Ind. Eng. Chem. Res. 1988,27, 803-810
803
Dautzenberg, F. M.; De Deken, J. C. Catal. Rev. Sei. Eng. 1984, 26, 421. Freeman, G. B.; Adkins, B. D.; Moniz, M. J.; Davis, B. H. Appl. Catal. 1985, 15, 49-58. Nomenclature Fuentes, G. A. Appl. Catal. 1985, 15, 33-40. Fuentes, S.; Figueras, F. J . Catal. 1978,54, 347-402. C = concentration of contaminant in solution inside the pellet, Garg, D.; Givens, E. N. Fuel Process. Technol. 1984, 9, 29-42. mol cm-3 Guin, J. A.; Tsai, K. J.; Curtis, C. W. Znd. Eng. Chem. Process Des. Deu. 1986, 25, 515-520. C , = concentration of contaminant in solution at the pellet Hegedus, L. Ind. Eng. Chem. Fundam. 1974, 13, 190. surface exterior, mol cm-3 Hiemenz, W. Discussion Section 3, paper 20, Sixth World Petroleum = Congress, Frankfurt, June 21, 1963. De = effective diffusivity, cm2 s-l Hughes, R. Deactivation of Catalysts; Academic: London, 1984. k , = volumetric reaction rate constant, (mol ~ m - ~s-l) ~ - ~ Kam, E. K. T.; Ramachandran, P. A,; Hughes, R. J. Catal. 1975,38, m = number of sites deactivated per molecule of contaminant 285. n = reaction order Kam, E. K. T.; Ramachandran, P. A.; Hughes, R. Chem. Eng. Sei. q = concentration of adsorbed contaminant inside the pellet, 1977, 32, 1307-1317. mol cmw3 Kovach, S . M.; Castle, L. J.; Bennett, J. V.; Schrodt, J. T. Znd. Eng. Chem. Prod. Res. Deu. 1978,17, 62-67. qo = concentration of adsorbed contaminant inside the pellet Levenspiel, 0. J. Catal. 1972, 25, 265-272. at saturation ( m > 0), mol cm-3 Lynch, A. W. Sandia Report SAND-85-0254C, 1985. 4* = 4 / q o Madsen, N. K.; Sincovec, R. F. Assoc. Comput. Mach. Trans. Math. r = radial position in the cylindrical pellet, cm Software 1979, 5, 326-351. r, = radius of cylindrical pellet, cm Masamune, S.; Smith, J. M. AIChE J. 1966,12, 384-394. r* = r / r o Murakami, Y.; Kobayashi, T.; Hattori, T.; Masuda, M. Ind. Eng. t = time, s Chem. Fundam. 1968, 7, 599-605. Newson, E. Ind. Eng. Chem. Process Des. Deu. 1975, 14, 27-33. Greek Symbols Ocampo, A.; Schrodt, J. T.; Kovach, S. M. Ind. Eng. Chem. Prod. B = dimensionless time, Det/r,2 Res. Deu. 1978, 17, 56-61. 6 = Thiele modulus, ro(k,/C,"-1/D,)1/2 Oyekunle, L. 0.;Hughes, R. Chem. Eng. Res. Des. 1984,62,339-343. Rajagopalan, K.; Luss, D. Ind. Eng. Chem. Process Des. Deu. 1979, 9, = initial Thiele modulus for metal poisoning, ro18,459-465. (k,,Con-1/Deo)1/2 Sagara, M.; Masamune, S.; Smith, J. M. A I C h E J. 1967, 13, 1226. x = "tolerance" to poisoning, qon/C, Stiegel, G. J.; Tischer, R. E.; Cillo, D. L.; Narain, N. K. Znd. Eng. Chem. Prod. Res. Deu. 1985,24, 206-213. Registry No. Ni, 7440-02-0; Mo, 7439-98-7; Fe, 7439-89-6; Ti, Stohl, F. V.; Stephens, H. P. Sandia Report SAND-85-0852C, 1985. 7440-32-6. Thakur, D. S.; Thomas, M. G. Znd. Eng. Chem. Prod. Res. Deu. 1984, 23, 349-360. Literature Cited Thakur, D. S.; Thomas, M. G . Appl. Catal. 1985, 15, 197-225. Thomas, M. G.; Sample, D. G. Fuel 1981, 60, 145-150. Adkins, B. D.: Davis, B. H. Prepr. Am. Chem. Soc., Petr. Chem. Diu. Tsotsis, T. T. DOE Report DOE/PC/41263-T1, 1982. 1985,475-487. Valdman, B.; Hughes, R. 6th Ibero Amer. Symp. on Catalysis, Rio Adkins, B. D.; Cider, K.; Davis, B. H. Appl. Catal. 1986,23,111-127. de Janeiro, 1978. Ahn, B. J.: Smith, J. M. AZChE J . 1984.30, 739-746. Valdman, B.; Ramachandran, P. A.; Hughes, R. J. Catal. 1976,42, Butt, J. B.' Adu. d h e m . Ser. 1970, 109, 259-496. 303. Cable, T. L.; Massoth, F. E.; Thomas, M. G. Fuel Process. Technol. Wheeler, A. Adu. Catal. 1951, 3, 250. 1981,4, 265-275. Received for review April 13, 1987 Cable, T. L.; Massoth, F. E.; Thomas, M. G. Fuel Process. Technol. 1985, 10, 105-120. Accepted January 22, 1988
tucky, Kentucky Energy Cabinet. The KECL is administered under contract by the University of Louisville.
c* c/c,
PROCESS ENGINEERING AND DESIGN Use of Dissociating Gases in Brayton Cycle Space Power Systems Hua-Min Huang and Rakesh Govind* D e p a r t m e n t of Chemical Engineering, University of Cincinnati, Cincinnati, Ohio 45221
In this paper an analysis of the use of dissociating gases in Brayton Cycle Space Power Systems has been presented. It has been shown that the development of higher efficiency cycles is necessary for minimizing isotope costs which have a dominant influence on total cycle economics. A dissociating gas Brayton cycle has been optimized for maximum efficiency and minimum radiator surface area. Results show that 40% higher thermal efficiencies and 25% reduction in radiator area can be achieved with a dissociating gas cycle when compared with a nondissociating gas cycle. Space power systems energized by the radioactive decay of plutonium 238 have been used with great success for 0888-5885/88/2627-0803$01.50/0
missions where more conventional solar arrays cannot be employed because of solar distance, hazardous environ0 1988 American Chemical Society
804 Ind. Eng. Chem. Res., Vol. 27, No. 5, 1988 SHADED AREAS INCLUDE
ALL COMBINATIONS O F S75OlLB C I I S 3 0 0 0 / L B J250/LB I h S S10001LB
-
300
150 200
f
f *
w' 1000
80-
50
-
30
MISSION DURATION, T = 10 YEARS
I
I
I
J
q , PERCENT
Figure 1. Plot of total power system cost per kilowatt hour versus the overall efficiency for f 2 variations in system cost parameters.
ment, or extended mission duration (Pietsch and Trimble, 1985). Thermal-to-electric energy conversion has utilized thermocouple junctions with greatly increased life, power, and reliability. However, as power requirements continue to increase and isotope costs comprise an increasing fraction of the total power system cost, the need for more efficient thermodynamic conversion becomes critical. Cost is a high-priority consideration for a space power system. It has been shown (Pietsch and Trimble, 1985) that the system cost can be expressed as COE =
(i+
Y ) / T
where COE = C / ( K T ) . Typical values (valid within a factor of 2) for the cost parameters are i = $2000/thermal W, 1 = $680/kg, and h = $227/kg. Curves which illustrate the effect of 4 (f2) variation in these parameters are shown in Figure 1. The curves show the tradeoff between the power system efficiency and its mass and demonstrate that isotope costa are the dominant influence for all parametric combinations considered. Isotope costa are minimized by maximizing the cycle efficiency. Hence, it can be concluded that future application of radioisotope power systems depends on the availability of higher efficiency cycles. High cycle efficiency also results in reduced collector area (if the cycle is solar powered) or radiator surface area for a nuclear source. The resultant decrease in drag due to size reduction is important for a low-orbit space station. This yields savings in orbit reboost costs, and flight stability is enhanced, resulting in reduced dynamic loads and thruster repositioning. For constant work output (10-400 kW), higher cycle efficiency also means the requirement of a lower turbine inlet temperature. High reliability and long life is important for space power systems with a reliability requirement in excess o€ 0.95 for missions of greater than 5-year duration. Lower turbine inlet temperature increases the l i e of equipment by lowering the potential for selective material sublimation in a vacuum by the separate constituents (e.g., chromium and aluminum).
Background Several publications on space station power cycles have appeared in the literature (Layton, 1980; Guentert and Johnson, 1971; English and Slone, 1960; Cheung, 1967; Kerwin, 1971; Gable and McCormick, 1978). These papers have assessed radioisotope cost/availability guidelines on space power system design and have described the development of the Brayton Isotope Power System (BIPS) based on these guidelines. The BIPS program was completed in 1978 with an intensive 1007-h test program including a continuous run of 402 h. A significant accomplishment was the refined definition of a BIPS flight system design. A solar-powered closed Brayton cycle (CBC) with potential applicability to the space station has also been studied (Pietsch and Trimble, 1985). A conceptual design of a 40-kw CBC system consisting of a 14.6-m-diameter parabolic collector has been presented. A 15-80-kw reactor-powered Brayton cycle has been studied where the working fluid is a mixture of helium and xenon (Kerwin, 1971). Performance characteristics were predicted over a range of power level, cycle temperature ratio, and turbine inlet temperature. Alkali metal Rankine cycles have also been studied for space power applications. The concept of an alkali metal-steam binary cycle was proposed in 1958, and a program to desige and develop an alkali metal Rankine cycle of 100-300-kw output, referred to as the Medium Power Reactor Experiment (MPRE), was conducted in the period 1959-1966 (Holcomb, 1985). System and component design studies were also done to compare the weight and size of the components for potassium versus cesium as the working fluid. Recent renewed interest in power requirements for space satellites and stations has led to new conceptual design studies of Rankine cycles with turbine inlet temperatures of 1365-1450 K. Critical questions which have been raised about the use of Rankine cycles are why (1)two-phase flow in the zero gravity environment of space to confirm proper design of the boiling reactor or external boiler and condenser is not well understood and (2) high-temperature performance, endurance, and reliability of a potassium turbine has not been demonstrated. Use of dissociating gases for space power plant applications in a Brayton cycle has also been investigated (Cheung, 1967). Substances which have been studied include phosphorus, iodine, chlorine, and selenium. Mass flow rate and radiator surface area required for a 1.25-MW plant were computed for an ideal Brayton cycle. However, optimization studies of the cycle operating conditions to minimize radiator area were not conducted. Dissociating gases as working fluids have been studied by several investigators (Blander et al., 1959; Kesavan, 1978; Huang, 1986). Brayton cycle studies demonstrated superior overall performance of dissociating gases as working fluids in conventional power plant cycles. The component sizes were smaller due to higher gas density, and heat transfer was increased due to reaction. Analysis of a Dissociating Gas Consider the reaction A2 ~t 2A. The analysis of this reversible reaction occurring during an isentropic change has been developed by using an equilibrium model (infinite reaction rates) and a nonequilibrium model (finite reaction rates). (a) Equilibrium Model for Cycle Efficiency. Suppose the reaction type is from A2 to 2A (such as A12C16s 2A1C13). A t temperature T and pressure P, the degree of dissociation is a. If ideal gas law is assumed, the equi-
Ind. Eng. Chem. Res., Vol. 27, No. 5, 1988 805 librium constant can be expressed as 4 d P = K p = e-AGo/RT = 1 - a2 exp[(-AGO + AHo)/RT]exp(-AHo/RT) ( 2 )
3
where AGO and AHo are the standard Gibbs free energy change and the standard heat of reaction at reference temperature To,respectively. The ideal gas law and enthalpy of a dissociating gas system are
W P V = -(1 M2
+ a)RT H = a m 0 + C P ( A P ) (-l a)T + 2LuC,(,)T
(3)
S Figure 2. Temperatureentropy diagram of a simple Brayton cycle. I
(4)
If constant specific heats are assumed for A2 and A, AGO can be expressed as AGO = AHo - AC,T In T - IRT (5)
= 2.0
where AC, = 2C (A) - Cp(A2). Equations 2-4 can be simplified to the folfowing dimensionless equations: P’ = p’T ’(1+ a ) (6)
0 .no 0.02
0.04
Ti
-
0.06
0.08
I
0.10
RTl/AHo
Figure 3. Equilibrium cycle performance A12Cle
f ( T ’) = T C>(A*)= C,(A~)/R J = AC,/R - 1 For an isentropic path as in turbines or compressors, the entropy change is zero. Hence, dh’ V‘dp‘ &=---= 0 T’ T‘ Equations 6, 8, and 9 may be combined to give the following differential equation relating the temperature and pressure along the isentropic p a t h
-
2AlC1,.
for the turbine. The procedure for solving the maximum efficiency, ,*-, is as follows: select a given temperature, T ’1, temperature ratio, T ’1, and initial pressure, p;. Pick a value of T ’2 between T ’1 and T $. Equation 10 is integrated by using the fourth-order Runge-Kutta technique, to determine the value of P i correspondingto the selected value of T ’2. Since P i = Pk and P i = P’l, eq 10 is again integrated to determine the value of T &‘ Enthalpies for each point in Figure 2 are calculated from eq 8. Enthalpies at points 2‘ and 4’ can be expressed as h2t = hi
+ (h2 - hi)/,c
h4’ = h3 - ~ t ( h 3- h4)
(14)
(15)
*, ,,,*, (10) Referring to the T-S diagram shown in Figure 2, the thermal efficiency *, of a cycle is given by h4, - h1 ,*=I-- h3 - hy (11) The component efficiencies are defined as h2 - hl qc =
a
for the compressor and
is calculated from eq 11. The maximum value of ,*, is obtained by direct search since the efficiency, ,*, is a unimodal function of T ‘2/ T This analysis can be extended to different types of chemical reactions (such as 2A 2B + C) and can be used when the specific heat, C,, is a function of temperature, m 1.
In Figures 3 and 4, the percentage change in maximum thermal efficiency, as compared with a nondissociating gas has been plotted versus T The working fluids studied are AlzC&belonging to the reaction A2 e 2A and NOCl corresponding to the reaction type 2A + 2B + C. The percentage change in maximum thermal efficiency increases as T $ / T decreases. This is due to the fact that, as T $IT 1’ approaches 1.0, the heat of reaction is a greater fraction of the heat absorbed, resulting in a higher thermal efficiency due to reaction. Also the increase in efficiency is higher for the A2 F! 2A reaction type, because the molar ratio is higher.
806 Ind. Eng. Chem. Res., Vol. 27, No. 5, 1988
are imposed: T,I300 K and T, I1000 K. (b) Nonequilibrium Model for Cycle Efficiency. In the previous analysis, chemical equilibrium was assumed in the cycles. Yet the reaction rates are not infinite. If the reaction rate is not fast enough, the equilibrium assumption may not hold. Hence, the reaction kinetics have to be studied. Consider a simple reaction A, ~t 2A (such as A12C1, .s 2AlC13). The equation of continuity for the ith chemical species in an n-component mixture can be written as follows: change due to the variation in temperature and pressure can be expressed as
I' 1
-
i = 1,2,...,n (16) RT / a n 1 0
Figure 4. Equilibrium cycle performance of 2 N O C l e 2 N 0 + C1,. Table I. Comparison of Brayton Cycles for Two Reaction Types with T 3 / T 1= 2.0, P I = 1 atm, T I = 500 K 2A s A, ~t 2A" 2B + Cb He 2.8703 X lo4 6.3453 X lo4 1.2149 X lo5 specific work, J/kg heat-transfer rate, 3.0012 X lo4 5.4396 X lo5 1.867 X lo6 Jlkg thermal efficiency useful work ratio turbine exhaust volume flow rate, m3/(s.MW)
since
pYi = CiMi Equations 16 and 17 can be combined to give
a
-
0.1166 0.2279 8.888
0.0650 0.1395 140.79
(CiUi,)
a (Ciu:,J + ci +az
ax
Mi (18) If the diffusion velocities are neglected, we get the continuity of component i in the absence of diffusion: dY
0.0957 0.2459 8.967
(17)
* 2NOC1 z 2 N 0 + C1,.
-DCi- _ -Ri + -Ci- Dp Dt Mi p Dt
The parameters that determine the efficiency of a Brayton cycle are C , AC,, PI,T $1T ',, and T ',. For both reaction types s t d i e d , the influence of C, and AC, on maximum efficiency is not significant. This is due to the fact that the term [Ci(A2)+ ( J + l ) a ] T 'in eq 8 is small, since T ' is small, usually in the range 0.01-0.1. For both reaction types, the lower the initial pressure P',, the higher the maximum thermal efficiency, and the maximum occurs a t a lower sink temperature, T This is due to the effect of pressure on the degree of dissociation. A lower pressure favors higher degree of dissociation, and the chemical reaction begins to play a dominant role on cycle efficiency. Further, the advantages of dissociating gases are not limited only to higher efficiencies. As shown in Table I, the useful work ratios (work output/turbine output) are higher for dissociating gases. A higher useful work ratio implies that, for a certain capacity of a power plant, the work used for compression is lower, and hence, the compressor can be smaller. Also the turbine-outlet volume flow rates are smaller for dissociating gases, resulting in a smaller turbine size. The above analysis is useful for screening dissociating gases categorized by the two reaction types, A, ~t 2A and 2A z 2B + C. For a specific case, given the heat of reaction, AHo, Figures 3 and 4 can be used to find the value of T = RT1/AHo, for a specific value of P', and temperature ratio, T $/T which corresponds to a maximum percentage change in thermal efficiency. If the sink temperature, T,, corresponding to the peak efficiency, is below ambient temperature, clearly the corresponding dissociating cycle is not economical. On the other hand, if the sink temperature is too high, for a fixed ratio, T $/T',, the maximum temperature (T,) in the cycle may increase beyond a reasonable bound, based on the material of construction. Usually the following constraints
Writing the rate equation for simple dissociation of A2 gives
a
A12C13s 2AlC1,.
',.
',
RA,/MA~ = -kfCA2 + k&A2
(19)
(20)
where kf and kb are forward and backward reaction constants. Equation 20 can be rearranged to give (21) RAJMA, = -kfCA2(1 - k*,/k,) where k , = kf/kb is equilibrium constant at temperature T, and k*, = C2A2!CA2. The concentration change due to the change of temperature and pressure can be expressed as
where t is time, and Ci is the concentration of component i. The adiabatic flow of ideal gases is characterized by (23) dp/dT = y - 1 where y is the instantaneous ratio of the specific heat at constant pressure to the specific heat at constant volume. If eq 22 and 23 are combined, the following equation is obtained: (dCi/dt), = (7 - l)-'(Ci/T) dT/dt (24) The concentration change due to chemical reaction is (dCNz04/dt)c= -kfCN1O4 + kbC2N02
(25)
where kf and kb are forward and backward reaction constants. Equation 25 can be rearranged to give (dCN,O,/dt), = k&~,o,(l- k * , / k J (26) where k , is the equilibrium constant at temperature T and c* = C2N02/CN204.For NO2, the concentration change is
Ind. Eng. Chem. Res., Vol. 27, No. 5 , 1988 807 dp
dP P
p
dT
da
T
l + a
(40)
Combining eq 39 and 40 we get P
”+ l+a
dT
T
or
where
c;
=
da c, + AlH+- RaT dT
From eq 19 and 41
and
k, = k(RT)-*’
where Au is change of number of moles in the chemical reaction. k*, can be approximated by k*, = k,
Ci _ Dp -- Ci _ (y’
(32)
dk, +( T * - T) dT
p Dt
DT - 1)-1 -
T
dt
To calculate d a / d T needed for Cl,, the following equations were derived: The concentration CA2can be expressed as
(33)
P (1 - 4 CAz = R T (1 + a )
where T * is the corresponding temperature to k*,, and hence k*, d In k, I--=-(T* - T) (34) k, dT
Hence
If eq 29-34 are combined, the following expression is obtained:
where Co = P/(RT). We know that
T*
--
1: e)( 1 - (y - 1)-1
1=
T
kf(
-
1-ff
+ kbC02(
RT1 CP dp - - d T AHP AH
where From eq 37 we get
+R
+
(43)
&) c01-~r l+(y dt dT
-
(44)
(35) Combining eq 43 and 44 leads to
(38)
AH=AE+RT
The overall concentration change with respect to time, from eq 19, 20, and 42, is
Noting that
we get DCA, = -k&O( Dt
G) 1-ff + kbCO(
&y
+
(1- a ) 1 DT (y’ - 1)-1Co--- - (47) (1 a ) T Dt
+
Combining eq 36 and 38 we get
Cp = C,
CO1 - C Yd T T 1 Q! dt
1)T
Differentiating eq 37 we get p dV + V d p = (1 + a)R d T + R T d a
l + a
2c0 d a dt
-
The equilibrium performance determined in the previous section would be a reasonable assumption if (T*/ T - 1) is small (less than 0.01) along the isentropic path. Hence, eq 35 can be used as a criterion in identifying equilibrium performance. In eq 19, the term C i / p Dp/Dt is calculated as follows: An energy balance on the adiabatic expansion of a system with chemical reaction is (1 + a)CUd T + AE d a = -p dV (36) The ideal gas law for the gas mixture is p V = (1 + a)RT (37)
d a- -=
~
(39)
808 Ind. Eng. Chem. Res., Vol. 27, No. 5, 1988 d
I
-0.85116
d
AREA = 168 b? RADIATOR 0.81
0.80 0.79
-
EOUlLlBRlU4 PERFORMANCE
,
\ ;
A t =
At *
1
I
T"
~ 1 6 1 0 IT2 ~
CWNGER 12.2 ATM '2
REACTOR
I
1600°1:
1,OE-2
6,OE-2
sc RECUPERATOR
Figure 6. A Brayton Cycle Power System. Table 11. Mass Summary of a 400-kWBrayton Power System kg
reactor reactor shield heat source heat exchanger cycle rotating unit (2) alternator radiator recuperators (2) radiator ducting and misc. power conditioning structure total
850 800 420 430 150 730 4040 400 150 300 8270
Table 111. Summary of Operating Parameters for an Aluminum Chloride (AlCl,) Cycle and a NOCl Cycle AlCl, NOCl A1C13 NOCl 1500.0 1500.0 1500.0 1500.0 Ti, K P,, atm 25.0 10.0 70.0 60.0 1095.1 1143.5 1034.8 T2 1212.9 p 2
T3 p3 T4 p 4
T5 p 5
T6 p6
A , m2 7, ?&
Results and Discussion Three Case studies will be presented in this paper. Both thermal efficiency and radiator area have been considered in each case. Case 1. Layton (1980) showed that, for a nondissociating gas (helium-xenon mixture) Brayton cycle with regeneration, 168 m2of radiator area is needed for generating 400 kW of power. The cycle thermodynamic efficiency was
2.5 817.0 2.375 500.0 2.375 592.6 25.83 1106.45 25.83 105 30.40
12.0 739.8 11.4 550.0 11.4 689.8 62.0 966.1 62.0 114 30.79
0.5 773.0 0.475 410.0 0.475 501.5 10.33 1008.2 10.33 158 32.60
10.0 639.6 9.5 440.0 9.5 581.7 72.3 916.7 72.3 158 40.06
30.5%. The operating parameters of the cycle are shown in Figure 6. The cycle peak temperature is 1500 K, and the theoretical radiator area calculated from eq 13 is 158.3 m2,which is 94.2% of the actual area. Table I1 summarizes the mass for each component in the cycle. It is clear that the radiator is the heaviest component in the power system. The two dissociating gases studied were A12C1, and NOC1. When the radiator surface area was minimized and the thermal efficiency was maximized, the minimum radiator area for AI&& was 105 m2 (efficiency = 30.4%) with a peak efficiency of 32.6% (radiator area = 158 m2) and 114 m2 (efficiency = 30.79%) with peak efficiency of 40% (radiator area = 158 m2) for NOC1. The operating parameters are summarized in Table 111. Case 2. Guentert and Johnson (1971) studied a Brayton cycle similar to case 1. The cycle peak temperature was 895 K. The cycle thermal efficiency was 19.5% with 362.3 m2 of radiator area needed to produce 60 kW of electricity. By use of NOCl as the dissociating gas, only 71 m2 of radiator area was needed with 23.6% thermal efficiency. If thermal efficiency was maximized, the peak efficiency obtained was 31.4% with 100 m2 of radiator area. The cycle operating parameters are summarized in Table IV.
Ind. Eng. Chem. Res., Vol. 27, No. 5, 1988 809 Table IV. Summary of Operating Parameters for a Nondissociating Gas (Helium-Xenon) Mixture and a Dissociating Gas (NOCl) Cycle nondissociating gas (He-Xe) NOCl 894.44 894.44 8.9 63.0 682.1 9.0 603.59 8.55 34.00 400.00 4.8 8.55 537.13 8.9 65.09 630.56 65.09 0.925 362.3 71 19.5 23.57
NOCl 894.44 63.0 682.1 9.0 549.45 8.55 325.0 8.55 462.09 65.09 632.2 65.09
Table V. Summary of Operating- Parameters for an AlCl, Cycle and a NOCl Cycle AICL NOCl 1365.0 7'1, K 1365.0 PI, atm 20.0 64.0 953.1 T2 1127.4 p2 2.5 8.0 712.7 T3 936.5 2.37 7.60 p3 500.0 T4 750.0 2.37 7.60 p4 854.0 662.68 T5 20.56 66.12 p5 853.5 T6 1056.6 66.12 p6 20.66 A, m2 47 140 26.45 v, 70 19.95
100 31.44
INERT GAS CYCLE TEMPERATURE RATIO 0.34
-
I
0.36
u
1' 14
'
16
/
NOCL CYCLE TEMPERATllRE RATIO 0.42
-
20
24
CYCLE THERMAL EFFICIENCY
Figure 7. A 300-kW alkali metal vapor Rankine cycle.
Case 3. Holcomb (1985) studied alkali metal Rankine cycles with a turbine inlet temperature of 1365 K. The cycle schematic with some operating parameters is shown in Figure 7. The radiator area needed was 42 m2 for generating 300 kW of electricity. The cycle thermal efficiency was 15.5% . By use of a A12C16Brayton cycle instead of the alkali metal Rankine cycle, for the same power generating capacity, the maximum efficiency was 19.5% with comparable radiator area. The cycle operating parameters are summarized in Table V. The above cases are summarized in Figure 8 where the optimized radiator areas and cycle thermal efficiencies have been plotted for a nondissociating gas (helium-xenon mixture) and NOCl at different cycle temperature ratios (compressor inlet temperature/turbine inlet temperature). Note that the efficiencies which can be attained for a dissociating gas, such as NOC1, are much higher than those possible with a nondissociating gas. Also, for the same efficiency, the radiator surface area needed for a dissociating gas is approximately one-fourth of that needed with a nondissociating gas. Further, the advantage of a dissociating gas over a nondissociating gas (lower radiator area and higher thermal efficiency) exists over a wide range of cycle temperature ratios, thereby providing flexibility in designing power systems. A dissociating gas due to ita higher specific heat and heat of reaction tends to have a higher turbine outlet temperature and lower compressor outlet temperature. Hence, in a dissociating gas Brayton cycle, the heat is rejected at a higher temperature, and since the radiation heat flux is proportional to the fourth power of the radiator surface temperature, consequently the radiator surface area is greatly reduced. Due to relatively large differences in the gas constant between the associated and dissociated states of the gas,
0 3q 3.36 28
7
32
(2)
Figure 8. Plot of optimized radiator area versus the cycle thermal efficiency for a nondissociating gas (helium-xenon mixture) and a dissociating gas (NOC1). The parameter is the cycle temperature ratio, i.e., compressor inlet temperature/turbine inlet temperature.
the ratio of turbine work to compressor work will be greater than for a nondissociating gas. Hence, for a certain power capacity, a dissociating gas cycle will need a smaller compressor than a nondissociating gas cycle. Also the turbine outlet volumetric flow rates will be smaller for a dissociating gas, resulting in smaller turbine sizes. Conclusions Analysis of a dissociating gas Brayton cycle has been presented in the paper. Equations were derived for computing reaction conversion and gas state during compression and expansion in the compressor and turbine. A criterion for checking the attainment of equilibrium condition at the turbine outlet was derived. Three cases were studied. It was shown that a dissociating gas such as NOCl yields about 40% higher thermal efficiency and results in one-fourth of the radiator area needed for a nondissociating gas. This finding has important implications for the design of a space station power system. Nomenclature A = radiator surface area C = total cost of a power system, $ Co = gas concentration, mol/cm3 COE = cost, $/(kW*h) Ci = molar concentration of component i, mol/cm3 Cp(A)= specific heat of molecule A, cal/(mobK) C p ( A 2 )= specific heat of molecule Az, cal/(moEK) C>(A21= dimensionless specific heat of molecule A2 = instantaneous specific heat at constant pressure, cal/(mol-K) C*, = instantaneous specific heat at constant volume, cal/ (mo1.K)
810 Ind. Eng. Chem. Res., Vol. 27, No. 5, 1988
(dC,/dt), = concentration change of component i due to chemical reaction, mol/(cm3.s) (dC,/dt), = concentration change of component i due to change of temperature or pressure, mol/ (cm3-s) h = hardware cost, $/kg h' = dimensionless enthalpy H = enthalpy, cal/mol i = isotope cost, $/W Z = integration constant K' = power system output, kW k = equilibrium constant kb = backward reaction constant at temperature T , cm3/ (mo1.s) k , = equilibrium constant based on concentration k*, = equilibrium constant calculated by existing concentrations kf = forward reaction constant at temperature T , s-l k , = equilibrium constant based on pressure 1 = launch cost, $/kg MI = molecular weight of species i, kg of i/mol of i M 2= molecular weight of A2, g/mol P = pressure, atm P' = dimensionless pressure R = gas constant R, = net mass rate of production of species i per unit volume by chemical reaction, kg of i/(m3-s) s = power system specific mass, W/kg t = time, s T = temperature, K T , = temperature of stream entering the radiator, K Tb= temperature of stream leaving the radiator, K T ' = dimensionless temperature T * = temperature correspond to k*, T , = sink temperature, K V = volume, cm3 V' = dimensionless volume VI? = vx + v:,x v,y = vy + v:,y vi,z
= u,
+ ur;,z
vx,vy, v, = mass weighted average velocities in the x , y , and z directions. respectively
u',,~,v ' ~ ,v',,~ ~ , = diffusion velocities of species i in x , y, and z
directions, respectively vi&, vIy,vIp = flow velocity of species i in x , y , and z directions, respectively W = mass flow, kg/s Yi = mass fraction of species i, kg of i/m3 Greek Symbols
a = degree of dissociation
y = instantaneous ratio of the specific heat at constant
pressure to the specific heat at constant volume = emissivity u = Boltzmann constant, cal/ (cm2.s.K4) AC, = specific heat difference between 2A and A,, cal/(mol.K) AE = change of internal energy of dissociating gases, cal/mol AG" = standard Gibbs free energy change of AP,cal/mol AH = heat of reaction of A2 at temperature T AHo = standard heat of reaction of A, at reference temperature
To,cal/mol AU = change of the number of moles in the chemical reaction p = gas density, g/cm3 p' = dimensionless gas density 7- = maximum efficiency of nondissociating inert gas cycles q* = thermal efficiency of dissociating gas cycles q*" = maximum efficiency of dissociating gas cycles 7 = combined cycle thermal efficiency q C = compressor efficiency qt = turbine efficiency Registry No. NOC1, 2696-92-6; Al,Cl,, 13845-12-0.
Literature Cited Blander, M.; Epel, L. G.; Fraas, A. P.; Newton, R. F. "Aluminum Chloride as a Thermodynamic Working Fluid and Heat Transfer Media". USAEC Report ORNL-2677; Oak Ridge National Laboratory, Oak Ridge, TN, 1959. Cheung, H. Presented at the 2nd Intersociety Energy Conversion Engineering Conference, Miami Beach, FL, 1967; paper 679003. J. 1960,30(11), 1097. English, R. E.; Slone, H. P. 0. Am. Rocket SOC. Gable, R. D.; McCormick, J. E. Presented at the 13th Intersociety Energy Conversion Engineering Conference, San Diego, CA, 1978; paper 789383. Guentert, D. C.; Johnson, R. L. Presented at the 6th Intersociety Energy Conversion Engineering Conference, Boston, MA, 1971; paper 7190855. Holcomb, R. S. Presented a t the 20th Intersociety Energy Conversion Engineering Conference, Miami Beach, FL, 1985; paper 859430. Huang, H. "Studies on Dissociating Gas Power Cycles". Ph.D. Dissertation, University of Cincinnati, Cincinnati, OH, 1986. Kerwin, P. T. Presented a t the 6th Intersociety Energy Conversion Engineering Conference, Boston, MA, 1971; paper 719054. Kesavan, K. "The Use of Dissociating Gases as the Working Fluid in Thermodynamic Power Conversion Cycles". Ph.D. Dissertation, Carnegie Mellon University, Pittsburgh, PA, 1978. Layton, J. P. Presented at the 15th Intersociety Energy Conversion Engineering Conference, Seattle, WA, 1980; paper 809140. Pietsch, A.; Trimble, S. Presented at the 20th Intersociety Energy Conversion Engineering Conference, Miami Beach, FL, 1985; paper 859443. Received for review December 29, 1986 Revised manuscript received November 20, 1987 Accepted December 11, 1987
