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Theoretical modeling of a directly heated solar-driven chemical reactor
Solar Energy' Vol. 45. No. 3, pp. 139-148, 1990
0038--092X/90 $3.00 + .00 Copyright ~ 1990 Pergamon Press pie
Printed in the U.S.A.
THEORETICAL MODELING OF A DIRECTLY HEATED SOLAR-DRIVEN CHEMICAL REACTOR EVA MEIROVITCH,AKIBASEGAL, and MOSHE LEVY Center for Energy Research, The Weizmann Institute of Science, Rehovot 76100, Israel Abstract--A theoretical formulation for calculating the performances of a solar-driven catalytic chemical reactor was developed. It accounts for the spatial distribution of the deposition of primary energy within the receiver, the heat transfer into the catalytic bed and the thermochemical endothermic reaction, chemical composition and flow distribution within the reactor. The theory set forth was applied to analyze results obtained in a solar furnace with a directly heated U-shapedtubular reactor, wherein catalyticcarbon dioxide reforming of methane occurred. We find that the receiver/reactor assemblyacts as a self-regulatingsystem. Beyond a fractional catalytic bed length of 0.14, solar energy can be converted primarily into chemical enthalpy. The fluid temperature gradient monitors the heat balanceby adjustingthe overallrate of conversion to the rate at which energy is being transferred through the reactor walls. Under certain circumstances, the process may be heat-transfer limited or controlled by chemical thermodynamics. A good fit between theory and experiment and accountability of all the intricate details in the various calculated performances of the receiver/reactor system support the theoretical model set forth in this study. We offer it as a tool for simulating future experimental results and for designing solar-driven reactors.
l. INTRODUCTION
Conventional solar receivers are enclosures built of refractory walls and absorbing devices acting as heat sinks[l]. The latter are often metal tubes, through which a "working fluid" is being passed. Usually one uses air, steam, molten salt, a liquid metal, etc., that enters cold and exits warm, having absorbed convectively the heat flux transmitted through the tube walls. Radiative solar energy has thereby been converted into sensible heat. Severe problems of storage and transportation are encountered with this method. For example, it is extremely inefficient, and at times impractical, to transport and store steam at 300°-400°C. A recently conceived variant of this concept is conversion of solar energy into chemical enthalpy, rather than sensible heat[2 ]. This can be brought about with endothermic gas-phase reactions. Solar heat is now stored in the chemical bond, the energy-rich products are gases that can be cooled to room temperature, pressurized and conveniently transported and stored. The energy can be recovered at will by performing the reverse reactions, and the products of the latter can, eventually, be recycled, with the process carried out in a closed loop. Alternatively, the solar driven reactors can be used to obtain products of industrial interest. Experimental work within this field has been performed so far in a few laboratories, using primarily sun-simulators[ 3 ]. Among the first studies using solar radiation were those performed at the Schaefer Solar Furnace of the Weizmann Institute [ 4]. Hence, the field being at its outset, many issues are still exploratory in nature, referring to configuration, limitations, and stability. For example, one could expose the reactors to direct illumination. Alternatively, one could think of integrated heating, with a screening fluid in between the site at which concentrated sunlight is being ab-
sorbed, and the site at which the process of conversion into sensible heat and chemical enthalpy unfolds, or of completely decoupling the sites at which energy is being absorbed and converted. Direct exposure to sunlight is necessarily of higher efficiency, due to reduced heat losses. On the other hand, directly illuminated surfaces are prone to disastrous local overheating. It is of paramount importance to have a proper understanding of the physical processes involved. Radiation and chemistry are coupled, interfaced by the convective transfer of heat through the reactor walls. Radiative heat transport and catalyzed gas-phase reactions have been treated so far independently. With thermochemical conversion of solar energy they become integrated into a self-regulatory system. Intricate details of the physical phenomena involved, limits of stability, constraints, etc. of the latter are to be investigated and comprehended in depth, in order to interpret experimental results obtained so far and design assemblies to be used in the future. These are the topics we focus on in this study. From a theoretical standpoint, we present below a comprehensive formulation, treating radiation and chemistry in an integrated form and using it to anal x~e experimental results obtained at the Schaefer Solar Furnace of the Weizmann Institute. The experiment is described in a separate publication[4]. We summarize below basic aspects essential to the forthcoming presentation. A U-shaped tubular reactor, filled up to 70% of its length with a catalytic bed, was placed horizontally in a cylindrical receiver with the bend facing the cavity aperture (see Fig. 1). Receiver wall, reactor wall and fluid temperatures were measured at the positions denoted in the figure, and the exit gas composition was determined. The reaction studied was carbon dioxide reforming of methane using a Rhodium catalyst. Kinetic aspects of this heterogeneously catalyzed gasphase reaction have been studied separately to find
139
E. MEIROVITCHel al.
140
ALUMINA RING
T8
TIO
TI2
TS~J
~T6 SOLAR RADIATION
T~2 /~T7
p.GA.S IN
T, /
T9 GAS OUT I
T4 4 6 cm
,-catalyst . - inert alumina Fig. I. Schematic drawing of the receiver/reactor system. The positions labeled T I-TI0 and T I2 denote experimental temperature measurement stations. The solid dots denote the catalytic bed and the circles, the alumina-filled tube segment.
that the results can be interpreted with a HinshelwoodLangmuir-type expression for the chemical rate [ 5,6 ]. The parameters determining the rate constant and the equilibrium constants for CO2 and CH4 absorption were obtained with a best-fit procedure [ 7 ]. Good agreement between theory and experiment was obtained with a literature correlation for the effective heat transfer coefficient of the catalytic bed, used previously with steam reforming[ 8,9 ]. As expected, high net heat fluxes are encountered at the tube inlet, where the temperatures are the lowest. These decrease subsequently as one proceeds along the reactor, and are again augmented at the tube segment facing the aperture. Intricate details that reflect the phenomena occurring inside the tube at the various positions are discussed below. Solar energy is converted into chemical energy and sensible heat. We elucidate the relative contribution of these two components along the reactor. Thermal conversion alone occurs at the two end segments: at the inlet to the reactor, the temperatures are too low to sustain the chemical reaction, whereas at the far end, there is no catalyst. Chemical and thermal conversion occur throughout the catalytic bed. Except for a small segment near the reactor inlet, solar energy is being converted primarily into chemical enthalpy, with the sensible heat complementing, as needed, the chemical conversion. Of particular interest is the response of the tube interior to the augmented flux impinging on the U-tube bend that faces the aperture. The overall rate ofenergy conversion emulates the enhanced rate of energy supply, mainly by adjusting the fluid temperature gradient. Within the last 15% of the catalytic bed, the theory predicts slight local cooling of the fluid to preserve energy and mass balance. Detailed calculations were performed for the individual experiments. Agreement between theory and experiment is good. As a general trend, reactor-wall and fluid temperatures are slightly overestimated, and the overall methane conversion at exit, somewhat un-
derestimated. Also, the fit is better in the middle than at the two ends. We assign the latter observation to a heat leak at the poorly insulated receiver back wall that was not accounted for in the calculations. We illustrate and discuss in depth the effect of the various operating parameters on the performance of the system, and assess limitations imposed by heat transfer and by chemical thermodynamics. The theoretical background is given in Section 2. Section 3 is devoted to results and discussion. Our conclusions appear in Section 4. 2. THEORETICAL BACKGROUND Illumination
Illumination at the Schaefer Solar Furnace is brought about with a flat 96 m 2 Arco heliostat tracking the sun, that reflects the radiation onto a spherical concentrator made of 590 concave mirrors with a rim angle of 65°[10]. The overall power entering the receiver and the efficiency of the latter were measured experimentally with calorimetry[ 11]. Flux distribution within planes parallel to the focal plane were measured with radiometry [ 12 ]. We performed an accurate calculation of the flux distribution in the focal plane, properly summing up the elliptical images generated by the individual mirrors. The accurate normalized (to a mean aperture flux) flux along the major axes of the elliptical sun image are shown by the dashed lines in Fig. 2. We found that direct solar flux distribution inside the cavity is quite insensitive to the precise flux distribution in the aperture plane. As the zonal method (wide infra) assumes that all surface elements are isothermal and equienergetic, we need to approximate the monotonically decreasing functions in Fig. 2 by step functions that correspond to the space-meshing pattern. We took the single average step function in Fig. 2 to represent the flux distribution along the radial dimension of the circular aperture plane. The low sensitivity of the flux distribution on the enclosure surfaces to the intricate
Theoretical modeling of a directly heated solar-driven chemical reactor
1.5
at and problem dimensionality is determined by the space meshing. Radiative losses through the aperture are accounted for accurately.
~
The chemical reaction The reactions taking place within the reactor are Rhodium catalyzed carbon dioxide reforming of methane:
O/O° 1.0 05
0'00
141
I
0.2
I
0.4
I
0.6
CO2 + CH4 ~ 2H2 + 2CO
I
0.8
r/R
Fig. 2. Normalized (to the average flux within the aperture, (2o)direct solar flux in the opening as a function of fractional radial distance along the two axes of the elliptical image (---). The step function (solid line) is taken to be the mean of dashed-line functions. details of the flux pattern in the aperture plane justifies this approximation. Should the radiation emitted by the sun image be diffuse, the flux distribution inside the receiver would be easily calculated using the proper shape factors between surface elements within the aperture and on the inner walls of the cavity. Figure 3 indicates that diffuseness is, indeed, an acceptable approximation. The solid line represents our calculation of the flux inside the receiver as a function of the normalized depth dimension for an empty cylindrical receiver, assuming diffuse radiation. The dotted line has been calculated with an exact formulation, according to Wan[13 ]. The small discrepancy between the two curves and the low sensitivity to the precise focal flux distribution corroborate our approximation.
with an enthalpy of 232 kJ/mol at ambient temperature, and the water gas shift ( W G S H ) reaction: H2 + CO: ~ H20 -I- CO with an enthalpy of 37.5 kJ/mol at ambient temperature. It is assumed that the WGSH reaction is at equilibrium [ 2,7 ]. The kinetic model used was a Hinshelwood-Langmuir-type expression for the chemical rate. suggested by J. T. Richardson from the University of Houston[6]. The kinetic parameters and the equilibrium constants for CO2 and CH4 adsorption were obtained with a statistical best-fit-type analysis [ 7]. The expression for the chemical rate is given by rc ~ ABC,
( 1)
with m
A KcH, A Pco,. PCH, = kKco2 D2
B = ( 1-
D2
D-I
rCOIHztCH,
D--I
IZ'--I
ICO2,-~
)
C = (1 + Kco2 A Pco2 + K~H, Pcm)-:.
Radiative heat transport At the high temperatures prevailing in the receiver, heat is being transported primarily by radiation. The zonal method is used to perform the radiative heat transport balance[14]. Namely, all inner surfaces of the receiver, and the outer surface of the reactor, are discretized into sufficiently small surface elements or zones, and the balance of heat transport among the individual zones is performed. Radiosity from element i includes thermal emission and reflected solar and thermal radiation. Irradiation on a given zone i includes the direct solar flux and solar and thermal radiation reflected by zones other than i. All zones are considered to be diffuse, isothermal and uniform in terms ofthe net heat flux experienced. Space meshing was performed in part automatically, using the package of programs called VIEW[15]. The net heat flux on all surface elements is calculated by solving a set of linear equations by Gaussian elimination. The input parameters to such a calculation are geometric characteristics, thermo-optical properties of the various zones and either the temperatures of, or the fluxes on any given surface element. The program calculates the fluxes on those elements with preset temperatures, and the temperatures of elements with preset fluxes. The preset constants have to be guessed
Term A is usually called the "kinetic" term, B the "break" term, and C the "adsorption" term. The various constants are k = 50e -s42°°/Rr, [(g - moles of CH~ converted) x (g - catalyst )- ~(sec)- t ] K~o~ = 0.0301e 4s°°°/Rr,
[1/bar]
K~H, = 0.038e 4s°°°/Rr, I
Q
[l/bar] I
I
I
//~// Iii II iI iI
0
/
i
0.2
1
0.4 Z/L
I
0.6
I
0.8
Fig. 3. Direct solar flux (in arbitrary units) as a function of normalized cylinder depth calculated with our model ( - - ) and according to Wan [ 13 ] (---).
142
E. MEIROVlTCH
Enthalpies are given in j / m o l , k is the chemical rate constant, K~o~ and K~H, the equilibrium constants for CO2 and CH, adsorption in units of l/bar. K is the equilibrium constant of the carbon dioxide reforming of methane reaction in l / ( b a r ) 2. Pco:, Pcm, Pco and PH2 are the partial pressures of carbon dioxide, methane, carbon monoxide and hydrogen, respectively, in atm. Pore diffusion was accounted for by multiplying r~ by an effectiveness factor n[ 3]. Mass transport between fluid and catalyst was accounted for using a literature mass transport coefficient ra[8]. Hence, the effective rate will be given by r,-~
=
r~-~rl-I + r~ l .
(2)
The convectively transferred heat flux Q is given by: Q = h a ( T w - TF).
(3)
hs is the effective heat transfer coefficients, T w and T F the reactor wall and fluid temperatures, respectively, at a given position along the reactor. hs was calculated according to refs. [9a] and [9b]. An effective conductivity k,fr given by the expression ketr = kmol + kturbulem +
k..~olid"
is calculated, k¢fris a function of the physical properties of the fluid, the geometric characteristics ofthe reactor and the catalytic bed and the flow regime, hs is given by 1
hB = [0.0305ReO._,6:dO.,dO.5
L
0.4046dt]
+
(4)
j
in units o f W / m 2 ( K ) , with kerr in units o f W / m ( K ) and dp and dt in meters. The energy balance at the axial position z is dT
dx
Q = rh6p ~z + Hrhx d---=."
(5)
rh denotes the mass flow rate of the gas mixture, ?p the specific heat of the gas mixture, H the heat of the reforming reaction, thx the molar flow rate of methane, and z the axial dimension. It is implicitly assumed that there is no mixing in the axial direction and complete mixing in the radial direction (i.e., this is a one-didT dx mensional model). ~'z and d"~-are the temperature and conversion gradients, respectively. Since both 6p and H are only marginally temperature-dependent, the heat flux absorbed within the reactor and converted to sensible heat and chemical enthalpy is determined dT dx mainly by ~zz and ~zz ' respectively. A small correction term accounting for the WGS reaction was also included in eqn (5).
el
al.
The mass balance is given by ~ A I , , , = mxdx
(6)
with Meat denoting the mass ofthe catalytic bed element in grams.
Integrated solution Equation (3) asserts that the rate at which energy impinges on unit area of wall (i.e., the heat flux) equals the rate at which it is transmitted convectively into the catalytic bed at each axial position z. Equation (5) assesses that the heat flux at each axial tube position z emulates the rate at which energy is being converted into chemical enthalpy and sensible heat. The mass balance at each location z is given by eqn (6). All three equalities have to be satisfied simultaneously at each axial position. In practice, the problem is solved numerically. The tube is divided into volume elements oftength Az, sufficiently small for the equal. . d T AT dx Ax rues ~ = ~== and = ~ z to hold true with acceptable :
accuracy. Also, we associate with each control volume thermodynamic ( TF and Tw) and kinetic (re~) parameters calculated to be the arithmetic means of the respective inlet (to the segment) and outlet values. The following procedure is carried through until the heat flux at each control volume A= emulates both the rate of convective heat transfer and the rate of conversion within the reactor. The calculation is initiated by guessing at the reactor wall temperature profile and specifying the working conditions (i.e., inlet gas temperature, pressure, flow rate, and composition). The program is made of two components: the radiative heat transfer program, RADSOLVER. and the chemical program. CHEM. With the initial reactor wall temperature guess, RADSOLVER calculates the flux profile Q along the axial tube dimension =. The chemical program commences by solving eqns (5) and (6) for the first control volume at the tube inlet, where gas temperature, pressure, reaction rate, and methane conversion are known (the last two parameters are zero) and the solution consists of determining the constants listed above at the outlet from the control volume. Knowing Tr, eqn (3) is now solved for Tw, which is the "new" wall temperature associated with the first control volume, presumably more trustworthy than the initial guess. Parameters associated with the exit from control volume i are taken to be "inlet values" for control volume i + 1. Hence, the procedure described above is followed by gradually proceeding along the reactor, until a new wall temperature profile is obtained. This is now fed into the program RADSOLVER as an improved boundary condition, and a new improved flux profile is calculated. The process is followed through anew until convergence is achieved, i.e., until two consecutive temperature profiles are close within a preset tolerance.
Theoretical modeling of a directly heated solar-driven chemical reactor The criterion for convergence is set by defining a parameter S as:
143
sion--conversion system and to address questions related to optimal reactor design and working conditions, we performed careful calculations, to be described N below. S = [ ~ (Twt+t(i)) - Twl(i))z] °5 < TOE (7) The theory underlying the simulations was prei= I sented in Section 2. Using a computer program to imwhere the index l counts the iterations, i denotes the plement it, we calculated the various performances of control volume and N is the number of control volumes the receiver/reactor system. associated with the reactor. The tolerance TOL is set As mentioned previously, the power allowed to enat 5 ~ degrees C. ter the receiver was controlled by means of shutters that screened part of the concentrator surface. Per3. R E S U L T S AND D I S C U S S I O N forming precise ray-tracing calculations that would Both the receiver and the reactor were described in properly account for this mechanical screening, and detail in ref. 4. Below is a brief description pinpointing any other perennial and ephemeral inaccuracies for such a small, and hence, sensitive system, would have basic features. The receiver, shown in Fig. 1, was made of a white been a very tedious and rather impractical task. An alternative to precise calculation would have alumina cylinder (Zircar ALC), 1.6 cm O.D., 1.3 cm I.D., and 46 cm long, insulated with 7 cm of alumina been to use the results of calorimetric measurements felt and encased in a l-cm thick aluminum cylinder performed using the same receiver as estimates of the acting as housing. The aperature was 8.5 cm in di- power captured by the latter. It turned out, however, ameter, cut in a high temperature-resistant alumina that uncertainties related to mirror reflectivity and board. Three thermocouples were inserted in the alu- other factors led to a substantial overestimation of the mina cylinder to monitor the temperature of the cavity. power input to the receiver/reactor set-up. The schematic in Fig. 1 illustrates the U-shaped We therefore varied the total power into the receiver tubular reactor used in this work. The Inconel tube until best fit between theory and experiment was obwas filled with catalyst up to 10 cm beyond the bend tained. The criterion for the "best" simulation was (to avoid reversing the reaction upon cooling at the closest correspondence between all the experimentally far end of the tube) and the rest was filled with inert measured temperatures (associated with the receiver wall, the reactor wall and the fluid) and their calculated alumina. Experiments were performed over a wide range of counterparts, and between the measured and calculated process conditions. Thus, the inlet gas temperature exit composition. ranged from 100 ° to 450°C, the inlet pressure from 2 Calculations were carried out for a large number of to 3 bar and the methane flow rate from 200 to 700 experiments covering the range of working conditions standard liters per hour (Is~ h), compatible with linear of the 55 experiments performed altogether[4]. We flow rates on the order of 2.5 to 3.3 m/s. The inlet gas show in Table I results obtained with four experiments, composition was given by a carbon dioxide to methane with inlet gas temperatures of 100 ° to 200°C, inlet gas ratio of approximately 1.2. pressures of 2 to 3 atm and methane flow rates at the The power impinging on the concentrating mirror inlet of 300 to 650 ls/h. The various trends in the assembly (hence, the power entering the receiver) was spatial (albeit temperature) dependence of the thercontrolled by the opening of the doors ofthe enclosure modynamic and kinetic parameters as well as the erhousing the concentrator. rors, both sporadic and systematic, are typical of all Receiver-wall, reactor-wall, and fluid temperatures the experiments. We discuss these below. were measured at several locations, as shown in Fig. As mentioned previously, the total power entering 1. Also, the overall methane conversion in the product the receiver was varied until best fit between the exgas was determined. Receiver-wall and reactor-wall perimentally measured temperatures and exit comtemperatures as high as 1130 ° and 1030°C, respecpositions and their calculated counterparts was obtively, and product gas temperatures in the range of tained. Since fluid temperatures and gas compositions 735 ° to 1030°C were measured. The overall methane determine the power converted within the reactor into conversions (in units of moles of methane converted sensible heat and chemical enthalpy, the power into per moles of methane in feed) were found to be within the reactor can be viewed as a qualifier of the fit. Althe range of 50% to 99 mol%. though this parameter cannot be determined experiOrder of magnitude estimates based on these data mentally by a direct measurement, it can be obtained point to solar power levels on the order of 0.5 to 2.5 by performing the heat and mass balance (see eqns kwatts converted to chemical enthalpy and sensible ( 5 ) and (6)) over the reactor with A T = T ExIt - T TM, V CH4 heat, net fluxes impinging on the reactor wall of 30 to Az --- L, Ax = -~ Exit, and taking average values for 120 kw/m 2, gas-hourly-space velocities of 8,000 to Tw, Te, 6p, and H. The results of such calculations 37,000 h -t (or contact times of 0.45 to 0.1 sec), ef- are listed in Table 1 under the heading "expt'l power fective heat transfer coefficients of 350 to 1 I00 W / m z into reactor." The "calc'd" counterpart was obtained (°C) and Reynolds numbers of 110 to 420[4]. by solving the set of eqns (5) and (6) for each control In order to understand the intricate details of op- volume, and adding up the corresponding powers. The eration of this integrated energy absorption-transmis- quite reasonable agreement between the calculated and
E. MEIROVITCHel al.
144
Table 1. Experimental and calculated temperatures, overall methane conversions and powers into reactor for experiments 1--4 Temperature(C) T10 TI2
Power into reactor (kw)
T2
T3
T4
1"5 T6
1"7 T9
1 Calc'd Expt'l
538 504
638 696
648 740
260 591 141 630
704 687 967 799 825 793 944 744
733 695
50.2 46.3
1.09 1.10
2 Calc'd Expt'I
590 595
730 763
767 835
325 659 173 731
g07 789 943 878 849 888 897 1037 865 802
64.0 71.5
1.38 1.50
3 Calc'd Expt'l
533 493
671 702
682 750
273 222
757 736 927 857 812 779 814 946 808 756
50.1 46.2
1.44 1.52
4Calc'd Expt'l
591 552
735 740
765 805
363 663 245 714
824 842
62.0 61.2
1.85 1.85
601 646
T8
Overall methane conversion (tool %)
815 1011 942 897 881 957 820 771
See Fig. l for identification of positions along the reactor where temperatures were measured. the experimental values warrant our best-fit criterion. We find that the calculated efficiency of the receiver is 66 +- 2%, taking into account only radiative losses via the aperture. The discrepancy between calculated and experimental overall methane conversions is at most 10%, and the errors are sporadic. On the other hand, the disagreement in temperatures is, in some cases, on the order of 16%. More importantly, systematic trends in the latter are reflected in Table 1. Thus, the experimental temperatures at T 12 and T5 are conspicuously overestimated by the calculations, whereas the temperatures measured at T3, T7 and T8 are underestimated. This, we think, is a consequence of the previously mentioned heat losses through the back wall of the receiver due to poor insulation, in view of which the thermodynamic equilibrium state of the experimental set-up differs from the one predicted for a perfectly insulated receiver. We refer below to experiment 1, and discuss in detail these general results using Figs. 4 and 5, wherein both the experimental and the theoretical performances of
Zw; 800
,,,.x .... , , . x
.....
.. So
/
T(*C)
,,
•
,,.....
•
40 xCH4
/
I'
40C
the reactor are depicted pictorially, and Table 1, wherein measured and calculated receiver wall temperatures are given. The calculated receiver wall temperature is 40°C lower than the experimental one at position T8, by 49°C higher at Tl0, and by 56°C higher at Tl2. We assign the experimental decrease in temperature upon approaching the back wall of the receiver, versus the calculated increase in temperature, to heat losses through the insulating material at the back that have not been accounted for in the calculation. As shown shortly this will also affect the location T5 on the reactor wall, adjacent to the receiver back wall. Reactor wall and fluid temperature and methane conversion profiles as a function of fractional tube length are shown in Fig. 4. In general, all the parameters displayed in Fig. 4 increase from inlet to outlet; in particular, the shape ofthese curves is of a very singular nature, to be discussed in detail below. The flux profile is shown by the solid line in Fig. 5. The rate of energy supply per unit area is high--78 k w / m 2 - - a t the reactor inlet, where the reactor-wall temperature, hence thermal emission, is low. Subsequently, it decreases up to a fractional catalytic bed length of 0.3, levels off and at z / L ~. 0.43 (with L denoting the length of the catalytic bed), increases again, first moderately, and then at - . / L ~ 0.6, abruptly. Between z / L -~ 0.6 and 0.86 the flux is high,
(mole%)
..".....
80
,
i
20 60 '
~
,/-
20C
0
.,.......... , 0.5
1.0
1.43
40
,,
-
Z/I.
Fig. 4. Calculated wall temperature profile (---) fluid temperature profile ( - - ) and methane conversion profile (. • • ) as a function of fractional catalytic bed length. L denotes the length of the bed. The fluid temperature at inlet was 102°C, the pressure 2.9 arm, the carbon dioxide to methane ratio I. 18, and the methane flow rate 651 Is~h. The power into the receiver was 1.44 kw. The x symbols represent experimental measurements of reactor-wall temperatures and the solid dots, experimental fluid temperatures. The symbol * denotes the overall methane conversion. The abscissa is given in units of fractional reactor length.
,I
(~
X
/".
I
I
I
05
I0
1.43
Z/L
Fig. 5. Heat flux (--), effective rate containing term (--.) and fluid temperature gradient containing term ( . . . . . ) as a function of fractional catalytic bed length, in units of (kw/ m:). Input data as described in the captions of Fig. 4.
Theoretical modeling of a directly heated solar-driven chemical reactor though nonuniform, with peaks at z / L ~ 0.64 and 0.86. A precipitous decrease is observed at z/L -~ 0.86, and from then on the flux decreases monotonically to 10 kw/m 2 at the end of the reactor. The intricate details of the heat flux profile reflect the self-regulating nature of the system, i.e., the mutual adjustment between the energy absorption and conversion processes. To follow and comprehend how this comes about, we plot in Fig. 5 the two functions that determine the heat flux absorbed at each position along the reactor (see eqn (5)): the term containing the fluid temperature gradient ( . . . . . ) and the term containing the effective rate (---), both in units of kw/m "-. As indicated in the theoretical section, the net heat flux impinging on the reactor wall at each axial location, i.e., the rate of energy absorption, must emulate the rate of energy conversion into sensible heat and chemical enthalpy. This is borne out by the three curves depicted in Fig. 5. It is quite obvious that up to - / L 0.15, the flux and the fluid temperature gradient term coincide, whereas the conversion gradient term is practically null. From z / L ~ O.15 onwards, the flux d T,, ,dx,, is given by' the sum of the ~ term and the d-'-zterm. It is illuminating to correlate local irregularities in the flux curve with those in the other two curves. Of particular interest are the extrema in the various curves at z / L = 0.47, 0.64, 0.86 and 1.14 (beyond z/ L = 1 the tube is filled with pure alumina), that reflect the coordinated coupling between absorption and conversion. For example, the flux balance at position z / L = 0.47 can only be achieved with a local minimum in the temperature gradient compensating for a local maximum in the rate, implied by eqn (6). The steep increase in the flux at z / L = 0.64 is brought about by the sudden exposure of the tube to direct insolation. Throughout the bend (0.64 < z / L < 0.86), the flux stays high. The shape of the flux profile between z/L = 0.64 and 0.86 reflects the fact that the corners of the bend receive thermal radiation from the receiver walls in addition to direct insolation, whereas the mid-bend segment only experiences the latter. The tube interior responds with a concave rate profile and a basically convex temperature gradient curve. As noted above, the bend in the reactor ends at z/ L = 0.86, and beyond that position, the tube is no longer exposed to direct solar radiation. Hence, we expect that the net heat flux, the reactor wall and fluid temperatures will all decrease abruptly at z / L = 0.86. This is, indeed, borne out by the curves in Figs. 4 and 5. As the temperature dependence of the effective rate is weaker than that of the net heat flux, the heat balance between z / L = 0.86 and 1.0 can only be preserved with a dramatic reduction in the fluid temperature gradient that would diminish both the chemical rate and the conversion to sensible heat. Eventually, the fluid temperature gradient becomes negative (see z~ L = 0.86 and vicinity in Fig. 5), i.e., the heat balance can only be achieved if the gas will cool down locally. At z / L = 1.0 the abrupt drop of the rate to zero, as the catalytic bed ends at this position, is compensated
145
for by a dramatic increase in the temperature gradient to absorb all the heat available as sensil~le heat. Beyond this position, the temperature gradient term and flux curves coincide. Two comments are noteworthy. First, the state of affairs at z / L = 0.86 uniquely reflects the coordination between absorption and conversion. With the boundary conditions of this particular experiment, the efficiency of the chemical heat sink at this position exceeds the efficiency of heat supply. Consequently, the fluid temperature decreases locally until the overall rate of conversion to both chemical enthalpy and sensible heat emulates the rate of heat transfer into the tube. Our second comment relates to the local minimum in the temperature profiles at z / L = 1.0 (see Fig. 4), not quite borne out by the experiment. As mentioned previously, heat is being drained from the system through the poorly insulated back wall of the receiver. Accounting for this phenomenon in the calculation would mitigate the constraint of the fluid temperature having to decrease in order to reduce the overall rate of conversion, and heat balance would be established by removal of excess heat by conduction through the insulator, rather than local cooling of the fluid. Obviously, agreement between theory, and experiment would be improved thereby. In summary., in up to one-fifth of the tube length. energy is being converted primarily into sensible heat. Between z / L = 0.43 and 1.0, solar energy is converted chiefly into chemical enthalpy. Beyond z/L = 1.0. where the catalyst bed ends, we again encounter conversion to sensible heat. For 0.3 < z / L < 0.43, about half of the solar radiation is converted into chemical enthalpy and the other half, into sensible heat. The self-regulatory characteristics of the receiver/ reactor assembly are also reflected in the reactor-wall and fluid temperature profiles (Fig. 4). The power entering the enclosure is redistributed quite uniformly by thermal emission and by reflected solar and thermal radiation. Since the lowest temperatures in the system are encountered at the reactor inlet, the net heat flux at that site will, due to reduced thermal emission, he high. The heat sink will respond with an increase in temperature of both the walls of the reactor and the fluid inside, as shown in Fig. 4. The rate at which these temperatures are enhanced is high, and heat is being absorbed efficiently as sensible heat. As one proceeds along the reactor, the net flux on its walls decreases (since the temperatures increase), entailing a decrease in the temperature gradients (see Fig. 5 ). Also, reactor wall and fluid are cooled by the onset of the endothermic chemical reaction at z / L = 0.15. At - / L = 0.43, chemical conversion assumes the main role as heat sink, with the sensible heat performing fine adjustments to establish the heat balance. Both temperature profiles in Fig. 4 increase monotonically. At z~ L = 0.64, the net flux (see Fig. 5) on the reactor (and hence the wall temperature) increases due to direct exposure of the latter to direct solar radiation. The fluid temperature in Fig. 4 follows suit, to both increase the effective rate and to augment heat conversion to sensible heat. The similarity between the heat flux (Fig.
146
E. MEIROVITCH et al.
5 ) and the fluid temperature (Fig. 4) profiles between
z/L = 0.64 and 0.86 (the portion at which the tube bends away from the opening) reflect just that. Beyond z/L = 0.86 the flux is back to normal, and both reactor wall the fluid temperature decrease. As mentioned above, in this region of the reactor the primary effect is conversion of solar into chemical energy. Yet, the effective chemical rate, due to its weak temperature dependence, does not cope with the relatively steep increase in heat flux, and cooling of the gas has to compensate for this: both reactor-wall and fluid temperature profiles in Fig. 4 bend over at z/L = 0.86, disrupting the monotonic increase of temperatures along the reactor. At z/L = !.0 the slope in the temperature profile changes sign back to normal (i.e., it becomes positive again) as the chemical reaction stops at the end of the catalytic bed. The increase in reactor-wall and fluid temperatures is quite dramatic, as the net flux is still quite high and the system has to switch instantaneously from a chemical heat sink to one based on conversion to sensible heat. Beyond z/L = 1.14, the fluid temperature tends asymptotically toward the reactor wall temperature. Radiative heat transport complies with this reality and the net flux impinging on the reactor tends to zero. The solid dots in Fig. 4 denote experimental measurements of the reactor-wall and fluid temperatures and of the exit methane conversion. The fit between theory and experiments at the fluid portions T2 and T3 and at the reactor wall positions T6 and T7 is good (see Fig. l for identifying these positions). The larger discrepancies at the reactor wall position T5, and at the fluid position T4, were discussed previously and assigned to heat losses at the back not accounted for in the calculations. The shape of the calculated methane conversion profile in Fig. 4 complies with the z~ L dependence of the fluid temperature. The solid dot at z/L = 1.0 denotes the experimental measurement. The calculated equilibrium methane conversions were at all temperatures higher than the respective calculated conversion values. Our simulations reproduce the experimental results with relatively high accuracy. This, as well as the accountability of all the details in the calculated heat
flux, temperature, methane conversion and rate profiles lend confidence to our model in general, and to the kinetic and heat transfer equations, in particular. The effect of flow rate, fluid temperature and pressure at the inlet to the reactor on its performances was investigated in depth. We performed extensive model simulations for two power levels--0.75 and 1.65 k w m received by the cavity, corresponding to 0.48-0.50 and 1.0- I. 1 kw, respectively, absorbed by the reactor. The methane flow rate varied from 250 to 650 Is~h, the inlet fluid temperature from 150" to 250°C, and the gas pressure at inlet from 2 to 3 bar. The feed composition was kept constant, with C O 2 / C H 4 -- 1.23. The results of 24 typical calculations are shown in Tables 2 and 3. For each calculation, we present the corresponding input parameters and the computed overall methane conversion, equilibrium conversion, fluid temperature and pressure at the exit. Table 2 illustrates the effect of the operating conditions--inlet temperature, pressure and flow r a t e - on the overall and equilibrium methane conversion and fluid temperature at exit. If the methane flow rate at inlet is set to be 650 lJh, 0.75 kwatts of solar power will be absorbed, generating overall methane conversions of 10%-13% and a temperature increase of 260 °3600C within the reactor. With 450 ls/h of methane passing through the reactor, the corresponding parameters are an approximately 20% overall methane conversion and a temperature increase of 300°-400°C and with 250 Is~h, the overall conversion is of the order of 35%--42% and the temperature increase is 3600--460°C. Pressure has a minor effect on the exit temperatures and a moderate effect on the overall and equilibrium methane conversions at low flow rates. At higher flows, the overall methane conversions and the fluid temperatures at exit are practically unaffected by increasing the pressure from 2 to 3 bars, whereas equilibrium conversions are affected slightly. With lower pressures and flow rates, equilibrium is attained at fractional reactor lengths less than unity. The last column in Table 2 specifics the location at which this happens. Beyond these positions, the operation of the chemical heat sink is "controlled by thermodynamics," i.e., the conversion of reactants into
Table 2. Input (columns 1-3) and output (columns 4-8) parameters for model calculations with 0.75 kW entering the receiver Exit Inlet Methane Fluid Ove~dl Fluid Exit Fractional equilibrium )ressure ~ow r~te temperature methane temperature )ressure robe length conversion , at inlet at inlet conversion at exit at which (bar) equilibrium (poles converted~ moles at equilibrium'~ (bar) (Is/h) (*C) \ moles in feed / ('C) is attained moles in f ¢ ~ ] 39.0 611.0 2.0 ' 0.51 2.0 250 150 39.0 34.0 3.0 250 150 34.0 611.0 3.0 0.45 19.8 20.0 535.0 1.9 '"2.0 450 150 18.0 18.0 3.0 450 150 545.0 2.9 0.84 10.0 15.0 511.0 1.8 '2.0 650 150 10.0 12.0 510.0 2.9 3.0 650 150 42.0 42.0 622.0 1.9 0.57 2.0 250 250 35.0 35.0 3.0 250 250 617.0 3.0 0.35 22.0 22.0 2.0 450 250 546.0 1.9 0.88 20.0 20.0 556.5 2.9 0.72 3.0 450 250 16.0 2.0 650 250 13.1 515.5 1.8 13.0 517.0 2.8 0.94 3.0 650 250 13.0 J
c: Calculation C; d: Calculation D.
Theoretical modeling of a directly heated solar-driven chemical reactor
147
Table 3. Input (columns 1-3) and output (columns 4-8) parameters for model calculations with 1.65 kw entering the receiver Overall Exit Inlet ~ethaze Fluid Fluid Exit Fractional equilibrium >ressure Ilow rate ~mperature methaae temperature )ressure tube length conversion at inlet at inlet conversion at exit at which (bar) equilibrium moles converted~ moles at,equilibrium'~ j b a r ) !is/h) ' (°Cl (*c) *s attained moles in feed ) mole~in feed ] e 2.0 250 150 • 3.0 250 150 54.5 57.0 673.5 1.8 2.0 450 150 694.0 2.9 0.80 55.0 55.0 a 3.0 450 150 34.0 42.0 621.0 1.7 2.0 650 150 34.9 35.0 615.5 2.8 b 3.0 650 150 e 2.0 250 250 e 3.0 250 250 57.0 60.0 684.0 1.8 2.0 450 250 695.5 2.9 55.0 55.0 3.0 450 250 0.84 37.0 45.0 630.0 1.7 2.0 650 250 38.0 38.0 625.5 2.8 3.0 650 250 a: Calculation A; b: Calculation B; e: Energy and mass balance cannot be performed at the far end of the reactor because of heat transfer limitations (see text).
products is monitored by the equilibrium constant. With 1.65 kwatts entering the receiver, proper functioning of the self-regulating system is, under certain circumstances, impeded by heat-transfer limitations. The system operates properly if, and only if, the rates of absorption of concentrated sunlight, the rate of heat transfer through the reactor wall and the rate of conversion into chemical enthalpy and sensible heat taking place within the reactor, are equal. With high powers and low flow rates, the fluid temperatures next to the outlet are high and may, eventually, approach the wall temperatures asymptotically. (Note that the latter cannot exceed a certain value above which the reactor would act as a heat emitter rather than a heat sink.) The temperature gradient between wall and fluid would tend to zero, and heat would no longer be transferred into the reactor. U n d e r these circumstances, the process is heat-transfer limited and susceptible to local overheating. Calculations labeled e in Table 3 that could not be completed for the reason mentioned above, pertain to this category. The situations depicted in Table 3 are, in actual fact, close to the stability limits of the self-regulatory system; hence, sensitivity is relatively low. The effect of increasing the pressure from 2 to 3 bar is minor. As expected, overall and equilibrium methane conversions and exit fluid temperatures are reduced by increasing the flow rate. The extent of these changes is, however, smaller than in the previous case, where 0.75 kw entered the receiver. Following this general discussion, we present below typical flux, reactor-wall temperature, fluid temperature, and rate profiles obtained with the model calculations A - D , labeled a - d in Tables 2 and 3. An inlet fluid temperature of 150°C, an inlet pressure of 3 arm, methane flow rates of 450 (A and C) and 650 (B and D) l,/h, and total powers of 0.75 (C and D) and 1.65 kw (A and B) were used. Figure 6 indicates that the flux profile along the reactor is determined primarily by the total power entering the receiver. Although the flux intensity at each position along the reactor length is reduced by decreasing the total power, the shape of the curve is preserved, being determined by geometric
characteristics of the receiver/reactor system. On the other hand, Fig. 7 shows that the conversion profile corresponding to a given power level depends to a large extent on the flow rate. The shapes of the curves in Fig. 7 are, in general, determined by the power level, whereas their amplitude and spread along the - / L axis are fixed by the flow rate. In particular, the m a x i m u m in the rate profile is shifted toward the exit by both increasing the flow rate, and lowering the power (the dip observed with A and B is due to the idiosyncratic geometry of the receiver/reactor). It is quite obvious that with A, the chemical heat sink operates most effectively, whereas with D, least efficiently. To illustrate heat transfer limitations, we present wall (---) and fluid ( - - ) temperatures and fluid temperature gradient profiles in Figs. 8 and 9, respectively. With both C and A, fluid and wall temperatures tend to merge close to the exit, where the tube is filled with pure alumina. In fact, the fluid temperature slightly exceeds the wall temperature at the exit of A, with the temperature gradient becoming negative (see curve A in Fig. 9 close to - / L = 1.43) and the flux becoming
6oF Ww
B
',
ol
,
,
0
05
1.0
1.43
Z/L Fig. 6. Heat fluxes as a function of fractional catalytic bed
length. An inlet fluid temperature of 150°C, an inlet pressure of 3 atm and a CO2/CH4 ratio of 1.23 were used for all the calculations (A-D). Additional input data w e r e : A: P ' = 1.65 kw, V(CH4) = 450 l,/h; B: P' -- !.65 kw, V(CI-h) = 650 l,/ h; C: P' = 0.75 kw, V(CH4) = 450 l,/h; D: P' = 0.75 kw, V(CH4) = 650 l,/h; with P' denoting the total power into the receiver and V(CH4) the inlet methane flow rate.
148
E. MEIROVITCH el al.
I
I P\ 2C"
AX
a-5
o.~
°°6
AT/AtI0i ~\\~'~. 0.5
,.o
O---
,.,~
Z/L Fig. 7. Effective rate of methane conversion as a function of fractional catalytic bed length for calculations A-D. Input parameters as described in the captions of Fig. 6. The ordinate units are moles ofCH4 converted/(cm-h). negative (see curve A in Fig. 6 close to z / L = 1.43). (Note that negative fluid temperature gradients are also encountered close to z / L = 0.86 in Fig. 9.) Yet, at that position, heat can, on the whole, still be transferred through the tube walls as the chemical heat sink is operative, with a large and positive conversion gradient (see curve A in Fig. 7).
Experimental results obtained with a directly illuminated tubular reactor mounted in a solar furnace were analyzed using a theoretical model that accounts for the radiative heat transport within the receiver and the endothermic chemical reaction occurring within the reactor. We find that throughout the reactor, the rates of net heat flux into the reactor emulate the rate at which energy is being converted, mainly into chemical enthalpy and partly into sensible heat. Under certain circumstances, depicted in the text, the process is controlled by heat transfer or by chemical thermodynamics. Convective heat losses through the insulating material at back are prone to reduce temperatures at the inlet to the reactor, and enhance them at the far end of the catalytic bed. The self-regulatory characteristics of this receiver/ reactor system are borne out by these results. We believe that the theoretical model set forth in
I
6OOl f~ss/s"'" "'"'~'- _. T(*C) / "" 4 0 0 ~ 0
0,5
0.5 I
1.0 I
Z/L
1.4:5 I
Fig. 9. Huid temperature gradients as a function of fractional catalytic bed length for calculations A and B. See captions of Fig. 6 for input parameters. The ordinate units are °C/cm. this study can be used with confidence to analyze experimental results and design solar-driven reactors.
Acknowledgments--The authors acknowledge helpful discussions with, and valuable comments by, Mrs. Rachel Levitan and Mrs. Hadassa Rosin. A.S. acknowledges financial support from the Ministry of Absorption of the State of Israel.
4. CONCLUSION
I
-IC
1.0
I.,,3
Z/L Fig. 8. Reactor wall temperature (---) and fluid temperature ( - - ) profiles versus the fractional catalytic bed length for calculations A and C. See captions of Fig. 6 for input parameters.
REFERENCES 1. K. Spindler, U. Gross, and E. Hahne, Int. J. Solar Energy I, 197 (1982). 2. D. Fraenkel, R. Levitan, and M. Levy, Int. J. Hydrogen Energy Ii, 207 (1986). 3. S. A. Paripatyadar and J. T. Richardson. Solar Energy 41,475 (1988). 4. R. Levitan, H. Rosin, and M. Levy, Solar Energy 42, 267 (1989). 5. J. T. Richardson, S. A. Paripatyadar, and J. C. Shen, AI ChEJ. 34, 743 (1989). 6. J.T. Richardson, private communication. 7. G. Adusei, R. Levitan, H. Rosin, E. Meirovitch, and M. Levy, Kinetics of CO2 reforming of methane and the reverse methanation reaction on a rhodium catalyst, The Weizmann Institute of Science, technical report (1990). 8. F. Kreith and M. S. Bohn, eds., Principlesof heat transfer. Harper and Row, New York (1986). 9. (a) A. D. DeWash and G. F. Froment, Chem. Eng. Sci. 27, 507 (1972). (b) Heat transfer characteristics of porous rocks, D. Kunii, J. M. Smith, (Northwestern Univ., Evanston, IL), A. I. Ch.E. Journal 6, 71-78 (1960). 10. I. Levy and M. Epstein, The solar furnace at the Weizmann Institute, The Weizmann Institute of Science technical report (1986). I I. H. Rosin, !. Levy, and R. Levitan, The solar furnace performance as measured by calorimetry, The Weizmann Institute of Science technical report (1986). 12. I. Levy, M. Epstein, and M. Meri, Solar furnace concentration ratio mapping by radiometric measurements, The Weizmann Institute of Science technical report ( 1987 ). 13. C. C. Wan, Solar thermal power programs, technical report SC5E-016, Ford Aerospace and Communications Corporation, Newport, CA (1980). 14. M. Abrams, SAND 81-8248, Thermal Sciences Division, Sandia National Laboratories, Livermore, CA ( 1981 ). 15. A. F. Emery, VIEW, version 5.5.3. Department of Mechanical Engineering, The University of Washington, Seattle, WA (1986). 16. R. B. Diver. D. E. E. Carlson, F. J. Macdonald, and E. D. Fletcher, J. Solar Energy Eng. 105, 291 (1983).
0038--092X/90 $3.00 + .00 Copyright ~ 1990 Pergamon Press pie
Printed in the U.S.A.
THEORETICAL MODELING OF A DIRECTLY HEATED SOLAR-DRIVEN CHEMICAL REACTOR EVA MEIROVITCH,AKIBASEGAL, and MOSHE LEVY Center for Energy Research, The Weizmann Institute of Science, Rehovot 76100, Israel Abstract--A theoretical formulation for calculating the performances of a solar-driven catalytic chemical reactor was developed. It accounts for the spatial distribution of the deposition of primary energy within the receiver, the heat transfer into the catalytic bed and the thermochemical endothermic reaction, chemical composition and flow distribution within the reactor. The theory set forth was applied to analyze results obtained in a solar furnace with a directly heated U-shapedtubular reactor, wherein catalyticcarbon dioxide reforming of methane occurred. We find that the receiver/reactor assemblyacts as a self-regulatingsystem. Beyond a fractional catalytic bed length of 0.14, solar energy can be converted primarily into chemical enthalpy. The fluid temperature gradient monitors the heat balanceby adjustingthe overallrate of conversion to the rate at which energy is being transferred through the reactor walls. Under certain circumstances, the process may be heat-transfer limited or controlled by chemical thermodynamics. A good fit between theory and experiment and accountability of all the intricate details in the various calculated performances of the receiver/reactor system support the theoretical model set forth in this study. We offer it as a tool for simulating future experimental results and for designing solar-driven reactors.
l. INTRODUCTION
Conventional solar receivers are enclosures built of refractory walls and absorbing devices acting as heat sinks[l]. The latter are often metal tubes, through which a "working fluid" is being passed. Usually one uses air, steam, molten salt, a liquid metal, etc., that enters cold and exits warm, having absorbed convectively the heat flux transmitted through the tube walls. Radiative solar energy has thereby been converted into sensible heat. Severe problems of storage and transportation are encountered with this method. For example, it is extremely inefficient, and at times impractical, to transport and store steam at 300°-400°C. A recently conceived variant of this concept is conversion of solar energy into chemical enthalpy, rather than sensible heat[2 ]. This can be brought about with endothermic gas-phase reactions. Solar heat is now stored in the chemical bond, the energy-rich products are gases that can be cooled to room temperature, pressurized and conveniently transported and stored. The energy can be recovered at will by performing the reverse reactions, and the products of the latter can, eventually, be recycled, with the process carried out in a closed loop. Alternatively, the solar driven reactors can be used to obtain products of industrial interest. Experimental work within this field has been performed so far in a few laboratories, using primarily sun-simulators[ 3 ]. Among the first studies using solar radiation were those performed at the Schaefer Solar Furnace of the Weizmann Institute [ 4]. Hence, the field being at its outset, many issues are still exploratory in nature, referring to configuration, limitations, and stability. For example, one could expose the reactors to direct illumination. Alternatively, one could think of integrated heating, with a screening fluid in between the site at which concentrated sunlight is being ab-
sorbed, and the site at which the process of conversion into sensible heat and chemical enthalpy unfolds, or of completely decoupling the sites at which energy is being absorbed and converted. Direct exposure to sunlight is necessarily of higher efficiency, due to reduced heat losses. On the other hand, directly illuminated surfaces are prone to disastrous local overheating. It is of paramount importance to have a proper understanding of the physical processes involved. Radiation and chemistry are coupled, interfaced by the convective transfer of heat through the reactor walls. Radiative heat transport and catalyzed gas-phase reactions have been treated so far independently. With thermochemical conversion of solar energy they become integrated into a self-regulatory system. Intricate details of the physical phenomena involved, limits of stability, constraints, etc. of the latter are to be investigated and comprehended in depth, in order to interpret experimental results obtained so far and design assemblies to be used in the future. These are the topics we focus on in this study. From a theoretical standpoint, we present below a comprehensive formulation, treating radiation and chemistry in an integrated form and using it to anal x~e experimental results obtained at the Schaefer Solar Furnace of the Weizmann Institute. The experiment is described in a separate publication[4]. We summarize below basic aspects essential to the forthcoming presentation. A U-shaped tubular reactor, filled up to 70% of its length with a catalytic bed, was placed horizontally in a cylindrical receiver with the bend facing the cavity aperture (see Fig. 1). Receiver wall, reactor wall and fluid temperatures were measured at the positions denoted in the figure, and the exit gas composition was determined. The reaction studied was carbon dioxide reforming of methane using a Rhodium catalyst. Kinetic aspects of this heterogeneously catalyzed gasphase reaction have been studied separately to find
139
E. MEIROVITCHel al.
140
ALUMINA RING
T8
TIO
TI2
TS~J
~T6 SOLAR RADIATION
T~2 /~T7
p.GA.S IN
T, /
T9 GAS OUT I
T4 4 6 cm
,-catalyst . - inert alumina Fig. I. Schematic drawing of the receiver/reactor system. The positions labeled T I-TI0 and T I2 denote experimental temperature measurement stations. The solid dots denote the catalytic bed and the circles, the alumina-filled tube segment.
that the results can be interpreted with a HinshelwoodLangmuir-type expression for the chemical rate [ 5,6 ]. The parameters determining the rate constant and the equilibrium constants for CO2 and CH4 absorption were obtained with a best-fit procedure [ 7 ]. Good agreement between theory and experiment was obtained with a literature correlation for the effective heat transfer coefficient of the catalytic bed, used previously with steam reforming[ 8,9 ]. As expected, high net heat fluxes are encountered at the tube inlet, where the temperatures are the lowest. These decrease subsequently as one proceeds along the reactor, and are again augmented at the tube segment facing the aperture. Intricate details that reflect the phenomena occurring inside the tube at the various positions are discussed below. Solar energy is converted into chemical energy and sensible heat. We elucidate the relative contribution of these two components along the reactor. Thermal conversion alone occurs at the two end segments: at the inlet to the reactor, the temperatures are too low to sustain the chemical reaction, whereas at the far end, there is no catalyst. Chemical and thermal conversion occur throughout the catalytic bed. Except for a small segment near the reactor inlet, solar energy is being converted primarily into chemical enthalpy, with the sensible heat complementing, as needed, the chemical conversion. Of particular interest is the response of the tube interior to the augmented flux impinging on the U-tube bend that faces the aperture. The overall rate ofenergy conversion emulates the enhanced rate of energy supply, mainly by adjusting the fluid temperature gradient. Within the last 15% of the catalytic bed, the theory predicts slight local cooling of the fluid to preserve energy and mass balance. Detailed calculations were performed for the individual experiments. Agreement between theory and experiment is good. As a general trend, reactor-wall and fluid temperatures are slightly overestimated, and the overall methane conversion at exit, somewhat un-
derestimated. Also, the fit is better in the middle than at the two ends. We assign the latter observation to a heat leak at the poorly insulated receiver back wall that was not accounted for in the calculations. We illustrate and discuss in depth the effect of the various operating parameters on the performance of the system, and assess limitations imposed by heat transfer and by chemical thermodynamics. The theoretical background is given in Section 2. Section 3 is devoted to results and discussion. Our conclusions appear in Section 4. 2. THEORETICAL BACKGROUND Illumination
Illumination at the Schaefer Solar Furnace is brought about with a flat 96 m 2 Arco heliostat tracking the sun, that reflects the radiation onto a spherical concentrator made of 590 concave mirrors with a rim angle of 65°[10]. The overall power entering the receiver and the efficiency of the latter were measured experimentally with calorimetry[ 11]. Flux distribution within planes parallel to the focal plane were measured with radiometry [ 12 ]. We performed an accurate calculation of the flux distribution in the focal plane, properly summing up the elliptical images generated by the individual mirrors. The accurate normalized (to a mean aperture flux) flux along the major axes of the elliptical sun image are shown by the dashed lines in Fig. 2. We found that direct solar flux distribution inside the cavity is quite insensitive to the precise flux distribution in the aperture plane. As the zonal method (wide infra) assumes that all surface elements are isothermal and equienergetic, we need to approximate the monotonically decreasing functions in Fig. 2 by step functions that correspond to the space-meshing pattern. We took the single average step function in Fig. 2 to represent the flux distribution along the radial dimension of the circular aperture plane. The low sensitivity of the flux distribution on the enclosure surfaces to the intricate
Theoretical modeling of a directly heated solar-driven chemical reactor
1.5
at and problem dimensionality is determined by the space meshing. Radiative losses through the aperture are accounted for accurately.
~
The chemical reaction The reactions taking place within the reactor are Rhodium catalyzed carbon dioxide reforming of methane:
O/O° 1.0 05
0'00
141
I
0.2
I
0.4
I
0.6
CO2 + CH4 ~ 2H2 + 2CO
I
0.8
r/R
Fig. 2. Normalized (to the average flux within the aperture, (2o)direct solar flux in the opening as a function of fractional radial distance along the two axes of the elliptical image (---). The step function (solid line) is taken to be the mean of dashed-line functions. details of the flux pattern in the aperture plane justifies this approximation. Should the radiation emitted by the sun image be diffuse, the flux distribution inside the receiver would be easily calculated using the proper shape factors between surface elements within the aperture and on the inner walls of the cavity. Figure 3 indicates that diffuseness is, indeed, an acceptable approximation. The solid line represents our calculation of the flux inside the receiver as a function of the normalized depth dimension for an empty cylindrical receiver, assuming diffuse radiation. The dotted line has been calculated with an exact formulation, according to Wan[13 ]. The small discrepancy between the two curves and the low sensitivity to the precise focal flux distribution corroborate our approximation.
with an enthalpy of 232 kJ/mol at ambient temperature, and the water gas shift ( W G S H ) reaction: H2 + CO: ~ H20 -I- CO with an enthalpy of 37.5 kJ/mol at ambient temperature. It is assumed that the WGSH reaction is at equilibrium [ 2,7 ]. The kinetic model used was a Hinshelwood-Langmuir-type expression for the chemical rate. suggested by J. T. Richardson from the University of Houston[6]. The kinetic parameters and the equilibrium constants for CO2 and CH4 adsorption were obtained with a statistical best-fit-type analysis [ 7]. The expression for the chemical rate is given by rc ~ ABC,
( 1)
with m
A KcH, A Pco,. PCH, = kKco2 D2
B = ( 1-
D2
D-I
rCOIHztCH,
D--I
IZ'--I
ICO2,-~
)
C = (1 + Kco2 A Pco2 + K~H, Pcm)-:.
Radiative heat transport At the high temperatures prevailing in the receiver, heat is being transported primarily by radiation. The zonal method is used to perform the radiative heat transport balance[14]. Namely, all inner surfaces of the receiver, and the outer surface of the reactor, are discretized into sufficiently small surface elements or zones, and the balance of heat transport among the individual zones is performed. Radiosity from element i includes thermal emission and reflected solar and thermal radiation. Irradiation on a given zone i includes the direct solar flux and solar and thermal radiation reflected by zones other than i. All zones are considered to be diffuse, isothermal and uniform in terms ofthe net heat flux experienced. Space meshing was performed in part automatically, using the package of programs called VIEW[15]. The net heat flux on all surface elements is calculated by solving a set of linear equations by Gaussian elimination. The input parameters to such a calculation are geometric characteristics, thermo-optical properties of the various zones and either the temperatures of, or the fluxes on any given surface element. The program calculates the fluxes on those elements with preset temperatures, and the temperatures of elements with preset fluxes. The preset constants have to be guessed
Term A is usually called the "kinetic" term, B the "break" term, and C the "adsorption" term. The various constants are k = 50e -s42°°/Rr, [(g - moles of CH~ converted) x (g - catalyst )- ~(sec)- t ] K~o~ = 0.0301e 4s°°°/Rr,
[1/bar]
K~H, = 0.038e 4s°°°/Rr, I
Q
[l/bar] I
I
I
//~// Iii II iI iI
0
/
i
0.2
1
0.4 Z/L
I
0.6
I
0.8
Fig. 3. Direct solar flux (in arbitrary units) as a function of normalized cylinder depth calculated with our model ( - - ) and according to Wan [ 13 ] (---).
142
E. MEIROVlTCH
Enthalpies are given in j / m o l , k is the chemical rate constant, K~o~ and K~H, the equilibrium constants for CO2 and CH, adsorption in units of l/bar. K is the equilibrium constant of the carbon dioxide reforming of methane reaction in l / ( b a r ) 2. Pco:, Pcm, Pco and PH2 are the partial pressures of carbon dioxide, methane, carbon monoxide and hydrogen, respectively, in atm. Pore diffusion was accounted for by multiplying r~ by an effectiveness factor n[ 3]. Mass transport between fluid and catalyst was accounted for using a literature mass transport coefficient ra[8]. Hence, the effective rate will be given by r,-~
=
r~-~rl-I + r~ l .
(2)
The convectively transferred heat flux Q is given by: Q = h a ( T w - TF).
(3)
hs is the effective heat transfer coefficients, T w and T F the reactor wall and fluid temperatures, respectively, at a given position along the reactor. hs was calculated according to refs. [9a] and [9b]. An effective conductivity k,fr given by the expression ketr = kmol + kturbulem +
k..~olid"
is calculated, k¢fris a function of the physical properties of the fluid, the geometric characteristics ofthe reactor and the catalytic bed and the flow regime, hs is given by 1
hB = [0.0305ReO._,6:dO.,dO.5
L
0.4046dt]
+
(4)
j
in units o f W / m 2 ( K ) , with kerr in units o f W / m ( K ) and dp and dt in meters. The energy balance at the axial position z is dT
dx
Q = rh6p ~z + Hrhx d---=."
(5)
rh denotes the mass flow rate of the gas mixture, ?p the specific heat of the gas mixture, H the heat of the reforming reaction, thx the molar flow rate of methane, and z the axial dimension. It is implicitly assumed that there is no mixing in the axial direction and complete mixing in the radial direction (i.e., this is a one-didT dx mensional model). ~'z and d"~-are the temperature and conversion gradients, respectively. Since both 6p and H are only marginally temperature-dependent, the heat flux absorbed within the reactor and converted to sensible heat and chemical enthalpy is determined dT dx mainly by ~zz and ~zz ' respectively. A small correction term accounting for the WGS reaction was also included in eqn (5).
el
al.
The mass balance is given by ~ A I , , , = mxdx
(6)
with Meat denoting the mass ofthe catalytic bed element in grams.
Integrated solution Equation (3) asserts that the rate at which energy impinges on unit area of wall (i.e., the heat flux) equals the rate at which it is transmitted convectively into the catalytic bed at each axial position z. Equation (5) assesses that the heat flux at each axial tube position z emulates the rate at which energy is being converted into chemical enthalpy and sensible heat. The mass balance at each location z is given by eqn (6). All three equalities have to be satisfied simultaneously at each axial position. In practice, the problem is solved numerically. The tube is divided into volume elements oftength Az, sufficiently small for the equal. . d T AT dx Ax rues ~ = ~== and = ~ z to hold true with acceptable :
accuracy. Also, we associate with each control volume thermodynamic ( TF and Tw) and kinetic (re~) parameters calculated to be the arithmetic means of the respective inlet (to the segment) and outlet values. The following procedure is carried through until the heat flux at each control volume A= emulates both the rate of convective heat transfer and the rate of conversion within the reactor. The calculation is initiated by guessing at the reactor wall temperature profile and specifying the working conditions (i.e., inlet gas temperature, pressure, flow rate, and composition). The program is made of two components: the radiative heat transfer program, RADSOLVER. and the chemical program. CHEM. With the initial reactor wall temperature guess, RADSOLVER calculates the flux profile Q along the axial tube dimension =. The chemical program commences by solving eqns (5) and (6) for the first control volume at the tube inlet, where gas temperature, pressure, reaction rate, and methane conversion are known (the last two parameters are zero) and the solution consists of determining the constants listed above at the outlet from the control volume. Knowing Tr, eqn (3) is now solved for Tw, which is the "new" wall temperature associated with the first control volume, presumably more trustworthy than the initial guess. Parameters associated with the exit from control volume i are taken to be "inlet values" for control volume i + 1. Hence, the procedure described above is followed by gradually proceeding along the reactor, until a new wall temperature profile is obtained. This is now fed into the program RADSOLVER as an improved boundary condition, and a new improved flux profile is calculated. The process is followed through anew until convergence is achieved, i.e., until two consecutive temperature profiles are close within a preset tolerance.
Theoretical modeling of a directly heated solar-driven chemical reactor The criterion for convergence is set by defining a parameter S as:
143
sion--conversion system and to address questions related to optimal reactor design and working conditions, we performed careful calculations, to be described N below. S = [ ~ (Twt+t(i)) - Twl(i))z] °5 < TOE (7) The theory underlying the simulations was prei= I sented in Section 2. Using a computer program to imwhere the index l counts the iterations, i denotes the plement it, we calculated the various performances of control volume and N is the number of control volumes the receiver/reactor system. associated with the reactor. The tolerance TOL is set As mentioned previously, the power allowed to enat 5 ~ degrees C. ter the receiver was controlled by means of shutters that screened part of the concentrator surface. Per3. R E S U L T S AND D I S C U S S I O N forming precise ray-tracing calculations that would Both the receiver and the reactor were described in properly account for this mechanical screening, and detail in ref. 4. Below is a brief description pinpointing any other perennial and ephemeral inaccuracies for such a small, and hence, sensitive system, would have basic features. The receiver, shown in Fig. 1, was made of a white been a very tedious and rather impractical task. An alternative to precise calculation would have alumina cylinder (Zircar ALC), 1.6 cm O.D., 1.3 cm I.D., and 46 cm long, insulated with 7 cm of alumina been to use the results of calorimetric measurements felt and encased in a l-cm thick aluminum cylinder performed using the same receiver as estimates of the acting as housing. The aperature was 8.5 cm in di- power captured by the latter. It turned out, however, ameter, cut in a high temperature-resistant alumina that uncertainties related to mirror reflectivity and board. Three thermocouples were inserted in the alu- other factors led to a substantial overestimation of the mina cylinder to monitor the temperature of the cavity. power input to the receiver/reactor set-up. The schematic in Fig. 1 illustrates the U-shaped We therefore varied the total power into the receiver tubular reactor used in this work. The Inconel tube until best fit between theory and experiment was obwas filled with catalyst up to 10 cm beyond the bend tained. The criterion for the "best" simulation was (to avoid reversing the reaction upon cooling at the closest correspondence between all the experimentally far end of the tube) and the rest was filled with inert measured temperatures (associated with the receiver wall, the reactor wall and the fluid) and their calculated alumina. Experiments were performed over a wide range of counterparts, and between the measured and calculated process conditions. Thus, the inlet gas temperature exit composition. ranged from 100 ° to 450°C, the inlet pressure from 2 Calculations were carried out for a large number of to 3 bar and the methane flow rate from 200 to 700 experiments covering the range of working conditions standard liters per hour (Is~ h), compatible with linear of the 55 experiments performed altogether[4]. We flow rates on the order of 2.5 to 3.3 m/s. The inlet gas show in Table I results obtained with four experiments, composition was given by a carbon dioxide to methane with inlet gas temperatures of 100 ° to 200°C, inlet gas ratio of approximately 1.2. pressures of 2 to 3 atm and methane flow rates at the The power impinging on the concentrating mirror inlet of 300 to 650 ls/h. The various trends in the assembly (hence, the power entering the receiver) was spatial (albeit temperature) dependence of the thercontrolled by the opening of the doors ofthe enclosure modynamic and kinetic parameters as well as the erhousing the concentrator. rors, both sporadic and systematic, are typical of all Receiver-wall, reactor-wall, and fluid temperatures the experiments. We discuss these below. were measured at several locations, as shown in Fig. As mentioned previously, the total power entering 1. Also, the overall methane conversion in the product the receiver was varied until best fit between the exgas was determined. Receiver-wall and reactor-wall perimentally measured temperatures and exit comtemperatures as high as 1130 ° and 1030°C, respecpositions and their calculated counterparts was obtively, and product gas temperatures in the range of tained. Since fluid temperatures and gas compositions 735 ° to 1030°C were measured. The overall methane determine the power converted within the reactor into conversions (in units of moles of methane converted sensible heat and chemical enthalpy, the power into per moles of methane in feed) were found to be within the reactor can be viewed as a qualifier of the fit. Althe range of 50% to 99 mol%. though this parameter cannot be determined experiOrder of magnitude estimates based on these data mentally by a direct measurement, it can be obtained point to solar power levels on the order of 0.5 to 2.5 by performing the heat and mass balance (see eqns kwatts converted to chemical enthalpy and sensible ( 5 ) and (6)) over the reactor with A T = T ExIt - T TM, V CH4 heat, net fluxes impinging on the reactor wall of 30 to Az --- L, Ax = -~ Exit, and taking average values for 120 kw/m 2, gas-hourly-space velocities of 8,000 to Tw, Te, 6p, and H. The results of such calculations 37,000 h -t (or contact times of 0.45 to 0.1 sec), ef- are listed in Table 1 under the heading "expt'l power fective heat transfer coefficients of 350 to 1 I00 W / m z into reactor." The "calc'd" counterpart was obtained (°C) and Reynolds numbers of 110 to 420[4]. by solving the set of eqns (5) and (6) for each control In order to understand the intricate details of op- volume, and adding up the corresponding powers. The eration of this integrated energy absorption-transmis- quite reasonable agreement between the calculated and
E. MEIROVITCHel al.
144
Table 1. Experimental and calculated temperatures, overall methane conversions and powers into reactor for experiments 1--4 Temperature(C) T10 TI2
Power into reactor (kw)
T2
T3
T4
1"5 T6
1"7 T9
1 Calc'd Expt'l
538 504
638 696
648 740
260 591 141 630
704 687 967 799 825 793 944 744
733 695
50.2 46.3
1.09 1.10
2 Calc'd Expt'I
590 595
730 763
767 835
325 659 173 731
g07 789 943 878 849 888 897 1037 865 802
64.0 71.5
1.38 1.50
3 Calc'd Expt'l
533 493
671 702
682 750
273 222
757 736 927 857 812 779 814 946 808 756
50.1 46.2
1.44 1.52
4Calc'd Expt'l
591 552
735 740
765 805
363 663 245 714
824 842
62.0 61.2
1.85 1.85
601 646
T8
Overall methane conversion (tool %)
815 1011 942 897 881 957 820 771
See Fig. l for identification of positions along the reactor where temperatures were measured. the experimental values warrant our best-fit criterion. We find that the calculated efficiency of the receiver is 66 +- 2%, taking into account only radiative losses via the aperture. The discrepancy between calculated and experimental overall methane conversions is at most 10%, and the errors are sporadic. On the other hand, the disagreement in temperatures is, in some cases, on the order of 16%. More importantly, systematic trends in the latter are reflected in Table 1. Thus, the experimental temperatures at T 12 and T5 are conspicuously overestimated by the calculations, whereas the temperatures measured at T3, T7 and T8 are underestimated. This, we think, is a consequence of the previously mentioned heat losses through the back wall of the receiver due to poor insulation, in view of which the thermodynamic equilibrium state of the experimental set-up differs from the one predicted for a perfectly insulated receiver. We refer below to experiment 1, and discuss in detail these general results using Figs. 4 and 5, wherein both the experimental and the theoretical performances of
Zw; 800
,,,.x .... , , . x
.....
.. So
/
T(*C)
,,
•
,,.....
•
40 xCH4
/
I'
40C
the reactor are depicted pictorially, and Table 1, wherein measured and calculated receiver wall temperatures are given. The calculated receiver wall temperature is 40°C lower than the experimental one at position T8, by 49°C higher at Tl0, and by 56°C higher at Tl2. We assign the experimental decrease in temperature upon approaching the back wall of the receiver, versus the calculated increase in temperature, to heat losses through the insulating material at the back that have not been accounted for in the calculation. As shown shortly this will also affect the location T5 on the reactor wall, adjacent to the receiver back wall. Reactor wall and fluid temperature and methane conversion profiles as a function of fractional tube length are shown in Fig. 4. In general, all the parameters displayed in Fig. 4 increase from inlet to outlet; in particular, the shape ofthese curves is of a very singular nature, to be discussed in detail below. The flux profile is shown by the solid line in Fig. 5. The rate of energy supply per unit area is high--78 k w / m 2 - - a t the reactor inlet, where the reactor-wall temperature, hence thermal emission, is low. Subsequently, it decreases up to a fractional catalytic bed length of 0.3, levels off and at z / L ~. 0.43 (with L denoting the length of the catalytic bed), increases again, first moderately, and then at - . / L ~ 0.6, abruptly. Between z / L -~ 0.6 and 0.86 the flux is high,
(mole%)
..".....
80
,
i
20 60 '
~
,/-
20C
0
.,.......... , 0.5
1.0
1.43
40
,,
-
Z/I.
Fig. 4. Calculated wall temperature profile (---) fluid temperature profile ( - - ) and methane conversion profile (. • • ) as a function of fractional catalytic bed length. L denotes the length of the bed. The fluid temperature at inlet was 102°C, the pressure 2.9 arm, the carbon dioxide to methane ratio I. 18, and the methane flow rate 651 Is~h. The power into the receiver was 1.44 kw. The x symbols represent experimental measurements of reactor-wall temperatures and the solid dots, experimental fluid temperatures. The symbol * denotes the overall methane conversion. The abscissa is given in units of fractional reactor length.
,I
(~
X
/".
I
I
I
05
I0
1.43
Z/L
Fig. 5. Heat flux (--), effective rate containing term (--.) and fluid temperature gradient containing term ( . . . . . ) as a function of fractional catalytic bed length, in units of (kw/ m:). Input data as described in the captions of Fig. 4.
Theoretical modeling of a directly heated solar-driven chemical reactor though nonuniform, with peaks at z / L ~ 0.64 and 0.86. A precipitous decrease is observed at z/L -~ 0.86, and from then on the flux decreases monotonically to 10 kw/m 2 at the end of the reactor. The intricate details of the heat flux profile reflect the self-regulating nature of the system, i.e., the mutual adjustment between the energy absorption and conversion processes. To follow and comprehend how this comes about, we plot in Fig. 5 the two functions that determine the heat flux absorbed at each position along the reactor (see eqn (5)): the term containing the fluid temperature gradient ( . . . . . ) and the term containing the effective rate (---), both in units of kw/m "-. As indicated in the theoretical section, the net heat flux impinging on the reactor wall at each axial location, i.e., the rate of energy absorption, must emulate the rate of energy conversion into sensible heat and chemical enthalpy. This is borne out by the three curves depicted in Fig. 5. It is quite obvious that up to - / L 0.15, the flux and the fluid temperature gradient term coincide, whereas the conversion gradient term is practically null. From z / L ~ O.15 onwards, the flux d T,, ,dx,, is given by' the sum of the ~ term and the d-'-zterm. It is illuminating to correlate local irregularities in the flux curve with those in the other two curves. Of particular interest are the extrema in the various curves at z / L = 0.47, 0.64, 0.86 and 1.14 (beyond z/ L = 1 the tube is filled with pure alumina), that reflect the coordinated coupling between absorption and conversion. For example, the flux balance at position z / L = 0.47 can only be achieved with a local minimum in the temperature gradient compensating for a local maximum in the rate, implied by eqn (6). The steep increase in the flux at z / L = 0.64 is brought about by the sudden exposure of the tube to direct insolation. Throughout the bend (0.64 < z / L < 0.86), the flux stays high. The shape of the flux profile between z/L = 0.64 and 0.86 reflects the fact that the corners of the bend receive thermal radiation from the receiver walls in addition to direct insolation, whereas the mid-bend segment only experiences the latter. The tube interior responds with a concave rate profile and a basically convex temperature gradient curve. As noted above, the bend in the reactor ends at z/ L = 0.86, and beyond that position, the tube is no longer exposed to direct solar radiation. Hence, we expect that the net heat flux, the reactor wall and fluid temperatures will all decrease abruptly at z / L = 0.86. This is, indeed, borne out by the curves in Figs. 4 and 5. As the temperature dependence of the effective rate is weaker than that of the net heat flux, the heat balance between z / L = 0.86 and 1.0 can only be preserved with a dramatic reduction in the fluid temperature gradient that would diminish both the chemical rate and the conversion to sensible heat. Eventually, the fluid temperature gradient becomes negative (see z~ L = 0.86 and vicinity in Fig. 5), i.e., the heat balance can only be achieved if the gas will cool down locally. At z / L = 1.0 the abrupt drop of the rate to zero, as the catalytic bed ends at this position, is compensated
145
for by a dramatic increase in the temperature gradient to absorb all the heat available as sensil~le heat. Beyond this position, the temperature gradient term and flux curves coincide. Two comments are noteworthy. First, the state of affairs at z / L = 0.86 uniquely reflects the coordination between absorption and conversion. With the boundary conditions of this particular experiment, the efficiency of the chemical heat sink at this position exceeds the efficiency of heat supply. Consequently, the fluid temperature decreases locally until the overall rate of conversion to both chemical enthalpy and sensible heat emulates the rate of heat transfer into the tube. Our second comment relates to the local minimum in the temperature profiles at z / L = 1.0 (see Fig. 4), not quite borne out by the experiment. As mentioned previously, heat is being drained from the system through the poorly insulated back wall of the receiver. Accounting for this phenomenon in the calculation would mitigate the constraint of the fluid temperature having to decrease in order to reduce the overall rate of conversion, and heat balance would be established by removal of excess heat by conduction through the insulator, rather than local cooling of the fluid. Obviously, agreement between theory, and experiment would be improved thereby. In summary., in up to one-fifth of the tube length. energy is being converted primarily into sensible heat. Between z / L = 0.43 and 1.0, solar energy is converted chiefly into chemical enthalpy. Beyond z/L = 1.0. where the catalyst bed ends, we again encounter conversion to sensible heat. For 0.3 < z / L < 0.43, about half of the solar radiation is converted into chemical enthalpy and the other half, into sensible heat. The self-regulatory characteristics of the receiver/ reactor assembly are also reflected in the reactor-wall and fluid temperature profiles (Fig. 4). The power entering the enclosure is redistributed quite uniformly by thermal emission and by reflected solar and thermal radiation. Since the lowest temperatures in the system are encountered at the reactor inlet, the net heat flux at that site will, due to reduced thermal emission, he high. The heat sink will respond with an increase in temperature of both the walls of the reactor and the fluid inside, as shown in Fig. 4. The rate at which these temperatures are enhanced is high, and heat is being absorbed efficiently as sensible heat. As one proceeds along the reactor, the net flux on its walls decreases (since the temperatures increase), entailing a decrease in the temperature gradients (see Fig. 5 ). Also, reactor wall and fluid are cooled by the onset of the endothermic chemical reaction at z / L = 0.15. At - / L = 0.43, chemical conversion assumes the main role as heat sink, with the sensible heat performing fine adjustments to establish the heat balance. Both temperature profiles in Fig. 4 increase monotonically. At z~ L = 0.64, the net flux (see Fig. 5) on the reactor (and hence the wall temperature) increases due to direct exposure of the latter to direct solar radiation. The fluid temperature in Fig. 4 follows suit, to both increase the effective rate and to augment heat conversion to sensible heat. The similarity between the heat flux (Fig.
146
E. MEIROVITCH et al.
5 ) and the fluid temperature (Fig. 4) profiles between
z/L = 0.64 and 0.86 (the portion at which the tube bends away from the opening) reflect just that. Beyond z/L = 0.86 the flux is back to normal, and both reactor wall the fluid temperature decrease. As mentioned above, in this region of the reactor the primary effect is conversion of solar into chemical energy. Yet, the effective chemical rate, due to its weak temperature dependence, does not cope with the relatively steep increase in heat flux, and cooling of the gas has to compensate for this: both reactor-wall and fluid temperature profiles in Fig. 4 bend over at z/L = 0.86, disrupting the monotonic increase of temperatures along the reactor. At z/L = !.0 the slope in the temperature profile changes sign back to normal (i.e., it becomes positive again) as the chemical reaction stops at the end of the catalytic bed. The increase in reactor-wall and fluid temperatures is quite dramatic, as the net flux is still quite high and the system has to switch instantaneously from a chemical heat sink to one based on conversion to sensible heat. Beyond z/L = 1.14, the fluid temperature tends asymptotically toward the reactor wall temperature. Radiative heat transport complies with this reality and the net flux impinging on the reactor tends to zero. The solid dots in Fig. 4 denote experimental measurements of the reactor-wall and fluid temperatures and of the exit methane conversion. The fit between theory and experiments at the fluid portions T2 and T3 and at the reactor wall positions T6 and T7 is good (see Fig. l for identifying these positions). The larger discrepancies at the reactor wall position T5, and at the fluid position T4, were discussed previously and assigned to heat losses at the back not accounted for in the calculations. The shape of the calculated methane conversion profile in Fig. 4 complies with the z~ L dependence of the fluid temperature. The solid dot at z/L = 1.0 denotes the experimental measurement. The calculated equilibrium methane conversions were at all temperatures higher than the respective calculated conversion values. Our simulations reproduce the experimental results with relatively high accuracy. This, as well as the accountability of all the details in the calculated heat
flux, temperature, methane conversion and rate profiles lend confidence to our model in general, and to the kinetic and heat transfer equations, in particular. The effect of flow rate, fluid temperature and pressure at the inlet to the reactor on its performances was investigated in depth. We performed extensive model simulations for two power levels--0.75 and 1.65 k w m received by the cavity, corresponding to 0.48-0.50 and 1.0- I. 1 kw, respectively, absorbed by the reactor. The methane flow rate varied from 250 to 650 Is~h, the inlet fluid temperature from 150" to 250°C, and the gas pressure at inlet from 2 to 3 bar. The feed composition was kept constant, with C O 2 / C H 4 -- 1.23. The results of 24 typical calculations are shown in Tables 2 and 3. For each calculation, we present the corresponding input parameters and the computed overall methane conversion, equilibrium conversion, fluid temperature and pressure at the exit. Table 2 illustrates the effect of the operating conditions--inlet temperature, pressure and flow r a t e - on the overall and equilibrium methane conversion and fluid temperature at exit. If the methane flow rate at inlet is set to be 650 lJh, 0.75 kwatts of solar power will be absorbed, generating overall methane conversions of 10%-13% and a temperature increase of 260 °3600C within the reactor. With 450 ls/h of methane passing through the reactor, the corresponding parameters are an approximately 20% overall methane conversion and a temperature increase of 300°-400°C and with 250 Is~h, the overall conversion is of the order of 35%--42% and the temperature increase is 3600--460°C. Pressure has a minor effect on the exit temperatures and a moderate effect on the overall and equilibrium methane conversions at low flow rates. At higher flows, the overall methane conversions and the fluid temperatures at exit are practically unaffected by increasing the pressure from 2 to 3 bars, whereas equilibrium conversions are affected slightly. With lower pressures and flow rates, equilibrium is attained at fractional reactor lengths less than unity. The last column in Table 2 specifics the location at which this happens. Beyond these positions, the operation of the chemical heat sink is "controlled by thermodynamics," i.e., the conversion of reactants into
Table 2. Input (columns 1-3) and output (columns 4-8) parameters for model calculations with 0.75 kW entering the receiver Exit Inlet Methane Fluid Ove~dl Fluid Exit Fractional equilibrium )ressure ~ow r~te temperature methane temperature )ressure robe length conversion , at inlet at inlet conversion at exit at which (bar) equilibrium (poles converted~ moles at equilibrium'~ (bar) (Is/h) (*C) \ moles in feed / ('C) is attained moles in f ¢ ~ ] 39.0 611.0 2.0 ' 0.51 2.0 250 150 39.0 34.0 3.0 250 150 34.0 611.0 3.0 0.45 19.8 20.0 535.0 1.9 '"2.0 450 150 18.0 18.0 3.0 450 150 545.0 2.9 0.84 10.0 15.0 511.0 1.8 '2.0 650 150 10.0 12.0 510.0 2.9 3.0 650 150 42.0 42.0 622.0 1.9 0.57 2.0 250 250 35.0 35.0 3.0 250 250 617.0 3.0 0.35 22.0 22.0 2.0 450 250 546.0 1.9 0.88 20.0 20.0 556.5 2.9 0.72 3.0 450 250 16.0 2.0 650 250 13.1 515.5 1.8 13.0 517.0 2.8 0.94 3.0 650 250 13.0 J
c: Calculation C; d: Calculation D.
Theoretical modeling of a directly heated solar-driven chemical reactor
147
Table 3. Input (columns 1-3) and output (columns 4-8) parameters for model calculations with 1.65 kw entering the receiver Overall Exit Inlet ~ethaze Fluid Fluid Exit Fractional equilibrium >ressure Ilow rate ~mperature methaae temperature )ressure tube length conversion at inlet at inlet conversion at exit at which (bar) equilibrium moles converted~ moles at,equilibrium'~ j b a r ) !is/h) ' (°Cl (*c) *s attained moles in feed ) mole~in feed ] e 2.0 250 150 • 3.0 250 150 54.5 57.0 673.5 1.8 2.0 450 150 694.0 2.9 0.80 55.0 55.0 a 3.0 450 150 34.0 42.0 621.0 1.7 2.0 650 150 34.9 35.0 615.5 2.8 b 3.0 650 150 e 2.0 250 250 e 3.0 250 250 57.0 60.0 684.0 1.8 2.0 450 250 695.5 2.9 55.0 55.0 3.0 450 250 0.84 37.0 45.0 630.0 1.7 2.0 650 250 38.0 38.0 625.5 2.8 3.0 650 250 a: Calculation A; b: Calculation B; e: Energy and mass balance cannot be performed at the far end of the reactor because of heat transfer limitations (see text).
products is monitored by the equilibrium constant. With 1.65 kwatts entering the receiver, proper functioning of the self-regulating system is, under certain circumstances, impeded by heat-transfer limitations. The system operates properly if, and only if, the rates of absorption of concentrated sunlight, the rate of heat transfer through the reactor wall and the rate of conversion into chemical enthalpy and sensible heat taking place within the reactor, are equal. With high powers and low flow rates, the fluid temperatures next to the outlet are high and may, eventually, approach the wall temperatures asymptotically. (Note that the latter cannot exceed a certain value above which the reactor would act as a heat emitter rather than a heat sink.) The temperature gradient between wall and fluid would tend to zero, and heat would no longer be transferred into the reactor. U n d e r these circumstances, the process is heat-transfer limited and susceptible to local overheating. Calculations labeled e in Table 3 that could not be completed for the reason mentioned above, pertain to this category. The situations depicted in Table 3 are, in actual fact, close to the stability limits of the self-regulatory system; hence, sensitivity is relatively low. The effect of increasing the pressure from 2 to 3 bar is minor. As expected, overall and equilibrium methane conversions and exit fluid temperatures are reduced by increasing the flow rate. The extent of these changes is, however, smaller than in the previous case, where 0.75 kw entered the receiver. Following this general discussion, we present below typical flux, reactor-wall temperature, fluid temperature, and rate profiles obtained with the model calculations A - D , labeled a - d in Tables 2 and 3. An inlet fluid temperature of 150°C, an inlet pressure of 3 arm, methane flow rates of 450 (A and C) and 650 (B and D) l,/h, and total powers of 0.75 (C and D) and 1.65 kw (A and B) were used. Figure 6 indicates that the flux profile along the reactor is determined primarily by the total power entering the receiver. Although the flux intensity at each position along the reactor length is reduced by decreasing the total power, the shape of the curve is preserved, being determined by geometric
characteristics of the receiver/reactor system. On the other hand, Fig. 7 shows that the conversion profile corresponding to a given power level depends to a large extent on the flow rate. The shapes of the curves in Fig. 7 are, in general, determined by the power level, whereas their amplitude and spread along the - / L axis are fixed by the flow rate. In particular, the m a x i m u m in the rate profile is shifted toward the exit by both increasing the flow rate, and lowering the power (the dip observed with A and B is due to the idiosyncratic geometry of the receiver/reactor). It is quite obvious that with A, the chemical heat sink operates most effectively, whereas with D, least efficiently. To illustrate heat transfer limitations, we present wall (---) and fluid ( - - ) temperatures and fluid temperature gradient profiles in Figs. 8 and 9, respectively. With both C and A, fluid and wall temperatures tend to merge close to the exit, where the tube is filled with pure alumina. In fact, the fluid temperature slightly exceeds the wall temperature at the exit of A, with the temperature gradient becoming negative (see curve A in Fig. 9 close to - / L = 1.43) and the flux becoming
6oF Ww
B
',
ol
,
,
0
05
1.0
1.43
Z/L Fig. 6. Heat fluxes as a function of fractional catalytic bed
length. An inlet fluid temperature of 150°C, an inlet pressure of 3 atm and a CO2/CH4 ratio of 1.23 were used for all the calculations (A-D). Additional input data w e r e : A: P ' = 1.65 kw, V(CH4) = 450 l,/h; B: P' -- !.65 kw, V(CI-h) = 650 l,/ h; C: P' = 0.75 kw, V(CH4) = 450 l,/h; D: P' = 0.75 kw, V(CH4) = 650 l,/h; with P' denoting the total power into the receiver and V(CH4) the inlet methane flow rate.
148
E. MEIROVITCH el al.
I
I P\ 2C"
AX
a-5
o.~
°°6
AT/AtI0i ~\\~'~. 0.5
,.o
O---
,.,~
Z/L Fig. 7. Effective rate of methane conversion as a function of fractional catalytic bed length for calculations A-D. Input parameters as described in the captions of Fig. 6. The ordinate units are moles ofCH4 converted/(cm-h). negative (see curve A in Fig. 6 close to z / L = 1.43). (Note that negative fluid temperature gradients are also encountered close to z / L = 0.86 in Fig. 9.) Yet, at that position, heat can, on the whole, still be transferred through the tube walls as the chemical heat sink is operative, with a large and positive conversion gradient (see curve A in Fig. 7).
Experimental results obtained with a directly illuminated tubular reactor mounted in a solar furnace were analyzed using a theoretical model that accounts for the radiative heat transport within the receiver and the endothermic chemical reaction occurring within the reactor. We find that throughout the reactor, the rates of net heat flux into the reactor emulate the rate at which energy is being converted, mainly into chemical enthalpy and partly into sensible heat. Under certain circumstances, depicted in the text, the process is controlled by heat transfer or by chemical thermodynamics. Convective heat losses through the insulating material at back are prone to reduce temperatures at the inlet to the reactor, and enhance them at the far end of the catalytic bed. The self-regulatory characteristics of this receiver/ reactor system are borne out by these results. We believe that the theoretical model set forth in
I
6OOl f~ss/s"'" "'"'~'- _. T(*C) / "" 4 0 0 ~ 0
0,5
0.5 I
1.0 I
Z/L
1.4:5 I
Fig. 9. Huid temperature gradients as a function of fractional catalytic bed length for calculations A and B. See captions of Fig. 6 for input parameters. The ordinate units are °C/cm. this study can be used with confidence to analyze experimental results and design solar-driven reactors.
Acknowledgments--The authors acknowledge helpful discussions with, and valuable comments by, Mrs. Rachel Levitan and Mrs. Hadassa Rosin. A.S. acknowledges financial support from the Ministry of Absorption of the State of Israel.
4. CONCLUSION
I
-IC
1.0
I.,,3
Z/L Fig. 8. Reactor wall temperature (---) and fluid temperature ( - - ) profiles versus the fractional catalytic bed length for calculations A and C. See captions of Fig. 6 for input parameters.
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