aid this, we define a term referred to as degree of reaction Rd.

Degree of reaction Rd can be defined as the ratio of static head to the total head in the energy transfer.

11-Impulse-and-Reaction (1)


Degree of reaction can be zero, positive or negative.

Rd=0, characterizes a close turbo machine for which a static head is equal to zero.

In the most general case, this will happen if U= U2 and Vr1 = Vr2.

These classes of turbo machines are referred to as impulse machines. In most practical situations Vr2 may be less than Vr1 even though r= r2.

This is generally due to frictional losses. Even then a machine is referred to as an axial flow turbines and pumps would have r= r2 and if Vr1 = Vr2, then they become examples of pure impulse machines.

Pelton Wheel, tangential flow hydraulic machines is also example of impulse machine.

Velocity Triangles for impulse machine: Velocity triangle for axial flow impulse machine is shown in the following figure.

11-Impulse-and-Reaction (2)


The velocity of whirl at exit is to be calculated by general expression,

11-Impulse-and-Reaction (3)

If the value obtained is negative, then it suggests that

11-Impulse-and-Reaction (4)

If Vw2 is positive, then OVT would appear as follows:

11-Impulse-and-Reaction (5)


If V2=0, then the OVT would look like

11-Impulse-and-Reaction (6)


Radial flow Pump and Compressors:

General analysis:

11-Impulse-and-Reaction (7)


11-Impulse-and-Reaction (8)

11-Impulse-and-Reaction (9)

11-Impulse-and-Reaction (10)

Most of the turbo machines belong to this class. In general, they have a restricted flow area for a given rotor diameter and have low to medium specific speed.

Significant aspects:

  1. Flow is outwards from the smaller to larger radius the Euler’s turbine equation. i.e.,11-Impulse-and-Reaction (11)  requires that 11-Impulse-and-Reaction (12) for pumps and compressors which are power absorbing machines. For this sake radial flow compressors and pumps generally have fluid entering at a smaller radius and leaving at a larger radius.
  2. The absolute velocity at inlet is oriented parallel to the axes of the shaft i.e., Va1 = V1 and hence there is no whirl component at inlet i.e.,Vw1 = 0.
  3. Since Vw1 = 0, the energy transferred is purely a function of exit condition i.e. 11-Impulse-and-Reaction (13)

Head-capacity relationship: 

11-Impulse-and-Reaction (14)


11-Impulse-and-Reaction (15)

11-Impulse-and-Reaction (16)

11-Impulse-and-Reaction (17)

From the velocity triangles for the 3 types of vanes it may be noticed that the whirl component at exit is least for backward curved vane (β<90° and most for a forward curved vane. When operating under similar condition of speed and cross section area. But from a practical view point a high value of exit velocity V2 is not desirable. This is because it becomes necessary to construct a diffuser of unreasonably large dimensions even for moderate sized rotors. Hence backward curved vane with β2 in the range of 20-25 degrees is preferred for radial flow pumps and compressors. Forward curved vanes are not preferred while radial vanes (β=90°) are used in select applications requiring very high pressure.

Expression for Degree of reaction in terms of rotor velocity and rotor blade angles:

We know that, Degree of reaction is given by,

11-Impulse-and-Reaction (18)


11-Impulse-and-Reaction (19)

11-Impulse-and-Reaction (20)

11-Impulse-and-Reaction (21)

For a pump it is generally acceptable to write degree of reaction as

11-Impulse-and-Reaction (22)

11-Impulse-and-Reaction (23)

We know that, Euler’s turbine equation for a pump may be written as

11-Impulse-and-Reaction (24)


11-Impulse-and-Reaction (25)

Degree of reaction is the ratio of suction head to the total head. Which may be written as

11-Impulse-and-Reaction (26)

11-Impulse-and-Reaction (27)

11-Impulse-and-Reaction (28)

11-Impulse-and-Reaction (29)

11-Impulse-and-Reaction (30)

General analysis of Turbines:

They are power generating turbo machines, which run on both incompressible fluids such as water as well as compressible fluids such as gases.

The efficiency of turbines may be defined as the ratio of actual work output to the fluid energy input.

This involves 2 types of efficiencies:

  1. Hydraulic efficiency /isentropic efficiency.
  2. Mechanical efficiency.

The mechanical efficiency takes care of all losses due to energy transfer between mechanical elements. In the turbines, mechanical efficiency is very high and of the order of 98 to 99%.

The hydraulic efficiency takes care of losses during flow.

We realize that, turbines must have a residual exit velocity so that flow is maintained.

However, this residual velocity so that flow is it represents a lot far as the rotor is concerned. Hence, even if we have idealized friction free flow it is not possible to transfer all the energy in the fluids due to the need to have the final residual exit velocity.

Hence, hydraulic efficiency is a product of 2 terms and is given by

ηH = ε*ηV

ηV - where is referred to as vane efficiency and takes care of frictional loss.

Utilization factor:

Utilization factor is defined as the ratio of the actual work transferred from the fluid to the rotor in an ideal condition to the maximum possible work that could be transferred in an ideal condition.

11-Impulse-and-Reaction (31)


11-Impulse-and-Reaction (32)

Relationship between ε and Rd:


Derive an expression for ε in terms of Rd:

11-Impulse-and-Reaction (33)


11-Impulse-and-Reaction (34)

11-Impulse-and-Reaction (35)

11-Impulse-and-Reaction (36)

11-Impulse-and-Reaction (37)

11-Impulse-and-Reaction (38)

Utilization factor may be written as

11-Impulse-and-Reaction (39)

Which gives,  11-Impulse-and-Reaction (44)

This expression holds good for Rd values between 0 and 1. This cannot be used for Rd=1 (100% reaction). Since, the expression becomes equal to 1 suggesting 100% utilization factor which could obviously lead to residual exit velocity V2 becoming zero.

General analysis of Axial flow turbines:

Most turbines involving compressible flow are axial turbines. Generally, steam and gas turbines are axial flow machines.

We know that in all axial turbine machines, U= U= U.

And hence the alternative form of turbine equation reduces to

11-Impulse-and-Reaction (45)


Degree of reaction, 11-Impulse-and-Reaction (46)

Change of fluid pressure in the rotor happens only due to change in the relative velocity component Vr, since, U remains constant.

Axial flow turbines are of 2 types:

  1. Impulse type for which Rd=0.since Vr1=Vr2 and hence power output 11-Impulse-and-Reaction (47)
  2. Reaction type: Generally any turbine which is not purely an impulse turbine is referred to as a reaction turbine. It is not a 100% reaction turbine. But, it is still referred to as a reaction turbine. Most reaction turbines are designed for 50% reaction which is found to be very advantageous from practical consideration. In the case of steam turbines it is implicit that a reaction turbine is 50% reaction turbine called as parson’s reaction turbine. 


If you are preparing for ESE/ GATE or other PSU Exams (Civil Engineering), then avail Online Classroom Program for ESE and GATE CE:

Online Classroom Program for ESE/GATE CE(20+ Courses and 180+ Mock Tests)

You can avail of BYJU'S Exam Prep Test Series specially designed for all AE & JE Exams:

BYJU'S Exam Prep Test Series AE & JE (160+ Mock Tests)

You can avail of BYJU'S Exam Prep Test Series specially designed for all Civil Engineering Exams:

BYJU'S Exam Prep Test Series ESE/GATE CE (180+ Mock Tests)


Sahi Prep Hai To Life Set Hai.



write a comment

AE & JE Exams


Follow us for latest updates