Ind. Eng. Chem. Res. 1987,26, 1356-1362
1356
Jordan shale but is equivalent to that for Colorado shale. A t temperatures higher than 475 "C, pyrolysis rate of Moroccan shale is lower than Colorado shale. Literature Cited Allred, V. D. Chem. Eng. Prog. Symp. Ser. 1966, 62(8), 55. Anthony, D. B.; Howard, J. B. AIChE J . 1976,22, 625. Behnisch, J.; Schaff, E.; Zimmermann, H. Thermochim. Acta 1980, 42, 65. Blazek, A. Thermal Analysis; Tyson, J. F., Transl. Ed.; Van Nostrand-Reinhold: London, 1973. Braun, R. L.; Rothman, A. J. Fuel 1975,57, 129. Campbell, J. H.; Koskinas, G.; Stout, N. Fuel 1978, 57, 372. Chen, W. J.; Nuttall, H. E. Paper presented at the 86th AIChE National Meeting, Houston, TX, 1979.
Coats, A. W.; Redfern, J. P. Nature (London) 1964, 201, 68. Dericco, L.; Barrick, P. L. Ind. Eng. Chem. 1956, 48, 1316. Galan, M. A,; Smith, J. M. AIChE. J . 1983, 29, 604. Granoff, B.; Nuttall, H. E. Fuel 1977, 56, 234. Haddadin, R. A.; Mizyet, F. A. Ind. Eng. Chem. Process Des. Deu. 1974, 13(4), 332. Haddadin, R. A.; Tawarah, K. M. Fuel 1980, 59, 539. Herrell, A. Y.; Arnold, C., Jr. Thermochim. Acta 1976, 17, 165. Hubbard, A. B.; Robinson, W. E. Rep. Invest.-L1.S. Bur. Mines 1950, 4744, 1. Nuttall, H. E.; Guo, T.-M.; Schrader, S.; Thakur, D. S. ACS Symp. Ser. 1983, 230, 269. Pan, Z.; Feng, H. Y.; Smith, J. M. AZChE. J . 1985, 31, 721. Rajeshwar, K. Thermochim. Acta 1981, 45, 253. Received for review March 10, 1986 Accepted March 16, 1987
Stability of Tubular and Autothermal Packed Bed Reactors Using Phase Plane Analysis Richard W. Chylla, Jr., Raymond A. Adomaitis, and Ali C h a r * Department of Chemical Engineering, Illinois Institute of Technology, Chicago, Illinois 60616
The regions of stability and parametric sensitivity of countercurrent reactor/heat exchangers are determined explicitly in the plane of inlet feed temperature-inlet coolant temperature. The concept of phase plane analysis is generalized to include all orders of reaction rate expressions, a broader range of system parameters, and is extended to the case of autothermal reactors. An industrial hydrocarbon oxidation reactor model and an autothermal CO oxidation reactor model have been used to illustrate and to evaluate the analysis method. The approach presented here is appealing since the region of safe inlet temperatures is determined explicitly and the region of safe operation can be optimized with respect to the reactor design parameters. 1. Introduction
Industrial packed bed reactors serve as the workhorse of the chemical industry, and yet there is much progress to be made in terms of obtaining reliable models for reactor design and control purposes. Autothermal reactors in particular suffer from large parametric sensitivities due to the inherent feedback mechanism. A large step fonvard in the design and operation of these reactors would be a diagnostic method to determine the region of safe operating temperatures for both the inlet coolant and the reactants. Accurate models of reactor temperature behavior are often complex and even numerical solution on a digital computer can be unwieldy and time consuming. Perhaps more importantly, the effect of a particular design parameter on the stability of the reactor is not always obvious and may often require repeated solutions. A major improvement in any control effort is to maximize the open-loopregion of stability. This work is aimed at providing a tool for the engineer designing the reactor and/or the control system to determine a priori how stable a given reactor configuration is and how the design parameters will affect the size of this region of stability. There have been numerous studies in the literature which offer criteria with which one may determine whether or not a set of inlet conditions leads to stable operation (van Welsenaere and Froment, 1970; Oroskar and Stern, 1979; Froment and Bischoff, 1979; Bilous and Amundson, 1956). The concept of using phase plane analysis to determine regions of safe operation has been proposed by Akella and Lee (1983) for reactors cooled by a countercurrent fluid, especially liquids. In this context, phase
* Author t o whom correspondence should be addressed. 0888-5885/87/2626-1356$01.50/0
plane analysis is used to denote the reactant temperaturecoolant temperature plane. Their algorithm is limited to positive-order reactions, and autothermal operation was not considered. The phase plane analysis in this paper: (1) extends analysis of Akella and Lee (1983) to a more general class of reactions, (2) permits the designer of an autothermal reactor to know how safe a set of inlet temperatures are for a given set of operation conditions, and (3) determines the necessity of a feed-effluent heat exchanger for autothermal operation. 2. Phase Plane Analysis of T u b u l a r Reactors The vast majority of industrial reactions take pldce in fixed bed catalytic reactors, and many of the bulk &emicals such as ammonia, methanol, and functionalized aromatics are produced by exothermic reactions. This requires large amounts of heat to be removed to prevent thermal runaway, undesirable side reactions, or unfavorable equilibrium conditions. In many reactors, this heat is removed by an external medium such as water or heat stable molten salts. In large scale operations, it is economical to use the heat of reaction to preheat the feed to desired reaction temperature. Since the coolant eventually enters the reactor as the feed stream, these reactors display high levels of parametric sensitivity and very narrow windows of safe operation. The following discussion reviews the development of an algorithm to produce a phase plane for a general countercurrent reactor/heat-exchanger configuration. Autothermal reactors are then treated as a subset of these reactors. Industrial exothermic reactions are commonly carried out in tubular packed beds filled with catalyst and cooled by flowing a cooling medium countercurrently. The 0 1987 American Chemical Society
Ind. Eng. Chem. Res., Vol. 26, No. 7, 1987 1357
n
t l
I"r'.:= I1 t,
c
Figure 1. (a, top) Countercurrent reactor/heat exchanger and (b, bottom) autothermal reactor operation.
bounded by these three lines is the region of safe operation. The safe operating zone is divided into an upper operating zone and a lower operating zone by the To = tl line. It has been shown (Akella and Lee, 1983) that the lower operating zone is insensitive to large fluctuations in the feed temperatures, while the same cannot be said of the upper operating zone. Expressions for the quenching line and the lateral ignition line have been given by Akella and Lee (1983) and will simply be stated here. An improved expression for the upper ignition line that is less conservative and valid for any kinetics will be developed in section 2.3. 2.1. Dimensionless Equations. Assuming constant physical properties, the material and energy balance equations may be rewritten in pseudodimensionless form (the temperatures could also be made dimensionless; however, it is more convenient in this case not to do so): dx = c1f dz dT _ c z f - CJT- t ) dz
-
-dt-- -c4(T - t )
(3)
dz
UPPER I G N I T I O N REGION
where c1
=
rR2L FAO
LATERAL IGNITION REGION
NEGLIGIBLE REACTION REGION Quench L i n e
L a t e r a l Ignition Line
/
/
cz =
LOWER OPERATING ZONE
/
7rR2L(-AH)
Mep
=
c1f
/
COOLANT
TEMPERATURE
t,
Figure 2. Reactor phase plane with definitions.
steady-state material and energy balances for the reactor shown in Figure l a are given elsewhere (Akella and Lee, 1983). Much information about the reactor behavior may be obtained without ever solving these nonlinear differential equations. The inlet reactor temperature-inlet coolant temperature phase plane shown in Figure 2 will be used for analysis. Three lines divide the plane into regions of safe operation, negligible reaction, and ignition. The line dividing the plane between regions of safe operation and negligible reaction is the quenching line. This line indicates the inlet coolant temperature below which no significant reaction takes place. The regions of safe operation and ignition for an inlet coolant temperature are divided by the lateral ignition line. This is the maximum allowable coolant temperature for which the reactor operation will remain stable. Between the lateral ignition line and the quench line are the feasible range of inlet coolant temperatures. For these feasible coolant temperatures, there will still be an inlet reactor feed temperature above which the reactor will ignite. This set of temperatures (one for each feasible inlet coolant temperature) is called the upper ignition line. The envelope
Mep
27rRL U c4 = -
(4)
hCP
2.2. Quenching Line. The quenc--ing line is the vertical line indicating the inlet coolant temperature below which no significant reaction takes place. This is determined as the inlet coolant temperature which would result in 1% conversion. The requirement for 1% conversion is given by
/
INLET
c3
27rRLU
= 0.01
(5)
and f is evaluated a t T1and x = 0 with Tl i= tl. 2.3. Upper Ignition Line (To> t l ) . This line determines the maximum allowable inlet feed temperature for a given inlet coolant temperature. Akella and Lee (1983) define this line using criteria on the sign of the first and second axial derivative of the reactant temperature. The necessary and sufficient conditions in the proof of this criteria are only valid for positive-order reactions, however. Many industrial reactions are of the Langmuir-Hinshelwood-Hougen-Watson (LHHW) type or have similar negative-order rate expressions. A different criterion must be used to generalize the analysis to account for all types of reaction rate expressions encountered. Regardless of the particular form of the rate expression used, insight to the behavior of the reactor in this region is illustrated by Figure 3. For the case of stable reactor operation, the reactant temperature profile goes through a maximum. Therefore, at one location z* in the reactor Qrem
=
Qgen
(6)
with
Q,,, = U(Tm,, - t(z*))27rR dZ Qgen
= (~*A)(-WTR dZ~
(7) (8)
A coolant stream heat balance dictates t o = tl
+ qc,
(9)
1358 Ind. Eng. Chem. Res., Vol. 26, No. 7, 1987
Substituting (14) in (12) yields an expression for the hot spot temperature in terms of the inlet coolant and feed temperatures: “1
Tmax =
“4
1+ c3/c4
where
(r*A)R(-Am (16) 2u The variable c6 is a function of the reaction rate at the hot spot, and hence, c13= cg(Tma~,x*) (17) c6
Z
DIMENSIONLESS DISTANCE
=
Therefore, for a given set of inlet temperatures (tl,To),one can iterate on T- until (15) is satisfied. If a Tmax cannot be found between To and some upper limit, say 700 OC, one can assume that a safe operating steady state does not exist for the given inlet conditions. Because x* is unknown, a conservative estimate must be made. For positive-order reactions, one may use x = o (18)
T >tl 0 STABLE CASE
instead since
rA(x = 0) > r(x*)
(19)
For negative-order reactions, although rA(x = 0 ) < r(x*)
I
Z
DIMENSIONLESS DISTANCE
Figure 3. Typical coolant and reactor temperature profiles.
where q is the fraction of heat generated that is transferred to the coolant stream, c5 is the adiabatic temperature rise c5
=
F A O ( - m = -c2c4 AC, clc3
(10)
A conservative heat balance on the reactant stream can be made:
Equation 11 simply states the “hot spot” temperature is equal to the inlet temperature plus the fraction of the adiabatic heat rise that was not transferred to the coolant stream. Solving (9) for q and inserting in ( l l ) ,
If tocan be estimated, (12) will be an expression for the maximum bed temperature, Tu, in terms of the phase plane variables Toand tl. Recalling that at the hot spot z* (7) is equal to (8),one can solve for t ( z * ) the coolant temperature at z*:
Examination of Figure 3 shows that the coolant temperature at the hot spot may closely be approximated by to (hot spot usually occurs near reactor inlet) so that (13) may be rewritten as
(20) the reaction rate is a much stronger function of T,, and we have found that the results are essentially unchanged when using rA(x = 0) in (15). Alternatively, one may be conservative and use the maximum r*A for a given Tmax found by an addition one-dimensional search. 2.4. Lateral Ignition Line (To C t l). As seen in Figure 2, this line separates the lower operating zone from the ignition region. In physical terms, this line is a vertical line indicating the maximum allowable inlet coolant temperature for the reactor to avoid ignition. Akella and Lee (1983) have shown that ignition will possibly occur for coolant inlet temperatures above a critical value of tl given by
such that dtl
=O
d Tmax Some reactions may instead by limited by unfavorable equilibrium or an undesired side reaction at a specific temperature. In that case, the maximum desired inlet coolant temperature is given by (21) with T, equal to the temperature at which unfavorable equilibrium occurs or the undesirable side reaction becomes significant. 2.5. Hydrocarbon Oxidation with Liquid Coolant. In order to illustrate the improved algorithm developed in this work, an example of an industrial hydrocarbon oxidation reactor that was cooled by a countercurrently flowing liquid was used. This example was first used by Froment and Bischoff (1979) and later by Akella and Lee (1983) in their phase plane work. In the phase plane for this reactor (Figure 4), the filled circles are the temperatures corresponding to ignition as calculated by integrating
Ind. Eng. Chem. Res., Vol. 26, No. 7, 1987 1359
\
.Numerically Simulated Ignition Points
U p p e r O p e r a t i n g Zone'
t
ignition point and, thus, undesirable. Therefore, if a suitable relationship To = TOG,) (25) for autothermal operation can be found and plotted in the phase plane, the feasibility of autothermal operation could be ascertained simply by observation of where in the phase plane this line falls. One could expect this line or autotherm to lie parallel to the To = t , (26) line since the capacity flow rates mc, for the coolant and reactant are equal (assuming cp is not a function of temperature); an increase in the inlet coolant temperature would cause a proportional increase in the exit coolant temperature. 3.1. Autotherm Algorithm. A simple algorithm may be proposed to determine the coolant exit temperature to = To for a given inlet coolant temperature, t,. A heat balance on the coolant stream dictates that to =
550 600 650 70 C 500 Figure 4. Phase plane for hydrocarbon oxidation with liquid coolant.
eq 1-3 using a digital computer. One can see how closely the algorithm predicts the actual behavior. The dashed upper operating line is calculated by using the algorithm of Akella and Lee. The algorithm presented in this work produces equal or better accuracy for a more general class of reaction kinetics and system parameters. 3. Extension to Analysis of Autothermal Reactors This concept of phase plane analysis can be readily adapted to autothermal reactors since they are a subset of the general class of wall-cooled tubular reactors. A schematic of a reactor under autothermal operation is reproduced in Figure lb. In this case, the feed serves the dual purpose of both coolant and reactant; the feed is passed through the annular space where it is preheated by using the heat of reaction and then turned around at z = 0 and sent through the reactor bed to produce products. Many important commercial processes such as methanol synthesis and ammonia synthesis operate autothermally. The phase plane analysis of autothermal reactors differs from the general case in the respect that the reactor inlet temperature, To,is now, for a purely autothermal process, goverened only by the temperature it entered the reactor as coolant tl and the reactor conditions. The existing phase plane may still be used with only the addition of a relationship coorelating the inlet coolant temperature with the exit coolant temperature since
To = to (23) For an autothermal process with no heat loss to the environment, To tl (24) for any significant conversion, and hence the lower operating zone would not exist. It will be shown later that heat losses can be incorporated and the lower operating zone will indeed exist. Consider the resulting Tofor an inlet t,. If this Towere to lie below the upper ignition line in the upper operating zone, one could assume that the reador operation would be safe. Likewise, a resulting Tothat fell above the upper ignition line could be considered an
'
t , + -- mcp mc, '?trans
(host
(27)
where qtrm denotes the heat transferred from the packed bed to the coolant and qloatdenotes the heat lost to the environment. The heat transferred to the coolant is simply given by qtrans= U(2rRL)AT (28) The heat lost to the environment may be approximated as a fraction of the heat generated qlost= (fraction)q,,, = (fraction)FAo(-AH) (29) Typically 30% of the heat generated will be lost, i.e., '?lost = O.~FAO(-W (30) For a rough estimate of the log mean temperature difference, we may use the arithmetic mean
Since as before
to == Ti
(32) (33)
where T, has already been determined from the upper operating line algorithm, (15). Restating in terms of the algorithm parameters,
3.2. Phase Plane Analysis of CO Oxidation Autothermal Reactors. As an illustration of the utility of phase plane analysis, an example is presented for a CO oxidation reactor operating in a purely autothermal manner. The phase planes for different sets of operating conditions will be compared with the numerical temperature and concentration profiles obtained by using a complex detailed model. The model derived by Adomaitis and Cinar (1986) is a heterogeneous dynamic model that accounts for the solid- and gas-phase concentrations, solidand gas-phase temperatures, inner wall temperature, coolant temperature, and outer wall temperature. The detailed model and solution techniques are given in Adomaitis and Cinar (1986). The rate equation for this reaction is given by rA
=
NCO) (1 + K(C0))Z
(35)
1360 Ind. Eng. Chem. Res., Vol. 26, No. 7, 1987
I
- I
Table I. CO Autothermal Reactor Parameters parameters value R, reactor diameter, cm 0.875 L , reactor length, cm 15.0 -AH,heat of reaction, cal/g-mol 67 000 1.34 x 10-3 TAo,molar flow rate of A, g-mol/min 6.47 M , mass flow rate of reactant, g/min riz, mass flow rate of coolant, g/min 6.47 0.247 c,,, heat capacity of coolant, cal/(g K) U,heat-transfer coefficient, cal/(cm2 7.67 X min K) mole fraction of CO, yco 0.006 1.3 X 10s(exp(-4155/T)) k , reaction rate constant, min-' K , adsorption rate constant, cm3/g-mol 1.95 X 105(exp(2125/T))
5000 SCCM YCO' 0 , ~ ' ' :
AModel i g n i t i o n P o i n t s OModel S t a b i r Points
@Psuedo Upper S t a b l e P o i n t
Y
I
A/ 4UTOTHERM
350
/
'
~
0
/ i-
w
/
i
/
/
/
300
/ I
300
350
400
INLET COOLANT
50C
450
TEMPERATURE
t,( K )
Figure 5. Phase plane for autothermal CO oxidation reactor. 500
-Y
A Model U p p e r S t a b l e P t s
5oCC SCcV
0 Model t o w e r S t a b l e P t s
vCo=o,G36
1
3
AUTOTHERY
k E 400 a I
,/'
W
t
0
c
0
L L
30 0
35C IN,ET
40C
45c
C C O ~ N T TEMPESATURE
5C0 T.,( 6 )
Figure 6. Phase plane for autothermal CO oxidation reactor.
and the values for the reactor parameters used in phase plane analysis are shown in Table I. Phase plane plots
for two different inlet concentrations are shown in Figures 5 and 6. The general shape of this phase plane is the same as the general case except for the linear nature of the upper ignition line. This is to be expected since c3 = c,, for autothermal operation and the two temperatures are now linearly related. In Figure 6 there are two upper ignition lines drawn corresponding to 100% adiabatic efficiency and 60% adiabatic efficiency which more closely corresponds to typical laboratory operating efficiencies. The dashed line is the autotherm obtained by using (34). Note that because of the simplifications assumed in the derivation of the autotherm, it does not lie quite parallel to the To = tl line. This can be compensated for by drawing an autothermal band parallel to the To= tl line enclosing both end points. Thus, a conservative prediction is made as to the approximate location of the autotherm. A more rigorous autotherm relationship may be derived if desired; however, we found this simple description to be quite accurate in predicting the reactor stability behavior as can be seen by noting the location of the upper steady-state model simulation points in Figures 5 and 6. The model points plotted in the phase planes were obtained by running the detailed reactor simulation and examining the resultant steady-state temperature and conversion profiles. There are two stable operating points in the lower operating zone of Figure 5 . Earlier it was stated that the lower operating zone would not exist, but this is only for the case of 100% adiabatic efficiency. In this case, these two points lie on the lower steady-state branch and the coolant stream loses more heat to the environment than is generated and transferred from the reactor bed. The upper steady state lies above the upper ignition line at an inlet coolant temperature of 400 K. Examination of the temperature and conversion profiles for this point shown in Figure 7a clearly shows this to be an "ignition" point. The temperature rise and conversion all occur in a very narrow section of the reactor bed. There are no thermal
Figure 7. (a, left) Profiles for Figure 5 ignition point, (b, middle) profiles for Figure 6 upper stable point, and (c, right) profiles for Figure 5 pseudoupper stable point.
Ind. Eng. Chem. Res., Vol. 26, No. 7, 1987 1361
1
PEF,i E NT C3 NVE R SIS N .Phase
Single Steady-State (Ignited)
Plane
Ignition Pts.
ranges of safe operation as well as characterize the temperature and concentration profiles that can be expected. Such a method is a invaluable design tool for determining optimum parameters as well as ranges of design temperatures and concentrations. 4. Comparison with Other Stability Criteria
0
OS
1 ,o
PERCENT
1,5
CO
2s
25
IN FEED
Figure 8. Comparison of model predicted multiplicity with phase plane analysis ignition points.
runaway conditions in the classical sense because of the negative-order kinetics, but these sharp conversion profiles are indicative of the ignited state as originally defined by Bilous and Amundson (1956). Note that both of the ignition points lie nearly on the predicted autotherm. For this particular case, no upper steady-state branch exists in the upper operating zone and so pure autothermal operation is not possible for these operating conditions. If the inlet reactor temperature could be manipulated independently using auxiliary cooling or bypass flow, however, the plot indicates the operation could be safe. This was simulated ( t l = 375 K, To= 420 K) and found to indeed be the case as seen in the profiles of Figure 7c. The phase plane for lower inlet CO concentration (Figure 6) is similar to Figure 5 except that two upper steady-state branch points lie in the upper operating zone and the profiles in Figure 7b show them to be safe operating points. It is worth noting that assuming 100% adiabatic efficiency would be a overly conservative estimate and some feasible operating points would be eliminated. A priori estimates of the thermal efficiency from operating experience would be valuable at the design stage so that overly conservative predictions do not exclude feasible operating points. Both of the phase planes shown in Figures 5 and 6 indicate that some additional preheating and/or additional feed-effluent heat exchange will be required since the inlet coolant temperature that quenches the reaction is around 350 K so that coolant cannot be fed at ambient conditions. Further accuracy of the phase plane approach for autothermal operation is demonstrated in Figure 8. This figure displays the upper and lower steady-state conversions plotted as a function of the inlet coolant temperature tl and the mole fraction of CO in the feed. The upper single steady-state region consists of operating points corresponding to 100% conversion and sharp temperature profiles, i.e., ignition points, as simulated by using the detailed reactor model. The filled squares are the ignition coolant temperatures as predicted by the lateral ignition line temperatures in phase plane analysis. The two methods agree within about 5 K. A remarkable amount of information about autothermal operation may be obtained without detailed reactor modeling. Phase plane analysis will indicate the temperature
Since Barkelew (1959) first proposed a criterion for a priori runaway estimation, there has been much activity in this area reported in the literature. Analytical expressions to approximate parametric sensitivity boundaries have been proposed by Thomas (1961), Gray and Lee (1965), Hlavacek and Marek (1969), van Welsenaere and Froment (1970), Morbidelli and Varma (1985), and recently by Henning and Perez (1986). An excellent review and comparison was recently given by Morbidelli and Varma (1985). Most of these criteria are restricted by one or more of the following: (i) only good for n = 1 and/or n 2 0; (ii) valid only for infinite activation energy; (iii) limited to inlet temperature equal to Twall. Virtually all criteria are presented as a line plotted in a plane of dimensionless reaction parameter vs. dimensionless heat-transfer parameter which divides the plane into regions of parametric sensitivity and parametric insensitivity, and runaway is inferred. Thus, stability is determined implicitly and, hence, is not a convenient diagnostic tool for design purposes. By comparison, in the phase plane analysis presented here, the region of stability for given reactor parameters is determined explicitly. The method is valid for both negative- and positive-order kinetics and does not have to be rederived for the particular kinetics used. Furthermore, phase plane analysis is not limited by assumptions on the inlet conditions or the range of activation energy. Phase plane analysis is therefore more amenable for design than other methods. 5. Conclusion
A major result of this work is the ability to extend the concept of phase plane analysis to all orders of reaction and a broader range of system parameters. The results obtained by using the algorithm developed here predict the regions of safe operation and ignition with better accuracy and less conservatism than other methods. In extending the analysis to autothermal reactors, an extremely valuable design tool has evolved which not only can determine the ranges of viable operating temperatures but also the feasibility of autothermal operation for a given set of design parameters and the necessity of feed-effluent heat exchange. Implementation of these algorithms on a computer is straightforward and ideally suited to industrial application in the design of wall-cooled tubular reactors and autothermal tubular reactors. Acknowledgment Partial financial support from the Department of Energy (DE-FG02-84ER13205)and the Petroleum Research Fund, administered by the American Chemical Society, is gratefully acknowledged. Nomenclature cIw4= constants defined in (4)
Ind. Eng. Chem. Res. 1987, 26, 1362-1365
1362 c5 = variable defined in (10) c6 = variable defined in (16) c = coolant heat capacity = reactant stream heat capacity
8 $ = molar flow rate of key species A
k = reaction rate constant K = adsorption rate constant L = reactor length m = mass flow rate of coolant M = mass flow rate of reactant rA = reaction rate per unit volume of reactor R = reactor radius T = temperature of reactant gas t = temperature of coolant U = overall heat-transfer coefficient x = conversion of key species A z = dimensionless length 2 = axial coordinate of reactor Greek S y m b o l
-AH = heat of reaction
Subscripts 0 = at the reactor inlet z = 0 1 = at the reactor exit z = 1
A = key reactant species max = maximum value Literature Cited Adomaitis, R. A.; Cinar, A. The Bifurcation Behavior of a n Autothermal Packed Bed Tubular Reactor; Illinois Institute of Technology: Chicago, 1986, submitted for publication. Akella, L. M.; Lee, H. H. AIChE J. 1983, 29(1), 87. Barkelew, C. H. Chem. Eng. Sci. 1959,25, 37-46. Bilous, 0.;Amundson, N. R. AIChE J. 1956, 2(1), 117. Froment, G. F.; Bischoff, K. B. Chemical Reactor Analysis and Design; Wiley: New York, 1979. Gray, P.; Lee, P. R. Combust. Flame 1965, 9, 201-203. Henning, T.; Perez, T. Chem. Eng. Sci. 1986, 41, 83-88. Hlavacek, V.; Marek, M.; John, T. M. Collect. Czech. Chem. Commun. 1969, 34, 3568-3880. Morbidelli, M.; Varma, A. Chem. Eng. Sci. 1985, 40, 2165. Oroskar, A.; Stern, S. A. AIChE J . 1979, 25(5), 903. Thomas, P. H. Proc. R. SOC.London, Ser. A 1961, A262,1922-1206. van Welsenaere, R. J.; Froment, G. F. Chem. Eng. Sci. 1970,25,1503. Received for review March 13, 1986 Revised manuscript received January 16, 1987 Accepted March 4, 1987
Superscripts * = at the hot spot location of reactor
A Group Contribution Method for Liquid Thermal Conductivity Manoj Nagvekar and Thomas E. D a u b e r t * Department of Chemical Engineering, The Pennsylvania S t a t e University, University Park, Pennsylvania 16802
An estimation method, based on the principle of second-order group additivity, has been developed for predicting the thermal conductivity of organic liquids. By use of experimental data, group contributions have been determined for two constants in the modified Reidel’s temperature correlation for liquid thermal conductivity. The proposed method is completely generalized and fairly easy to use as the only input parameter required; besides, the molecular structure is the critical temperature. In terms of accuracy, it compares favorably with the existing prediction methods. The average deviation for 226 compounds, for which experimental data are available, is 5.9%. Thermal conductivity is a very important transport property, especially for liquids, since it is required for most heat-transfer calculations, including forced convection. Unfortunately, it is an extremely difficult property to measure, primarily due to the development of convective currents and heat losses during the course of the experiments. Under these circumstances, it is necessary to turn to an empirical method to estimate the liquid thermal conductivity. In this paper, we propose a prediction method which is based on the group contribution approach. Group Contribution Methods When the contribution of an “element” of a molecule (atom, bond, group, etc.) to a particular property is constant regardless of the nature of the rest of the molecule, the property is termed an additive one. Many molecular properties have been shown to obey additivity rules, to varying degrees of accuracy. Knowing the structure of the molecule, an additive property may be easily and quickly estimated by summing up the contributions corresponding to the “elements” of the molecules. Benson and Buss (1958) have presented a very good discussion of additivity rules and classified them in a hi-
erarchical manner. The primary assumption in constructing additivity rules is that the influence of neighboring elements decreases rapidly with distance in a molecule. This assumption determines the definition of what constitutes a “group”, since the distance at which groups cease to interact must be assumed before the groups can be specified. In this study, a second-order contribution scheme, which is based on nearest-neighbor interactions, was used. In the second-order scheme, a group is defined as a polyvalent atom together with its ligands, a t least one of which must also be polyvalent. Thus, this scheme is restricted to compounds having at least two polyvalent atoms. The nomenclature followed in this study is to first identify the polyvalent atom and then its ligands. Thus, C-(C)(H), represents a C atom attached to another C atom and three H atoms, i.e., a primary methyl group. Cd represents a double bonded carbon atom, C , a triple bonded carbon atom, and Cb a carbon atom in the benzene ring. For multiple-bonded carbon atoms, the other multiplebonded carbon atom is implied since these atoms must always occur in pairs. This scheme of classification for group additivity can only distinguish among isomers when
0888-5885/87/2626-1362$01.50/0 0 1987 American Chemical Society
