# Quasi-Steady Analytical Model Benchmark of  an Impulse Turbine for Wave Energy Extraction

Authors: A. Thakker; J. Jarvis; A. Sahed
Journal: International Journal of Rotating Machinery
 (2008)

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## Abstract

This work presents a mean line analysis for the prediction of the performance and aerodynamic loss of axial flow impulse turbine
wave energy extraction, which can be easily incorporated into the turbine design program. The model is based on the momentum
principle and the well-known Euler turbine equation. Predictions of torque, pressure drop, and turbine efficiency showed favorable
agreement with experimental results. The variation of the flow incidence and exit angles with the flow coefficient has been reported
for the first time in the field of wave energy extraction. Furthermore, an optimum range of upstream guide vanes setting up angle
was determined, which optimized the impulse turbine performance prediction under movable guide vanes working condition.


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## Body

## 1. Introduction

The
oscillating water column (OWC) wave energy harnessing method is
considered as one of the best techniques of converting wave energy into
electricity. It is an economically viable design due to its simple geometrical
construction, and is also strong enough to withstand against the waves with
different heights, periods, and directions. The design (see Figure1) consists
of an OWC chamber and a circular duct, which reciprocally moves the air
from and into the chamber as the wave enters and intercedes from the chamber.
The wave energy is converted into air pneumatic energy inside the chamber. A
special turbine mounted inside the duct converts the air pneumatic energy to a
mechanical power. A matching generator is coupled to the turbine to produce
electricity [1].Figure 1
Impulse turbine
power take-off unit with OWC.In order to use the potential wave energy resource,
efficiently, turbine design/operation with low losses, high efficiency, and
desirable performance is needed. The efficiency is a measure of performance and
a poorly performing turbine becomes unavailable for power plant. Therefore a
sound knowledge of the turbine efficiency limits is necessary for the power
take-off design/operation.The impulse turbine discussed here is one of a class
of turbines called self-rectifying turbines, that is, turbines that rotate in
the same direction no matter what the direction of the airflow is.
Self-rectifying turbines are a response to the need for turbines to extract
power from bidirectional airflows that arise in wave power applications such as
the OWC. The basic  turbine design parameters were based on the optimum
design parameters given by Setoguchi and Takao [2], but with a H/T ratio of 0.6.
The details are given in Table 1 and a 2D sketch at mid radius is shown in
Figure 2. The rotor consists of 30 blades with a chord length, Lr=100mm and pitch, Sr=50mm.
There are 26 fixed angle mirror image guide vanes on both sides of the rotor.
The guide vanes inlet/outlet angle is fixed at 30°. The turbine
was tested at a constant axial air velocity of 7.22m/s.
Data were collected with the help of data acquisition system to minimize the
errors. Experiments were performed by varying the rotational speed from 1300 to
100 rpm, thus giving a flow coefficient range of 0.22–2.90 under
unidirectional steady flow conditions. The peak efficiency of 44.6% was
achieved at ϕ=0.88,
corresponding to a rotational speed of 300 rpm. The Reynolds number at the peak
efficiency point was 0.92×105.Table 1
Rotor and guide vanes geometry.
ParameterSymbolH/T=0.6Blade profile: ellipticalNumber
of bladesz30Tip
diameterD600.0 mmChord
lengthLr100.0 mmBlade
passage flowTa20.04 mmPitchSr50 mmBlade
inlet angleγ60°Guide
vanes profile: plate typePitchSg58.0 mmChord
lengthLg131.0 mmNumber
of guide vanesg26Guide
vane inlet/outlet angle30°Figure 2
Impulse
turbine geometry and velocity vectors.The state-of-the-art of the wave energy impulse
turbine is crucially getting closer to an actual prototype and an analytical
model of the turbine derived from first principles would prove necessary. Thus
far, the highest efficiency reported by model testing has been 50% at most,
which makes this particular impulse turbine underperforming the conventional
axial turbine by a wide margin. Partly, this is due to the fact that designing
a turbine for a wave power application requires that a design range be taken
into account, rather than a single design point [3]. Furthermore,  the
operational  environment to which the turbine must be designed is inherently
random, constantly varying, and difficult to predict. Therefore, the prediction
of operational flow coefficients is potentially problematic. The flow when
passes through rotor stagnation and static properties changes. If there is no
static pressure drop in a rotor, the turbine is called impulse turbine.
Experimental studies showed that in this turbine, though typified under “Impulse," a substantial degree of reaction is present [4], which tells about
the amount of losses generated in the turbine blade passage.Using simple performance prediction of the flow at the
mean line of blade height combined with numerical method, an explicitly new
analytical model for estimating the maximum efficiency of wave energy
extraction impulse turbine of any size is presented and compared favorably to a
set of experimental data. The theoretical analysis which, based on the angular
momentum principle and Euler turbine equation, yields the turbine performance
parameters previously obtained by model testing under quasi-steady state. The
model yields equations that predict the shape of the curves obtained by
plotting the experimental data.


## 2. Turbine Working Principle

The working principle of the impulse turbine should be
well understood in order to optimize the design of the power take-off unit. In
order to achieve this, velocity triangles can be used to calculate the basic
performance of the turbine. Figure2 shows the relevant velocities along with
the turbine blades and guide vane geometry.The air issued from the OWC is constrained to
exit to the atmosphere through the turbine duct. In the annular duct, the air
flows axially with a velocityVa before it gets diverted with an angle α2 with respect to the turbine axis by the inlet
guide vanes located upstream of the turbine rotor. Besides the introduction of
the prewhirl angle α2,
the inlet guide vanes play a role of stationary nozzles for which the purpose
is to accelerate and guide the flow smoothly into the rotor. The air exits the
upstream guide vanes at absolute velocity V2 which is higher than Va.
A certain amount of static pressure drop is achieved through the nozzle.
Therefore, the air flow gains in dynamic pressure and looses in static pressure
when passing the inlet guide vanes. The rotor rotates at velocity UR and relative to the rotor, the velocity of the
air as it impinges on the rotor entrance is W2.
The air is turned by the rotor and exits, relative to the rotor, at velocity W3.
However, in absolute terms the rotor exit velocity is V3.


### 2.1. Turbine Losses

Flows through an axial turbine blade passage are
always three-dimensional, viscous, and unsteady. Both the geometric description
of the fluid flow domain and the physical processes present are extremely
complicated [5]. Nevertheless, the flow through the wave energy impulse turbine
is considered incompressible with subsonic regime as the flow rates generated
by the OWC are relatively low. But regions of laminar, transitional, and
turbulent flows, separated flows, and fully developed viscous profiles may all
be present simultaneously due to the complicated geometry of the flow field.In general the losses generated within the turbine
passage consists of profile loss, secondary flow loss, tip clearance loss, and
mechanical loss. The latter could be reduced with improved manufacturing and
assembling technology, and profile and secondary flow loss (60%) could be
reduced by designing a turbine to operate at optimum design parameters.
However, there are many design parameters for minimizing aerodynamic losses
within the turbine passage. Among them, the incidence angle is the most
important as it is immediately related to the aerodynamic losses [6]. The guide
vanes and blade profile losses can be significant if the blade shapes are not
optimized for the local operating conditions. Profile losses are driven by
surface finish, total blade surface area, blade shape and surface velocity
distributions, and proper matching between guide vane and blade to minimize
incidence losses. Equally significant losses can be caused by the complex
secondary flows generated as the viscous boundary layers along the pressure and
suction of the air path are turned through the blade passage.The clearance between the blade tips and casing
end-wall in a turbine induces leakage flow, which arises due to pressure
difference between the pressure surface and suction surface of the blade. The
leakage flow emerging from the clearance interacts with the passage flow (main
stream) and rolls up into a vortex known as tip leakage vortex. Although the
size of the clearance is typically about 1% of the blade height, the leakage
flow through this small clearance has a significant effect on the aerodynamics
of the turbine. For example, the tip clearance loss of a turbine blade can
account for as much as one-third of the total losses [7]. The tip leakage loss
is driven by the higher reaction levels at the blade tip, which increases the
pressure drop across the blade tip.In contrast to the general design problem familiar in
industry (e.g., steam and gas turbines), where a turbine would be expected to
operate at a single design point for the majority of the time, the performance
of a turbine intended for use in a wave energy application must be considered
over a range of flow coefficient. This is a consequence of the constantly
changing and bidirectional nature of the airflow (varying loads), thus
designing a turbine to a wave power application requires that a design range be
taken into account, rather than a single design point [3]. Nevertheless, a
certain similarity can be found between the steam impulse turbine and the
self-rectifying impulse turbine as for both the fluid's pressure is changed to
velocity by accelerating the fluid with a nozzle and upstream guide vane,
respectively. Also, the steam rotor blade can be recognized by its shape, which
is symmetrical about the rotor midchord and has inlet and exit angles around 20°.
It has constant cross section from hub to tip, which is almost similar to that
of the wave energy impulse turbine.


### 2.2. Flow Incidence and Deviation Angle

The relative inlet flow vector at blade leading edge (W2) is not a simple geometric value but a
measured parameter; that is, it depends on the rotational speed and the
absolute flow velocity. The incidence i on the rotor is defined as an angle calculated
from the relative inlet flow vector to the blade inlet angle (see Figure 3).
Optimizing the incidence would minimize pressure losses in the blade passage,
which the turbine efficiency is directly related to. The optimum incidence
depends on the input power as well as the blade profile. The range of
applicable incidence becomes narrow when the turbine operates at high input
power [8]. Also, due to Cho and Choi [8] the optimum is for small negative
values (around −20°) but the efficiency quickly drops as the
incidence grows to negative over the range of applicable incidence, in which
case the flow tends to strike the blade leading edge axially and beyond.Figure 3
Velocity
triangles.The angle of the flow leaving the blade at the
trailing edge (β3) is of great importance as it determines the
magnitude of relative exit flow at blade trailing edge (W3). The deviation angle (d) from the rotor is defined as an angle
calculated from the relative exit flow vector to the blade exit angle. Assuming
that the axial velocity and the relative tangential velocity remain the same
(for optimum design flow coefficient), the angle β3 is dependant on the geometry profile of the
blade's trailing edge and the pressure difference between the suction side and
pressure side [4, 9].The basic objective of the turbine design is that the
blade angles at inlet and exitγ must be matched properly with the fluid flow
angles (β2, β3), respectively. They need not necessarily be
equal, but should be matched properly to minimize losses [4].The absolute angle (α3) at the blade trailing edge is equally
important as it determines the absolute velocity (V3) by which the air flow enters the inlet of
the downstream guide vanes. If the angle α3 is close to 60°,
the metal blade angle, then the air would enter the downstream guide vanes
swiftly as the air flow direction would be parallel to the straight portion of
the guide vanes.  Under this condition, the wave energy impulse turbine would be
working at maximum efficiency because of the reduced losses through the downstream guide vanes
[4, 9]. These losses are
known, from experimental work, to be substantial and directly affect the
turbine efficiency.  Outside the above working condition, the air will
strike the guide vanes more or less axially, causing bigger losses. In the
latter case, the turbine is said to be off-designed. For an ideal wave energy impulse turbine, α3 should also match the setting up angle of the downstream guide vanes, θ2, which adds to the complexity of the design (see Figure 2).


### 2.3. Turbine Performance Evaluation

Thus far the turbine characteristics under steady flow
conditions have been obtained by model testing and were evaluated according to
the torque coefficientCt,
input power coefficient Ca and flow coefficient ϕ,
which are defined by Setoguchi and Takao [2]:(1)Ct=To(1/2)ρa(Va2+UR2)bZLrrR,(2)Ca=ΔPQ(1/2)ρa(Va2+UR2)bZLrVa,(3)ϕ=VaUR,(4)Re=ρaVa2+UR2Lrμ,(5)η=CtCaϕ, where Re is the Reynolds  number based on the chord
length, To:
measured output torque, ρa:
density of air, b:
rotor blade height, UR:
circumferential at rR, Va:
mean axial flow velocity, rR:
mean radius, Z number of rotor blades, ΔP:
measured total pressure drop between settling chamber and atmosphere, and Q:
flow rate. The test Reynolds number based on the chord was 0.4×104.On the other hand, it was reported in [10] that the
quasi-steady torque and input coefficient were higher, especially at high flow
coefficient than those found when the turbine is operated under unsteady flow
conditions. Also, the drop-off in efficiency under steady state at high flow
coefficient was not seen under sinusoidal flow pattern. Furthermore, the
sinusoidal testing showed that if the efficiency is considered through the
complete sinusoidal wave, it is relatively constant. Therein, it is therefore
important that future turbines are designed with this in mind due to the
difference between the results found from unsteady testing and those predicted
using fixed flow testing.Velocity triangles can be constructed at any section
through the blades (e.g., hub, tip, midsection, etc.) using the various
velocity vectors, but are usually shown at the mean radius. The obvious snag is
that of the reduction in aerodynamic design work. At other radii velocity
triangles will vary, demanding the introduction of either blade or guide vanes
twisting [11]. Nevertheless, it is known that the efficiency of a properly
designed axial flow turbine can be predicted with fair accuracy (1 or 2%) by
the adoption of simple mean line analysis methods which incorporate proper loss
and flow angle correlations [12].Similar to simple theory for ordinary turbo-machine,
the following assumption can be given to analyze the turbine performance.(i)
Turbine performance is estimated from
condition at mean radius.(ii)
Absolute nozzle exit flow angleα2,
the complement of θ1,
is constant.(iii)
Relative rotor exit flow angle is constant,β3.
Due to blade symmetry of this particular design of
self rectifying turbine we can state that(iv)
the angles between the relative flow vector
and the absolute velocity vector at inlet and outlet of the rotor are identical
(ϵ).


## 2.1. Turbine Losses

Flows through an axial turbine blade passage are
always three-dimensional, viscous, and unsteady. Both the geometric description
of the fluid flow domain and the physical processes present are extremely
complicated [5]. Nevertheless, the flow through the wave energy impulse turbine
is considered incompressible with subsonic regime as the flow rates generated
by the OWC are relatively low. But regions of laminar, transitional, and
turbulent flows, separated flows, and fully developed viscous profiles may all
be present simultaneously due to the complicated geometry of the flow field.In general the losses generated within the turbine
passage consists of profile loss, secondary flow loss, tip clearance loss, and
mechanical loss. The latter could be reduced with improved manufacturing and
assembling technology, and profile and secondary flow loss (60%) could be
reduced by designing a turbine to operate at optimum design parameters.
However, there are many design parameters for minimizing aerodynamic losses
within the turbine passage. Among them, the incidence angle is the most
important as it is immediately related to the aerodynamic losses [6]. The guide
vanes and blade profile losses can be significant if the blade shapes are not
optimized for the local operating conditions. Profile losses are driven by
surface finish, total blade surface area, blade shape and surface velocity
distributions, and proper matching between guide vane and blade to minimize
incidence losses. Equally significant losses can be caused by the complex
secondary flows generated as the viscous boundary layers along the pressure and
suction of the air path are turned through the blade passage.The clearance between the blade tips and casing
end-wall in a turbine induces leakage flow, which arises due to pressure
difference between the pressure surface and suction surface of the blade. The
leakage flow emerging from the clearance interacts with the passage flow (main
stream) and rolls up into a vortex known as tip leakage vortex. Although the
size of the clearance is typically about 1% of the blade height, the leakage
flow through this small clearance has a significant effect on the aerodynamics
of the turbine. For example, the tip clearance loss of a turbine blade can
account for as much as one-third of the total losses [7]. The tip leakage loss
is driven by the higher reaction levels at the blade tip, which increases the
pressure drop across the blade tip.In contrast to the general design problem familiar in
industry (e.g., steam and gas turbines), where a turbine would be expected to
operate at a single design point for the majority of the time, the performance
of a turbine intended for use in a wave energy application must be considered
over a range of flow coefficient. This is a consequence of the constantly
changing and bidirectional nature of the airflow (varying loads), thus
designing a turbine to a wave power application requires that a design range be
taken into account, rather than a single design point [3]. Nevertheless, a
certain similarity can be found between the steam impulse turbine and the
self-rectifying impulse turbine as for both the fluid's pressure is changed to
velocity by accelerating the fluid with a nozzle and upstream guide vane,
respectively. Also, the steam rotor blade can be recognized by its shape, which
is symmetrical about the rotor midchord and has inlet and exit angles around 20°.
It has constant cross section from hub to tip, which is almost similar to that
of the wave energy impulse turbine.


## 2.2. Flow Incidence and Deviation Angle

The relative inlet flow vector at blade leading edge (W2) is not a simple geometric value but a
measured parameter; that is, it depends on the rotational speed and the
absolute flow velocity. The incidence i on the rotor is defined as an angle calculated
from the relative inlet flow vector to the blade inlet angle (see Figure 3).
Optimizing the incidence would minimize pressure losses in the blade passage,
which the turbine efficiency is directly related to. The optimum incidence
depends on the input power as well as the blade profile. The range of
applicable incidence becomes narrow when the turbine operates at high input
power [8]. Also, due to Cho and Choi [8] the optimum is for small negative
values (around −20°) but the efficiency quickly drops as the
incidence grows to negative over the range of applicable incidence, in which
case the flow tends to strike the blade leading edge axially and beyond.Figure 3
Velocity
triangles.The angle of the flow leaving the blade at the
trailing edge (β3) is of great importance as it determines the
magnitude of relative exit flow at blade trailing edge (W3). The deviation angle (d) from the rotor is defined as an angle
calculated from the relative exit flow vector to the blade exit angle. Assuming
that the axial velocity and the relative tangential velocity remain the same
(for optimum design flow coefficient), the angle β3 is dependant on the geometry profile of the
blade's trailing edge and the pressure difference between the suction side and
pressure side [4, 9].The basic objective of the turbine design is that the
blade angles at inlet and exitγ must be matched properly with the fluid flow
angles (β2, β3), respectively. They need not necessarily be
equal, but should be matched properly to minimize losses [4].The absolute angle (α3) at the blade trailing edge is equally
important as it determines the absolute velocity (V3) by which the air flow enters the inlet of
the downstream guide vanes. If the angle α3 is close to 60°,
the metal blade angle, then the air would enter the downstream guide vanes
swiftly as the air flow direction would be parallel to the straight portion of
the guide vanes.  Under this condition, the wave energy impulse turbine would be
working at maximum efficiency because of the reduced losses through the downstream guide vanes
[4, 9]. These losses are
known, from experimental work, to be substantial and directly affect the
turbine efficiency.  Outside the above working condition, the air will
strike the guide vanes more or less axially, causing bigger losses. In the
latter case, the turbine is said to be off-designed. For an ideal wave energy impulse turbine, α3 should also match the setting up angle of the downstream guide vanes, θ2, which adds to the complexity of the design (see Figure 2).


## 2.3. Turbine Performance Evaluation

Thus far the turbine characteristics under steady flow
conditions have been obtained by model testing and were evaluated according to
the torque coefficientCt,
input power coefficient Ca and flow coefficient ϕ,
which are defined by Setoguchi and Takao [2]:(1)Ct=To(1/2)ρa(Va2+UR2)bZLrrR,(2)Ca=ΔPQ(1/2)ρa(Va2+UR2)bZLrVa,(3)ϕ=VaUR,(4)Re=ρaVa2+UR2Lrμ,(5)η=CtCaϕ, where Re is the Reynolds  number based on the chord
length, To:
measured output torque, ρa:
density of air, b:
rotor blade height, UR:
circumferential at rR, Va:
mean axial flow velocity, rR:
mean radius, Z number of rotor blades, ΔP:
measured total pressure drop between settling chamber and atmosphere, and Q:
flow rate. The test Reynolds number based on the chord was 0.4×104.On the other hand, it was reported in [10] that the
quasi-steady torque and input coefficient were higher, especially at high flow
coefficient than those found when the turbine is operated under unsteady flow
conditions. Also, the drop-off in efficiency under steady state at high flow
coefficient was not seen under sinusoidal flow pattern. Furthermore, the
sinusoidal testing showed that if the efficiency is considered through the
complete sinusoidal wave, it is relatively constant. Therein, it is therefore
important that future turbines are designed with this in mind due to the
difference between the results found from unsteady testing and those predicted
using fixed flow testing.Velocity triangles can be constructed at any section
through the blades (e.g., hub, tip, midsection, etc.) using the various
velocity vectors, but are usually shown at the mean radius. The obvious snag is
that of the reduction in aerodynamic design work. At other radii velocity
triangles will vary, demanding the introduction of either blade or guide vanes
twisting [11]. Nevertheless, it is known that the efficiency of a properly
designed axial flow turbine can be predicted with fair accuracy (1 or 2%) by
the adoption of simple mean line analysis methods which incorporate proper loss
and flow angle correlations [12].Similar to simple theory for ordinary turbo-machine,
the following assumption can be given to analyze the turbine performance.(i)
Turbine performance is estimated from
condition at mean radius.(ii)
Absolute nozzle exit flow angleα2,
the complement of θ1,
is constant.(iii)
Relative rotor exit flow angle is constant,β3.
Due to blade symmetry of this particular design of
self rectifying turbine we can state that(iv)
the angles between the relative flow vector
and the absolute velocity vector at inlet and outlet of the rotor are identical
(ϵ).


## 3. Torque Analysis

Applying “momentum principle—Newton's second law,”
the torque generated by the turbine shaft due to the tangential momentum change
of air passing through the turbine rotor can be evaluated as follows [11]:
Generated torque = rate of change of moment of
momentum:(6)To=ddt(mrRΔV).Replacing  the expression of change of air whirl
velocity in (6), we get (7)To=m˙rRUR(Vθ2UR+Vθ3UR).From the velocity diagram of Figure3, we have (8)Vθ2UR=ϕ(Wθ2+URVa)=ϕtanβ2+1,Vθ3UR=ϕtanα3.Replacing (8) in (7), the torque is expressed
as (9)To=m˙rRURϕ(tanβ2+tanα3+1ϕ).By the replacement for the expression ofm˙ and using blade height, chord length, and flow coefficient definition,
we get from (9) (10)To=12ρUR2bZLrrR2(πDrZLr)ϕ2(tanβ2+tanα3+1ϕ).Multiplying and dividing (10) through (1+ϕ2),
we get (11)To={12ρUR2(1+ϕ2)bZLrrR}2(πDrZLr)(ϕ21+ϕ2)×(tanβ2tanα3+1ϕ).Now, we define the theoretical torque coefficient(Ct)Th as (12)(Ct)Th=To{(1/2)ρUR2(1+ϕ2)bZLrrR},(13)(Ct)Th=2(πDrZLr)(ϕ21+ϕ2)(tanβ2+tanα3+1ϕ).The term contained in the first brackets of (13) is
the inverse of the turbine rotor solidity σr; (14)(Ct)Th=2σr(ϕ21+ϕ2)(tanβ2+tanα3+1ϕ).From (8) we have (15)⇒tanα2=tanβ2+1ϕ,β2=arctan(tanα2−1ϕ).Defining the incidence on the rotori,
as an angle calculated from the relative inlet flow vector W2 to the blade angle α2; (16)i=β2−γ=arctan(tanα2−1ϕ)−γ.The incidence is negative as shown in Figure3.The angleβ3 is dependant on the geometry profile of the
blade's trailing edge and the pressure difference between the suction side and
pressure side. Hill and Peterson (1992, 1994) [12] reported that β3 can be evaluated as (17)β3=arccos(TaSr).From which we can evaluate the deviation angled as (18)d=β3−γ.The deviation angle is positive as shown in Figure3.From assumption 3 and velocity diagram Figure3, (19)α3=β3−ϵ=arccos(TaSr)−ϵ.


## 4. Pressure Drop Analysis

From Figure3, we have (20)Vθ3=Wθ3−UR.Using (7), we get the output power as (21)Po=Toω=m˙UR(Vθ2+Wθ3−UR),dPodUR=m˙(Vθ2+Wθ3−2UR)=0.Therefore when(22)UR=Vθ2+Wθ32⇒Po=Po,max=ρ4(Vθ2+Wθ3)2Q.Given that the maximum power output equals the
available power to the turbine rotor at blade inlet(23)⇒Po,max=ΔPthQ, where ΔPth is the pressure gradient across the turbine
rotor, which can be estimated as (24)ΔPth=ρ4(Vθ2+Wθ3)2.On the other hand,(25)ΔPQ=m˙I, where ΔP is the actual pressure gradient across the
turbine (including the guide vanes) and I is the actual enthalpy drop through the
turbine. The latter equals the theoretical enthalpy drop of the turbine (Ith) plus enthalpy loss through the turbine (ΔI1) which in turn is related to the total loss
coefficient ζ by [13] (26)ζ=ΔI1(1/2)Va2.Using (25) and the definition of the actual enthalpy (I=Ith+ΔI1), we get (27)⇒ΔP=m˙QIth+m˙QΔI1=ΔPth+ΔPL, where ΔPL is a measure of pressure losses through the
turbine (including the guide vanes and tip clearance losses): (28)ΔPL=12ζρVa2,(29)ΔP=12ρ(Vθ2+Wθ3)22+12ζρVa2.Multiplying both sides of (29) by Q,
we get the input power as (30)ΔPQ=12ρVa2{12(Vθ2Va+Wθ3Va)2+ζ}bZLr(πDrZLr)Va.Multiplying and dividing through(1+ϕ2),
(30) becomes (31)ΔPQ={12ρUR2(1+ϕ2)bZLrVa}(πDrZLr)(ϕ21+ϕ2)×{12(tanβ2+tanα3+2ϕ)2+ζ}.Now, we define the theoretical input coefficient(Ca)Th as (32)(Ca)Th=ΔPQ{(1/2)ρUR2(1+ϕ2)bZLrVa},(Ca)Th=1σr(ϕ21+ϕ2){12(tanβ2+tanα3+2ϕ)2+ζ}.


### 4.1. Loss Coefficientζ

Losses in turbines are used to be expressed in terms
of loss coefficients. They are manifested by a decrease in stagnation enthalpy,
and a variation in static pressure, compared to the isentropic flow. The
commonly used loss coefficient is the stagnation pressure loss coefficient,
which is convenient for experimental work and especially for this particular
impulse turbine. Wei [14] has reported in his doctoral thesis by referring back
to Horlock [15] that the incidence is the most important parameter to predict
the off-design profile loss. The stagnation pressure loss coefficient in the
relative frame is defined as [16] (33)ζR=P02,rel−P03,rel(1/2)ρW22, where P0,rel=P+(1/2)ρW2 is the relative total pressure at blade inlet
and outlet (2,3), respectively.From (33) we have (34)ζR=(P2+(1/2)ρW22)−(P3+(1/2)ρW32)(1/2)ρW22,(35)ζR=(P2+(1/2)ρW22)−(P3+(1/2)ρW32)(1/2)ρW22+(1/2)ρ(V22−V22)+(1/2)ρ(V32−V32)(1/2)ρW22,(36)ζR=(P2+(1/2)ρV22)−(P3+(1/2)ρV32)(1/2)ρW22+(W22−V22)+(V32−W32)W22.Multiplying and dividing throughV22 (36), rearranging we get (37)ζR=(V2W2)2{(P2+(1/2)ρV22)−(P3+(1/2)ρV32)(1/2)ρV22︸A+(W2V2)2−1+(V3V2)2−(W3V2)2︸B}.Applying Euler turbine equation [10], (38)(P2+12ρV22)−(P3+12ρV32)=ρUR(Vθ2+Vθ3)(39)⇒A=ρUR(Vθ2+Vθ3)(1/2)ρV22(40)A=2(URV2)2(VaUR)(Vθ2Va+Vθ3Va).From Figure3 we have V2=Va/cosα2, tanα2=Vθ2/Va,
and tanα3=Vθ3/Va(41)⇒A=2ϕ(cosα2)2(tanα2+tanα3).Also, from Figure3 we  have W2=Va/cosβ2, W3=Va/cosβ3,
and V3=Va/cosα3(42)⇒B=(cosα2)2{1(cosβ2)2−1(cosα2)2+1(cosα3)2−1(cosβ3)2}.By replacing (39) and (42)
in (37), simplifying, and rearranging, we get (43)ζR=(cosβ2)2{2(tanα2+tanα3)ϕ+1(cosβ2)2−1(cosα2)2+1(cosα3)2−1(cosβ3)2},ζR accounts only for aerodynamic losses (profile
and secondary flow losses, which cannot be separated) through the blade
passage. These are greatly affected by the flow incidence [7, 11]. Therefore the
tip clearance loss and the loss in the turbine stationary guide vanes are not
accounted for. For this, an underestimation of the actual losses is to be
expected when only using the losses in the rotor (43). The actual losses would
be accurately simulated if we could have an expression of ζ which accounts for all the losses in the
turbine (including tip clearance leakage, guide vanes, near hub, and casing
walls losses). For the particular WERT test rig, UL, the data logger
acquisition compensates the torque measurement for the mechanical and windage
losses.It was reported from model testing that substantial
stagnation pressure drop across the downstream guide vane exists causing
losses. To take these into account, estimation from three-dimensional
computational fluid dynamics (CFD) of the losses across the downstream guide
vanes was utilized.The variation of pressure loss coefficient through the
downstream guide vane at various stations from hub to tip was computed for the
flow coefficientsϕ=0.45,
0.67, 1, 1.35, and 1.68, respectively. The total pressure loss coefficient has
been defined as follows [17–19]: (44)ζGV=Poi−PoPoi−Psi, where P is the pressure and the subscripts i, o, and s denote inlet conditions, total and static
conditions, respectively.Then, the arithmetic mean of the computed values from
hub to tip for each flow coefficient were calculated, plotted and an optimum
curve fit correlation (ζGV=f(ϕ)) was established as illustrated in Figure 4; (45)ζGV=5.78ϕ−1.85.Figure 4
Correlated downstream guide vane loss coefficient
versus flow coefficient [10].The correlation of the downstream guide vane loss (45)
derived from validated computational (CFD) work should be considered
specific for the design of impulse turbine wave energy
extraction; (46)ζ=ζR+ζGV.


## 4.1. Loss Coefficientζ

Losses in turbines are used to be expressed in terms
of loss coefficients. They are manifested by a decrease in stagnation enthalpy,
and a variation in static pressure, compared to the isentropic flow. The
commonly used loss coefficient is the stagnation pressure loss coefficient,
which is convenient for experimental work and especially for this particular
impulse turbine. Wei [14] has reported in his doctoral thesis by referring back
to Horlock [15] that the incidence is the most important parameter to predict
the off-design profile loss. The stagnation pressure loss coefficient in the
relative frame is defined as [16] (33)ζR=P02,rel−P03,rel(1/2)ρW22, where P0,rel=P+(1/2)ρW2 is the relative total pressure at blade inlet
and outlet (2,3), respectively.From (33) we have (34)ζR=(P2+(1/2)ρW22)−(P3+(1/2)ρW32)(1/2)ρW22,(35)ζR=(P2+(1/2)ρW22)−(P3+(1/2)ρW32)(1/2)ρW22+(1/2)ρ(V22−V22)+(1/2)ρ(V32−V32)(1/2)ρW22,(36)ζR=(P2+(1/2)ρV22)−(P3+(1/2)ρV32)(1/2)ρW22+(W22−V22)+(V32−W32)W22.Multiplying and dividing throughV22 (36), rearranging we get (37)ζR=(V2W2)2{(P2+(1/2)ρV22)−(P3+(1/2)ρV32)(1/2)ρV22︸A+(W2V2)2−1+(V3V2)2−(W3V2)2︸B}.Applying Euler turbine equation [10], (38)(P2+12ρV22)−(P3+12ρV32)=ρUR(Vθ2+Vθ3)(39)⇒A=ρUR(Vθ2+Vθ3)(1/2)ρV22(40)A=2(URV2)2(VaUR)(Vθ2Va+Vθ3Va).From Figure3 we have V2=Va/cosα2, tanα2=Vθ2/Va,
and tanα3=Vθ3/Va(41)⇒A=2ϕ(cosα2)2(tanα2+tanα3).Also, from Figure3 we  have W2=Va/cosβ2, W3=Va/cosβ3,
and V3=Va/cosα3(42)⇒B=(cosα2)2{1(cosβ2)2−1(cosα2)2+1(cosα3)2−1(cosβ3)2}.By replacing (39) and (42)
in (37), simplifying, and rearranging, we get (43)ζR=(cosβ2)2{2(tanα2+tanα3)ϕ+1(cosβ2)2−1(cosα2)2+1(cosα3)2−1(cosβ3)2},ζR accounts only for aerodynamic losses (profile
and secondary flow losses, which cannot be separated) through the blade
passage. These are greatly affected by the flow incidence [7, 11]. Therefore the
tip clearance loss and the loss in the turbine stationary guide vanes are not
accounted for. For this, an underestimation of the actual losses is to be
expected when only using the losses in the rotor (43). The actual losses would
be accurately simulated if we could have an expression of ζ which accounts for all the losses in the
turbine (including tip clearance leakage, guide vanes, near hub, and casing
walls losses). For the particular WERT test rig, UL, the data logger
acquisition compensates the torque measurement for the mechanical and windage
losses.It was reported from model testing that substantial
stagnation pressure drop across the downstream guide vane exists causing
losses. To take these into account, estimation from three-dimensional
computational fluid dynamics (CFD) of the losses across the downstream guide
vanes was utilized.The variation of pressure loss coefficient through the
downstream guide vane at various stations from hub to tip was computed for the
flow coefficientsϕ=0.45,
0.67, 1, 1.35, and 1.68, respectively. The total pressure loss coefficient has
been defined as follows [17–19]: (44)ζGV=Poi−PoPoi−Psi, where P is the pressure and the subscripts i, o, and s denote inlet conditions, total and static
conditions, respectively.Then, the arithmetic mean of the computed values from
hub to tip for each flow coefficient were calculated, plotted and an optimum
curve fit correlation (ζGV=f(ϕ)) was established as illustrated in Figure 4; (45)ζGV=5.78ϕ−1.85.Figure 4
Correlated downstream guide vane loss coefficient
versus flow coefficient [10].The correlation of the downstream guide vane loss (45)
derived from validated computational (CFD) work should be considered
specific for the design of impulse turbine wave energy
extraction; (46)ζ=ζR+ζGV.


## 5. Results and Discussion

Figure5 shows the theoretical and experimental torque
coefficient versus flow coefficient. The torque coefficient follows an
exponential trend up to flow coefficient (ϕ=0.8) then a short transition occurs and the trend
becomes logarithmic in nature for the rest of the flow coefficient. There is no
apparent stall, as opposed to the Wells turbine. It can be seen that the
model's torque coefficient predicts well the shape of the torque coefficient
obtained by plotting the experimental data. The prediction seems to be almost
perfect at low flow coefficient (up to ϕ=0.8) and then a small discrepancy appears for the
rest of the flow coefficient.Figure 5
Turbine torque coefficient versus flow
coefficient [10].Figure6 shows  the variation of the incidence angle (i) and absolute exit angle (α3), calculated from (16) and (19),
respectively, with the flow coefficient. The incidence angle, directly related
to the profile geometry [9], is very high (negative) at low flow coefficients
(high rotational speed) making the air flow relatively impinging at the suction
side of the leading edge. In other word, the stagnation point is located at the
suction side of the leading edge and the air flow acts as a break on the
turbine. This would be the main reason of the negative torque obtained
experimentally [4]. As the flow coefficient increases a rapid decline of the
incidence occurs from −120° at ϕ=0.29 up to ϕ=1.1 at which it reaches −20°, where the turbine efficiency reaches a plateau. The optimum incidence angle for
wave energy impulse turbine is for small negative values (around −20°) similar to that reported in [5]. At this
condition the stagnation point is located close on the end profile camber line.
As the flow coefficient continues increasing the incidence angle is decreasing
almost steadily to reach a value of −4.5° at the highest flow coefficient of ϕ=3.57.
At this low incidence angle the relative flow angle (β2,
(16)) is almost aligned with the blade angle (γ) presenting a supposedly optimum working
conditions, where the incidence angle ensures the smoothest entry condition
into the turbine rotor blade.Figure 6
Flow incidence and absolute exit angle versus flow coefficient.The trend of the exit angle is similar, but varies in
phase (66°) with the incidence angle, Figure 6. At very
low flow coefficient ϕ=0.29,
the absolute exit flow angle is about −55° causing the flow to impinge onto the downstream
guide vane straight portion vertically (the ideal is when the flow direction is
parallel to the downstream guide vane straight portion, smooth entry). Then, a
rapid decrease (from high negative value) of the exit flow angle is shown with
the increase of flow coefficient, reaching 46° at ϕ=1.1.
At this flow coefficient the flow entering the inlet guide vane is relatively
smooth. In between the above two flow coefficients, there was a condition where
the exit flow vector V3 became aligned with the axial flow vector Va,
precisely at a value of flow coefficient ϕ=0.53.
As the flow coefficient continues beyond ϕ=1.1,
a steady increase of exit flow angle is shown up to the highest flow
coefficient of ϕ=3.57 at which it reaches 62.8°.
During the steady increase of the absolute exit flow angle, a favorable working
condition sets up, where the exit flow vector becomes closer to the alignment
with the downstream guide vane. This will reduce greatly the losses through
downstream guide vanes as will be shown below.Figure7 displays  the input coefficient theoretical
and experimental data versus flow coefficient in the case of the losses through
the rotor passage and the downstream guide vanes were taken into account (based
on experimental analysis the upstream guide vane losses were found to be low
[4, 9]). It can be seen that the discrepancy is smaller relative to the earlier
case especially at low flow coefficient. Also the slope of the model input
coefficient at high flow coefficient is almost zero (no change). Whereas at
high flow coefficient, the experimental input coefficient seems to increase
with the flow coefficient. This could be explained by the boundary layer
separation losses at the blade suction side caused by the high inlet relative
flow angle (low incidence angle) and higher loading. Also the contention for
the front part of the blade tip could be blocked by the inlet boundary layer
“aerodynamically closed” and therefore could be sustaining the horseshoe
vortex system [18]. Moreover the tip leakage vortex, which  is not incorporated
in the model, increases separation. Previous author's results [19] obtained
from CFD and validated experimentally showed up to 4% of losses could be generated by the gap
existing between the blade tip and the duct inner surface for the specific impulse
turbine wave energy extraction.Figure 7
Turbine input coefficient versus flow
coefficient [10].Figure8 shows the turbine efficiency from the
experimental as well as theoretical data, which is typical of the impulse
turbine used for wave energy conversion [2]. At low levels of flow coefficient
it can be seen that the efficiency is very sensitive to changes in the flow
coefficient. Furthermore, the experimental efficiency was maximum (η=44.6%) at flow coefficient (ϕ=0.88) which is known as the optimum flow
coefficient and the Reynolds number was (Re=0.92×105). Subsequently, the efficiency decreases as
the flow coefficient is further increased, although its sensitivity to changes
in flow coefficient diminishes. Low flow coefficient thus greatly reduces the
efficiency to an extent that the high flow coefficient does not. This is due to
the incidence angle which is very high (negative) at low flow coefficients,
making the air flow relatively impinging at the suction side of the leading
edge. This is shown experimentally as a small positive value or even negative
torque in the region of very low coefficient. Also the low flow coefficient
region is characterized with instable working conditions where the efficiency
could drop or rise sharply for a small turbine angular velocity increase or
decrease, respectively. Nevertheless, as the flow coefficient starts increasing
from minimum the turbine picks up in performance sharply to reach the optimum.
The reason is that the boundary layers are of type turbulent, due to higher
Reynolds number in which case the flow separation does not exist. Also at low
flow coefficient the turbine inertia is high helping the turbine to reach the
optimum performance quickly. The conjugation of these two effects overcomes the
very high flow incidence angle effect.Figure 8
Turbine efficiency versus flow coefficient
[10].At high flow coefficient though the incidence angle is
optimum because of the low turbine rotational speed, still we notice a slight
drop in the turbine performance. This can be explained by the effect of
boundary layers of type laminar, due to lower Reynolds number, which are prone
to separation especially in the tip region [14]. In the boundary layer regions,
the velocity deficit and the total pressure loss increase as the Reynolds number
decreases (high flow coefficient).  Also, here
the inertia is not helping to recover the turbine performance as
it is low because of the low rotational speed. At moderate flow
coefficient the turbine performance is at maximum and the working condition is
stable, that is, there is no sudden change in turbine characteristics within
this region of flow coefficient. However, the incidence angle and the boundary layers are not favorable in this moderate flow conditions. This high  turbine efficiency working
region can be characterized as the compromising region, between the incidence
angle getting closer to the optimum and the boundary layer approaching the
turbulent state. Furthermore, this region is characterized as stable due to the
impulse turbine reaching the maximum efficiency at low level of torque
coefficient.The model efficiency predicts the experimental data
with fair accuracy (2%) overall, though the input coefficient, which
does not take into account the tip clearance losses, underpredicts the
experimental input coefficient by an average error of (4%). The reason is that the torque coefficient
is also underpredicted by the model with an average error of (2%) as can be seen in Figure 5, therefore the
ratio of torque coefficient (Ct) and the input coefficient (Ca), which is proportional to the efficiency
(5), is lower than it would be if the torque was perfectly predicted.Figure9 shows  the variation of the experimental and
theoretical losses in the blade passage and the downstream guide vane with the
flow coefficient. As can be seen the losses through the guide vane decrease
rapidly with the increase of the flow coefficient. The model guide vane losses
predict well the experimental data at low flow coefficient and a discrepancy
appears gradually as the flow coefficient increases. Similar trend can be seen
for the experimental losses through the blade passage up to the flow
coefficient of (ϕ=1.1) and subsequently follows a straight line
with a small slope for the rest of the flow coefficient. Whereas the model
losses through the blade passage generally decreases following a straight line
with a small slope up to the flow coefficient of (ϕ=1.1) and then remains almost constant for the
rest of the flow coefficient. Particularly, the model predicted
loses better through the rotor at high flow
coefficient than in the low flow coefficient range. Also, we notice that the
level of losses through the guide vane outweighs those of the blade from very
low flow coefficient up to (ϕ=2.7) at which point the two curves cross each
other. Beyond this point, the losses through the guide vanes continue
decreasing below zero at high flow coefficient. The small negative value of the
losses has been interpreted from model testing of the impulse turbine as a
static pressure recovery at outlet downstream guide vanes [4, 18]. Therefore,
the behavior of the downstream guide vane in the range of high flow coefficient
is similar to that of the diffuser.Figure 9
Turbine loss coefficient versus flow coefficient
[10].Figure10 depicts the
variation of the  rotor losses versus the incidence angle. It can be seen from
the figure that the losses through the blade passage
generally decrease uniformly as the incidence angle decreases from high
negative value. In other words, the loss through the rotor is minimized when
the incidence angle approaches zero, for which the inlet-flow path is online
with the blade angle (see Figure 3).Figure 10
Rotor loss coefficient versus incidence angle
[10].Figure11 shows  the variation of the experimental and
theoretical total losses in the turbine with inlet angle. As can be seen the
predicted total losses trend is similar to the experimental data. The level of
losses decreases with the increase of relative inlet flow angle following two
different curvatures  with an inflexion point that is located at β2=0°. The latter angle does not represent the minima point as seen from Figure 11. The minima scenario would be typical to a reaction turbine with an optimum angle not necessarily 0° [11]. Therefore, though the reported
experimental pressure drop across the turbine rotor was substantial [4, 18],
the loss reported in the present study reinforces the impulse characteristic of
this particular turbine for which there is no negative or positive stall. In
other word, there is no high positive or negative incidence for which the air
flow separates catastrophically from the blade surface resulting in sudden
collapse of torque and increase in pressure drop and therefore of loss.Figure 11
Turbine loss coefficient versus inlet flow
angle (β2) [10].In order to identify the optimum inlet flow angle for
which the impulse turbine efficiency is maximum, a semilog graph was used.As we can see from Figure12, the total loss
coefficient follows a straight portion with a moderate slope from low inlet
angle up to around β2=35° from which point the loss starts decreasing
rapidly for the rest of the turbine operational range. As a result the optimum
inlet flow angle, at which the turbine efficiency is maximum (the optimum
performance of this particular turbine is reached at lower level of flow
coefficient), was found to be β2=35°.
The experimental value reported in [4] was β2=33.8°.Figure 12
Turbine loss coefficient versus inlet flow angle
(β2) [10].Figure13 shows variation of the optimum upstream
guide vane angle with the flow coefficient while keeping the relative inlet
flow angle optimum β2=35°.
For lower flow coefficient, the guide vane angle is small because of the lower
input power, whereas at higher flow coefficient region, where the impulse
turbine is known to handle larger input power without drastic decrease of
efficiency (unlike the Wells turbine), the guide vane angle is larger. This is
in order to satisfy the turbine optimum relative inlet angle.Figure 13
Upstream guide vane angle versus flow
coefficient.Figure14 shows the model prediction of the impulse
turbine efficiency when the upstream guide vanes setting up angle is changed
from 13° to 46° while keeping the relative inlet flow angle
optimum β2=35° (see Figure 12). As we can see the efficiency
at each flow coefficient hence corresponding upstream guide vanes setting up
angle is improved for the majority of flow coefficient especially at lower flow
coefficient range.Figure 14
Efficiency versus upstream guide vane setting
up angle.


## 6. Conclusions

An explicit new quasi-steady analytical model based on
the well-known angular momentum principle and Euler turbine equation for
predicting the impulse turbine performance was presented. The accuracy of the
model has been verified using previously carried out experimental study. The
predicted torque coefficient was found to fit well the experimental data. The
input coefficient, which is directly related to the pressure drop across the
turbine which in turn is greatly affected by the generated losses, is also well
predicted up to high region of flow coefficient. Furthermore, the model
predicted the turbine efficiency with a fair accuracy (2%) overall.The evolution of the flow incidence and absolute exit
angle on the rotor blade leading edge and trailing edge, respectively, with the
flow coefficient has not been reported up till now in the literature of this
type impulse turbine. This gave a deep insight in understanding the behavior of
impulse turbine wave energy extraction. Furthermore, it has been elucidated
that the downstream guide vane plays an important role in the impulse turbine
efficiency.However, the proposed model for performance prediction
could be further improved by incorporating tip clearance and viscous losses in
order to predict the turbine total loss more accurately.The usefulness of the presented model consists of its
capability to quantify and provide a basis for comparing performance of the
self-rectifying impulse turbines. This would allow the designer to size an
impulse turbine to a given wave power application.



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*Source: 536079-2008-12-02.xml*