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The catalytic porous wall reactor with axial flow of the reacting mixture
151
The Chemical Englneertng Journal, 23 (1982) 151 - 160 0 Elsewer Sequoia S A , Lausanne - Prmted m The Netherlands
The Catalytic Porous Wall Reactor with Axial Flow of the Reacting Mixture
R MIHAIL and C TEODORESCU Polytechmc (Recewed
Institute
Bucharest,
17 December
Department
of Chemwol
Engmeermg,
Poltzu
1, 7000 Bucharest
(Romanro)
1979, In fmal form 5 May 1981)
AbSt7YlC.t
A detailed mathematrcal model of a porous wall tubular reactor 1spresented, takrng rnto account the superposed effects of heat and mass transfer m two space drrectlons DynamK profiles of conversion and temperature are also obtarned and the influence of various parameters on reactor performances 1s drscussed, together with an analysrs of the e ffec tweness fat tor The ma thema trcal model encompasses a broader class of heterogeneous contactors, mcludmg the monolrthlc structures Some advantages of the porous wall reactor are discussed and the efflcaency rmprovement attamable m such reactors IS evaluated
INTRODUCTION
Over the past years, several new types of reactors have been suggested to meet the demand for carrying out a mde variety of chemical reactions m an economic way The pints of interest are to get lower pressure drop m the reactor m order to mmlmlze the energy required by the fluid transport and to obtain a better catalyst efficiency thus reducing the reactor volume and the catalytic mass used by the process A reactor type which could possibly meet the above requirements 1s the porous wall reactor (PWR) This reactor type consists of a hollow cylinder havmg the walls made of a porous matenal The reacting fluld enters the tube or flows along the outer wall, parallel to the cylinder axis The type mentioned operated mth suction or inJection has already been considered m literature [l, 51 Such systems could offer several advantages over the solid wall reactor,
because the driving force of the reactlon can be maintained at a nearly constant level along the entire length of the tube However, Its practical form 1s difficult to construct for industrial use In contrast, a PWR with axial flow only, without suction or mlectlon, can easily be realized by some regular superposltlon of Raschlg-type catalyst particles Such a reactor 1s dlstmgulshed by advantages over the conventional fixed-bed catalytic systems, notably, (1) removing the catalyst particle core, as compared to a solid cylinder of the same volume (and hence contammg the same amount of catalyst), hollow cylinders present lower dlffuslonal hmltatlons mth high levels of stablhty and control of the system, (ii) use of catalytically non-uniform active materials simply by dlstrlbutmg active zones along the inert porous support In a hmltmg case of the catalyst deposlted as a thin layer on the inner surface of the tube, the porous wall reactor reduces to the tube-wall reactor, now widely used as a monolithic converter [2 - 4, 91 Theory and experiments on the catalytic converter [ 2 - 4, 7, 81 have shown that the transfer properties for the catalytic zone and for the mert support differ substantially. Mass transfer 1s limited to the thin layer of the catalyst, where chemical reaction takes place, while heat transfer 1s important m the inert substrate only Such conslderatlons induced Young and Fmlayson [ 3, 4, 81 to develop a surface mass transfer model and a heat transfer model for the inert solid support This model has been solved for a mde variety of condltlons However, simultaneous mass and heat transfer accompanied by a chemical reactlon has not been considered In the present paper, we consider all the interacting phenomena mentioned m a cata-
152
of anlsotroplc properties within the wall, resulting m different values for mass and thermal dlffuslvltles in radial and axial dlrectlons, as discussed by Gunn [lo], this amsotropy may result from different tortuosity factors of the pores, due to the specific technology for the manufacturing of the hollow cylmders In the general form of the model both surfaces of the porous wall are accessible to the flowmg gas, this leads to diffusion coupling of the outer and inner surfaces of the hollow cylinders Cases of insulating one side of the solid for heat and/or mass transfer will be treated as particular sltuatlons derived from the general model by equating some parameters to zero (111)Changes of the physical properties of the sohd or fluid with temperature and conversion are neglected (IV) Unsteady state condltlons occur at the solid wall only The main geometnc and physical parameters of the model are summarized m Fig 1 The differential mass and heat balances and interphase transfer equations for the fluid and solid body, converted to the dlmenslonless form constitute the basic set of equations of the present model Though unsteady state condltlons involve the physical system as a whole, only the solid model contains the time derlvatlves This slmphflcatlon does not affect the validity of the mathematical description, proof of the above assumption follows from the so-called ‘quasi-static’ hypothesis [3, 4, 7, 81 In most gas-solid catalytic systems, hydrodynamic, chemical and temperature changes take place at very wdely differing rates Dlstmctlon can also be made for the rate of
lytlc PWR for all the length and thickness of the reactor The differences between the models developed by Smith and Carberry [ 21, Young and Fmlayson [3, 41, Votruba et al [15] and the one presented here are given in Table 1 A model of the PWR has been investigated and its performances have been established for a variety of operating condltlons of practical relevance, mcludmg the transient behavlour
DEVELOPMENT MODEL
OF
THE
MATHEMATICAL
Consider a porous tube wth axial symmetry, immersed m a flowmg fluid Heat. mass transfer and a chemical reaction occur simultaneously m all the points of the porous body, access to the Internal core and on the tube surface 1s diffusion limited The assumptions used m the derivation of the mathematical model to the PWR are summarized as follows (1) The fluid is SubJect to convective transport only The flow IS considered to be turbulent and the fluid velocity IS restricted only to the axial component The assumptions on turbulence are valid for most mdustrlal cases of interest apart from the commonly used automobile exhaust afterburners where lammar flow prevails due to very small hydrodynamic length [3] Therefore the mathematical model will be unldlmenslonal for the fluid flowing in a PWR, (11) Axial and radial heat and mass transfer m the solid phase are assumed The model consider the occurrence TABLE 1 Comparison
of exammed
physlcal models for catalytic
reactors
Reference
Smith and Carberry [ 21
Young and Fmlayson [ 3 ]
Votruba eta1 [15]
Present paper
FluId phase
Convective and dIffusIonal transport Re = 680
Convective and dlffuslonal transport Re > 2000
Convective transport only Re > 2000
Convective transport only Re > 2000
Solrd body
Surface reactton and Surface reactlon and thermal conductlon m dlffuslon In a thm the sohd layer Thermal conductlon only In the sohd
Surface reactlon Longltudmal heat conductlon within the sohd structure
Dlffuslon, thermal conductlon and chemical reaction throughout the sohd body
153
catalytic
porous solid body
a2w,
aw,
ap2
ap
-+c(p)-+ll-
a2w, az2
aw,
+
sat =JwW,L
gas flowing outside the tube (p = 1)
(3)
dW L-F(W,,-WW,)=O, dp gas flowing outslde the tube (bulk condltlon)
dWs.e - d.2 + G( W,.e-W,)=O, inlet condltlon
z+A(W,.Fig 1 Physwal statement
inlet condltlon
of the problem
the same process (e g , heat accumulation) in the gas and m the solid material For a resldence time value of 0 15 - 0 20 s, the transient response of the velocity profile 1s 10 - 20 times faster, heat accumulation term m the gas equation 1s neglected since the ratio of gas to solid thermal capacities 1s around 7 X 105, mass accumulation term 1s neglected m the fluid model because of the large dlscrepancy between the residence time and the mass transfer time constant (about 20 times larger) The study of the dynamic behavlour of the PWR 1s restricted to the transient response of the solid body because the time necessary for the wall to approach the steady-state 1s the longest time constant of the entire system Equations (1) to (9) below have been obtamed on this assumption The dlmenslonless model equations for chemical conversion and temperature are structurally identical, the term W denotes either conversion or temperature Coefficients m these equations have different expressions for conversion and for temperature parts of the mathematical model and they are hsted in Table 2 The standard form of the mathematical model is as follows
for the solid (z = 0)
W,)= 0, for the gas (z = 0)
W g-1 = W,.c? = W&O, exit condltlon
(7)
for the solid (z = 1)
dW,_ -0 dz
(8)
mltlal condltlon
W,=W,
(6)
att=
for the solid body 0
(9)
Equations (2), (4), (6), (8) and (9) constitute the set of the mltlal and boundary condltlons for the partial differential eqn (3) and are usually encountered m the description of analogous physical systems The above equations are written m the order m which they perform, starting from a point in the fluid bulk flowing through the tube, contmumg with the transfer process on the inner solid surface, the balance equation for the porous wall, the transfer process on Its outer surface and ending with the convective transport m the fluid flowmg outslde the hollow cylinder Processes taking place in the solid body are governed by eqn (3) Equations (1) and (5) constitute the fluid model together with the corresponding mltlal condltlon (eqn (7))
gas flowing inside the tube (bulk condltlon)
dW,
L
dz
1
MODEL SOLUTION
+ A(Wg.1-iv,)=
0,
(1)
inner surface of the tube (p = 0)
d W, -+wW,,,-wW,)=O, dp
(2)
A primary analysis of the set of equations presented above reveals the followmg features (1) The essential element of the model 1s the second order parabohc partial differential
154
TABLE 2 Expresstons for the coefflclents m the mathematical model Corwersion
equahon
Temperature
equation
A
B
k,.,(Rz
1
-RI
kt,A&
h eff.r
D eff.r R2
C(P)
R,
R2 -RI
-RI
+P&
- RI)
D eff,z(Ra
- RI j2
RI +P(RP-R~) h.zVh
D eff.r fl -(R2
-
RI J2
k m/AR2
-_(--aH,)(R2
ke(R2
D eff.r 2~R2Hkm
-
Rl J2
T0
RI )
h?ff.r
2nR2Hkt.e
.e
Aextwgwz.e
A extw2.e -RI
-
Aeff.r
-RI)
-
- R, J2 h eff.r H2
D eff.rCO
-(R2
- RI)
J2
D eff.1
-tR2
-RI x eff.1
j2 (P&g
+ Pkck)
equation descrlbmg processes In the solid body I e eqn (3) The method of solvmg the eqn (3) ~11 influence the solution of the
entxre model (11) The three dlstmct physical sectlons of the model (inside fluid, catalyst wall, outslde fluid), linked by eqns (2) and (4), must be solved simultaneously (111)Converslon and temperature equations of the type (3) are coupled by a strongly nonlmear source term, this generating a nonlinear set of simultaneous partlal and ordinary dlfferentxtl equations The Peaceman-Rachford solution procedure [ll - 131 transforms the dlfferentlal problem in a sequence of algebraic linear equations, using a known dlscretlsatlon technique of the denvatlves in the time-space domam shown in Fig 2 The time Increment 1s halved and the difference equations are solved first for the radial terms then for the axial ones, for the first and second halves of
Fig 2 NumerIcal grid over the trme-space domam
the tune increment respectively, thus completmg one Iteration The radial part of the dlscretlsed model follows from
155
Steady state IS considered reached when vanatlons m conversion and temperature become negllgble (0 0001 relative change for two successive time increments) Finally, let us consider the method of solving the linear systems of sunultaneous equations The matrices of these systems are tndlagonal so that they can be easily solved by a sunphfled algorithm (Thomas-Doohttle, Forz=l,
,N.,=2,
t1111
,M-1
For the axial section of the model
Table 2 contains the expressions of the coefflclents m the mathematical model, the values of several parameters of interest, used m setting up these coefflclents are summarized m Table 3 Though the thermodynamic properties of the system were considered constant throughout the mtegratlon, then variation wth temperature and composltlon can be easily implemented TABLE 3 Ranges of numerlcal values of data used for computatlons
+ERff+' For1=2
,.., N-1,1=1,.
,M. Symbol
Replication of these formulas for all the indicated index values results m four systems of linear equations for a smgle tune Increment (1 e conversion and temperature for the first and second halves of the time Increment) Equations (1) and (5) are solved straghtforwardly by a simple Euler backward formula Special attention IS given to the couphng boundary condltlon eqns (2) and (4), completing m this way the algorithm for sunultaneous computation of gas and solid profiles Though the Peaceman-Rachford procedure IS recommended as stable regardless of time Increment values used m computation for partial dlfferentlal equations not contammg source terms strongly non-hnear, a tune increment control device was imposed at the end of each iteration, to prevent error accumulation, this control conslsts of (I) lnnltmg the relative changes of conversion and temperature to less than 1% (n) mtroducmg maxlmal values of conversion (less than 2 0) and temperature (less than the adlabatlc temperature rise) Once one of these restnctlve condltlons 1s encountered, the computation 1s reiterated with a reduced time step, reduction of this parameter under a certain Imposed lower hmlt is slgnalled by an emergency message
of the parameter
h m.it h,,e kt,,. b Rl*R2
H :;;I?
Zf;’
pre-eiponentlal Re = (~,.PI)/(~,)
factor
Ranges of values used O-OlmCl O-Olkcal(msK)-l 0 003 - 0 08 m Ol-5m 0-08m2h-l 0 - 0 001 kcal (m s K)-l 10 - 100 kcal mole1 105 - 1011 s-l 2000 - 5000
The dynamics of the temperature proflles m the sohd body, plotted m Fig 3, mdlcates a tune migration of the hot spot from the inlet zone to a point equldlstant from the surfaces exposed to the flowmg fluid. The value of the hot spot temperature 1s m the vlcmlty of the adlabatlc temperature rise This 1s to be expected because of the important dlffuslonal llmltatlons occurnng m the porous wall, which are essentially the same as encoutered m a single catalyst particle or m a fmed catalytic bed The relatively thm porous wall of the PWR enables better heat transfer and leads to smaller radial temperature gradients For usual kmetlcs and operating condltlons, most spectacular changes in temperature values take place m the first segment of the reactor length (no catalyst decay was considered), the rest of the porous
156
ature spot, whereas m the case of a PWR this maximum is singular, its location is easily detected and adjusted Figure 4 presents conversion and temperature profiles for different values of the gasto-solid transfer coefflclents The insulated outer surface of the cylinder modifies the boundary condltlon (4) as follows
d W, -=0
atp=l
(12)
dp
T
T
I.2
l!23 a
cd
I.1
‘0
z
1
Fig 3 Dynamics of temperature profdes r, actual time, T,, time required to reach the steady state (a) Gas proflles, (b) InsIde and outslde surface profiles, (c) profdes atp = 0 2 and p = 0 8, (d) profdes atp=05 wall IS practically at the same temperature This 1s m agreement with the pseudo- constant solid temperature assumption made by Young and Fmlayson [ 3, 41 for the radial coordmates of the system This very lmportant conclusion 1s presented by the authors cited for a honeycomb afterburner m which heat and mass transfer do not take place simultaneously at the same point of space The model exammed overcomes this slmphficatlon, but as the consequences are the same, the slmpllflcatlon mentioned seems Justified In order to check the validity of a cntenon devised by Young to test whether the radial temperature can be regarded as constant, the ratio (RhXeff/RhsXg) 1s computed For a standard case, the ratio of solid to fluid thermal conductlvltles IS m the range 20 - 80, and the ratio of hydraulic radu lies between 1 8 - 2 5, hence the proposed cnterion is 30 - 150, much larger than 20, the limit suggested by Young, above which the assumption of a constant radial temperature holds The improved heat transfer performance of the PWR, as compared to the fured bed reactor is due to the fact that in a fuced bed, each pellet reproduces to a certam degree a maximum shaped temperature profile and the thermal control adjusts the overall temper-
‘0 Fig 4 (A) Temperature and (B) converslon profdes for permeable and Insulated external sufaces (a) Gas proflles, (b) profiles at p = 0, (c) profdes at p = 0 5, (d) profiles at p = 1
This causes a lower performance of the PWR since the insulated wall really doubles the path of dlffuslonal processes The maximum temperature 1s now located near the insulated wall, that IS, m a region where coolmg of the flowing fluid IS msufflclent When trying to draw quantitative conclusions about the influence of the catalyst core removal on the performance of the catalytic process, an effectiveness factor has been derived, using the obvious assumptions for catalytic systems As mentioned earlier, it IS expected that hollow particles show lower dlffuslonal resistance, as a material urlth large pores This IS caused by shorter paths for dlffuslonal transport and better
157
access of the fluld to the internal surface and m the reverse direction The local non-isothermal effectiveness factor 1s the ratio of the rate of the chemical process at an arbitrary point of the system to the rate of chemical reaction computed on the basis of bulk fluid concentration and temperature (m the absence of dlffuslonal resMances) Conversion and temperature in the gas phase vary only along the reactor axis At each axial level, the reference rate of reaction will be that corresponding to the flowmg fluid at the same axial level The numerator of the effectiveness factor 1s the reactlon rate at each point of the solid, for each posltlon in the radial and axial dlrectlons Plots of the values computed m this way are shown m Fig 5 Figure 6 contains the plot of a slmllar local effectiveness factor for a cylmdrlcal pellet (RI = 0) Because the shape of the profiles 1s the same, an overall effectiveness factor defined as 7)=
V(RJ)
d.R dz
(13)
1s computed for two geometries, cyhndrlcal pellet (6 = 16 92) and hollow cyhnder having the same volume ({ = 22 11) A sensible
7 50
40 1
n
n
0.5
0
Fig 6 Effectweness (RI = 0)
3
factor for cyhndrlcal
'
pellet
unprovement of the effectiveness factor 1s detected ({ 30% larger) For hollow cylinders, which 1s the case of the PWR, this enhancement does not increase mdefmltely as the walls get thinner This leads to an optimum seeking problem, first mentioned by BasmadJian [ 141 In order to fmd the upper limit of this enhancement effect, we have compared several hollow cylinders with different geometnes and a solid cylinder v&h the same volume, under isothermal condltlons For solid porous cylinder and for a fixed bed reactor having a void fraction E, the effectiveness factor for a first order kinetics, written in terms of the Thlele modulus IS qSc =(I-r)itanh#
(14)
For a PWR of inner and outer radu R, and this expression becomes,
R2, respectively, ~17PWR= (l--~)(l
+R,IR,)(l
-R,IW+R))
x
G X tanh
Fig 5 Effectweness factor for the PWR
(I- RIIRz) 1 - R,/(H + R2)’
(15)
The ratio of these two factors, compared to unity is a measure of the mentioned enhancement effect of core removal A plot of this ratio us the sohd cylinder Thlele modu-
158 Ius,
mth (RJR*) as a parameter, shows that for $J up to 10 the solid cylinder IS more efficient than the PWR geometry This seems obvious smce small values of I$ correspond to large values of diffusion coefflclents or small kmetlc constants, m either case the chemical process 1s the hmltmg step, its rate 1s not mfluenced by the geometry of the solid body As the kinetic constant increases 1 e the dlffuslon resistance prevals, hollow cylinders become more efficient but this tendency IS limited, for large Thlele modulus values, as can be seen from Figs 7a and 7b The efficiency of PWR is, m these cases 30 - 90% higher, it increases rapidly towards a maximum value as the ratio of reactor length to the external radius of the tube mcreases (Fig 8) 1oc
‘01
1
2
3 4.
5
6
7
8
9
II
o@dR,IR2)
Fig 8 Optlmal geometry of the Isothermal PWR IO
d
I
0.1
L
lb Ftg 7a Isothermal effectiveness factors ratlo (PWR us sohd cyhnder), H/R* = 5
Fig 7b Isothermal effectweness factors rat.10 (PWR us sohd cylmder), H/R, = 100
Another important conclusion follows from this plot the high sensitivity of the ratio of optunal radn to small varlatlons of the Thlele modulus. These profiles can serve for a rough estimation of the dlmenslons of the PWR To end the dlscusslon about the effectlveness factor, we emphasize that the value of this factor, computed for the non-isothermal case 1s greater than unity, that 1s because the rate of the chemical process IS linear m conversion but varies exponentially with the temperature, enabling that m a pomt of the solid, the rate of the process is greater than that m the bulk gas For isothermal condltlons the effectiveness factor varies only between 0 and 1 The mathematical model presented testifies that the PWR can represent a good alternative to the classical fured-bed system m some of Its most important characterlstlcs uzz improved thermal sensltlvlty, increased effectlveness factors for catalysts v&h large dlffuslonal resistance and reduced pressure drops The Peaceman-Rachford algorithm used to solve the system of equations proved a powerful procedure, stable and efflclent even with highly non-linear source terms and for a reduced number of grid points (8 - 10) m both space du-ectlons It 1s a reliable way of
159
solving such probltbms because implemented
It IS easily
Wz.,%e
z z
fluid velocity inside and outside the tube dimensional axial coordinate dlmenslonless axial coordinate
NOMENCLATURE
coefficient m the mathematical model cross area for the gas flowmg A ext outside the tube coefficient m the mathematical B model coefflclent m the mathematical C(P 1 model inlet concentration co specific heats of the gas and cg yck catalyst, respectively coefficient m the mathematical D model D eff.rr Dewr effective dlffuslvlty m radial and axial directions coefflclents In the mathematical E, F, G model H total height of the PWR axial increment h thermal effect of the chemical -AH, reaction radial posltlon index 1 axial posltlon index I k time increment index k In,,-k m,e mass transfer coefficients to the inner and outer surface of the PWR, respectively heat transfer coefflclents to the klrke inner and outer surface of the PWR, respectively M number of grid points m the axial direction N number of grid points m the radial dlrectlon rate expression R(W,) hydraulic radu for the cross area Rhv Rh.s of the flow and of the wall, respectively R dimensional radial coordinate inner and outer radn of the PWR, RI, Rz respectively Re Reynolds number S coefflclent m the mathematical model t time coordinate Inlet temperature TO W conversion or temperature m the standard model A
Greek symbols radial increment void fraction of the porous solid E effectiveness factor r) effectiveness factor for a solid vsc cylinder effectiveness factor for the PWR t)PWR overall effectiveness factor r, effective thermal conductivity i eff.r> Lff,z gas thermal conductlvlty h3 gas kinematic vlscoslty “8 radial dlmenslonless coordmate P density of the gas and catalyst, P&k respectively time increment Thlele modulus 6
Subscnp ts e eff g 1
m r S t
external side condltlon effective value hydraulic parameter internal side condltlon mass coefficient radial direction solid condltlon thermal coefflclent axial direction inlet condltlon
REFERENCES V HlaviEek and M KubRek, Chem Eng SCI , 27 (1972) 127 T G Smith and J J Carberry, Chem Eng SCI , 30 (1975) 221 L C Young and B A Fmlayson, AIChE J. 22 (1976) 331 L C Young and B A Fmlayson, AlChE J, 22 (1976) 343 Y T Shah, T Remmen and S H Chlang, Chem Eng &I, 25 (1970) 1947 J G Knudsen and D L Katz, Fluid Dynamics and Heat Transfer, McGraw-Hdl, New York, 1958, p 105 T G Smith and J J Carberry, Chem Eng SCI , 32 (1976) 1071
160 8 N B Ferguson and B A Fmlayson, AIChE J, 20 (1974) 539 9 D J Gunn, Chem Eng &I, 22 (1967) 1439 10 D Peaceman and H Rachford, J Sot Ind Appl Math, 3 (1955) 28 11 L Lapldus, DIgItal Computatrons for Chemrcal Engmeers. McGraw-Hlll, New York, 1962, p 150 12 B Carnahan, B A Luther and J 0 Wilkes.
Apphed Numerical Methods, Whey, New York, 1967, p 508 13 K Hartmann and M G Slmko, Kybernetlsche Methoden m der Chemle und Chemlscher Tech nologre. Akademle Verlag, Berlm, 1972, p 105 14 D Basmadjlan, J Catat, 2 (1963) 440 15 J Votruba, J Smkule, V HlavPEek and J Sknvanek, Chem Eng SCI . 30 (1975) 117
The Chemical Englneertng Journal, 23 (1982) 151 - 160 0 Elsewer Sequoia S A , Lausanne - Prmted m The Netherlands
The Catalytic Porous Wall Reactor with Axial Flow of the Reacting Mixture
R MIHAIL and C TEODORESCU Polytechmc (Recewed
Institute
Bucharest,
17 December
Department
of Chemwol
Engmeermg,
Poltzu
1, 7000 Bucharest
(Romanro)
1979, In fmal form 5 May 1981)
AbSt7YlC.t
A detailed mathematrcal model of a porous wall tubular reactor 1spresented, takrng rnto account the superposed effects of heat and mass transfer m two space drrectlons DynamK profiles of conversion and temperature are also obtarned and the influence of various parameters on reactor performances 1s drscussed, together with an analysrs of the e ffec tweness fat tor The ma thema trcal model encompasses a broader class of heterogeneous contactors, mcludmg the monolrthlc structures Some advantages of the porous wall reactor are discussed and the efflcaency rmprovement attamable m such reactors IS evaluated
INTRODUCTION
Over the past years, several new types of reactors have been suggested to meet the demand for carrying out a mde variety of chemical reactions m an economic way The pints of interest are to get lower pressure drop m the reactor m order to mmlmlze the energy required by the fluid transport and to obtain a better catalyst efficiency thus reducing the reactor volume and the catalytic mass used by the process A reactor type which could possibly meet the above requirements 1s the porous wall reactor (PWR) This reactor type consists of a hollow cylinder havmg the walls made of a porous matenal The reacting fluld enters the tube or flows along the outer wall, parallel to the cylinder axis The type mentioned operated mth suction or inJection has already been considered m literature [l, 51 Such systems could offer several advantages over the solid wall reactor,
because the driving force of the reactlon can be maintained at a nearly constant level along the entire length of the tube However, Its practical form 1s difficult to construct for industrial use In contrast, a PWR with axial flow only, without suction or mlectlon, can easily be realized by some regular superposltlon of Raschlg-type catalyst particles Such a reactor 1s dlstmgulshed by advantages over the conventional fixed-bed catalytic systems, notably, (1) removing the catalyst particle core, as compared to a solid cylinder of the same volume (and hence contammg the same amount of catalyst), hollow cylinders present lower dlffuslonal hmltatlons mth high levels of stablhty and control of the system, (ii) use of catalytically non-uniform active materials simply by dlstrlbutmg active zones along the inert porous support In a hmltmg case of the catalyst deposlted as a thin layer on the inner surface of the tube, the porous wall reactor reduces to the tube-wall reactor, now widely used as a monolithic converter [2 - 4, 91 Theory and experiments on the catalytic converter [ 2 - 4, 7, 81 have shown that the transfer properties for the catalytic zone and for the mert support differ substantially. Mass transfer 1s limited to the thin layer of the catalyst, where chemical reaction takes place, while heat transfer 1s important m the inert substrate only Such conslderatlons induced Young and Fmlayson [ 3, 4, 81 to develop a surface mass transfer model and a heat transfer model for the inert solid support This model has been solved for a mde variety of condltlons However, simultaneous mass and heat transfer accompanied by a chemical reactlon has not been considered In the present paper, we consider all the interacting phenomena mentioned m a cata-
152
of anlsotroplc properties within the wall, resulting m different values for mass and thermal dlffuslvltles in radial and axial dlrectlons, as discussed by Gunn [lo], this amsotropy may result from different tortuosity factors of the pores, due to the specific technology for the manufacturing of the hollow cylmders In the general form of the model both surfaces of the porous wall are accessible to the flowmg gas, this leads to diffusion coupling of the outer and inner surfaces of the hollow cylinders Cases of insulating one side of the solid for heat and/or mass transfer will be treated as particular sltuatlons derived from the general model by equating some parameters to zero (111)Changes of the physical properties of the sohd or fluid with temperature and conversion are neglected (IV) Unsteady state condltlons occur at the solid wall only The main geometnc and physical parameters of the model are summarized m Fig 1 The differential mass and heat balances and interphase transfer equations for the fluid and solid body, converted to the dlmenslonless form constitute the basic set of equations of the present model Though unsteady state condltlons involve the physical system as a whole, only the solid model contains the time derlvatlves This slmphflcatlon does not affect the validity of the mathematical description, proof of the above assumption follows from the so-called ‘quasi-static’ hypothesis [3, 4, 7, 81 In most gas-solid catalytic systems, hydrodynamic, chemical and temperature changes take place at very wdely differing rates Dlstmctlon can also be made for the rate of
lytlc PWR for all the length and thickness of the reactor The differences between the models developed by Smith and Carberry [ 21, Young and Fmlayson [3, 41, Votruba et al [15] and the one presented here are given in Table 1 A model of the PWR has been investigated and its performances have been established for a variety of operating condltlons of practical relevance, mcludmg the transient behavlour
DEVELOPMENT MODEL
OF
THE
MATHEMATICAL
Consider a porous tube wth axial symmetry, immersed m a flowmg fluid Heat. mass transfer and a chemical reaction occur simultaneously m all the points of the porous body, access to the Internal core and on the tube surface 1s diffusion limited The assumptions used m the derivation of the mathematical model to the PWR are summarized as follows (1) The fluid is SubJect to convective transport only The flow IS considered to be turbulent and the fluid velocity IS restricted only to the axial component The assumptions on turbulence are valid for most mdustrlal cases of interest apart from the commonly used automobile exhaust afterburners where lammar flow prevails due to very small hydrodynamic length [3] Therefore the mathematical model will be unldlmenslonal for the fluid flowing in a PWR, (11) Axial and radial heat and mass transfer m the solid phase are assumed The model consider the occurrence TABLE 1 Comparison
of exammed
physlcal models for catalytic
reactors
Reference
Smith and Carberry [ 21
Young and Fmlayson [ 3 ]
Votruba eta1 [15]
Present paper
FluId phase
Convective and dIffusIonal transport Re = 680
Convective and dlffuslonal transport Re > 2000
Convective transport only Re > 2000
Convective transport only Re > 2000
Solrd body
Surface reactton and Surface reactlon and thermal conductlon m dlffuslon In a thm the sohd layer Thermal conductlon only In the sohd
Surface reactlon Longltudmal heat conductlon within the sohd structure
Dlffuslon, thermal conductlon and chemical reaction throughout the sohd body
153
catalytic
porous solid body
a2w,
aw,
ap2
ap
-+c(p)-+ll-
a2w, az2
aw,
+
sat =JwW,L
gas flowing outside the tube (p = 1)
(3)
dW L-F(W,,-WW,)=O, dp gas flowing outslde the tube (bulk condltlon)
dWs.e - d.2 + G( W,.e-W,)=O, inlet condltlon
z+A(W,.Fig 1 Physwal statement
inlet condltlon
of the problem
the same process (e g , heat accumulation) in the gas and m the solid material For a resldence time value of 0 15 - 0 20 s, the transient response of the velocity profile 1s 10 - 20 times faster, heat accumulation term m the gas equation 1s neglected since the ratio of gas to solid thermal capacities 1s around 7 X 105, mass accumulation term 1s neglected m the fluid model because of the large dlscrepancy between the residence time and the mass transfer time constant (about 20 times larger) The study of the dynamic behavlour of the PWR 1s restricted to the transient response of the solid body because the time necessary for the wall to approach the steady-state 1s the longest time constant of the entire system Equations (1) to (9) below have been obtamed on this assumption The dlmenslonless model equations for chemical conversion and temperature are structurally identical, the term W denotes either conversion or temperature Coefficients m these equations have different expressions for conversion and for temperature parts of the mathematical model and they are hsted in Table 2 The standard form of the mathematical model is as follows
for the solid (z = 0)
W,)= 0, for the gas (z = 0)
W g-1 = W,.c? = W&O, exit condltlon
(7)
for the solid (z = 1)
dW,_ -0 dz
(8)
mltlal condltlon
W,=W,
(6)
att=
for the solid body 0
(9)
Equations (2), (4), (6), (8) and (9) constitute the set of the mltlal and boundary condltlons for the partial differential eqn (3) and are usually encountered m the description of analogous physical systems The above equations are written m the order m which they perform, starting from a point in the fluid bulk flowing through the tube, contmumg with the transfer process on the inner solid surface, the balance equation for the porous wall, the transfer process on Its outer surface and ending with the convective transport m the fluid flowmg outslde the hollow cylinder Processes taking place in the solid body are governed by eqn (3) Equations (1) and (5) constitute the fluid model together with the corresponding mltlal condltlon (eqn (7))
gas flowing inside the tube (bulk condltlon)
dW,
L
dz
1
MODEL SOLUTION
+ A(Wg.1-iv,)=
0,
(1)
inner surface of the tube (p = 0)
d W, -+wW,,,-wW,)=O, dp
(2)
A primary analysis of the set of equations presented above reveals the followmg features (1) The essential element of the model 1s the second order parabohc partial differential
154
TABLE 2 Expresstons for the coefflclents m the mathematical model Corwersion
equahon
Temperature
equation
A
B
k,.,(Rz
1
-RI
kt,A&
h eff.r
D eff.r R2
C(P)
R,
R2 -RI
-RI
+P&
- RI)
D eff,z(Ra
- RI j2
RI +P(RP-R~) h.zVh
D eff.r fl -(R2
-
RI J2
k m/AR2
-_(--aH,)(R2
ke(R2
D eff.r 2~R2Hkm
-
Rl J2
T0
RI )
h?ff.r
2nR2Hkt.e
.e
Aextwgwz.e
A extw2.e -RI
-
Aeff.r
-RI)
-
- R, J2 h eff.r H2
D eff.rCO
-(R2
- RI)
J2
D eff.1
-tR2
-RI x eff.1
j2 (P&g
+ Pkck)
equation descrlbmg processes In the solid body I e eqn (3) The method of solvmg the eqn (3) ~11 influence the solution of the
entxre model (11) The three dlstmct physical sectlons of the model (inside fluid, catalyst wall, outslde fluid), linked by eqns (2) and (4), must be solved simultaneously (111)Converslon and temperature equations of the type (3) are coupled by a strongly nonlmear source term, this generating a nonlinear set of simultaneous partlal and ordinary dlfferentxtl equations The Peaceman-Rachford solution procedure [ll - 131 transforms the dlfferentlal problem in a sequence of algebraic linear equations, using a known dlscretlsatlon technique of the denvatlves in the time-space domam shown in Fig 2 The time Increment 1s halved and the difference equations are solved first for the radial terms then for the axial ones, for the first and second halves of
Fig 2 NumerIcal grid over the trme-space domam
the tune increment respectively, thus completmg one Iteration The radial part of the dlscretlsed model follows from
155
Steady state IS considered reached when vanatlons m conversion and temperature become negllgble (0 0001 relative change for two successive time increments) Finally, let us consider the method of solving the linear systems of sunultaneous equations The matrices of these systems are tndlagonal so that they can be easily solved by a sunphfled algorithm (Thomas-Doohttle, Forz=l,
,N.,=2,
t1111
,M-1
For the axial section of the model
Table 2 contains the expressions of the coefflclents m the mathematical model, the values of several parameters of interest, used m setting up these coefflclents are summarized m Table 3 Though the thermodynamic properties of the system were considered constant throughout the mtegratlon, then variation wth temperature and composltlon can be easily implemented TABLE 3 Ranges of numerlcal values of data used for computatlons
+ERff+' For1=2
,.., N-1,1=1,.
,M. Symbol
Replication of these formulas for all the indicated index values results m four systems of linear equations for a smgle tune Increment (1 e conversion and temperature for the first and second halves of the time Increment) Equations (1) and (5) are solved straghtforwardly by a simple Euler backward formula Special attention IS given to the couphng boundary condltlon eqns (2) and (4), completing m this way the algorithm for sunultaneous computation of gas and solid profiles Though the Peaceman-Rachford procedure IS recommended as stable regardless of time Increment values used m computation for partial dlfferentlal equations not contammg source terms strongly non-hnear, a tune increment control device was imposed at the end of each iteration, to prevent error accumulation, this control conslsts of (I) lnnltmg the relative changes of conversion and temperature to less than 1% (n) mtroducmg maxlmal values of conversion (less than 2 0) and temperature (less than the adlabatlc temperature rise) Once one of these restnctlve condltlons 1s encountered, the computation 1s reiterated with a reduced time step, reduction of this parameter under a certain Imposed lower hmlt is slgnalled by an emergency message
of the parameter
h m.it h,,e kt,,. b Rl*R2
H :;;I?
Zf;’
pre-eiponentlal Re = (~,.PI)/(~,)
factor
Ranges of values used O-OlmCl O-Olkcal(msK)-l 0 003 - 0 08 m Ol-5m 0-08m2h-l 0 - 0 001 kcal (m s K)-l 10 - 100 kcal mole1 105 - 1011 s-l 2000 - 5000
The dynamics of the temperature proflles m the sohd body, plotted m Fig 3, mdlcates a tune migration of the hot spot from the inlet zone to a point equldlstant from the surfaces exposed to the flowmg fluid. The value of the hot spot temperature 1s m the vlcmlty of the adlabatlc temperature rise This 1s to be expected because of the important dlffuslonal llmltatlons occurnng m the porous wall, which are essentially the same as encoutered m a single catalyst particle or m a fmed catalytic bed The relatively thm porous wall of the PWR enables better heat transfer and leads to smaller radial temperature gradients For usual kmetlcs and operating condltlons, most spectacular changes in temperature values take place m the first segment of the reactor length (no catalyst decay was considered), the rest of the porous
156
ature spot, whereas m the case of a PWR this maximum is singular, its location is easily detected and adjusted Figure 4 presents conversion and temperature profiles for different values of the gasto-solid transfer coefflclents The insulated outer surface of the cylinder modifies the boundary condltlon (4) as follows
d W, -=0
atp=l
(12)
dp
T
T
I.2
l!23 a
cd
I.1
‘0
z
1
Fig 3 Dynamics of temperature profdes r, actual time, T,, time required to reach the steady state (a) Gas proflles, (b) InsIde and outslde surface profiles, (c) profdes atp = 0 2 and p = 0 8, (d) profdes atp=05 wall IS practically at the same temperature This 1s m agreement with the pseudo- constant solid temperature assumption made by Young and Fmlayson [ 3, 41 for the radial coordmates of the system This very lmportant conclusion 1s presented by the authors cited for a honeycomb afterburner m which heat and mass transfer do not take place simultaneously at the same point of space The model exammed overcomes this slmphficatlon, but as the consequences are the same, the slmpllflcatlon mentioned seems Justified In order to check the validity of a cntenon devised by Young to test whether the radial temperature can be regarded as constant, the ratio (RhXeff/RhsXg) 1s computed For a standard case, the ratio of solid to fluid thermal conductlvltles IS m the range 20 - 80, and the ratio of hydraulic radu lies between 1 8 - 2 5, hence the proposed cnterion is 30 - 150, much larger than 20, the limit suggested by Young, above which the assumption of a constant radial temperature holds The improved heat transfer performance of the PWR, as compared to the fured bed reactor is due to the fact that in a fuced bed, each pellet reproduces to a certam degree a maximum shaped temperature profile and the thermal control adjusts the overall temper-
‘0 Fig 4 (A) Temperature and (B) converslon profdes for permeable and Insulated external sufaces (a) Gas proflles, (b) profiles at p = 0, (c) profdes at p = 0 5, (d) profiles at p = 1
This causes a lower performance of the PWR since the insulated wall really doubles the path of dlffuslonal processes The maximum temperature 1s now located near the insulated wall, that IS, m a region where coolmg of the flowing fluid IS msufflclent When trying to draw quantitative conclusions about the influence of the catalyst core removal on the performance of the catalytic process, an effectiveness factor has been derived, using the obvious assumptions for catalytic systems As mentioned earlier, it IS expected that hollow particles show lower dlffuslonal resistance, as a material urlth large pores This IS caused by shorter paths for dlffuslonal transport and better
157
access of the fluld to the internal surface and m the reverse direction The local non-isothermal effectiveness factor 1s the ratio of the rate of the chemical process at an arbitrary point of the system to the rate of chemical reaction computed on the basis of bulk fluid concentration and temperature (m the absence of dlffuslonal resMances) Conversion and temperature in the gas phase vary only along the reactor axis At each axial level, the reference rate of reaction will be that corresponding to the flowmg fluid at the same axial level The numerator of the effectiveness factor 1s the reactlon rate at each point of the solid, for each posltlon in the radial and axial dlrectlons Plots of the values computed m this way are shown m Fig 5 Figure 6 contains the plot of a slmllar local effectiveness factor for a cylmdrlcal pellet (RI = 0) Because the shape of the profiles 1s the same, an overall effectiveness factor defined as 7)=
V(RJ)
d.R dz
(13)
1s computed for two geometries, cyhndrlcal pellet (6 = 16 92) and hollow cyhnder having the same volume ({ = 22 11) A sensible
7 50
40 1
n
n
0.5
0
Fig 6 Effectweness (RI = 0)
3
factor for cyhndrlcal
'
pellet
unprovement of the effectiveness factor 1s detected ({ 30% larger) For hollow cylinders, which 1s the case of the PWR, this enhancement does not increase mdefmltely as the walls get thinner This leads to an optimum seeking problem, first mentioned by BasmadJian [ 141 In order to fmd the upper limit of this enhancement effect, we have compared several hollow cylinders with different geometnes and a solid cylinder v&h the same volume, under isothermal condltlons For solid porous cylinder and for a fixed bed reactor having a void fraction E, the effectiveness factor for a first order kinetics, written in terms of the Thlele modulus IS qSc =(I-r)itanh#
(14)
For a PWR of inner and outer radu R, and this expression becomes,
R2, respectively, ~17PWR= (l--~)(l
+R,IR,)(l
-R,IW+R))
x
G X tanh
Fig 5 Effectweness factor for the PWR
(I- RIIRz) 1 - R,/(H + R2)’
(15)
The ratio of these two factors, compared to unity is a measure of the mentioned enhancement effect of core removal A plot of this ratio us the sohd cylinder Thlele modu-
158 Ius,
mth (RJR*) as a parameter, shows that for $J up to 10 the solid cylinder IS more efficient than the PWR geometry This seems obvious smce small values of I$ correspond to large values of diffusion coefflclents or small kmetlc constants, m either case the chemical process 1s the hmltmg step, its rate 1s not mfluenced by the geometry of the solid body As the kinetic constant increases 1 e the dlffuslon resistance prevals, hollow cylinders become more efficient but this tendency IS limited, for large Thlele modulus values, as can be seen from Figs 7a and 7b The efficiency of PWR is, m these cases 30 - 90% higher, it increases rapidly towards a maximum value as the ratio of reactor length to the external radius of the tube mcreases (Fig 8) 1oc
‘01
1
2
3 4.
5
6
7
8
9
II
o@dR,IR2)
Fig 8 Optlmal geometry of the Isothermal PWR IO
d
I
0.1
L
lb Ftg 7a Isothermal effectiveness factors ratlo (PWR us sohd cyhnder), H/R* = 5
Fig 7b Isothermal effectweness factors rat.10 (PWR us sohd cylmder), H/R, = 100
Another important conclusion follows from this plot the high sensitivity of the ratio of optunal radn to small varlatlons of the Thlele modulus. These profiles can serve for a rough estimation of the dlmenslons of the PWR To end the dlscusslon about the effectlveness factor, we emphasize that the value of this factor, computed for the non-isothermal case 1s greater than unity, that 1s because the rate of the chemical process IS linear m conversion but varies exponentially with the temperature, enabling that m a pomt of the solid, the rate of the process is greater than that m the bulk gas For isothermal condltlons the effectiveness factor varies only between 0 and 1 The mathematical model presented testifies that the PWR can represent a good alternative to the classical fured-bed system m some of Its most important characterlstlcs uzz improved thermal sensltlvlty, increased effectlveness factors for catalysts v&h large dlffuslonal resistance and reduced pressure drops The Peaceman-Rachford algorithm used to solve the system of equations proved a powerful procedure, stable and efflclent even with highly non-linear source terms and for a reduced number of grid points (8 - 10) m both space du-ectlons It 1s a reliable way of
159
solving such probltbms because implemented
It IS easily
Wz.,%e
z z
fluid velocity inside and outside the tube dimensional axial coordinate dlmenslonless axial coordinate
NOMENCLATURE
coefficient m the mathematical model cross area for the gas flowmg A ext outside the tube coefficient m the mathematical B model coefflclent m the mathematical C(P 1 model inlet concentration co specific heats of the gas and cg yck catalyst, respectively coefficient m the mathematical D model D eff.rr Dewr effective dlffuslvlty m radial and axial directions coefflclents In the mathematical E, F, G model H total height of the PWR axial increment h thermal effect of the chemical -AH, reaction radial posltlon index 1 axial posltlon index I k time increment index k In,,-k m,e mass transfer coefficients to the inner and outer surface of the PWR, respectively heat transfer coefflclents to the klrke inner and outer surface of the PWR, respectively M number of grid points m the axial direction N number of grid points m the radial dlrectlon rate expression R(W,) hydraulic radu for the cross area Rhv Rh.s of the flow and of the wall, respectively R dimensional radial coordinate inner and outer radn of the PWR, RI, Rz respectively Re Reynolds number S coefflclent m the mathematical model t time coordinate Inlet temperature TO W conversion or temperature m the standard model A
Greek symbols radial increment void fraction of the porous solid E effectiveness factor r) effectiveness factor for a solid vsc cylinder effectiveness factor for the PWR t)PWR overall effectiveness factor r, effective thermal conductivity i eff.r> Lff,z gas thermal conductlvlty h3 gas kinematic vlscoslty “8 radial dlmenslonless coordmate P density of the gas and catalyst, P&k respectively time increment Thlele modulus 6
Subscnp ts e eff g 1
m r S t
external side condltlon effective value hydraulic parameter internal side condltlon mass coefficient radial direction solid condltlon thermal coefflclent axial direction inlet condltlon
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160 8 N B Ferguson and B A Fmlayson, AIChE J, 20 (1974) 539 9 D J Gunn, Chem Eng &I, 22 (1967) 1439 10 D Peaceman and H Rachford, J Sot Ind Appl Math, 3 (1955) 28 11 L Lapldus, DIgItal Computatrons for Chemrcal Engmeers. McGraw-Hlll, New York, 1962, p 150 12 B Carnahan, B A Luther and J 0 Wilkes.
Apphed Numerical Methods, Whey, New York, 1967, p 508 13 K Hartmann and M G Slmko, Kybernetlsche Methoden m der Chemle und Chemlscher Tech nologre. Akademle Verlag, Berlm, 1972, p 105 14 D Basmadjlan, J Catat, 2 (1963) 440 15 J Votruba, J Smkule, V HlavPEek and J Sknvanek, Chem Eng SCI . 30 (1975) 117