## Velocity Diagrams

Velocity diagrams for different values of Rd appear as shown below:

(i) V_{1}>V_{2}, V_{r1}>V_{r2}; R_{d}<0 (R_{d} is negative)

(ii) V_{r1}=V_{r2}; R_{d}=0

(iii) V_{1}=V_{r2}; V_{r1}=V_{2}; R_{d}=0.5

(iv) V_{1}=V_{2 }; R_{d}=100%

(v) V_{2}>V_{1}; V_{r2}>V_{r1} ; R_{d}>100%

We know that for utilization factor ε to be maximum, the exit velocity V2 should be minimum.

For a given rotor speed U, the minimum value of V2 is obtained only if V22 is axial and the velocity triangles would look as shown:

Velocity triangle for maximum utilization factor condition:

Form the expression, it is clear that ε_{maximum} will have the highest value if α_{1 }= 0.

But α_{1} = 0, results in V2 = 0 which is not a practically feasible condition. The zero angle turbines which would have α_{1} = 0 appears as shown:

Though the zero angle turbines are not practically feasible it represents the ideal condition to be aimed at. In a Pelton wheel, we arrive at a condition wherein the jet is deflected through an angle of 165 to 170 degrees. Though an angle of 180 degrees would be the ideal condition as in case of a zero angle turbine. Impulse turbine designed for maximum utilization.

The ratio is referred to as a blade speed ratio φ which will have limiting value of 0.5 for a zero angle turbine. But in practical situation, α_{1 }is in between 20 to 25 degrees. But φ varies from 0.45 to 0.47. The blade speed ratio is very useful performance parameter and it may be noted that the closer its value is to 0.5, the better it is.

**Expression for power output:**

**Reaction turbine:**

We know that,

For a fixed value of α_{1}, as Rd increases ε_{maximum}.

But for Rd=1 (100% reaction turbine), this equation doesn’t holds good.

Let us examine how ε_{maximum} is affected by Rd.

**Case (1):** Rd=1,

V_{w1}=V_{w2}

Hence by Euler’s turbine equation

For maximum utilization V_{2} needs to be axial. If V_{2} is to be axial, then V_{1} also should be axial which means that the denominator of the expression becomes equal to infinity which reduces to to zero. This only means that α_{1} should be as low as possible to get meaning full values of. This represents contradicting condition and hence Rd = 1 is not preferred.

**Case (2):** Rd>1

V2>V1, Vr2>Vr1

As Rd> ε tends to zero.

In this case, V_{2}>V_{1 }and hence V_{2} can never be axial and hence the condition for ε_{maximum} [An axial orientation for V2 can never be met]

The utilization factor ε is given by

As Rd increases, ε decreases.

This means that the stator has to function to not only diffuse V_{2} to as low a value as possible but also turn the fluid through a very large angle. This results in the poor flow efficiency and hence Rd greater than 100% is not practically preferred.

**Case (3):** Rd<0 [negative Rd]

V_{r1}>V_{r2}

For this condition, it is noticed that rd is negative denominator increases, ε decreases.

V_{r2}<V_{r1}, also means that the pressure is increasing as fluid passes through the rotor.

i.e. the rotor is acting like a diffuser. This is not preferred since pressure always has to decrease along the flow path for good flow efficiency. Hence, Rd < 0 is not practically preferred.

**Case (4):** Rd = 0.5,

We know that for a 50% reaction turbine, velocity triangles are similar and for maximum utilization condition the triangle would appear as shown.

We notice,

The angles are identical but reversed for the rotor and the stator. From the practical view point, the manufacturing of blades becomes simple. Since the same blade can be used for either the stator or the rotor by merely reversing the direction. It can also be shown that in a multistage turbines 50% reaction gives maximum stage efficiency. Since V_{r2}>V_{r1}, pressure reduces along the flow path in the rotor resulting in high flow efficiency. In general, n Rd value between 0 and 1 is preferred due to practical considerations.

From the velocity triangle it can be noted that V_{w1}=U

V_{w2} = 0 (for maximum utilization factor condition)

*∴ P = U ^{2}*

Comparing the energy transfer achieved by 50% reaction turbine with an impulse turbine when both are designed for ε_{maximum} condition and operating with the same rotor velocities. We notice that an impulse turbine transfers twice as much energy as 50% reaction turbine gives the better flow efficiencies.

If multi staging is attempted, then for a given value of energy transfer, a 50% reaction turbine would need twice the number of stages as that of impulse turbines. In actual practice, when multistage is attempted, the initial stages are designed for an impulse turbine when maximum fluid velocity is available. The subsequent stages are 50% reaction stages.

## 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.

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_{1 }= U_{2} and V_{r1 }= V_{r2}.

These classes of turbo machines are referred to as impulse machines. In most practical situations V_{r2} may be less than V_{r1} even though r_{1 }= r_{2}.

This is generally due to frictional losses. Even then a machine is referred to as an axial flow turbines and pumps would have r_{1 }= r_{2} and if V_{r1 }= V_{r2}, 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.

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

If the value obtained is negative, then it suggests that

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

If V2=0, then the OVT would look like

**Radial flow Pump and Compressors:**

General analysis:

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:**

- Flow is outwards from the smaller to larger radius the Euler’s turbine equation. i.e., requires that 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.
- The absolute velocity at inlet is oriented parallel to the axes of the shaft i.e., V
_{a1 }= V_{1}and hence there is no whirl component at inlet i.e.,V_{w1 }= 0. - Since V
_{w1 }= 0, the energy transferred is purely a function of exit condition i.e.

**Head-capacity relationship:**

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 *β*_{2} 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,

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

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

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

**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:

**Hydraulic efficiency /isentropic efficiency.****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

*η _{H} = ε*η_{V}*

*η _{V}* - 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.

Relationship between *ε* and *R _{d}*:

OR

Derive an expression for *ε* in terms of Rd:

Utilization factor may be written as

Which gives,

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_{1 }= U_{2 }= U.

And hence the alternative form of turbine equation reduces to

Degree of reaction,

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:

- Impulse type for which Rd=0.since V
_{r1}=V_{r2 }and hence power output - 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.

## Turbomachinery

Turbomachine is defined as a device in which energy transfer takes place between a flowing fluid and a rotating element resulting in a change of pressure and momentum of the fluid. Energy is transferred into or out of the turbomachine mechanically by means of input/output shafts.

**Principal Parts of a Turbo Machine**

- Rotating element consisting of a rotor on which are mounted blades.
- A stationary element in the form of guide blades, nozzles, etc.
- Input/output shafts.
- Housing

Schematic cross sectional view of a steam turbine showing the principal parts of a turbo machine.

**Functions:**

- The rotor functions to absorb/deliver energy to the flowing fluid.
- The stator is a stationary element which may be of many types:-
- Guide blades which function to direct the flowing fluid in such a way that energy transfer is maximized.
- Nozzles which function to convert pressure energy of the fluid to kinetic energy
- Diffusers which function to convert kinetic energy to pressure energy of the fluid.

- The input /output shafts function to deliver/receive mechanical energy to or from the machine.
- The housing is a protective enclosure which also functions to provide a path of flowing fluid. While a rotor & input /output shaft are essential parts of all turbo machines, the stator & the housing are optional.

**Classification of Turbo Machines:**

- According to the nature of energy transfer:
- Power generating turbo machines: In this, energy is transferred from the flowing fluid to the rotor. Hence, enthalpy of the flowing fluid decreases as it flows across. There is a need for an output shaft.
- Ex: Hydraulic turbines such as Francis turbine, Pelton wheel turbine, Kaplan turbine, steam turbine such as De-Laval turbine, Parsons Turbine etc, Gas turbines etc,
- Power absorbing Turbo machines: In this, energy is transferred from the rotor to the flowing fluid. The enthalpy of the fluid increases as it flows there is a need for an input shaft.
- Ex: Centrifugal pump, Compressor, blower, fan etc,
- Power transmitting turbo machines: In this energy is transferred from one rotor to another by means of a flowing fluid. There is a need for an input / output shafts. The transfer of energy occurs due to fluid action.
- Ex: Hydraulic coupling, torque converter etc,

**Schematic representation of different types of turbo machine based on fluid flow:**

- Axial flow fan.
- Radial outward flow fan.
- Mixed flow hydraulic turbine.

**Based on the type of fluid flow:**- Tangential flow in which fluid flows tangential to the rotor Ex: Pelton wheel etc,
- Axial flow in which the fluid flows more or less parallel to the axes of the shafts /rotors. Ex: Kaplan turbine, Axial flow compressor.
- Radial flow in which fluid flows along the radius of the rotor this is again classified as:
- Radially inward flow. Ex: Old fancies turbine.
- Radially outward flow. Ex: Centrifugal Pump

- Mixed flow which involves radius entry & axial exit or vise-versa. Ex: Modern francises turbine & Centrifugal Pump

**Based on the type of Head:**- High head &low discharge. Ex: Pelton wheel.
- Medium head &medium discharge. Ex: Francis turbine.
- Low head & high discharge. Ex: Kaplan turbine.

**Application of 1 ^{st} & 2^{nd} law of thermodynamics of turbo machines:**

In a turbo machine, the fluctuations in the properties when observed over a period of time are found to be negligible. Hence, a turbo machine may be treated as a steady flow machine with reasonable accuracy & hence, we may apply the steady flow energy equation for the analysis of turbo machine.

Hence we may write

Where, subscript ‘1’ is at the point of entry & subscript ‘2’ is at point of exit.

It is also true that, thermal losses are minimal compared to the amount of work transferred & hence may be neglected. Hence we may write,

Where, h_{02 }& h_{01} are stagnation exit & entry respectively.

w = ∆h_{0.}

In a power generating turbo machine, ∆h_{0} is negative (since h_{02} < h_{01}) & hence w is positive.

On the same line, for a power absorbing turbo machine, ∆h_{0} is positive (since h_{02 }> h_{01}) & hence w is negative.

From the 2^{nd} law of Thermodynamics:

In the above relation, we note that vdp would be a negative quantity for a power generating turbo machine & positive for power absorbing turbo machine.

Hence Tds which is always a positive quantity would reduce the amount of work generated in the former case & increase the work absorbed in the later case.

**Efficiency of a turbo machine:**

Generally, we define 2 types of turbo machine .in case of turbo machine to account for various losses 2 type of efficiency is considered:

- Hydraulic efficiency/isentropic efficiency
- Mechanical efficiency.

**Hydraulic efficiency/isentropic efficiency:**

To account for the energy loss between the fluid & the rotor

2. **Mechanical efficiency:**

To account for the energy loss between the rotor & the shaft.

**Schematic representation of Compression & Expansion process:**

(a) Power absorbing machine. (b) Power generating machine.

**Analysis of Energy Transfer in turbo machines:** Analysis of energy transfer in turbo machines requires a consideration of the kinematics and dynamic factors involved. The factors include changes in the fluid velocity, rotor velocity and the forces caused due to change in the velocity.

We apply Newton’s second law of motion as applicable to rotary movement. i.e., Torque is proportional to the rate of change of angular momentum.

Another important consideration is the treatment of a turbo machine as a steady flow machine.

- This involves following assumptions:
- Mass-flow rate is constant.
- State of fluid at any given point does not change.
- Heat and Work transfer are constant.
- Leakage losses are negligible.
- Same steady mass of fluid flows through all section.

Velocity Components:

Fig. velocity components through a rotor

The fluid enters the rotor with an absolute velocity say V_{1} and leaves with an absolute velocity say V_{2.}

The absolute velocity of the fluid will have components in the axial, radial and tangential direction which may be referred to as V_{a},V_{w} and V_{f }respectively.

The axial components do not participate in the energy transfer but cause a thrust which is borne by the thrust bearings. The radial components also do not participate in the energy transfer but cause a thrust which are borne by the journal bearings. The only components which participate in the energy transfer is the tangential component Vw.

V_{a1} and V_{a2} : Axial components of V_{1} and V_{2} respectively.

V_{f1 }and V_{f2} : Radial components of V_{1} and V_{2 }respectively.

V_{w1} and V_{w2 }: Tangential components of V_{1} and V_{2 }respectively referred to as whirl velocity, flow velocity. Let the rotor move with an angular velocity ω.

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