Single Arm, Centrifugal, Water Turbine for Low Head and Low Flow Application: Part 1- Theory and Design

Transcription

1 Energy an Power 2018, 8(2): DOI: /j.ep Single Arm, Centrifugal, Water Turbine for Low ea an Low Flow Application: Part 1- Theory an Design Kiplangat C. Kononen 1, Augustine B. Makokha 1,*, Sitters W. M. Cox 2 1 Department of Energy Engineering, Moi University, Eloret, Kenya 2 Department of Civil & Structural Engineering, Moi University, Eloret, Kenya Abstract Many parts of sub-saharan Africa have small streams an rivers that have a potential to prouce hyropower on small scale for rural an off-gri applications. owever, the high cost of commercially esigne micro-hyro turbines limits the exploitation of the available hyropower. Thus any efforts to improve the unerstaning of the funamental principles involve in the esign, manufacture, an operation of simple micro-hyro turbines that coul be constructe from reaily available low cost materials an construction methos woul be invaluable. This paper presents the funamental theory an the methoology for esign of a single arm centrifugal reaction water turbine that woul allow efficient extraction of hyro-power. Continuum mechanics approach was aopte where the esign of the turbine was erive from a mathematical moel that was base on the balance equations formulate in a rotating control volume. The moel showe that the theoretical power output is equivalent to the prouct of the following: ensity of water, volume flow rate, angular velocity of the turbine, raial length of the turbine arm an the absolute velocity of water as they exit the turbine. Further, the analysis showe that the theoretical efficiency of the turbine is irectly proportional to the ratio of angular acceleration to the gravitational acceleration. The experimental test results of the moel turbine are presente in our subsequent work (Part 2). Keywors Turbine, Low hea, Low flow, Centrifugal, Micro-hyro 1. Introuction yro energy contributes the highest percentage towars electricity generation in the renewable realm an is evenly istribute across the worl [1]. For the last one an half centuries hyroelectric prouction comes mostly from large projects, with ams that require a large reservoir of water to supply the turbine generator. Since most of the best locations for this type of hyro power aroun the worl an even in Africa are alreay being use, research an evelopment for hyro power is moving towars micro-hyro systems, which require less water to operate an can supply electricity to smaller communities or iniviual homes. This means that there is nee for the most efficient an reliable machines for this application. Turbines are more often than not, use for the transformation of hyro energy to mechanical energy. Most historians consier turbines as the prime haulers of civilization for the last four centuries. Even so currently there are a few companies who * Corresponing author: augustine.makokha@gmail.com (Augustine B. Makokha) Publishe online at Copyright 2018 The Author(s). Publishe by Scientific & Acaemic Publishing This work is license uner the Creative Commons Attribution International License (CC BY). manufacture turbines an supply equipment for generation of electricity from micro hyro sources. Most of the low hea micro hyroelectric units available use impulse type turbines for energy conversion [2]. From the turbines literature review, it is clear that impulse turbines are suitable for high hea an low volume flow rate applications only. When impulse turbines are use for low hea operation their energy conversion efficiency rops rastically to even less than 60% [3, 4]. Furthermore the current turbines are complex to manufacture, more so in the orinary workshops since most of them require mouling processes an eicate machines to fabricate [5]. This alone, contributes a lot to their extremely high cost. To curb this barrier the turbine esigne in this paper emphasises on low-cost fabrication which means the use of locally available material. Some of the previous turbines that have attempte to solve the structural impeiment in micro-power generation inclue the following: ero s turbine, Barker s mill, Ján Anrej Segner turbine, Pupil s turbine, Whitlaw s turbine an cross-flow turbine. The latest simple reaction water turbine is the split reaction turbine by Abhijit Date [6]. As much as subsequent years have seen moifications to increase the efficiency of the Barker s mill turbine esign, some scientist regar it as obsolete an not economically viable to be use any more [7]. owever some writers like Akbarzaeh et al [8] seem to

2 52 Kiplangat C. Kononen et al.: Single Arm, Centrifugal, Water Turbine for Low ea an Low Flow Application: Part 1- Theory an Design ispute such views. Their analysis showe that to large extent the simple reaction water turbine is misunerstoo. In their paper Akbarzaeh et al [op cit] have ientifie the main geometric an operational parameters. They conclue that the energy conversion efficiency of a simple reaction water turbine increases with the increase in the rotational spee among other factors. Despite recent researches on the simple reaction turbine there is still some gap on the effects of hea variation to the performance of the simple reaction turbine. Also the effect of using a single nozzle outlet or a single arm structure on the performance has not been well articulate. This paper aims at presenting a continuum analysis on the consequence of the hea variation as well as the variation of raial arm length on the performance of a simple reaction turbine in relation to its esign. Moreover the corollary of nozzle losses on the turbine performance has been presente here. The analysis of these key factors will help solve some of the impeiments face when using a simple reaction turbine for micro hyro generation. 2. Conceptual Design of the Turbine Figure 1 shows the graphical esign concept of the single arm centrifugal reaction water turbine. The intake stationary pipe is connecte on one sie to the penstock via a flow control valve an on the other sie is joine to the intake rotary pipe using a special mechanical rotary seal joint evice. Bearings are mounte onto the central output shaft such that the arm is at the mile an the intake rotary pipe on one en. Figure 1. Graphical representation of the single arm centrifugal reaction turbine esign 3. Continuum Analysis of the Turbine 3.1. Continuum Theory Most of the past researches on the simple reaction water turbine use a rather a-hoc analysis to give a mathematical moel of the turbine [6, 8, 10]. In this paper, the analysis presente, uses a continuum mechanics approach to erive a mathematical moel of a single arm reaction water turbine. The balance equations for continuity, moment of momentum an energy have to be evaluate to erive the relations for the generate torque, spee an efficiency of the turbine as a function of rotational spee an other input parameters such as rate of flow an supplie water hea. These equations have to be erive for the moving turbine arm, which necessitates the use of a transport theorem for a moving control volume. Figure 2 shows the configuration of a rotating one arm reaction turbine. At the pivoting centre water is flowing into the turbine an at the exit nozzle the water leaves the turbine. The control volume coincies with the circular tube at the entry an exit areas an thus rotates with the same spee as the turbine. Figure 2. Flow in a one arm reaction turbine control volume The various notations are explaine: Symbol v is use to inicate the absolute flow velocity with respect to fixe coorinate system { e y, e x, e z }. v i represents the absolute flow velocity of water flowing into the moving control volume via the inlet pipe. While v e represents the absolute flow velocity of water exiting the moving control volume via the nozzle. n i is an outwarly irecte unit normal to the inlet cross-sectional area. While n p is an outwarly irecte unit normal to the surface of the semi-circular conuit. Furthermore n e is an outwarly irecte unit normal to the outlet cross-sectional area. The cross-sectional area at the inlet of the one arm reaction turbine is represente by A i. Conversely A p represents the surface area along the semicircular conuit. While A e represents the outlet cross-sectional area at the en of the semi-circular conuit. V c enotes the volume occupie by water within semicircular conuit of the one arm reaction turbine, also referre as a moving control volume. While capital V is use to represent volume in general. x is the istance of a flui particle from the centre of rotation to a certain position in the control volume. R is the istance of nozzle from the centre of rotation.

3 Energy an Power 2018, 8(2): u e is the linear velocity at the en of the turbine arm as the turbine rotate an is given by u e = ω R. Angle β is the angle between the vectors u e an v e, usually referre as the angle of jet projection. The internal iameter of the semicircular conuit is enote by. The velocity of the control volume V c is equal to: u = ω x. The angular velocity vector an the raial vector are given by: ω = ωe y an R = Re x. Therefore; u e = ω R = ωe y Re x = ωr e y e x = u e e z The next step in the analysis is to formulate the balance equations for the moving control volume, in orer to obtain the algebraic equations governing the problem Mass Conservation Using continuum mechanics approach for a moving control volume, the continuity equation for a one arm reaction turbine becomes: t ρv + ρ(v u ). n A = 0 (1) V c Figure 3 shows the velocity iagram at the nozzle for v r, v e an u e. Figure 3. Velocity iagram of the nozzle Assuming velocities are constant over cross-section, the continuity equation becomes: ρ(v u ). u A = ρ(v e u e cosβ)a e ρv i A i = 0 (2) Using Q = v i A i the continuity equation finally results into: v i A i = (v e u e cosβ)a e = Q (3) 3.3. Moment of Momentum (Angular Momentum) The moment of momentum equation for a one arm reaction turbine is given by: t ρx v V + ρx v (v u ). n A V c = ρx g V + x t A (4) V c Again the first term is equal to zero, since the integral is constant. Furthermore the effect of gravity within the turbine balances to approximately zero an can be neglecte. Therefore: ρx v (v u ). n A = x t A (5) Solving the left han sie an applying the relevant limits, with e x n e = sinβe x e x + cosβe x e z = cosβe y it is finally foun: ρx v (v u ). n A = ρrv e (v e u e cosβ)a e cosβe y (6) Again solving the right han sie of the equation an simplifying becomes: ( x t )A = (x t )A A P = +Te y (7) The integral over A p is equal to the torque applie on the flui an is enote by +Te y where T > 0. Consequently, the torque applie by the flui on the arm is equal to Te y. Equating the left-han sie an right-han sie of the moment of momentum equation, elivers: Te y = ρrv e (v e u e cosβ)a e cosβe y (8) With u e = ωr, an using the continuity equation (3) the following expressions for the moment of momentum equation can be rewritten as: T = ρrv e Qcosβ (9) 3.4. Energy Conservation The general continuum energy equation for the turbine reas; t ρgz ρv2 V + V c ρgz ρv2 (v u ). n A = ( t v )A (10) Solving the left an right han sie of the equation an equating the two sies an further simplifying, the energy equation becomes; 1 2 ρv e 2 (v e u e cosβ)a e = p i ρv i 2 v i A i Tω (11) 3.5. Bernoulli Equation for Supply Pipe Figure 4. Velocity iagram of the penstock an the tank In this section the Bernoulli equation for the supply pipe is erive. This equation helps in etermining v i an p i.

4 54 Kiplangat C. Kononen et al.: Single Arm, Centrifugal, Water Turbine for Low ea an Low Flow Application: Part 1- Theory an Design Figure 4 shows the moelling for both the penstock an the tank. Bernoulli s equation reas: p i v i = p ρg 2g i + 1 ρv 2 i 2 = ρg (12) Substituting this result into the energy equation an making use of the continuity equation an the moment of momentum equation, the following equation is arrive at; v r = (v e cosβωr) = 2g + ω 2 R 2 cos 2 β (13) Where v r represents relative flow velocity. It shoul be note that an ieal turbine is consiere in the above erivations, which means that friction an other losses in the nozzle are not taken into consieration Calculation of Power an Efficiency without Nozzle Losses The power prouce by the turbine (i.e. power performe by flui on the tube) is calculate from the energy equation (11); P = Tω = p i 1 2 ρv i 2 v i A i 1 2 ρv e 2 (v e u e cosβ)a e (14) Using the result of Bernoulli s equation (12) an the continuity equation (3), the power equation becomes; P = ρq g 1 2 v e 2 (15) Introucing equation (13) an substituting it into the power equation elivers the following; P = ρqωrcosβ 2g + ω 2 R 2 cos 2 β + ωrcosβ (16) The power available is P = ρqg so that the hyraulic efficiency is given by; η = P = (ωrcosβ)( 2g+ω2 R 2 cos 2 β+ωrcosβ) (17) P g Introuction of hea ue to centrifugal pumping effect as iscusse by A. Akbarzaeh [8]; c = ω2 R 2 cos 2 β (18) 2g Or: η = 2. c + c 2 c (19) η = 2 K + K 2 K with K = c (20) Figure 5 shows how efficiency varies with K, as compute from equation (20). It can be seen that as K increases the efficiency approaches one Calculation of Power an Efficiency Factoring in Nozzle Losses In the previous section it was assume that an ieal flui without friction losses was use. owever, at high rotational spees the relative velocity between nozzle an flui becomes very high introucing energy losses which are proportional to v 2 r. To account for these losses the power equation (15) is aapte as follows: P = ρq g 1 2 v e kv r 2 (21) Where k is a proportionality factor epening on the roughness an other losses in the nozzle. The introuction of these losses will affect the exit velocity v e. owever, the moment of momentum equation remains unchange. Using equation (13) an knowing that P = Tω an with T equals to the thir relation in equation (9), equation (21) transforms into: ρqv e (ωrcosβ) = ρq g 1 v 2 e 2 1 k(v 2 e ωrcosβ) 2 (22) Solving the quaratic equation (22) in an making v e the subject leas after some mathematical manipulation to equation (23): v e = 2g+ω2 R 2 cos 2 β + ωrcosβ (23) (1+k) Equation (23) is therefore the equation of the water jet velocity (v e ) after introuction of the frictional losses at the nozzle. Using equation 23 the power equation 21 can be rewritten to look like: P = ρqωrcosβ 2g+ω2 R 2 cos 2 β + ωrcosβ (24) (1+k) Comparison of this power equation (i.e. incluing friction losses) with equation (16) (i.e. without friction losses) reveals that the only ifference is the factor (1+k). owever, as will be shown later, the factor k has a huge impact on the efficiency. In this case the hyraulic efficiency after some mathematical manipulation becomes; Figure 5. Efficiency η against K Further simplification of equation (17) an taking into account that cos β < 0, the efficiency equation becomes; η = 2 K+K2 (1+k) K ; K = c (25) Figure 6 shows the tren of efficiency (η) against K while incorporating the effect of factork variation. The efficiency

5 Energy an Power 2018, 8(2): curves in figure 6 isplay a clear maximum. Further, it can be seen that an increase in factor k penalizes the hyraulic efficiency. In an ieal situation where there are no losses ue to friction the factor k is taken to be zero. owever in real life application the factor k is a value higher than zero. In our subsequent work, experiments shall be conucte to stuy the effects of factor k by varying the geometry an size of the nozzle. efficiency of the turbine is solely affecte by the value of K (i.e. ratio of hea ue to angular acceleration to the hea ue to gravitational acceleration). owever in a non-ieal case (hyraulic frictional losses are consiere), efficiency is not only affecte by K but also parameter k in an inverse proportion, which means that to achieve a higher turbine efficiency, k must be minimise as much as possible. The pumping effect of the turbine on the flui (ue to centrifugal force), implies that the turbine can raw in as much water as it requires for optimum operation without necessarily requiring a natural high input flow rate. From the theoretical analysis presente here, it can conclue that the single arm centrifugal reaction turbine possesses goo qualities for micro-hyro power generation where low hea an low flows present barriers to achieving esirable turbine efficiency. Results of the performance evaluation of the esigne turbine are presente as part 2 of this research. Figure 6. Effect of factor k on turbine efficiency It shoul be clear that capital letter K is use to enote a non-imensional factor representing the ratio of the hea ue to angular acceleration to the hea ue to gravitational acceleration such as shown by equation (20). Conversely the small letter k is use to enote the coefficient of friction or rather a proportionality factor epening on the roughness an other losses in the nozzle Determination of k The continuity equation (equation 3) along with the equation of v e (equation 23) were use to erive the expression for k which is presente as: k = (2g + ω 2 R 2 cos 2 β) A e Q 2 1 (26) It is clearly seen that parameter k is a function of, ω, R, Q an more eviently the orifice cross-sectional area A e which are all known uring the experiments. Equation 26 is similar to a less general equation presente by Akbarzaeh an Date [6]. 4. Conclusions The aim of this paper was to present the funamental theory an esign of a single arm centrifugal micro-hyro turbine that can operate uner relatively lower heas as well as low to meium flow rates. By making use of the continuum mechanics approach, the water flow rate within the turbine, the turbine torque, the turbine power an the turbine efficiency was theoretically analyse. The analysis showe that in an ieal case (with no turbine losses) the REFERENCES [1] Sternberg, Rolf. "yropower: imensions of social an environmental coexistence." Renewable an Sustainable Energy Reviews 12.6 (2008): [2] A. Date, Low ea Simple Reaction Water Turbine, Ph.D. Dissertation, RMIT University, Melbourne, [3] arvey, Aam. Micro-yro Design Manual: a guie to small-scale water power schemes. No /341. Intermeiate Technology Publications, [4] Waell, R., an P. Bryce. "Micro-hyro systems for small communities." Renewable energy (1999): [5] Toshiba Corporation, Toshiba yraulic Turbines, Company Catalogue, , 00-07T1 (2007). [6] A. Date an A. Akbarzaeh, Design an Analysis of a Split Reaction Water Turbine, Renewable Energy, Vol. 35, No. 9, 2010, pp [7] Duncan, W. J., Thomas, A.S., Young. Mechanics of Fluis, Ewar Arnol, [8] A. Akbarzaeh, C. Dixon an P. Johnson, Parametric Analysis of a Simple Reaction Water Turbine an Its Application for Power Prouction from Low ea Reservoirs, Proceeings of Fluis Engineering Division Summer Meeting, New Orleans, 29 May-1 June [9] A. Date an A. Akbarzaeh, Design an Cost Analysis of Low ea Simple Reaction yro Turbine for Remote Area Power Supply, Renewable Energy, Vol. 34, No. 2, 2009, pp [10] Calvert J. B. (n..). Turbines. Retrieve September 12, 2015, from istory an Moeling of turbines.