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19 ПРИЛИВНАЯ ЭНЕРГЕТИКА И ЭНЕРГЕТИКА МОРСКИХ ТЕЧЕНИЙ
TIDE ENERGY AND SEA TIDE ENERGY
Статья поступила в редакцию 25.12.10. Ред. рег. № 917 The article has entered in publishing office 25.12.10. Ed. reg. No. 917
УДК 621.224
ВЫСОКОЭФФЕКТИВНЫЙ СПОСОБ ИЗВЛЕЧЕНИЯ ЭНЕРГИИ ИЗ БЕЗНАПОРНОГО ПОТОКА ТЕКУЩЕЙ ЖИДКОСТИ НА ОСНОВЕ СПЕЦИФИЧЕСКОГО ГИДРОДИНАМИЧЕСКОГО ЭФФЕКТА
Г. В. Трещалов
Инженерно-исследовательская группа по разработке альтернативных источников энергии «ЭРГ» Ташкент, Узбекистан, 100098, Кара-Камыш-2/1-3-43 Тел./факс: (99871) 2790590, E-mail: t-ger@mail.ru
Заключение совета рецензентов: 16.01.11 Заключение совета экспертов: 20.01.11 Принято к публикации: 25.01.11
Настоящая статья относится к разработкам в области альтернативной энергетики и анализирует материал, опубликованный в номере № 3 за 2005 г. журнала «Альтернативная энергетика и экология». А именно статью «Бесплотинные ГЭС нового поколения на основе гидроэнергоблока». [1]
В настоящей статье мы попытались дать объяснение, специфического гидродинамического эффекта, возникающего при ускорении потока жидкости и выяснить условия его возникновения, руководствуясь только законом сохранения энергии и балансом энергии в живых сечениях потока.
Также здесь приведен пример математического расчёта энергетических характеристик высокоэффективных свободно-поточных гидротурбин особой конструкции, использующих этот гидродинамический эффект.
Ключевые слова: энергия, мощность, турбина, свободно-поточный, поток, спокойный, бурный, глубина, критическая, гидродинамический, эффект, обратная связь, гидравлика, гидравлический прыжок, эжекция.
A HIGHLY EFFICIENT METHOD FOR DERIVING ENERGY FROM A FREE-FLOW LIQUID ON THE BASIS OF THE SPECIFIC HYDRODYNAMIC EFFECT
G.V. Treshchalov
Engineering and Research Group ("ERG") Kara-Kamish-2/1-3-43, Tashkent, 100098, Uzbekistan Tel./fax: (99871) 2790590; e-mail: t-ger@mail.ru
Referred: 16.01.11 Expertise: 20.01.11 Accepted: 25.01.11
Traditional free flow turbines slow down water stream using its kinetic energy. This article describes a method for deriving energy from a free water stream using a unique hydraulic turbine which unlike traditional hydraulic turbines speeds up the water stream using potential energy thanks to a specific hydraulic effect.
The article also describes a method of mathematic calculations of output capacity of this kind of turbines. The design of a turbine which also uses this special hydraulic effect is described in the article entitled "A new generation of damless hydroelectric stations based on hydro-energy units" published in the third issue of "Alternative Energy and Ecology" in 2005.
Keywords: energy, hydrodynamics, turbine, free flow, calm, turbulent, critical, sub critical, super critical, hydraulic jump, dept, effect, feedback, "Froude number", efficiency, ejection effect.
1
Organization(s): PhD Candidate, Energy Globe Awards - 2008, 2010 renewable energy scientific prizes.
Education: Tashkent Technical University, Faculty of Hydropower (1981-1986). Experience: Ust-Ilimsk HPP, Engineer (1986-1989). ORGRES, Chief engineer (1990-1993), Tashkent Telephone Company, main specialist (1998-2000). BBC, Senior specialist (2000-2006). "Engineering and Research Group" ("ERG") - head (2006 - now). Main range of scientific interests: renewable energy, hydropower, wind energy units. Publications: patent application WIPO W0/2007/131246.
German Treshchalov
Introduction
A group of engineers has constructed a hydraulic turbine to receive energy from a free flow of water (a free flow hydraulic unit). However, when its capacity was measured it was established that it generated more energy than it was designed for. It is well-known that a flow of water has kinetic energy that can be extracted (which is what free-flow turbines do [2]). However, it is impossible to extract all of its kinetic energy. In order to do this, the flow should be stopped completely and then it would cease to be a flow. That is why the velocity of water flow at the exit from a working unit of turbine is slower than its flow at the entrance - it is precisely this difference that defines the efficiency of any facility. Considering the fact that the kinetic energy is known as being proportional to the square of the speed, and that the energy decreases by four times when the speed decreases, it will be easy to calculate that, let's say, when the water flow speed at the turbine input and output is equal to 1m/sec and 0.5 m/s, respectively, we will be able to extract 75% of the kinetic energy from the flow.
Strictly speaking, the power of the free-flow turbine is calculated by a semiempirical formula (1) (this formula can also be applied to calculate the power of wind turbines)
P = KV3Sp,
(1)
where V - incoming flow speed; S - the square of the turbine's effective cross section across the flow; p -moving medium density; K - constant coefficient that depends on a turbine type and is usually equal to 0.10.35.
This formula represents the very kinetic energy of the flow per a time unit, because VSp is right the water mass that goes through the turbine at one second and the formula (1) takes on the following form, which is familiar to us:
E =
mV2 (VSp)V2
However, it should be considered that, according to the flow continuity condition, the flow's square must increase when the outward flow's speed drops. This leads to degradation in the flow evenness at the turbine's outlet and an increase in turbulence, which negatively affects the unit's efficiency. In order to decrease these factors' adverse effect in traditional turbines, expanding cones are installed at their outlets, which partly increases the efficiency.
Because empirical coefficient K of the formula (1) includes the twain from the kinetic energy formula denominator, the hydraulic and mechanical efficiency coefficient of the turbine, losses per irregularity and turbulence in the incoming flow and so on, it accepts values of less than 0.3. This coefficient is measured through an empirical way by means of natural tests of a specific turbine.
This coefficient is often called the WEUC, the watercourse energy utilization coefficient, by an analogy with the wind turbine WEUC - the wind energy utilization coefficient.
Theoretical Analysis
But let's get back to our machine. As we have already mentioned, this facility produced even a greater amount of energy than the total kinetic energy of the flow.
Where does this additional energy received from the facility come? Does the flow of water have kinetic energy only?
(Here we do not consider the internal (thermal) energy of water or the energy of the intermolecular and interatomic bonds of water as a substance.)
Let us try to answer these questions.
Let us take one ton of water (1 m3) flowing with a velocity of 1 m/s.
There is no doubt about its kinetic energy, which is:
Ek =
mV2 1000 (kg) • 1 (m2 /s2)
= 500 (Joule). (2)
However, there is also pressure by the top layers of water on the bottom ones (potential energy). If we let this cube of water spread, then we can extract it. Considering that the gravity centre of the cube is at the middle of its height, that is h = 0.5 m, it is equal to:
Ep = mgh = 1000 (kg) • 9.8 (m /s2) • 0.5 (m) = 4900 (Joule)
(3)
This means that the potential energy of this cubic meter of water is up by almost 10 times on its kinetic energy. It is easy to calculate that, at a speed of 0.5 m/s, this difference will amount to almost 40 times!
It should be noted that in the formula (3) a half of the water column height is taken as h because a separate water volume's height will decrease from the overall to zero as it will flow. For an infinite water flow with constant depth, which will be reviewed later, the incoming flow's full depth is taken as water column height.
Now let us imagine that we are extracting part of kinetic energy from a cubic meter of water, which is flowing within a current, and use it to "move aside" the cubic meter of water that follows it (downstream). That is we will speed up the downstream cubic meter of water by slowing down the upstream volume of water. As a result, a level difference arises between them and potential energy emerges in the difference between these levels, which can be extracted from the current. The following question arises: will the amount of the extracted potential energy be more, less or equal to the energy used to speed up the second cubic meter of water - or, in other words, the energy expended to increase its kinetic energy?
International Scientific Journal for Alternative Energy and Ecology № 12 (92) 2010
© Scientific Technical Centre «TATA», 2010
Let us resort to mathematics.
As an example, we will consider a machine that is shown as a diagram on Fig. 1, which makes it possible to speed up the outflowing stream of water by extracting part of the inflowing stream's energy - that is, a machine with positive feedback between the energies of the inflowing and outflowing streams. By the way, a machine that works on this very principle has been invented. It is this machine that our story started with.
The device's energy balance is as follows:
E = Ki + Ph - K2. (4)
The total output of energy from the device is equal to the potential energy of the difference between the marks plus the kinetic energy of the inflowing stream and minus the kinetic energy of the outflowing stream. After omitting all the computations, we have:
E = M
gh +
V2
1 -
H
K Hi - hj
(5)
or
E = M
gHi
i - VL
V
(V2 - V22)'
2 у
(6)
Рис. 1. Схема устройства: 1 - рабочие элементы входного потока воды; 2 - рабочие элементы выходного потока воды;
3 - рабочие элементы, обеспечивающие положительную обратную связь между входным и выходным потоками воды;
4 - отметка уровня горизонта входного потока воды;
5 - отметка уровня горизонта выходного потока воды;
6 - дно русла;
H1 - эффективная глубина входного потока воды;
Н2 - глубина выходного потока воды; V1 - скорость входного потока воды; V2 - скорость выходного потока воды; h - перепад уровней входного и выходного потоков воды
Fig. 1. Scheme of the device: 1 - working parts of the inflowing stream of water; 2 - working parts of the outflowing stream of water; 3 - working parts ensuring positive feedback between the inflowing and outflowing streams of water; 4 - mark showing the level of the inflowing stream of water; 5 - mark showing the level of the outflowing stream of water; 6 - channel bed; H1 - actual depth of the inflowing stream of water; H2 - depth of
the outflowing stream of water; V1 - velocity of the inflowing stream of water; V2 - velocity of the outflowing stream of water; h - drop between the levels of the inflowing and outflowing streams of water
The device works based on the following principle:
The working parts of the inflowing stream 1 extract part of the kinetic energy from the stream and transmit it - with the help of the positive feedback 3 - to the working parts of the outflowing stream 2, which give the outflowing stream additional acceleration.
Because the amount of water entering the device is equal to the amount of outflowing water, and the speed of the outflowing stream is higher than that of the inflowing stream, then the sectional area of the outflowing stream will be less than that of the inflowing stream.
Therefore, its depth H2 will be less than the depth of the inflowing stream Hi by the value h. As a result of this, potential energy appears between the different levels of the inflowing and outflowing streams.
where M is the weight of the water entering the device in a unit of time, which is equal to the density of water multiplied by the active area of the inflowing stream and multiplied by its velocity.
Then the most interesting aspect occurs. It can be seen that the left side of the equation, which is in brackets, will increase in a linear fashion when it depends on h or in a hyperbola when it depends on V2, whereas the right part will decrease, and in a parabola at that. Which side will gain the upper hand?
Let us plot a graph showing energy's dependence on the drop between the levels h (Fig. 2). The graph will be plotted to show the various levels of the inflowing stream's velocity V1 after designating it as a constant.
Рис. 2. Зависимость энергии от перепада уровней при различных значениях скорости входного потока Fig. 2. Energy's dependence on the difference in the levels and the input flow velocity
It is remarkable that the graph showing the energy's dependence on the drop between the levels h has an extremum. On the rising branch of the graph, the energy balance will be positive (the power factor > 1), i.e. the extracted potential energy will be mostly expended as kinetic energy on speeding up the outflowing stream, and the device will self-accelerate until it reaches the maximum.
The energy produced by the device at this point will be several times the kinetic energy of the inflowing stream - and under certain conditions, tens and even hundreds of times!
The speed of the outflowing stream will be significantly higher (2 to 3 times as higher at times) than the speed of the inflowing stream. Therefore, the kinetic energy of the outflowing stream is 4 to 9 times the kinetic energy of the inflowing stream.
Furthermore, the graphs show that not everything appears to be quite right with the inflowing speed. It also has an extremum. To see this better, let us plot a 3D diagram (Fig. 3).
However paradoxical this may seem at first glance, but the diagrams show there is an optimal speed for the inflowing stream. When it is exceeded, the device's power capacity will sharply fall. This is due to the fact that a significant amount of energy needs to be spent on speeding up a stream that is flowing fast already.
One parameter has been left unaccounted for in these diagrams, the entry depth H1 to be precise.
But because the diagrams are three-dimensional now, to plot the output energy's dependence on this parameter, too, we will show the sequence of 3-D graphs for various values of the flow's entry depth (Fig. 4).
The diagrams show that depending on the entry depth, the machine's energy output grows in a non-linear fashion, almost in quadratic dependence. Below we will examine what exactly this dependence looks like.
The question may arise: "How does the outflowing stream, which has a shallower depth, interact with the water flow around it, which has a normal constant depth?" Here we have to recall that the velocity of the outflowing stream is higher than that of the surrounding medium and this creates what is called in hydraulics "hydraulic jump" as a result of ejection effect, which equalises the discrepancy between the kinetic and potential energies of the two flows. This "jump" is in essence surf, a vortex in the flow.
Рис. 3. Зависимость энергии от разницы уровней (вверху)
и выходной скорости (внизу) Fig. 3. Energy's dependence on the difference in the levels (up) and exit speed (down)
Рис. 4. Зависимость энергии от трех параметров: разницы уровней h, входной скорости V
и входной эффективной глубины H1 (0,9; 1,2; 1,5 и 1,8 м) Fig. 4. Energy's dependence on three parameters: the difference in levels h, the entry speed V1 and the effective entry depth H1 (0.9, 1.2, 1.5 and 1.8 m)
International Scientific Journal for Alternative Energy and Ecology № 12 (92) 2010
© Scientific Technical Centre «TATA», 2010
Let us examine in detail what happens with the flow, what the depth and velocity of the outflowing stream depend on, how long the hydraulic jump is and what conditions need to be met to produce such an effect.
In all of the aforementioned computations, only the Bernoulli equation (energy conservation law) and the flow continuity equation (mass conservation law) are used.
Considering the fact that the turbine, which is located on the water flow, extracts some energy from this flow, the Bernoulli generalized equation for two cross-sections of the free voluntary flow - the first one (before incoming the unit) and the second one (at the unit's outlet), without taking into consideration losses, will take on the following form:
MgHi +
mv;2
= MgH 2 +-
MV2
- + E,
where E - is energy that the turbine takes from the flow. Therefore, the energy released at the turbine is equal to
E = MgH 1 - MgH 2 +
mv;2 mv22
(7)
Let us define H2 = k^ or k = H2/H;, where k is a dimensionless coefficient. Then
MV2 MV2 E = MgH1 - MgkH1 + MV±- - MV^. (8)
Let us express V2 in terms of V1 taking into account the flow continuity equation, namely HV = const (when the flow's width is constant in two clear sections). We will get the following:
or
H1V1 = H2V2
H V = H2V2 and V2 = V1/k .
(9)
(10)
So the ratio of the velocities of the inflowing and outflowing streams depends only on the ratio of the height (depth) of the streams (with the width being the same).
Accordingly, the formula (8) assumes the form:
E = MgH1 - MgkH1 +
mv;2 mv;2
2k2
(ii)
or
MV., 2
E = MgH. (1 -k) + —^(1 -1/k ). (12)
Let us find the extreme of the energy in relation to k. To do this, let us differentiate the formula (11) on k
E' = 0-MgH, + 0 + M- = -MgHx + MV-. (13)
By equating (13) to zero, we get
= MV1- _
MgH1 = ~kr. Hence, k = 3V? /gH1 . (14)
Conclusion: The turbine generates the maximum energy when the ratio of the levels of inflowing and outflowing streams is
k = H2 /H1 = VVTgH therefore H2 = 3Vf H2Jg .(15)
If we look up any textbook on hydraulics, for example Hydraulics by R.R Chugayev [3] or Hydraulics by I.I. Agroskin [4], we will see that formula (15) above agrees with formulas (7-49) [3] or (15-13) [4], which correspond to the so-called "critical depth" of the flow, the depth at which a flow is in the border state between being calm (sub critical) and turbulent (super critical).
But why will the depth of the outflowing stream be equal to the critical depth? The thing is that the stream's energy density is minimal at the critical depth (this is exactly why it is called critical), and, as one can observe, an increase in the velocity of the outflowing stream -with the unit rate of flow being constant and, therefore, its depth decreasing - yields a positive power factor (above 1).
That is to say, the stream releases energy, which is partially spent on the additional acceleration of the outflowing stream through the [positive] feedback ensured by the machine. This process will continue until the power factor becomes equal to 1, that is to say until the stream enters the critical state.
It is thus possible to conclude that the device described above extracts all the additional energy from a flow by bringing the outflowing stream to the critical state, that is to say to the border state where the flow turns from being sub critical to being super critical.
In accordance with [3] the specific velocity head of the flow in the critical state is equal to half of its depth and the energy density of the flow is equivalent to its gross head (the sum of its potential and velocity heads):
VI 2 g
H
V/
■ = or Hk ;
V/
E' + Hk =- Hk , k 2 g k 2
(16)
(17)
where E' is the flow's energy density at the critical depth and velocity.
The flow strength at the effective cross-section is equal to the flow's energy density multiplied by the weight of the water passing through the effective cross section per unit of time, or unit rate of flow, i.e. V\H\.
Taking into account (15) and (17), formula (7) can be re-written as follows:
E' = E'-Ek = H + V--3H2 = H + V--HV 1 k 1 2g 2 2 1 2g 2У
2
g
And the flow strength at the effective cross section per unit of time is
E = HVg ^H1+ Vg-] • (18)
Or, ultimately,
E = HV ^Hig+V- -f (HVg)2)
or
V3 3 i-
E = HXg + Hi -y- - (HVi)) g2. (19)
Knowing that the depth of the outflowing stream is equal to the critical depth, it is now possible to plot, as a function, the dependence of the generator's output power on the depth and velocity of the inflowing stream.
The critical flow will be determined as a ratio of the velocity and depth and is shown as the graph (a parabola) lying at the "gulley" of the 3-D diagram (Fig. 5). The flow's specific energy density on this graph has been accepted as zero. The total energy of the sub critical and super critical flows are calculated relative to it. In this diagram, all the sub critical flows are on the left and the all the super critical flows are on the right.
The graph and formula (19) show that the output power's dependence on the flow's entry depth is pretty complex. It grows in a quadratic fashion if it depends on the first term in the polynomial formula and in a linear fashion if it depends on the second term. It decreases by the power 5/3 if it depends on the third term
Let us now try to calculate the length of the resulting hydraulic jump. At the moment there is no precise formula for determining it. All available formulas are empirical and very often there is a considerable variance between the results obtained with their help. We will use energy balance for the calculation. Let us examine Fig. 6.
Рис. 5. Энергетическая диаграмма спокойного и бурного состояний потоков относительно критического Fig. 5. The energy diagram of the sub critical and super critical states in relation to the critical state
Fig. 6 shows that there is a lack of energy density in the hydraulic jump area between the original and the emerging (after the jump) flow regimes.
The total energy taken away from the flow in this manner is equal, in accordance with formula (18), to the difference of energy density multiplied by the unit rate of flow and multiplied by the gravitational acceleration:
E = dE 'HV g.
Therefore, it is evident that the length, in meters, of the hydraulic jump emerging after the device will be numerically equal to the unit rate of flow multiplied by the gravitational acceleration.
Рис. 6. Гидравлический прыжок на выходе аппарата: H1, H2 - глубина (потенциальный напор) входного и выходного потоков соответственно; V|2/2g, V22/2g - скоростной напор входного и выходного потоков соответственно; AE' - разница удельной энергии входного и выходного потоков; Lj - длина гидравлического прыжка Fig. 6. Hydraulic jump at the exit from the device: H1, H2 - the depth (potential head) of the inflowing and outflowing streams respectively; Vt2/2g, V22/2g - the velocity head of the inflowing and outflowing streams respectively; AE' - the difference between the energy density of the inflowing and outflowing streams; Lj - the length of the hydraulic jump
International Scientific Journal for Alternative Energy and Ecology № 12 (92) 2010
© Scientific Technical Centre «TATA», 2010
Conclusion
To summarize all of the above, it can be concluded that such a machine creates a head of water for itself and is able to extract potential energy from an evenly-flowing stream of water [5].
In addition, analyzing the diagram on Fig. 3, 4 and 5, a few important conclusions can be drawn.
First of all, one can see on these diagrams that this effect exists and it is very "capricious" - a number of conditions for the flow's efflux must be strictly met, namely the proportion between the flow's incoming speed and its depth. Only with the specific combination of these parameters, we can get to the diagram's peak and extract the maximum power from the flow. At these parameters' insignificant deviation from the optimal values, the effect either "becomes blurred", or disappears at all and it will be very difficult to find it, and it can well be mistaken for measuring errors.
Secondly, it is interesting that the effect disappears when the flow's kinetic energy increases (when its speed increases), or in other words, when it approaches critical or super critical state. Judging by the diagrams, the optimal speed is the speed of 1-1.5 m/s. However, because the water flow with this speed is considered to be of low potential and it is not often used for extracting energy by free-flow turbines, there have been carried out just few experiments under such conditions and, therefore, it has not been possible to reveal this kind of effect.
Thirdly, the flow's incoming effective depth is very critical to the appearance of this effect. It is obvious (the left diagram on Fig. 4), that if the depth is less than one meter (the overall dimensions of most of the free-flow turbines are usually less than these values) this effect is barely noticeable, commensurate with measurement errors and "spreads" in turbines' hydraulic and mechanical efficiency.
Fourthly, in order to reveal this kind of effect it is necessary to use special machines that have feedback between the incoming and outward flows.
Also, another interesting aspect should be noted. Unlike traditional free-flow turbines, a machine working on a similar principle does not slow down the outward flow by extracting its kinetic energy, but speeds it up by extracting its potential energy.
The advantages of this technology before the conventional hydropower engineering are the following:
- As low investment cost as 50-150 USD/kW -which is 800/1300 USD/kW in case of hydroelectric dams.
- The shortest commissioning terms (60-180 days after the start of construction). It takes years and decades to build a hydroelectric dam.
- No water reservoir is required (ecological effect).
- No costs of flood damages as there will be no water reservoir.
- No auxiliary mechanisms and equipment are required (such as oil and compressor units, servomotors and etc.), which increases reliability.
- Low maintenance costs.
- No need to create infrastructure around the hydropower plant (motor and rail roads, settlements of constructors and operators and etc.).
- No need to select a dam location as the unit is mobile and can be mounted in any suitable location.
- Its proximity to power consumers (no need to build power transmission lines and high-voltage transformers).
- No risk of station flooding because of the absence of a station.
- No risk of dam destruction because of the absence of one (there have been cases of this kind of catastrophes in the world).
- Power density is up by 5-100 times on conventional damless hydropower stations.
- The possibility of operating in the wide range of flow speeds, starting from 0.1 m/s, at which conventional hydropower plants cannot operate,
therefore, water power resources can be fuller utilized.
***
It should be noted in conclusion that a similar effect - the effect of power amplification during acceleration of liquids and gases - was also discovered during experiments conducted by other scientists and researchers.
An example of such experiments can be the tests conducted by L.S. Kotousov, who researched into conditions affecting jets of water flowing out of nozzles, the results of which have been published in Journal of Technical Physics [6].
References
1. Lenev N.I. Damless hydroelectric power station of new generation based on a hydrogen power unit // Alternative Energy and Ecology - ISJAEE. 2005. № 3 (23). P. 76-79.
2. Shchapov N. Turbine Equipment for Hydropower Stations. M.: Gosenergoizdat, 1961.
3. Chugayev R.R. Hydraulics. M.: Energoizdat, 1982.
4. Agroskin I.I., Dmitriyev G.T. and Pikalov F.I. Hydraulics. M.: Gosenergoizdat, 1954.
5. Treshchalov G.V. Verfahren Zur Gewinnung Von Energie Aus Der Stromung Eines Fliessenden Mediums. EP2019202 WIPO W0/2007/131246.
6. Kotousov L.S. Research into the speed of water jets coming out of nozzles with different geometry // Journal of Technical Physics (JTF). 2005. Vol. 75, Iss. 9.
