I
W. J. THOMAS and STANISLAW PORTALSKI Battersea College of Technology, London, England
ynamics in Methanol Synthesis
s
OMETIMES optimum temperature and pressure for a reaction at elevated temperature and pressure can be advantageously predicted thermodynamically. Such calculations yield not only valuable information on the optimum operational conditions but they may also indicate sensitivity of the process to changes in such conditions. An example is the synthesis of methanol from hydrogen and carbon monoxide.
AC, = Aa
+ A b . T + A-y, T 2 +. . . . + m . a +. ~ . . .] + b.as +....I [ u . ~ A
or,
2200-1470 p.s.i. used History of process, kinetics of reaction, and plant details; temp. and press., 300-375' C. and 270-350 atm. ,300-400' C. a s optimum temp. Methanol plants in Germany
(1)
(2) (3) (6)
Little literature has appeared on optimum conditions for the methanol synthesis and the diversity of opinion is surprising when a t least a dozen plants are operating in the United States. The method described here for thermodynamically establishing optimum conditions for methanol synthesis is based on determination of pressure and temperature effects on heat of reaction and chemical equilibrium.
+ [AcY.T + '/z
A general equation for chemical reaction is aA 6B = 1L mM (1)
+ +
+
+
Because heat capacity is an extensive property, its change in the system is
+ m ( C p ) ~ . . . .I + + . . . .I
[u(C~)A
b(Cp)B
ACp d T
(3)
ST:
where AH" is the heat of reaction a t To. Variation of heat capacity may be
C,
= a
' / Z A P .T 2
+ . . . . IF,
+ '/3Ay. T 3 - I o
(7)
where I,, is the temperature integration constant. If AH is known at one temperature, the constant of integration can be combined with the constant value of AH, to give a constant (AHo - I o ) . which can then be evaluated. In this way AH at any value of T may be obtained from Equation 7. These derived equations can be applied to the methanol reaction,
+ 2Hz(,)
GO@)
(8)
CH30H(,)
The heat of reaction at standard temperature and pressure is first calculated from published values (4) of heats of formation for the components : A H z ~ Ofor CH,OH(,, = -48100 cal.; for COe), -26400 cal.; and for H2(K),0 (by
convention).
+ BT + y T z + 6T3+
+ 24.274 X
CpcH,oH= 4.394
4.81
(4)
where a,fi, y, and 6 are empirical constants. C, for the reaction may then be expressed as a function of temperature:
10-77-2
+
x
10-7
T~
+ 1.836 X 10-3T + 2.80
x
-2
C3cHtOH
-15.842
+
- C~co 22.84 X 10-3 T 80.97 X 10-'T2
and Aa
= -15.842; AB = 22.84 X AT = -80.97 X 10-7
Substituting these values in Equation 7. AHT = ( A H o 11.42 X
Heat of Reaction vs. Temperature (Constant press., 1 atm.) AHT,
Temp., OX. 298.2 300 400 500 600 700 800
900 1000 1100
I200
Cal./ G.-Mole
Io) - 15.842 T
AHT? Gal./
Temp.,
G.-Mole
C. 25 100 200 300 400
O
-21700 -21710 -22600 - 23320 - 23900 - 24343 - 24670 - 24890 - 25040 - 25130 -25150
-21700
-22390 - 23200 - 23750 - 24230 - 24570 - 24870 - 25030 -25100 - 25145
500 600 700 800 900
Derivation of Equation
A thermodynamic equation of state (7) expressed as
(E)
= V
(10)
-T(Z)
may be applied to each component in the methanol reaction, and an algebraic summation can be made, followed by integration. Thus, AH = AH,
+
where
spt
AV.dP -
+
TiM
-
VA
-
VB
(12)
and where the standard state of each component is the pure substance a t Po. Equation 10 indicates that if ideal conditions apply, then pressure will not affect heat of reaction because
10-7~2
These values were obtained from Dodge (2) and Ewe11 ( 3 ) . Thus Acp =
A H , = -17920 - 15.84 T 11.42 X 10-3 T 2- 26.99 X 10-7 T3 ( 9 )
A v = VL
10-377 - 68.55 x CpH,= 6.947 - 0.1995 X lO-3T
(2)
where C, refers to molar heat capacities of the components. By considering the change in heat content of the system, the Kirchhoff equation (8) can be obtained. which on general integration gives
+
A?. T 3
+ Aa.T +
Cpco = 6.342
( c p ) ~
AHT = AHo
AHT = AH,
AB. T 2 f
Thus for the methanol reaction at 25' C. and 1 atm., AHz6' = -21700 cal. AC, for the reaction is determined:
Temperature and Heat of Reaction
ACp =
(6)
Similar equations can be obtained for the terms, a, p, y, 6. When Equation 5 is substituted in Equation 3 and integrated, A H , = AH,
the last equation to determine the conThus, stant (AH,, - I,) as -17920.
+
Aa = [ I . ~ L
'/3
Literature on Methanol Plants Subject Ref. Temp. and press., 245-400' C. and
(5)
where
+
T2 - 26.99 X 10-7T3
At 25' C . and 1 atm.. AHo = -21700 cal. This value can be substituted in
so that
At elevated pressures, the conditions will not be ideal, and, therefore, pressure will affect heat of reaction. T o determine this effect, an equation of state must be used in conjunction with Equation 11. For this study, the equation of Berthelot has been chosen. This equation explicit in P is P.V=R.T
9 P.Tc
[1 + -128 P,.T ~
(1 VOL. 5 0 , NO. 6
-6
x
g)]. . .
JUNE 1 9 5 8
(13)
967
where Po, T,, and V , are the critical pressure, temperature, and molar volume, respectively. NOWA V = VCH,OB - Vco - 2. V s , (14) Combining Equations 13 and 14, AV =
AHr,," is a constant corresponding to 1 atm. of pressure, but having a different value a t each temperature. This value is calculated at 1 atm. pressure by Equation 9. AHr,p0's are calculated at different temperatures in this way and when substituted in Equation 19, give a series of equations: AHrp = A H T , ~ ' - mP
A general thermodynamic equation using van der Waal's equation has also been derived, and although it has not been used for methanol calculations, it should lead to the same result as the Berthelot equation ( 7 7). Values of AH,, are given in Figure 1 which shows clearly a range of stable optimum working conditions.
Using Equation 15,
S
(20)
AV.dP =
Combining Equation 22 and 25, AFo = AH, - ACZ.T I n . T 2 A @ . T2 - -61 ' A y . T3 f Z.T (26)
1
where Z = -R.C. Both Z and C are integration constants and their values are determined from one known value of K , or F." If the specific heats of all the reacting substances are known accurately as a function of temperature, then one value of K , will suffice to determine C and Z. Thus, K , can be expressed as a function of temperature. For the methanol reaction, neglecting side reactions, K , and T are related by [Standard entropies in gram-tal./' K.
Equilibrium Constant
( 5 , 6 , 9,
Equilibrium constant as a function of temperature can be calculated by AF' = AH' - T.AS" (21 1 AF' = -RT.ln K , (22) AH
d_l n_K= , -
and
By combining Equations 11, 16, and 17;
(23) dT RT2 where AF" is free energy change of reaction; AS, entropy change of reaction; AH, heat capacity change in the reaction; T , absolute temperature; and K,, equilibrium constant in terms of activities. Superscript zeros are change in free energy, heat content, and entropy when all reactants and products are in a standard state of unit activity. Substituting Equation 7 in Equation 23.
The critical constants are given (5, 8) and by substituting the values in Equation 18, 0.5411 P -
3255000
7 P (19)
In.K, =
AH^ f -RT
+
A c z .T ~ ~
___
R
1
-
-33,003
-32,000
-31.000
CO
7o)l
-
47.32 EU; HB- 21.23 E L
ASo
Thus
=
-53.12 EL-
Using Equation 21 at 298.2 AFO = AH'
" K.,
- T.AS' = -21700 f (298.2 X 53.12) = -5870
From the heat capacity equations and previously calculated AH at 25' C.: AH,
=
-17920:
22.84 X
Aoc = -1584: A y = -80.97
A@ =
X IO-'
Substituting these values, and the calculated value of AF in Equation 26, I = -46.59. The final equation for K , as a function of temperature then is
0.002499 T - 2.953 X 10-7 ~2 10.20 ( 2 7 ) Substituting values for T in Equation 27, the results in Table I are obtained. Because the reaction is exothermic, the equilibrium becomes more unfavorable for methanol as the temperature increases. The considerations described apply to the reacting mixture at 1 atm. pressure. Although the equilibrium constant K , is a function of temperature only, the equilibrium state of the system is strongly dependent on pressure.
7
Pressure Equilibrium Constants
Considering Equation 1, the equilibrium constant in terms of activities is
-30,000
a.
x'
- 56.66 EU;
+
Integrating Equation 24,
AHT,= ~ AHT.~' -
CHSOH
-29,000
-28,000
If the standard state of unit activity is the state of unit fugacity, then activity of each reacting component can be replaced by its fugacity, so that the equation of the equilibrium constant becomes
-27,000
-26,000
-25.003
-24,000
1
I PRESSURE, ATM
1W
Figure
968
1.
Effect
of
pressure on m e t h a n o l h e a t
INDUSTRIAL AND ENGINEERING CHEMISTRY
I
3w
403
of
reaction
where f is fugacity. Nowf/@ = y where y is fugacity coefficient, and p is partial
METHANOL S Y N T H E S I S pressure. Thus substituting for the fugacities in Equation 29:
(30)
Data for determining fugacities of individual gases in mixtures are generally not available; an approximation is: f nf', where f is fugacity of the component in the gaseous mixture; f', fugacity of the pure component at the total pressure; and n, mole fraction of the component in the mixture. If y' = f'/P, then the expression for the equilibrium constant becomes
Table 1.
ToC. K. ToC. Kll
+ .. ) -
+VI
(a
+ b + .. )
Kfl =
Kp'
X
Kyf
These values were used for hydrogen. Knowing T and P i n each case, the values of y t were found from Newton's universal graphs. In this way, the effect of pressure on K.,! at any temperature can be determined. K y is a function of temperature only, and independent of pressure. However, the equilibrium state of a system is strongly dependent on pressure as is reflected in the values of K,.. As K,t/Kyi
700
T,OC. 100
P. .Atmos.
Pr
100 200 300 400 500 600
1.01 2*.02 3.03 4.04 5.05 6.06
co
Hz
0.73
9.05 Y
700
7.07
800 900 1000
8.08 9.09 10.10
...,
.. .. .. .... .... ..
1.01 2.02 3.03 4.04 5.05 6.06
7.07
Y
4.81 9.62 14.43 19.24 24.05 28.86 33.67 38.84 43.29 48.10
1.06 1.11 1.17 1.24 1.32 1.36 1.44 1.53 1.60 1.70
4.81 9.62 14.43 19.24 24.05 28.86 33.67 38.48 43.29 48.10
Y
pr
11.49
0.52 0.34 0.26 0.22 0.19 0.18 0.17 0.16 0.16 0.17
8.08 9.09 10.10
2.78
p,
Tr
0.92
200 100 200 300 400 500 600 700 800 900 1000
2.89 5.78 8.67 11.56 14.45 17.34 20.23 23.12 26.01 28.90
..
.. KY KP ( X 10-2) ( X 10-2)
3.52
1.05 1.08 1.13 1.18 1.23 1.29 1.35 1.40 1.46 1.54
1.01 2.02 3.03 4.04 5.05 6.06
2.89 5.78 8.67 11.56 14.45 17.34 20.23 23.12 26.01 28.90
1.04 1.09 1.15 1.29 1.38 1.46 1.58
....
..
7.07 8.08 9-09 10.10
4.81 9.62 14.43 19.24 24.05 28.86 33.67 38.48 43.29 48.10
100 200 300 400 500 600 700 800 900 1000
500
1.01 2.02 3.03 4.04
1.04 1.07 1.11 1.15 1.20 1.24 1.29 1.34 1.38 1.44
5.05 6.06
7.07 8.08 9.09 10.10
0.77 0.68 0.62 0.58 0.56 0.55 0.55 0.55 0.56
1.01 2.02 3.03 4.04 5.05 6.06
700
7.07
800 900 1000
8.08 9.09 10.10
5.01
4.81 9.62 14.43 19.24 24.05 28.86 33.67 38.48 43.29 48.10
1.04 1.07 1.10 1.14 1.17 1.22 1.25 1.29 1.33 1.38
T,
1.50 100 200 300 400 500 600
1.04 1.08 1.13 1.20 1.27 1.34 1.44 1.52 1.62 1.72
2.89 5.78 8.67 11.56 14.45 17.34 20.23 23.12 26.01 28.90
16.35 0.88
18.80 0.92 0.86 0.80 0.76 0.74 0.72 0.71 0.71 0.72 0.73
4.81 9.62 14.43 19.24 24.05 28.86 33.67 38.48 43-29 48.10
17.7 13.0 9.75 7.83 6.40
5.17
... ...
( x 10-2)
4.26
Tr 1.31
400
45.3 29.2
KY
13.92 0.76 0.60 0.47 0.40 0.37 0.34 0.33 0.34 0.34 0.35
2.89 5.78 8.67 11.56 14.45 17.34 20.23 23.12 26.01 28.90
1.04 1.08 1.12 1.19 1.26 1.34 1.40 1.48
1.57 1.65
5.75 5.78 8.67 11.56 14.45 17.34 20.23 23.12 26.01 28.90
VOL. 50, NO. 6
... ...
(x
KP 10-4)
3.58 4.97 7.15 9.60 11.98 14.65 17.54 19.35 22.00 24.60
Kr ( X 10-2)
KP ( X 10-6) 1.378 1,726 2.075 2.695 3.210 5.780 4.295 4.830 5.450 6.050
78.2 62.5 50.2 40.0 33.6 28.5 25.1 22.3 19.8 17.8
KY
1.04 1.07 1.14 1.19 1.25 1.30 1.36 1.42 1.50 1.56
4.21 6.53 IO.80 14.67 19.60 24.35 29.80 36.85
67.6 48.6 33.8 25.2 20.2 16.5 13.8 12.5 11.0 9.84
( X 10-2)
5.75 1.03 1.06 1.09 1.11 1.14 1.18 1.21 1.24 1.27 1.34
KP'
KY
1.02 1.05 1.10 1.17 1.24 1.33 1.42 1.56
T, 1.12
300 100 200 300 400 500 600 700 800 900 1000
500 1.015 X
400 1.079 X 10-5 900 4.052 X IO-@
T,
(33)
At low pressures where the gases may be assumed as ideal, then KYi = 1, and Kpl = Kfl. At higher pressures, however, KPl is not constant, and to calculate it, both K f t and Kyt must be known. Newton (8) evaluated y' for a number of gases from experimental PVT data, and published a set of curves of universal activity coefficients. As these curves were based on experimental data, they provide a convenient and accurate source of fugacities and activity coefficients for gases. T o calculate the effect of pressure a t any temperature on the methanol reaction, it is necessary to evaluate the reduced temperatures and pressures. The critical values are given (4) for methanol, hydrogen, and carbon monoxide. From these values the reduced temperatures and pressures were calculated in each case for each component. Newton (8)suggested that for some gases, such as hydrogen, better universal reduced curves could be obtained if the reduced values were calculated :
300 2.42 X IO-' 800 1.112 X 10-8
3.642 X 10-8
CHsOH
(32)
and K,f shows that Equation 31 is based on an approximation. And hence K f j cannot be exactly constant combining the first and third factors of Equation 31, K,l = K,, X PAN. Thus
200 1.909 X 10-a
K,) as a Function of Pressure and Temperature
Table II.
where AN is the change in the number of moles in the reaction. AN = ( I
0 100 677300 12.92 600 1.610 X 10-7
K, ds a Function of Temperature
KP ( X 10-8)
83.5 71.5 59.0 51.9 45.5 39.8 35.7 32.5 29.8 26.1
JUNE 1 9 5 8
1.215 1.420 1,720 1.960 2.230 2.550 2.842 3.120 3.400 3.890
969
Table II.
K,!
CHIOH
T,’C.
1.70
600 Atmos.
200 300 400 500 600 700 800 900 1000
21.20
800
Y
p,
Y
p,
Y
0.96 0.92 0.88 0.86 0.84 0.83 0.83 0.83 0.84 0.85
4.81 9.62 14.43 19.24 24.05 28.86 33.67 38.48 43.29 48.10
1.02 1.05 1.08 1.11 1.14 1.17 1.20 1.24 1.28 1.32
2.89 5.78 8.67 11.56 14.45 17.34 20.23 23.12 26.01 28.90
1.01 1.07 1.11 1.16 1.21 1.26 1.32 1.37 1.45 1.54
7.07 8.08 9.09 10.10
1.01 2.02 3.03 4.04 5.0s 6.06 7.07 8.08 9.09 10.10
23.65 0.97 0.94 0.92 0.90 0.89 0.89 0.96 0.97 0.91 0.93
1.01 2.02 3.03 4.04 5.05 6.06 7.07 8.08 9.09 10.10
4.81 9.62 14.43 19.24 24.05 28.86 33.67 38.48 43.29 48.1
1.02 1.04 1.06 1.08 1.11 1.13 1.14 1.16 1.22 1.24
2.89 5.78 8.67 11.56 14.45 17.34 20.23 23.12 26.01 28.90
0.98 0.97 0.96 0.95 0.95 0.96 0.96 0.97 0.98 1.00
4.81 9.62 14.43 19.24 24.05 28.86 33.67 38.48 43.29 48.10
( X 10-2) 1.04 1.07 1.11 1.15 1.19 1.24 1.28 1.32 1.40 1.44
-
7.97
1.02 1.03 1.05 1.07 1.09 1.12 1.14 1.16 1.18 1.21
2.89 5.78 8.67 11.56 14.45 17.34 20.23 23.12 26.01 28.90
1.03 1.06 1.10 1.14 1.18 1.23 1.28 1.32 1.37 1.43
rp
900
2.29 100 200 300 400 500 600 700 800 900
1000
1.01 2.02 3.03 4.04 5.05 6.06 7.07 8.08 9.09 10.10
0.99 0.99 0.99 0.99 0.99 0.99 1.00 1.02 1.04 1.06
4.81 9.62 14.43 19.24 24.05 28.86 33.67 38.48 43.29 48.10
= K p f ,and if K y , which is a constant is determined for one case, then once the values of K y ( are determined by means of Newtons universal graphs, it will be possible to calculate K,, as a function of pressure at any temperature (Table 11). I n some cases, the value of y’ could not be obtained without introducing uncertainty, and blank spaces have been left in the table when such situations arise,
Conclusions Heat of the methanol reaction increases with pressure, and because the reaction is exothermic, then by the LeChatelier-Braun effect, temperature should be kept as low as is practical for a maximum yield of methanol. For large converters with integral heat exchange systems, hot spots may develop in the
970
1.02 1.03 1.05 1.07 1.08 1.10 1.12 1.15 1.17 1.19
2.89 5.78 8.67 11.56 14.45 17.34 20.23 23.12 26.01 28.90
89.7 81.2 73.7 67.1 60.7 56.1 51.5 48.3 43.7 42.0
1.03 1.06 1.10 1.13 1.17 1.21 1.25 1.29 1.33 1.36
K,
10
Nomenclature 7)
1.81 2.065 2.365 2.675 3.000 3.345 3.680 4.080 4.550 5.070
(X
91.5 85.8 78.7 72.7 67.8 61.8 57.7 54.7 51.4 47.8
92.4 88.0 81.6 76.6 72.5 67.6 63.8 59.8 57.2 55.0
KP
10-9
4.06 4.48 4.94 5.43 6.00 6.49 7.07 7.53 8.34 8.67
ICY
( X 10-2)
IC7 ( X lo-%)
8.72
2s 5 0
(X
a 88.8 78.0 68.0 60.1 53.6 48.1 43.7 39.4 35.3 31.7
K,
7.24
TT 26.10
2.10 100 200 3 00 400 500 600 700 800 900 1000
( X 10-21
p,
1.90
700 800 900 1000
KY
6.49
1.01 2.02 3.03 4.04 5.05 6.06
‘700 100 200 300 400 500 600
60
Hz TI
P, 100
be useful also for checking operating conditions of existing processes.
a s a Function of Pressure and Temperature (Continued)
(X
KP
10-8)
1.215 1.295 1.412 1.530 1.640 1.800 1.928 2.035 2.325 2.325
KP
( x 10-9) 4.39 4.61 4.97 5.29 5.59 6.00 6.35 6.77 7.08 7.36
catalyst mass, and become hot enough (about 400’ C.) to produce methanization, which may cause both product losses and enforce shutdown by converter runaway. Conversion increases rapidly with pressure. K , increases rapidly with pressure, but the effect is enhanced by the fact that K,! becomes smaller with increasing pressure. At about 250 atm. of pressure, heat of reaction is virtually independent of temperature. Fluctuations around a working temperature of 300’ C. would produce little effect on heat of reaction. This is significant because a process operated in this region may be more easily controlled. These calculated values agree with values quoted ( 2 ) under practical conditions. This thermodynamic study for methanol synthesis should be useful for predicting optimum values of temperature and pressure for new processes. I t can
INDUSTRIAL AND ENGINEERING CHEMISTRY
b
C CP
= moles of component .4 = moles of component B = integration constant
heat capacity, cal./grammole/’ K. F = free energy, cal./grammole AH = heat of reaction, cal./ gram-mole AHo,AH, = heat of reaction a t TooK. and T oK., respectively, cal./gram-mole 4HV0 = heat of formation a t T oIC. Io = temperature integration constant I = integration constant K,, K , K,, K,, K y = equilibrium constant, in terms of activities, fugacities, moles, partial pressures, and fugacity coefficients, respectively I, m = moles of componenr: L, and M , respectively n = niole fraction of any component in a mixture = partial pressure, arm. = pressure, atm. = critical pressure, atm. pc PT = reduced pressure, atm. R = gasconstant,cal./ OK./mole S = entropy, cal./’ K. T = absolute temperature, K. Tc = critical temperature, O I
