THERMAL REGIMES IN CONTINUOUS A Institute of Chemical
Physics
G
OF EXOTHERMIC PROCESSES STIRRED TANK REACTORS
MERZHANOV
and V
(Branch), U S S R Academy USSR
G
ABRAMOV
of Sciences,
Chernogolovka,
Moscow
Regron 142432,
(Received 20 March 1976,accepled m revised form 13 August 1976) reames of a CSTR are analysed from the standpomt of the thermal explosion theory Crrtlcal phenomena and time charactenstlcs of the process are considered It 1s shown that for appropnately chosen dlmenslonless parameters the cnWal comhhons of the thermal explosion and mduction penod m a CSTR shghtly differ from those for nonflow (batch) systems These corrections may be performed by usmg an ad&honal parameter w&h IS characteristic for tlow systems In the remon of multtphclty of steady states, the cnhcal con&tion of thermal explosion 1s characterrzed by a jump of selfheatmg, whde m the nonflow systems such phenomenon occurs only for a zero order reaction For a umque steady state the transItIon through the cnhcal AbstractThermal
con&tion 1scontinuous
1 INTRODUCTION
A stured tank reactor model, widely used m chenucal engmeenng conslderatlons, gves nse to a very simple mathematical descnphon If mucmg 1s supposed to be perfect An analysis of this model yields a number of Important results, which can be utilzed for more complicated sltuahons The first mveshgahon of thermal regunes m perfectly nuxed tank reactors of batch and contmuous types were reported by Semenov[l], MacMulhn and Weber[21, Zeldovlch[31, and Denblgh141 some decades ago However, an mtenslve development of the theory was catalyzed by the papers by Van Heerden [S] and Amundson [6,7] The aim of this paper IS to bndge the gap between the results obtamed m the region of reactor and thermal explosion theory The paper makes an attempt to apply the methods used m the explosion theory toward detied analysis of thermal regnnes m a CSTR 2 GOVERNING
EQUATIONS
DImensionless cntena The heat and mass balances descnbmg transient behavior of a CSTR are
dT
CPZ
=
$(T Qpko epJ”‘(p(x) F(T
dx _d~
=

 To)
TV)
kOemmRTq(x) $(x
 x,,)
The following analysis 1s performed m dlmenslonless variables makmg use of the sumlanty cntena whch Frankfrom the transformahons by follow Kamenetsky [8]
fl =
& * (T
T = k,, emelRT* t,
 T,),
v=.
Da = Q k0eEIRT*, **=Ly+cpu
x0x Cl
S
The temperature of the reactor operatmg m the absence of a reachon 1s chosen as a scale temperature T
= *
aSTo aS
CPUT,.,
+
+
Cpu
(3)
whuzh makes It possible to reduce the number of dunensionless parameters to four in the set of equahons (l), (2) The basic parameter 1s the Semenov number, Se, which expresses the ratio of charactenstlc tnue of heat transfer (tune of thermal relaxahon) to that of heat release (tune of reaction under the a&abahc condltlons) The Damkohler number, Da, expresses the ratlo of the residence tune m the reactor and the tune of chemical reachon The parameter y represents tbe duuenslonless adlabahc temperature rise The parameter /3 IS the Qmensionless achvahon energy With a properly chosen scale temperature, the parameter /3 plays a correction role The parameters Se, y, /3are common in the classical thermal explosion theory [S], while the parameter Da IS the spectic for flow systems In an tiabahc reactor Y = (Da/Se) = 7, and hence the number of parameters reduces to three
It 1s well known that the set of equations (1) (2) admits multiple steady states[9, lo] It 1s convenient to study
A
476
G
MERZHANOV and V
these states m a diagram which IS often referred to as the Semenov diagram (Fig 1) This diagram shows the dependences of the rate of reactlon heat release as well as the rate of heat removal under the steadystate conditions on the temperature
Fig
G
ABRAMOV
l(a) and described
by
where q1 =
eel’
[email protected]@ f( 4) ,,2=1ww~144(8+~))
A
xv + P2)
”
42 = s
The intersections of the curves q, and the striughtlmes q2 (points l5) determine the steadystate temperature 8, and the rates of heat release and heat removal at these temperatures for the gven values of the parameters Se, Da, /3 The conversion under the steadystate comhtions 1s ?~s= v& The states correspondmg to the pomts 1 and 2 on Fig l(a) are called the lowtemperature whrle those correspondmg to the points 4 and 5 are referred to as the huhtemperature regimes The results obtamed from the analysis of the steadystate conditions are drawn m the parametric plane “Se Da” (/3 = 0, firstorder reaction), see Fig 2 The curves 1 and 2 on this figure confine the region of multlphclty of steady states They are determined as the condltlons of contact between the curve q, and the strwhtline q2 on
, 1
44 3
d
6
8 (a)
05
4
010
025p
The dashed lme 4 1s given from the defimtlon of dlmensionless parameters
and It corresponds to an adiabatic reactor (a = 0. a, = (CPU/S)) The region to the left of this hne has no direct physical sense m the given model Multiphclty of the steady states 1s possible only for low values of both parameters p and y For instance, for a firstorder reaction, the steady state 1s always umque If f3 + y > 0 25 This mequahty may be easily obtamed after combmatlon of eqns (4) and (5) 4 STABILITY TOWARDS SMALL PERTURBATIONS
s
005
&Da
results of analysis of stability of steady states towards small perturbations can be represented m Fig 2 by the characterlstlc parametnc curves which are gven by the set of equations The
Fig 1 Determrnahonof steady states m a CSTR in the Semenov &gram The straight hnes represent the rates of heat removal for various values of the parameter Se, the curves represent the rates of reaction heat evolution (The parameters B and Y are fixed) (a) Mulhple steady states are powble (b) A umque steady state occurs
0
Bj2 determme the lower and upper boundaries of smgular points of the “saddle” type The dashed lme 3 on Fig 2 separates the regions of umque or multiple solutions As follows from the above expresston, this lme IS described by 1=4(/3+v)or
0 15
Da
[email protected] 2 Remans of mulhphclty and mstabdlty of steady states for the first order reaction Curves l3 fl =O, curves 4. 5 fi =0, y=oOo5
obtained analytically from eqns (l), (2) by the Lyapunov method [9111 Values of @,, determme the boundarres of the regon of unstable steady states on the temperature axis In Fig 2 the curve 5 corresponds to y = 0 005 In Ref [ll] the slmdar curves are shown m the coordinates Da, l/r In Fig 2 the region of mstablhty IS hatched by slant lines for h&temperature steady states, by vertical lines for the lowtemperature remme, and by horizontal lmes for the unique remme In the region with double hatching all steady states are unstable With decreasing values of the parameter y the remon of mstablllty expands while for increasing values It contracts and finally disappears The values of y at which the mstablhty reDon disappears are given m Fig 3
Thermal regimes of exothermlc
processes
m contmuous
0
stured tank reactors
477
010
005
Da 0
Wg 3
IO
&e Regions of mstabdlty
of steady states (,3 = 0, f(v) =
1 q)
by the curves 1 and 2 The curve 3 IS a boundary above which all the lowtemperature steady states are stable towards small perturbations The curve 1 IS described by the equation
y = (1 
4
v)’
determmmg the boundary of the reson where the solution of the above equation yields the real values of &, The curves 2 and 3 are described by the equations
and y = z”‘l d(l4V)) respectively, which represent the condlttons for equahty of &, and 8, or 0, (1 e the condltlons for comcldence of boundarles of the region of mstablllty and that of saddles on the temperature axis) The vertical lme 4 m Fig 3 separates the regons of multlple and umque steady states (expression 4), the dashed lme 5 expresses the condltlon Se = Da/y (an adlabatlc reactor) Unhke Fig 2, the hatched regions m Fig 3 define only necessary condltlons of mstablllty The s&Went condltlons are defined by the values of the parameters Se and Da In the Ref [121, the regons of possible mstabdtty of steady states are presented m the coordmates l/r, Da/&y 5
UNSTEADY THERMAL
REGIMES OF CSTR
A steadystate analysis of eqns (1) and (2) does not answer the questlon which regme will occur m the reactor d for the particular values of the parameters multiple steady states may exist Even m the case of one stable regime. it wdl not necessardy be mamtamed, smce the final state of the reactor may be a nondamping oscillatory process The question of the final state and the ways of Its approachmg may be solved only by numerrcal solution of eqns (l), (2) The results of such analysis are presented m the parametic plane “SeDa” m Fig 4 The CES Vol 32. No 5B
FIN 4 Regions of various transient thermal regimes of CSTR, curves l3 fi = 0 02, curves 4, 5 /3 = 0 02, y = 0 01
curves l4 m Fig 4 are slmllar to those m Fig 2 The curve 5 IS the crltlcal condltlon of the thermal explosion (see Sectlon 6) Several characteristics regions can be specdied m this plane If the values of the parameters he m regions IaIc the process proceeds with small selfheating and a lowtemperature steady state 1s mamtamed (pomts 1, 2 m Fig la, point 1 m Fig lb) In Region IIaIId the process occurs with large selfheatmg effects Of special interest IS the region IIc where the final state of the reactor 1s the lowtemperature one (pomt 1 Fig la), however m the course of its approachmg high temperatures are attamed In the region IIa the final state of the reactor IS the hightemperature (point 5 m Fig la), m the region IIb both a h&temperature and a lowtemperature states are possible (points 2, 4 rn Fig la) The region IId IS located m such part of the parametnc plane where steady states are unique, approaching a partrcular steady state IS here possible (pomts 1, 2, 3 m Fig lb) In the cases consldered the final states of the reactor may also be a nondamping oscdlatory process Some temperaturetime curves m the reactor are given m Fig 5 The curve 1 m Fig 5(a) corresponds to the case where after a small selfheating process a lowtemperature steady state IS established (region Ia) The curve 2 represents the sltuatlon where a strong selfheating process precedes the approaching of a lowtemperature repme (region IIb) In the thud case (Fig 5b) a perlodlc regune with a large amplitude results Osclllatlons occur near a hightemperature steady state (region IIa) In the phase plane “67)” smgular pomts correspond to steady states of the reactor, m the first two cases the points of a type “stable focus” result, while for the third case “unstable focus” exists The curves m Fig 5(c) represent the h&temperature regime (region IIa) The smgular point on the phase plane 1s a “stable node” 6 CRITICAL CONDITION OF TFIERMAL EXPLOSION
The curve which separates the regions Isc and 1Iad (Fig 4) 1s very unportant In
[email protected] 1ac the process with low selfheatmg effects occurs, here the rate of heat hberation IS close to that of heat removal tnto surroundmgs In the upper reeons the rate of heat hberatlon
A
478
G
MERZHANOV and V
G
ABRAMOV
prevads the rate of heat removal which leads to a strong selfacceleration phenomena and a high selfheatmg results Frequently ths process 1s referred to as a thermal explosion regme[ 1,8] For high values of heat of reactlon and actlvatlon energy, I e for small values of the parameters fi and 7, It 1s Important to know the boundary between the explosion and nonexplosion reglans which 1s called the crItIcal con&tion of thermal explosion At high values of the parameter Da (1 e the convection term m eqn (2) can be neglected) the reactor regime may be approximated by a batch operation In this case the cntlcal condition for thermal explosion IS defined by
se= i
20
i IO
(l+ B)F(y)
:
30
T/r Fig 5(c) Fig 5 Possible nonsteady thermal remmes of CSTR (a) B = 0 02, (1)Se=048,(2)&=049,(b),9=002,y=005, ~=005,v=O2
v=02,Se=050,(c)f3=002,v=005,&=05 y=oo5
2
20
T'Y
Fig 5(a)
(l)y=OO4,(2)
This relationship gives a satisfactory accuracy for /3 < 0 05, y < 001[8, 13151 The function F(r) which accounts for concentration changes m the preexplosion period 1s called a correction for the mass consumption For the firstorder reaction, F(r) = 1 + 2 4 ~*‘~[8,14] At small values of p and 7, F(y) = 1 and (1t @) = I For small values of the parameter Da the concentration of reactants m the reactor 1s practically at its nutlal value, 1 e the mode1 of the zeroorder reactlon may be adopted [l] Here agam F(y) = 1 At intermediate values of the parameter Da, the critical condition of the thermal explosion 1s to be determined by a numerical integration of eqns (1) and (2) The curve 5 m Fig 4 represents the critical condltlon for p = 0 02 and y = 0 01 This curve may be described approximately [ 161 by
IO
Se=;(l+p)(I+24y2’3~)
8 5
/ (
20 r/r Fux
5(b)
L
v+oo5
The relation descrlbmg cntlcal condltlons of thermal explosion m the form (7) 1s shghtly dependent on v This IS because of the fact that the parameter Se successfully mcorporates the parameters which are characterlstlc for the flow systems For small values of y, due to small values of F(r). the crltlcal condltlon of thermal explosion does not depend practically on the mltlal temperature If the latter IS lower than 13,and therefore the analysis may be carried out for constant values of @,. In this paper it was assumed 0,. = V,” = 0 Near the critical condltlon of the thermal explosion, the reactor exhibits a high sensltlvlty to perturbations The character of transltlon through the cntlcal condition depends on the region where it occurs Transition
Thermal
of exothernuc processes m contmuous stied
regimes
from the region Ia m the reeon IIa or IIb proceeds m such a way that the dependence of the maximum temperature, reached m the nonstationary regime, on the parameter Se has a dlscontmulty at Se = Se,,, This 1s displayed m Fig 6 where several dependences &,(Se) are drawn for various values of y and Y A Jump from the subcntlcal to the supercntlcai reames IS drawn by dash lines In the phase plane “0  q” a dlscontmuous change of the maximum temperature IS caused by the fact that the curves e(v) cannot exhlblt a maximum m the region between the mrddle and upper singular pomts (Fig 7a) The cntical condition of the thermal explosion comcldes here with the condltlon of passage of a saddle separatrlx through the point &, TJ,= The transltlon from regions Ib and Ic to reaons IIc and IId occurs m such a way that the dependence &(Se) has no Jump As m this case there 1s only one singular point m the phase plane, the turning “downwards” of the curve e(q) IS posslbIe at any temperature which exceeds the temperature of the steady state (Fig 7b) However, at small values of the parameters #I and 7, a change of the value of the maximum selfheating occurs m such narrow range of Se, that It may be constdered as a Jump (see curve 2 m Fig 6) Evidently, the cnt~cal condition of the thermal explosion IS clearly defined m all cases discussed above With increasing values of the parameters /ii and y the phenomenon of thermal explosion degenerates In the IC
tank reactors
479
remon of multiphclty of steady states a clearlydefined boundary exists between the hightemperature and the lowtemperature regimes, but the temperatures attained at these repmes are close and d&erence between them becomes mslgmficant In the region of unique steady states, transition from one regune to another becomes so smooth that It 1s almost lmposslble to determine the crltlcal conditions [ 173 As noted above, a convenient form of the parameter Se slmphfies slgndicantly the mathematical analysis of the model After simple rearrangements we may easily find a relationship between Se and the parameter Se, used m the thermal explosion theory of batch systems se, =
Se ]
se
YDU
As can be inferred from this expresslon for high values of the residence time, the dtierence between Se and Se* 1s not slgmiicant When approachmg the adlabatlc con&tlons, (Da/Se)+ y and Se, + 00 In tlus case the crmzl condition of the thermal explosion m the CSTR IS Da=ySe,,=z(l+p)F(v)
(8)
and deternunes the critical residence ttme m the reactor If the residence time 1s less, the thermal explosion does not occur 7 INDUCTION PERIOD
em5
04
se
06
Fig
6 Dependence of maximum selfheating on the number Se p = 0 02 The results are obtamed by a numerical solution of eqns (l)and(2) (1)y=OO1,v=OO1,(2)y=OOl,~=O3,(3)y=O1, v=O1,(4) 7=01,v=02,(5)y=01, v=o3
(a) (ti Possible nonsteady thermal regnnes of CSTR m the phase plane “8  TJ”(sohd hnes) The dashed hnes represent (1) Isoclme of honzontals (de/d? = 0) (2) lsoclme of vertwals (dv/
[email protected] = 0) (a) Mulhple steady states (b) a umque steady state Fig 7
A very important charactenstlc of the supercntical regimes IS the mductlon penod which should be conveniently defined as the time of thermal acceleration of the process (tune of attammg the maximum value of rate of heat hberation)[l4] In the subcritIca regunes the tune of attammg the maximum temperature is important Several curves which represent the dependence of the induction penod and time of attammg the maximum temperature on the number Se are shown m Fig 8 These dependences have been obtamed by numerical mtegration of the eqns (l), (2) The figure reveals an essential increase of time at Se * Se,,, which IS spectic for the thermal explosion This results from the fact that when approachmg the cmtical state, the rate of heat evolution and the rate of heat removal are close to each other for higher values of the time variable and the process proceeds m a quasisteady state reame For Se+m all curves approach one asymptotethe adiabatic mductlon period of the thermal explosion of batch systemsdefined by T. = ~(1 + 28) [ 14,181 For v > 0 3 the mductlon periods of thermal explosion occurmg m a CSTR are m a good agreement with those for batch systems [16] To calculate the induction period, analytical methods can be also used[8, N223 8 OSCILLATIONS A numerical solution of the set of eqns (1) and (2) makes it possible to investigate the regime of the undamped osclllatlons The temporal profile of the tem
A G MERWANOV and V G ABRAMOV
480
Q
heat of reactlon gas constant heat transfer surface t tune T temperature Tml inlet temperature TO ambient temperature T* scale temperature volumetric flow rate ; reactor volume x concentration initial concentration X0 Da Damkohler number Se Semenov number
R S
Fg 8 Dependence of the mductton penod of the thermal explosion and time of attammg the maxumnn selfheatmg on the numberSe#t=OM (1)~=001,~=001,(2)y=001,~=03,(3) y=Ol, v=O1,(4) y=Ol, v=o3 perature IS drawn m Fig 5(b) The process shown m tis figure LS of a relaxation character, I e the thermal exploslon leadmg to high selfheatmg effects and practically complete converslon IS replaced by the period of coohng and relatively slow fillmg the reactor with a fresh component, followed by a new thermal explosion etc A comprehensive analysis of osclllatmg regnnes has been carried out recently [ 121 9 CONCLUSION It should be noted that the “thermal explosion” concept used m the present paper and generally adopted m the thermal explosion theory does not necessarily mean that the process must be of destructive character If a small quantity of gaseous products 1s generated by the reaction, the process may look hke a shght flash, however, external effects need not occur Of course, If the quantity of the gaseous products IS high, the thermal explosion may result in a reactor fadure If the reactor approaches the supercntlcal state, it IS necessary that the time of operation m thus regme should be shorter than the induction period of the thermal explosion Though the numerical analysis was camed out m the present paper for the firstorder reaction the results may be adopted also for the reactlons of other types AcknowledgementsThe authors would hke to thank Dr Hlavacek for his Interest m the work and helpful remarks
V
NOTATION c E
k0 41992
thermal capacity activation energy preexponential factor rates of heat evolution respectively
and heat removal,
Greek s ymbo Is heat transfer coefficient m surroundmgs effective heat transfer coefficient dunenslonless parameter see eqn (2) dimensionless parameter see eqn (2) conversion conversion in the steady state regune conversion at 7 = 0 dunenslonless temperature dlmenslonless temperature at 7 = 0 dlmenslonless temperature m the steadystate repme boundary temperature of the reaon of unstable steady states &mensionless parameter density dlmenslonless hme mductlon period of the thermal explosion time of attammg the maximum temperature REFERENCES [II Semenov N N Z Phys 1928 48 571 [23 MacMullm R B and Weber M , Trans Am Inst Chem Engng 1935 31 409 01 Zeldovlch Ya B Zh Tekhn fiz 1941 11 493 (m Russlan) [41 Denbgh K G , Trans Faraday Sot 1944 40. 352 [53 Van Heerden C , Industr Engng Chem 1953 45 1242 [61 Amundson N R , De lngenrelcr 1955 37 8 171 Amundson N R and Bdous 0 1, Am Inst Chem Engng J 1955 1 513 D A , D~ffusron and Heat Exchange m iSI FrankKamenetsky Reactron Kmetrcs (Translated by Thon) Pnnceton Uxuversity Press 1955 to the Analysrs of Chemical Reactors [91 Ans R , Introductron New York I%5 of the Chemrcal Cl01 Volter B V and Sahukov I E , Stab&y Reactors Moscow Izd “Khlmla”, 1972 (m Russian) 1111 Hlavacek V , Kublcek M and Jehnek J , Chem Eng Scr 1970 25 1441 [121 Uppal A , Ray W H and Poore A B , Chem Engng Scl 1974 29 967 [131 Merzhanov A G and Dubovrtsky F I, Uspekhl Khdm 1966 35 4 (m Russian) 1141 Barzykm V V , Gontkovskaya V T , Merzhanov A G and Khudyaev S I, Zh Pnkl Mekh Tekh FU 1964 3 118 (m Russian) [I51 Parks J R, J Chem Phys 1961 34 46 t161 Abramov V G and Merzhanov A G , Theor Osnovy Khlm Tekhn, 1975 9 863 (m Russian)
Thermal regunes of exothenmc
processes
[17] Merzhanov A G , Zehkman E G and Abramov V Dokl Akad Nauk S S S R 1968 180 639 (m Russian) [18] Todes 0 M Zh AZ Khrm 1939 13 868. Todes 0 M Melent’ev P V , Zh FIZ Khrm 1939 13 1594, 1940 I4 (III Russian) 1193 Gray P and Harper M J , Tram Faraday Sac 1959 55
G, and 1026 581
III contmuous
stured tank reactors
481
[20] Kmbara T and Ak~ta K , Comb and Flame, 1960 2 173 [21] Thomas P H , Tram Faraday Sot 1960 56 833 [22] Merzhanov A G and Gngor’ev Yu M , DOW Akad Nauk SSSR 1967 176 1344, Fsnka Goremya I Vzryva 1%7 3. 371 (m Russlan)