- Email: [email protected]
Heat and mass transfer in monolithic honeycomb catalysts—III
mo%2309nm7014839/302
Ckmrcd &gmemn# Sc~encc Vol 33 pp 839445 @ Pcrpmon Press Ltd 1978 Rmed Mloreal Bntam
00/o
HEAT AND MASS TRANSFER IN MONOLITHIC HONEYCOMB CATALYSTS-III RADIATION
MODEL
J&f SINKULE and VLADIMfR HLAVAEEK Department of Chemical Engmeenng, Institute of Chemical Technology, 166 28 Prague 6, Czechoslovakia (Recemd 6 June 1977, accepted in rewed form 12 October 1977) Abstract-An analysis ISmade of the effect of radlatton exchange upon the behavior of an exothermlc catalytic reactlon occurring on a surface of a cyhndncal passage An n pnon model was developed for calculation of temperature and conversion profiles m the passage and sohd phase as well The model postulates plug flow and gas-to-sohd heat and mass transfer The gas ISassumed to be transparent but radial exchange wlthm the passage, lonfltudmal conductlon of the solid phase and exothermlc reactlon occumng on the catalyst surface are taken mto account The governmg equations descnbmg this model represent a nonhnear boundary value problem for ordmary dtfferentlal equations and mtegro-ddFerentml equation of Fredholm type subJect to Integral boundary condttlons The resultmg set of nonhnear drfferentml and mtegrodlfferenttal equattons was approxtmated by a specud fimtedlfference scheme and quadrature formulas The set of nonlmear algebrluc equations was solved by the Newton-Raphson method It was found that radlatlon exchange can cause slgmficant changes of temperature dlstrlbutlon m the passage Calculated examples mdlcated that the model suggests the existence of multiple steady states 1 NlltOlWClTON
The performance of monohthtc reactors depends on the combmed effects of heat and mass transport and exothermlc chemical reactlon An assumption which hmlts the apphcablltty of all models pubhshed 1s the neglect of the radlatlon on converslon for an exothermlc gas-phase reactlon Smce the temperature rtse wlthm the monohth may amount to several hundred degrees m a few centimeters of the passage, radiation effects may strongly affect the heat transmlsslon m the ngd structure Most recently, Lee and Ar1s[21 have considered a model using two space dlmenslons with strongly no&near boundary condltlons tncludmg exothermrc chemtcal reaction and radlatton effects Thm paper IS noteworthy in that they address the problem systematic analysis of temperature profiles m the monohthlc structure Lee and Ans did not calculate multtple steady states, however, with respect to the recent work of Young and Fmlayson[6] multiple steady states may be expected also for a two-dlmenslonal radiation model Prehmtnary results obtamed for the one-dtmenaonaI radlatlon model have been reported by us recently [l] Chen and Churchdl[3] have analyzed a homogeneous chemlcai reaction occumng m a tube takmg mto account radlatlon effects They have reported, for this parttcular case, the existence of multiple steady states The purpose of this paper IS to study the one-dlmenslonal model of a monohth mcludmg convectlon of heat and mass, sohd conduction, gas-to-solid transfer, surface exothermlc chemical reactlon and heat transmlsslon by
J//,,‘/“‘,/“’ *=+ y,e 4 _-_ /////I////////j R=O r--
radlatlon The present work extends the mvesugation of multiple steady state behavior of flow tubular systems to the troublesome problems mcludmg important radlatlon effects 2
GOVERNING BQVATIONS
followmg assumptions are used m the development of heat and mass balance equations -Average values of physical properties for gas and ---sohd phase are used (I e p. C,, AH, A,) -Gas-to-sohd heat and mass transfer coefficients, k, and o. are represented by average values -Flowmg reaction gas ts transparent and the emlsslvlty of the solid surface IS 1 Simple exothermlc surface catalytic reaction IS consrdered, A + B -Steady state condltlons are attamed -One srngle passage IS analyzed The mass balances m the gas phase and at the sohd surface may be wntten m the form
The
dC v-&-a&&-c)=o UC
Uf
- C,) - k,C,” exp ( - E/K&)
The enthalpy balances are respectively
=0
(2)
m the gas phase and on the wall
vpC,.$$ra(T.
- T)=O
(3)
S&,s-tm(T,-T)+(-AH)kG”exp(-EIRT.)
‘L. ---I f34& __J 1=/_
Here the term H(x) radlatlon
Fig 1 Schema of a passage
839
represents
+H(x)=O
(4)
the heat transmltted
by
J
840
SINKULE and
V
HLAVACEK
Referring to Rg 2 and using the Lamberts law the heat emltted from the element dA, which IS absorbed by the element dA IS
(11) vPC, [ TI - T(L)1 =
(5) Smce the temperature T, depends coordmate x, an mtegratlon around may be performed151 d*Q = uEK(x Here the geometric
-
only on the axial the elementar rmgs
x,)Ts”(x,) dx, dx
configuratlon
+
dy @-J&o dg @-
(7)
Here dQE IS the energy emltted by the elementar nng, d& represents the energy transmttted from the whole tube to the elementar rmg and finally, d& IS the flux from the two end cawtles For these contnbutlons we may write
To and
K(x-x,)dxr
1
cavittes, respectively The boundary condmons
x=O
C=C,,
vpC, (To - Tm) = cm
of the fore
(16)
~~-h(d-8)+DaAT..(1-ru)“exp *
and
(8) .
aft
are
T=T,,
J,.,(t9-@=O
The dlmenstonless heat flux W(e) ISthe dtierence the heat emttted and that whtch IS absorbed
dx
dx
T, are the temperatures
- y) = 0
+ E.H({) = 0
1
L
makrng
Jo(u-y)-~(l-_O)nexp[~]=O
dQ~=u~[T~4(_D_K(x-x,)dx,
I
dlmenslonless
(12)
IS
H(x) dx = - dQE + dQr + dQc
+ T,4
Ts4(x) - T,4] dx, dx
These equattons may be rendered use of the followmg substltutlons
which IS dependent on the difference of axtal coordmates of both rings x - x, The radtatton term H(x) dx conststs of three contnbutions
_
[
(6)
factor K(x -xl)
dQE = vu&Ts4(x) dx L K(x - x,)Ts4(xa) d x, d&= WE [I 0
K(x -x1)
UE
s=O
0
+
-W’(v) drl + FtWo4 (18)
F(S - &e,’ - S4(&)
Here F IS a view factor for an element ~Ddx at a distance down the passage and K IS a view factor between two elements ?rD dx at a dtstance r) - 6 from one another,
(9)
K(x - XI) ETl”(x) - To’) dx, dx
H(S) = j-- K(r, -
between
K(x)= The relatlonshtp
between
1-w F(x) and K(x) IS gven
(19) by
(10)
(20)
I
=I
Rg 2
x Radlatron flux m a passage
The first term on the r&t hand side of eqn (18) represents a contrtbutton from every part of the cylmdecal surface whtle F(&)&’ and F(S - &%4 are conmbuttons from the two end cavltles Fmally, the last term of the r&&t-hand side of eqn (18) IS the total energy emItted from the elementar nng ?rD dx The boundary condttlons may be formulated assummg “black holes”. I e the ends of the monohth are exposed
841
Heat and mass transfer m monohtluc honeycomb catalysts-111 to cavltles which are supposed to be black body radiator The radlatlon energy emltted from the tube IS absorbed here by the flowmg gas
.f=o
e=eO
y=o
80-l-E,
$=o
- &‘I dl, = 0
’ F(aW’(v) I0
8, - g(S) - E, 10’ F(S - q)[#(v)-
(21)
,
&‘I ds = 0
Integratton
Q + E,H(@ = 0 de
(24)
(25)
(26) The last term m eqn (26) can be wrltten m the form
Ths
I’0
F@ - WI’ d& -
relatron can be rearranged
+
(27)
to
I
“K(cp)dp 0
-
loaa’(e) de
=2
I-0
1 + AT&y(S) - 8, = 0 Instead of adlabatx condltlons adlabatx condltlon may be used 80= 8, = 1
of (25) yields
+
and with respect
(22) and
to (21)_(24) a
i3W (24) a non-
(22)
It IS assumed that the ends of the ceramic matenal are msulated After msertmg eqns (14)-(16) mto eqn (17) we have I d26 d8 pe,B-@+ATad
On usmg thrs equation simple equatron results
K(P) ds
(31)
Here we have assumed that the energy emltted by radlatlon IS absorbed by the end cavltes and LS not transnutted to the gas phase G EQUATIONS 3. NuMglllCALSOUlTIONOFGOVERNIN The set of governing equations, eqns (14)-(24), IS represented by a not&near boundary value problem for ordmary dlfferentml equations and an uttegro-dlfferen’tlal equation of Fredholm type subject to rntegral boundary con&tons For numertcal solution of these equations firutedtierence approxlmatlons are used, I e the derlvatlves and the mtegrals are approximated by difference and quadrature formulas, respectively The resulting set of nonlinear algebraic equations IS solved by the NewtonRaphson techmque For approxlmatton of dertvatlves the three-pomt and five-point formulae have been used Trapezoukd rule proved to be better than the Simpson quadrature formula, apparently because of dlscontuuuty of the first denvatlve of the kernel However, the trapezoidal Integration formula can be successfully used only d the number of md pomts 1s high smce the profile of K(x), m compartson with the dependence 6*(x), IS very steep m the vlcmrty of the pomt x = 0 To reach a high accuracy of the calculation a more comphcated procedure has to be adopted The gist of this procedure 1s to use three auxthary grid pomts laymg between the points q and xi-i The values of the vartable 6 at these pomts have been evaluated by mterpolatlon from the points a,-,, a,, 6,+, and &+z Makrng use of auxlhary pomts the value of the Integral In the interval (v,, q,+,) have been calculated by the five-pomt Newton-Cotes quadrature formula
K(P) dv
K(v) ds -
=2F(O)-F(S-q)-F(q) (28)
=l-F(6-~)-F(?p) here we have used the relations K(V)
= K(V)
and F(O) = l/2 After msertmg eqn (28) Into the first term In (27) we have j-’ H(t) d Z = j-08 F(rl)[&‘-
0
+
S4(d1 dl,
Fts - a)b%4- S*(v)1 drl
Here Q = vi + h/4 J. wj are weights in the quadrature formula and L(q) IS the mterpolation polynomial for 6* The Integrals occurrmg m the boundary condltlons have been approxunated m the same way The accuracy of the calculation has Improved considerably If instead of eqn (24), eqn (30) has been used For problems with nonadlabatlc boundary condltrons (31) the calculauon procedure IS the same The resultmg set of no&near algebrsllc equations has been solved by makmg use of the Newton-Raphson method For a unrque solution of governing equations convergence was always attamed On the other hand, for multiple solutions a very good uutlal profile must be selected m order to attam convergence
842
J SINKULEand V
The resulting Jacobian matrix does not exhibit band structure, however, after an appropriate ordering of equations only the nonzero elements can be stored The calculations have been performed for 50 gnd pomts One Iteration on the computer ICL 4-72 amounts to 35 set The calculation with radiation model 1s slower by a factor 50 m comparison with the non-radiative case Six iterations were necessary to reach the tolerance of calculatlon lo-’ Detads of calculation wrll be reported elsewhere [4]
HLAVA~EK
8 .+
125
4 RESULTS OF CALCULATION
The parameter Es8 1s a measure of the importance of radiation This parameter represents a ratio between radiative heat flux from the Internal surface of the passage and convective flux for Inlet conditions ‘O-
,
IO
20
E
The magnitude of the parameter Es for important exothermic reactions IS supposed to be In the range 0 OOOI0 IS The magnitudes of the variables entering into E, are reported m Table 1 To demonstrate the differences between radiation and non-radiation models the value Es = 0 1 has been chosen for calculations In Fig 3 the temperature profiles m the gas and solid phases are drawn Three cases are considered (A) radiatmn neglected, (3) radiation with adiabatic condltlons, (C) radiation wrth non-adiabatic conditions Evidently, the operation of the monolith IS m the “dtffusion regime” For the adiabatic condltlons the inlet gas IS preheated because of strong radiation effect The heat loss by radiation m the mlet section cannot be compensated by higher gas temperature, heat conduction m the solid phase and radiation from the reactor section having high temperature As a result the radiation model gives nse to lower temperature of the solid phase m the inlet section m comparison with the non-radiation model The same effect can be observed at the monolith outlet, however, here the temperature difference between the monohth and gas IS very small and hence the increase of B, IS not important Since the view factor K(x) decreases rapidly for higher values of x, the gas preheating effect IS important only for reaction zones occurrmg m the Inlet section For nonadiabatic conditions the shape of temperature profiles IS strongly affected by the radiation cooling at both ends of the passage Evidently a hot spot on the dependence 6 = 9(e) exists The temperature profile S(e) may be roughly divided mto three parts In the first part of the monolith the amount of the heat evolved by chemical reactlon and the heat transmitted IS m eqmhbrrum wurth the heat lost and the temperature profile 1s roughly constant FInally, m the last section the heat
Table
D(mm) I-20
Kn[“Cl 100-300
1 Magmtudes
Rg 3 Temperature profiles m the monohth y=20. Da=OOfM, JD=&=02S. S=20) Ol,C, E.=Ol,Bo=81=1.B.---,6,-
(AT.,, = 0 5, Pe, = 5, A, E,=O. B, Es=
I
0
IO
6 Wg 4 Conversion profiles m a monolith y=20, Da=OOO4. JD=JH=OZS, 6=20) 01,c,Es=oI,e~=8,=l
(AT,,d = 0 5, Pe, = 5, A, E,=O, B. E,=
generated 1s very low and the radlatton coohng IS htgher than the heat supply The conversion profiles are drawn in FQ 4 Wtth respect to the values of the parameters Do and 6 the reaction IS carried out in the “diffusion regtme” As a result of a shift of the reacuon zone the radlatron models predict lower values of conversion This concluston IS m agreement with results calculated from the two-dimensional model [21 In Fig 5 the dependence Da = f(y) IS shown For the
of variables
In E,
Re
(pC,MJlm3Kl
u~CW/m2K’l
200-2500
500-1000
5 67 x IO-’
LID l&50
Heat and mass transfer m monohthw
”
843
cataIysts--III
I
“3
Y,
honeycomb
8
Ftg 5 Dependence of Damkijhler number Do on exit conversLon y(8)(ATod=05. y=20. JD=J”=l, Pe,=5, Da=OO103, 8= 3 4)
govermng parameters selected multiple steady states occur Even for a low value of 6, (1 e the effect of boundary conditions cannot be neglected) both dependences are very smular The lgmtton bound calculated from both models IS almost the same This fact can be easily explained smce m the lower steady state the temperature IS low and the radiation effect IS not important The region of extinction 1s also not affected by rtiation The inlet gas ISpreheated because of rtiafion effect, however, the heat loss from the hot catalytic surface caused by gas-to-solid heat transfer 1s lower since the gas has a higher temperature In Fig 6 axial temperature and conversion profiles are drawn for the case of multiple steady states It IS obvious that for the upper steady state the ddferences between the surface and gas temperatures are very lugh For the parameters presented in l+g 7 five steady
I 05 Y,
8
Fig 7 Dependence of Damkiihler number Da on exit conversion Y(g) -3 &=Ol.--, E,=Ol. &,=B,=l,------, E,=O Parameters see Fii 7(a)
0003
B
c
Y 0t j-
/ (3
0002
I
1’
1’ _./-rc
I-
-- --
I-
/
_//* 2
.-’
-
/
25
m r’
l
*_*_*-I-. I
E
r
-2
Fii 6 Temperature and conversion profiles IIIa monohth (AT&, = 05. 7=20, JD=Js=l, Pe,=5,~=0.0103, 8=34) -, y, ----.e,---,6,E,=o1 ClISVd
33 No
7-D
Y, 8 Ftg 7(a) Remon of high converSlons
from
Rg
7 AT,,
= 0 5,
/’
/I
-’
I
09
f
states are possible for a model without radiation Radlabon model prechcts also five steady states, however, only m a narrow range of Du, Du ~(2 31 x lo-‘, 2 38x lo-‘) So far, we did not perform stabWy analysa, however, we assume that the 1st. 3rd and 5th profiles are stable These profiles are drawn m Fe 8 In the lower steady state the exothernuc reaction occurs m the “kmetlc reame” whlk for the upper steady state external heat
J S~NWWE and V. HLAVCSEK
Y
Fig 8 Axial temperature profiles m a monolith MultIpIe solutrons (AT&=O5, y=20, n=l, S=20. &=&=O& &0003,Pe.=5) -, E,=Ol,----,E,=Ol, f&=&=1.----. E, =0
and mass transfer Is rate-controlhng For the lower as well as upper steady state profiles the axlaI temperature profile wlthm the sohd phase IS almost constant whde for the mlddle steady state (the 3rd profile) a steep temperature gradlent m the sohd phase occurs The dependence of Do = f(y) for higher values of Pe. IS drawn m FQ 9 The nonradlatlve model predicts five steady states m the regon D&3 1 x lo-“, 3 47 x lo-‘) The upper temperature profiles calculated from radlatlon models are drawn m Fig 10 We may notlce that for nonadlabatlc boundary condltlons for lower value of Do
000
FIB IO Axml temperature profiles m a monolith (AT.,, = 0 5. I = 20, II = 1, JO = JH = 0 25, 7 = 20. Pe. = 50)
-,
E. = 0 1,
---.E,=O~.B~=~,=~.A,D~=OOO~,B,~~I=OOO~,C.L~= 0002lS
the hot spot temperature is higher This sltuatlon can be explamed by the fact that for lower values of Do the hot spot zone IS stufted downstream and hence the heat loss by rachatlon coohng IS lower 5 coNcLustoN ANDU!X~ Based on results of numerIcal calculation followmg conclusions may be drawn -The exit conversion IS not affected by radlatlon -The shape of temperature profiles m the passage IS dependent on the type of boundary condttlons The differences between radiation and non-radlatlon models are important -Radlatlon effect for adlabatlc boundary conditions does not affect essentmlly the lgnltlon and extmctlon bounds -Radlatlon effect dlmuushes the remon of five steady Ac&nowJe&emeti-The helpful comments of Prof H Hofmann Erlangen are gratefully acknowledged NUl.ATtON concentration of the component A m gas and on solid surface C, heat capacity D duuneter of the passage Do Damkiihler number see eqn (13) E actrvatlon energy ES dlmenslonless number, see eqn (13) ( - AH) heat of reactlon Qmenstonless gas-to-sohd heat and mass transfer coefficients pre-exponential factor mass transfer coeffictent axml dutance length of the passage order of the reactlon heat flux P&let number for solid, see eqn (13)
C, C,
83 000
Y,
-6.
I
FIB 9 Dependence of Damkshler number Do on exrt conversIon -, E, = 0, --, E.=Ol, &,=B,=l AT,=OS. JD=
Y(8)
JH=025.~=20.Pes=50,n=1.6=20
Heat and mass transfer m monollttw honeycomb catalysts-III
R gas constant S sobd cross-sectmn T, Ts temperature m gas and on sold To. TI temperature of the cavities dimensronless adrabatx temperature AT,,
WE StephawBoltzmann
constant E dunenslonless axial dwtance w conversion on sohd surface
rise,
see
eqn (13) volumetric flow rate in a stngIe passage conversion m gas, see eqn (13)
tt Y
Greek svmbnts gas-to-solid heat transfer coef%xent a actwation energy, see eqn (13) Y dtmenslonless s dlmenstonless length of the Passage, see eqn (13) 8 dtmenslonless sohd temperature, see eqn (131 # dimensionless gas temperature, see eqn (131 hr heat conductivity of the sohd phase P gas density ir wetted penmeter I
--
---
845
Subs~npts m related to rnput condltlons
BEFERMCB Hlav&Eek V et 01. Heidelberg Prpc 4th ISCRE D VI-240 (Dcchema) 1976 121 Lees T andAnsR.Ckefzz &gegSe; 1977x827 [3] Chen J L P and Church11S W, Comb F/ 1972 18 27 [4] Smkuk 1 and Htav&%k V Insr J Heur MOSSrr~n~, m preparauon [Sl Hottet H C , Radtant Heat Tmnsmtsston McGraw HIM.New York 1950 161 Young L C and Rnlayson B A , Pmc 2nd Int Symp 1;7nrte Element Methods Flow Problems Santa Margbenta Llgure. Italy, June 1976 [l]
.
Ckmrcd &gmemn# Sc~encc Vol 33 pp 839445 @ Pcrpmon Press Ltd 1978 Rmed Mloreal Bntam
00/o
HEAT AND MASS TRANSFER IN MONOLITHIC HONEYCOMB CATALYSTS-III RADIATION
MODEL
J&f SINKULE and VLADIMfR HLAVAEEK Department of Chemical Engmeenng, Institute of Chemical Technology, 166 28 Prague 6, Czechoslovakia (Recemd 6 June 1977, accepted in rewed form 12 October 1977) Abstract-An analysis ISmade of the effect of radlatton exchange upon the behavior of an exothermlc catalytic reactlon occurring on a surface of a cyhndncal passage An n pnon model was developed for calculation of temperature and conversion profiles m the passage and sohd phase as well The model postulates plug flow and gas-to-sohd heat and mass transfer The gas ISassumed to be transparent but radial exchange wlthm the passage, lonfltudmal conductlon of the solid phase and exothermlc reactlon occumng on the catalyst surface are taken mto account The governmg equations descnbmg this model represent a nonhnear boundary value problem for ordmary dtfferentlal equations and mtegro-ddFerentml equation of Fredholm type subJect to Integral boundary condttlons The resultmg set of nonhnear drfferentml and mtegrodlfferenttal equattons was approxtmated by a specud fimtedlfference scheme and quadrature formulas The set of nonlmear algebrluc equations was solved by the Newton-Raphson method It was found that radlatlon exchange can cause slgmficant changes of temperature dlstrlbutlon m the passage Calculated examples mdlcated that the model suggests the existence of multiple steady states 1 NlltOlWClTON
The performance of monohthtc reactors depends on the combmed effects of heat and mass transport and exothermlc chemical reactlon An assumption which hmlts the apphcablltty of all models pubhshed 1s the neglect of the radlatlon on converslon for an exothermlc gas-phase reactlon Smce the temperature rtse wlthm the monohth may amount to several hundred degrees m a few centimeters of the passage, radiation effects may strongly affect the heat transmlsslon m the ngd structure Most recently, Lee and Ar1s[21 have considered a model using two space dlmenslons with strongly no&near boundary condltlons tncludmg exothermrc chemtcal reaction and radlatton effects Thm paper IS noteworthy in that they address the problem systematic analysis of temperature profiles m the monohthlc structure Lee and Ans did not calculate multtple steady states, however, with respect to the recent work of Young and Fmlayson[6] multiple steady states may be expected also for a two-dlmenslonal radiation model Prehmtnary results obtamed for the one-dtmenaonaI radlatlon model have been reported by us recently [l] Chen and Churchdl[3] have analyzed a homogeneous chemlcai reaction occumng m a tube takmg mto account radlatlon effects They have reported, for this parttcular case, the existence of multiple steady states The purpose of this paper IS to study the one-dlmenslonal model of a monohth mcludmg convectlon of heat and mass, sohd conduction, gas-to-solid transfer, surface exothermlc chemical reactlon and heat transmlsslon by
J//,,‘/“‘,/“’ *=+ y,e 4 _-_ /////I////////j R=O r--
radlatlon The present work extends the mvesugation of multiple steady state behavior of flow tubular systems to the troublesome problems mcludmg important radlatlon effects 2
GOVERNING BQVATIONS
followmg assumptions are used m the development of heat and mass balance equations -Average values of physical properties for gas and ---sohd phase are used (I e p. C,, AH, A,) -Gas-to-sohd heat and mass transfer coefficients, k, and o. are represented by average values -Flowmg reaction gas ts transparent and the emlsslvlty of the solid surface IS 1 Simple exothermlc surface catalytic reaction IS consrdered, A + B -Steady state condltlons are attamed -One srngle passage IS analyzed The mass balances m the gas phase and at the sohd surface may be wntten m the form
The
dC v-&-a&&-c)=o UC
Uf
- C,) - k,C,” exp ( - E/K&)
The enthalpy balances are respectively
=0
(2)
m the gas phase and on the wall
vpC,.$$ra(T.
- T)=O
(3)
S&,s-tm(T,-T)+(-AH)kG”exp(-EIRT.)
‘L. ---I f34& __J 1=/_
Here the term H(x) radlatlon
Fig 1 Schema of a passage
839
represents
+H(x)=O
(4)
the heat transmltted
by
J
840
SINKULE and
V
HLAVACEK
Referring to Rg 2 and using the Lamberts law the heat emltted from the element dA, which IS absorbed by the element dA IS
(11) vPC, [ TI - T(L)1 =
(5) Smce the temperature T, depends coordmate x, an mtegratlon around may be performed151 d*Q = uEK(x Here the geometric
-
only on the axial the elementar rmgs
x,)Ts”(x,) dx, dx
configuratlon
+
dy @-J&o dg @-
(7)
Here dQE IS the energy emltted by the elementar nng, d& represents the energy transmttted from the whole tube to the elementar rmg and finally, d& IS the flux from the two end cawtles For these contnbutlons we may write
To and
K(x-x,)dxr
1
cavittes, respectively The boundary condmons
x=O
C=C,,
vpC, (To - Tm) = cm
of the fore
(16)
~~-h(d-8)+DaAT..(1-ru)“exp *
and
(8) .
aft
are
T=T,,
J,.,(t9-@=O
The dlmenstonless heat flux W(e) ISthe dtierence the heat emttted and that whtch IS absorbed
dx
dx
T, are the temperatures
- y) = 0
+ E.H({) = 0
1
L
makrng
Jo(u-y)-~(l-_O)nexp[~]=O
dQ~=u~[T~4(_D_K(x-x,)dx,
I
dlmenslonless
(12)
IS
H(x) dx = - dQE + dQr + dQc
+ T,4
Ts4(x) - T,4] dx, dx
These equattons may be rendered use of the followmg substltutlons
which IS dependent on the difference of axtal coordmates of both rings x - x, The radtatton term H(x) dx conststs of three contnbutions
_
[
(6)
factor K(x -xl)
dQE = vu&Ts4(x) dx L K(x - x,)Ts4(xa) d x, d&= WE [I 0
K(x -x1)
UE
s=O
0
+
-W’(v) drl + FtWo4 (18)
F(S - &e,’ - S4(&)
Here F IS a view factor for an element ~Ddx at a distance down the passage and K IS a view factor between two elements ?rD dx at a dtstance r) - 6 from one another,
(9)
K(x - XI) ETl”(x) - To’) dx, dx
H(S) = j-- K(r, -
between
K(x)= The relatlonshtp
between
1-w F(x) and K(x) IS gven
(19) by
(10)
(20)
I
=I
Rg 2
x Radlatron flux m a passage
The first term on the r&t hand side of eqn (18) represents a contrtbutton from every part of the cylmdecal surface whtle F(&)&’ and F(S - &%4 are conmbuttons from the two end cavltles Fmally, the last term of the r&&t-hand side of eqn (18) IS the total energy emItted from the elementar nng ?rD dx The boundary condttlons may be formulated assummg “black holes”. I e the ends of the monohth are exposed
841
Heat and mass transfer m monohtluc honeycomb catalysts-111 to cavltles which are supposed to be black body radiator The radlatlon energy emltted from the tube IS absorbed here by the flowmg gas
.f=o
e=eO
y=o
80-l-E,
$=o
- &‘I dl, = 0
’ F(aW’(v) I0
8, - g(S) - E, 10’ F(S - q)[#(v)-
(21)
,
&‘I ds = 0
Integratton
Q + E,H(@ = 0 de
(24)
(25)
(26) The last term m eqn (26) can be wrltten m the form
Ths
I’0
F@ - WI’ d& -
relatron can be rearranged
+
(27)
to
I
“K(cp)dp 0
-
loaa’(e) de
=2
I-0
1 + AT&y(S) - 8, = 0 Instead of adlabatx condltlons adlabatx condltlon may be used 80= 8, = 1
of (25) yields
+
and with respect
(22) and
to (21)_(24) a
i3W (24) a non-
(22)
It IS assumed that the ends of the ceramic matenal are msulated After msertmg eqns (14)-(16) mto eqn (17) we have I d26 d8 pe,B-@+ATad
On usmg thrs equation simple equatron results
K(P) ds
(31)
Here we have assumed that the energy emltted by radlatlon IS absorbed by the end cavltes and LS not transnutted to the gas phase G EQUATIONS 3. NuMglllCALSOUlTIONOFGOVERNIN The set of governing equations, eqns (14)-(24), IS represented by a not&near boundary value problem for ordmary dlfferentml equations and an uttegro-dlfferen’tlal equation of Fredholm type subject to rntegral boundary con&tons For numertcal solution of these equations firutedtierence approxlmatlons are used, I e the derlvatlves and the mtegrals are approximated by difference and quadrature formulas, respectively The resulting set of nonlinear algebraic equations IS solved by the NewtonRaphson techmque For approxlmatton of dertvatlves the three-pomt and five-point formulae have been used Trapezoukd rule proved to be better than the Simpson quadrature formula, apparently because of dlscontuuuty of the first denvatlve of the kernel However, the trapezoidal Integration formula can be successfully used only d the number of md pomts 1s high smce the profile of K(x), m compartson with the dependence 6*(x), IS very steep m the vlcmrty of the pomt x = 0 To reach a high accuracy of the calculation a more comphcated procedure has to be adopted The gist of this procedure 1s to use three auxthary grid pomts laymg between the points q and xi-i The values of the vartable 6 at these pomts have been evaluated by mterpolatlon from the points a,-,, a,, 6,+, and &+z Makrng use of auxlhary pomts the value of the Integral In the interval (v,, q,+,) have been calculated by the five-pomt Newton-Cotes quadrature formula
K(P) dv
K(v) ds -
=2F(O)-F(S-q)-F(q) (28)
=l-F(6-~)-F(?p) here we have used the relations K(V)
= K(V)
and F(O) = l/2 After msertmg eqn (28) Into the first term In (27) we have j-’ H(t) d Z = j-08 F(rl)[&‘-
0
+
S4(d1 dl,
Fts - a)b%4- S*(v)1 drl
Here Q = vi + h/4 J. wj are weights in the quadrature formula and L(q) IS the mterpolation polynomial for 6* The Integrals occurrmg m the boundary condltlons have been approxunated m the same way The accuracy of the calculation has Improved considerably If instead of eqn (24), eqn (30) has been used For problems with nonadlabatlc boundary condltrons (31) the calculauon procedure IS the same The resultmg set of no&near algebrsllc equations has been solved by makmg use of the Newton-Raphson method For a unrque solution of governing equations convergence was always attamed On the other hand, for multiple solutions a very good uutlal profile must be selected m order to attam convergence
842
J SINKULEand V
The resulting Jacobian matrix does not exhibit band structure, however, after an appropriate ordering of equations only the nonzero elements can be stored The calculations have been performed for 50 gnd pomts One Iteration on the computer ICL 4-72 amounts to 35 set The calculation with radiation model 1s slower by a factor 50 m comparison with the non-radiative case Six iterations were necessary to reach the tolerance of calculatlon lo-’ Detads of calculation wrll be reported elsewhere [4]
HLAVA~EK
8 .+
125
4 RESULTS OF CALCULATION
The parameter Es8 1s a measure of the importance of radiation This parameter represents a ratio between radiative heat flux from the Internal surface of the passage and convective flux for Inlet conditions ‘O-
,
IO
20
E
The magnitude of the parameter Es for important exothermic reactions IS supposed to be In the range 0 OOOI0 IS The magnitudes of the variables entering into E, are reported m Table 1 To demonstrate the differences between radiation and non-radiation models the value Es = 0 1 has been chosen for calculations In Fig 3 the temperature profiles m the gas and solid phases are drawn Three cases are considered (A) radiatmn neglected, (3) radiation with adiabatic condltlons, (C) radiation wrth non-adiabatic conditions Evidently, the operation of the monolith IS m the “dtffusion regime” For the adiabatic condltlons the inlet gas IS preheated because of strong radiation effect The heat loss by radiation m the mlet section cannot be compensated by higher gas temperature, heat conduction m the solid phase and radiation from the reactor section having high temperature As a result the radiation model gives nse to lower temperature of the solid phase m the inlet section m comparison with the non-radiation model The same effect can be observed at the monolith outlet, however, here the temperature difference between the monohth and gas IS very small and hence the increase of B, IS not important Since the view factor K(x) decreases rapidly for higher values of x, the gas preheating effect IS important only for reaction zones occurrmg m the Inlet section For nonadiabatic conditions the shape of temperature profiles IS strongly affected by the radiation cooling at both ends of the passage Evidently a hot spot on the dependence 6 = 9(e) exists The temperature profile S(e) may be roughly divided mto three parts In the first part of the monolith the amount of the heat evolved by chemical reactlon and the heat transmitted IS m eqmhbrrum wurth the heat lost and the temperature profile 1s roughly constant FInally, m the last section the heat
Table
D(mm) I-20
Kn[“Cl 100-300
1 Magmtudes
Rg 3 Temperature profiles m the monohth y=20. Da=OOfM, JD=&=02S. S=20) Ol,C, E.=Ol,Bo=81=1.B.---,6,-
(AT.,, = 0 5, Pe, = 5, A, E,=O. B, Es=
I
0
IO
6 Wg 4 Conversion profiles m a monolith y=20, Da=OOO4. JD=JH=OZS, 6=20) 01,c,Es=oI,e~=8,=l
(AT,,d = 0 5, Pe, = 5, A, E,=O, B. E,=
generated 1s very low and the radlatton coohng IS htgher than the heat supply The conversion profiles are drawn in FQ 4 Wtth respect to the values of the parameters Do and 6 the reaction IS carried out in the “diffusion regtme” As a result of a shift of the reacuon zone the radlatron models predict lower values of conversion This concluston IS m agreement with results calculated from the two-dimensional model [21 In Fig 5 the dependence Da = f(y) IS shown For the
of variables
In E,
Re
(pC,MJlm3Kl
u~CW/m2K’l
200-2500
500-1000
5 67 x IO-’
LID l&50
Heat and mass transfer m monohthw
”
843
cataIysts--III
I
“3
Y,
honeycomb
8
Ftg 5 Dependence of Damkijhler number Do on exit conversLon y(8)(ATod=05. y=20. JD=J”=l, Pe,=5, Da=OO103, 8= 3 4)
govermng parameters selected multiple steady states occur Even for a low value of 6, (1 e the effect of boundary conditions cannot be neglected) both dependences are very smular The lgmtton bound calculated from both models IS almost the same This fact can be easily explained smce m the lower steady state the temperature IS low and the radiation effect IS not important The region of extinction 1s also not affected by rtiation The inlet gas ISpreheated because of rtiafion effect, however, the heat loss from the hot catalytic surface caused by gas-to-solid heat transfer 1s lower since the gas has a higher temperature In Fig 6 axial temperature and conversion profiles are drawn for the case of multiple steady states It IS obvious that for the upper steady state the ddferences between the surface and gas temperatures are very lugh For the parameters presented in l+g 7 five steady
I 05 Y,
8
Fig 7 Dependence of Damkiihler number Da on exit conversion Y(g) -3 &=Ol.--, E,=Ol. &,=B,=l,------, E,=O Parameters see Fii 7(a)
0003
B
c
Y 0t j-
/ (3
0002
I
1’
1’ _./-rc
I-
-- --
I-
/
_//* 2
.-’
-
/
25
m r’
l
*_*_*-I-. I
E
r
-2
Fii 6 Temperature and conversion profiles IIIa monohth (AT&, = 05. 7=20, JD=Js=l, Pe,=5,~=0.0103, 8=34) -, y, ----.e,---,6,E,=o1 ClISVd
33 No
7-D
Y, 8 Ftg 7(a) Remon of high converSlons
from
Rg
7 AT,,
= 0 5,
/’
/I
-’
I
09
f
states are possible for a model without radiation Radlabon model prechcts also five steady states, however, only m a narrow range of Du, Du ~(2 31 x lo-‘, 2 38x lo-‘) So far, we did not perform stabWy analysa, however, we assume that the 1st. 3rd and 5th profiles are stable These profiles are drawn m Fe 8 In the lower steady state the exothernuc reaction occurs m the “kmetlc reame” whlk for the upper steady state external heat
J S~NWWE and V. HLAVCSEK
Y
Fig 8 Axial temperature profiles m a monolith MultIpIe solutrons (AT&=O5, y=20, n=l, S=20. &=&=O& &0003,Pe.=5) -, E,=Ol,----,E,=Ol, f&=&=1.----. E, =0
and mass transfer Is rate-controlhng For the lower as well as upper steady state profiles the axlaI temperature profile wlthm the sohd phase IS almost constant whde for the mlddle steady state (the 3rd profile) a steep temperature gradlent m the sohd phase occurs The dependence of Do = f(y) for higher values of Pe. IS drawn m FQ 9 The nonradlatlve model predicts five steady states m the regon D&3 1 x lo-“, 3 47 x lo-‘) The upper temperature profiles calculated from radlatlon models are drawn m Fig 10 We may notlce that for nonadlabatlc boundary condltlons for lower value of Do
000
FIB IO Axml temperature profiles m a monolith (AT.,, = 0 5. I = 20, II = 1, JO = JH = 0 25, 7 = 20. Pe. = 50)
-,
E. = 0 1,
---.E,=O~.B~=~,=~.A,D~=OOO~,B,~~I=OOO~,C.L~= 0002lS
the hot spot temperature is higher This sltuatlon can be explamed by the fact that for lower values of Do the hot spot zone IS stufted downstream and hence the heat loss by rachatlon coohng IS lower 5 coNcLustoN ANDU!X~ Based on results of numerIcal calculation followmg conclusions may be drawn -The exit conversion IS not affected by radlatlon -The shape of temperature profiles m the passage IS dependent on the type of boundary condttlons The differences between radiation and non-radlatlon models are important -Radlatlon effect for adlabatlc boundary conditions does not affect essentmlly the lgnltlon and extmctlon bounds -Radlatlon effect dlmuushes the remon of five steady Ac&nowJe&emeti-The helpful comments of Prof H Hofmann Erlangen are gratefully acknowledged NUl.ATtON concentration of the component A m gas and on solid surface C, heat capacity D duuneter of the passage Do Damkiihler number see eqn (13) E actrvatlon energy ES dlmenslonless number, see eqn (13) ( - AH) heat of reactlon Qmenstonless gas-to-sohd heat and mass transfer coefficients pre-exponential factor mass transfer coeffictent axml dutance length of the passage order of the reactlon heat flux P&let number for solid, see eqn (13)
C, C,
83 000
Y,
-6.
I
FIB 9 Dependence of Damkshler number Do on exrt conversIon -, E, = 0, --, E.=Ol, &,=B,=l AT,=OS. JD=
Y(8)
JH=025.~=20.Pes=50,n=1.6=20
Heat and mass transfer m monollttw honeycomb catalysts-III
R gas constant S sobd cross-sectmn T, Ts temperature m gas and on sold To. TI temperature of the cavities dimensronless adrabatx temperature AT,,
WE StephawBoltzmann
constant E dunenslonless axial dwtance w conversion on sohd surface
rise,
see
eqn (13) volumetric flow rate in a stngIe passage conversion m gas, see eqn (13)
tt Y
Greek svmbnts gas-to-solid heat transfer coef%xent a actwation energy, see eqn (13) Y dtmenslonless s dlmenstonless length of the Passage, see eqn (13) 8 dtmenslonless sohd temperature, see eqn (131 # dimensionless gas temperature, see eqn (131 hr heat conductivity of the sohd phase P gas density ir wetted penmeter I
--
---
845
Subs~npts m related to rnput condltlons
BEFERMCB Hlav&Eek V et 01. Heidelberg Prpc 4th ISCRE D VI-240 (Dcchema) 1976 121 Lees T andAnsR.Ckefzz &gegSe; 1977x827 [3] Chen J L P and Church11S W, Comb F/ 1972 18 27 [4] Smkuk 1 and Htav&%k V Insr J Heur MOSSrr~n~, m preparauon [Sl Hottet H C , Radtant Heat Tmnsmtsston McGraw HIM.New York 1950 161 Young L C and Rnlayson B A , Pmc 2nd Int Symp 1;7nrte Element Methods Flow Problems Santa Margbenta Llgure. Italy, June 1976 [l]
.