Analyse the electrical characteristics of an MagnetoHydroDynamic generator for maximizing the thermal efficiency

TAÏP CHÍ PHAÙT TRIEÅN KH&CN, TAÄP 17, SOÁ K1- 2014 Analyse the electrical characteristics of an MagnetoHydroDynamic generator for maximizing the thermal efficiency • Le Chi Kien Ho Chi Minh City University of Technology and Education (Manuscript Received on February 10 th , 2014; Manuscript Revised August 13 th , 2014) ABSTRACT: In this study, a Faraday type maximizes the thermodynamic MagnetoHydroDynamic (MHD) generator is efficiency, is independent of the studied to consider the effect of electrical regenerator efficiency, but dependent on characteristics to the thermal efficiency. The Mach number and the compressor generator performance is specified by efficiency. It can also be seen that there optimizing the cycle efficiency with respect to is no need for a high entrance Mach the load parameter and by optimizing output number more than 5 because the power density with respect to seed fraction increases in thermal efficiency are and operating pressure. As the calculation insignificant. results, the value of load parameter, which Keywords: MHD generator, thermal efficiency, electrical characteristic, load parameter, output power density. 1. INTRODUCTION electron temperature. Each of these studies considers a particular noble gas and seed for Techniques of Magnetohydrodynamic which high conductivity was attained. In these (MHD) power generation are being studied with studies, however, no attempt has been made to increasing interest for the development of high consider the effect of electrical characteristics temperature materials and high field strength such as load parameter, electrical conductivity magnets progresses. Devices using these of MHD generator for a specified generator techniques are to take the place of the turbo operating under conditions appropriate for generator in a conventional power generation maximizing the thermal efficiency. cycle. Several studies have been proposed that combine Rankine, Brayton, or hybrid cycles In this study a constant area linear duct with with liquids, vapors, and mixtures of these two segmented electrodes operating as a Faraday as proposed working fluids [1-4]. Some of these type MHD generator is studied to consider the studies may be used in a Brayton cycle where effect of electrical characteristics to the thermal the working fluid is an alkali metal vapor seeded efficiency. The magnetic field is constant and in a noble gas. These studies utilize the induced unaffected by the fluid. The current through electric field of the plasma to increase the each pair of electrodes is adjusted so that the Trang 37 SCIENCE & TECHNOLOGY DEVELOPMENT, Vol 17, No.K1- 2014 generated voltage is constant. The working fluid η η = conv (6) is a noble gas seeded with cesium, and the S 1−η 1− conv 2 1 γ effects of viscosity and heat conduction are U ()γM 0 P neglected. The comparison between different Heater 4 seeded noble gas working fluids will be Compressor examined for the optimum conditions to be 2′ obtained. 3 2. THERMODYNAMIC CYCLE 1 6 5′ MHDGenerator EFIFICIENCY Cooler Regenerator A Brayton cycle is considered with (a) Schematic diagram temperatures defined as shown in figure 1. The T4 compressor efficiency, generator (isentropic) T5′ efficiency, and the generator efficiency are T3 T5 defined as follows: Temperature T2′ T − T =η (T ′ − T ) (1) T 2 1 comp 2 1 T2 6 ηS (T4 −T5 ) = T4 −T5′ (2) T3 − T2′ = T5′ − T6 =ηreg (T5′ − T2′ ) (3) T1 where the primed subscripts denote actual state Entropy points in figure 1. It is of interest to relate the (b) Temperature-entropy diagram generator efficiency to the variables defined in Figure 1. Brayton cycle temperature definitions the text and to discuss some of the implications It should be noted that this isentropic of the concept. The efficiency ηS can be efficiency is based on total properties. An expressed in terms of the solution to the isentropic change in total enthalpy that is not generator equations as follows: From the zero can occur if the work is being done. This definition of ηS: can be illustrated as follows. T −T η η η = 4 5′ = conv = conv (4) S T The momentum and energy equations of the T4 −T5 5 1−Y 1− MHD generator are T4 However, Y=( T /T )=( p /p )(γ-1)/ γ must also du dp 5 4 L H ρu + + jB = 0 (7) be expressed in terms of the generator variables. dx dx The ratio of total pressures pL/pH is expressed in dh 2 du terms of the dimensionless exit static pressure P, ρu + ρu − jE ⊥ = 0 (8) dx dx the exit gas velocity U, and the total temperature Multiplying equation (7) by u and subtracting ratio T5'/T4: γ (γ −1) from the equation (8) yield p PγM 2  T  L 0  5′  = γ γ −1   (5) 2 p 2 () T  dh 1 dp  j H ()PγM 0U  4    ρu −  = j()E⊥ + uB = (9) dx ρ dx σ so that   Trang 38 TAÏP CHÍ PHAÙT TRIEÅN KH&CN, TAÄP 17, SOÁ K1- 2014 From the Second Law of Thermodynamics, The area Ar (radiator area), required for a fixed however, the left side of this equation can be maximum temperature T 4 can be obtained from written as A T  4 ε eff σ SB r 4 1−ηth T4 2 =   (16) ds j W η  T  ρuT = (10) th th  ave  dx σ where εeff is effective emissivity of radiator, σSB so that a constant entropy process can occur if σ is the Stefan-Boltzmann constant. Equation (15) approaches infinity. Hence, for an MHD is rewritten in terms of generator, the isentropic efficiency compares the actual generator to a generator using an a = (1−ηreg )[1− (1−Y )ηS ]  infinitely conducting working fluid. ηreg  (17) b = []Yηcomp + ()1−Y  The parameters Y and Z (Y, Z ≤1) are defined as ηcomp  (γ − )1 γ 4  p  T by using equation (15) to evaluate (T 4/T ave ) and  L  2 Y =   , Z = (11) equation (2) to eliminate the temperature terms.  pH  T4 Here, a, b are machine efficiency parameters, The thermodynamic efficiency for zero pressure then the area per unit power output becomes drops through the heater, regenerator, and cooler ε σ A T 4 1−η 1 may be expressed in terms of these parameters eff SB r 4 = th . W 3η a + ()b − Y Z as th th (18)  1 1  η η − Z 1−Y . −  ( S comp )( ) (12) 3 3 3 ηth = Y Z ()a + bZ  Z()()1−ηreg []1− 1−ηcomp Y + ηcomp −   +ηreg ηcomp []1− ()1−Y ηS  Differentiation with respect to Z produces the following equation for Z, which minimizes A , If the cycle is to be used in the space r in equation (15): environment, then it is desirable to minimize radiator area. The temperature ratio Z, which  aY 3Z 3  Z 4 − 4  = 3ηSηcomp (19) minimizes the area, can now be determined. The  ()a + bZ − bY 3Z 4  heat radiated per unit electric power developed can be expressed as The solution to this fifth-degree polynomial can be obtained in two special cases. The parameters Q 1−η rad = th (13) ω and v are defined as Wth ηth YZ  ω = where Wth is thermodynamic work delivered by  1− ()1− Y ηS  cycle,  (20) 3YηSηcomp  v =  4 1− ()1− Y ηS  Qrad = ε eff σ SB ArTave (14) and Equation (16) then becomes 3 3 4 3T6 T1 Tave = 2 2 (15) T6 + T6T1 + T1 Trang 39 SCIENCE & TECHNOLOGY DEVELOPMENT, Vol 17, No.K1- 2014   1.0  3  ηreg =0  ()1−η reg ω  0.8 ] ω 4 − 4 = v (21) S   η  b  b 4 )  1−η reg + ω  − ω  Y   Y  Y  0.6 ηreg =1 /[1-(1- It may be seen that when ηreg =1 0.4 YZ = v ω 0.2 ω = (22) 4 0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 and when ηreg =0 (and b=0), v=3 YηSηcomp /[1-(1-Y)ηS] ω(4 − ω 3 )= v (23) Figure 2. Effect of regenerator efficiency ηreg These two solutions, which are plotted in figure 3. ANALYSIS OF GENERATOR 2, are nearly the same for v≤2. As a matter of CHARACTERISTICS fact, there is a condition for which the solutions will all be the same, namely, when the second A linear MHD generator is analyzed using term in the brackets of the equation (16) is small the fluid flow equations. The fluid is considered compared to 4. It can be shown that if to be a perfect gas, and the effects of heat conduction and viscosity are neglected. The 1−ηreg −ηconv electrical conductivity is to be calculated using ηS ≥ηconv + (24) ηcomp ()1+ηreg the concept of magnetically induced ionization [5,6], which implies an elevated electron where η (1− Y ) =η (25) S conv temperature. This elevated temperature is the then the second term will be less than 0.4. If result of an energy balance between the energy ηreg =1, the inequality is always true. For the added to the electrons by the induced electric remainder of the analysis, it will be assumed field and the energy lost by the electrons upon that the parameters are chosen such that this collision with the other particles. inequality is satisfied. Then, the value of Z that 3.1. Development of MHD Equations minimizes Ar is The continuity, momentum, energy, and 3 state equations for the MHD generator are the Z = ηSηcomp (26) 4 following [7]: and the thermodynamic cycle efficiency may be d ()ρu = 0 (28) written as dx 1 η du dp 4 conv ρu + + jB = 0 (29) ηth = (27) dx dx ηreg ηconv + ()1−ηreg . dh du   Y  2 3 ρu + ρu − jE ⊥ = 0 (30) .1− ηconv 1+ηcomp  dx dx  4  1− Y  γ p h = (31) γ −1 ρ Trang 40 TAÏP CHÍ PHAÙT TRIEÅN KH&CN, TAÄP 17, SOÁ K1- 2014 where ρ is density, u is fluid velocity, p is equation (31) and the momentum and energy pressure, j is current density, B is magnetic field equations. The resulting expression can be strength, h is enthalpy, γ is ratio of specific heat, integrated to obtain the following relation E⊥ is the transverse component of electric field. between the pressure and velocity: The restriction imposed by Maxwell's Bw ()()ρu 2 + p − ρ u 2 + p = . equation, curl E=-∂B/∂t, for a constant magnetic 0 0 0 V (35) field and a one-dimensional problem require  u 3   ρ u 3   γ ρ   γ 0 0  . up +  −  u0 p0 +  that be a constant, equal to -V/w (V is voltage  γ −1 2   γ −1 2  and w is the distance between electrodes), throughout the channel. This constant can be At this point, it is convenient to introduce expressed as some fraction of the entrance open- the following non-dimensional variables and parameters: circuit field u0B as V u γ −1 p K = (32) U = , K L = K , P = , u γ ρ u 2 u0 Bw 0 0 0 2  1  ρ u 2 where K will be called the load parameter. M = 1− 1−  , M 2 = 0 0 , L  2  0 γ +1 M 0  γp0 The generator is assumed to be segmented, τ 2 = 1− K M − K and the segments are assumed to be infinitely ( L )( L L ) thin, so that no axial currents flow. The proper where U is non-dimensional fluid velocity, KL is Ohm's Law is load voltage parameter, P is non-dimensional  V  pressure, ML is Mach number parameter, M0 is j = σ uB −  (33)  w  entrance Mach number, τ is a parameter. where σ is electrical conductivity includes Hall Equation (35) may then be expressed as effects and ion slip, and j is parallel to u×B . The  2  γ +1 τ  restriction that K is a constant places a γP =U − U − K L −  (36) 2  U − K L  restriction on the load resistance RL: (Aej)RL = (Aej)0RL,0 = constant Equation (36) represents the relation between pressure and velocity. Since the duct is where A is the electrode area, R is load e L segmented with infinitely thin segments, the resistance, and the subscript zero denotes power developed in the generator can be entrance values. If all electrodes are given the obtained by integrating the product of voltage same area A , the current can be eliminated as e and current VjH dx over the length of the follows: generator:  u  L σ  − K   1     Π = VjH dx = ρ u3KwH 1+  − ()U + P  (37) RL 0, j  u0  ∫ 0 0  2   γM 0   = = (34) 0   RL j0 σ 0 ()1− K where H is the height of electrodes. To solve the system of equations (28) to (31), the enthalpy h can be eliminated by using Trang 41 SCIENCE & TECHNOLOGY DEVELOPMENT, Vol 17, No.K1- 2014 This power can be compared to the total  2  τ  enthalpy flux entering the generator: 1 1−   γ +1 U − K  ξ = ∫ L dU (44)  γ p 1  2γ ΩU ()U − K  0 2  (38) U Total enthalpy flux = ρ0u0wH  + u0   γ −1 ρ0 2  It is noticed that if the conductivity is constant The ratio of these terms is called the (=l), equation (44) can be integrated: conversion efficiency ηconv and may be written 2   γτ    1 − K   as 1 −    ln   +    K    U − K     K (1−U ) (U − M ) 2 η = L L (39) γ + 1   γτ   1 − K   (45) conv ξ = +   ln  L  − ()U − K L M L     2γ   K   U − K L    2  The power output of a generator with a   γτ  K  1 1  −    −    K  γ 1 − K U − K  specified inlet condition can now be determined.   L L  In order to calculate the output power density, however, a relation between velocity and which is in agreement with the results of other generator length must be determined. The two investigations [8,9]. variables, non-dimensional conductivity =σ/σ0 By using equation (44) for interaction length, it and dimensionless interaction length ξ, defined is possible to express the output power density by ℘ as follows: 2 σ 0 B x Π 2 2 K L (γ +1)(1−U )(U − M L ) ξ = (40) ℘= = σ 0u0 B (46) ρ0u0 wHL 2(γ −1 )(U − KL )ξ are introduced. Equation (29) can then be This is the power density for a constant-area written as generator. It is of interest to gage the effect of velocity variation as well as conductivity d ()()U + P + Ω U − K = 0 (41) variation. The power density at the entrance to dξ the generator is which can be expressed as 2 2 ℘0 = σ 0u0 B K(1− K ) (47) ∂P 1 1+ The ratio of equation (46) to equation (47) ξ = ∂U dU (42) ∫ ΩU ()U − K U ℘ (γ +1)(1−U )(U − M ) = L (48) ℘ 2γξ U − K 1− K Equation (42) provides a relation between U and 0 (L )( ) the interaction length. An expression for ∂P/∂U will be used for comparison. This ratio will be can be obtained by differentiating equation (36): calculated for the constant conductivity case, 2 where ξconst is given by equation (45), and for ξ ∂P 1 γ +1   τ   = − 1+    (43) as determined from equation (44) by use of the ∂U γ 2γ  U − K L     non-equilibrium conductivity. so that equation (42) becomes The cycle thermodynamic efficiency may be conveniently expressed in terms of a generator Trang 42 TAÏP CHÍ PHAÙT TRIEÅN KH&CN, TAÄP 17, SOÁ K1- 2014 (isentropic) efficiency. This efficiency, which is 1.0 defined as the actual change in total enthalpy of K the working fluid in the generator compared to 0.8 the change in total enthalpy for an isentropic Kmax process between the same total pressure 0.6 conditions, is derived in section 2 above. The Load parameter, parameter, Load thermodynamic cycle efficiency for the Brayton K∞ 0.4 cycle under conditions appropriate for space 0 1 2 3 4 5 6 7 8 9 application is also calculated in section 2. Entrance Mach number, M0 Certain limiting values for ηconv , however, can Figure 3. Load parameters K (ratio of voltage to be obtained without specifying the conductivity. open-circuit voltage) for maximum thermal efficiency and infinite choking length for initial compressor 3.2. Limiting Case efficiency of 0.8 From equation (42) it can be seen that, as U 0.28 ηreg =0.99 approaches K, ξ will approach infinity; 0.24 th obviously, this is a limiting value for U. This η 0.20 ηreg =0.9 situation represents the maximum interaction 0.16 ηreg =0.8 length and, consequently, the maximum amount 0.12 of energy that can be taken from the fluid. In 0.0 8 Thermal efficiency, efficiency, Thermal η =0 some cases, however, the interaction length reg 0.0 4 cannot become indefinitely large. It is limited by 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 the phenomenon called “choking”, which can be Load parameter, K (a) Entrance Mach number of 2.0 characterized by the criterion that the local Mach number reaches 1. In the dimensionless 0.14 symbols defined previously, this condition is 0.12 50.0 th η 0.10 10.0 equivalent to 3.0 0.08 U = γP (49) 0.06 2.0 This condition, when substituted into the 0.0 4 Thermal efficiency, efficiency, Thermal equation (36), leads to the following 1.5 0.0 2 specification of U at choking: M0=1.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Load parameter, K U = K +τ (50) ch L (b) Regenerator efficiency of 0.0 It is noticed that this is the value of the velocity for which the integrand in equation (44) is zero; that is, Uch is the condition that makes ∂ξ/∂U= 0. Trang 43 SCIENCE & TECHNOLOGY DEVELOPMENT, Vol 17, No.K1- 2014 0. 28 The quantity ηconv can be calculated from th η =0.9 0. 24 reg η equation (39) and ηS from section 2 for a 0. 20 0.8 specified γ and Mach number as a function of K. 0. 16 Therefore, the thermal efficiency ηth can be 0. 12 0.0 calculated by means of equation (27) for a 0. 08 specified compressor efficiency and regenerator Thermal efficiency, efficiency, Thermal 0.04 efficiency. In figure 4(a) this efficiency is 0 1 2 3 4 5 6 7 8 9 plotted for γ=5/3, M0=2.0, and ηcomp =0.8 with Entrance Mach number, M0 regenerator efficiency as a parameter. Two (c) Maximum load parameter items should be noted: first, the efficiency has a Figure 4. Thermal efficiency for limiting solution maximum at some values of K, and second, this with compressor efficiency of 0.8 value of K is independent of ηreg even though Two different operation limits have been the efficiency varies with ηreg (this is true for all described: first, when U=K and the duct is supersonic Mach numbers). The value of K also infinitely long, and second, when U=Uch and the depends on ηcomp but that dependency will not duct is choked. For any generator operation the be investigated. proper limiting value can be determined by In figure 4(b), the efficiency is plotted again as a considering the case where the duct is choked at function of K with γ=5/3 and ηcomp =0.8, but with infinity. Formally, this occurs when Uch =K. This ηreg =0 and Mach number as the parameter. It can condition can be substituted into equation (50) be seen that the K for the optimum efficiency and the K for which this occurs (call it K ) can ∞ does depend on the Mach number. The value of be determined from the following: K for which the thermodynamic efficiency is γ −1 optimized is called Kmax and is shown in figure 3. K = K +τ (51) ∞ γ ∞ In figure 4(c), the efficiency at K=Kmax and which may be written as ηcomp =0.8 is plotted as a function of Mach number with regenerator efficiency as a 2 2 ()()γ −1 1− M L (γ −1 )(1+ M L ) + M L − parameter. It can be seen that when M0>5, the K = γ 4 2 (52) ∞ 1− ()γ −1 2 increases in thermal efficiency are insignificant. Therefore, there is no need for a high entrance The criterion for distinguishing between the Mach number more than 5. two limiting cases may therefore be stated as For the limiting values of U, ηconv in equation follows: For K>K∞, the duct will not choke and (39) becomes U will approach K, while for K<K∞, the duct will be choked and U will approach Uch . The γ −1 η = (1− K )(K − M )K>K (53) duct is infinitely long and choked for K=K . conv max max L ∞ ∞ M L When γ=5/3, K∞, is as shown in figure 3. It is or noted that for M0<1 the duct will always choke, if sufficiently long, since K must be less than 1; K L (1−U ch )(U ch − M L ) ηconv = K<K∞ (54) whereas, as shown in equation (52) K∞, must be ()U ch − K L M L greater than unity. Trang 44 TAÏP CHÍ PHAÙT TRIEÅN KH&CN, TAÄP 17, SOÁ K1- 2014 4. CONCLUSIONS load parameter, but this value of load parameter is independent of the regenerator efficiency In conclusion, it may be stated that a value even though the thermal efficiency varies with of the load parameter which maximizes the the regenerator efficiency. From the calculations, thermodynamic efficiency of the limiting the load parameter for the optimum thermal solution has been calculated. This value is efficiency clearly depends on the Mach number. independent of the regenerator efficiency, but dependent on Mach number, and the compressor When the entrance Mach number is more efficiency (assumed to be 0.8 for all calculations than 5, the increases in thermal efficiency are presented herein). insignificant. Therefore, there is no need for a high entrance Mach number. For the limiting solutions the efficiency is independent of the form of the electrical The conductivity to be used in the calculation conductivity. Of course, the electrical of output power density is that which is conductivity of the plasma is of great practical determined on the basis of the theory of importance in that it largely determines the magnetically induced ionization. This generator length required to extract power, conductivity depends on the velocity as well as which in turn determines the output power the usual parameters. All results obtained from density of the generator. It is natural, then, to this study will be much more significant for use the generator output power density as a optimizing the efficiency of the MHD generator means of comparing the usefulness of various in the future works. working fluids (the larger the better, of course). It is concluded that, if the duct is sufficiently long, for the entrance Mach numbers smaller than 1, the duct will always choke. The thermal efficiency has a maximum at some values of Phân tích các ñc tính ñin c a máy phát t th y ñ ng ñ c c ñ i hi u su t nhi t • Lê Chí Kiên Tr ưng ði h c S ư ph m K thu t TP.HCM TÓM T T Bài báo này nghiên c u máy phát T nhi t c a h th ng. Ho t ñng c a máy th y ñng lo i Faraday và xem xét nh phát T th y ñng ñưc ch rõ b ng hưng c a các thu c tính ñin ñn hi u su t cách t i ưu hóa hi u su t nhi t có xét Trang 45 SCIENCE & TECHNOLOGY DEVELOPMENT, Vol 17, No.K1- 2014 ñn tham s ti và t i ưu hóa m t ñ công nhi t nh ưng l i ph thu c vào s Mach su t phát ra có xét ñn t l ch t c y và áp và hi u su t máy nén khí. K t qu cũng su t làm vi c. Theo k t qu phân tích, giá tr cho th y r ng không c n thi t s Mach tham s ti mà làm c c ñi hi u su t nhi t, ca vào l n h ơn 5 vì khi ñó hi u su t không ph thu c vào hi u su t b tái sinh nhi t t ăng không ñáng k . T
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