a kinetic study of ammonia synthesis: modeling high-pressure steady-state and forced-cycling...

Chemical Engmeering Scw~ce, Vol. 44, No. 1, pp. 9-19, 1989 Printed in Great Britain.

ooo9-2509/89 $3.00+0.00 0 1989 Pergamon Press plc

A KINETIC STUDY OF AMMONIA SYNTHESIS: MODELING HIGH-PRESSURE STEADY-STATE AND FORCED-CYCLING

BEHAVIOR

LEROY CHIAO and ROBERT G. RINKER’ Department of Chemical and Nuclear Engineering, University of California, Santa Barbara, CA 93106,

U.S.A.

(Received 1 August 1987; accepted 13 June 1988)

Abstract-The kinetics of ammonia synthesis under non-steady conditions on promoted industrial iron catalysts are not weil understood. Although classic models, such as the one proposed by Temkin and Pyzhev [Acra Physicochem. 12, 327-356 (1940)], fit steady-state data and are widely used in commercial reactor design, they fail to describe cycling kinetic behavior. A phenomenological adsorption/desorption (A/D) model is developed, the parameters of which are fit to experimental data from a differential reactor. This model predicts observed experimental integral reactor forced-cycling and steady-state behavior for several temperatures and pressures. Moreover, the model obeys thermodynamic equilibrium constraints. The existence of nitrogen storage in the bulk phase of iron catalysts is confirmed, but we show that it does not directly affect either steady-state or time-dependent ammonia production. Also, several potential pitfalls in kinetic modeling are exposed.

INTRODUCTION

It has been shown both theoretically (for example, Bailey and Horn, 1971; Lynch, 1983; Grabmuller et al.,

1985; Thullie et al., 1986, 1987a, 1987b) and experimentally (for example, Cutlip, 1979; Wilson and Rinker, 1982; Barshad and Gulari, 1985; Chiao et al., 1986) that forced concentration cycling can result in time-averaged production rates that are higher than the optimal steady-state rate. In the case of ammonia synthesis, the kinetics under cycling conditions are not well understood. Steady-state kinetic models, such as those proposed by Temkin and Pyzhev (1940), Ozaki et al. (1960) and Brill et al. (1967), were shown by Wilson (1980) to be incapable of describing cycling kinetic behavior. Wilson (1980) proposed a two-site kinetic model which successfully fit his low-pressure (1.14 MPa) forced-cycling data. However, the Wilson model agrees only qualitatively with high-pressure (4.24 MPa) data (Chiao et al., 1986). Recently, Li et al. (1985a, 198513) proposed a model which fits moderate pressure (2.38 MPa) ammonia synthesis forced- cycling data. The dynamic model includes a nitrogen storage term, written such that the stored nitrogen in the bulk catalyst contributes directly to dynamic ammonia production. Unfortunately, since Arrhenius parameters are. unavailable, we cannot check the performance of the Li model with our experimental data.

The goal of the present study was to develop a phenomenological kinetic model for high-pressure ammonia synthesis, which fits dynamic and steady- state data, as well as conforms to thermodynamic equilibrium limits. Moreover, the kinetic mechanism must consist of realistic elementary steps. The kinetic

?Author to whom correspondence should be addressed.

parameters are fit, whenever possible, to individual experiments designed for single-parameter identifi- cation. Thus we avoid optimizations with many de- grees of freedom, results of which often have no physical basis.

The experimental apparatus is as described pre- viously (Chiao et al., 1986) with a few modifications. In the interest of brevity, a detailed description (Chiao, 1987) is not included here.

At this point, we introduce some important defi- nitions related to concentration forcing. As the cycle time z approaches zero, a plug-flow reactor (PFR) system approaches the relaxed steady state (RSS) as introduced by Bailey and Horn (1971). Here, the concentration oscillations occur so rapidly that the catalyst surface concentrations cannot respond to the gas-phase changes and thus attain an invariant relaxed state, different in general from steady state. It has been shown that, for a catalyst of constant site density and activity, a PFR under relaxed steady state should yield the best time-averaged production rates (Thullie et al., 1987a). In order to operate near the relaxed steady-state region, z must be of the same order or less than the catalyst relaxation time, t,.

The quasi-steady state (QSS) is defined as the limit when the cycle time approaches infinity. The situation here is that steady state is achieved during each portion of the cycle, and the time-averaged reaction rate (rpss) is simply the weighted average of the two steady states, r,, r,,. Consequently,

ross=yr,+(I --Y)r,, (I)

where the cycle split y is defined as the fraction of the cycle that the stream containing the higher concentration of nitrogen flows to the reactor. Note that, for a catalyst of constant site density and activity, rQSS is always less than the optimal steady-state rate ross. Other important cycling parameters to consider are

9

10 LEROY CHIAO and ROBERT G. RINKER

the compositions of the alternating feed streams (y&, y:). By convention, gN>y:. Cycling amplitudes are sometimes used and are defined by:

Ai = yh - yg” for ygss I yh (2a)

Ai = ygss - yh for ygss 2 yh (2b)

where y, Oss is the mole fraction of feed-nitrogen at optimal steady state. The cycling operation is sym- metric if A, = A,, = A. Finally, in comparing the limr- aueraged production rates (PR) of ammonia between forced-cycling (FC) and optimal steady-state (OSS) operation, we define a production-rate enhancement factor:

‘P = {PR,c)I{PRoss). (3)

CATALYST STATE CHANGES

Chiao et al. (1986) report catalyst state changes during various modes of reactor operation and ident- ify three operating regimes: The reduced state, which occurs immediately after initial catalyst reduction by hydrogen at elevated temperatures; the actioe state, which is attained after long exposure of the catalyst to a steady-state reaction mixture; and the super-actiue state which results from FC operation with favorable cycling parameters. These observations are based on experimental integral reactor data. At each successive state, the reactor gave higher production rates than at the previous state. Also, Chiao et al. (1986) report that conversion from one state to another is reversible. Although several other investigators using promoted iron catalysts do not report catalyst state changes (for example, Wilson and Rinker, 1982; Jain et al., 1982), Coucouvinous (1983) alludes to the existence of an active state, and Rambeau and Amariglio (1978) also show experimentally that an active state does exist. It would be easy to miss observation of the state change from reduced to ache, for example, since the process is gradual and very slow, on the order of 30 to 300 hours, using steady-state mixture flow.

The observed production rate differences (catalytic state changes) can be attributed to one or both of two explanations: The actual catalytic activity changes (i.e. the turnover frequency changes), or the number of active sites changes. The first explanation is favored by Rambeau and Amariglio (1978). Chiao et al. (1986) also lean in this direction, but the opinion is strictly an intuitive one since no data were available for site density estimation.

Most recent experiments performed in our laboratories provide evidence for the second explanation. Nitrogen A/D experiments show that the number of active sites varies according to catalyst state when the data are fit to dissociative Langmuir isotherms. We suspect that the change in the number of sites is due, at least in part, to reconstruction of the catalyst at the microcrystalline level. Brill et al. (1967) and Bozso et al. (1977) both report that the (Ill) crystal face of iron

is orders-of-magnitude more active for nitrogen chemisorption than the (100) face, suggesting that the (111) sites can be considered as the only active sites. Moreover, they both report that formation of Fe,N on the surface of, and in the bulk-phase of, the iron catalyst orients the crystal structure of the iron atoms into the (111) configuration. Thus, if Fe,N is being formed, the (100) sites are being transformed into the more active (111) form.

Frankenburg (1955) established that direct Fe,N formation on irori at 673 K occurs only if the nitrogen partial pressure is greater than 100 MPa, far above our operating pressures. However, Kemball (1966) and Boudart and Dje’ga-Mariodassou (1984) report that, if the nitrogen is replaced with 0.10 MPa of ammonia under the same conditions, then the nitride is formed. They both use virtual pressure arguments to explain this phenomenon. Steady-state ammonia synthesis can result in high virtual pressures of nitrogen, thus facilitating the formation of the iron nitride and subsequent (100) to (111) site reconstruction. Virtual nitrogen pressures from direct ammonia exposure are higher still and should therefore also convert the reduced state to the active state. To test this theory, we exposed catalyst which had been in the reduced state for several weeks to a mixture of 5% ammonia in helium in an experiment performed at 703 K and 0.65 MPa (compare to steady-state mixture activation at 703 K and 4.24 MPa). We found that the steady- state ammonia production rate came up to the same level as that for the steady-state mixture activation case, providing strong evidence that the formation of Fe,N plays an important role in the reduced-to-active state transition. We suspect that the transition from. the active to super-active state follows the same mechanism: during FC operation, some conditions can cause transient regions of very high virtual nitrogen pressure in the reactor bed, thus creating an environ- ment for even more iron nitride formation and consequently more (111) site construction. At present, there is no evidence to suggest that both the number of sites and the turnover frequency are different upon catalyst state change. As discussed later, nitrogen A/D experiments on the active and super-active states, for ob- taining and comparing rate constants with those obtained for the reduced state, were essentially impossible in our laboratories due to out-diffusion of stored nitrogen in the catalyst bulk phase.

Consequently, our modeling strategy has been to examine all three observed catalyst states and to assume that the state transitions are due only to changes in the total number of active sites, presumably through Fe,N formation. Moreover, the state transitions are treated as reversible as has been experimentally confirmed in our laboratories. We do not attempt to model the actual kinetics of the catalyst state transitions. All kinetic modeling experiments in our laboratories were performed in a differential plug- flow reactor with an inlet Row rate of 8.34 x 10m5 standard m”/s and a catalyst mass of 1.5 x LOe3 kg.

Kinetic study of ammonia synthesis 11

NITROGEN ADSORPTION/DESORPTION

Many researchers have observed that the presence of hydrogen has an accelerating effect on the chemisorption of nitrogen on promoted iron catalysts (for example, Joris et al., 1939; Kummer and Emmett, 1951; Tamaru, 1963; Takezawa and Toyoshima, 1970; Wilson, 1980; Coucouvinous, 1983) which is in agreement with experimental observations in our laboratories (Chiao, 1987). We performed two types of nitrogen A/D experiments: experiments on hydrogen pre-treated catalyst and on helium pre-treated catalyst. In every case, the hydrogen pre-treated catalyst yielded larger rate constants and surface coverages.

Boreskova et al. (1982) claim that the accelerating effect of hydrogen is due simply to the removal of adsorbed oxygen which has a poisoning etTect on iron catalysts. However, calculations show that the maximum possible contamination of our catalyst by oxygen and water is well below the level which Boreskova et al. call “poisoned.” It has been suggested (Ozaki et al., 1960; Morikawa and Ozaki, 1971) that pre-adsorbed hydrogen accelerates nitrogen adsorption via the formation of an adsorbed NH inter- mediate. An effective driving pressure (EDP) concept (Chiao, 1987) suggests that the presence of hydrogen reacting with the adsorbing nitrogen induces a virtual pressure which is thought to contribute to the overall adsorption driving pressure of nitrogen, thus accelerating chemisorption. These various theories are considered in the phenomenological modeling.

Nitrogen adsorption experiments on the reduced catalyst were conducted according to the following procedure: After the appropriate catalyst pre-treatment (at least 120s exposure), the feed stream to the reactor was switched to pure nitrogen at 4.24 MPa for a specific adsorption time. The flow was then immediately switched to pure hydrogen at the same temperature and pressure, and the instantaneous outlet ammonia production was monitored. Integration of the ammonia production vs time curve gave the total amount of ammonia produced for a given nitrogen exposure. Assuming that each ammonia molecule occupied one active site, the number of sites covered was calculated. In this manner, adsorption curves were obtained for a variety of temperatures.

It was determined that, within our operating ranges of temperature and pressure, adsorption steady state is achieved after approximately 180 s. Thus, for the desorption experiments on the reduced catalyst, the catalyst, after an appropriate pre-treatment, was exposed to pure nitrogen at 4.24 MPa for at least 3&s. The feed stream was then switched to pure helium for a specific desorption time, after which pure hydrogen was run through the reactor. In a manner identical to the adsorption calculations, integration of the instantaneous ammonia production curves allowed construction of desorption curves for the same temperatures explored in the adsorption case.

Langmuir isotherms were used to model our ni-

trogen A/D experiments. This resulted in fewer parameters to be fit numerically. Numerical least-squares procedures were performed using,the LMDIF routine (Sandia National Labs). The data fit the dissociative Langmuir models very well, in agreement with Bozso et al. (1977). The reduced catalyst A/D parameters were fit as follows: From the desorption data, initial estimates of the desorption rate constants kdes and the initial adsorbed amounts S$ were made using least- squares fits to the appropriate form of the Langmuir desorption equation, i.e.

l/S, = 2k,,,t + 1 /S; (4)

where S, is the kgmols of adsorbed nitrogen per unit weight of catalyst; SE is the initial value of S,; and t is the elapsed desorption time. Next, initial estimates of the adsorption rate constants kads were made using the Langmuir adsorption equations to fit the adsorption data using the initial estimates of kdes and Sg. Thus,

S, = C + Dl” coth {coth - 1 ( - C/D”*) - 2BD”‘t) (5a)

where

B = kadsPN - kdes

C = k,ds&QrlB

D=C(C-QQ,)

(5b)

(W

(54

Qr = So,/@ (W

0 = (I&PNy/( 1 + (KadP#2) (W

Kad = kadslkdes. (3%)

In eqs (5), PN is the nitrogen partial pressure; Q, is the total kgmols of active sites per unit weight of reduced catalyst; K,, is the A/D equilibrium constant; and 0 is the fractional surface coverage. With these initial estimates of kads, kdes and SE, eqs (5e, f, g) were used to calculate initial estimates of Q,. Finally, these initial estimates were fine-tuned by using the A/D data and performing a three-parameter least-squares optimization on kads, k,,, and Q,, using eqs (4) and (Sa, b, c, d). Typical optimization results are shawn in Figs l(a, b).

After all the rate constants were optimized, Arrhen- ius plots were prepared and correlations fit (Table 1). The value of Q, was observed to be a function of temperature. This is not surprising, if one recalls that the (111) crystal planes are believed to be the active sites. It is reasonable that the equilibrium between the relative amounts of the different crystal planes should at least depend upon temperature. Because the calcu- lation of Q, in this manner contained significant error, the Q, values here were not correlated with temperature. The values of Q, used for the correlation were fitted to reduced catalyst steady-state data, along with the forward surface reaction rate constants, and are discussed later.

Nitrogen storage in the bulk-phase of iron catalysts has been reported by many researchers (for example, Grabke, 1968; Jain et al., 1982). Work in our laboratories confirms that storage exists in the active and


(4

10.0

('3

4.0

0 100 200 300 400 500 600 700 600 900 TIME (s)

Hydroge; Pre-Treated Catalyst D

.

Helium Pre-Treated Catalyst

100 200 300 TIME (s)

Fig. 1. (a) Typical nitrogen desorption isotherms with optimized fits. Pressure and temperature are 4.24 MPa and 703 K, respectively. Catalyst was exposed to nitrogen for 300 s after 120 s pre-treatment (He or HZ). Desorption was in He. Solid lines are dissociative Langmuir isotherm fits. (b) Typical nitrogen adsorption isotherms with optimized fits. Pressure and temperature are 4.24 MPa and 703 K, respectively. Catalyst was exposed to pre-treated He or H, for 120 s.

Solid lines are dissociative Langmuir isotherm fits.

Table 1. fiitrogen A/D Arrhenius parameters

ki=k?exp(-EJRT)

ln(ky)=4.259 kg/kgmol-s-MPa u,,, = 0.283 EJR = 1.027 x lo4 K- ’ 6, = 1.207 x lo3

ln(k’? 1) = 0.403 kg/k$mol-s a, = 4.748 x 1O-3 E_ ,/R = 1.168 x lo4 K-’ 6, = 1.999 x 102

In(k2) = 3.168 kg/kgmol-s-MI% ci,,, = 0.380 E,/R = 5.677 x lo3 K-’ cm = 1.621 x lo3

In(k’? 3) = 0.342 kg/kgmol-s-MPa 0,=2.000x 10-z E_,/R=8.939 x 10” K-l ci,,, = 8.416 x 10’

i= 1, - 1: Helium pre-treated catalyst. i=3, - 3: Hydrogen pre-treated catalyst.

super-active catalyst states. However, we also found that it does not exist in the reduced state. The following experiment was performed over the active catalyst: After a thorough hydrogen sweep of the reactor (no outlet ammonia detected), the feed stream was switched to pure helium for 600 s. Then the feed was switched back to pure hydrogen, and ammonia production was observed. The only source of adsorbed nitrogen was from diffusing bulk-phase nitrogen. Note that this’ experiment, performed on a reduced catalyst

resulted in no ammonia production. These results are physically reasonable: under conditions of the active and super-active states, high virtual pressures of nitrogen exist for long times. Thus we expect diffusion of nitrogen into the bulk to occur. The reduced catalyst exposure to long-term hydrogen removes any stored nitrogen. We have observed that prolonged exposure of the catalyst in the more active states to pure hydrogen returns it to the reduced state.

The stored nitrogen diffusion renders it impossible to obtain nitrogen A/D data for the active and super- active states using the described procedure. After adsorbed nitrogen is hydrogenated or desorbed, the bulk nitrogen begins to diffuse to the surface to occupy the empty sites.

For the helium pre-treated A/D data, the modeled mechanism is:

N, + 2S* = 2N-S* with rate constants k,, k _ 1. (6)

For the hydrogen pre-treated A/D data, an EDP model uses k, , k _ 1, following the example developed by Chiao (1987). Alternatively, Ozaki et al. (1960) suggest the following adsorption mechanism:

N, + 2H-S* = 2NH-S* with rate constants k,, k- 3.

(7)

We are not equipped to accurately measure hydrogen A/D in our laboratories, but at our operating conditions ( T> 673 K, Pt 2 4.24 MPa) hydrogen A/D should be very rapid. Thus we assume that hydrogen A/D is essentially instantaneous, for modeling purposes:

H, + 2S* = 2H-S* essentially instantaneous. (8)

Also, we assume that the dimensionless equilibrium constant for hydrogen A/D is unity. Finally, we can model the hydrogen pre-treated A/D data with a mechanism written like eq. (6) using the rate constants in eq. (7):

N, + 2S* = 2N-S* in the presence of hydrogen with rate constants k,, k _ J (9)

where the effect of hydrogen is uncertain. Modeling of nitrogen A/D in this manner is consistent with arguments by Boreskova et al. (1982).

PHENOMENOLOGICAL MODELING

Ammonia synthesis investigators frequently dis- agree on whether the hydrogenation of adsorbed nitrogen and nitrogen intermediates proceeds via Eley-Rideal or Langmuir-Hinshelwood steps. Our experiments, although not conclusive, suggest that at least one Eley-Rideal step occurs. When hydrogen is passed over pre-adsorbed nitrogen, a large pulse of ammonia is formed. However, when nitrogen is passed over pre-adsorbed hydrogen, only very small trace amounts of ammonia result, slightly above the detec- tor noise level. These experiments were performed at 703 K and 4.24 MPa. Although hydrogen desorption is very rapid, we expect that it would not all desorb


before some measurable ammonia synthesis occurs, if the hydrogenation steps are of the Langmuir-Hinshelwood type. With this background, including the dissociative Langmuir isotherms for nitrogen A/D, we propose two potential mechanisms which can also be expressed in terms of EDP kinetics:

Kinetic model Ml:

H, + 2S* = 2H-S* essentially instantaneous (lOa)

N, + 2H-S* = 2NH-S* (lob)

NH-S* + H, = NH, +S* rate constants c4, k_ 4.

(1Oc)

Kinetic model M2:

N, + 2S* = 2N-S* (1 la)

2N-S* + 3H, = 2NH, -t- 2S* rate constants k,, kp4_

(1 lb)

For the classical kinetic cases, the nitrogen adsorption steps, eqs (lob), (lla), are governed by the rate constants kJ, /c_~; in the EDP kinetic cases, the rate constants k,, k_ 1 are used. The overbar rate constants are compatible with per-unit-fluid-volume rate expressions, whereas the non-overbar rate constants refer to per-unit-catalyst-weight rate expressions. The A/D constants can be converted as follows, assuming ideal gas law:

k&R V~ca,)exp = k,d, (12a)

kdes(llP&ex~ = k,,, (12b)

PC.M = WL/&VR_ (12c)

The “exp” subscript indicates that the constants used should be those for the experiments to which the rate constants were fit. The transient kinetic rate expressions and dimensionless parameters are defined for the Ml and M2 models in Tables 2 and 3.

SURFACE-REACTION PARAMETERS

Steady-state data from the differential reactor were obtained experimentally for all three catalyst states at various temperatures. The surface-reaction forward rate constants and Q, values were fit, in two-parameter optimizations, to reduced catalyst data using the LMDIF least-squares routine. Initial values for Q,

were obtained from the nitrogen A/D optimizations. The surface-reaction reverse rate constant is a dependent variable since we require that the kinetic models obey thermodynamic equilibiium. For example, in the case of kinetic model Ml:

and

K,=K,K,K~, where K,= 1

Ki=~i/~-i

(13a)

(13W

where K~, K_~ are the dimensionless rate constants of step i, which are defined later. The parameter K, is the

Table 2. Transient Ml and EDP-Ml reaction rate expressions

I-N= -ra,+r32

rH=-r31+rr32-r41+rr42

r,=r,, -rd2

re=W3,--r,,)--r,, +r,z A=- ‘al+‘32

rS1 =~~y~(f-@)*

r 32 =K_30z

r41 ‘GY”@

r 42=K-&Y,4(1-@)

ICY = TQ~’ p:=, .&

~_~=RTsQfp&,ii-~,lP,

‘h = rQi~eat&

K-,=rQi~o,tE-,

M 1 equilibrium condition:

(2.1)

(2.2)

(2.3)

(2.4)

(2.5)

(2.6)

(2.7)

(2.8)

(2.9)

(2.10)

(2.11)

(2.12)

(2.13)

K_.+=K~(K,/K,)“* {see eqs (13)} (2.14)

i = r, a, s for reduced, active, super-active catalyst states.

For EDP-Ml (Chiao, 1987): 1. Substitute subscript “1” for subscript “3”

in eqs (2.1)-(2.11). 2. In eq. (2.6), replace Y, with (edp):

tedp) = YN + { Y:/(YN Vi) 1 (2.15)

3. Replace eq. (2.14) with:

K_d=Kq(2K3/Ky)1’2 (2.16)

4. Use remaining equations as written.

dimensionless overall equilibrium constant, and Ki is the dimensionless equilibrium constant of step i. As was discussed earlier, K, is set to unity for modeling purposes. Rearranging eqs (13):

IC_~=IC,(K,/K,)~‘~. (14)

Since K,, K, are known, we see that K --4 is a dependent variable of K.,.

Thus, from reduced catalyst data, Arrhenius correlations were developed for ICY and Q,. Since catalyst state transitions were modelled as changes only in the number of active sites, the rate constants developed from the reduced catalyst apply to the uctive and super-

active catalyst states as well. Therefore, Arrhenius correlations were developed for the active site density Q. by single-parameter least-squares optimizations to the steady-state differential reactor data. Unfortu- nately, it proved impossible to estimate the super- active site density Q, at the higher temperatures, apparently due to the transition kinetics. We found

that the site density would change at the higher temperatures before our experiments could be com-

pleted. However, good data were obtained at 673 K

[Fig. 2(b)], and we were able to determine values for Q.. These results were applied to the higher temperatures by using the multiplier (Qs/Qa)673.

Although differential plug-flow reactor data can be


Table 3. Transient M2 and EDP-M2 reaction rate expressions

I-,= -rj, fr,,

l-H=3(-*41 +r,*1

FA=2(rd1 -r42)

F~=2(r31-rJz-r41+r42)

A= --r,,+r,,-r,, +rb2

rll,=K,y,(l-C9)2

rsz=L3@2

r4, = KAYO@

r42=rc_4y:(1-@)2

~3 = TQ? P% K_3=RTsQfp:a,E-JP,

KL=~(P,IRT)~Q?P:.,& K_~=~(P,/RT)QFP~ A_ L CM 4

(3.1)

(3.2)

(3.3)

(3.4)

(3.5)

(3.4)

(3.7)

(3.8)

(3.9) (3.10)

(3.11)

(3.12)

(3.13)

M2 equilibrium condition:

K_a=Kq(K~/~y) {analogous to Ml treatment) (3.14)

i = r, a,.s for reduced, active, super-active catalyst states.

For EDP-M2 (Chiao, 1987): 1. Substitute subscript “1” for subscript “3”

in eqs (3.1)-(3.11). 2. In eq. (3.6), replace y, with (edp):

(edp)=~~+~Llk~Z,) (3.15)


K-4 = ~(2&lK,) (3.16)

4. Use remaining equations as written.

treated algebraically, we decided to integrate the full set of ordinary differential equations in order to obtain more rigorous results. We used the RKF45 routine (Sandia National Labs), a 4th, 5th order Runge-Kutta method with sixth order extrapolation.

Figure 2(a, b) shows the differences between the catalyst states. The solid lines are Ml optimizations to the steady-state data. The differences in production rates are very clear. All models fit the differential reactor data well and yielded good Arrhenius correlations (Table 6). For example, Fig. 3 contains the EDP-Ml optimization results for the reduced catalyst.

At this point, we have four very different kinetic models all of which contain independently-fit A/D kinetics; all fit steady-state data equally well; all yield very reasonable Arrhenius parameters; and all satisfy chemical equilibrium. Moreover, no unreasonable multi-parameter optimizations were used. Except for fine-tuning the single-parameter A/D optimizations (3-parameter), the other optimizations were all either single- or two-parameter. Thus, even though our modeling approach has been conservative, several kinetic models still fit the data equally well. Rigorous testing is required to discriminate between them. This illustrates some of the perils involved in kinetic modeling.

(4 FEED NITRCGEN MOLE FRACTlON

W FEED MOLE FRACTION OF NITROGEN

Fig. 2. (a) Comparison of active catalyst and reduced catalyst at steady state. Pressure and temperature are 4.24 MPa and 703 K, respectively. Solid lines are optimized fits of model Ml. (b) Comparison of super-active catalyst and active catalyst at steady state. Pressure and temperature are 4.24 MPa and 673 K, respectively. Solid lines are optimized

fits of model Ml.

ae 0.5, I

0.0 0.2 0.4 0.6 0.6 1.0 FEED MOLE FRACTION OF NITROGEN

Fig. 3. Reduced catalyst behavior at steady state for various temperatures and at a pressure of 4.24 MPa. Solid lines are

optimized fits of model ED&‘-Ml.

Kinetic model simulations are usually compared directly to the data to which they were optimized but should apply to other reactor configurations as well. Thus, the next test was to see how well our four models (which were fit t6 differential reactor data) agree with integral reactor steady-state data.

SCALE-UP TO INTEGRAL REACTOR STEADY STATE

After the governing equations were changed to simulate the integral reactor, both EDP-Ml and EDP-M2 failed. Both models predicted steady-state curves that were much lower than experimentally observed. All steady-state comparisons presented here were for the active catalyst state. Kinetic model M2 gave an acceptable fit to the integral reactor data, but


Ml is in better agreement [Fig. 4(a)]. Figure 4(b) shows Ml performance over the pressure range stud- ied, and we see that it fits the data well. It should be emphasized here that the parameters for these models were all fit to entirely different data, i.e., these plots do not compare the models to the data to which they were optimized. Furthermore, the rate constants used in these figures and the figures which follow are the correlated values (Arrhenius parameters), not the directly optimized ones.

The scale-up test shows that, under steady state, M 1 and M2 kinetics are transfercable to integral reactor data but the EDP models are not. That Ml model fits the data better than M2 and is more reasonable physically. However, it is premature at this point to designate one as the final kinetic model because the final model must correctly predict dynamic behavior as well.

CATALYST STORAGE CONSIDERATIONS

As metioned earlier, Li et al. (1985a, b) have developed a transient kinetic model for ammonia synthesis based on the concept of nitrogen storage within the bulk-phase of the iron catalyst. They show that, without including bulk storage, their model fails to describe their FC data. Li et al. use the film theory of mass transfer, and develop a 2-parameter lumped mass transfer equation of the form:

dC,/dt=(p,,,lW,)~(crP,-CC,) (15)

where p is a lumped mass transfer coefficient and 0: is a

2.5, 1

0.0 0.2 0.4 0.6 0.6 1.0

FEED MOLE FRACTION OF NITROGEN

PRESSURE (MPa)

Fig. 4. (a) Kinetic model Ml scale-up from differential reactor to integral reactor at steady state for active catalyst. Pressure and temperature are 4.24 MPa and 703 K, respectively. Rate constants for model Ml were fit to differential reactor data. (b) Integral reactor performance at optimal steady state for active catalyst at a temperature of 703 K.

Solid line is prediction of model Ml.

lumped solubility parameter. The variable P, is the nitrogen partial pressure and C, is the concentration of nitrogen in the catalyst bulk. The mass flux expression is adapted for, and added to, the transient mass-balance equation for adsorbed nitrogen. A dimensionless form of eq. (15) is also included in the model.

Though an interesting idea, it seems that the time scale for nitrogen diffusion into the catalyst bulk

should be very large compared to the reaction and FC

time scales. Also, the question arises as to whether the catalyst can hold enough nitrogen to generate a mass flux which directly contributes, in a significant manner, to ammonia synthesis. Since eq (15) equates a

steady-state mass transfer expression to a time deriva- tive, such lumped-capacitance methods are good approximations only when the internal resistance to transport is much lower than that at the boundary. Normally, however, nitrogen diffusion through a catalytic iron matrix should encounter much larger re- sistances than at the catalyst surface.

Despite these reservations, we felt that the concept should be investigated. Using a correlation developed by Grabke (1968), we first estimated the initial amount of bulk nitrogen in the iron catalyst at steady state. Thus

CN (mol/cm3) = (pNH,/piy) 1.1 x lo2

x exp[- 17,75Ocalmol-‘/RT]. (16)

Also, since aP, = C, at steady state (eq. 15) we calculated LX From experiments performed in our laboratories, we estimated the maximum mass transfer rate of nitrogen from the bulk catalyst. After switching the feed to the reactor from a steady-state mixture (actioe

catalyst state) to pure hydrogen, we monitored the

ammonia production until a pseudo-steady-state con-

dition existed. Assuming diffusion of bulk nitrogen to

be the slow step in the production of ammonia under these conditions, the maximum mass transfer rate was calculated. Using the appropriate form of eq. (15), we then calculated p. These values are compared with

those reported by Li et al. (1985b) in Table 4 by converting their “gatom” units to “gmol” units. The

storage parameter values which we report here were

calculated individually based on individual exper-

Table 4. Nitrogen bulk-storage parameters

Li et al. (1985b) Chiao et al., this paper

0.5882 1.823 x lo-” 3.050 7.313 x lo-”

Conditions for Li et al.: P,=2.38 MPa, T=673 K. Par- ameters optimized to FC data. Errors in the units used for certain variables in the Li et al. paper have been corrected in the above table.

Conditions for Chiao et al.: P, = 4.24 MPa, T= 703 K. Parameters calculated directly from experimental data using experimental correlation by Grabke (1968).


iments and an experimental correlation; the values reported by Li et al. (1985b) were fitted directly to their FC data.

differential equations to ordinary differential equa-

Nitrogen storage should affect dynamic operation only, and the Li model correctly predicts this. Im- plementation of the storage term into Ml (active catalyst state) produced measurable but insignificant differences in the dynamic simulations compared to Ml (active catalyst state) without the storage term. This is true regardless of which parameter set in Table 4 was used. The transient equations for Ml and storage-Ml appear in Table 5. Even when the storage parameters were arbitrarily increased by an order of magnitude, the storage model still made no difference. In addition, the storage model cannot predict super- ache performance when the catalyst is in the active state. Thus, we conclude that, while bulk nitrogen storage definitely exists, it does not contribute directly to ammonia synthesis, at least under high-pressure

conditions.

tions, were not used since it has been shown in general that they are unsatisfactory form modeling PFRs under FC operation (Chiao and Rinker, 1987).

DYNAMIC RESULTS

Figure 5 shows the enhancement factor dependency on the cycle time for pure-component cycling, and clearly the Ml simulation for the active catalyst state fits the experimental data well. Figure 6 shows the same type of plot, but for (0.40, 0.20) cycling at 4.24 MPa. Similar results were obtained at 5.54 MPa but are not plotted in Fig. 6. The simulation point at T = 0 is obtained through a relaxed steady-state calcu- lation as described by Bailey and Horn (197 1) at the reduced catalyst state. The simulation points at t =30 s and t=40 s are calculated for the active state. The solid line is an interpolation fit, and corresponds to the assumption that the catalyst transforms smoothly from the active to the reduced state as =

approaches zero. The point at z=4 s is explained by Chiao et al. (1986): plots of experimental input wave-

form show that the square-wave input is seriously damped by mixing in the switching-valve system.

Kinetic model M2 failed preliminary dynamic simulations, and neither of the EDP models survived the scale-up test. Consequently, only Ml was put through rigorous dynamic testing. The partial differential equations (Table 5) were solved numerically using the method of characteristics, as described by Acrivos (1956). Simulations were run on a DEC VAX 1 l/780. Simplifying approximations, such as the differential reactor approximation or the staged-CSTR approximation, whieh convert the transient PFR partial

Thus, the actual cycling amplitudes are very small, and the catalyst apparently transforms to the super-active state. Thus the model Ml simulation is in excellent agreement with the experiment.

Figure 7 shows enhancement factor dependence on

Cl.50 3

Table 5. Transient plug-flow reactor equations

Dimensionless groups: Z=XjL (5.1) <=t/s (5.2)

f = FIFO (5.3)

a=F,r/V, (5.4)

b = R Trip, (5.5)

c=ptlWTQ,~,,,) (5.6)

k = r, a, s for reduced, active, super-active catalyst states.

Mass balance equations:

ay,iz + af(ayl/W = r, -_y,A (5.7)

i = N, H, A for nitrogen, hydrogen, ammonia.

df/dz = A/a (5.8) dQ/dt = cl-, (5.9)

Storage equations:

r,,,=(b~lEV~)(CIP,yN--XNCOU) (5.10)

XN = C,IC$ (5.11)

dx,/de=(t~cl&V,C~)(aP,y,~x,C~) (5.12)

To incorporate storage equations:


dO/dc = c(r, - 1-,,,) (5.13)

2. Include eq. (5.12) in the mass balance equation set.

CYCLE TIME (s)

Fig. 5. Enhancement factor dependency on cycle time for active catalyst with pure-component cycling (l,O). Cycle split is 0.25. Pressure and temperature are 4.24 MPa and 703 K,

respectively. Solid line is prediction of model Ml.

0.50 1 a I 0 10 20 30 40

CYCLE TME (5)

Fig. 6. Enhancement factor dependency on cycle time for active catalyst with mixture cycling (0.40,0.20). Cycle split is 0.50. Pressure and temperature are 4.24 MPa and 703 K, respectively. Solid line and point at r =4s are predictions

of model Ml.


1 ;I;;; \* i-5 ijil

Y 0.95 - -

5 0.90 - . .

z w 0.65 ’ 1 ’ 8 ’ ’

4.00 4.50 5 00 5.50 6.00 6.50 7.00

PRESSURE (MPa)

Fig. 7. Enhancement factor dependency on pressure and cycle split for active catalyst with mixture cycling (0.40,0.20). Cycle time is 30 s and temperature is 703 K. Solid lines are

predictions of model Ml.

total pressure. In general, we see that higher pressures bring the enhancement down. This is consistent with extrapolations made by Wilson and Rinker (1982). Model Ml simulation fits the experimental data reasonably well. The simulation point at 4.24 MPa and y = 0.8 is for the super-active state, and the rest are at the active state. Again, the solid lines are model predictions based on the assumption that catalyst transformations are continuous.

A comparison between steady-state and pure-component cycling is shown in Figure 8. The Ml simulations are for the active state. Note that the pure- component cycling curve is always below the steady- state curve. Again, the simulation agrees with the data. Figure 9 contains the high-improvement region described by Chiao et al. (1986). The Ml simulation is conducted for the super-active state, and it predicts the correct behavior.

An ammonia-production curve for two cycles in an experimental differential reactor at optimal conditions is shown in Fig. 10, and the Ml simulation at the super-active state is superimposed. The M 1 simulation fits the data well, and predidts the correct responses, which lends support to the model. The fit is even better than it appears: dispersion and mixing in the reactor and sample lines cause widened peak widths and attenuated peak heights, so that the actual experimental curve should be sharper and closer in shape to the Ml simulation curve.

We see that kinetic model Ml is the only model of

the original four which satisfies all of the testing. A summary of the surface parameters, correlated from Arrhenius expressions, appears in Table 6.

CONCLUSIONS

A physically reasonable phenomenological kinetic model for ammonia synthesis has been developed. Kinetic model Ml was narrowed from an original group of four through testing under various steady- state and dynamic operating conditions, including testing in different reactor configurations. The mechanism is based on extensive and reproducible experimental work as well as realistic parameter estimations. The kinetic model fits steady-state and dynamic data over a wide pressure and temperature range and

TIME-AVERAGED FEED NITROGEN MOLE FRACTION

Fig. 8. Comparison of performance of active catalyst between steady-state and pure-component cycling (1,O). Press- ure and temperature are 4.24 MPa and 703 K, respectively.

Solid lines are predictions of model Ml.

Kinetic Model Ml

CYCLE SPLIT

Fig. 9. Enhancement factor dependency on cycle split for FC operation of super-active catalyst with a symmetrical amplitude of 0.10 and a cycle time of 30 s. Pressure and temperature are 4.24 MPa and 703 K, respectively. Solid line is

prediction of model Ml.

s c q Experimental

f! - Kin. Model Ml

p 0.61

9 p 0.5

a

Fii 0.4

$

0’ O.“,‘-’

10 20 30 40 50 60

TIME (s)

Fig. 10. Instantaneous ammonia production during FC operation of super-active catalyst with mixture cycling (0.40, 0.20) for a cycle time of 30 s and a cycle split of 0.80. Pressure and temperature are 4.24 MPa and 703 K, respectively. Solid

line is prediction of model Ml.

also conforms to thermodynamic equilibrium limits. Moreover, the parameters for the model are fitted to separate steady-state and nitrogen A/D data obtained in a differential reactor. We compare simulation results not only with these data but also with indepen- dent integral reactor data using correlated parameter values from Arrhenius expressions.

The issue of nitrogen storage in the bulk-phase of iron catalysts is also addressed. We show that, although the storage definitely exists, it does not


Table 6. Kinetic model Ml surface Arrhenius parameters kads

ki = I?? exp (- EJRT)

ln(~~) = 0.117 m3/kgmol-s E,fR = 1.752 x 10’ K - ’ ~II(E?~)P:‘* =0.106 MPa’12-

m3/kgmol-s E-./R= 1.168 x lo-’ K-’ ln(Qf)= -4.401 kgmol/kg E,/R=6.44Ox 1Oj K-l ln(QP) = - 4.520 kgmol/kg E,fR=4.6OOx 10’ K-’

(Q&a&,3= 1.1181 Q,cr,=Q,(T)x(Q,/Q,),,,

Qi=QFexp(-EJRT)

u, = 1.773 x 10-Z

CT, = 3.730 x 102

o,= 2.476 x lO-2 0, = 1.999 x 102

q,,=O.716 om = 3.924 x 102

urn = 0.745 orn = 5.235 x lo2

contribute directly to ammonia synthesis at high- pressures ( Pl 2 4.24 MPa) under either steady-state or dynamic conditions. Its presence may influence the catalyst activity with respect to turnover frequency, but we have no evidence to support or discredit this conjecture.

Throughout this work, we have illustrated some of the potential pitfalls of kinetic modeling. Researchers often rely on least-squares optimizations involving large numbers of parameters. This practice tends to yield results with very little physical significance, and often the models break down under rigorous testing. Even if unrealistic parameter estimations are not used, the need for thorough testing cannot be overstated. Had we not applied scale-up and dynamic testing of the models in this work, four kinetic models would have resulted, all of which contain nitrogen A/D constants fit to separate A/D data, yield good Arrhenius expressions, fit steady-state data and com- ply with equilibrium.

IL4

Ki

4

SE

t T

Acknowledgements-Support for this work by the Division of tR Chemical Sciences (Office of Basic Energy Sciences) of the u Department of Energy (U.S.A.) under grant number DE- Va FG03-84ER13300 is gratefully acknowledged. w,

CN

G

El/R f F

F0

k,

k-1

k,

k-3

xN NOMENCLATURE

cycling amplitude of cycle interval i, dimen- yi sionless. concentration of nitrogen stored in the cata- Greek lyst bulk, kgmol/m3. a

initial concentration of nitrogen stored in the catalyst bulk, kgmol/m’. activation energy of step i, l/K. dimensionless flowrate, F/F,.

volumetric flowrate, m3/s. inlet volumetric flowrate, m3/s. He pre-treated per-catalyst-weight N, adsorption rate constant, kg/kgmol-s-MPa. He pre-treated per-catalyst-weight N, desorption rate constant, kg/kgmol-s. Hz pre-treated per-catalyst-weight N, adsorption rate constant, kg/kgmol-s-MPa. Hz pre-treated per-catalyst-weight N, desorption rate constant, kg/kgmol-s.

l-i

A E

0

Ki

5

per-catalyst-weight N, adsorption rate constant, kg/kgmol-s-MPa. per-catalyst-weight N, desorption rate constant, kg/kgmol-s. He pre-treated per-fluid-volume N, adsorption rate constant, m’/kgmol-s-MPa. He pre-treated per-fluid volume N, desorption rate constant, m3/kgmol-s. H, pre-treated per-fluid-volume N, adsorption rate constant, m3/kgmol-s-MPa. H, pre-treated per-fluid-volume N, adsorption rate constant, m3/kgmol-s. per-fluid-volume N, adsorption rate constant, m3/kgmo1-s-MPa. per-fluid-volume N, adsorption rate constant, m3/kgmol-s. per-fluid-volume forward surface reaction rate constant, dimensions depend on kinetic model; e.g. for Ml, m3/kgmol-s. per-fluid-volume reverse surface reaction rate constant, dimensions depend on kinetic model; e.g. for Ml, m3/kgmol-s. dimensionless equilibrium constant for elementary reaction step i.

dimensionless overall reaction equilibrium constant. total reactor pressure, MPa. concentration of total sites, kgmol/kg. universal gas constant. dimensionless rate of elementary reaction i.

concentration of adsorbed nitrogen atoms, kgmol/kg. initial concentration of adsorbed nitrogen atoms, kgmol/kg. time, s temperature, K. catalyst relaxation time, s. superficial gas velocity, m/s. reactor volume, m3. weight of catalyst, kg. dimensionless stored nitrogen concentration, normalized to Cg. mole fraction of species i.

symbols

solubility parameter, kgmol/m3-MPa. cycle split, dimensionless. dimensionless reaction rate. dimensionless mole balance rate. reactor void fraction, dimensionless. surface coverage, dimensionless. dimensionless rate constant of elementary process i.

lumped mass-transfer coefficient, m3/s. dimensionless time. catalyst density with respect to fluid volume, kg/m3. standard deviation of parameter estimate, same dimensions as parameter. cycle time, s.


* production-rate enhancement factor, dimen-

sionless.

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a kinetic study of ammonia synthesis: modeling high-pressure steady-state and forced-cycling...

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