-A study on the placement of turbines downstream of a hydroelectric power station
-En studie av placeringen av turbiner nedströms från ett vattenkraftverk
Marie Stjernquist Desatnik Supervisors:
Nenad Glodic Raad Qassim
AJ125X Examensarbete i Energi och miljö, grundnivå
Stockholm, Augusti 2016
ABSTRACT
Human energy related activity has increased the carbon dioxide concentration in the atmosphere by 40
% since the preindustrial area. A leading contributor to this is the burning of fossil fuels in order to extract energy. Due to this, we are today facing many issues following climate change and global warming. The search for renewable energy sources is therefor of outmost importance. The biggest renewable energy source today is hydropower. It is big in newly industrialized countries such as China and Brazil, where in the latter it covers over 70% of the electricity supply. A relatively unexplored area is the implementation of turbines in the downstream of a hydroelectric power station. The risk here is that the turbine[s] will increase the water depth and hence impact the hydraulic head i.e. the potential power of the hydro power station negatively This report explores the potential and optimal placement of a turbine/ turbine fences in the downstream by using the Bernoulli equation. Two different cases are examined. The geometry in the first one is a finite uniform channel whereas in the second one it is a finite divergent channel. The equations for the two different cases will also vary, mainly in the expression of the friction loss. It is shown that the choices of expression for the friction loss and geometry are important for the results as the two cases gave different results. However, the second approach is supposed more accurate and shows that the closer the beginning of the channel the more power can be extracted but this also means an increase in water depth. For some flows and turbine diameters a turbine fence was also placed further on, hence departed from the expected result. This shows the complexity of the problem and the importance of an optimization model.
SAMMANFATTNING
Atmosfärens koldioxidhalt har ökat med 40% sedan förindustriell tid till följd av mänsklig aktivitet. En ledande orsak till detta är förbränningen av fossila bränslen för att utvinna energi. Detta har lett till global uppvärmning och klimatförändringar. Utvecklingen av förnybara energikällor är därför ytterst viktig. Idag är vattenkraft den största källan för förnybar energi och täcker hela 70% av elektricitetförsörjningen i Brasilien.
Ett relativt outforskat område är att implementera turbiner nerströms efter ett vattenkraftverk, risken med detta är att turbinen/turbinerna ökar vattendjupet och således påverkar fallhöjden dvs den potentiella energin från vattenkraftverket negativt. I denna uppsats används Bernoullis ekvation för att undersöka potentialen för metoden samt optimala placeringen av turbinen/turbinerna. Två olika metoder appliceras. Valet av geometri och uttryck för friktionsförlusten visar sig vara mycket viktiga då de två metoderna ger olika resultat.
Resultaten från den andra metoden antas mest riktiga, dessa visar att desto närmare kanalens början desto mer energi kan utvinnas, dock betyder detta också ett ökat vattendjup. Det visade sig att turbinerna också
placerades ut längre fram i kanalen för vissa volymflöden samt turbindiametrar dvs. avskilde från det förväntade resultatet. Detta visar på problemets komplexitet samt betydelsen av en optimeringsmodell.
ACKNOWLEDGEMENT
I am most thankful to my supervisor Raad Qassim at Universidade Federal do Rio de Janeiro for introducing me to the problem and guiding me along the way. I also want to thank my supervisor in Sweden Nenad Glodic. At last I want to thank Alex Lopes Ancora for using his expertize on the coding in Lingo.
1 INTRODUCTION ... 1
1.1 THEORETICAL BACKGROUND ... 3
2.3 TEST CASE 1: Simple geometry ... 5
2.4 TEST CASE 2: Divergent channel ... 8
2 METHODS………..4
2.1 TEST CASE 1: Simple geometry………..5
2.2 TEST CASE 2: Divergent geometry………...8
3 RESULTS ... 9
3.1 TEST CASE 1 ... 9
3.2 TEST CASE 2 ... 12
4 DISCUSSION ... 14
5 CONCLUSION ... 16
NOMENCLATURE ... 17
LIST OF TABLES ... 17
LIST OF FIGURES ... 18
6 REFERENCES ... 19
7 APPENDIX ... 20
7.1 LINGO CODE ... 20
1 INTRODUCTION
Human activity has increased carbon dioxide concentration in the atmosphere by 40% since the pre- industrial era and has now reached 400 ppm, this is higher than earth has experienced in 800 000 years.
Before the industrial revolution the CO
2concentration in the atmosphere was more or less constant at 280 ppm as the carbon dioxide released into the atmosphere matched the one being absorbed. Since then the burning of fossil fuels in order to extract energy has released carbon dioxide that has been stored for millions of years in the sea bottom into the atmosphere. As the effects of climate change are becoming a growing global concern the search for renewable energy sources is a topic of increasing interest. Having spent the year in Brazil, this report will have some focus on the country.
1Access to water and energy are two of the main keys to global development. The interaction between these two are therefor of interest, how can existing water resources be used in order to extract energy for electricity? For most people the answer is simple, hydropower. Although hydropower is not entirely free from GHG emissions (considering land use and damming) it is today classified as the largest renewable energy resource.
2At COP21 Brazil signed some ambitious agreements; The country will cut GHG emissions by 37 % below 2005 levels by 2030 and increase the percentage of renewables in the energy mix to 45 % in 2030.
The country has also agreed to promote clean technology and restoring and reforesting 12 million
hectares of forest by 2030.
370% of Brazils electricity is today generated from Hydropower
4and Brazil is currently building Belo Monte. This will be the third largest hydropower plant in the world in terms of installed generating capacity. The potential for hydropower in the country is massive but expanding further means deforestation and environmental impacts on especially the Amazons.
5It is therefor of interest to examine alternative ways to increase the amount of energy extracted from already existing hydropower plants.
Downstream of a hydroelectric power station, water velocity typically exceeds 1.5m/s. In the last few years, a number of projects have been carried out with a view to the assessment of the recovery of the kinetic energy embedded in such fluid streams so as to generate electrical power for use in the operation of the hydroelectric power station itself and/or for commercialization in the electricity market.
6The presence of a hydrokinetic turbine in an open channel, such as that downstream of a hydroelectric power station, alters the fluid flow therein as compared with that in the absence of the turbine. The major change is that the fluid level at the entry of the open channel downstream of the power station increases, so as to
1
J.T Mathis et al. , ‘Ocean acidification risk assessment for Alaska’s fishery sector’, (2014), www.sciencedirect.com, (accessed 3 May 2016)
2
Vilanova Mateus, Balestieri José, ”Hydropower recovery in water supply systems: Models and case study”
(2014), sciencedirect.com
3
Brazil’s INDC, COP21, 2015
http://www4.unfccc.int/submissions/INDC/Published%20Documents/Brazil/1/BRAZIL%20iNDC%20english%20 FINAL.pdf
4
BEN, BALANCO ENERGÉTICO NACIONAL 2015, Governo Federal BRASIL
5
GREENPEACE ”Damning the Amazon”, 2016
6
Shafei. et al. “Novel approach for hydrokinetic turbine applications” (2014), sciencedirect.com
overcome the additional resistance provided by the turbine.
7Different types of turbines exist, the placement also differs. This report examines first a simpler version where one single turbine covers the whole cross sectional area. Secondly a channel is examined where the turbines are placed as a fence and four different typical diameters are examined. The turbines have just like the channels a rectangular cross sectional area.
The Swedish Centre for renewable Electric Energy Conversion at Uppsala University made a study on the effects on the upstream hydropower station by placing energy converters in the river. The authors state that the method of placing in-stream energy converters in rivers is a research area that is still pretty unexplored. Traditionally what is taken advantage of is the potential power that creates the driving force for the river flow, it is this energy that is converted by hydropower stations. Damns are used to increase the potential power. Due to the river flow after the power station there is potential to extract energy from the kinetic energy through in-stream energy converters, in this case turbines. The aim of this report is to examine the maximum amount of energy that can be extracted depending on the placement of the turbine(s). Fluid mechanical theory, supported by experiment, shows that as resistance to flow increases in an open channel, the fluid level at the entry of the channel tends to increase. In the operation of a hydroelectric power station, this may cause an adverse effect on the power generation capacity of the station, this is shown in the case study by Lalander. E and Leijon. M. They found that in the specific case study turbines should not be implemented since the energy extracted form the turbines was less than the energy lost from the hydropower station due to the presence of the turbines.
8All former studies examined agree that there is more potential for energy extraction in the downstream of a power station than is being taken advantage of at the moment. Access to energy is absolutely vital in order for countries and societies to develop, for a sustainable development clean energy is the key issue.
This method could therefor be of utter importance. Yüksel I states that the capacity of small hydropower plants in combination with other water infrastructure facilities will enable the dissemination of such technology in developing countries as improvements in their water supply systems are made. This technology would hence be useful in countries with a big share of hydropower, especially in rapidly developing countries with a fast growth in energy demand, such as Brazil and China.
9Liu Yue and J. Packey Daniel examined two different options of where to place the turbines, the first is to place the turbine behind the existing conventional hydropower plant and the second to place the turbine at sites close to the powerhouse. Here they mention that the concept of hydrokinetic energy has been
investigated since 1973 and the potential of extracting clean energy may be high but that not enough research has been done yet and that it is not yet economically feasible. The report stages however that if it could be proven cost beneficial this would be very helpful for the implication of the technology. The majority of the studies on the issue bring up the economical aspect, it is relatively difficult for new energy technologies to enter the market and hence difficult to get investments for research. However, as countries
7
Powell, D.M. (2014) Flow resistance in gravel-bed rivers: Progress in research. Earth-Science Reviews, vol.136, pp.301-228.
8
Lalander, E., Leijon, M. (2011) “In-stream energyconverters in a river – effects on upstream hydropower station”.
Sciencedirect.com
9
Liu Yue, J.Packey, Daniel ”Combined-cycle hydropower systems e The potential of applying hydrokinetic turbines in the tailwaters of existing conventional hydropower stations”. (2014) sciencedirect.com
worldwide agree to cut GHG emissions one can expect that more attention will be drawn towards this kind of technology.
The turbine power can according to the Swedish study be expressed as a variable dependant on the number of turbines and location along the channel and for numerical simulation using the power
coefficient Cp. Tidal power has been studied earlier quite a lot, extracting power from rivers differs from tidal in that sense that tidal is more predictable with their periods. River flow may vary a lot. The aim here is to examine an ideal flow in channels of different geometries.
As can be seen above studies have been done on the issue but a single expression for the relation between the placement of the turbine and the energy extracted yet to be made.
1.1 THEORETICAL BACKGROUND
Liquid flow can be expressed by the Bernoulli equation or by the Saint Venant. The Bernoulli will be used in this study. The Bernoulli equation [1] is based on the energy principle namely the conservation of energy in an isolated system.
𝐏
𝟏+ 𝛂
𝟏𝟐𝛒𝐔
𝟏𝟐+ 𝛒𝐠𝐡
𝟏= 𝐏
𝟐+ 𝛂
𝟏𝟐𝛒𝐔
𝟐𝟐+ 𝛒𝐠𝐡
𝟐[1]
Equation [1] denotes the Bernoulli equation without the friction loss. The first terms on both sides are the pressure energy, the second the kinetic energy per unit volume and the third the potential energy per unit volume.
When devices are placed along the channel bed, this will naturally affect the flow, this must therefor be taken into account in the equation. Without devices there will also be a friction lost. Maria Kartezhnikova and Thomas M. Ravens, choose to represent the friction slope by the Manning (bottom) roughness
coefficient. The authors state that as the number of devices along the river increases the flow rate of the channel decreases and there will be a peak of energy possible to extract followed by a decline. They note that the turbines will have their own manning coefficient.
10According to Ali R. Vatankhah the manning roughness coefficient will also vary due to change in water depth. Note that this has nothing to do with the placement of the devices. Variation of the manning coefficient due to water depth in an open circular channels is given by Camps curve;
n=n
!(−0.8627X
!+ 0.4281X
!+ 0.7626X
!− 1.02X
!+ 0.8057X + 1)
11[2]
where;
n= the variable Manning roughness coefficient (variable with the height) 𝑛
!= Manning roughness coefficient under the full flow condition
X=1-Z Z=y/D
10
Maria Kartezhnikova, Thomas M. Ravens, (2013) ”Hydraulic impacts of hydrokinetic devices”.
Sciencedirect.com
11
Ali R. Vatankhah, (2014) ”Analytical solution of gradually varied flow equation in circular channels using
variable Manning coefficient” Sciencedirect.com
y=flow depth D=pipe diameter
The problem is that equation [2] is designed for a circular pipe whereas the channel examined here is rectangular. The following simplifications are made; the pipe diameter equals our channel width and the flow depth y will be the same as the h in the rectangular channel.
2 METHODS
Fluid flow in a river channel is considered as gradually varied open channel flow and is described by the Bernoulli equation. Two test cases of the Bernoulli equation will be examined. In the first one the friction losses will solely be expressed by the manning roughness coefficient n and the geometry is a finite
uniform channel. See equation [3] and figure [1]. Excel will be used for this geometry in order to find the optimal placement of the turbine.
h
!!!+
!∗!!!! !∗
!!!!!!
= h
!+
!∗!!!! !∗
!!!!
− ∆X1 ∗ S
![3]
The first terms on both sides in equation [3] are the water depths ( [3] is based on [1]. The pressure P divided by the gravity g and the density ρ gives the height h), the second terms are the volumetric flow Q divided by the gravity g and the channel width W. The fourth term on the right side of the equality is the distance between the beginning of the channel and the placement of the turbine, ∆X1 multiplies by the friction slope S
!. This is given by;
S
!=
!!∗!!!!!/!∗!!!
[4]
Where Q is the volumetric flow, n the manning roughness coefficient, R the hydraulic radius and A the cross-sectional area.
Figure 1. Finite uniform channel seen from above. The blue cylinder illustrates the turbine.
In the second test case a turbine drag coefficient will be added to express the friction loss due to the presence of the turbine(s). The optimization program LINGO will be used for this geometry.
h
!!!+
!!"!!!!= h
!+
!"!!!− (∆x
!∗ S
!+½ C
DU
i2W
i/g) [5]
The first terms on both sides in equation [5] are the water depths and the second are the the velocities divided by the gravity. The third term on the right side of the equality is the distance between section i in the channel and section i+1 multiplied by the friction slope. The other part represents the friction caused by the turbine which is given by the turbine drag coefficient C
!multiplied by the velocity U and a binary value W. This is divided by the gravity g. The binary value depends on whether there is a turbine fence or not in the section i.
Figure 2. Divergent finite channel seen from above. The black lines illustrate the turbine fences.
2.3 TEST CASE 1: Simple geometry
Equation 2 expresses the channel with the friction slope without the turbine. The hydraulic radius, R is defined as; R =
!!= (w ∗ h)/(2h + w), for a wide rectangular channel where w>>h, R=h this
simplification will be used in [3]. The turbine channel will be discretized for four sections and the turbine will be placed between section two and three (see figure 1). The idea is that ℎ
!is known. The manning coefficient for the turbine𝑛
!will be defined as 𝑛
!= (
!!!). [ 6]
Where ℎ
!is the water depth with the turbine fence placed at and h the water depth without the turbine fence at the same cross-section i.
ℎ
!will be defined as
!!!!! !.
12The varying Manning coefficient [2] will be applied for this test case.
Replacing the velocity U by Q/h*w and adding using the assumption that R=h ; h
!!!+
!∗!!!! !∗
!!!!!!
= h
!+
!∗!!!! !∗
!!!!
− ∆X1 ∗
!!∗ (
!!∗!!!!^(!"!)∗!^!
+
!!∗!!!!!!^(!"!)∗!^!
) [7]
Looking at four sections in this geometry we get the three equations:
h
!− h
!+
!∗!!!! !∗ (
!!!!
−
!!!!
) − ∆𝑋
!∗
!!∗!!!"^!
∗ (
!!!^(!"!)
+
!!!^(!"!)
) = 0 [8]
h
!− h
!+
!∗!!!! !∗ (
!!!!
−
!!!!
) − ∆𝑋
!∗
!!"^!!∗!!∗ (
!!!^(!"!)
+
!!!^(!"!)
) = 0 [9]
12
Maria Kartezhnikova, Thomas M. Ravens, (2013) ”Hydraulic impacts of hydrokinetic devices”.
Sciencedirect.com
h
!− h
!+
!!!∗!!!
∗ (
!!!!
−
!!!!
) − ∆𝑋
!∗
!!∗!!!"^!
∗ (
!!!^(!"!)
+
!!!^(!"!)
) = 0 [10]
Introducing the following constants in the case where h4 is known;
C1= Q^2/2*g*w^2 [11]
C2= Q^2*n^2/2*w^2 [12]
C3= h4+ (Q^2/(2gw^2)*1/h
!^2) [13]
C4=
!!!!!/!
[14]
[7]-[9] become;
h
!− h
!+ C1 ∗ (
!!!!
−
!!!!
) − ∆X1 ∗ C2 ∗ (
!!!^(!"!)
+
!!!^(!"!)
) = 0 [15]
h
!− h
!+ C1 ∗ (
!!!!
−
!!!!
) − ∆X2 ∗ C2 ∗ (
!!!^(!"!)
+
!!!^(!"!)
) = 0 [16]
h
!− C3 + C1 ∗ (
!!!!
) − ∆X3 ∗ C2 ∗ (
!!!^(!"!)
+C4) = 0 [17]
To improve the expression camps curve [1] will be applied for the variable of the manning equation due to the height. When a variable manning coefficient is introduced the constant C2 and C4 will change whereas C1 and C3 remain the same.
C1= Q^2/2*g*w^2 [18]
C2= Q^2/2*w^2 [19]
C3= h4+ ((Q^2/2gw)*1/h4^2) [20]
C4 =(−0.8627 1 − hw!
!
+ 0.4281 1 − hw!
!
+ 0.7626 1 − hw!
!
− 1.02 1 − hw!
!
+ 0.8057 1 − hw + 1)! ! h!
!"
!
[21]
At this point it is time to add the turbine to the equations. The turbine will have their own Manning coefficient.
The volumetric flow Q [m^3/s] is assumed a constant due to the volumetric flow balance. When the turbine is placed in the channel, the area in which the water can flow will be less. Q is defined as wh*U. Hence, if wh (area) becomes smaller this means that the velocity through the turbine must be higher in order to give the same volumetric flow.
Next step is to introduc α, this was earlier set to 1. Ifε is the porosity then:
U= Q/whε. In the Bernoulli equation we look at U^2, hence in the following equations α=1/ε^2
Note: A simplification will be made here; α will be applied for the total height at the place of the turbine.
Correctly it should only be applied for the height of the turbine. The velocity through the turbine and over
the turbine will be different; this will not be taken into account.
The manning coefficient for the turbine is given by [6], here h is the height without the turbine present and will be and average of
!!"!!! !"where h
!"and h
!"are the heights of the water at the two points 2,3 before the turbine is placed there.
0= h
!− h
!+ C
! !!!!
−
!!!!!
− ∆X
!∗ C
! !!!(!!.!"#$ !!(!!)!
!
!!.!"#$ !!(!!)!
!
!!.!"#" !!(!!)!
!
!!.!" !!(!!)!
!
!!.!"#$ !!(!!)! !!)!
!!
!"
!
+
+
!!!(!!.!"#$ !!!!!!! !!!.!"#$ !!!!!!! !!!.!"#" !!!!!!! !!!.!" !!!!!!! !!!.!"#$ !!!!!!! !!)!! !!!!!
!!"!!!"
!
!!
!"
!
[22]
Note that the last part of [21],
!!!!!!!"!!!"
represent the manning coefficient for the turbine 𝑛
!.
[23]
0=h
!− h
!+ C
! !!!!!
−
!!!!!
− ∆X
!∗ C
! !!!(!!.!"#$ !!!!!!! !!!.!"#$ !!!!!!! !!!.!"#" !!!!!!! !!!.!" !!!!!!! !!!.!"#$ !!!!!!! !!)!! !!!!!
!!"!!!"
!
!!
!"
!
+ +
!!!(!!.!"#$ !!!!!!! !!!.!"#$ !!!!!!! !!!.!"#" !!!!!!! !!!.!" !!!!!!! !!!.!"#$ !!!!!!! !!)!! !!!!!
!!"!!!"
!
!!
!"
!
[24]
0=h
!− C
!+ C
! !!!!
− ∆X
!∗ C
! !!!(!!.!"#$ !!!!!!! !!!.!"#$ !!!!!!! !!!.!"#" !!!!!!! !!!.!" !!!!!!! !!!.!"#$ !!!!!!! !!)!! !!!!!
!!"!!!"
!
!!
!"
!
+
n
!!C
!The power extracted by the turbine is given by Betz’ law
13; P =
!!
∗ C
!∗ ρ ∗ A
!∗ U i
![25]
To assess the model above the following values will be used for this case:
L= 1000 m B=10 m
∆X
2= 2 m h
4= 0.75 (m) n= 0.03 ht = 2 m Q = 20 m
3/s hmax= 12 m 𝜀 = 0,5
ρ = 1000 (kg/m
3) D = 0,5 (m) At = 0,1963 (m
2) Cp= 0.5
13
F. Behrouzi, A. Maimum, M. Nakisa (2014) ”Review of Various Designs and Development in Hydropower Turbines” http://waset.org/publications/9997410/review-‐of-‐various-‐designs-‐and-‐development-‐in-‐
hydropower-‐turbines
2.4 TEST CASE 2: Divergent channel
The idea for the second approach is to discretize for i sections instead of only 4. The channel will be bigger and hence the flow will be much larger compared to approach 1. The number of turbines will increase from 1 to 100 and will be placed as fences cross-sectional along the channel. The energy extracted is given by equation [25] below. This is the function that is to be maximized.
P =
!∈!!!∗ C
!∗ ρ ∗ A
!∗ U i
!∗ w(i) ∗ Nt
![26]
The eauqlities and constraints for [26] are given by equations [27]- [38]. The interpretation of the Bernoulli equation for this case is given by [27].
h
!!!+
!!"!!!!= h
!+
!"!!!− ∆x
!∗ S-½ C
DU
i2W
i*Nt
!/g , ∀i∈I [27]
Constraint [28] ensures that the fluid level at the channel entry does not exceed that one given to be max in order to not affect the potential power from the hydropower station
h
!< h
!"#$[28]
Equality [28] ensures mass balance, hence, Q is a constant.
U
1h
1b
1= Q , ∀i∈I [29]
The average friction slope between two sections is given by the equation below.
S = 0.5 ∗ S
!+ S
!!!, ∀i∈I [30]
The friction slope at section I is given by [31].
S
!=
!!∗!!!!!/!∗!!!
, ∀i∈I [31]
The hydraulic radius is given by the cross-sectional area A over the wetted perimeter P.
R
i=
i !!∗!!!∗!!∗!! ,
∀i∈I [32]
The cross-sectional area is given by the width b multiplied by the water height h.
A
i= b
i*h
i, ∀i∈I [33]
Constraint [34] states that the number of turbines placed do not exceed the number of turbines (M) available.
W
!!∈!
≤ M [34]
To allow a turbine in a section the waterdepth must exceed that of the turbine, constraint [35] ensures That the depth is sufficiente.
h
i< C
t*D*W
i ,∀i∈I [35]
Both the water height and the velocity must be positive values, this is expressed as:
h
i, U
i> 0 , W
i∈ (0,1) , ∀i∈I [36]
The divergence is 10 degrees meaning the width will increase downstream of the channel by:
b
i= b
i+ sin(10)*X
i, ∀i∈I [37]
Constraint [37] ensures that the Froude number remains equal or lower than 1 for all cross sections i.
U
i/(g*h
i) < 1, ∀i∈I [38]
To assess the model, the following values will be used for this case:
L = 1000 m g = 9.81 m/s^2 Cp = 0.5 Ro = 1000 Cd = 0.6 At = 1m^2 B1 = 200 Ct = 1.5
Angle (in degrees) = 10 n = 0.025
M = 100
The values for the diameter of the turbine D and the volumetric flow Q will vary. The following volumetric flows will be examined; 4000, 6000, 8000 and 10000m^3/s and for each of the following turbine diameters; 2, 3.2, 4.1 and 5 m.
3 RESULTS
In this section the results from the two different geometries will be presented. One should have in mind that it is difficult to compare the two different results since they are based on different geometries and the Bernoulli equation for the two cases are interpreted differently.
3.1 TEST CASE 1
Table one shows how the three different heights vary with the placement of the turbine. ∆𝑋
!marks the
distance from the initial of the channel (i.e. right under the hydroelectric power station) to the placement
of the turbine. The figures below (3,4,5) show the variation of the water depth along the channel with the
different placements of the turbine. Figure 3 illustrates the channel without the turbine and with the
variable manning coefficient.
Figure 3. Open channel flow with variable manning coefficient, no turbine placed.
Graph 4 and 5 show the flow with a constant respectively variable manning coefficient and with the turbine placed in the channel. It is clear that as a variable manning coefficient is applied and the turbine placed the complexity of the flow increases.
Figure 4. Open channel flow with constant manning coefficient and turbine placed.
The legend shows different Δ𝑋
!Figure 5. Open channel flow with variable manning coefficient and turbine placed.
The legend shows different 𝑋
!Table 1 shows that the power extracted increases with ∆𝑋
!. As to not affect the potential power of the hydroelectric power station ℎ
!"#may not exceed 12 m, hence the boxes marked in red below are
considered as invalid and the maximum power that can be extracted is 0.8824 W. Even though being the maximum value it is very low. The numerical values are of secondary interest here, examining equation 25 it is proportional to the value of the cross-sectional area and the velocity. In this case both these values are very small, having an area under 0.2 m and the velocities shown in Table.1 below.
m m m m m/s W
Table 1. Results from excel show how the heights (m) varies depending on the distance
between the beginning of the channel and the turbine.
3.2 TEST CASE 2
The table below (Table 2) shows the results from the lingo code. Turbine fences are placed in the sections illustrated below. For full information on number of turbines, friction slope etc. see code in the appendix.
Table 2 below shows the flow velocity and water depth/height in the different sections. The numbers marked in yellow are the max velocities for each flow and diameter whereas the green numbers show the section and velocity where the turbine fences are placed. The boxes marked in red show the heights (water depths) that exceeds 12 m, this was not included in the code as to not limit the results.
Q=4000
Q=6000
Q=8000
Q=10 000
[
[i] Velocity
m/s Height Velocity
m/s Height
m Velocity
m/s Height m Velocity
m/s Height
m
D=2 1 1.50 13.3 1.73 17.38 1.90 21.02 2.08 24.00
2 1.68 11.42 1.93 14.88 2.13 18.0 2.35 20.36
3 2.08 8.84 2.41 11.45 2.67 13.81 3.01 15.26
4 4.35 4.07 5.56 4.77 6.85 5.17 2.89 15.32
5 4.23 4.023 5.39 4.75 6.52 5.23 2.78 15.33
6 4.13 3.98 5.23 4.72 6.27 5.24 2.68 5.56
7 4.04 3.93 5.09 4.68 6.07 5.23 7.133 5.61
8 4.95 3.88 4.97 4.63 5.89 5.21 6.84 5.62
9 3.90 3.83 4.86 4.59 5.74 5.17 6.61 5.60
10 3.81 3.77 4.76 4.54 5.60 5.13 6.42 5.57
11 3.75 3.72 4.67 4.48 5.48 5.09 6.25 5.54
12 3.70 3.66 4.59 4.42 5.38 5.03 6.11 5.50
13 3.66 3.59 4.53 4.36 5.29 4.97 5.99 5.43
14 3.63 3.53 4.47 4.29 5.21 4.91 5.88 5.37
15 3.60 3.46 4.43 4.22 5.14 4.84 5.79 5.30
16 3.58 3.39 4.40 4.14 5.08 4.76 5.71 5.22
17 3.57 3.31 4.37 4.06 5.04 4.68 5.65 5.14
18 3.57 3.22 4.35 3.96 5.01 4.59 5.60 5.04
19 3.59 3.13 4.36 3.86 5.00 4.49 5.58 4.92
20 3.63 3.92 4.39 3.75 5.02 4.37 5.57 4.78
21 3.71 2.89 4.45 3.61 5.07 4.23 5.60 4.60
22 3.85 2.72 4.58 3.42 5.18 4.04 5.70 4.16
23 4.41 2.32 5.12 3.00 5.68 3.60 6.15 4.13
D=3.2 1 1.49 13.40 1.71 17.50 1.89 21.17 2.04 24.54
2 1.52 12.23 1.75 15.96 1.93 19.30 2.08 24.54
3 1.60 10.93 1.84 14.25 2.03 17.22 2.19 22.36
4 1.74 9.42 2.01 12.25 2.22 14.77 2.40 19.94
5 2.10 7.38 2.44 9.53 2.71 11.46 2.93 17.10
6 3.87 3.80 4.85 4.54 5.74 5.12 6.63 13.24
7 3.76 3.71 4.69 4.46 5.52 5.05 6.31 5.54
8 3.67 3.62 4.55 4.37 5.34 4.98 6.07 5.52
9 3.60 3.52 4.45 4.27 5.19 4.88 5.87 5.47
10 3.55 3.42 4.37 4.16 5.07 4.77 5.72 5.39
11 3.52 3.30 4.31 4.04 4.99 4.65 5.60 5.30
12 3.52 3.17 4.29 3.90 4.94 4.51 5.52 5.18
13 3.56 3.01 4.31 3.73 4.94 4.34 5.49 5.05
14 3.67 2.81 4.41 3.51 5.01 4.11 5.54 4.88
15 4.25 2.34 4.97 3.00 5.53 3.59 6.00 4.65
D=4.1 1 1.56 12.79 1.81 16.59 1.98 20.21 infeasable 2 1.54 11.81 1.79 15.29 1.96 18.65
3 1.55 10.79 1.81 13.91 1.97 17.00
4 1.61 9.66 1.88 12.39 2.04 15.20
5 1.74 8.29 2.06 10.51 2.22 13.00
6 2.06 6.54 2.52 8.04 2.66 10.13
7 3.59 3.53 2.36 8.06 5.21 4.86
8 3.52 3.40 2.22 8.07 5.04 4.73
9 3.48 3.25 4.26 4.00 4.93 4.58
10 3.48 3.07 4.24 3.79 4.88 4.39
11 3.57 2.85 4.31 3.54 4.92 4.14
12 4.14 2.34 4.86 3.00 5.42 3.58
D=5 1 4.76 4.20 1.87 16.09 2.06 19.38 2.23 22.50 2 4.38 4.12 1.80 15.02 2.00 18.08 2.15 21.00 3 4.08 4.02 1.77 14.0 1.96 16.74 2.12 19.40 4 3.86 3.91 1.79 12.67 1.99 15.20 2.14 17.61 5 3.68 3.79 1.85 11.28 2.07 13.48 2.23 15.61 6 3.55 3.65 2.01 9.66 2.26 11.46 2.44 13.26 7 3.45 3.51 2.40 7.56 2.75 8.81 2.98 10.16 8 3.40 3.35 2.25 7.57 2.58 8.82 2.79 10.19 9 3.39 3.15 4.14 3.88 2.42 8.83 2.62 10.21 10 3.48 2.91 4.20 3.61 4.80 4.21 5.33 4.75 11 4.06 2.36 4.77 3.01 5.34 3.59 5.81 4.12
Table 3. Velocity and water depth for each volumetric flow and diameter in each section i.
As can be seen in the table above, the turbine fences never interact with the maximum velocity. The turbine fences are throughout the 16 different cases (except the infeasible ones) placed in the beginning of the channel i.e. before the friction losses start to have severe impact on the kinetic energy. A majority of the placements result in a water depth that exceeds 12 meters, see numbers marked in red next to the ones marked in green in table 3. A turbine fence is placed in a section that may seem a bit random in 5 cases.
See the numbers marked in red in table 1 and the numbers marked in green in table 4 below.
Table 4. Power extracted for each volumetric flow and diameter.
Table 4 shows the power (W) that can be extracted for each case. The diameter 3.2 gives the highest power extracted for all flows. A maximum amount of power can be extracted when the flow equals 10000 m^3/s and the turbine diameter is 3.2 m, in this case the turbine fences will be placed at sections 1- 5 and a total of 97 turbines (see NT in code) out of 100 available will be placed. Since the objective function [25] depends on kinetic energy it is proportional to the velocity, as the volumetric flow increases so does the velocity, see table 3, the energy extracted increases with Q for all diameters. Due to the channel being divergent and the volumetric flow constant, as the width b increases the height and the velocity will decrease. The height and velocity are also affected by the friction slope and when turbines are placed the friction loss increases, see equation [3], since 𝑊
!is a binary value the equation will look different without turbines. The friction slope [30] depends on the volumetric flow Q, the manning coefficient n, the hydraulic radius R and the cross sectional area A. Since Q and n are both constants the friction slope will differ with R and Ai. Both R and Ai are proportional to the height hi, this will naturally decrease along the channel, given this the friction slope will increase along the channel. Following the law of conservation of energy, as more energy is lost to friction there will be less kinetic energy available to extract for the turbines. That is why the green marked boxes in table 3 are marked interesting results.
These five turbine fences are all placed quit far downstream the channel and do not directly follow the other turbine fences. This shows the complexity of the problem.
Examining the number of turbines, NT in the code the results show that a maximum number of turbines is placed in all cases. Taking a look at [25] this is obvious since the power extracted, P is proportional to NTi.
4 DISCUSSION
This is a complex problem of many dimensions. First of all, a model will always be a model, this is an idealised channel and the results will therefor naturally be quite far from reality. Secondly the choice of mathematical equation may be discussed, is the Saint Venant equation be more accurate? As shown in the report the friction loss in the Bernoulli equation may be expressed in many different ways. So far no superior truth exists. Two different approaches were examined. The two cases gave different results, the first one showed that the power extracted by the turbine increases as the turbine is placed further from the beginning of the channel whereas the second approach showed that the closer the turbine is placed the more energy can be extracted. The power extracted depends on the velocity since it is the kinetic energy that is taken advantage of. The volumetric flow Q is constant, this meaning that as the height decreases the velocity will increase, this is the explanation to why the turbine should be placed as far away down the channel as possible in the first case. The friction loss seems not to make that much of a difference in the first case. In the second approach Q is still constant but the width of the channel increases, this in combination with the friction loss results in a more complex pattern of velocity change along the channel (see table 3). Note that the numeric values are of secondary interest, especially the first test case. This since only ideal channels are examined and the values for cross-sectional area and velocities are very small. The main aim of this report is to see how the values differs with each other, not the actual value.
The first approach examines a very small channel (10 m broad) with a small flow and only one turbine. It
is discretized in four sections and the model is simplified such that ℎ
!is set. The friction loss is more
complex in the first case as it varies along the channel bed, however I do not consider the results to be in
sync with reality. This since ℎ
!is set and some of the constants may be questioned such as the manning
coefficient, the volumetric flow etc. However I do believe that approach one was necessary in order to
understand the equation and the concept of friction loss.
Approach one expresses the friction loss by two different manning coefficients, one that varies with the flow due to the water depth and one for the turbine itself. These equations became too complicated for me to solve in lingo or matlab. Instead ℎ
!was set and different distances for ∆𝑋
!were examined in excel. It is possible to continue with these equations to improve the accuracy, that would be adding a different manning coefficient for the turbine and for the free space between the turbine and the surface.
The latter way of expressing the friction loss was a combination of a constant manning coefficient for the free flow and a drag coefficient Cd for the turbines. This model proved easier to solve in Lingo. This is a simplified way but is widely used. To match reality one would have to examine a real channel and include drag from stones, mud, trees etc. The program Sisbahia could be used to model this, a modeling program for mainly coats areas and rivers. This is not considered in this study.
The values chosen for the constants are the ones most widely used in the literature studied. These may be questioned, how valid is the value of the manning coefficient? How much does the coefficient vary?
Would it be more accurate to have a variable one along the channel? Probably. As stated by Ali R.
Vatankhah the manning coefficient will (without turbines placed in the channel) vary due to change in water depth. This was included in the first case but not in the second.
The maximum height was set to 12, this is an approximation done by Prof. Raad Qassim who has a broad experience in hydropower. This was used in the first approach but not in the second. The reason why ℎ
!"#was not set in the second approach was in order to not limit the results since no study on the value of ℎ
!"#was made. Hence, the next step in the model would be to include the cost benefit from the hydropower station, how much does an increase in ℎ
!affect the potential power of the hydropower station? When does it become counterproductive? Given an hydroelectric power station with installed capacity of 1000 MW (the installed capacity in Brazil varies from 250 MW Balbina dam to 11 000 MW Belo monte) given a power factor of 0.5 this will provide 500 MW on an average.
14The best result from the geometry considered generates 0.34. Since this is an ideal channel reality will most likely generate less. It is not a lot in comparison but in a country like Brazil where 70% of the electricity is generated by hydropower and both population and demand is growing it may very well be worth installing turbine fences in the downstream. The next step after the energy cost benefit analysis would be to do an economical cost benefit analysis. What are the cost of the turbines and the installation etc.
It has ben proven that this is an optimization problem since the results do not follow a straight line. It is clear that the turbine fences should be placed close to the beginning of the channel where the velocity is at its´ highest, even though the channel is at its’ narrowest here and hence does not fit as many turbines as further down. However, in five cases a turbine fence was placed at what seems a random place, not following the other turbine fences. It is somewhat of a mystery but ought to be due to the change of friction loss. This shows the complexity of the problem but also the need of a model. Another approach may be to set the Bernoulli equation [26] to an inequality instead of an equation, the physical justification being that there are turbine wake losses in addition to turbine drag.
14
