KINETIC PROPERTIES AT HIGH PRESSURE
3.2.6 Change of reaction rate constant with pressure
The sign of the volume of activation, Av #, determines whether the rate constant, k, increases or decreases with the pressure. The value of Av# is positive when the volume of the activated complex is larger than the volume of the initial reactants. Thus the rate constant, k, decreases when the pressure is increased. When the volume reduces in the transition state, Av # is negative and k will increase with increasing pressure (Table 3.2-2).
The values for Av # range between +30 and -60 cm3/mol.
Within a small range of pressure it can be assumed that Av # does not change with pressure.
The integration of eqn. 3.2-7 gives:
(3.2-53)
Table 3.2-2
How activation volume affects rate constant
Activation volume, Av # Change of rate constant, k, with pressure, p
positive + decreases $
negative- increases 1'
,~30
<
~ 20
r ~
~, 10
~
,..: 0
r
0
1.0
a ~ 0.8
t~
0.6
b ~
0.4 9
~ 0.2
~. 0.0
1000 2000 3000 ~
Pressure [bar]
h
0 10'00 20'00 3000
Pressure [bar]
Figure 3.2-6. Relative reaction rate constant kp/kl bar. Activation volume: a, -30 cm3/mol, 298 K; b, -30 cm3/mol, 373 K; c, -10 cm3/mol, 298 K; d, -10 cm3/mol, 373 K; e, +30 cm3/mol, 298 K; f, +30 cm3/mol, 373 K; g, +10 cm3/mol, 298 K; h, +10 cm3/mol, 373 K.
The change in rate constant at various pressures and temperatures, calculated from eqn. 3.2- 53 for several values of Av # is presented in Fig. 3.2-6. The rate-constant changes exponentially with the pressure. The effect is steeper when the activation volume is large and the temperature is low, and vice versa.
3 . 2 . 7 P r o b l e m s
1) Evaluation of activation volume from experimental data:
The rate constant of a polymerization reaction was determined at 433.8 K and various pressures (Table 3.2-3).
In order to evaluate Av # the rate constants are plotted on a logarithmic scale versus the pressure. The slope of the resulting straight line is ct = 9.10 -9 Pa -1. A value of Av #= -32.5-10 -6 m 3 mo1-1 can be obtained from Av # = a.R.T, with R = 8.314 J mo1-1 K -1 and T = 433.8 K.
Table 3.2-3
Rate constant of a polymerization reaction at various pressures
Pressure, p [MPa] Relative reaction rate constant, kp,/klbar
120 140 160 180
0.041 0.050 0.061 0.067
2) Change o f rate constant with pressure:
The half-life time, rl/2, of decomposition of di(t-butyl)peroxide at a temperature o f 463 K and ambient pressure is 50 s. One may calculate the rate constant and the half-life time for decomposition at 300 MPa and the same temperature, when the activation volume is Av # = +13 cm 3 mol 1.
Rate constant k = ln___2_2 = ln___22 = 0.014 s-1 rl/2 50 s
Av#.(p-po)
R.T = 0.014 s -1 .e kp = k(p=300 MPa) "e
13.10 -6 .(300-0.1).10 -6
8,314.463 = 5 . 1 . 1 0 - 3 s - 1
The half-life time for decomposition at 300 MPa is then: rl/2 = k ln2 _ l n 2 - 136 S -1 5.1.10-3s
3.3 M e a s u r e m e n t o f c h e m i c a l k i n e t i c d a t a at h i g h p r e s s u r e
The main aim of the methods described in this chapter is to obtain data for the design o f chemical reactors, for the simulation of their operation behaviour, and, last but not least, to evaluate the influence of temperature and pressure on reaction rate. For this purpose, the techniques for measuring reaction rates at high pressures are presented. The details o f the apparatus are mentioned in Chapter 4.3.4.
3.3.1 M e a s u r e m e n t o f r e a c t i o n rates
Generally, the same methods can be used to measure reaction rates at high pressure as at low pressure. Some of them are more suitable than others for use at high pressure. The selection depends whether a homogeneous or a heterogeneous reaction should be investigated, whether it is a gas- or liquid-phase reaction, or a catalyst is used.
The apparatus used are mostly stirred-tank-, tubular-, and differential recycle reactors. Also, optical cells are used for spectroscopic measurements, and differential thermal-analysis apparatus and stopped flow devices are applied at high pressures.
The stirred-tank reactor or stirred autoclave is continuously operated when rapid changes in concentration do not allow samples to be taken from the reactor for analysis. In the stirred- tank reactor complete uniformity of concentration and temperature throughout the reactor is assumed. The exit stream from the reactor has the same composition as the fluid within the reactor (Fig. 3.3-1). The rate of reaction can be evaluated from the material balance
1. (c c 0) (3.3-1)
0 = r _ m B 7"
When the residence time, r, is known from the reactor volume and the volumetric throughput, 9, and the temperature inside the reactor is measured, the reaction rate can be determined from the concentration, c, of a component in the reactor, and Co in the feed.
When a solid catalyst is required the autoclave must be equipped with a rotating basket in which the catalyst is placed.
Figure 3.3-1. Continuously operated stirred autoclave.
In a tubular reactor the concentration varies from point to point along a flow-path, as shown by the solid curve in Fig. 3.3-2, right. The material balance
dc d 2 c
0 = r - p. w . - - + Dax . ~ (3.3-2)
dz dz 2
is a differential equation, and the concentration of a component must be measured at different locations to determine the reaction rate. A further term, Dax (d2c/dz2), must be considered in the material balance if the flow is not plug-flow. Dax is the effective diffusivity in the axial direction, to be determined from measurement of the residence-time distribution. When the tubular reactor could not be run under isothermal conditions, an energy balance
O= r . ( - A H R ) - P.C p . ~ d T d 2 T F
+ h . ~ - k w 9 . (T - T K ) (3.3-3)
dz dz 2 --V
must be taken into account and also the course of temperature, T, must be measured together with the temperature, TK, of the coolant or of the heating fluid. Thus, the rate of reaction cannot be determined directly as a function of concentration and temperature. Material- and energy-balances must be solved by numerical methods and the kinetic parameters must be
t><t---
t><l----
?
Figure 3.3-2. Tubular reactor.
c , T
... i ...
... x~ ...
evaluated by least-squares techniques [16].
On the other hand, the tubular reactor is a simple and inexpensive apparatus. It's small inner diameter requires a low thickness of the tube-wall to resist high pressure, and facilitates the removal or heating of the feed in order to operate the reactor under isothermal conditions.
Solid catalyst can easily be placed in the tubular reactor.
The d i f f e r e n t i a l r e a c t o r uses a thin catalyst bed in which only small changes of concentration and temperature occur (Fig. 3.3-3). The rate of reaction, r, can be obtained from the difference in concentration, Ac, over the catalyst bed or its thickness, Ax, the volumetric throughput, 9, or the molar throughput, nges , and the quantity of catalyst, WK, using the material balance:
Ac Ax
0 = r - 9. or 0 = r - nges 9 (3.3-4)
wK wx
The measurement of a small concentration gradient requires more analytical work, and often gives less accurate kinetic data. For this reason, in the d i f f e r e n t i a l r e c y c l e r e a c t o r a fraction of the reaction mixture leaving a thin catalyst bed is recycled and added again to the feed (Fig. 3.3-4). This results in a larger difference of concentration, Co, or mole fraction, Xo, between the feed and c or x at the reactor outlet, which is used to determine the reaction rate from the material balance:
1 90
0 = r - - . (c o - c) or 0 = r - 9 (2:0 - x) (3.3-5)
7 wK
Again, WK is the quantity of catalyst and x is the average residence time. The recycle ratio, 11 = 9rec/90, which is determined by the volumetric stream, 9rec, of the recycled reaction mixture and the feed, 90 , should be large enough that the concentration- and temperature- gradient over the catalyst bed are only small. Criteria for selecting the recycle ratio are presented in References 17 and 18. Appropriate ~7-values are in the range of 20 - 400,
WK
h 0 , 90 , xo
X XO
Figure 3.3-3. Differential reactor.
X Ii,
~kr = r/. ~0
X XO" XO
no, f~O , xo
J . . . -
z[
X
Figure 3.3-4. Differential recycle reactor.
depending on the desired conversion, reaction enthalpy, and heat capacity of the reaction mixture.
On the one hand, the differential reactor with recycle permits kinetic measurements of high accuracy. On the other hand, a transfer equipment is required to recycle a fraction of the reaction mixture. This can be difficult when the pressure is high. For this purpose, a jet loop reactor was developed which is equipped with an ejector to recycle the fluid. The design of the jet loop reactor is described in Chapter 4.3.4.
Problems can arise in the investigation of rapid reactions if the reactants are not heated sufficiently fast to the desired temperature, and if the samples from the reactor are not cooled rapidly to stop the reaction. A more sophisticated approach consists of monitoring the changes in concentration in an optical cell, in situ, by means of spectroscopy. Both infra-red and Raman spectroscopy can be used, depending on the sensitivity of characteristic bonds and the wave-number range of interest.
A n optical cell for pressures of up to 200 MPa and temperatures to 200~ is presented in Chapter 4.3.4. The cell can be coupled with a commercial Raman spectrometer to measure the course of the intensity of a bond's signal with time. By calibration, the intensity versus time curve can be converted into a concentration versus time curve, from which the rate of reaction and kinetic parameters can be evaluated. The method is explained in Chapter 3.3.2, considering the decomposition of an organic peroxide.
Optical cells can also be used to investigate the kinetics of radical polymerization reactions under high pressure by means of the rotating sector method. Again, the apparatus is presented in Chapter 4.3.4. An example of the method for the evaluation of individual rate constants in radical polymerization of ethylene is given below.
The stopped-flow method uses syringe-type pumps, (a), to feed the components, A and B, through a mixing cell, (c), into the reaction cell, (d), which can be an optical cell (Fig. 3.3-5).
The pumps, mixing cell, and reactor are well thermostatted. The flow is stopped when the syringe, (e), is loaded and operates a switch, (f), to start the monitoring device. The change in concentration is detected either by spectroscopy or conductivity measurement.
Figure 3.3-5. Stopped flow apparatus [19]. a, Syringe type pump; b, thermostat; c, mixing cell; d, reaction cell; e, stop syringe; f, switch; g, photo multiplier; h, monochromatic filter; i, lamp; j, controller; k, transducer; 1, computer.
The heat effects during physical or chemical conversion are used in differential thermo analysis (DTA) to evaluate the reaction rate together with kinetic and thermodynamic parameters. The apparatus (Fig. 3.3-6) consists of two identical high-pressure cells which are arranged in an oven, a. In one cell the components to be investigated are placed, the other cell
g h g
f
, e -
!--
E
I T I J
I I
I / \ A T I
I I
, / \ , ,,,
I I
I I
I I
I I
Time t, Temperature 1" 2
L__
S.-- I 1 )
E ID
, . . a
._=
i11
r
o
a
Figure 3.3-6. DTA-device and typical AT-curve [20]. a, Oven; b, amplifier; c, potentiometer;
d, synchron motor, e, ignition; f, thyristor; g, recorder; h, integrator, i, differentiator; T1, temperature of the sample cell; AT, difference in temperature between sample and reference.
is empty or contains a reference with similar thermodynamic properties as those of the sample. The course of the temperature of the sample cell is measured and compared to the temperature of the reference cell. The course of the temperature of the sample cell in which a physical or chemical conversion takes place is shown in Fig. 3.3-6, right (upper curve),together with the course of the temperature difference between the cells (lower curve).
The shape of the AT-curve depends on the kinetics and the heat of reaction.
For simple reactions, expressions have been derived to determine the kinetic parameters together with the order of the reaction from the AT-curve [21 ]. A DTA-cell for pressures of up to 300 MPa and temperatures to 300~ is shown in Chapter 4.3.4.
3.3.2 Examples
In this chapter, the application of the apparatus mentioned before to measure reaction rates under high pressure and to evaluate kinetic data, is described. As examples, the decomposition of organic peroxides, the radical polymerization of ethylene, and the synthesis of methanol are selected.
Organic peroxides, which readily decompose into free radicals under the effect of thermal energy, are used under high pressures as initiators for radical polymerizations. The measurement of the influence of pressure on the rate of decomposition gives rise to the determination of the activation volume, which, in turn, allows conclusions to be drawn on the decomposition mechanism and the transition state.
Owing to the very high rate of decomposition, in-situ measurement of concentration by means of Raman spectroscopy was applied. The peroxide used was t-butylperoxy pivalate (see Chapter 5.1, Table 5.1-2) dissolved in n-heptane at a concentration of 1 wt.%. In order to observe the change in intensity of absorption of the O-O bond at 861 cm -~, the spectrometer was adjusted to this wave number. The change of intensity is an indication of the reduction in the peroxide concentration, and was recorded as a function of time. The apparatus was calibrated before measuring the intensity of peroxide solutions of different concentrations [22].
In Fig. 3.3-7 the intensity versus time plot is shown for an experiment at 180 MPa and 120~ It can be seen that the intensity decreases exponentially within 100 s to 58% of its initial value.
By means of the calibration, the absorption intensity curve is converted into a concentration versus time curve. The rate of decomposition is then obtained by differentiating this curve and plotted on a logarithmic scale versus the logarithm of the concentration. From the slope of the resulting straight line, an order of unity for the decomposition is evaluated, and from the intersection at the ordinate the rate-constant is obtained.
In order to determine the activation volume, the rate constants from tests at different pressures are plotted on a logarithmic scale versus the pressure (Fig. 3.3-8). As outlined in Chapter 3.2.3, from the slope of the resulting straight line an activation volume of +7 ml/mol is obtained.
The rotating-sector method was applied to determine the individual rate constants of chain propagation and chain termination of the radical polymerization of ethylene [23,24]. The photo-initiator was diphenyldisulfide. First, the overall rate of polymerization was measured under steady illumination at pressures of 50 - 175 MPa and 132 - 199~ (Fig. 3.3-9). It increases first steeply and then less steeply with increasing pressure. At 175 MPa the rate of polymerization is ten times higher than at the low pressure of 50 MPa.
110~.
100~~- I
_ 90 9
~ 80
~ 7o 60
50 . . . 0
0 20 40 60 80 1 O0
200 ,---, rj
o
150 100 o 5o :~
Time [s]
Figure 3.3-7. Peroxide decomposition, change of intensity with time. Solid line, intensity;
dashed line, pressure; dotted line, temperature; initial peroxide concentration: 1 wt.%.
The ratio kp/kt ~ of the rate constants of chain propagation and termination was determined from the overall rate, r, the known concentration, M, of the monomer and the rate of decomposition, ri, of the photo-initiator, which was measured separately, using the expression:
r = kp/kt ~ [M].ri (3.3-6)
The ratio kp/kt can be obtained from the expression:
k d k t --" 2 . r . I M ] -1. r ( 3 . 3 - 7 )
,---, -8.2 ~ ~ . , ~ d lnk Av# I
-
2 ~ d p T = R T I
"~o = - 8 . 4 " " ' ~ ' ~
O
-8.6
0 ' 5'0 ' 100 150
Pressure [MPa]
200
Figure 3.3-8. Determination of the activation volume.
10 .5
o
= 1 0 .6
9 I , , , , 4 o
/
5'0 ' 100 ' 1:50 ' Pressure [MPa]
200
Figure 3.3-9. Overall rate of polymerization, e , 132~ O, 144~ x, 153~ A, 164~ *, 172~ o, 181 ~ [3, 189~ v, 199~
The life-time, r, of the radicals can be determined from the ratio of overall rates of polymerization measured at the steady- and unsteady state as a result of intermittent illumination by the rotating sector. In Fig. 3.3-10 the rate constant, kp, of chain propagation (left) and kt, that of termination (fight), are plotted versus the pressure. Both rate constants increase with increasing temperature. The energy of activation of chain propagation is
Ep = 37 kJ/mol, and that of chain termination is Et = 9.9 kJ/mol. The influence of pressure is different, kp increases with increasing pressure. An activation volume of AVp # = -25.5 ml/mol can be obtained when kp is plotted on a logarithmic scale versus the pressure. The value of kt
16- E
o E o.15-
v E
14-
/ x f / x - "
/ x "
x ~ / x ~ /
/ / o
~ ~ I O ~
/ O ~ 9
~o ~ /
~ 2 8 -
27-
e -
11
5s 1C)0 150 2()0 26- 5,0 160 1~i0 260
Pressure [MPa] Pressure [MPa]
Figure 3.3-10. Rate constants of chain propagation (left) and termination (fight) of radical polymerization of ethylene. X, 189~ O, 153~ ~, 132~
decreases with increasing pressure, because of diffusion-controlled chain termination. On average, a value of Avt # = 7 cm3/mol is obtained.
The advantage of the jet-loop reactor for kinetic measurements is demonstrated in the investigation of the synthesis of methanol from hydrogen and carbon monoxide at pressures of 2 - 8 MPa and temperatures of 225 - 265~ A copper catalyst in the form of cylindrical pellets, with both diameter and length of 5 mm, was used.
The reaction rate was determined from the difference between the methanol concentrations in the reactor outlet and inlet. It increases steeply with increasing pressure and also with increasing temperature, but tends to level off at high temperatures (Fig. 3.3-11).
A rate equation was derived on the assumption that in the first step formaldehyde is formed on the catalyst surface from adsorbed carbon monoxide and hydrogen. The subsequent conversion of formaldehyde to methanol was assumed to be the rate-determining step. The experimental data were best expressed by a rate equation of the Langmuir-Hinshelwood type:
dPCO2 " ~)22 - ~CH3OH / K e q
rCH 3 0 H - (A + B . dpc 0 + C . dpH2 + D . dPCH 3 OH + E . dPC02 )2
(3.3-8)
In this equation, fugacities were used instead of partial pressures, to take into account the non- ideal behaviour of gases at high pressure. The coefficients, A to E, were determined by means of non-linear regression calculation by a method of Marquardt [25]. From the measurements at various temperatures, the frequency factor, ko, and the activation energy, E, were evaluated.
The data are collected in Table 3.3-1.
0.09
0.08 ,~d
~ 0 . 0 7 c
o o 6
0.050.04 / 5 , ~
0 0 3
~ 0.02 ,; a
0.01 , ~ ~ i "-:-:z:~~ "
. . . . . .
o.oo +
Pressure [MPa]
Figure 3.3-11. Rate of methanol formation. Composition of feed: 13 vol.% CO, 82 vol.% H2, 4 vol.% CO2. temperature: a, 225~ b, 235~ c, 245~ d, 255~ solid lines, experimental;
dashed lines, calculated from eqn. 3.3-8.
Table 3.3-1
Coefficients of the rate equation for formation of methanol [26]
ko [bar 3/2 for A; bar 1/2 for B to E] E [kJ/mol]
A 6.33.10 TM 128.3
B 2.28.10 -3 -39.4
C 2.12.10 -6 -65.0
D 8.14.100 3.9
E 2.03.10 -11 -116.0
References of sections 3.1, 3.2, 3.3
1. M. Baerns, H. Hofmann and A. Renken, Chemische Reaktionstechnik, Bd.1, 100, G.Thieme Verlag, Stuttgart, New York, 1987.
2. O. Levenspiel, Chemical Reaction Engineering, 2nd ed., 349, J.Wiley, New York 1962.
3. S.D. Hamann, Chemical Kinetics, in R.S. Bradley (ed.), High Pressure Physics and Chemistry, Vol. 2, 163, Academic Press, London 1963.
4. H. Eyring, J. Chem. Phys. 3 (1935) 107.
5. M.G. Evans and M. Polanyi, Trans. Faraday Soc. 31 (1935) 875.
6. S. Glasstone, K.J. Laidler and H. Eyring, Theory of Rate Processes, McGraw Hill, New York 1941.
7. G. Luft, P. Mehrling and H. Seidl, Angew. Makromol. Chem. 73, Nr. 1118 (1978) 95.
8. B. Raistrick, R.H. Sapiro and D.M. Newitt, J. Chem. Soc. (1939) 1761.
9. C.K. Ingold, Structure and Mechanism in Organic Chemistry, Bell, London 1953.
10. G. Luft and Y. Ogo, Activation Volumes of Polymerization Reactions, in J. Brandrup and E.H. Immergut (eds.), Polymer Handbook, 3rd ed. J. Wiley, New York 1989.
11. P.-Ch. Lim and G. Luft, Makromol. Chem. 184 (1983) 849.
12. M.G. Gonikberg, Chemical Equilibria and Reaction Rates at High Pressures, Israel Program for Scientific Translations, Jerusalem 1963.
13. G.F. Froment and K.B. Bischoff, Chemical Reactor Analysis and Design, J. Wiley, New York 1979.
14. K.E. Weale, Chemical Reactions at High Pressures, E. and F.N. Spon, London 1967.
15. H.R. Hunt and H. Taube, J. Am. Chem. Soc. 80 (1958) 2642.
16. D.M. Himmelblau, Process Analysis by Statistical Methods, J. Wiley, New York 1969.
17. I.I. Ioffe and L.M. Pissmen, Heterogene Katalyse, Akademie Verlag, Berlin 1975.
18. G. Luft and H.A. Herbertz, Chem.-Ing. Tech. 41, No. 11 (1969) 667.
19. B. Fechner, Doctor Thesis, University of Darmstadt 1995.
20. J. Szabo, G. Luft and R. Steiner, Chem.-Ing.-Tech. 41, No. 18 (1969) 1007.
21. H.J. Borchardt and F. Daniels, J. Am. Chem. Soc. 79 (1957) 41.
22. W. Kessler, G. Luft and W. ZeiB, Ber. Bunsenges. Phys. Chem. 101, No. 4 (1997) 698.
23. G. Luft, P.-Ch. Lim and M. Yokawa, Makromol. Chem. 184 (1983) 207.
24. P.-Ch. Lim and G. Luft, Makromol. Chem. 184 (1983) 849.
25. D.W. Marquardt, J. Soc. Ind. Appl. Math. 11 (1963) 431.
26. O. Schermuly and G. Luft, Ger. Chem. Eng. 1 (1978) 222.