(5) Heerden, C. van, Conference on Fluidization Technology, Soc. Chem. Ind. (London), June 1952. (6) Heerden, C. van, Nobel, A. P. P., and Krevelen, D. W. van, Chem. Eng. Sci., 1, 51 (1951).
(7) Heerden, C. van, Nobel, A. P. P., and Krevelen, D. W. van, (8) (9)
(10) (11)
General Discussion on Heat Transfer, Inst. Mech. Engrs. (London) and Am. Soc. Meoh. Engrs. (1951). Jolley, L. J., Fuel, 28, 114 (1949). Leva, M., General Discussion on Heat Transfer, Inst. Mech. Engrs. (London), and Am. Soc. Mech. Engrs. (1951). Leva, M., and Grummer, M., Chem. Eng. Progr., 48,307 (1952). Leva, M., Grummer, M., and Weintraub, M., Ibid., 45, 563 (1949).
(12) Levenspiel, O., and Walton, J. S.,Chem. Eng. News, 27, 1999 (1949). (13) Logwinuk, K., thesis, Cae Institute of Technology, 1948. (14) Mickley, H. S., and Trilling, C. A,, IXD.EKG.CHEM.,41, 1135 (1949). (15) Philips Tech. Ret., 11, 91 (1949). (16) Vreedenberg, H . A , , Conference on Fluidization Technology, Soc. Chem. Ind. (London), June 1952. (17) Vreedenberg, H. h., General Discussion on Heat Transfer, Inst. Mech. Engrs. (London) and Am. Soc. Mech. Engrs. (1951). RECEIVED f o r review January 3, 1953.
ACCEPTEDMarch 30, 1953.
Autothermic Processes Properties and Reactor Design C. VAN HEERDEN STAATSMIJNEN IN LIMBURG, GELEEN, THE NETHERLANDS
1 n autothermic processes t h e temperature level a t which the reaction proceeds is maintained by t h e heat of reaction alone, It is shown in this paperthat these processesare characterized by a simple diagram consisting of two curves which give the production and the consumption of heat as functions of some reference temperature. From this diagram t h e typical need of a n ignition by external heating being t h e properties of autothermic reactions-the most peculiar o n e - c a n easily be understood. This type of diagram i s indispensable for t h e calculation of industrial converters i n which autothermic processes are carried out. As a n illustration of t h e principles involved, the temperature and t h e concentration distributions i n a n ammonia synthesis converter are calculated.
W
HEN the temperature level a t which an exothermic chemi-
cal reaction proceeds is above room temperature, this level is often maintained by the heat of reaction alone. The combustion of fuels belongs to this group of autothermic processes. Besides, it is common practice in chemical industry to make a conversion proceed autothermically if possible, in order t o avoid expensive heating by external means. Well-known examples are the Haber-Bosch ammonia synthesis and the shift-reaction of carbon monoxide with steam. I n these processes a steady state must be established a t which the heat consumption is balanced by the heat production. As the rate of reaction generally varies very rapidly with temperature, the fractional conversion will change from near zero to near unity within a relatively small temperature region. However, the heat consumed-which mainly consists of the sensible heat of the reaction products leaving the system and of the heat losses to the surroundings-will change approximately linearly with temperature. From the general behavior of heat consumption and heat production, the peculiar properties of autothermic properties can be easily understood. The most characteristic feature is the necessity of an ignition by external heating before a steady state a t which the reaction processes can be established. r h e principles discussed in this paper are generally applicable t o all autothermic processes and are applied to a practical example, the ammonia synthesis column. GENERAL D I A G R A M
OF A U T O T H E R M I C PROCESSES
The simplest diagram of an autothermic process is shown in Figure 1. It is assumed that the reaction proceeds isothermally a t a temperature, T,, that the reaction products leave the reactor a t the same temperature, T,, and that the reactants enter a t a temperature, To, normally room temperature. For the moment, it is assumed that t,he reaction also proceeds adiabatically, so t h a t the heat consumed consists solely of the sensible heat of the
1242
reaction product. To simplify the discussion, all heat quantities will be expressed per gram mole of a suitably chosen reaction component. At a given residence time of the reactants in the reactor, the heat, QI,produced by the reaction will depend on the reaction temperature in a way schematically represented by curve a in Figure 2. Starting a t low temperatures the reaction will a t first be so slow that Q, is practically zero. ilt a certain temperature level the reaction rate starts rising rapidly with temperature; as a result, the heat produced will arrive a t a constant maximum value within a relatively small temperature interval. When the value of Qr remains constant the conversion is complete. The heat consumption is given by the relation &E
=
c(T,
- To)
where c is the heat capacity of the reaction products per gram mole of one of the reaction components. If it is assumed that c is independent of temperature and of the degree of conversion, Qcis represented by a straight line, b, which intersects the temperature axis a t Ti. At points of intersection 0, I , and S of curves a and b the production and consumption of heat are e,qual. A t point I this equilibrium is unstable. With a small rise in temperature the heat production increases more rapidly than the heat consumption and the temperaturewill continue to rise untila stableequilibrium a t ASis reached. In the opposite case of a small temperature drop a t I the temperature will continue to fall until it reaches the value Toa t 0. Point I corresponds to a state of ignition and Ti is the ignition temperature. The equilibrium a t 0 correspond with the stable nonreacting state before ignition, while at the temperature T , the stationary reacting state is established after ignition by external heating t o temperature T,. Of course it is possible for curves a and b to have no points of
INDUSTRIAL A N D ENGINEERING CHEMISTRY
Vol. 4.5, No. 6
intersection other than 0 (b' in Figure 2). This will occur when the reaction is too slow or the heat of reaction too small. In that case autothermic reaction is not possible. One important property of autothermic reactions can be deduced from Figure 2. Because of the characteristic S-shape of the heat production curve, the stationary state, S, will lie in the region of almost complete conversion. This conclusion is in full agreement with practical experience. For instance, when burning solid fuels the oxygen content of the exit gases is always very low, and in flames either the fuel or the oxygen is completely consumed, even a t the limits of flammability. Stability of Autoreactants
n
thermic Practical Reactions. examples of autothermic processes are normally much more complicated than the
7 1Reo;;or
} -
U
Figure 1. Simple Diagram Autotherm io Processes
of
schematic approach outlined. Nevertheless, it is possible to draw some general conclusions from the diagram of Figure 2 about the way in which the stable state, represented by S, is effected by a variation in operating conditions, The most important variable in this respect is the feed rate of the reactants, If the feed rate is increased, the fractional conversion a t any value of the reaction temperature will decrease. This means that curve a will shift to the right, while the constant value, approached a t high temperature, will remain the same (see Figure 3). I n the absence of heat losses to the surroundings, the slope of the straight line is independent of the feed rate. Thus, a t a continued increase of the feed rate, curve a will ultimately arrive at position a'. A t a still further increase a stable equilibrium no longer exists and the reaction cuts out.
and b" and a t a further decrease of the feed rate, the reaction will be quenched. Thus it can be understood t h a t for any autothermic process there exist an upper and a lower limit for the feed rate. Beyond these limits the reaction cannot proceed autothermically. Because of the rapid change of reaction rate with temperature, these limits are generally very wide apart, while in the whole region between these limits the conversion will be almost complete. A typical example of quenching due to a too low value of the feed rate is the extinction of a stove when the air is throttled down too much. The blowoff limit of flames is an example of the upper limit of feed rate. Dilution of the reactants with-some inert component has an effect similar to the reduction of the feed rate. If it is remembered that all heat quantities in Figures 2 and 3 are taken per gram mole of one of the reactants, it is clear that the slope of the heat consumption line increases proportionally with the degree of dilution. If it is assumed that the reaction is of the first order, the acurve will not change for a given value of the residence time. Thus it follows that there is an upper limit of dilution above which the reaction is quenched. Examples of this phenomenon are the limits of flammability of a mixture of air and, say, methane. Here too, the conversion between the limits is almost completein poor flames the conversion of the methane fuel, in rich flames that of the oxygen. f
//
i
Tr
Figure 3.
Figure 2. Schematic Diagram of Heat Production and Heat Consumption as Functions of Reaction Tern peratu re
Consequently, there exists an upper limit for the feed rate of autothermic processes above which the reaction is blown out. If, on the other hand, the feed rate is reduced, curve a shifts to the left and the conclusion might be drawn that a t a sufficiently low feed rate every exothermic reaction can be carried out autothermically. Here, however, heat losses t o the surroundings must be taken into account and a t very low values of the feed rate these losses will outweigh the sensible heat of the reaction products. In this region the slope of the straight line will increase and, because of the rapid change of reaction rate with temperature, this effect will ultimately outweigh the shift of curve a to the left. The curves will finally arrive a t the positions a" June 1953
Influence of Variation i n Feed Rate
Recovery of Heat. I n the preceding sections it was pointed out that the main heat consumption is the sensible heat of the reaction p,roducts leaving the reactor at the reaction temperature. It is possible to recover an important part of this heat by heat exchange between the hot outlet and the cold inlet. It can easily be shown that the amount of heat recovered in any type of heat exchanger is approximately proportional to the temperature difference between flows entering the heat exchanger. Thus, the application of a heat exchanger between inlet and outlet of an autothermic process corresponds to a decrease in the slope of the straight line, b, in the diagrams. Heat recovery may be applied t o carry out a process autothermically, if this is impossible without recovery of heat, and to heighten the stationary reaction temperature, T,. To carry out exothermic catalytic processes autothermically, recovery of heat is very often necessary. The ammonia synthesis and t h e shift-reaction of carbon monoxide with steam belong to this group. An example, where preheating of the feed with the outlet gases is only applied to heighten the reaction temperature, is the blast furnace. Reactions Limited by Equilibrium Conditions. For reactions in which a t the temperatures involved the chemical equilibrium
INDUSTRIAL AND ENGINETRING CHEMISTRY
1243
between reactants and reaction products limits the degree of conversion attainable, the a-curve in the diagram will have a different shape. As autothermic reactions are always exothermic, at increasing temperature the equilibrium composition will shift t o the side of the reactants. Consequently, a t high temperatures, where the conversion is almost complete, the production of heat will decrease and the a-curve will be of the shape shown in Figure 4. Point S, corresponding to the stationary reaction state, lies on the falling part of the curve and a reduction of the slope of the heat consumption line, b, not only increases the reaction tempera-
@?\.g ~~
Y
c
.O n
55 :g CL"
c 0
*
7z
t
L TO
Figure 4.
which means that the heat content of the outlet gases above the inlet tempkrature equals the heat of reaction produced in the column. This is approximately equivalent to the statement that the temperature difference between inlet and outlet gases is proportional to the fractional conversion. To facilitate the calculations, the following assumptions are made: 1. There is no temperature difference between the catalyst granules and the interstitial gas. 2. The temperature in the catalyst bed is constant in any cross section of the converter. 3. As a consequence of the second assumption, the rate of heat exchange between the ascending and descending currents of gas is represented by an over-all heat transfer coefficient, U , which is constant throughout the converter. In the following calculations this rate of heat exchange will be expressed in terms C of the height of transfer unit, H = 7 ,where C is the feed rate Ila expressed as heat capacity per unit time and a is the total heat exchanging surface per unit of length of the converter. 4. The heat capacity of the gas is independent of temperature and conversion; the temperature rise, 7, of the reacting gas corresponding to the adiabatic formation of 1% of ammonia is constant and equals 15" C. 5. The reaction velocity satisfies the well-known relation of Temkin ( 2 ) .
Schematic Diagram of Reaction Limited by Equilibrium Conditions
ture but also decreases the reaction yield. Thus, for those processes where this slope can be varied-for instance, by a variation of the degree of heat exchange between outlet and inlet-it is advantageous to choose the conditions such that the slope of b is as great as possible (b' in Figure 4). Under these conditions, at the limit of stability the conversion will reach its maximum. This important point will be discussed more fully in the next section, which deals with the calculation of ammonia synthesis converters.
catalyst bed heat
tubes
exchanger
PRACTICAL APPLICATION
Calculation of Ammonia Synthesis Converter. The general principles of the preceding sections may be applied to the calculation of the temperature and concentration distribution in an ammonia synthesis converter. The main purpose of this practical example is to illustrate how the properties of this much more complicated system can be completely understood from a modified diagram like Figure 2 , and how the use of such a diagram is indispensable in technical calculations. To simplify this calculation, it is based on the most simple construction of a synthesis converter, consisting of a cylindrical catalyst vessel with a large number of countercurrent heat exchanger tubes inside (see Figure 5 ) . The cold nitrogen-hydrogen mixture enters the column a t the bottom, Aon~supward through the heat exchanger tubes and downward through the catalyst bed, and leaves a t the bottom. The degree of heat exchange can be varied by by-passing the upn-ard flow in the heat exchanger tubes through a central tube of large diameter. With some modifications the following calculation method can also be applied to more complicated constructions, including the use of a separate heat exchanger and the application of cold gas injection somewhere in the top of the column. The characteristic temperature distribution in a converter of the type shown in Figure 5 is given in Figure 6. Following the path of the gas stream, the temperature rises gradually inside the heat exchanger tubes. I n the upper part of the catalyst bed vihere the heat of reaction exceeds the heat removed by the cooling tubes, the temperature rises and passes through a maximum value. I n the lower part of the converter the reaction rate decreases gradually, so that the bottom section mainly Serves as a heat exchanger. The process as a whole is assumed to be adiabatic,
1244
inlet Figure 5.
Ammonia Synthesis Converter
The original relation of Temkin reads:
where I< is the equilibrium constant and IC the temperature-dependent rate constant. After some modifications, substituting a nitrogen to hydrogen ratio of 1 to 3 and the temperature dependency of k , this relation can be written as: dz
2.4 X 1012 = F(2,
T)=
0
+ z)(l
P-3/*(1
INDUSTRIAL AND ENGINEERING CHEMISTRY
20,000 -_ _ X
- z)'&
T
Vol. 45, No. 6
This relation was used in the numerical calculations in this paper, where v is the linear gas velocity in the empty vessel and P is the total pressure. 6. N o heat is lost t o the surroundings. 7. The numerical data are as follows: 12 Height of converter, L, meters Diameter of converter, meters 0.7 Pressure, atmospheres 300 1to3 N Zto Hz ratio % 1.5 Inlet "3, 50 Inlet temperature, O C. Linear gas velocity in empty vessel, meters per second 0.16 3 Height of transfer unit (H), meters The value of the height of the transfer unit, H,was estimated by a rough calculation, using literature data on heat transfer rate in packed columns. The value of 3 meters must be considered as the minimum value obtained if the column is operated a t its maximum heat exchanging capacity. By-passing of the heat exr
of maximum reaction rate, T,, the temperature, To.lwhere the reaction rate is 0.1 times the maximum rate, and
2
T,,O C.
0.02 0.05 0.10 0.15 0.20
950 760 650 590 540
T,,
O
c.
To.l,O C.
840 680 580 530 490
I
-L
i
Distance from top, x
L
Figure 6. Temperature Distribution in Synthesis Converter with Internal Heat Exchange
T , for
500 420 370 340 320
Mole Fraction/M. 1.6 0.23 0 048 0 017 0.007
In the last column the maximum reaction rate a t T , is given, Because of the complexity, of the function F ( z , Tz) an analytical solution of the Differential Equations 1, 2, and 3 is impossible. A numerical stepwise integration is the usual line of attack for this kind of problem. However, besides the Differential Equations 1, 2, and 3, the solution must satisfy the following boundary conditions (see Figure 6):
Tl = To when x =
\Y
- at
dx
some values of z :
T2 = T I when x = 0
Outlet
dz
L
(4) (5)
When solving such a problem by the method of stepwise integration, one starts a t one end of the column with given values of T I , TB,and z and integrates step by step until the other end is reached. For the present problem this procedure is not possible, as the two boundary conditions (Equations 4 and 5) are given a t different ends of the column. Thus for x = L,z or Tz is unknown. For 2 = 0, temperature Tt ( T I = T 2 )a t the top of the column is unknown. This difficulty was solved in the following way: Starting from an arbitrary value of the top temperature, T t , at x = 0, the stepwise integration was carried out until x = L. It is clear that this arbitrary choice will not lead t o the correct value TOof T I (2)at x = L. The integration is repeated for a number of different values of T,,and in this way the temperature difference between the ends of the heat exchanger tubes, T 2 T I ( L ) = 9,as found by stepwise integration, is calculated as a function of Tt. The relation found is represented by curve a in Figure 7 and the general shape of the curve can be understood easily.
-
changing tubes increases the value of the height of the transfer unit. The results of these calculations will prove that the values used are reasonable. Starting from the above assumptions, the concentration and temperature distribution 1000. must satisfy the following differential equations: 800.
9 6oo. I
I 400.
(3)
F(z, T z ) is a modified formulation of the re200. action velocity, obtained from the original rate equation of Temkin. After substitution of the given conditions, F was calculated as a function of T2 for a number of z values and plotted as a set of curves with z as the parameter. The nuFigure 7. mericalvaluesneeded ware found by interpolation. For any value of z there is some corresponding temperature, T,,a t which the gas mixture is in chemical equilibrium. Far below this equilibrium temperature the reaction rate increases rapidly with the temperature. Approaching T , the function, F , passes through a maximum to ' 'become zero at T , and negative above T,. The following tabulation, prepared for orientation purposes, gives values of the equilibrium temperature, T,,the temperature
.
June 1953
Tt P U
Relation between 9 and Tt for Varying Height of Transfer Unit
Because a t low values of Tt the reaction is slow, the temperature difference, TZ T I ,remains small, and according to Equation 1 the total change, 9, in T1 during the integration will also be small. For Tt = T,, the equilibrium temperature for the inlet gas, the reaction rate is zero and T2 and T I are constant throughout the integration. For a nitrogen t o hydrogen ratio of 1 to 3 and an
-
INDUSTRIAL AND ENGINEERING CHEMISTRY
1245
inlet percentage of 1.5% of ammonia, the equilibrium temperature equals 1080' C. Thus, in all calculated curves like t h a t in Figure 7, 0 is zero for Tg = 1080" C. In the region between low temperature where the reaction rate is practically zero and the equilibrium temperature where the reaction rate becomes zero again, the 0, Tt curves will show a maximum. For those points of the $-curve which satisfy the condition 0 = Tt - Tothe corresponding solutions satisfy the boundary condition of Equation 5 . This means t h a t curve a in Figure 7 must be intersected by straight line b. The diagram of Figure 7 is then equivalent to t h e diagram of Figure 4. Point of intersection 0 corresponds with the nonreacting state, 1 with the state of ignition, and S with the steady reaction state.
9 M ~
stability is reached and at still higher values autothermic reaction is no longer possible. The temperature and concentration distributions a t H = 7.5 meters are given in Figure 9, and from a comparison with Figure 8 it is clear t h a t it is much better to operate the converter a t the limit of stability. The temperature level is then much lower and the outlet ammonia concentration higher. The decrease in stability is demonstrated by the fact t h a t the region PQ,where the reaction rate has a negative temperature coefficient, is much smaller in Figure 9 than in Figure 8. These results are in complete agreement with practical experience. In practice, synthesis converters are always operated in such a way t h a t some reference temperature-e.g., the temperature at the top of the column-is kept as low as possible, continuously taking care t h a t the reaction does not cut out. The influence of a decreasing activity of the catalyst is illustrated in Figure 10. Here the a-curves are given for different values of the catalyst activity, accounted for in activity coefficient, f, by which the reaction rate-F(z, T2)in Equation 3-is multiplied. The curves in Figure 10 correspond with f = I, 0.5, 0.25, and 0.15 and were calculated with H = 3 meters. At this value of H the limit of stability is reached for f = 0.15-that is, for a catalyst activity about seven times lower than the original 900
700
OJ 0
2
4
6
e
)
o
1
2
Distonce from top (n-)
Figure 8. Temperature and Concentration Distribution Corresponding w i t h Stationary State, S, i n Figure 7 Reaction rate has negative temperature coefficient between P and Q Equilibrlum temperature T. corresponds with local ammonia cAnceAtration, z
If 0 is multiplied by the heat capacity of the feed the equivalence between the diagram of Figure 7 and Figure 4 becomes more direct. Yet the a-curve in Figure 7 does not correspond with the heat of reaction, as it does in Figure 4, but with the heat transferred from the reaction space to the heat exchanging tubes. Generally, it can be said that any specified problem will have its own diagram, depending on the way in which the integration of the differential equations is carried out. Such a diagram will always consist of a n a-curve, giving some heat quantity determined by the reaction rate function and a b-curve, giving some heat quantity determined by the boundary conditions, both as functions of some reference temperature. In Figure 8 the complete temperature and concentration distribution is given corresponding to the steady state, S, in Figure 7. Moreover, the variation of the equilibrium temperature throughout the converter is given. From this figure it is clear that the converter owes its stability to the fact t h a t for an important part the reaction proceeds in the direct vicinity of chemical equilibrium, where (between points P and &) the reaction rate has a negative temperature coefficient. The complete calculation was repeated for higher values of the H.T.U.-that is, for lower values of the heat exchanging capacity of the converter. The resulting a-curves for H 6 and 7.5 meters are also given in Figure 7. 4 t H = 7.5 meters the limit of
-
1246
01 0
2
4 6 8 1 Distance from top (m)
0
1
I
2
Figure 9. Temperature and Concentration Distribution Corresponding w i t h Stationary State a t L i m i t of Stability, S', i n Figure 7 Reaction rate has negative temperature coefficient between P and Q Equilibrium temperature, Te, corresponds w i t h local ammonia concentration, z
value. Comparing the limit of stability a t f = 1 with t h a t a t J = 0.15, it was found that the top temperature increases from 400' to 570" C., the H.T.U. value decreases from 7.5 meters to 3 meters, and the exit ammonia concentration decreases from 21.5 to 12%. These results are also in agreement with practice, where because of the decreasing catalyst activity, the heat exchange capacity must be enlarged gradually. This is accompanied by a continual rise of the temperature level and a decrease in the degree of conversion. Finally, the results obtained with different catalyst activities can be used aB an orientation about the influence of a change in the feed rate. If it is assumed, t h a t the H.T.V. is independent of the gas velocity (because H =
c;
-,Ua C
INDUSTRIAL AND ENGINEERING CHEMISTRY
N
V , and U
-
d.8,
this aa-
Vol. 45, No. 6
sumption is approximately correct), it can be deduced from the Differential Equations 1, 2, and 3 t h a t an increase in feed rate is equivalent with a proportional decrease in catalyst activity.
fixed beds. A paper on this subject will be published in the near future, the contents of which may be found in the thesis of van Loon (1). NOMENCLATURE
1000.
1
a
= heat exchanging surface per unit of
C
= feed rate expressed as heat capacity
f
= activity coefficient, dimensionless
length of column
f7 I
per unit of time
800
I
H
=
K
""I
c Ua
= height of transfer unit (H.T.U.), m.
= eauilibrium constant of the reaction
\\\
1 N~ + 3 H* + N H ~
2 reaction rate constant height of column, m. total pressure, atm. quantity of heat produced, respectively, consumed per gram mole of one of the reactants Tt 1%) U = over-all heat transfer coefficient Figure 10. Relation between 6 and Tt for Different Values of Catalyst 5 = distance from the top of the colActivity umn, m. = linear gas velocity in empty catalyst u vessel, m./sec. z = mole fraction of ammonia, dimensionless Thus the four curves of Figure 10 also apply by approximation to 0 16 = F(z, T ) = modified reaction rate function, m.-l the conditions f = 1 and v = - meters per second. Again in f T = temperature, O C. agreement with practice, it follows that a n increase in the gas To = inlet temperature of reactants velocity necessitates a n increase in the heat exchanging capacity TI = temperature in heat exchanger tubes T z = temperature in catalyst bed and, consequently, operation at a higher temperature level. T I = equilibrium temperature T , = temperature of maximum reaction rate CONCLUSION To.r = temperature (
