Fluid Machines Notes

Fluid Machines (EG 616 ME)

Ram C. Poudel
Department of Mechanical Engineering, Pulchowk Campus
7 December 2012

Chapter 1: Introduction

1.1 Turbo-machines
1.2 Hydraulic Machines
1.3 History of Development of Water Wheels & Water Turbines
1.4 Water Wheels

Turbomachines

Definition: A device in which energy is transferred either to or from a continuously flowing fluid by the dynamic action of one or more moving blades rows.
Etymology: Turbo - turbinis - latin: which spins or whirls around.

Categories of Turbomachines

Power Absorbing:
- Absorb power to increase fluid pressure and head.
- Examples: Fans, compressors, pumps, etc.
Power Producing:
- Produce power by expanding fluid to a lower pressure or head.
- Examples: Wind, hydraulic, steam, and gas turbines.

Categories Based on Flow Direction

Axial Flow Turbo-machine:
- Path of through-flow is wholly or mainly parallel to the axis of rotation.
Radial Flow Turbo-machines:
- Path of through flow is wholly or mainly in a plane perpendicular to the rotational axis. [Figure 1.1c]
Mixed flow Turbo-machines:
- The direction of through flow at the rotor outlet when both radial and axial velocity components are present in significant amount [Figure 1.1b]

Examples of Turbomachines

Single stage axial flow compressor or pump
Mixed flow pump
Centrifugal compressor or pump
Francis turbine (mixed flow type)
Kaplan turbine
Pelton wheel

Coordinate Systems

Cylindrical polar coordinate system is used for description and analysis due to the cylindrical shape of turbomachines.
Axes:
- Axial (x)
- Radial (r)
- Tangential (or circumferential) ( $r\theta$ )

Simplifications and Meridional Velocity

Simplification: Flow does not vary in the tangential direction.
Axi-symmetric stream surfaces: Flow moves through the machine on these surfaces.
Meridional velocity ( $C_m$ ): Component of velocity along an axi-symmetric stream surface.
Equation: $Cm = \sqrt{cx^2 + c_r^2}$
Purely axial-flow machines:
- Radius of flow path is constant, radial flow velocity is zero.
- $Cm = Cx$
Purely radial flow machines:
- Axial flow velocity is zero.
- $Cm = cr$
Total flow velocity: $c = \sqrt{cx^2 + cr^2 + c{\theta}^2} = \sqrt{cm^2 + c_{\theta}^2}$
Swirl angle: $\alpha = tan^{-1}(c{\theta} / Cm)$

Relative Velocities

Analysis within rotating blades is performed in a frame of reference stationary relative to the blades.
Flow appears steady in this frame of reference.
Relative velocity is absolute velocity minus the local velocity of the blade.
- $w{\theta} = c{\theta} - U$
- $wx = cx$
- $wr = cr$
Relative flow angle: $\beta = tan^{-1}(w{\theta} / cm)$
Relationship: $tan \beta = tan \alpha - U/c_m$

Fundamental Laws

Continuity of Flow Equations
First Law of Thermodynamics and Steady Flow Energy Equations
Momentum Equation
Second Law of Thermodynamics

Definition of Efficiency for Turbines

Overall Efficiency ( $\eta_o$ ):
- $\eta_o = \frac{\text{mechanical energy available at coupling of output shaft in unit time}}{\text{maximum energy difference possible for the fluid in unit time}}$
Mechanical energy losses occur due to friction at bearings, glands, etc.
- Small machines: 5% or more.
- Medium and large machines: as little as 1%.

Isentropic Efficiency

Isentropic or hydraulic efficiency ( $\etai$ or $\etah$ ):
- $\etai \text{ (or } \etah) = \frac{\text{mechanical energy supplied to the rotor in unit time}}{\text{maximum energy difference possible for the fluid in unit time}}$
Mechanical efficiency ( $\eta_m$ ):
- $\etam = \frac{\text{shaft power}}{\text{rotor power}} = \frac{\etao}{\etai} \text{ (or } \frac{\etao}{\eta_h})$
Isentropic efficiency in terms of work:
- $\etat \text{ (or } \etah) = \frac{\text{actual work}}{\text{ideal (maximum) work}} = \frac{\Delta W}{\Delta W_{max}}$
For an adiabatic turbine:
- $\Delta W = \frac{Wx}{\dot{m}} = (h{01} - h{02}) + g(z1 - z_2)$

Steam and Gas Turbines

Figure 1.9(a) shows a Mollier diagram representing the expansion process through an adiabatic turbine.
Line 1-2 represents the actual expansion and line 1-2s the ideal or reversible expansion.
Actual turbine rotor specific work:
- $\Delta Wx = \frac{Wx}{\dot{m}} = h{01} - h{02} = (h1 - h2) + \frac{1}{2}(c1^2 - c2^2)$
Ideal work output:
- $\Delta W{max} = \frac{W{max}}{\dot{m}} = h{01} - h{02s} = (h1 - h{2s}) + \frac{1}{2}(c1^2 - c{2s}^2)$
Total-to-total efficiency ( $\eta_t$ ):
- $\etat = \frac{\Delta Wx}{\Delta W{max}} = \frac{h{01} - h{02}}{h{01} - h_{02s}}$
If the difference between the inlet and outlet kinetic energies is small:
- $\etat \approx \frac{h1 - h2}{h1 - h_{2s}}$
Exhaust kinetic energy is not wasted in:
- The last stage of an aircraft gas turbine, where it contributes to jet propulsive thrust.
- One stage of a multistage turbine where it can be used in the following stage.

Hydraulic Turbines

Hydraulic efficiency ( $\eta_h$ ) is a form of the total-to-total efficiency.
Steady flow energy equation in differential form for an adiabatic turbine:
- $dW_x = \dot{m}[dh + d(\frac{c^2}{2}) + gdz]$
For an isentropic process, $Tds = 0 = dh - \frac{dp}{\rho}$ .
Maximum work output for an expansion to the same exit static pressure, kinetic energy, and height as the actual process:
- $W{max} = \dot{m} \int1^2 \frac{dp}{\rho} + (\frac{c1^2 - c2^2}{2}) + g(z1 - z2)$
For an incompressible fluid, maximum work output:
- $W{max} = \dot{m} [\frac{p1 - p2}{\rho} + \frac{c1^2 - c2^2}{2} + g(z1 - z2)] = \dot{m}g(H1 - H_2)$
- Where $gH = \frac{p}{\rho} + \frac{c^2}{2} + gz$ and $\dot{m} = \rho Q$
Turbine hydraulic efficiency:
- $\etah = \frac{W}{W{max}} = \frac{\Delta W}{g[H1 - H2]}$

Compressors and Pumps

Isentropic efficiency of a compressor or hydraulic efficiency of a pump ( $\etac$ or $\etah$ ):
- $\etac \text{ (or } \etah) = \frac{\text{useful (hydrodynamic) energy input to fluid in unit time}}{\text{power input to rotor}}$
Overall efficiency:
- $\eta_o = \frac{\text{useful (hydrodynamic) energy input to fluid in unit time}}{\text{power input to coupling of shaft}}$
Mechanical efficiency:
- $\etam = \frac{\etao}{\etac} \text{ (or } \frac{\etao}{\eta_h})$

Compressors and Pumps Specific Work

For a complete adiabatic compression process from state 1 to state 2, the specific work input is:
- $\Delta W = (h{02} - h{01}) + g(z2 - z1)$
Total-to-total efficiency:
- $\etao = \frac{\text{ideal (minimum) work input}}{\text{actual work input}} = \frac{h{02s} - h{01}}{h{02} - h_{01}}$
If the difference between inlet and outlet kinetic energies is small:
- $\etac \approx \frac{h{2s} - h1}{h2 - h_1}$
For incompressible flow, the minimum work input is given by:
- $\Delta W{min} = \frac{W{min}}{\dot{m}} = [\frac{p2 - p1}{\rho} + \frac{c2^2 - c1^2}{2} + g(z2 - z1)] = g[H2 - H1]$
For a pump the hydraulic efficiency is therefore defined as:
- $\etah = \frac{W{min}}{\Delta W} = \frac{g[H2 - H1]}{\Delta W}$

Introduction and Classification of Fluid Machines

Positive Displacement
Turbomachines
- Radial-Flow (Centrifugal)
- Axial-Flow
- Mixed-Flow

Machines for Doing Work on a Fluid

Pumps
Fans
Blowers
Compressors

Machines for Extracting Work (Power) from a Fluid

Hydraulic Turbines (Impulse and Reaction)
Gas Turbines (Impulse and Reaction)

Turbomachinery Analysis: The Angular Momentum Principle

$\tau \times \overrightarrow{Fs} + \int{cv} \overrightarrow{r} \times \overrightarrow{g} \rho dV + T{shaft} = \int{cs} \overrightarrow{r} \times \overrightarrow{V} \rho \overrightarrow{V} \cdot d \overrightarrow{A}$

Momentum Equation

Newton's second law of motion relates the sum of external forces to the rate of change of momentum.
Equation:
- $\sum Fx = \frac{d}{dt}(mcx)$
For steady flow with uniform velocities:
- $\sum Fx = \dot{m}(c{x2} - c_{x1})$

Moment of Momentum

Vector sum of moments of external forces equals the time rate of change of angular momentum.
Equation:
- $TA = \frac{d}{dt} (mr c\theta)$
- where r is the distance from the axis of rotation and $c_\theta$ is the velocity component perpendicular to both the axis and radius vector r.

Law of Moment of Momentum

For a control volume enclosing the rotor of a turbomachine with swirling fluid entering at radius $r1$ with tangential velocity $c{\theta 1}$ and leaving at radius $r2$ with tangential velocity $c{\theta 2}$ :
- $TA = \dot{m}(r2 c{\theta 2} - r1 c_{\theta 1})$

Euler Work Equation: Pump

For a pump or compressor rotor running at angular velocity $\Omega$ , the rate at which the rotor does work on the fluid is
- $TA \Omega = \dot{m}(U2 c{\theta 2} - U1 c_{\theta 1})$
- where $U = \Omega r$
Specific work (work done on the fluid per unit mass):
- \Delta W = \frac{TA \Omega}{\dot{m}} = U2 c{\theta 2} - U1 c_{\theta 1} > 0
This is Euler's pump equation.

Euler Work Equation: Turbine

For a turbine, the fluid does work on the rotor, and the sign for work is reversed. Thus, the specific work is
- \Delta Wt = \frac{Wt}{\dot{m}} = U1 c{\theta 1} - U2 c{\theta 2} > 0
This is Euler's turbine equation.

Euler Work Equation - General Forms

For any adiabatic turbomachine (turbine or compressor), applying the steady flow energy equation gives
- $\Delta W = (h{01} - h{02}) = U1 c{\theta 1} - U2 c{\theta 2}$
This can be written as
- $\Delta h0 = \Delta(Uc{\theta})$
For a stationary blade row, $U = 0$ and therefore $h_0 = constant$ . This is expected since a stationary blade cannot transfer any work to or from the fluid.

Euler Turbomachine Equation

$ T{shaft} = (r2 V{t2} - r1 V{t1}) \dot{m}$ $Wm = (U2 V{t2} - U1 V{t1}) \dot{m}$
$ H = \frac{(U2V{t2} - U1V{t1})}{mg}$

Rothalpy

The Euler work equation can be rewritten as
- $I = h0 - Uc\theta$
- where I is a constant along the streamlines through a turbomachine.
The function I is called rothalpy, a contraction of rotational stagnation enthalpy.
Rothalpy can also be written in terms of the static enthalpy
- $I=h + \frac{1}{2}V^2 - Uc_\theta$

Turbomachinery Analysis Velocity Diagrams

Concepts of absolute velocity, velocity relative to blade, and rotor velocity.

Turbomachinery Analysis: Idealized Centrifugal Pump

Assumptions:
- Negligible torque due to surface forces (viscous and pressure).
- Inlet and exit flow tangent to blades.
- Uniform flow at inlet and exit.
- Zero inlet tangential velocity.

Turbomachinery Analysis: Idealized Centrifugal Pump Head Equation

Head Equation: $H = C1 - C2Q$
Shutoff Head:
- $C1 = \frac{U2^2}{g}$
- $C2 = \frac{U2 cot \beta2}{\pi D2 b_2 g}$

Turbomachinery Analysis: Idealized Centrifugal Pump (Continued)

Relationship between head and volume flow rate for centrifugal pump with forward-curved, radial, and backward-curved impeller blades.
Equation:
- $H = \frac{\omega^2 R^2}{g}$

Turbomachinery Analysis: Machines for Doing Work on a Fluid

$ WH = \rho Q g Ha$
$ Ha = (\frac{P}{\rho g} + \alpha \frac{V^2}{2g} + Z){discharge} - (\frac{P}{\rho g} + \alpha \frac{V^2}{2g} + Z)_{suction}$

Pump Efficiency:
- $\etaP = \frac{\rho Q g H}{\dot{Wm}}$

Turbomachinery Analysis: Machines for Extracting Work (Power) from a Fluid

$ WH = \rho Q g Ht$
$ Ht = (\frac{P}{\rho g} + \frac{V^2}{2g} + Z){inlet} - (\frac{P}{\rho g} + \frac{V^2}{2g} + Z)_{outlet}$

Turbine Efficiency:
- $\etat = \frac{\dot{Wm}}{ \rho Q g H_t}$

Performance Characteristics

Machines for Doing Work on a Fluid
Comparison of ideal and actual head-flow curves for a centrifugal pump with backward-curved impeller blades.
Typical pump performance curves from tests with three impeller diameters at constant speed.

Performance Characteristics (Machines for Extracting Work)

Performance curves for an impulse turbine showing ideal and actual torque and power vs. ratio of wheel speed to jet speed.
Performance of typical reaction turbine as predicted by model tests and confirmed by field test.

Performance Characteristics: Dimensional Analysis and Specific Speed

Flow Coefficient:
- $\Phi = \frac{Q}{A2 U2}$
Head Coefficient:
- $\Psi = \frac{gH}{U_2^2}$
Power Coefficient:
- $\Pi = \frac{\dot{W}}{\rho Q U2^3} = \frac{\dot{W}}{\rho \omega^2 Q R2^5}$

Performance Characteristics: Torque Coefficient

Torque Coefficient:
- $\Tau = \frac{T}{\rho A2 U2 R_2} = \psi \Phi$

Performance Characteristics: Specific Speed

Specific Speed:
- $N_S = \frac{\omega Q^{1/2}}{h^{3/4}}$
Specific Speed (Customary Units):
- $N_{Scu} = \frac{N (rpm) [Q (gpm)]^{1/2}}{[H (ft)]^{3/4}}$
Specific Speed (Customary Units):
- $N_{Scu} = \frac{N (rpm) [P (hp)]^{1/2}}{[H (ft)]^{5/4}}$

Performance Characteristics Geometric Proportions

Typical geometric proportions of commercial pumps as they vary with dimensionless specific speed.
Typical geometric proportions of commercial hydraulic turbines as they vary with dimensionless specific speed.

Performance Characteristics: Similarity Rules

For Dynamic Similarity:
- $\frac{Q1}{\omega1 D1^3} = \frac{Q2}{\omega2 D2^3}$
- $\frac{g h1}{\omega1^2 D1^2} = \frac{g h2}{\omega2^2 D2^2}$
- $\frac{\dot{W1}}{\rho1 \omega1^3 D1^5} = \frac{\dot{W2}}{\rho2 \omega2^3 D2^5}$

Applications to Fluid Systems: Machines for Doing Work on a Fluid

Superimposed system head-flow and pump head-capacity curves.
Equation:
$ \frac{P1}{\rho} +\alpha1 \frac{V1^2}{2} + g z1 - (\frac{P2}{\rho} + \alpha2 \frac{V2^2}{2} + g z2 ) = h_{all pump} $

Applications to Fluid Systems : Pump Wear

Effect of pump wear on flow delivery to system.

Applications to Fluid Systems: Pumps in Series

Operation of two centrifugal pumps in series.

Applications to Fluid Systems: Pumps in Parallel

Operation of two centrifugal pumps in parallel.

Applications to Fluid Systems: Fans, Blowers, and Compressors

Exploded view of typical centrifugal fan.

Applications to Fluid Systems: Fans, Blowers, and Compressors

Typical types of blading used for centrifugal fan wheels

Applications to Fluid Systems: Fans, Blowers, and Compressors

Typical characteristic curves for fan with backward-curved blades

Applications to Fluid Systems: Positive-Displacement Pumps

Schematic of typical gear pump

Applications to Fluid Systems

Machines for Doing Work on a Fluid
Propellers
Speed of Advance Coefficient: Thrust, Torque, Power Coefficients, and Propeller Efficiency

Applications to Fluid Systems

Machines for Extracting Work (Power) from a Fluid
Hydraulic Turbines
Wind-Power Machines