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Heat transfer from exothermically reacting fluid in vertical unstirred vessels—I. temperature and flow fields
Chemical Engineering Science. Vol. 42, No. 9. pp. 2183-2192. Printed in Great Britain.
HEAT
1987. 0
ooO9-2509187 1987 Pergamon
$3.00 + 0.00 Journals Ltd.
TRANSFER FROM EXOTHERMICALLY REACTING FLUID IN VERTICAL UNSTIRRED VESSELS-I. TEMPERATURE AND FLOW FIELDS J. R. BOURNE+ Technisch-chemisches
Laboratorium
ETH, CH-8092
Zurich, Switzerland
and F. BROGLI, F. HOCH and W. REGENASS Ciba-Geigy AG, CH-4002 Etasel, Switzerland (Received 13 August 1986; in revised form 29 December 1986; accepted 15 February 1987) Abstract-Exothermically reacting fluid poses the danger of temperature run-away (thermal explosion) in some circumstances (e.g. interruption of stirring in a reactor and during storage). If the explosion time is longer than that required to establish free convection in a heat-generatingfluid,convectiongreatlymodifies
the internal temperature distributions relative to those developed when only conduction operates. Experiments on laminar free convection, using flow visualisation and temperature profiles, showed a
thermally stratified core moving slowly upwards and a thin thermal boundary layer on the wall of an unstirred vertical cylindrical vessel. An analogy between the temperature distributions in active (exothermitally reacting) and passive (cooling without reaction) systems at equal heat removal rates was established. It offers a basis for safer and cheaper experiments, whereby information from passive tests may be applied to active systems, provided their rates of heat generation are stationary and uniform. Criticality will occur in the upper layers of fluid when free convection operates and a method is given for predicting it.
INTRODUCTION Exothermically reacting fluids are often processed in stirred tank reactors, where turbulent mixing ensures a uniform temperature in the fluid, except for temperature gradients in the thermal boundary layers on cooling surfaces. Semenov [l] provided the criterion for thermal stability, whereby a temperature run-away (thermal explosion) could be avoided. Sometimes no motion is present in exothermically reacting matter, e.g. during the storage of a powder or a solidified product. Internal temperature gradients develop, the distribution being parabolic if the rate of heat generation and the thermal conductivity are uniform. Frank-Kamenetski [2] worked out the corresponding stability criterion for heat transfer by conduction. Exothermically reacting fluids in unstirred vessels can exhibit fluid motion. Internal temperature gradients cause density differences, which then induce free convection whose flow regime can be laminar, transitional or turbulent. Free convection does not always develop and stable thermal stratification is also possible. Particularly since the serious accident in Seveso ([3]-[S]) it is essential to assess the safety risk following an interruption of mechanical stirring as well as during the storage of exothermically reacting fluids. Typical questions include: Does free convection develop? Are internal temperature gradients formed? How large are inside film heat transfer coefficients?
Can they be obtained from generalised correlations? Is the system safe? A passive system contains no heat source (or sink) and the characteristic temperature relevant to free convection is the difference between the externally imposed temperature at the system boundary (e.g. a cooling surface) and the temperature (at some point) within the system. An active system contains a heat source (or sink) and the characteristic temperature difference is that developing-in the sense of a dependent variable-between the system and its surroundings. It needs to be related to the strength of the source, which may be regarded as an independent variable, and introduced into the dimensionless groups Gr and Ra, which apply to free convection, heat and mass transfer etc. Kee [6] defines this characteristic temperature difference, when the heat source is uniform (e.g. radioactive decay), as given in eq. (1):
+To whom correspondence should be addressed. 2183
AT, = L,2Qp/&
(1)
whereas for an Arrhenius-like, temperature-dependent reaction rate [k = k, exp (- E/RT)] Merzhanov et al. [7] and Jones [S] give AT, = RT,2/E.
(2)
In eq. (2) T, is the temperature of the surroundings and RTz/E is called the pre-explosion temperature difference. The definition and use of AT, will be con-
sidered subsequently. Valuable contributions to the study of free convection in active systems, particularly in the context of thermal explosions, have been made by Merzhanov
J. R. BOURNE et al.
2184
and his school. The Frank-Kamenetski given by 6
=
E
L:Q,(T,)
RT;
1
’
Derivation from a steady-state heat balance shows that 6 is proportional to the rate of internal heat generation divided by the rate of heat removal by conduction. For a given shape of the reaction zone, critical values of S have been worked out [9] above which the internal temperature distribution becomes unstable and thus time-dependent. Working with Arrhenius-like decomposition reactions (e.g. 3C&80 W/kg at 60°C) in various solvents, Merzhanov et al. [7] observed 6 values greatly in excess of those based on eq. (3) and attributed this to the higher rates of heat transfer caused by free convection. For small, vertical tubes this increase was correlated by Ra and three regimes were identified (no convection, heat conduction; free convection, thermal convection; free convection, thermal explosion). The concept of an induction time to develop free convection, rc was introduced. If the time to explosion, r cx,was shorter than r c, a further regime (rapid explosion in quiescent fluid) resulted. Ra was defined by
Ra=&Pr=@L3RT’ va
scales etc. An alternative approach must be found.
parameter 6 is
‘-5
and covered approximately the range lo’-10’. When Ra = IO’, the measured critical value of 6 was roughly 15 times that calculated for heat removal by conduction alone. The additional heat removal by free convection thus stabilized the system against runaway. Limitations of this work include:
(4 Flow regime. Experiments in passive systems,
for which RTz/E in eq. (4) is replaced by the imposed temperature difference, indicate the transition from laminar to turbulent free convection when Ra is near 109-109.‘. The experiments of Merzhanov ez al. [7] probably referred to laminar flow, whereas the values of Ra in industrial-scale reactors often imply turbulent free convection. (b) Shape/boundary conditions. Free convection is sensitive to shape (e.g. change from vertical to horizontal cylinder or to sphere) and boundary conditions (e.g. whether walls are at same or different temperatures, whether base of container is adiabatic or isothermal [lo]). (c) Experimental method. Merzhanov determined criticality in the presence of free convection by permitting thermal explosion. This was acceptable on an approximately 200-ml scale, but would become too expensive and dangerous at the scale of industrial reactors. Their criticality must be assessed by sound physical principles and calculations based upon free convection heat transfer characteristics, obtained at various
The present paper addresses itself to limitations (a) and (c).
PRINCIPLE
OF EXPERIMENTAL
METHOD
following analogy between active and passive systems is postulated and will subsequently be justified. For a given heat flux out of a system, the local temperature gradient with respect to time and position in The
(a) a passive system, which is being cooled down; and (b) an active system in the steady state, which has the same physical properties as the passive system and whose internal heat generation rate is uniform, will be equal. Provided this is valid, free convection heat transfer in active systems can be studied by means of the corresponding passive system. The following development proves the validity of the analogy for the limiting cases mentioned in the Introduction, namely forced convection and conduction.
Temperature
distribution
during forced
convection
A well-stirred liquid of mass m and temperature TR(t) is within a cylinder of radius R and length L, having an adiabatic lid and isothermal walls of area A, overall heat transfer coefficient U and uniform jacket temperature T,(t). Passive system:
dT, 0)
=
"CP y&--
-
UA[T,(t)
t=O
TR = TJ = T,
t>o
T,(t)
= T, -
-
T,(r)]
(5)
i-t.
For such linear reduction in the cooling jacket temperature, eq. (5) gives AT(t) = TR(t) - T,(t)
= h(1
- e-“‘),
(6)
where r is the time constant mc,/UA. Active system: The heat source Q, is taken to be steady: this corresponds to the chemical reaction rate of zero order of thermal explosion theory [9], i.e. to the “worst case” of negligible consumption of reagents. mc,?
= Q,&‘--
UA[T,(t)-
t=O
T,
= T,
t>o
T, = steady.
TJ]
(7)
Heat transferfrom exothermicallyreactingfluid in verticalunstirredvessels-I Solution of eq. (7) gives QPV T, = =(l
AT(t) = T,&)-
-e-‘I’).
(8)
Applying eqs (6) and (8) to the same reactors and keeping the same physical properties of their contents, then T(t) will be the same in the unsteady and also in the pseudo-steady [t -P 00, AT(t) ---, constant] states provided pc$
= Q p.
(9)
Temperature distribution during free conuection In so far as free convection heat transfer is intermediate in magnitude between the cases of forced convection and conduction, it might be anticipated that eq. (9) would be fulfilled for identical temperature distributions when free convection operates. It has, however, not been possible to prove this. It will be tested experimentally here.
Equation (9) proves the analogy for forced-convection heat transfer.
Temperature distribution during conduction The situation is basically the same as for forced convection. The temperature depends upon time and the radial position CT,&-, t)]. The external temperature TAt) will be taken to be that on the outer surface of the fluid, i.e. wall and jacket-side heat transfer resistances will for simplicity be considered to be negligible (Bi + a3). Passive system: L’AT at=*
(10)
AT = T,(r, t) - T,(R, t) t=O
AT(r, 0) = 0
t>OAT(r,t)=OandT,=T,(R,t)=T,-pt. The temperature distribution in the fluid when the boundary temperature is decreasingly linearly is given by eq. (11) [ll]:
ATO-, t) =
p(R” - r2)
2F
*
(11)
4a
(J, and J1 are zero- and first-order Bessel functions and a” the corresponding positive, real roots). Active system: aAT PC,Y&- = AT(r, 0) = 0
t=a t>o
AT(R, t) = 0, i.e. T, = steady aAT(O, t)
ar
=
o
2185
vided that eq. (9) is satisfied. (NB u = L/PC,.) Thus the analogy also holds for conductive heat transfer.
.
The heat source is taken to be steady and uniform, i.e. QP = constant. The solution of eq. (12) is then eq. (3):
ATtr, tj = QpW2 - r2) 41 For same reactors and constant physical properties, comparison of eqs (11) and (13) reveals the same temporal and spatial temperature distributions pro-
EXPERIMENTAL
A 2-dm3 glass heat flow calorimeter (Cl23 and [13]) was used consisting of a vertical cylinder (D = 0.11 m) with a dished base, all enclosed in a cooling jacket. Jacket temperatures were constant (50°C) for active runs and were reduced at linear ramp rates up to 40”C/h for passive runs. The wall of the calorimeter had a thickness of 0.005 m and the thermal resistance of this wall and the coolant (silicone oil) was measured [ 143. Chromel-alumel thermoelements (Thermocoax, Philips), having a diameter of 0.5 mm (time constant 25 ms), were used in the bulk of the liquid, whilst to explore temperature distributions in the thermal boundary layer the diameter was reduced to 0.25 mm (7 ms). These elements were positioned vertically (to within f 0.5 mm) and radially (to within f 1 mm in the bulk and f 0.1 mm in the boundary layer). The temperature difference AT between the local fluid temperature CT,&, z, t)] and the jacket temperature [Tdt)] was recorded to within f 0.02”C as a function of time. Before each run temperature was made uniform by agitating the liquid with a stream of nitrogen bubbles and brought to within 0.2”C of the jacket temperature. For the passive runs water, ethylene glycol, glycerol and toluene were employed. Approximate ranges of relevant physical properties were: conductivity (A) 0.1 (toluene)-0.6 (water) W/m K; viscosity (q) 0.4 (toluene)- lo3 (glycerol) kg/m s; heat capacity (c,) 1800 (toluene)-4200 (water) J/kgK; density (p) 850 (toluene) - 1250 (glycerol) kg/ma; thermal expansion coefficient (/3) 0.4 x 10m3 (water and glycerol)- lo-’ (toluene) K-‘; and Pr 5 (water and toluene)-104 (glycerol). The temperature dependence of all these properties was considered. The active system was the isomerisation of trimethylphosphite to dimethylmethanephosphonate, catalysed by methyliodide at or near 50°C: 0 (CH,O),-P (TMP)
-
CH,l
CH3 0)2-!-CH3. (DMMP)
This reaction runs to complete conversion without forming by-products. It can be repeatedly started and stopped by addition of methyliodide and triethylamine respectively. The physical properties of an equimolar mixture at various temperatures are available [14].
J. R.
2186
BOURNE
reaction enthalpy is significant (- 180.9 kJ/mol [IS]) and the adiabatic temperature rise is 570°C. The reaction kinetics have been studied [lS] and in the absence of solvent exhibit a region of conversion near 50% where the rate of heat generation is steady, despite the consumption of TMP. At 50°C this period persisted for some 2 h, which proved convenient for studying free convection in an exothermically reacting fluid. By varying the catalyst concentration in the range 0.07 1-O. 139 mol CH,I/kg reaction mixture, various powers could be developed, e.g. at the highest CHpI level 17.8 W/kg reaction mixture at 50°C for over 2 h The
c141. RESULTS Passive
system:
The
reduced at a constant
jacket
rate *and
temperature
T,
was
the fluid temperature
er al.
at any point [T,Jr,
z, t)] started to fall at the same rate
some 30-90 min (glycerol needed 90 min, the other fluids responded faster). This quasi-steady state, defined by dAT/dt = 0, reflected the time lag needed to develop a steady Aow field in the liquid (and not a measuring lag). distributions measured in an Temperature equimolar TMP-DMMP mixture without catalyst (Fig. l), ethylene glycol (Fig. 2) and glycerol (Fig. 3) show radially a thermal boundary layer, whilst axially the temperature indicates strata with the highest temperature just below the liquid surface. Film thicknesses fall roughly in the range 1 mm (TMP-DMMP)-12 mm (glycerol): AT, is defined as the fluid temperature at any point T(r, z) minus the inside wall temperature T(R, z) at the same time. The radial distribution is not parabolic, which is characteristic of conduction [eq. (1 l)], even for the most viscous after
AT PC1 6-
cu, PC1 2
lZ-
f-
0
-F ;
--Y? 31
;
3
x hmll
Fig. 1. Vertical and horizontal tempe.ra_turedistributions during passive, quasi-steadycooling. Medium: TMP-DMMP (equimolar). T = 19”C/h. H/D = 1.08. TJ = 51°C. (0) AT, (0) AT,
10 ATPCI_
72
z hml
x Imml
Fig. 2. Vertical and horizontal temperafure distributions during passive, quasi-steady cooling. Medium: ethylene glycol. T = 20”C/h. H/D = 1.0. (0) AT, (0) AT,
L
2187
Heat transferfrom exothermicallyreacting fluid in verticalunstirredvessels-1
0
z Imml
150 0
2
x Imml
L
0
0.2
0.1
0.6
0.8 1.0 r/R
Fig. 3. Vertical and horizontal temperature distributions during passive, quasi-steady cooling. Medium: glycerol. T = 3O”C/h. H/D = 1.0. (0) AT, (0) AT,.
fluid (glyceroI), denoting the action of free convection. The vertical distributions were determined outside the boundary layer (i.e. in the core) and z = 0 refers to the tank base. AT = TR(z) - T, and that part of AT taking place in the outside coolant film and the glass wall is whilst the difference between the bulk fluid AT, temperature at any height and the inside wall temperaThe thermal resistance of ture is AT, = ATAT, especially the glass (5 mm thick) was evidently considerable. In Figs l-3 El marks the surface: the liquid temperature fell towards the surface as well as in the gas space (z > Zf). z(AF) is the position where AT is equal to its vertical-averaged mean value AT. The vertical AT profiles were increased and decreased by increasing and decreasing p (in the range 1040°C/h), respectively. Figure 4 is an example. Liquid heights in the range W = 0.5-1.5D were also studied [14]. Injection of dye (methylene blue) in the base of the tank just off-centre helped to visualize the flow pattern. During the quasi-steady state cooling of water at 2O”C/h, the tracer rose vertically without radial dispersion through the core region at approximately l-2 cm/min. The core region exhibited essentially plug flow upwards, the flow turning radially outwards within less than 1 cm of the surface to feed the downflow in the boundary layer. Thus the observed vertical flow in the core agrees well with the stratified temperature distributions (Figs 14). Heat exchange (cooling) occurred radially between the cool vessel wall and the fluid flowing downwards in the boundary layer. (In the experiment cited, the thickness and velocity of this layer were roughly l-2 mm and 15-20 cm/min). There was some tendency for the boundary layer to separate over the dished base, but most of the cooler fluid followed the base contour to
10
AT
, 0
sb
Id0
z Imml
150
Fig. 4. Vertical temperature distributions during passive, quasi-steady cooling at three rates. Medium: TMP-DMMP (equimolar). H/D = 1.0. T, = 50°C. T(“C/h): (0) 19, (m) 28, (0) 34.
rejoin the ascending core. Analogous results during heating have been reported [16]. The local heat Aux density increased in all cases with the height in the vessel. Figure 5 gives one example. Combined with the local inside film temperature differences [AT,(r)], local film heat transfer coefficients h&) could be calculated: Figure 6 gives examples for TMP-DMMP, glycerol and toluene. From these and other results [14] it was concluded that the film coefficient was uniform. Active system: Steady-state vertical temperature distributions T,(z) were attained 20-90 min after ad-
J. R. BOURNE
2188
et al.
(b) The temperature in the upper part of the stratified core increases faster than in the passive case (Fig. 1; see Figs 9 and lo), which is due to the Arrhenius-like effect of temperature on reaction rate.
40
0
2
rmm1
40
Fig. 5. Local heat flux densities as functions of height and cooling rate during passive. quasi-steady cooling. Medium: TMP-DMMP (eouimolar). H/D = 1.0. T W3h): (0) 19, (0) * . 28, (0) 36.
dition of catalyst to the reaction mixture. Figure 7 includes radial temperature distributions in the thermal boundary layer, vertical temperature distributions in the core and local heat flux densities, and film heat transfer coefficients at various heights. The runs used liquid depths of 0.120 and 0.114 m, respectively, and good reproducibility is indicated_ Several similarities with the results for passive systems (Figs 1-6) are evident. Thus, for example, local heat fluxes and film heat transfer coefficients were evaluated from the temperatures in the boundary layer and are compared in Fig. 8 with values from passive runs (no catalyst). The two significant differences are: (a) The temperature in the boundary layer exhibits a weak maximum (see also Figs 9 and 10). In the core the temperature at a given height was independent of radial position [14].
hR IW/td
Figure 9 compares active and passive runs, which according to eq. (9) should be identical. The temperature in the upper part of the core is some 5°C above that in the jacket (50°C) and in the lower core. Temperatures in the active case rise above those in the passive run in this region; this reflects faster reaction with increasing temperature. This becomes still clearer when the rate of heat generation (at 50°C) is raised from 9.1 to 17.8 W/kg reaction mixture (Fig. 10). The observed vertical temperature profile is consistent with 38.3 W/kg, which is double the value given by eq. (9) C48.9 compared with 22.8 W (Fig. 1 l)]. Because higher temperature differences can occur in active runs than in the comparable [according to eq. (9)] physical runs, the film heat transfer coefficients (Fig. 8) are also higher than expected: free convection offers therefore some degree of internal compensation against rising heat generation rates. The values of Ra [see eq. (4)], where Lc = H and the characteristic temperature difference is AT,, ranged Prom lo*.’ to 109.1, whilst in the passive experiments the range was 106~2-10g~‘. The flow regime was likely to have been laminar in almost all runs. APPLICATION
TO AVOIDANCE
OF THERMAL
EXPLOSION
Potentially dangerous situations arise with exothermically reacting fluids when, for example, the mechan-
KI
Glycerol
lOO-
0
200 100 -
200
0 Fig. 6. Local inside
5il
’
40
140
z Imml
film heat transfer coefficients as functions of height and cooling rate (passive runs).
+ (“C/h) H/D
= 1.0 1.5
=
10
20
30
40
Cl I
0
n A
0 +
Heat transfer from exothermically reacting fluid in vertical unstirred vessels-1
2189
AT [%I 6-
q [W/m21
hR [W/m2 Kl ZOO-
0
Go
2 lmml
0
160
55
2 Imml
nxl
;
o-
i
x [mm1
Fig. 7. Exothermic isomerisation of TMP: reproducibility of results. T, = 50°C. QP = 9.1 W/kg at 50°C. (0) 0.120 m, (0) 0.114 m. (0) and (I) = corresponding values of AT,.
H:
ical agitation of a reaction vessel fails or thermally unstable fluids have to be stored. It is required to calculate the critical masses below which thermal runaway will not occur. The information collected here on free convection and temperature distributions in active and passive systems refers in particular to cylindrical, vertical containers. The way in which it can be applied to assessing safety will now be outlined. In all experiments the fluid near the tank base had a temperature equal to or slightly in excess of the jacket coolant T,. The temperature of the rising fluid increased to a maximum To + AT,, in the top layer of the core. The criterion (9) to avoid a thermal explosion in this region (dQ,/dT -C dQ/dT when QP = Q) is
Loo hR fW/m2K1 200 -
0 1500 -
q lWlm21
AT,,
1000-
< RT,2/E,
(14)
and the corresponding induction time [9] is r,
= 0.64pc,RT;t/EQ,(T,).
It was found experimentally that AT_
(15) was
in the
range 2-3 AT, the higher ratio 3 being more characteristic of the active runs. As a first estimate AT_ can be found from
500-
AT_ 0
3
sb
z Imml
160
Fig. 8. Heat flux densities and film heat transfer coefficients for TMP in passive and active runs. QP (W/kg at 50°C): (0) 9.1, (0) 13.3, (A)17.8. T (“C/h): (I) 19, (0) 28, (A) 36.
= 3AT,
(16)
and it will be shown in Part II how AT can be related to the reactor size and the thermal properties of the reaction.
The above analysis assumes that free convection has time to develop before thermal explosion occurs. This
J. R. BOURNE
2190
et al.
6-
1.0 o.5-mb
60
0
2 tmml
0
1
; xhml
3
Fig. 9. Vertical and horizontal temperature distributions for TMP in passive and active runs. TJ = 50°C. H/D = 1.0. (0) Q p = 9.1 W/kg at 5O”C, active. (0) T = 19”C/h, passive. (0) and (I) = corresponding values of AT,.
AT
PC1
20
2-
15 -
I-
Fig. 10. Vertical and horizontal temperature distributions for TMP in passive and active runs. T, = 50°C. H/D = 1.0. (0) QP = 17.8 W/kg at 5O”C, active. (0) T = 36.2”C/h, passive. (e) and (I) = corresponding values of AT,
Heat transfer from exothermically reacting fluid in vertical unstirred vessels-I 60
2191
The way is shown in which the information gained in this study can be applied to avoiding thermal explosion in exothermically reacting fluids undergoing free convection. Generally it may be concluded that the physical situation during free convection in passive and active systems has been clarified.
Qp IWl _ LO -
NOTATION
A a CP D 50
0
E
100
z
lmml
Fig. 11. Cumulative thermal power during TMP isomerisation as function of vertical position in liquid. QP at 50°C WI
Q p effective w
11.6 17.0 22.8
16.0 28.2 48.9
0 0 a
k passive (“C/h) : C
19 28 36
9 H h, J k Lc m 4 QP
Q r R
implies [ 71: ‘5, 4 zex.
(17)
and a sufficiently high value of Ra (e.g. in excess of 103)
C71 and C93)-
T
=.I TO
CONCLUSIONS
Vertical and radial temperature profiles measured during free convection in liquids in a vertical cylindrical container having a dished base revealed: (a) a thermally stratified core with laminar upward fluid flow and a uniform radial temperature, which increased from the base towards the surface; and (b) a thin thermal boundary layer on the vessel wall with downward fluid flow and steep radial temperature gradients. These deductions were confirmed by flow visualisation. An analogy was postulated and proved mathematically for the limiting cases of thermal conduction and forced convection (turbulent mixing), whereby the temperature distributions developed in passive and active systems become identical. This permits safer and cheaper experimentation on passive systems in order to gain insight into active (exothermically reactive) ones. Passive and active experiments involving free convection confirmed the analogy with respect to horizontal and vertical temperature distributions, flow patterns and local heat transfer conditions, provided that the heat source was stationary and uniform. When, however, large temperature differences developed within the fluid, the effect of an acceleration of reaction rate was noted in the active case.
AT
ATmax AZ
u V v X
z
TMP DMMP Bi Gr Pr Ra
heat transfer area root of Bessel function heat capacity vessel diameter activation energy gravitational acceleration liquid height inside film heat transfer coefficient Bessel function reaction rate constant characteristic length mass heat Aux density specific rate of internal heat generation specific rate of heat removal radial coordinate radius, gas constant time temperature jacket temperature temperature of surroundings, initial temperature rate of temperature change temperature difference over inside thermal boundary layer temperature difference over wall and outside thermal boundary layer overall temperature difference maximum value of AT characteristic temperature difference overall heat transfer coefficient volume vertical (axial) velocity horizontal position in thermal boundary layer vertical position (axial coordinate) trimethylphosphite dimethylmethanephosphonate Biot number Grashof number Prandtl number Rayleigh number thermal diffusivity (L/PC,) coefficient of thermal expansion Frank-Kamenetski parameter dynamic viscosity thermal conductivity kinematic viscosity density characteristic time time to establish free convection adiabatic induction time (explosion
time)
2192
J. R. BOURNE REFERENCES
Semenov N. N., Some Problems ofChemical Kinetics and Reuctioity 2. Pergamon Press London 1959. [2] Frank-Kamenetski D. A., D@ision and Heat Transfer in Chemical Kinetics. Plenum New York 1969. [3] Sambeth J., Chimia 1982 36 128. [4] Salomon C. M., Chimia 1982 36 133. [S] Theofanus T. G., Chem.. Engng Sci. 1983 38 1615,1631. [6] Kee R. J., Landram C. S. and Miles J. C., J. Heat Transfer, Trans. ASME 1976 783 55. [7] Merzhanov A. G., Pribytkova K. V. and Shtessel E. A. Combust. Explos. Shock Waves 1971 7 58. 137. [1]
et al. [S]
[9] [lo] [1 l] [12] [13] [14] [15] [16]
Jones D. R., In&. J. Heat Mass Transfer 1974 17 11. Gray P. and Lee P. R., Oxidation Combust. Rev. 1967 2 1. Hiddink J., Schenk J. and Bruin S., Appl. Sci. Res. 1976 32 217. Carlslaw H. S. and Jaeger J. C., Conduction of Heat in Solids. Clarendon Press Oxford 1959. Regenass W., Thermochim. Acta 1977 20 65. Regenass W., Am. them. Sot. Symp. Ser. 1978 65 37. Hoch F., Dissertation No. 8001 ETH Zurich 1986. Martin H.. Doctoral thesis University Base1 1973. Evans L. B., Reid R. D. and Drake E. M., A.I.Ch.E. J. 1968 14 251.
HEAT
1987. 0
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$3.00 + 0.00 Journals Ltd.
TRANSFER FROM EXOTHERMICALLY REACTING FLUID IN VERTICAL UNSTIRRED VESSELS-I. TEMPERATURE AND FLOW FIELDS J. R. BOURNE+ Technisch-chemisches
Laboratorium
ETH, CH-8092
Zurich, Switzerland
and F. BROGLI, F. HOCH and W. REGENASS Ciba-Geigy AG, CH-4002 Etasel, Switzerland (Received 13 August 1986; in revised form 29 December 1986; accepted 15 February 1987) Abstract-Exothermically reacting fluid poses the danger of temperature run-away (thermal explosion) in some circumstances (e.g. interruption of stirring in a reactor and during storage). If the explosion time is longer than that required to establish free convection in a heat-generatingfluid,convectiongreatlymodifies
the internal temperature distributions relative to those developed when only conduction operates. Experiments on laminar free convection, using flow visualisation and temperature profiles, showed a
thermally stratified core moving slowly upwards and a thin thermal boundary layer on the wall of an unstirred vertical cylindrical vessel. An analogy between the temperature distributions in active (exothermitally reacting) and passive (cooling without reaction) systems at equal heat removal rates was established. It offers a basis for safer and cheaper experiments, whereby information from passive tests may be applied to active systems, provided their rates of heat generation are stationary and uniform. Criticality will occur in the upper layers of fluid when free convection operates and a method is given for predicting it.
INTRODUCTION Exothermically reacting fluids are often processed in stirred tank reactors, where turbulent mixing ensures a uniform temperature in the fluid, except for temperature gradients in the thermal boundary layers on cooling surfaces. Semenov [l] provided the criterion for thermal stability, whereby a temperature run-away (thermal explosion) could be avoided. Sometimes no motion is present in exothermically reacting matter, e.g. during the storage of a powder or a solidified product. Internal temperature gradients develop, the distribution being parabolic if the rate of heat generation and the thermal conductivity are uniform. Frank-Kamenetski [2] worked out the corresponding stability criterion for heat transfer by conduction. Exothermically reacting fluids in unstirred vessels can exhibit fluid motion. Internal temperature gradients cause density differences, which then induce free convection whose flow regime can be laminar, transitional or turbulent. Free convection does not always develop and stable thermal stratification is also possible. Particularly since the serious accident in Seveso ([3]-[S]) it is essential to assess the safety risk following an interruption of mechanical stirring as well as during the storage of exothermically reacting fluids. Typical questions include: Does free convection develop? Are internal temperature gradients formed? How large are inside film heat transfer coefficients?
Can they be obtained from generalised correlations? Is the system safe? A passive system contains no heat source (or sink) and the characteristic temperature relevant to free convection is the difference between the externally imposed temperature at the system boundary (e.g. a cooling surface) and the temperature (at some point) within the system. An active system contains a heat source (or sink) and the characteristic temperature difference is that developing-in the sense of a dependent variable-between the system and its surroundings. It needs to be related to the strength of the source, which may be regarded as an independent variable, and introduced into the dimensionless groups Gr and Ra, which apply to free convection, heat and mass transfer etc. Kee [6] defines this characteristic temperature difference, when the heat source is uniform (e.g. radioactive decay), as given in eq. (1):
+To whom correspondence should be addressed. 2183
AT, = L,2Qp/&
(1)
whereas for an Arrhenius-like, temperature-dependent reaction rate [k = k, exp (- E/RT)] Merzhanov et al. [7] and Jones [S] give AT, = RT,2/E.
(2)
In eq. (2) T, is the temperature of the surroundings and RTz/E is called the pre-explosion temperature difference. The definition and use of AT, will be con-
sidered subsequently. Valuable contributions to the study of free convection in active systems, particularly in the context of thermal explosions, have been made by Merzhanov
J. R. BOURNE et al.
2184
and his school. The Frank-Kamenetski given by 6
=
E
L:Q,(T,)
RT;
1
’
Derivation from a steady-state heat balance shows that 6 is proportional to the rate of internal heat generation divided by the rate of heat removal by conduction. For a given shape of the reaction zone, critical values of S have been worked out [9] above which the internal temperature distribution becomes unstable and thus time-dependent. Working with Arrhenius-like decomposition reactions (e.g. 3C&80 W/kg at 60°C) in various solvents, Merzhanov et al. [7] observed 6 values greatly in excess of those based on eq. (3) and attributed this to the higher rates of heat transfer caused by free convection. For small, vertical tubes this increase was correlated by Ra and three regimes were identified (no convection, heat conduction; free convection, thermal convection; free convection, thermal explosion). The concept of an induction time to develop free convection, rc was introduced. If the time to explosion, r cx,was shorter than r c, a further regime (rapid explosion in quiescent fluid) resulted. Ra was defined by
Ra=&Pr=@L3RT’ va
scales etc. An alternative approach must be found.
parameter 6 is
‘-5
and covered approximately the range lo’-10’. When Ra = IO’, the measured critical value of 6 was roughly 15 times that calculated for heat removal by conduction alone. The additional heat removal by free convection thus stabilized the system against runaway. Limitations of this work include:
(4 Flow regime. Experiments in passive systems,
for which RTz/E in eq. (4) is replaced by the imposed temperature difference, indicate the transition from laminar to turbulent free convection when Ra is near 109-109.‘. The experiments of Merzhanov ez al. [7] probably referred to laminar flow, whereas the values of Ra in industrial-scale reactors often imply turbulent free convection. (b) Shape/boundary conditions. Free convection is sensitive to shape (e.g. change from vertical to horizontal cylinder or to sphere) and boundary conditions (e.g. whether walls are at same or different temperatures, whether base of container is adiabatic or isothermal [lo]). (c) Experimental method. Merzhanov determined criticality in the presence of free convection by permitting thermal explosion. This was acceptable on an approximately 200-ml scale, but would become too expensive and dangerous at the scale of industrial reactors. Their criticality must be assessed by sound physical principles and calculations based upon free convection heat transfer characteristics, obtained at various
The present paper addresses itself to limitations (a) and (c).
PRINCIPLE
OF EXPERIMENTAL
METHOD
following analogy between active and passive systems is postulated and will subsequently be justified. For a given heat flux out of a system, the local temperature gradient with respect to time and position in The
(a) a passive system, which is being cooled down; and (b) an active system in the steady state, which has the same physical properties as the passive system and whose internal heat generation rate is uniform, will be equal. Provided this is valid, free convection heat transfer in active systems can be studied by means of the corresponding passive system. The following development proves the validity of the analogy for the limiting cases mentioned in the Introduction, namely forced convection and conduction.
Temperature
distribution
during forced
convection
A well-stirred liquid of mass m and temperature TR(t) is within a cylinder of radius R and length L, having an adiabatic lid and isothermal walls of area A, overall heat transfer coefficient U and uniform jacket temperature T,(t). Passive system:
dT, 0)
=
"CP y&--
-
UA[T,(t)
t=O
TR = TJ = T,
t>o
T,(t)
= T, -
-
T,(r)]
(5)
i-t.
For such linear reduction in the cooling jacket temperature, eq. (5) gives AT(t) = TR(t) - T,(t)
= h(1
- e-“‘),
(6)
where r is the time constant mc,/UA. Active system: The heat source Q, is taken to be steady: this corresponds to the chemical reaction rate of zero order of thermal explosion theory [9], i.e. to the “worst case” of negligible consumption of reagents. mc,?
= Q,&‘--
UA[T,(t)-
t=O
T,
= T,
t>o
T, = steady.
TJ]
(7)
Heat transferfrom exothermicallyreactingfluid in verticalunstirredvessels-I Solution of eq. (7) gives QPV T, = =(l
AT(t) = T,&)-
-e-‘I’).
(8)
Applying eqs (6) and (8) to the same reactors and keeping the same physical properties of their contents, then T(t) will be the same in the unsteady and also in the pseudo-steady [t -P 00, AT(t) ---, constant] states provided pc$
= Q p.
(9)
Temperature distribution during free conuection In so far as free convection heat transfer is intermediate in magnitude between the cases of forced convection and conduction, it might be anticipated that eq. (9) would be fulfilled for identical temperature distributions when free convection operates. It has, however, not been possible to prove this. It will be tested experimentally here.
Equation (9) proves the analogy for forced-convection heat transfer.
Temperature distribution during conduction The situation is basically the same as for forced convection. The temperature depends upon time and the radial position CT,&-, t)]. The external temperature TAt) will be taken to be that on the outer surface of the fluid, i.e. wall and jacket-side heat transfer resistances will for simplicity be considered to be negligible (Bi + a3). Passive system: L’AT at=*
(10)
AT = T,(r, t) - T,(R, t) t=O
AT(r, 0) = 0
t>OAT(r,t)=OandT,=T,(R,t)=T,-pt. The temperature distribution in the fluid when the boundary temperature is decreasingly linearly is given by eq. (11) [ll]:
ATO-, t) =
p(R” - r2)
2F
*
(11)
4a
(J, and J1 are zero- and first-order Bessel functions and a” the corresponding positive, real roots). Active system: aAT PC,Y&- = AT(r, 0) = 0
t=a t>o
AT(R, t) = 0, i.e. T, = steady aAT(O, t)
ar
=
o
2185
vided that eq. (9) is satisfied. (NB u = L/PC,.) Thus the analogy also holds for conductive heat transfer.
.
The heat source is taken to be steady and uniform, i.e. QP = constant. The solution of eq. (12) is then eq. (3):
ATtr, tj = QpW2 - r2) 41 For same reactors and constant physical properties, comparison of eqs (11) and (13) reveals the same temporal and spatial temperature distributions pro-
EXPERIMENTAL
A 2-dm3 glass heat flow calorimeter (Cl23 and [13]) was used consisting of a vertical cylinder (D = 0.11 m) with a dished base, all enclosed in a cooling jacket. Jacket temperatures were constant (50°C) for active runs and were reduced at linear ramp rates up to 40”C/h for passive runs. The wall of the calorimeter had a thickness of 0.005 m and the thermal resistance of this wall and the coolant (silicone oil) was measured [ 143. Chromel-alumel thermoelements (Thermocoax, Philips), having a diameter of 0.5 mm (time constant 25 ms), were used in the bulk of the liquid, whilst to explore temperature distributions in the thermal boundary layer the diameter was reduced to 0.25 mm (7 ms). These elements were positioned vertically (to within f 0.5 mm) and radially (to within f 1 mm in the bulk and f 0.1 mm in the boundary layer). The temperature difference AT between the local fluid temperature CT,&, z, t)] and the jacket temperature [Tdt)] was recorded to within f 0.02”C as a function of time. Before each run temperature was made uniform by agitating the liquid with a stream of nitrogen bubbles and brought to within 0.2”C of the jacket temperature. For the passive runs water, ethylene glycol, glycerol and toluene were employed. Approximate ranges of relevant physical properties were: conductivity (A) 0.1 (toluene)-0.6 (water) W/m K; viscosity (q) 0.4 (toluene)- lo3 (glycerol) kg/m s; heat capacity (c,) 1800 (toluene)-4200 (water) J/kgK; density (p) 850 (toluene) - 1250 (glycerol) kg/ma; thermal expansion coefficient (/3) 0.4 x 10m3 (water and glycerol)- lo-’ (toluene) K-‘; and Pr 5 (water and toluene)-104 (glycerol). The temperature dependence of all these properties was considered. The active system was the isomerisation of trimethylphosphite to dimethylmethanephosphonate, catalysed by methyliodide at or near 50°C: 0 (CH,O),-P (TMP)
-
CH,l
CH3 0)2-!-CH3. (DMMP)
This reaction runs to complete conversion without forming by-products. It can be repeatedly started and stopped by addition of methyliodide and triethylamine respectively. The physical properties of an equimolar mixture at various temperatures are available [14].
J. R.
2186
BOURNE
reaction enthalpy is significant (- 180.9 kJ/mol [IS]) and the adiabatic temperature rise is 570°C. The reaction kinetics have been studied [lS] and in the absence of solvent exhibit a region of conversion near 50% where the rate of heat generation is steady, despite the consumption of TMP. At 50°C this period persisted for some 2 h, which proved convenient for studying free convection in an exothermically reacting fluid. By varying the catalyst concentration in the range 0.07 1-O. 139 mol CH,I/kg reaction mixture, various powers could be developed, e.g. at the highest CHpI level 17.8 W/kg reaction mixture at 50°C for over 2 h The
c141. RESULTS Passive
system:
The
reduced at a constant
jacket
rate *and
temperature
T,
was
the fluid temperature
er al.
at any point [T,Jr,
z, t)] started to fall at the same rate
some 30-90 min (glycerol needed 90 min, the other fluids responded faster). This quasi-steady state, defined by dAT/dt = 0, reflected the time lag needed to develop a steady Aow field in the liquid (and not a measuring lag). distributions measured in an Temperature equimolar TMP-DMMP mixture without catalyst (Fig. l), ethylene glycol (Fig. 2) and glycerol (Fig. 3) show radially a thermal boundary layer, whilst axially the temperature indicates strata with the highest temperature just below the liquid surface. Film thicknesses fall roughly in the range 1 mm (TMP-DMMP)-12 mm (glycerol): AT, is defined as the fluid temperature at any point T(r, z) minus the inside wall temperature T(R, z) at the same time. The radial distribution is not parabolic, which is characteristic of conduction [eq. (1 l)], even for the most viscous after
AT PC1 6-
cu, PC1 2
lZ-
f-
0
-F ;
--Y? 31
;
3
x hmll
Fig. 1. Vertical and horizontal tempe.ra_turedistributions during passive, quasi-steadycooling. Medium: TMP-DMMP (equimolar). T = 19”C/h. H/D = 1.08. TJ = 51°C. (0) AT, (0) AT,
10 ATPCI_
72
z hml
x Imml
Fig. 2. Vertical and horizontal temperafure distributions during passive, quasi-steady cooling. Medium: ethylene glycol. T = 20”C/h. H/D = 1.0. (0) AT, (0) AT,
L
2187
Heat transferfrom exothermicallyreacting fluid in verticalunstirredvessels-1
0
z Imml
150 0
2
x Imml
L
0
0.2
0.1
0.6
0.8 1.0 r/R
Fig. 3. Vertical and horizontal temperature distributions during passive, quasi-steady cooling. Medium: glycerol. T = 3O”C/h. H/D = 1.0. (0) AT, (0) AT,.
fluid (glyceroI), denoting the action of free convection. The vertical distributions were determined outside the boundary layer (i.e. in the core) and z = 0 refers to the tank base. AT = TR(z) - T, and that part of AT taking place in the outside coolant film and the glass wall is whilst the difference between the bulk fluid AT, temperature at any height and the inside wall temperaThe thermal resistance of ture is AT, = ATAT, especially the glass (5 mm thick) was evidently considerable. In Figs l-3 El marks the surface: the liquid temperature fell towards the surface as well as in the gas space (z > Zf). z(AF) is the position where AT is equal to its vertical-averaged mean value AT. The vertical AT profiles were increased and decreased by increasing and decreasing p (in the range 1040°C/h), respectively. Figure 4 is an example. Liquid heights in the range W = 0.5-1.5D were also studied [14]. Injection of dye (methylene blue) in the base of the tank just off-centre helped to visualize the flow pattern. During the quasi-steady state cooling of water at 2O”C/h, the tracer rose vertically without radial dispersion through the core region at approximately l-2 cm/min. The core region exhibited essentially plug flow upwards, the flow turning radially outwards within less than 1 cm of the surface to feed the downflow in the boundary layer. Thus the observed vertical flow in the core agrees well with the stratified temperature distributions (Figs 14). Heat exchange (cooling) occurred radially between the cool vessel wall and the fluid flowing downwards in the boundary layer. (In the experiment cited, the thickness and velocity of this layer were roughly l-2 mm and 15-20 cm/min). There was some tendency for the boundary layer to separate over the dished base, but most of the cooler fluid followed the base contour to
10
AT
, 0
sb
Id0
z Imml
150
Fig. 4. Vertical temperature distributions during passive, quasi-steady cooling at three rates. Medium: TMP-DMMP (equimolar). H/D = 1.0. T, = 50°C. T(“C/h): (0) 19, (m) 28, (0) 34.
rejoin the ascending core. Analogous results during heating have been reported [16]. The local heat Aux density increased in all cases with the height in the vessel. Figure 5 gives one example. Combined with the local inside film temperature differences [AT,(r)], local film heat transfer coefficients h&) could be calculated: Figure 6 gives examples for TMP-DMMP, glycerol and toluene. From these and other results [14] it was concluded that the film coefficient was uniform. Active system: Steady-state vertical temperature distributions T,(z) were attained 20-90 min after ad-
J. R. BOURNE
2188
et al.
(b) The temperature in the upper part of the stratified core increases faster than in the passive case (Fig. 1; see Figs 9 and lo), which is due to the Arrhenius-like effect of temperature on reaction rate.
40
0
2
rmm1
40
Fig. 5. Local heat flux densities as functions of height and cooling rate during passive. quasi-steady cooling. Medium: TMP-DMMP (eouimolar). H/D = 1.0. T W3h): (0) 19, (0) * . 28, (0) 36.
dition of catalyst to the reaction mixture. Figure 7 includes radial temperature distributions in the thermal boundary layer, vertical temperature distributions in the core and local heat flux densities, and film heat transfer coefficients at various heights. The runs used liquid depths of 0.120 and 0.114 m, respectively, and good reproducibility is indicated_ Several similarities with the results for passive systems (Figs 1-6) are evident. Thus, for example, local heat fluxes and film heat transfer coefficients were evaluated from the temperatures in the boundary layer and are compared in Fig. 8 with values from passive runs (no catalyst). The two significant differences are: (a) The temperature in the boundary layer exhibits a weak maximum (see also Figs 9 and 10). In the core the temperature at a given height was independent of radial position [14].
hR IW/td
Figure 9 compares active and passive runs, which according to eq. (9) should be identical. The temperature in the upper part of the core is some 5°C above that in the jacket (50°C) and in the lower core. Temperatures in the active case rise above those in the passive run in this region; this reflects faster reaction with increasing temperature. This becomes still clearer when the rate of heat generation (at 50°C) is raised from 9.1 to 17.8 W/kg reaction mixture (Fig. 10). The observed vertical temperature profile is consistent with 38.3 W/kg, which is double the value given by eq. (9) C48.9 compared with 22.8 W (Fig. 1 l)]. Because higher temperature differences can occur in active runs than in the comparable [according to eq. (9)] physical runs, the film heat transfer coefficients (Fig. 8) are also higher than expected: free convection offers therefore some degree of internal compensation against rising heat generation rates. The values of Ra [see eq. (4)], where Lc = H and the characteristic temperature difference is AT,, ranged Prom lo*.’ to 109.1, whilst in the passive experiments the range was 106~2-10g~‘. The flow regime was likely to have been laminar in almost all runs. APPLICATION
TO AVOIDANCE
OF THERMAL
EXPLOSION
Potentially dangerous situations arise with exothermically reacting fluids when, for example, the mechan-
KI
Glycerol
lOO-
0
200 100 -
200
0 Fig. 6. Local inside
5il
’
40
140
z Imml
film heat transfer coefficients as functions of height and cooling rate (passive runs).
+ (“C/h) H/D
= 1.0 1.5
=
10
20
30
40
Cl I
0
n A
0 +
Heat transfer from exothermically reacting fluid in vertical unstirred vessels-1
2189
AT [%I 6-
q [W/m21
hR [W/m2 Kl ZOO-
0
Go
2 lmml
0
160
55
2 Imml
nxl
;
o-
i
x [mm1
Fig. 7. Exothermic isomerisation of TMP: reproducibility of results. T, = 50°C. QP = 9.1 W/kg at 50°C. (0) 0.120 m, (0) 0.114 m. (0) and (I) = corresponding values of AT,.
H:
ical agitation of a reaction vessel fails or thermally unstable fluids have to be stored. It is required to calculate the critical masses below which thermal runaway will not occur. The information collected here on free convection and temperature distributions in active and passive systems refers in particular to cylindrical, vertical containers. The way in which it can be applied to assessing safety will now be outlined. In all experiments the fluid near the tank base had a temperature equal to or slightly in excess of the jacket coolant T,. The temperature of the rising fluid increased to a maximum To + AT,, in the top layer of the core. The criterion (9) to avoid a thermal explosion in this region (dQ,/dT -C dQ/dT when QP = Q) is
Loo hR fW/m2K1 200 -
0 1500 -
q lWlm21
AT,,
1000-
< RT,2/E,
(14)
and the corresponding induction time [9] is r,
= 0.64pc,RT;t/EQ,(T,).
It was found experimentally that AT_
(15) was
in the
range 2-3 AT, the higher ratio 3 being more characteristic of the active runs. As a first estimate AT_ can be found from
500-
AT_ 0
3
sb
z Imml
160
Fig. 8. Heat flux densities and film heat transfer coefficients for TMP in passive and active runs. QP (W/kg at 50°C): (0) 9.1, (0) 13.3, (A)17.8. T (“C/h): (I) 19, (0) 28, (A) 36.
= 3AT,
(16)
and it will be shown in Part II how AT can be related to the reactor size and the thermal properties of the reaction.
The above analysis assumes that free convection has time to develop before thermal explosion occurs. This
J. R. BOURNE
2190
et al.
6-
1.0 o.5-mb
60
0
2 tmml
0
1
; xhml
3
Fig. 9. Vertical and horizontal temperature distributions for TMP in passive and active runs. TJ = 50°C. H/D = 1.0. (0) Q p = 9.1 W/kg at 5O”C, active. (0) T = 19”C/h, passive. (0) and (I) = corresponding values of AT,.
AT
PC1
20
2-
15 -
I-
Fig. 10. Vertical and horizontal temperature distributions for TMP in passive and active runs. T, = 50°C. H/D = 1.0. (0) QP = 17.8 W/kg at 5O”C, active. (0) T = 36.2”C/h, passive. (e) and (I) = corresponding values of AT,
Heat transfer from exothermically reacting fluid in vertical unstirred vessels-I 60
2191
The way is shown in which the information gained in this study can be applied to avoiding thermal explosion in exothermically reacting fluids undergoing free convection. Generally it may be concluded that the physical situation during free convection in passive and active systems has been clarified.
Qp IWl _ LO -
NOTATION
A a CP D 50
0
E
100
z
lmml
Fig. 11. Cumulative thermal power during TMP isomerisation as function of vertical position in liquid. QP at 50°C WI
Q p effective w
11.6 17.0 22.8
16.0 28.2 48.9
0 0 a
k passive (“C/h) : C
19 28 36
9 H h, J k Lc m 4 QP
Q r R
implies [ 71: ‘5, 4 zex.
(17)
and a sufficiently high value of Ra (e.g. in excess of 103)
C71 and C93)-
T
=.I TO
CONCLUSIONS
Vertical and radial temperature profiles measured during free convection in liquids in a vertical cylindrical container having a dished base revealed: (a) a thermally stratified core with laminar upward fluid flow and a uniform radial temperature, which increased from the base towards the surface; and (b) a thin thermal boundary layer on the vessel wall with downward fluid flow and steep radial temperature gradients. These deductions were confirmed by flow visualisation. An analogy was postulated and proved mathematically for the limiting cases of thermal conduction and forced convection (turbulent mixing), whereby the temperature distributions developed in passive and active systems become identical. This permits safer and cheaper experimentation on passive systems in order to gain insight into active (exothermically reactive) ones. Passive and active experiments involving free convection confirmed the analogy with respect to horizontal and vertical temperature distributions, flow patterns and local heat transfer conditions, provided that the heat source was stationary and uniform. When, however, large temperature differences developed within the fluid, the effect of an acceleration of reaction rate was noted in the active case.
AT
ATmax AZ
u V v X
z
TMP DMMP Bi Gr Pr Ra
heat transfer area root of Bessel function heat capacity vessel diameter activation energy gravitational acceleration liquid height inside film heat transfer coefficient Bessel function reaction rate constant characteristic length mass heat Aux density specific rate of internal heat generation specific rate of heat removal radial coordinate radius, gas constant time temperature jacket temperature temperature of surroundings, initial temperature rate of temperature change temperature difference over inside thermal boundary layer temperature difference over wall and outside thermal boundary layer overall temperature difference maximum value of AT characteristic temperature difference overall heat transfer coefficient volume vertical (axial) velocity horizontal position in thermal boundary layer vertical position (axial coordinate) trimethylphosphite dimethylmethanephosphonate Biot number Grashof number Prandtl number Rayleigh number thermal diffusivity (L/PC,) coefficient of thermal expansion Frank-Kamenetski parameter dynamic viscosity thermal conductivity kinematic viscosity density characteristic time time to establish free convection adiabatic induction time (explosion
time)
2192
J. R. BOURNE REFERENCES
Semenov N. N., Some Problems ofChemical Kinetics and Reuctioity 2. Pergamon Press London 1959. [2] Frank-Kamenetski D. A., D@ision and Heat Transfer in Chemical Kinetics. Plenum New York 1969. [3] Sambeth J., Chimia 1982 36 128. [4] Salomon C. M., Chimia 1982 36 133. [S] Theofanus T. G., Chem.. Engng Sci. 1983 38 1615,1631. [6] Kee R. J., Landram C. S. and Miles J. C., J. Heat Transfer, Trans. ASME 1976 783 55. [7] Merzhanov A. G., Pribytkova K. V. and Shtessel E. A. Combust. Explos. Shock Waves 1971 7 58. 137. [1]
et al. [S]
[9] [lo] [1 l] [12] [13] [14] [15] [16]
Jones D. R., In&. J. Heat Mass Transfer 1974 17 11. Gray P. and Lee P. R., Oxidation Combust. Rev. 1967 2 1. Hiddink J., Schenk J. and Bruin S., Appl. Sci. Res. 1976 32 217. Carlslaw H. S. and Jaeger J. C., Conduction of Heat in Solids. Clarendon Press Oxford 1959. Regenass W., Thermochim. Acta 1977 20 65. Regenass W., Am. them. Sot. Symp. Ser. 1978 65 37. Hoch F., Dissertation No. 8001 ETH Zurich 1986. Martin H.. Doctoral thesis University Base1 1973. Evans L. B., Reid R. D. and Drake E. M., A.I.Ch.E. J. 1968 14 251.