Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 4, 1979
L = load variable disturbance m = process input variable P, = ultimate period Rf = f2,man/fl,max R, = w2Iw1 s = Laplace transform operator S = static contribution to the interaction between loops T l . . . .T6 = six time constants of the process TR = controller reset time T CCM = recommended controller gain by Continuous Cycling kethod y 1 = integral square of the error for Loop 1 y z = integral square of the error for Loop 2 Y = Y1 + Y2 yls* = optimum value of y 1 if only Loop 1 is present Greek S y m b o l s CY = constant defined by eq 37 @ = constant defined by eq 38 w = critical frequency
607
Literature Cited Bhalodia, M., Ph.D. Thesis, State University of New York at Buffalo, Buffalo, N.Y., 1973. Bristol, E. H. I€€€ Trans. Autom. Control 1966, AC- 1 7 , 133. Carroll, G. W. J . Oper. Res. SOC.Am. 1961, 9 , 169. Chen, K.; Mathias, R. A.; Sauter, D. M. Trans. Am. Inst. Nectr. Eng., Part 2 1962, 82, 336. Falb, P. L.; Wolovich, W. A. I€€€ Trans. Autom. Control 1967, AC- 12, No. 6, 651. Freeman, H. Trans. Am. Inst. Electr. Eng., Part 2 1957, 76,28. Freeman, H. Trans. Am. Inst. Necfr. Eng.. Part 2 1958, 77,1. Harriott, P. "Process Control", McGraw-Hill: New York, 1964. Hooke, R.; Jeeves, T. J . Assoc. Comput. Mach. 1961, 8 , 212. Kavanagh, R. J. Trans. Am. Inst. Electr. Eng., Part 2 1957, 76,95. Kavanagh. R. J. Trans. Am. Inst. Nectr. Eng.. Part 2 1958, 77,425. Jackson, R. Trans. SOC.Inst. Tech. 1958, IO, 68. MacFarlane, A. G. J. Proc. I€€ 1970, 117, 1039. Morgan, B. S.,Jr. Proc. JACC, 1964, 468. Newton, G. C., Jr.; Gauld, L. A.; Kaiser, J. F. "Anawical Design of Linear Feedback Controls", Wiiey: New York, 1957; pp 366-381. Niederlinski, A. Automatica 1971, 7,691. Rekasius, 2 . V. "Proceedings, 3rd Annual Allerton Conference on Circuit and System Theory", Urbana, Ill., 1965, p 439. Rijnsdorp, J. E. Automatica 1965, 7, 15. Rosenbrock, H. H. Comput. J . 1960, 3 , 175. Weber, T. W.; Bhalodia, M. Ind. Eng. Chem. Process Des. Dev. 1979, 18, 217. Weigand, W. A.; Kegerreis, J. E. Ind. Eng. Chem. Process Des. Dev. 1072, 1 1 , 86. Ziegler, J. G.; Nichols, N. B. Trans. ASME 1942, 6 4 , 759.
Superscripts
* = optimum overbar = average value Subscripts
a = actual controller setting 1 = Loop 1 (except for G's and h's) 2 = Loop 2 (except for G's and h's) Alphabetical Constants
Received for review May 27, 1977 Accepted June 15, 1979
The Rate and the Fundamental Mechanisms of the Reaction of Hydrogen Sulfide with the Basic Minerals in Coal Amir Attar' and Francols Dupuis Department of Chemical Engineering, University of Houston, Houston, Texas 77004
The basic minerals in coal are calcite, aragonite, dolomite, and siderite. Ion-exchangeable calcium, in the form of salts of organic acids, is also present in coals. These salts decompose upon heating and release calcium oxide. The network or reactions among H2S and the various minerals is
,2-
C02
+
M O (porous oxide)
MCO
\"
MO sintered oxide
\
HZS
MS
+
HzO
HZS
MS + C O , -+ HzO The rates of reaction of siderite, calcite, and dolomite with H2S have been studied in the temperature range of 400-800 OC. The rate of the chemical reaction (RCR) controls the rate of consumption of siderite at temperature below 495 OC. However, above 500 OC the rate of diffusion (ROD) in the gas-film controlled the rate of reaction of decomposed siderite (porous FeO) with H2S. The RCR controlled the rate of reaction of dolomite and half-calcined dolomite at temperatures below 670-700 OC. The RCR controls the rate of consumption of fresh calcite below 460 OC. However, as soon as a layer of Cas is formed on the surface, the reaction stops and its rate becomes controlled by the ROD of gases in the layer of Cas. At temperatures around 640 OC, the surface thermal diffusion is fast enough and separate crystals of Cas appear to be formed on the surface of the CaC03. New surface is exposed and the reaction commences. The rates of reaction of calcite and dolomite become controlled by the ROD in the gas around 700 OC. A pulsed differential reactor (PDR) has been used to derive the kinetic data on the various systems. A detailed mathematical analysis of PDR's and their use to obtain kinetic data is presented with rate data and rate constants for the reaction of H2S with the mineral matters in coal.
Introduction
The inorganic matter in coal can be classified into three groups of minerals according to their reactions with hy0019-7882/79/1118-0607$01.00/0
drogen sulfide (H,S): (1)basic minerals, ( 2 ) minerals with catalytic activity, and (3) inert minerals. I n the temperature range of 200-900 "C, most of the minerals are 0 1979 American
Chemical Society
608
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 4, 1979
inert or have a slightly catalytic activity on the rate of decomposition of H2S. However, the basic minerals react with H2S and the corresponding sulfides are formed. The most important minerals in this category are calcite (trigonal CaCO,), aragonite (orthorhombic CaCOJ, dolomite (CaC0,.MgC03), siderite (FeC03), and to some extent montmorillonite (clay). Several alkaline compounds which can react with H2Sare formed during the processing of coal, e.g., CaO, FeO, MgO, and Fe203. During coal pyrolysis or hydrogasification, some of the sulfur is released as HzS. Although the H2S resides in the reactor for only a short time, part of the sulfur is retained in the solid as alkaline sulfide due to the reaction of H2S with the alkaline minerals. The chemistry and thermodynamics of these processes have recently been reviewed by Attar (1977, 1978). The objective of this study was to derive kinetic data on the rate of reaction of H2S with the alkaline minerals in coal when short contact times are available. The systems of reaction between H2S and a metal carbonate MCO, involve four reactions (Glund et al., 1930; Stinnes, 1930; Bertrand, 1937; Parks, 1961; and Squires, 1972): (1)direct reaction of the carbonate with H2S MC03 + H2S
-
MS
+ H20+ C 0 2
(1)
(2) decomposition of the carbonate to the oxide MO and
cos
MC03
-
MO
+ C02
(2)
(3) reaction between the oxide and H2S MO
+ H2S
MS
+ H2O
(3)
(4) sintering of the oxide and the formation of nonporous materials porous MO
A
nonporous MO
(4)
Reactions 1 and 2 are parallel, and reactions 3 and 4 are parallel. The last two reactions are in series with reaction 2.
, i nonporous MO
f-
Co2
MCO
\?
HZS
+
MS +
\
Mo
HZS
MS
+
H20
(5)
CO2 + H 2 0
H2S is formed in coal by the reactions of FeS, with hydrocarbons and H 2 (Powell, 1921) and by the decomposition of the organic sulfur compounds (Thiessen, 1945). In particular, H2S is formed when the sulfidic functional groups decompose to HzS and an olefin. The H2S can react with the basic minerals and thus the sulfur is trapped in the char in the form of the sulfide (Armstrong and Himms, 1939). The kinetics of the reactions of H2S with fully calcined and semicalcined dolomite were studied by Ruth et al. (1971, 19721, Pell et al. (19711, and Squires (1972). In the present work, the mechanism of the reaction, the rates, and rate constants were derived for the reactions of H2S with calcite, siderite, dolomite, and montmorillonite. Several studies were conducted in The City College of New York by Pell et. al. (19711, Ruth et al. (19721, and Squires (1972), who published data on the rate of reaction
of H2S with semi- and fully calcined solomite. In our work, only a limited number of tests on half-calcined dolomite were conducted; however, some of the phenomena which they reported were not observed by us. However, we used dolomite from different sources. Direct comparison between data from The City College and from this study is impossible because we could not confirm some of the assumptions which they used in their mathematical model and because different experimental techniques were used over a different range of parameters. We had to propose a different reaction model since we could not confirm the existence of a reversible “surface complex” of CaO and H2S. (CaCO,-MgO) + H2S * [(CaC03.MgO)H2S] [(CaC03.MgO)H2S] * CaS + MgO + HzO + COz (6) No data were found on the rate and the mechanism of the reaction of H2S with siderite, calcined siderite, and montmorillonite. The main observations and conclusions from this study are summarized below. A full elaboration of these results is presented in the Discussion section. (A) Complete conversion of dolomite to CaS and MgC03 is obtainable a t 570 “C, but only 1.8-2.9 wt % of -200 mesh calcite reacts a t these conditions (0.77 m2/g surface area). (B) At the conditions of the test, the rates of reaction of calcite and dolomite at 700 “C are essentially identical and controlled by the rate of mass transport of H2S in the gas. (C) The rate of reaction of dolomite is very sensitive to impurities; i.e., different rocks react a t different rates a t temperatures where the rate of the chemical reaction controls. (D) The apparent rate of consumption of H2S a t different temperatures is limited by different ratecontrolling steps. The rate-controlling step depends also on the particular materials, particular sample, and on its crystalline structure. The available data are consistent with the following model. (A) At low temperatures and when the surface is fresh, the rate is controlled by the rate of the chemical reaction. However, as soon as a layer of MS is formed the rate of diffusion of C 0 2 in MS controls. (B) At intermediate temperatures, the rate of consumptions of H2S is controlled by the availability of free surface of MO or MCO,. In this range of temperature, the carbonates decompose according to reaction 2. A new surface is thus exposed which is not covered by MS; thus, the effective rate of consumption of H2S increases. (C) The available surface decreases by sintering when the temperature is too high; thus the apparent rate of consumption of H2S may decrease. (D) When the rate of decomposition of MCOB becomes very large, the rate of consumption of H2S may be limited by mass transfer in the gas. The crystalline structures of calcite and dolomite are trigonal (Bragg and Claringbull (1965)) and are basically identical (Table I), except that in dolomite alternate Ca2+ ions are replaced by Mg2+ions. The spacial dimensions are different because the ionic radius of Mg2+is 0.65 A and that of Ca2+is 0.99 A (Greenwood, 1970). The mechanism of their reaction is, however, very different. One possible explanation for the difference in their behavior, which is consistent with our kinetic tests, with our scanning electron microscopy (SEM) observation, and with our differential scanning calorimetry (DSC), is presented below. At 570 O C the rate of reaction of pure dolomite with HzS is almost constant and does not vary with the conversion (up to 30 wt % conversion). Impure dolomite reacts initially a t a much larger rate than after some of it has been
Ind. Eng. Chem. Process Des. Dev., Vol. 18,No. 4, 1979
809
Table I. Crystals, Structures, and Dimensions material
crystal system triogonal
calcite, CaCO,
dimensions, A a = 4.983;
(Y
OUTLET
=
46” 7‘ orthorhombic a = 5.72; b = 7.94; c=4 trigonal FCC a = 5.76 a = 4.78 FCC trigonal a = 4.30 FCC
aragonite, CaCO, dolomite, CaMg(CO,), Cas Ca 0 FeCO, Fe 0 (nonstoichiometric) FeS (nonstoichiometric)
a = 3.43; c = 5.79
hexagonal
INLET
Figure 2. Schematic diagram of the differential reactor.
(1) a differential reactor, (2) a pulse injector, (3) a gas chromatograph with a TC detector, (4) an integrator with a data system, and (5) a gas feed and control system. Figure 2 shows the differential reactor. The shell (l), the tube (2), and the filter (3) are made out of quartz. The gas inlet and outlet are through a SS connector (4). The quartz reactor is inserted through the wall of a high temperature furnace where temperature is controlled and monitored. Powder (0.14.5 g) of samples with particle size of -200 270 mesh is packed in the reactor in the annulus between the shell (1)and the inner tube (2) and held in place with plugs of quartz wool. The gas flows from the tube through the packed bed of particles into the annulus and out. The specific surface area and the pore volume of the powdered samples were measured using nitrogen porosimetry. Selected solid samples of reactant and products have been $ examined ~ using ~ X-ray ~ diffraction ~ ~ differential ~ and scanning calorimetry (DSC). Pulses of the reactive gas, H2S,are fed using a microprocessor-controlled gas chromatograph injector. The size of the injector loop determines the size of the pulse of HzS which is introduced into the stream of helium. The helium flows through the reactor, a chromatographic column, and a TC detector. Each pulse of H2S which is injected into the reactor results in a pulse which consists of the reaction products plus the unreacted H2S. The mixture of gases is separated on the column and detected by the TC detector. The signal from the detector is integrated by a microprocessor which multiplies the areas by the proper calibration factors and prints the amounts of each component in the pulse of products of the reaction. The integrator and the injector are synchronized so that automatic periodic operation is possible. The repeatability of the injections was 0.05% or better. The overall accuracy of the analysis was at least 10% and usually better than 2% based on material balance. Mixtures of H2S, C02, and H 2 0 were separated on a 6 ft X in. column of Chromosorb 105 80/100 mesh at 90 OC. The helium flow as 75 mL/min. The various minerals that were tested were NBS standard minerals, except for the calcite which was purchased from Fisher Scientific. The major advantage of the pulsed differential reactor method is that it permits direct measurement of the amounts of all the gaseous reactants and products that participated in all the reactions that took place in the reactor. Therefore, since several concentrations and variables are measured, the method permits the calculation of the rates of individual reactions. Temperature programmed tests also permit establishment of the mechanism of the reaction. Mathematical Analysis Pulsed differential reactors (PDR) were used extensively in studies of catalytic reaction. The mathematical treatment of data from PDR has been reviewed recently by Furusawa et al. (1976). Most of the studies used PDR to derive information on the mechanism of reactions and on their activation energies. Since very little has been found in the literature on the derivation of rate constants for gas-solid reactions from PDR data, a brief description
+
)INJECTION
iNJECTION
V4LVE
- - + - J
. ’1
~E~~b?~~Rf-
s
~
MICROPROCESSOR
~
!
~
RECORDER
~
f JRNACE
0
0
0
Figure 1. Schematic diagram of the experimental system.
converted. A t 570 “C the rate drops to about half the initial rate after about 4% of the material has been converted. At 570 “C calcite reacts initially at about the same rate as pure dolomite; however, the rate of reaction drops to nil very rapidly. We estimated that the apparent rate of consumption of H2S by calcite stops when a layer of about 78 molecules of Cas, on the average, is formed on its surface. Such a “thin” layer of (Ca2+S2-)is apparently sufficient to block the diffusion of C02, HzS, or H20 and the reaction stops. The surface layer of (Ca2+S2-) rearranges at around 635-650 “C and forms small crystallites of the thermodynamically stable salt Cas. Apparently, the rate of diffusion of ions on the surface becomes fast at 635 “C and the time scale for the crystalline rearrangement becomes of the order of the time scale of the experiment. (Ca2+,S2-)(amorphous)
-
C a s (crystalline)
(7)
When the Ca2+S2-layer breaks and crystalline CaS is formed, a new surface of CaO or CaC03 is exposed and the reaction can proceed. Sulfidized crystals of calcite have the same morphology as that of calcite. However, their surface develops “bumps” when heated above 635 “C. The “bumps” are believed to be small crystals of Cas. DSC of the sulfidized calcite shows a small “shoulder” around 640 “C, which could be attributed to the energy needed to rearrange the surface (Ca2+,S2-) to crystalline Cas. Since kinetic and DSC tests as well as SEM observations support the proposed mechanism, we believe that it is the actual reaction mechanism. Around 700 “C no difference was detected between the rate of reaction of calcite and dolomite; however, at this range of temperature the rate of consumption of H2S appeared to be controlled by the rate of gas-phase mass transport in the particular apparatus used. Experimental Section Figure 1 is a schematic flow diagram which shows the relations among the various parts of the system. The experimental system consists of five major components:
610
Ind. Eng. Chem. Process Des.
Dev., Vol. 18, No. 4,
1979
of the method that we used seems appropriate. The methodology and one kinetic model are analyzed following this section and other models are analyzed in the Appendix. PDR's operate in an inherently unsteady-state mode. Therefore, they do not permit obtainment of the absolute value of the rate constant of a reaction but rather an upper and a lower bound on the value. However, since the experimental parameters can be chosen so that the interval of reliability is as small as desired, utilization of PDR's does not construe a fundamental difficulty. First, a method is described which permits calculation of the upper and lower bounds on the rate constants of a reaction, then applications to specific cases are discussed. Definition of the R a t e of Reaction. The rate of a reaction is traditionally defined as the rate of disappearance or formation of a material per unit volume, mass, or surface area of reactant or catalyst. It is convenient to define the average rate of disappearance of the reagent gas in the ith pulse, ri, as
d W 0 - WJ (8) W ,WO and the rate of formation of product due to the jth reaction in the ith pulse as
ri =
The value of Wo,the size of the pulse, is determined by the size of the injection loop chosen and is independent of the shape of the pulse. Obviously, if the value co is substituted in rate equations instead of the real concentration cr, the value of the calculated rate of reaction will be larger than the real rate since co > c,. Thus for an nth-order reaction
kA"c0 > kA"cr (11) The concentration of reactive gas at the outlet of the reactor, c, is equal to 0 at t = ti, passes through a maximum, and becomes 0 again after a sufficiently long time. However
Io
qc dt =
The number of moles of reactive gas that leave the reactor, Wi,can be determined experimentally from the detector output and its value is independent of its shape. Substitution of c instead of c, in rate equations results in calculated values smaller than the real rates, since c < c,, thus for an nth-order reaction
kA"c < kAnc, (13) From inequalities (11) and (13) one concludes that kA"c < kA,c, < kA"co
QPij
r , .= 'I W,W,
(9)
The quantities ri and rij have the units of volume of gas disappeared (or formed)/(time X moles of solid). ri and rij have to be multiplied by the molar volume of the reagent gas at the temperature and pressure in the reactor in order to be transformed to the common units, moles of gas disappeared/(time X moles of solid). The rate of consumption of the solid in the reactor depends on the amount of solid, A , and on the instantaneous concentration of gas, c , around it. Now, since the reactor is differential and contains a very small sample and since only a small fraction of each pulse is consumed, it is reasonable to assume that at any instant the concentration of gas that all the particles face is the same. Under such circumstances, the apparent rate of reaction becomes independent of the actual shape of the pulse, as provided in the next section. When the rate of the chemical reaction controls, the rate of conversion of the reactive gas with the solid depends on the concentration of gas in contact with the solid and on the "level" of contact. For example, if the solid is "infinitely porous" and all its volume is available for this reaction, chances are that the rate will be first order with respect to the solid. However, if reactions occur only on the surface of the unreacted solid, and the solid consists of isomorphous crystals which geometry do not change upon the conversion, chances are that the rate of the reaction will be of order 2 / 3 with respect to the solid. Two cases are differentiated, the general case where the rate of reaction depends on the n power of the number of moles of solid, n # 1, and the case n = 1, where all the volume of the solid is equally available. The largest concentration of reactive gas that the solid can encounter is the instantaneous concentration of gas a t the inlet, cos The inlet concentration varies from co = 0 at the instant of injection of the ith pulse, passes through some nonzero values and becomes, after a sufficiently long time, zero again. However
Wi
(14)
The value of c can be estimated using a material balance on the differential reactor. It is assumed that the concentration of reactive gas in the reactor is uniform and equal to the concentration at the exit (pseudo backmixed reactor)
Coq - Cq = ak,Anc
(15)
therefore
Rate of Reaction Depends on n-Power of the Solid. When the rate of reaction depends on the nth power of the number of moles of solid in the reactor, A , and only one reaction takes place, then _ -dA = kA"cr dt If we multiply eq 17 by q and rearrange the equation we obtain -- 9
d(A1-") = kqc, dt 1-n Integration of the last equation for the ith pulse yields Ai-ll-")=
im kqc, dt (19)
If the reaction is isothermal and only a small fraction of the pulse is consumed, then the lower bound on k , kl, can be obtained by taking c, = co
io klqc, d t k l A mqco dt =
= klWo
where Wois the number of moles of H2Sthat were injected. Combining the last two equations yields
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 4, 1979 611
If Ai Ai-l = Ao, namely the amount of solid does not change much, then
where
Equation 22 can be used to evaluate B from which the rate constant kl can be determined. When a small fraction of the material is converted, Ai = Ao, a plot of log (Ai/Ao)vs. i yields a straight line with the slope (1/1 - n ) log [l - (B/Aol-n)]. To evaluate n, several experiments should be carried out with different initial amounts of the solid, Ao, or with pulses of a different size, W,. The upper bound on the rate constant k , k,, can be estimated by substituting for c, the value of c at the outlet. Substitution of eq 16 in eq 19 yields A 1-1 . )=From stoichiometry a(Ai-1 -- Ai) = (WI-l- W,) Therefore
(25)
Equation 27 can be used to evaluate k , using data on consecutive pulses, kui can be evaluated from
A better estimate of k, is then calculated by averaging several values of kUi i
The average rate of consumption of gas per unit mass of solid is
The rate of the particular reaction which produces the j t h product is (32) Note that if only part of solid, e.g., only a surface layer on the top of each crystal reacts, then SThas to be determined experimentally using the relation
or
h, =
ith pulse be denoted by Sji. The relative response factor of the detector to the j t h product is denoted kj, where the response factor to the reactive gas is taken as a unity. The total weight of solid in the reactor can react with the stoichiometric amount of gas, Wg;the area that a peak of magnitude W , would have had is denoted by ST. It is assumed that the detectors are linear; therefore, when all the weight of the solid is available for the reaction
m
h,, m i=l A very important special case is when n = 2 / 3 . This occurs when the rate of reaction depends on the available surface area. When the solid is “infinitely porous” it may be 1. The case where n = 1 is discussed in the Appendix. Evaluation of the Rate Constants from Experimental Data The data that are derived in each experiment include the initial condition of‘the sample, its weight, or number of moles, W,, and its specific surface area, sa. The length of the cycle, 6,, is usually determined by the difficulty of the separation of the products. The number of moles of reactive gas per injection, W,, is determined by the fineness of the resolution which is required or by the sensitivity of the experimental system. The value of q can be used to modify B or y; however, it is usually dictated by the separation procedure. Denote by So the area that the recorder will plot when a pulse of gas and no reaction occurs. Let the area of the peak of unreacted gas from the ith pulse be denoted by S,, and the area of the peak of the j t h product from the
m
ST =
1 (SO - si)
(33)
1=1
Arguments of Mass and Heat Transfer in PDR’s Gas-solid reactions proceed in a sequence of steps which involve: (1)diffusion of the reactive gas (RG) through the gas film around the particles; (2) diffusion of the RG through the ash layer of the solid; (3) chemical reaction; (4) diffusion of the gaseous products in the ash layer; and ( 5 ) diffusion of the products in the gas film around the particles. Adsorption of the reactive gas on the active core or on the layer of ash or desorption of the gaseous products could provide additional resistance to mass transport. Detailed discussion of the various steps is present in many textbooks, e.g. Levenspiel (1962) and Szekely et al. (1976). Since each step could control the apparent rate of reaction, it is desired to know the controlling step in each experimental condition. PDR’s allow the determination of the rate controlling step in three ways: (1)changing the flow rate q , ( 2 ) changing the pulse size W,, and (3) temperature programing the reactor. When the rate of the chemical reaction controls, the value of r, should be invariant with respect to the flow rate or to the pulse size and should increase with the increase in the temperature according to the Arrhenius equation. We found that when PDR’s are used, the easiest way to find the controlling mechanism at a given set of conditions is by programing the temperature and determining the apparent activation energy of the reaction. When the rate of the chemical reaction controls, the values of the activation energies that will be observed are of the order of 6-45 kcal/mol. When diffusional processes control, the values of the activations energies will be in the range of 0-5 kcal/mol. ,Several examples of the application of this technique are discussed in the following sections. Slow adsorption and desorption steps will result in “tailing injection” into the GC columns and in poor GC separation. The shape of the peaks that result when pulses of gaseous products are introduced directly into the TC detector, without first being separated, gives a gross indication on the relative importance of the sorption and desorption steps. Figure 3 shows a typical output from the TC detector which shows that the sorption steps were
612
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 4, 1979
n A plot of log (1- X ) vs. A, will yield a straight line of slope equal to n - 1. 2. Determination of the Isothermal Rate Constant. The upper and lower bounds on the value of the constant are determined by eq 21 and 27 or their analogues. SIGNAL
KNEW
(1 - n)Wo
S T A R T RUN
RUN t T ! M E
5
4
[ ,r
1
qAi-ll-n
(ONE
START~I
CYCLE)
H-S PEAKS
3 Of SUCCESSiVE
PULSES
CO PEAKS OF’SUCCESSIVE
PULSES
Figure 3. Superimposed chromatograms of five pulses. Top: a chromatogram of one cycle; bottom: superimposed chromatograms of five cycles. Pulses of 0.517 mL of HzS were injected at 0.974 bar. The mineral was 0.1000 g of calcite at 570 “C. The peaks to the right are COz peaks and the peaks to the left are the corresponding peaks of the unreacted HzS. Note that when more HzS is consumed, less H2S leaves the reactor and more COSis formed.
not important in the case studied. The COzand HzS peaks remained narrow and sharp even after they were separated on the GC column. The HzO peak is a typical “tailing peak”, however, since the HzO has been injected together with the C 0 2 and H2S the quality of the injection could not have caused the peak to tail. A separate test has shown that the water peak tails due to the interaction of HzO with the GC packing material. Control of the temperature of the reaction in P D R s is simpler than in plug or CSTR reactors because the heat capacity of the reactant is small relative to that of the reactor. Moreover, large heat transfer coefficients are realized in the reactor since the carrier gas helps to dissipate the heat of reaction and equilibrate the reactants temperature with that of the reactor. Therefore, the accuracy of the assignment of the temperature of the reaction is determined by the quality of temperature control available for the reactor. Experimental data from PDR’s permit the study of the mechanisms of reactions and to determine various rate constants associated with gas-solid reactions. Several applications are discussed in this section and actual cases are analyzed in the following sections. 1. Determination of the O r d e r of a Reaction w i t h Respect to t h e Solid. Isothermal tests using different amounts of solid in the reactor can usually yield data which allow the calculation of the order of the reaction. For example, where eq 21 applies, plots of log di vs. i yield a straight line whose slope is
[l - 41’-”] e ki
e
qAi-ll-n [ l - 4i1-y (1- n)Wi
(35)
The bounds on the value of ki depend on the value of i and on the size of the pulses. Smaller pulses will result in 4i 1 - t, t 0, or; 1 - 4i1-n 0. However, also Woand Widecrease, and since Wi/Wo< 1,the effect of decreasing the pulse size is larger on the upper bound than on the lower bound of k. Small conversions of the reactants will result in a narrower bounds on the calculated rate constant. However, smaller conversions require more accurate chemical analysis of the gaseous products. The following step-by-step procedure was found useful in calculations of rates and rate-constants. (1)Calculate STfor stoichiometry or from eq 33. (2) Evaluate 4i for each pulse using eq 30. (3) Estimate the upper and lower bounds on the rate constant using inequality 8, eq 21 and 27 or their analogues as appropriate for the kinetic model chosen. In order to calculate the specific rate of reaction, eq 36 and 37 have to be used.
-
-
-
(37)
V, is the specific molar volume of the reactive gas at the temperature and pressure of the reaction. 3. Determination of the Specific R a t e Constant. The specific conversion, R, can be calculated using eq 38.
The specific conversion [no. of molecules/((second X no. of sites)] gives a crude measure of the activity of the surface. However, the value of R, is well defined only at the very beginning of each run when the value of Sa,the specific surface area, is well known. Sachanges due to the progress of the main reaction, due to sintering and due to side reactions. 4. Determination of the Rate-Controlling Step. Three methods were described for the determination of the controlling step under “arguments of mass and heat transfer”. Several sets of data are analyzed in the Results section. 5. Determination of t h e Activation Energy. The activation energy of a reaction can be calculated from the slope of the curve of plots of log ri vs. l/Ti. By programming the temperature upwards one can obtain ri = ri(T). Straight lines are obtained which slope is E J R . As long as the rate of the chemical reaction controls, 3000 < E a / R 23000 K. However, when the rate of diffusion begins to control, one obtains 0 C E,/R < 2500 K. It is recommended to use relatively large samples, e.g., 0.4-0.5 g in the case of dolomite, and small pulses of reactive gas, e.g. lo4 mol/pulse in the case of H2S. In this way the available surface area of solid will not change significantly during the run. However, too large samples
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 4, 1979
7
1.4
1
-
12
AGE0 5 l/4 HR SlDRlTE 635' C
09
1
PURE DOLOMITE 57OOC 0 CALCITE 570'C A CALCITE 710'C 0 DOLOMITE 5 7 O O C X SIDERITE 570' C +
613
4
$2 0 4 -
0
0
02 01
O
4
8
.+ x
k v
,'. IO
2b
30
40 50 60
7b
80
do
12
I6
20
24
28
I O z w t % CONVERTED
Figure 5. The effect of the calcination time on the rate of reaction of siderite with H2S;0.1 g of -200 mesh siderite was reacted a t 635 "C with pulses of 0.517 mL of H2S. The pressure was 0.974 bar.
Yo CONVERTED
Figure 4. Isothermal rates of reaction of various minerals. Each sample was 0.100 f 0.005 g, -200 mesh. The pressure was approximately 0.974 bar. Additional details are given in the Experimental Section.
.
401
cannot be used because the reactor will not be "micro" anymore. The sensitivity of the analytical techniques determines the size of the reactive gas pulse that can be used. Results The most important variables that determine the rate of the consumption of H2S by the mineral samples are the temperature, the time-temperature history of the samples, the initial conditions of the samples, and the conversion. Two types of kinetic experiments were conducted: (1) "isothermal", and (2) "temperature programmed'. In each experiment, pulses of H2S with a fixed size were injected, and the amounts of COP,H20,and unreacted H2S were determined. The pressure was 0.942 bar in all cases. In addition, the raw carbonates were tested by DSC and SEM, and their specific surface area was determined by N2 porosimetry. Arguments of material balance can be used to deduce the following conclusions. (1)The total number of moles of MS that are formed is equal to the total number of moles of H2S that are consumed and also to the number of moles of water that are produced. (2) The number of moles of H20 and of C 0 2 that are produced in reaction 1 as a result of a given pulse of H2S is equal to the number of moles of H2S that are consumed in the reaction. The number of moles of H 2 0 , in excess to the number of moles of C02, is found by reaction 3. Figure 4 shows the isothermal rate of consumption of H P S by calcite, pure dolomite, dolomite, and siderite at 570 "C, as a function of the conversion of the solid. The rate of the reaction with H2S behaves according to one of three modes: (1) rate independent of the conversion, (2) rate decreasing with the conversion, but complete conversion is obtainable, (3) rate decreasing very rapidly with the conversion, and complete conversion is not obtainable. Figure 4 shows that the rate of reaction of dolomite at 570 "C increases somewhat as a function of the conversion. However, a t 700 "C the rate decreases slowly with the conversion. (The curve for dolomite at 700 "C is not shown.) The rate of reaction of calcite at 570 "C decreases very sharply and becomes essentially zero after about 2.8% of the material is converted. At 700 "C the rates of conversion of dolomite and calcite to MS are the same. The specimen of dolomite that was used in this study continued to react a t 570 "C and the formation of solid
..
3 '
2
I
A
c
SIDERITE
Y
I-
I
09
I
I I
' IO
" / I
"
12
'
1 I3
I4
I5
16
IO^,^., T
Figure 6. Rate of consumption of H2S by siderite, calcite, and dolomite in a temperature-programmed test; 0.1 g, -200 mesh samples were reacted with H2S pulses of 0.517 mL each. The pressure was 0.974 bar; the rate of temperature programming was approximately 3.3 OC/min. The curve become flat when the rate of mass transfer becomes the controlling mechanism.
Cas had little effect on the rate. The reaction was stopped artificially after 30% of the solid was converted. Impure dolomite reacts 2-2.5 times faster than pure dolomite at 570 "C. Half calcined dolomite and calcined calcite continued to react at 570 "C at least until 15 w t 7'0 of the sample was converted. The rate gradually decreased but was about 75% of the value of the initial rate. Siderite decomposes at much lower temperatures than CaC03 or dolomite, with FeO and C 0 2being formed. The decomposition of siderite is complete at 495 "C. Since FeO sinters much more rapidly than CaO, the available surface area of FeO decreases when the siderite is heated for prolonged times above 500 "C. Figure 5 shows the rate of consumption of H2S by decomposed siderite that was kept 2 l l 4 and 5lI4 h at 625 "C. Figure 6 shows the rate of consumption of H2Sby calcite, dolomite, and siderite whose temperature was increased at about 3.3 "C/min. The data were not corrected for the conversion of the material; the effect of the correction is more important at the higher temperatures where the rate is large. The rate of consumption of H2S by calcite and dolomite seems to be very similar above 650 "C; however, different rates are observed below 650 "C. Figure 7 shows the rate
614
Ind. Eng.
Chem. Process Des. Dev., Vol. 18, No. 4, 10
.L
1979
1 +DOLOMITE 0 CALCITE
+'
13-CALCITE 17K lmm=O.O59p .02
/
0 II
1.0
0.9
1.1
1.2 1.3 lO'/T *K
1.4
1.6
1.5
Figure I . Rate of evolution of C 0 2 from calcite and dolomite in a temperature-programmed test; 0.100 g, -200 mesh mineral samples were reacted with pulses of H2S,0.517 mL each; pressme: 0.914 bar; rate of temperature programming: 3.3 OC/min.
14- CALCITE+HeS 17K lmm=588Ao 700OC Figure 9. SEM pictures of dcite. Top: calcite crystals magnified 8500 times. When T < 634 O C such crystals are inert to H2S except for a surface layer of thiclmessof @FQ molecules. Bottom: reactivated crystals of sulfided calcite, magnified 8500 times. Activation was accomplished by heating the solid above 650 "C.
.01
ma
t
W8
mo
I
I
,,,
,f7
I
t
70,
t
607
ex
197
7 ,711
Y O
,
,
I
*c
I.I 1.2 IO~/T ' i ' Figure 8. Temperatureprcgrm test. The rates of formation of CO,, H,O, and the rate of consumption of H2S by CaCO. in a temperature-programmedexperiment, where a layer of Cas was formed on the calcite crystals. The rate of temperature programming was approximately 3.3 OC/min. The pressure was approximately 0.914 bar.
0.9
IO
of evolution of COz from calcite and dolomite as a result of reaction 1. The data show very clearly that the rate of evolution of COz from calcite decreases with the temperature up to 635 "C but then it increases monotonically with the temperature. The decrease observed in the initial rate of reaction of calcite is due to the coverage of the surface by CaS and the creation of resistance to mass transfer of COz. The plotted rate is not the initial rate of reaction of clean surfaces. However, as long as the rate of the chemical
reaction controls, the MS layer will make no difference by definition. Experimentally the rate of evolution of COz that results from the thermal decomposition of the carbonate shows as a continuous drift in the detector base line. However, COz that is formed due to the reaction of H2S with calcite appears as a peak prior to the H2S peak (Figure 3). The magnitude of the "drift" allows the calculation of the rate of the decomposition of calcite to CaO and COP The decomposition of calcite and dolomite to CaO were complete at 810 OC and the decomposition of siderite was complete at 495 "C. Figure 8 shows the rate of formation of water by reaction 3. The data show that the rate of evolution of water becomes almost constant at temperature above 685 'C. At temperatures above about 685 OC, the rate of reaction 3 is controlled by the rate of mass transport in the gas phase. Examination of the surface of calcite and calcite which has been reacted with HzS at 570 "C using SEM have shown that the surface remained very smooth and that the crystalline form did not change due to the reaction. However, the surface of calcite crystah which were reacted with O2 or heated above 640 "C became "wavy", "wrinkled and seems to have "bumps". Palladium coating was used in the tests and the magnification was 1.7 X lo4 (Figure 9). DSC of "sulfidized calcite showed a small shoulder around 640 "C which could correspond to the rearrangement of surface (Caz+Sz-)to small nuclei of stable Cas. Rate data on some selected systems are presented in Table 11. The data were derived on samples of particles
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 4, 1979
Table 11. Activation Energies for the Rate of Consumption of H,S b y Minerals
material
T, ”C range
calc:te
770
770
480
Fa, kcall controlling mechanism mol diffusion of gas through C a s rate of reaction with a fresh surface rate of reaction
0.0
17.3 37.0 4.0
rate of gas-phase mass transfer rate of reaction rate of gas-phase mass transfer rate of reaction
18.9 2.0 15.2
rate of gas-phase + mass transfer and sintering
0.0
-200 mesh, of about 0.1-0.2 g each. The pressure in the reactor was 0.942 bar unless otherwise specified. The solid material was permitted to equilibrate 21/4 h at the reaction temperature. Equation 22 with n = 2 / 3 has been used to evaluate the lower bound on the rate constant. The upper and lower bounds on the rates of the reactions of H2S with various minerals are summarized in Table 111. The calculations were done using eq 30 and 34. The rate constants can be converted to more conventional units by multiplying their value by the specific molar volume of HzS at the reaction temperature. Several sets of such data were published, e.g., West (1948). Note that most of the rate constants are based on the weight of the mineral. The authors suggest, however, that the rate data be converted to surface base whenever extrapolations to different particle sizes are needed.
Discussion of the Results The discussion is divided into two parts: discussion of the mechanism of the reactions, and discussion of the rates of the reactions. The Mechanism of the Reaction. The rate of consumption of H2Sis influenced by the rates of reactions 1-4, which take place simultaneously, and by the rate of mass transfer of H2S, H20,and C 0 2through the layer of product MS which is formed on the surface of the MC03 or MO crystals. The rate of the chemical reactions is a strong function of the temperature; therefore, the rate of different steps may control at different temperatures. In particular, the rate of decomposition of CaC03 and of dolomite (reaction 2) becomes very large at temperatures above 739 OC. Therefore, large amounts of MO are produced which compete with the MCOBfor the available HzS. Since CaO
reacts with H2S more rapidly than CaC03, the apparent rate of consumption of H2Sby the mixture of CaO-CaC03 at temperatures above 730 OC is significantly larger than at lower temperatures. In addition, the decomposition of the carbonates yields a fine powder of oxide which has a significantly larger specific surface area than the original crystalline carbonate (approximately 2-4 times in this case). Therefore, the apparent rate of consumption of H2S by the solid oxide will appear to be much larger. Mass transfer in the gas may also become the limiting step at very high temperatures. Figure 6 shows that in the experimental setup of this study the rate of mass transport in the gas film became controlling at about 500 OC in the case of siderite and around 780 “C in the case of calcite and dolomite. These observations correlate extremely well with the temperature of decomposition of the carbonates to the oxides. In other words, the rate of reaction of FeO, CaO, and half-calcined dolomite with H2S is so much larger than that of the corresponding carbonate that as soon as the oxides are formed the diffusion through the gas film becomes the controlling step. The superficial velocities of the helium when the gas film diffusion begins to control the rate of mass transport are approximately 290,403, and 420 cm/s for siderite, dolomite, and calcite, respectively. The rate of mass transfer in the solid may limit the apparent rate of reaction by preventing the reagent H2S from reaching the reacting material or by preventing the products COz and H20 from escaping. The role that the product MS plays depends on the crystalline structure of MS and that of “host” crystal, the MO or the MC03. If the crystalline structures of MS and the host are “compatible” so that the layer of MS can adhere to the surface of the host, as a “solid solution”, for example, then resistance to mass transport will be created as a result of the reaction. The application of the macroscopic concept of “solid solution” cannot be made directly in the case of a very thin surface layer (5-100 molecules thickness) since such a layer is no doubt in a metastable state. The microstructure of the surface of the solid that forms when the first S2- ions replace the CO2- or the 02-ions is not the same as that of MS or MO crystals since the M2+ions are still in the electrostatic field of the bulk of the crystal, namely the MO or MCOBcore. However, the Gibbs free energy of the system can be reduced if the surface layer will reach equilibrium. If the temperature is sufficiently high, the rate of diffusion of ions is fast and they may form a solid solution or nucleate as separate crystals, MS and MO, whose structures will be identical to the macroscopic structure of the corresponding salts. However, if the temperature is not sufficiently high, an unstable layer with an ill-defined crystalline structure will be formed. Such a surface layer will be unstable thermodynamically and its rate of “stabilization” will be limited by the rate of dif-
Table 111. Rate Constants for the Reactions of H , S with t h e Alkaline Minerals in Coal
material calcite calcite (CaO) sideriteb siderite siderite dolomite (pure) dolomite calcined calcite calcined dolomite calcined calcite calcined dolomite a
T,“ C 570 7 00 570 635 410 570 570 570 570 7 00 700
lower bound, (g)” Limo1 s 0.089 0.439 0.39 0.27 0.232 0.315 0.357 0.204 0.345 0.305 0.37 1
’
E , = dimensionless standard deviation = (u’)/av.
615
* Era 0.044 0.153 0.135 0.093 0.081 0.110 0.119 0.07 0.12 0.072 0.092
Equilibrated 511, h.
upper bound, (g)”’ Limo1 s 0.163 0.662 0.89 0.513 0.372 0.447 0.536 0.425 0.739 0.567 0.489
r Era 0.325 0.197 0.32 0.39 0.21 0.223 0.108 0.09 0.175 0.162
specific surface, m’lg 0.77
2.27 5.71 18.0
616
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 4, 1979
fusion of the ions on the surface. Any process that could destroy the surface layer or enhance the rate of diffusion of the ions on the surface will enhance the rate of formation of separate stable crystalline materials. For example, increasing the temperature and possibly water vapors enhance the rate of diffusion of ions. Oxidation of the sulfide ion S2-to the sulfate SO:- results in “breaking” the surface layer since the molar volume of SO-: is larger than that of S2-. Squires and coworkers (Pell et al., 1971; Ruth et al., 1972; Squires, 1972) observed an increase in the rate of reaction of half-calcined dolomite with HzS as a result of oxidation of the surface by oxygen. The peculiar dependence of the rate of reactions of calcite and H2S on the temperature seems to be unique to calcite. We did not observe similar phenomena in the case of siderite and semi- and fully calcined dolomite at 570 “C up to conversion of 30 wt ’70.Squires (1972), Pell et al. (19711, and Ruth et al. (1972), who used thermogravimetry to examine half-calcined dolomite, observed similar behavior to what we observed in the case of calcite. Pell et al. (1971) and Ruth et al. (1971) observed that the reaction of HzS with calcined and semi-calcined dolomite a t temperatures below 570 “C stopped after a given conversion, which depended on the reaction parameters. Squires and coworkers (Squires, 1972; Ruth et al., 1971) proposed another mechanism to explain the peculiar behavior of half-calcined dolomite. They postulated that a “surface complex” of CaO (solid) with H2S (surface) is formed on the surface of half-calcined dolomite and blocks the progress of the reaction. In their mathematical model (Ruth et al., 1972) it is implied that the “surface complex” is formed in a reuersible process, namely, the “surface complex” can decompose and expose fresh reactive surface again. We attempted to determine whether a “surface complex” is indeed formed by injecting small pulses of H2S and observing the shape of the pulse of gaseous products that exit the reactor. The peaks had no “tail” whatsoever, which indicates that reversible adsorption of H2Sdoes not occur at temperature above 400 “C. Squires (1972) pointed out that he had results which “shaked his faith” in the reversible adsorption theory for half-calcined dolomite; however, he did not present any detailed information. The crystalline structures of the most important crystals are summarized in Table I. Figure 4 shows that a 570 “C calcite stops to react after about 2.8 wt 70of the -200 mesh material was converted. Dolomite, which has the same crystalline configuration, continues to react although the initial rate of the reaction depends on the impurities in the material. Dolomite and calcite have the same crystalline structure, except that in dolomite every alternate Ca2+ion is replaced by a Mg2+ion. Had all the components of dolomite been reactive, one would expect that MgS and Cas would have formed as a result of the reactions with H2S. However, at 570 “C only Cas is formed since MgCO, does not react with H2S at 570 “C. Therefore, a “continuous” layer of Cas cannot be formed on the surface of dolomite but it can be formed on the surface of calcite. Therefore, once a layer of average thickness of about 78 molecules is formed on the surface of the calcite, at least one of the gases C02,H 2 0 , or COz cannot diffuse through it anymore and reaction 1 stops. It is plausible that the reaction between H2S and calcite stops because C 0 2 cannot diffuse through a layer of Cas. One piece of evidence which supports this assumption is that CaO reacts with H2S according to reaction 3 and is completely converted to Cas. Had Cas been impermeable to H 2 0 or to H2S, the reaction of CaO whould have also stopped before complete conversion to Cas. The dif-
1
)
,INJECTION NUMBER
Figure 10. The rate of reaction of FeO with H2S. In the reactor there was 0.100 f 0.005 g of siderite, -200 mesh. The data were analyzed assuming n = 2/3 and a first-order reaction with respect to the gas. The pressure was approximately 0.974 bar.
ference in the behavior of CaO and CaC03 cannot be used, however, as conclusive evidence in support of the assumption that the diffusion of C 0 2is blocking the progress of the reaction since CaO and CaC03 have different crystalline structures and the structure of a thin surface layer of (Ca2+,S2-)will not be necessarily the same in both cases. The latter theory is supported by the experimental evidence on the rate of evolution of C02by reaction 1and by the rate of absorption of H2S by reactions 1 and 3, as a function of the temperature (Figure 8). The experiment was conducted as follows: first the surface of the calcite was reacted with H2Sat 590 “C until the reaction stopped, and then the temperature was programmed up slowly while injecting small pulses of H2S to study the reactivity of the material. The data show clearly that around 640 “C the layer of C a s breaks, new surfaces of CaO CaC03 are exposed, and reactions 1 and 3 can commence. The equilibrium pressure of C 0 2 a t 640 “C is estimated to be about 0.02 atm. Such a pressure is not sufficiently large and cannot break the layer of Cas. It is very likely that at 640 “C the rate of surface diffusion of ions is sufficiently large and permits the surface ions to reorganize in the thermodynamically stable crystalline form of C a s and CaO. The crystalline structures of FeO and FeS are different and apparently FeS does not adhere to the surface of FeO. Indeed, even if it adheres, FeS seems to permit diffusion of H2S since reaction 3 can proceed at 570 “C into completion. However, limited resistance to mass transfer is observed when 3-4 wt 7% of the material is converted. FeO sinters at a much larger rate than CaO, and at high temperatures its available surface area decreases very rapidly with time. Figure 10 shows that at 570 “C, eq 22 with n = 2 / 3 fits the data very adequately, but at 635 “C, or 700 “C the description is less adequate. Equation 22 was derived from eq 17 in which the sintering effect which reduces the surface area was not taken into consideration. Note that an increase in the temperature results in an increase in the rate constant of reactions 3 and 4. The latter reaction reduces the surface area which is available for reaction with H2S according to reactions 1 and 3. Figure 6 shows that the rate of reaction of siderite with H2S above 500 “C is controlled by the rate of gas diffusion through the gas film.Therefore, the results
+
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 4, 1979 617
plotted in Figure 9 are consistent with the assumptions that the gas film diffusion controls and that the available surface shrinks according to a reaction with a 2/3 order with respect to the unreacted material. Note, however, that if the time scale for sintering was of the order of the time scale of the reaction, the 2 / 3 power law would have failed. Quantitative R a t e Data. The evaluation of the activation energies was done with caution, since they have greatly different values in different temperature regions. A summary of the data is given in Table 11. Squires (1972) reported that the activation energy of the reaction of half-calcined dolomite with gaseous H2S is about 23 kcal/mol. Pel1 et al. (1971) reported activation energy of 22.9 kcal/mol for the reaction of half-calcined dolomite with H2S. The agreement with the value estimated in this work is reasonable, considering that different mathematical models and different experimental systems were used in each study. We estimate that at 640 "C the value of E , is about 18.9 kcal/mol. In general, the rate of mass transfer controls when the reacting material is a carbonate, and the rate of reaction 2 or the rate of the gas-phase mass transfer controls when the reacting solid is the oxide. Small activation energies (0-4kcal/mol) are observed when the carbonate reacts and 10-25 kcal/mol is observed when the oxide reacts. However, note should be made of the decomposition reaction 2 in which the carbonate is transformed into an oxide. This reaction cannot be controlled and it tends to activate the solid even when the carrier gas contains C 0 2 (the rate of reaction 2 is suppressed in the latter case). Calculations show that the lower bound on the conversion of half-calcined dolomite at 570 "C in 5 vol % H2S, 0.942 bar total pressure, is 8.92% after 100 s. Ruth et al. (1972) reported conversion of about 10 wt % after 100 s a t 550 "C. The agreement is excellent although different dolomite samples were used with different particle size distribution. Our sample was -200 mesh 18 m2/g and the sample of Ruth et al. was -250 +270 mesh. The data of Ruth et al. (1972) show that the reaction stops after about 12 w t 70of the solid is converted, while our data show that dolomite keeps reacting until a t least 30 wt 5% of the material is converted. Extrapolation of our data suggests that complete conversion will be achieved after 180 s at 570 "C and 5 vol % H2S. Appendix R a t e of Reaction Proportional to t h e Number of Moles of t h e Solid-"Infinitely Porous Solid". If the entire amount of the solid in the reactor is equally available for reaction, the rate may depend on the first power of the number of moles of solid or on the weight of the solid. _ _dA = kAc dt The lower bound on the rate constant can be evaluated from the equation
The upper bound on k , k,, can be evaluated using the equation
or Ai =
exp[:
A0
+ wi]
Rate of Reaction of Simultaneous Reactions. Very often a solid can react with a gas in more than one way. Moreover, the solid may decompose or change phase. The latter types of reactions may be independent of the sequence of pulses. For example, the decomposition of a carbonate to the oxide and C 0 2 occurs almost independently from the reactions of the oxides and the carbonates with pulses of H2S. Consider the case where the solid is consumed by a first-order decomposition reaction which occurs continuously, e.g., decomposition, while equal pulses of gas of magnitude W o are introduced every 6, seconds.
_ -dA
= kA"c6 + kdA dt where 6 assumes the value 1 when a pulse of gas is present in the reactor and 0 the rest of the time. In most practical cases, the residence time of the pulse in the reactor, T , is short compared with the cycle time, BC. Therefore, it is possible to develop an approximate solution for the rate of depletion of the solid. Two cases should be differentiated
I.
kdA
>> kA"c
(A@
and
II. kdA
