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### Chapter 2

## 2.

## PERFORMANCE ANALISYS

Turbomachine are devices where energy is transferred to or from a working fluid by the action of one or more moving blade cascade. The rotor objective is to change the kinetic energy, stagnation enthalpy and stagnation pressure of the working fluid. In a turbine, the energy is extracted from the fluid whereas in a pump (or compressor) the energy is imparted to the fluid.

In axial flow turbomachinery the flow path is axial whereas in radial (or centrifugal)
turbomachinery it is mainly radial. A third option is the one in which the flow path is not strictly radial
*or axial: these are the so called mixed-flow turbomachines. *

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### 2.1.

### TURBOMACHINES GEOMETRY

The general geometry of a turbomachine rotor is shown in Figure 2.2.

**Figure 2.2 Cross-section of a pump impeller. Brennen [2] **
In the figure the following notation has been used:

•* RT1*: inlet blade tip radii,

•* RH1*: inlet blade hub radii,

•* RT2*: discharge blade tip radii,

•* RH2*: discharge hub radii,

The discharge angle θ is close to 90° in centrifugal pumps and close to 0° in axial pumps. In practice, a mixed flow configuration is usually used and θ is at an intermediate value 0° < θ < 90° which is simply given by:

tan *vr*

*u*

θ = (2.1)

*where vr is the radial velocity and u is the axial velocity of the flow. *

Developing the meridional surface (see Figure 2.3) the fluid velocity in a nonrotating reference
*frame v(r) and the velocity relative to the rotating blades w(r) can be seen. *

*v*θ* and w*θ* are the corresponding components in the circumferential direction whereas vm and wm* are

the components in the meridional direction.

It is important to know the angle α (incidence angle) at which the flow arrives at the leading edge of the blades:

1 1

( )*r* *b* ( )*r* ( )*r*

α =β −β (2.2)

where β*b(r) is the blade angle and is defined as the inclination of the tangent to the blade in the *

meridional plane and the plane perpendicular to the axis of rotation; β*(r) is the flow angle and it is the *
angle between the relative velocity vector in the meridional plane and a plane perpendicular to the axis
of rotation.

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This parameter shall not be confused with the angle of attack that is the angle between the incoming flow and the chord line. However in an axial flow pump they are equal to each other if the blades are straight.

Another important parameter is the deviation angle δ which is the difference between the flow angle and the blade angle at the trailing edge.

2 2

( )*r* * _{b}* ( )

*r*( )

*r*

δ =β −β (2.3)

**Figure 2.3 Development of the meridional surface (top-left) and velocity triangle (bottom-left) with definitions of incidence and **
**deviation angles (right). Brennen [2] **

Other important angles are the deflection angle (β2-β1) and the camber angle (βb2-βb1). The former is the angle of rotation of the flow whereas the latter is the corresponding angle through which the blades have turned.

*The parameters that characterize the flow are the static pressure p, the total pressure pT*_{, and the }

*volumetric flow rate Q . *

*The total pressure pT*_{ is defined by: }

2
1
2
*T*
*p* = +*p* ρ*v* (2.4)
2 2
1
2 (v v )
*T*
*m*
*p* = +*p* ρ + ϑ (2.5)
2 2 2
1
2 ( 2 r )
*T*
*p* = +*p* ρ *w* + Ω − Ω*r v*ϑ (2.6)

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*It is clear that, in an incompressible flow, pT*_{ corresponds to the total mechanical energy per unit }

volume of working fluid and the variation of this value between the inlet and the outlet of a pump represents the mechanical energy imparted to the fluid or extracted from the fluid.

Instead of the total pressure rise the total head rise is usually used:
2 1
(_{p}T* _{p}T*)

*H*

*g*ρ − = (2.7)

### 2.2.

### NONDIMENSIONAL PARAMETERS AND PUMP

### PERFORMANCE

Dimensional parameters like total head rise and volumetric flow rate are nondimensionalized by defining the head coefficient

2 1
2 2 2 2
( *T* *T*)
*T* *T*
*p* *p* *gH*
*R* *R*
ρ
−
Ψ = =
Ω Ω (2.8)

and the flow coefficient

*T*
*Q*
*AR*
Φ =
Ω
(2.9)

Both parameters can be expressed for inlet or discharge areas though the latter is usually used, thus

*RT2* is usually exploited for their evaluation.

A third important nondimensional parameter is the Reynolds number that in turbomachines is
defined as
2
2
2
Re *T*
*L*
*R*
ν
Ω
= (2.10)

where ν*L* is the kinematic viscosity of the fluid.

For Reynolds number values higher than 106_{ (turbulent flux), the characteristic curves become }
independent from it and thus two different turbopumps can be considered fluid dynamically similar to
each other if they operates at same Φ and Ψ.

Despite this, at the beginning of the design process of a turbomachine, the values of flow
coefficient and head coefficient are unknown due to their dependency from size and shape. Indeed the
fixed parameters are the rotating speed, the volumetric flow rate and the head rise. For these reasons
*another nondimensional parameter is defined, the specific speed, N (often indicated also with ΩS*):

1/2 1/2
3/4 _{(} _{)}3/4
*T*
*L*
*Q* *Q*
*N*
*gH*
*p*
ρ
Ω Ω
= =
∆
(2.11)

For different values of the specific speed parameter, the architecture of the turbomachine will
change to obtain the best possible efficiency (Figure 2.4 and Figure 2.5) and any pump geometry’s
*diagram (η,N) will show a maximum corresponding to the optimum specific speed (Figure 2.6). *
*Hence, for low values of N, the radial configuration is used but for high N the axial configuration is *
preferred.

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**Figure 2.4 Typical specific speeds in turbomachinery. Lakshminarayana [3] **

**Figure 2.5 Typical pump geometries for different design speeds. Brennen [2] **

From Figure 2.6 it is clear that the best efficiency range is obtained between 0.4 and 1.2 of the specific speed and 90% efficiency can be reached.

To obtain an expression from which the efficiency can be evaluated, it is possible to exploit an energy balance between inlet and discharge flow. As a first step it is possible to find out the basic thermodynamic measure of the energy, the total specific enthalpy:

2 2
1 1
2 2
*T* *p*
*h* *h* *u* *gz e* *u* *gz*
ρ
= + + = + + + (2.12)

where the parameters in the expression are
•* the specific internal energy e; *

•* the magnitude of the fluid velocity |u|; *
•* the vertical elevation z; *

The expression does not take into account the energy associated with other type of external forces such as magnetic field forces and it is valid if no chemical reactions are present.

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**Figure 2.6 Maximum efficiencies for different type of pumps. Brennen [2] **

From the first law of thermodynamics is possible to find a relation between the discharge and inlet
specific enthalpy, *h and *2*T* *h respectively, the mass flow rate *1*T* *m*, the net rate of heat addition *Q and *
the net rate of work *W : *

2 1
( *T* *T*)

*m h* −*h* = + *Q W* (2.13)

For incompressible and inviscid flow the fluid mechanical problem can be decoupled from the heat transfer one as follows:

2 1
(_{p}T* _{p}T*)

_{W}*m*ρ −

_{=} (2.14) 2 1 /

*e e Q m*− = (2.15)

The rate of work done on the fluid is

*W P T* = = Ω (2.16)

*where T is the torque applied to the fluid. From equation (2.14) and (2.16) it is possible to obtain the *
following expression

2 1
(_{p}T* _{p}T*)

*m* _{ρ}− = Ω*T* (2.17)

As a consequence of the second law of thermodynamics, the equation (2.17) has to be replaced by the following:

2 1
(_{p}T* _{p}T*)

*m* _{ρ}− ≤ Ω*T* (2.18)

Therefore some of the energy transferred to the fluid is transformed in heat instead of increasing
the stored energy inside the fluid. For this reason it is possible to define an hydraulic efficiency of the
*pump, ηP*, that is the quantity of work that increase the mechanical energy stored in the fluid:

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*T* *T*
*i*
*P* *P _{P}*

*m*

_{T}p*Q p*η ρ ∆ ∆ = = = Ω Ω (2.19)

_{T}The noncavitating performance of a pump can be represented in a nondimensional form by means
of a (Φ,Ψ) diagram, with the design conditions at Ψ(Φ*D*). A typical curve for this representation can be

seen in Figure 2.7, where also the efficiency η*P*(Φ) is presented. The head coefficient depression

visible at flow coefficient values between 0.08 and 0.12 is typical of axial pumps where flow separation and stall occur.

**Figure 2.7 Typical noncavitating performance characteristics for an axial flow pump. Brennen [2] **

*The hydraulic efficiency is different from the global efficiency ηS* of the turbomachine which is

lower due to the mechanical losses that may occur at the bearings or as result of the disk friction caused by the fluid interacting with other nonactive surfaces.

### 2.3.

### FLOW FEATURES

In this chapter an identification of the main deviations from an ideal flow will be presented. A
detailed analysis of these secondary flows is beyond the scope of the present work but the knowledge
of the main features here presented are important to understand how the flow phenomena may affect
the hydraulic efficiency. For a detailed analysis the reader is referred to Brennen2_{ and }
Lakshminarayana3_{. }

At high Reynolds numbers most of the hydraulic losses are due to the turbulent mixing and secondary flows which arises due to wall boundary layers or as a consequence of an upstream flow.

The presence of viscous effects may develop nonuniformities of radial gradient profile in velocity, stagnation pressure and stagnation enthalpy. Some of these nonuniformities may also be present in case of an inviscid flow and they may manifest as a direct consequence of the rotation of the impeller.

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To understand the origin of this kind of phenomena it is possible to consider a two dimensional cascade blade row with flow velocity profile as in Figure 2.8, where AAA is the streamline in uniform region whereas BBB is the streamline in shear layer:

*a. Neglecting the viscous effects and the velocity gradient in normal direction, with *

incompressible and steady flow hypothesis, the pressure gradient (∂ ∂*p n*/ ) is balanced by
the centripetal acceleration:

2
*A*
*A* *A*
*u*
*p*
*n* *R*
ρ
∂
_{ =}
_{∂}
(2.20)

*with RA* the radii of streamline at point A.

*b. Applying the boundary layer approximation, the pressure gradient for AAA and BBB *

should be the same, obtaining

2 2
*A* *B*
*A* *B* *A* *B*
*u* *u*
*p* *p*
*n* *n* *R* *R*
ρ ρ
∂ ∂
_{=} _{=} _{>}
_{∂} _{∂}
(2.21)

Hence, there is an imbalance due to the velocity gradient and streamline BBB will deviate
*from its ideal path towards BB’B’’, leading to the presence of a cross flow (v) that *
therefore is a deviation from main flow and is known as secondary flow.

*c. For the continuity equation there is also a spanwise velocity w given by *

*w* *v*

*b* *n*

∂ _{= −}∂

∂ ∂ (2.22)

(neglecting streamwise pressure gradient, ∂*u s*/∂ =0)

The secondary flow will thus induce secondary vortices and due to this a vorticity,

/ /

*S* *v b* *w n*

ω = −∂ ∂ + ∂ ∂ , will be present.

As shown before, the presence of secondary flows affects the performance of turbomachines. Their effects can be classified in the following way:

*a. 3D flow field due to the cross-flow, u and w. *

b. Creation of a vortex that can produce separation regions (it can become meaningful). c. Influence on the pressure difference between inlet and discharge due to the flow turning. d. Losses tends to decrease the efficiency up to 2-4%.

e. Change in incidence angle for downstream flow. Hence the interaction between rotor and stator is affected and development of vibration, flutter and noise are possible. Off-design corrections needed.

f. Influence on the temperature field and cooling system or requirement.

g. Development of cavitation due to the influence on the pressure field leading to possible damages and losses in performance.

The viscosity and the turbulence influences the boundary layer growth and the former tends to decrease streamwise normal vorticity, diffusing and dissipating the vortices.

It is possible to describe the most common secondary flows which manifest in different type of pumps. Secondary flows in unshrouded axial pumps and in radial impellers are represented in Figure 2.9, Figure 2.10 and Figure 2.11.

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**Figure 2.8 Origin of a secondary flow in a single blade. Lakshminarayana [3] **

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**Figure 2.10 Secondary flows in a cross-section of an unshrouded axial flow impeller. Brennen [2] **

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As shown in Figure 2.10, secondary flows in unshrouded axial flow impellers can be identified by three principal deviations from the ideal operative condition.

• First of all, an important tip leakage flow is present at the blade tips due to the pressure difference between the suction and pressure surfaces. Because of its presence the secondary flow will be entrained on both surfaces of the blade. When the blade is more loaded, as in the case of flow rates lower than the design condition, this flow becomes more important. Bhattacharyya observed a backflow associated with this leakage flow and a backflow at the hub.

• The boundary layer will generate an outward radial flow on both surfaces but the one on the suction surface can be stronger due to the leakage flow.

• Lakshminarayana noticed the presence of secondary flows at the hub and at the casing. For unshrouded centrifugal impellers it is possible to make a similar reasoning. On the other hand a secondary flow that develops a disturbed and separated flow may arise in the vaneless spiral volute of a centrifugal pump. In particular, the volute is planned to collect the discharging flow, providing a circumferentially uniform velocity and pressure, hence, it has to operate at the design flow coefficient. Therefore, in off-design conditions, disturbed and separated flows can occur in correspondence of the cutwater or tongue as shown in Figure 2.12. In this figure, the situations in which the flow coefficient is below (left) or above (right) the design condition are represented.

**Figure 2.12 Secondary flow in a vaneless volute of a centrifugal pump with flow rate below (left) and above(right) the design **
**condition. Brennen [2] **

### 2.4.

### PREROTATION AND DISCHARGE FLOW

The phenomena of prerotation belongs to the secondary flows category and, like the other phenomena of this class, affects the performance of the turbomachine, interacting also with cavitation.

In prerotation a swirling flow is present at the inlet of the pump and therefore the flow has axial vorticity. This swirl velocity will change the incidence angle and so the performance of the pump.

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From the vorticity transport theorem (in steady flow condition) the property of the prerotation flow is derived, consisting in the dependence of the vorticity from the axial position due to:

*a. Viscosity tends to diffuse the vorticity into the flow. *

*b. Cross-sectional area variation that accelerates or decelerates the flow, leading to an *

increase in the swirl velocity as a consequence of the vortex stretching. Hence the presence of a nozzle at the inlet tends to increase the magnitude of the preexisting swirl.

*The condition a is first considered when a uniform and symmetric cross-section of the duct is *
present. In inviscid flow (no viscosity), if there is a position at which the swirl is zero, then the swirl is
equal to zero everywhere (Kelvin’s theorem). In prerotation, even in presence of viscous effects, this
result is not altered since the fact that from an axial motion, axial vorticity cannot be generated.
Despite this, there are two conditions in which prerotation can be present.

The first one is due to the backflow, given by the tip leakage flow. As the flow coefficient is lowered under a certain critical value, the pressure difference at the tip produces a large leakage flow which leads to an annular region of backflow which can penetrates upstream for many diameters of the inlet plane where it is entrained back into the primary flow. Therefore the Kelvin’s theorem is not violated since the vorticity is imparted to the backflow from the impeller and then spread in the main flow.

It is important to note that the backflow occurs also in radial impellers with either shrouded or unshrouded configuration.

**Figure 2.13 Backflow as a consequence of tip leakage flow. Brennen [2] **

The second condition in which prerotation takes place is when the suction line is fed by means of a reservoir that present a free surface. In this case an axial vorticity is created as a consequence of the nonuniformity that the reservoir has. The magnitude of the vorticity increases as the flow enters in the duct and from that moment it becomes quite steady. As the vortex is stretched from the inlet duct to

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the free surface, air can be present in the core, leading to a two-phase flow at the inlet of the pump and decreasing the performance. Even in the absence of air, the vortices would affect the pump performance.

As the prerotation concerns the inlet flow, also the discharge flow requires important considerations.

At the discharge of a pump, a diffuser is often used to preserve the gains of energy obtained by the impeller, preventing hydraulic losses. To do this the swirl velocity has to be recovered.

In axial pumps a stator row of blades is usually used whereas in centrifugal pumps a volute, with
vanes or not, is present at the discharge (Figure 2.12). In radial pumps, the volute is an important part
of the pump’s design because its task is to minimize losses, uniforming the pressure circumferentially
*at impeller’s discharge where a tangential velocity v*θ*2 and a radial velocity vr2* are present.

### 2.5.

### REFERENCES

[1] http://turbo.mech.iwate-u.ac.jp/

*[2] C.E. Brennen, Hydrodynamics of Pumps, Oxford University Press, 1994. *

*[3] B.Lakshminarayana, Fluid Dynamics of Heat Transfer of turbomachinery, John wiley & *