UNIVERSITÀ DEGLI STUDI DI CAGLIARI FACOLTÀ DI INGEGNERIA

DIPARTIMENTO DI INGEGNERIA MECCANICA MACCHINE A FLUIDO

PROJECT FOR ERASMUS STUDENTS

ANGEL ANDRES GARCIA MORALES FEBRUARY 2015

Dimensional Analysis

Dimensional Analysis

GENERAL CONCEPTS

Incompressible flow In fluid mechanics, incompressible flow is understood as a flow in which the material has a constant

density within an infinitesimal volume that moves with flow velocity. This volume is known as a fluid

parcel.

The fact that a flow is incompressible doesn’t necessarily imply that the fluid itself is incompressible. In

gases, the distinction between compressible and incompressible flows depends on the Mach number. The

Mach number must be greater than 0,3 for a gas flow before significant compressibility occurs.

Flow, head and power coefficients The performance of turbomachinery can be expressed in terms of control variables, geometric variables

and fluid properties. The power (P) of fluid machinery working with incompressible flows, such as pumps,

depends upon the density of the fluid (ρ) the rotational speed (N), the characteristic diameter (D), the

head change (ΔH), the volume flow rate (Q) and the gravitational constant (g). Since the gravitational

constant and the head change are normally considered as one quantity, the general equation, can

therefore be written as

𝑃 = 𝑓(𝜌, 𝑁, 𝐷, (𝑔∆𝐻), 𝑄)

Out of this equation, 3 dimensionless groups can be separated:

Frist group, taking P and 𝜌𝑁𝐷

𝑃 = 𝜑(𝜌𝑁𝐷) = Π1𝜌𝑁𝐷 And in order to eliminate the dimensions in the group, knowing that Power is composed by mass

by length squared, over the cubic power of time (see Appendix, Table N.1), we obtain the

following dimensionless equation, known as the Power Coefficient

Π1 =𝑃

𝜌𝑁3𝐷5= 𝑃𝑜𝑤𝑒𝑟 𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡

Second group, taking Q and 𝜌𝑁𝐷

𝑄 = 𝜑(𝜌𝑁𝐷) = Π2𝜌𝑁𝐷 And in order to eliminate the dimensions of Q, the cube of mass over time (see Appendix, Table

N.1), the equation would transform itself into the Flow Coefficient

Π2 =𝑄

𝑁𝐷3= 𝐹𝑙𝑜𝑤 𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡

Third group, taking 𝑔∆𝐻 and 𝜌𝑁𝐷

𝑔∆𝐻 = 𝜑(𝜌𝑁𝐷) = Π3𝜌𝑁𝐷 And in order to eliminate the dimensions of 𝑔∆𝐻, a unit of length squared over the time squared

(see Appendix, Table N.1), the equation would change into the Head Coefficient

Π3 =𝑔𝐻

𝑁2𝐷2= 𝐻𝑒𝑎𝑑 𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡

See Table N.1 for a detailed chart of the dimensional formula of each parameter.

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Dynamic Similarity Dynamic similarity exists between a model and a prototype that are geometrically similar between them.

If this dynamic similarity indeed exists, then each independent dimensionless parameter for the model

will be equal to those of the prototype, and the dependent dimensionless parameters for the model will

be equal to those of the prototype.

Simply put, the dynamic similarity exists between a model and a prototype, when the force vectors at

corresponding points are similar. The numbers relevant to the flow must be preserved between model

and prototype. Mathematically it could be expressed by

Π1 𝑝𝑟𝑜𝑡𝑜𝑡𝑦𝑝𝑒 = Π1 𝑚𝑜𝑑𝑒𝑙 , Π2 𝑝𝑟𝑜𝑡𝑜𝑡𝑦𝑝𝑒 = Π2 𝑚𝑜𝑑𝑒𝑙 , 𝑒𝑡𝑐.

Depending on the speed of the flow, especially at lower speeds, it’s the Reynold’s Number that become

important, and so it is simple to preserve the Reynolds number.

Specific Speed and Specific Diameter

Specific Speed The operating conditions for turbines and other kinds of fluid machinery are important parameters that

make easier the process of equipment selection, which is why they often serve as a starting point in the

design phase. The basic parameter, independent of the size of the machine, can be constructed by either

using the values N, P and H, leaving D aside, or by using N, Q and H.

The parameter Ksp built with N, Q and H, is then obtained by dividing the Flow Coefficient by the Head

Coefficient, is used for pumps and compressors, and takes the following shape

𝐾𝑠𝑃 =(𝑄 𝑁𝐷3⁄ )

12⁄

(𝑔𝐻 𝑁2𝐷2⁄ )3

4⁄=

𝑁𝑄1

2⁄

(𝑔𝐻)3

4⁄

While the parameter KsT is used for turbines, and uses the following

𝐾𝑠𝑇 =𝑁𝑃

12⁄

𝜌1

2⁄ (𝑔𝐻)5

4⁄

Thanks to the nature of these dimensional parameters, they remain always the same for similarity of flow

in machines of homologous series. Both Ks parameters correspond to the parameters of maximum

efficiency of the machines they refer to. The Ks parameter is often known as the shape factor of machines

the machines. The specific speed of both turbines and pumps uses the same parameters as the shape

factor, but leaving the gravity and density parameters out of the equation. We therefore obtain

𝑁𝑠𝑇 = 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑠𝑝𝑒𝑒𝑑 𝑓𝑜𝑟 𝑡𝑢𝑟𝑏𝑖𝑛𝑒𝑠 =𝑁𝑃

12⁄

𝐻5

4⁄

𝑁𝑠𝑃 = 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑠𝑝𝑒𝑒𝑑 𝑓𝑜𝑟 𝑝𝑢𝑚𝑝𝑠 =𝑁𝑄

12⁄

𝐻3

4⁄

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Specific Diameter The specific diameter is another dimensionless parameter used in the construction of turbomachinery, it

is obtained by the relation between the Head Coefficient and the Flow Coefficient, and is represented as

𝐷𝑠 = 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 =𝐷(𝑔𝐻)

14⁄

𝑄1

2⁄

Cordier Diagram The Cordier Diagram was born as an effort by Cordier to correlate the data for turbomachines in terms of

Ns and Ds. Cordier found that turbomachines tend to group along a definable curve when plotter with

their values of Ns and Ds. He further found that the machines could be classified into a definable curve in

function of Ds, and that the machines that scattered away from said curve where those with less

efficiency. The Cordier Diagram is used as a guideline for the design of turbomachinery.

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Graph N.1: Cordier Diagram

Effects of Reynolds and Froude The Reynolds number is defined as the ratio of inertial force to viscous force. Large Reynold’s numbers

indicate that, for the flow’s internal forces, is the viscous force that dominates. Large Reynold’s numbers

also mean the viscosity has an important effect in a restricted region around the body. Smaller Reynold’s

numbers mean the viscous forces are important in the regions near the body. The Reynold’s number is

obtained with the following equation

𝑅𝑒 = 𝑅𝑒𝑦𝑛𝑜𝑙𝑑′𝑠 =𝜌 𝑉 𝐿

𝜇

On the other hand, the Froude Number is the ratio of inertial forces to gravity forces. It’s especially

important for flows with a free surface, and is given by

𝐹𝑟 = 𝐹𝑟𝑜𝑢𝑑𝑒 =𝑉2

𝑔𝐿

Froude’s Number is a parameter that, when preserved between model and prototype, dictates the scaling

of other variables.

Compressible Flow In fluid mechanics, compressible flows refer to those flows who suffer significant changes in fluid density

when moving at a certain speed. The only flows that show these changes in density are gasses, which is

why this discipline deals only with gases, leaving liquids aside. Compressible gas flows are, as stated in the

previous section, those with a Mach number greater than 0,3.

Dimensional analysis for compressible flows and determination of dimensionless parameters Dimensional analysis refers to the way of simplifying the solution of a physical problem by appealing to a

dimensional homogeneity and so reducing the number of relevant variables.

The dimensional analysis for compressible flows is more complex to that of incompressible flows. It uses two extra characteristics in respect to the incompressible flow, these are the stagnation speed of sound at entry to the machine (𝑎01), and the ratio of specific heats of the gas (𝛾 = 𝐶𝑝/𝐶𝑣). The analysis takes gases as either perfect gases, or dry vapors that approximate in behavior to perfect gases. Other differences include the replacement of the volume flow rate for the mass flow, seeing as the density changes importantly; likewise, the head change is replaced by the isentropic stagnation enthalpy change (∆ℎ0𝑠). The performance parameter for a turbomachine can be expressed in function of 8 variables as

∆ℎ0𝑠, 𝜂, 𝑃 = 𝑓(𝜇, 𝑁, 𝐷, ṁ, 𝜌01, 𝑎01, 𝛾) However, by selecting 𝜌01, 𝑁, 𝐷 as common factors, the three parameters can be reduced to

dimensionless groups

∆ℎ0𝑠

𝑁2𝐷2, 𝜂,

𝑃

𝜌01𝑁3𝐷5= 𝑓(

ṁ

𝜌01𝑁𝐷3,𝜌01𝑁𝐷2

𝜇, 𝑁𝐷

𝑎01, 𝛾)

The former equation can, however, be changed according to the needs one may have, and as such, some variables will be dropped and changed for more adequate forms depending on the need.

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Reduced parameters in compressible flow For compressors handling a perfect gas, the isentropic stagnation enthalpy rise can be written as

𝐶𝑝(𝑇02𝑠 − 𝑇01) for the perfect gas. Figure (a) illustrates a compression process where the stagnation

point changes at constant entropy between the entrance and exit pressures. Figure (b) illustrates the

equivalent process for a turbine.

Applying both 𝑝/𝜌𝛾 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 and 𝑝/𝜌 = 𝑅𝑇, the following expression is obtained

𝑇02𝑠

𝑇01= (

𝑝02

𝑝01)

(𝛾−1)/𝛾

And so,Δℎ0𝑠 = 𝐶𝑝𝑇01[(𝑝02/𝑝01)(𝛾−1)/𝛾 − 1] Since 𝐶𝑝 = 𝛾𝑅 (𝛾 − 1)⁄ and 𝑎012 = 𝛾𝑅𝑇01 , then the flow

coefficient can be expressed as

ṁ

𝜌01𝑎01𝐷2=

ṁ𝑅𝑇01

𝜌01√𝛾𝑅𝑇01𝐷2=

ṁ√𝑅𝑇01

𝜌01√𝛾𝐷2

As ṁ ≡ 𝜌01𝐷2(𝑁𝐷), the power coefficient can be alternatively written as

Р = 𝑃

𝜌01𝑁3𝐷5=

𝐶𝑝 ΔT0

(𝑁𝐷)2=

ΔT0

T01

And by plugging all the new forms of non-dimensional groups in the original equation, and cancelling the gamma terms, as gamma is also an independent variable, we obtain

𝑝02

𝑝01, 𝜂,

ΔT0

T01= 𝑓(

ṁ√𝑅𝑇01

𝜌01𝐷2,

𝑁𝐷

√𝑅𝑇01

, 𝑅𝑒, 𝛾)

For industrial applications of machines that have a specific size and handle a single gas, it’s usual to ignore parameters 𝛾, 𝑅, 𝐷 and Re, in case the machine operates at a high Reynold’s number. However, by omitting D and R, the equation is no longer dimensionless

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Graph N.2: Compression process

Characteristic curves of compressor and turbine expressed in terms of reduced parameters In the characteristic curves for both compressors and turbines, the pressure ratio across the machine is

plotter as the function of ṁ√𝑇01 𝜌01⁄ for values of 𝑁 √𝑇01⁄ . A big difference between turbines and

compressors, expressed in the curves, is how strongly dependent is the performance of a compressor

upon 𝑁 √𝑇01⁄ , while the turbine shows a weaker one.

Figures N.3 and N.4 illustrate the overall characteristics of compressors and turbines, respectively

Graph N.3: Characteristic curve

Graph N.4: Characteristic curve

Turbomachinery

Turbomachinery

FUNDAMENTAL EQUATIONS Turbomachines are devices in which energy is transferred to or from a continuously flowing fluid by dynamic action of one or more blade rows. This transfer of energy also changes the angular momentum of the fluid.

In the compressor illustrated in Figure N.1, we can observe which are the elements of the compressor that experience a transfer of either mass, momentum or energy. Both the intake and outtake of the compressor have specific radii measured from the axis. The inlet has an axial component of velocity, as well as a tangential component, and the resulting vector is shown and marked with V1, as does the outlet. In an assumed steady flow, both inlet and outlet have properties such as velocity and its components, enthalpy, temperature, and density. The energy transfer takes place thanks to the shaft power input, which is in the form of torque, considering the Force as the tangential blade force.

Graph N.1: Analyzed Compressor

In order to make a quantitative analysis it’s necessary to expand the equations used for control volume. As the diagram illustrates, there’s no energy, mass or momentum transfer between the surfaces I and O. Therefore, we can integrate just at the inlet and outlet surfaces. The continuity equation is then given by

∬ 𝜌1𝑉1 ∗ 𝑑𝐴1

𝑟𝑖𝑛𝑔 𝐺𝐻

= ∬ 𝜌2𝑉2 ∗ 𝑑𝐴2

𝑟𝑖𝑛𝑔 𝐽𝐾

The momentum equation is given by the following equation, applied in the ϴ-direction.

𝐷𝜔

𝐷𝑡= 𝑣∇2𝜔

Applying the previous equation to the control volume in Figure N.1, is given by

∭ 𝑟𝐹𝜃𝜌𝑑𝑣

𝑐.𝑣

= ∬ (𝑟2𝑉𝜃2)𝜌2𝑉2 ∗ 𝑑𝐴2

𝑟𝑖𝑛𝑔 𝐽𝐾

− ∬ (𝑟1𝑉𝜃1)𝜌1𝑉1 ∗ 𝑑𝐴1

𝑟𝑖𝑛𝑔 𝐺𝐻

Which, after the assumption of uniform entry and exit velocity, and after integrating the left hand side of the equation, process which represents the sum of the moments of all body forces exerted inside the control volume, knowing this sum is equal to the torque transmitted by the shaft, can be reduced to

𝑑𝑇 = 𝑑ṁ(𝑟2𝑉𝜃2− 𝑟1𝑉𝜃1

)

Where dT is the torque applied by the shaft and dm is the mass flow through the control volume. The term inside of the parentheses represents the change in angular momentum of the absolute flow. The shaft power input to the control volume is given by

−𝑑𝑃𝑠ℎ𝑎𝑓𝑡 = Ω(𝑑ṁ)(𝑟2𝑉𝜃2− 𝑟1𝑉𝜃1

)

And from here, the general Euler equation for turbomachinery relating the input shaft power to the change in angular momentum can be written as

−𝑃𝑠ℎ𝑎𝑓𝑡 = Ωṁ(𝑟2𝑉𝜃2− 𝑟1𝑉𝜃1

)

The previous equation is valid only in the analysis of a two-dimensional strip. In machinery such as pumps, compressors and fans, due to the constant radii, the equation changes to

−𝑃𝑠ℎ𝑎𝑓𝑡 = ṁ𝑈(𝑉𝜃2− 𝑉𝜃1

)

In centrifugal turbomachines the equation changes to

−𝑃𝑠ℎ𝑎𝑓𝑡 = ṁ(𝑈2𝑉𝜃2− 𝑈1𝑉𝜃1

)

The energy equation applied to the same control volume is given by

�̇� = 𝑃𝑠ℎ𝑎𝑓𝑡 + 𝑃𝑠ℎ𝑒𝑎𝑟 + ṁ(ℎ02− ℎ01

)

Neglecting the shear forces on the control surface and assuming adiabatic flow, we obtain an equation valid for viscous, compressible flows that includes all losses associated with viscous flows over blades

𝑃𝑠ℎ𝑎𝑓𝑡

ṁ= ℎ02

− ℎ01

If we relate P shaft to the flow and thermodynamic properties, we obtain

−𝑃𝑠ℎ𝑎𝑓𝑡

ṁ= (ℎ02

− ℎ01) = (𝑈2𝑉𝜃2

− 𝑈1𝑉𝜃1)

When applied to turbomachinery that handles liquids, we obtain

−𝑃𝑠ℎ𝑎𝑓𝑡

ṁ= 𝑔(𝐻2 − 𝐻1) = (𝑈2𝑉𝜃2

− 𝑈1𝑉𝜃1)

Where (𝐻2 − 𝐻1), is the ideal head rise in units of length.

ABSOLUTE AND RELATIVE MOTION Understanding how the flow appears from the point of view of stationary components, versus that of

those rotating, is a key concept in turbomachinery. In order to understand said movement, it is wise to

first analyze the nature of movement in situations where two forces act upon the same subject.

For any moving subject, we can identify three kinds of velocity: the frame velocity (U), the absolute

velocity (V), and the relative velocity (W). The three velocities can be related in something called the

velocity triangle, and also by the following equation

𝑊 = 𝑉 + 𝑈

In turbomachinery, there are three states worth analyzing: the entry to the stator, the gap between the

stator and the rotor, and the exit from the rotor.

At point one, we have an incoming velocity but since the stator is fixed, there’s no relative motion, and therefore no velocity triangle to be drawn. At point 2 we have the flow viewed from the stator and the flow viewed from the moving rotor as two different frames of reference, which means we can draw a velocity triangle. At point 3, with two frames of reference, a velocity triangle can be drawn.

Then, with trigonometry we can solve for any of the three variables.

DERIVATION OF EULER EQUATION The Euler equation applies to all types of turbomachinery, without making distinctions about the design of type of flow. The equation brings three important points to light:

The equation will give a negative value for turbines, and a positive value for compressors. This is because it is affected by the direction of the flow.

The power input or output is given by the change in angular momentum.

Radial flow machines tend to deliver or absorb more power per stage, compared to an axial flow device.

Angular momentum is given, at any moment, by

𝐿 = 𝑚𝑉𝜃𝑟 Where m represents the mass. So the change of angular momentum between the entry and the exit in a turbomachine, is

𝑚2𝑟2𝑉𝜃2− 𝑚1𝑟1𝑉𝜃1

And the time rate of change is

𝑚2𝑟2𝑉𝜃2− 𝑚1𝑟1𝑉𝜃1

𝑡= ṁ(𝑟2𝑉𝜃2

− 𝑟1𝑉𝜃1)

The time rate of angular momentum becomes equal to the torque, and we obtain,

𝑇 = ṁ(𝑟2𝑉𝜃2− 𝑟1𝑉𝜃1

)

And as we know power is torque times rotational speed, then

𝑃 = 𝑇𝜔 = ṁ𝜔(𝑟2𝑉𝜃2− 𝑟1𝑉𝜃1

)

Where 𝜔 is the rotational speed of the device. By dividing the last equation by the mass, we obtain the Euler Turbomachinery Equation 𝑤 = 𝜔(𝑟2𝑉𝜃2

− 𝑟1𝑉𝜃1)

ROTHALPY EQUATION The specific work equation can be given by

𝑤 = ℎ02− ℎ01

Given this, and using the Euler equation, we can apply both to the steady flow energy equation applied to a turbine, obtaining the following

𝑤 = 𝑈2𝑉𝜃2− 𝑈1𝑉𝜃1

= ℎ02− ℎ01

⟹ ℎ01− 𝑈1𝑉𝜃1

= ℎ02− 𝑈2𝑉𝜃2

And so we can define the quantity called rothalpy

𝐼 = ℎ0 − 𝑈𝑉𝜃 The rothalpy is a quantity conserved over a blade row, in either the stator or rotor, and not a stage.

RADIAL FLOW COMPRESSORS Radial flow compressors achieve a pressure rise by adding kinetic energy to a continuous flow of fluid, this kinetic energy is converted to an increase in potential energy by slowing the flow using a diffuser. The flow enters the compressor and moves the centrifugal impeller, which forces the flow to pass through the compressor. After exiting the centrifugal impeller, the flow reaches a speed near to that of sound, and is then slowed down by a stationary compressor. This reduction in velocity causes the pressure to rise, which makes the flow receive the name of compressed fluid.

Components The components of a simple, basic centrifugal compressor are:

1. Inlet: typically found as a simple pipe. May include valves, and both pressure and temperature instrumentation.

2. Centrifugal impeller: it’s the compressor’s rotating set of blades that raises the energy of the working fluid. Impellers are designed open, covered, with splitters, or without splitters. Impellers are analyzed using Euler’s equation.

3. Diffuser: the diffuser converts the kinetic energy of the fluid into pressure, by slowing down its velocity. The working performance of the diffuser can be analyzed using Bernoulli’s principle.

4. Collector: gathers the flow from the diffuser and delivers it to a downstream pipe.

Applications The most common applications for radial flow compressors are:

Gas turbines and auxiliary power units

Turbochargers and superchargers for diesel automotive engines

Natural gas pipelines, to move gas from the production site to the consumer.

Oil refineries, natural gas processing, petrochemical and chemical plants.

Air conditioning and refrigeration, and HVAC

Air separation processes.

Performance For radial flow compressors, performance at specific operating points can be read by using both the

pressure-specific volume, and temperature-entropy plots. By studying the plots, we can see how the

pressure rises between inlet and exit.

As a way to analyze performance for turbomachinery, the use of performance maps is preferred, as it

shows flow and pressure ratio for each of the speed lines. The analysis of performance maps is often more

convenient and cost effective than performing the different needed tests in order to find the desired

information. Performance maps are also known as characteristic curves.

IDEAL AND REAL CHARACTERISTIC CURVES The performance characteristics of centrifugal compressors is graphically presented in diagrams that plot

a family of curves, and are known as performance map or operating envelope.

The figure represents a common performance map, where the inlet volume flow is plotted against the pressure ratio. Within the map we can locate many useful performance parameters of turbomachinery. Some of these specific points are:

Approximate surge limit: defines the minimum flow necessary to avoid a potentially damaging surge condition.

Stonewall limit: defines the flow at which the gas velocity approaches the velocity of sound. Each of the curves in the family, located between the surge and the stonewall limits, represent the flow-pressure for a given compressor speed. The elliptical curves denote compressor efficiency.

SLIP FACTOR The slip factor is a measure of the fluid slip of compressors or turbines. The fluid slip is the name given to the times when the angle at which the fluid leaves the impeller is not the same as the actual blade angle. In the course of flow through the impeller blades, occurs a difference of pressure and velocity between the leading and trailing faces of the impeller blades, having the leading face a high pressure and low velocity, and being the opposite for the trailing face. This phenomena results in a circulation around the blade, and a non-uniform velocity distribution. Under this situation, the mean direction of flow at outlet changes from 𝛽2 to 𝛽′2, and therefore the tangential velocity at outlet is reduced from 𝑉𝑤2 to 𝑉′𝑤2, and the difference ∆𝑉𝑊 is then defined as the slip. Therefore, the slip factor 𝜎𝑠 is defined as

𝜎𝑠 =𝑉′𝑤2

𝑉𝑤2

The following figure demonstrates the phenomena that occurs in this case

Appendix

Table N.1: Variable physical parameters of Fluid Machine

Variable Physical Parameter Dimensional Formula

D = any physical dimension of the machine, as a measure of size L

Q = volume flow rate L3T-1

N = rotational speed (rev/min) T-1

H = difference in head across the machine L

𝝆 = density of the fluid ML-3

g = acceleration due to gravity LT-2

P = power transferred between the fluid and rotor ML2T-3

Sources of Information

Auld, D., & Srinivas, K. (2006, October 10). Aerodynamics for Students : A Web Site dedicated to

Theoretical Aerodynamics. Retrieved January 11, 2015, from http://www-

mdp.eng.cam.ac.uk/web/library/enginfo/aerothermal_dvd_only/aero/contents.html

Biswas, D., Sarkar, D., & Som, D. (2005, June 15). NPTEL Online-IIT Kanpur. Retrieved January 11,

2015, from http://nptel.ac.in/courses/112104117/ui/Course_home-1.htm

Dixon, S. (2005). Fluid mechanics, thermodynamics of turbomachinery (5th ed.). Amsterdam:

Elsevier-Butterworth-Heinemann.

Wright, T. (1999). Fluid machinery: Performance, analysis, and design. Boca Raton, Fla.: CRC Press.