PREMIXIN(
NOT POSSIBLE
m > 1.0 t
P
Mn-,
t
HV*
-
m
> 1.0
0.3 + SEVERAL
UMINAR REACTOR til
> 1.0
t = 0.3-+HV*
REACTORS FOR KINETIC STUDIES
. reaction rates at very high temperatures Determining becoming increasingly important. In propellant
Isothermal Jlow reactors can be considered f o r
'
,
IS ,
temperahre fast-reaction kinetics.
De-
sign limits up to 3000" K. are gioen here
.
combustion research, temperatures generally exceed 2000' K. and often exceed 3000' K. At such temperatures the shock wave technique has been used most frequently for the determination of gas phase reaction rates (17). Isothermal flow reactors offer greater simplicity and better control over reaction variables. While isothermal systems have been used successfully at lower temperatures (5, 7, lo), they have not been reported for kinetic studies at temperatures characteristic of propellant combustion. Although few structural materials are suitable at 3000" K., several refractories have properties that would enable construction of kinetic reactors for certain specific reactions at this temperature (1, 2). However, such materials are lknited to rather simple shapes, and special fabrication techniques must be used. Isothermal flow reactor design then becomes a problem of determining a practical geometr/- that will meet heat transfer and flow requirements, and also give data in meaningful kinetic ranges. shown above, certain designs, although they seem desirable for kinetic studies, just will not work. This chart is based on nitrogen, but the transport properties of simple gases are quite similar at 3000' K., and the conclusions are approximately valid for other gases. This has been verified'by calculation. The kinetic time ranges that
e,
42
INDUSTRIAL AND ENGINEERING CHEMISTRY
.
LAMINAR MIXING
POSTMIXING
PREMIXING
P = 1 ATM lil
> 10-2
0.4 + HV'
I
P = lOJATM NOT PRACTICAL
KINETIC REQUIREMENTS
B E R N A R D S I - - :L
rn Gases must be mixed rapidly to eliminate dependence on interdiffusion o f reactants
J. W. C O N A N T
W. T. S H A T Z E R
rn Where reactants are mixed prior to elevation to high temperature, heating time must be o small. fraction of total reaction time
rn Appreciable pressure gradients cannot be tolerated might be expected at 3000" K. are indicated in Table I. In considering the practical limits outlined, it must be remembered that these results are for a temperature which is believed to represent the upper limit of workable container materials. Also, flow restrictions would be reduced for temperatures approaching 2000O K. Further, it should be borne in mind that these conclusions resulted from a reasonable estimate of the dimensions of reactors and heating devices possible with refractory substances. When the flow is turbulent, mixing requires no special apparatus. However, for laminar flow, if the gases cannot be premixed, a mechanical mixer must be incorporated into the flow system beyond the heating chamber. In cases where the g a m can be mixed prior to elevation to 3000' K., the mixing can be accomplished by any one of a number of published methods (77). Since refractory materials are usually difficult to fabricate into complex shapes, only simple geometries can be considered as postmixing devices. Advantage may be taken of the swirling motion that accompanies the emergence of a gas stream from an orifice. This swirling motion resolves to the linear. motion usually associated with laminarity at very short distances from the orifice. Effective use of this principle for rapid mixing thus requires calculation of the relations between orifice radii and the distance through which swirling motion persists. A special advantage of high temperature kinetics is that
along reaction path
rn
Reaction poth volume must be considerably smaller than volume flow rate
REACTOR DESIGN CONSIDERATIONS
rn Any set of parameters that results in a calculated gas velocity in excess tainable
of the acoustic velocity is unat-
rn Any set of parameters that results in a pressure gradient which i s large compared to the poth length is unottainable
rn Residence time in the heating chamber must be sufficient for the desired increase in gas temperature
rn Pocked heater tubes can be used to increase heat transfer rates but lead to pressure gradient restrictions
VOL 5 4
NO. 6 JUNE I962
43
homogeneous gas phase reactions can be studied for reactants that are nonvolatile at ordmary temperatures. Since this may require the diusion of a vapor into a stream of preheated carrier gas, it is necessary to calculate the diffusion barrier for various flow conditions.
Pressure gradient for gas Bow through packed tubes
is shown in Figure 2. Empirical friction values were calculated (3) and used in conjunction with the treatment of reference (79). To make these factors applicable, a transformation equation was used:
To design for turbulent flow, a Reynolds number of 4000 was used as the criterion. It was convenient to express this factor as the maximum radius which would assure turbulent flow for each m a s rate: r-x.
=
m 6280 p
Pressure gradient due to gas Bow through simple tubes was calculated by a treatment for one-dimensional compressible flow (79). Adiabatic conditions were assumed for the constant temperature regions while a combined adiabatic-isothermal treatment was used for the heating zones. The friction factors, f, needed for these calculations were obtained from reference (72). In certain cases of very low pressure where free-molecule flow was operative, the proper corrections were applied (73). These data are summarized in Figure 1, in terms of h,9, the length of tube at which the pressure falls to 90% of its entrance value.
Whm this article was prcjared, Bernard Sicgel was a Research Chemist with Aerojet Gnural Corp., Amsa, Calif. Dr. Siege1 is now Head of the Propellant Chemistry Section of the Aerodynamics and Propulsion Laboratory, Anospacc Cor!., El Segundo, Calif. W . T . Shatzer is Managn of Enginening Staff, Space Propulsion Division of Anojel Gmnal Corp., and John W . Conant is Rescarch C h i s t with Anojet. AUTHORS.
44
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
[- -7 3 APN2' 1
METHODS FOR REACTOR DESIGN
f . = f'
c
A value of 0.2594 was computed for the fraction of free space, based on smooth, nonporous spheres of 2mm. diameter and closest packing for a face-centered cubic lattice. A aitical Bow oriIice was considered for the laminar postmixing problem. The radius of such an orifice was calculated using a constant of 0.6, selected as representative of poorly constructed orifices that would probably result from the use of refractory materials (76). ro =
The minimum heating path length required to bring cold gases to the desired high temperature, for a fixed wall temperature, was calculated for both simple and packed tubes. For simple tubes, the equations below, derived from Newton's law of cooling and appropriate heat transfer coefficients (74, apply: & = - mCp
R
[
In T, - TI]""laminar flow (4) T, - Ts T, - TI m In T, - TI turbulent flow (5)
1 . 5 ~
-
10
e f
-
2
P
5
Figure 2. Prassure gradient of nitrogen in 4 packed heater. Tube backed with s&heru,2.0 mm. in diomcfar
10-1
-
10-2
-
10-8
10-3
For packed heating tubes of the geometry described above, a factor was derived from the ratio of heat exchange areas of packed and unpacked t u b a : n L
-
L., (packed) . = Ln. (unpacked) X
1.o
10-3
10.0
M*SS FLOW, G./SEC
rl r
__ 1.41 r,
+ r,2-
(6)
selected because of the low molecular weight of its vapor, which is essentially atomic boron ( I ) . All other nonvolatile materials should present greater diffusion barriers. Since boron Kas a normal boiling point of 4200' K. and a vapor pressure of 1.6 mm. at 3000" K. the calculations were for partial vapor pressures of boron of 1 mm. The diffusion constant for boron into nitrogen at atmospheric pressure and 3000° K. was
An empirical treatment for packed heating tubes is given by McAdams (75) but the temperature range in the latter is far below the present range, and extrapolation gave considerably lower values. Equation 6 was believed to be more accurate. The high temperature viscosity and thermal conductivity data required for the computations of Figure 3 were taken from the most recent National Bureau of Standards tables (77). ,This required some extrapolation of data, but these were consistent with more recently calculated values of transport data. The &,values are shown only at atmospheric presswe because they vary with pressure only to the slight extent that viscosity is pressure dependent. Diffusion barricm also enter into reactor design. The addition of a component of low vapor pressure into a carrier gas stream is essentially a problem of evaporation from a condensed phase and the subsequent diffusion of the vapor into the carrier gas. In a steady state flow system, the vaporization rate must equal the diffusion rate into the carrier gas and also the flow rate of the vapor in the subsequent mixed stream. From Fick's law of diffusion and the Knudsen-Hertz rate law for evaporation (S),one can obtain Equation 7.
Lo
=
DIAP',
- Ab', - how (2nnkT)'''I kTGm
(7)
For the diffusion calculations, the model system chosen was boron diffusing into nitrogen. Boron was
MASS FLOW G./S€C.
Figure 3. Minimum length of hcol ixchnngn medad fm nitrogen V O L 5 4 NO. 6 JUNE 1 9 6 2
45
This procedure can be used to investigate any reaction of interest, turbulent or laminar flow, and any reactor geometry
determined (6) as 19.3 square cm. per second. This leads to a maximum diffusion path length range of 0.0004 to 0.007 cm. for a nitrogen mass flow of 2.0 grams per second for the range of surface areas from 1 to 10 square cm. At the lower mass flow of 10-1 gram per second the comparable LD values are 0.014to 0.14 cm. ALLOWABLE RANGES OF DESIGN VARIABLES
Iurbulent Flow Systems
'1 1
I
I
1 I
Turbulence is desirable because of the greater simdicity of mixing reactant gases and the simpler kinetic system resulting from the relatively uniform gas flow compared to laminar flow. From the Mach number and maximum radii for turbulence data, it was seen that either very high pressures or high mass flows are required for turbulence. The small radii associated with turbulence at low mass flows require velocities that far exceed the acoustic velocity at normal pressures. At 3000° K. a mass flow of lo-' gram per second would require pressures in excess of 500 atm., but lo-' gram per second would require pressures in excess of 5 atm. Atmospheric pressure becomes feasible at 1 .O gram per second, but only for short path lengths; thus, a mass flow approaching 2.0 grams per second is necessary to maintain a reasonably constant pressure for the path length corresponding to the reaction zone. At low temperatures this restriction is considerably reduced-Le., at 300' K. turbulence can be achieved with low pressure gradient at a mass flow of lo-' gram per second.
Premixed Gases Reactants that are gases at ordinary temperatures could be mixed prior to elevation to high temperatures. 'If flow in the heater were turbulent, mixing could be accomplished by simply bringing the reactants together for a short path length. The limiting factor in such a system would be the requirement for low residence time in the heating chamber. It was shown by the following analysis that this could not be accomplished at ordinary pressures for systems in which the flow was also turbulent in the reactor. If simple unpacked tubes are used, heat exchange data showed that excessively long heating paths would be necessary at the high mass flow required for turbulence at atmospheric pressure. For a mass flow of 2 grams per second, the m i n i u m length of unpacked heat exchanger which is necessary to attain 3000' K. is 46
INDUSTRIAL AND ENGINEERING CHEMISTRY
several orders of magnitude greater than the at the maximum radius corresponding to turbulence. Further, the minimum residence time in the heater would be 20 msec., which would require that the reactor itself be enormously long to make the residence time in the heater small by comparison. The requirement for a short residence time in the heater can be evaluated by restricting it to some fraction of the minimum reaction time. If 0.05 is accepted as this fraction, then not only is the lower kinetic time limit at a mass flow of 2.0 grams per second restricted to 400 msec. for turbulent premixing but a minimum reactor path length of the order of 11,000cm. is required. Padked heat exchangers were considered next, but it was shown that heater radii in excess of 5.0 un. (at 1.5 atm.) were required to obtain an L,, that was compatible with the pressure gradient restrictions. Since this would lead to a minimum residence time of 22 msec. in the heater, the reactor length would again reach a value that was not only excessively long from the viewpoint of construction but also incompatible with pressure gradient restrictions. On the other hand, turbulent premixing is feasible if the turbulence requirement is removed for the reactor. By using a packed heater and a reactor radius greater than r,,,, one can obtain reasonably short reaction paths even at a mass flow of 2.0 grams per second. However, the residence time requirement in the heater fixes the minimum kinetic time that could be studied by this system at 440 msec.
Pornnixed Gases The above conclusions show that it is not pwible to study the kinetics of the decomposition of single reactants at ordinary pressures in turbulent reactors. However, reactions among several reactants can be studied under these conditions if the mixing is accomplished after the gases have been heated, as shown in Figure 4. For conditions shown, the kinetic time range corresponds to 0.3 to 2.0 msec.; with a reactor path length of 5 to 30 cm. An upper limit of 30 un. is considered reasonable for refractory materials, and this range was used in all subsequent calculations. Unfortunately the upper limit of the kinetic time range cannot be significantly ex-
I ,
HEATED GAS
I ,
7, HEATED GAS
Fiprr 5. Plates of onies can be used for mixing
tended if turbulence is to be maintained. Increasing the mass flow would increase the kinetic range, but as the mass flow is already exceedingly high, this is not a fruidul approach. The kinetic time range can be readily extended if turbulence in the reactor is surrendered. The kinetic time limit is then bounded only by the limits of construction of radii in refractory materials. A reactor radius of 5 cm. would lead to an upper time limit of 64.6 msec. Reactants of low vapor pressure at 3000' K. can be treated like the postmixed cases already discussed, except that a preheated inert camer gas is necessary because of the high pressure requirement for turbulence. For a nitrogen mass flow of 2 grams per second, the diffusion barrier to attaining a boron partial pressure of 1 mm. at atmospheric total pressure is such that no appreciable distance between the condensed phase and the carrier gas is possible. The evaporating liquid must therefore be directly in the path of the gas stream, and a vaporizing chamber at point A in Figure 4 cannot be considered. The kinetic time range is substantially unaffected by dilution of the diffusing vapor with an inert gas because of the reduction of the reactant mass flow as well as the pressure. laminar Flow Systems
If laminar flow systems are used for kinetic studies, high m a s flows are not necessary at ordinary pressures, and low total pressures are possible. 'Because of the large increase in volume flow corresponding to lowered pressures, it would appear that with constant mass flow lower pressures would result in lower residence times. However, the higher velocities at low pressures require tubes of larger radii and the kinetic time ranges are similar for different pressures, if mass flow and temperature are constant. Premixed Gases
In laminar premixed systems, high masa flows result in higher rather than lower kinetic time ranges. At atmospheric pressure, capillary radii are possible at low masa flows, and thus at a masa flow of loJ gram per second the minimum reaction time that could be studied
is 4.5 msec. for a reactor radius of 0.05 cm.; the range could be extended to one second by increasing the radius to 0.3 cm. However at the higher mass flow of 10" gram per second, the longer heater necessary to attain 3000' K. does not permit the use of a reactor radius of 0.1 cm., although this radius presents no flow restrictions and would result in an even lower kinetic time range. T o obtain a total residence time sufficiently greater than the heater residence time, the reactor radius must be 0.5 cm., corresponding to a kinetic time range of 47 to 269 msec. To obtain the lower kinetic time range corresponding to the smaller radius, a capillary radius of 0.01 cm. would be necessary for the heater. This would require pressures in excess of 20 atm. At lod atm., a mass flow of lo-" gram per second leads to a kinetic time range of 7 to 43 msec. for a radius of 2.0 cm. This system is easily applicable to kinetics. As was true at atmospheric pressure, at the higher mass flow of lo-* gram per second the minimum kinetic times are not compatible with short residence times in the heater, and the lowest reaction time is 24 msec. Postmixed Gases
A very simple mixing procedure would be to pass the separate preheated gases through orifices which face each other at a distance such that the emerging gases have intersecting swirling motions. The limiting criterion for such a mixer would be the effect on flow properties of the distance between facing orifices. From data on velocity relationships (78), it can be seen that for effective mixing, the distance between facing orifices cannot exceed approximately twice the orifice diameter. Orifice diameter for critical flow becomes exceedingly small at low mass flows, and therefore moderately high mass flows, are required if flow restrictions are to be avoided in the mixing region. In Figure 5, the dashed lines represent plates of orifices and the volume bounded by these plates is the mixing region. At atmospheric pressure and 3000' K., a satisfactory system can be obtained for a masa flow of lo-' gram per second for each reactant. If each orifice plate contains 10 orifices, the mass flow through each orifice is 10" gram per second and the spacing between orifice plates cannot be more than 0.10 cm. Setting the diameter of the orifice plate at 0.36 cm. results in an equivalent radius for the mixing region of 0.081 cm., which is quite compatible with pressure gradient restrictions. Residence time in the mixing region is well below the required minimum value. This system corresponds to a kinetic time range of 0.4 to 2.2 msec. for a reactor radius of 0.2 cm., and this range can be extended to higher values by increasing the reactor radius. However, if the mass flow is reduced to lo-' gram per second for each reactant, the orifice plates must be spaced at a vanishingly small distance (less than 0.033 cm.) from each other if 10 orifices are used on each plate, and it is clear that fewer orifices must be used. Since a network of orifices would lead to more complete mixing than a single orifice, it can be concluded that mass flows much below V O L 5 4 NO. 6 JUNE 1 9 6 2
47
. . ..... .. J
?
'
. . . ..
..
.
.. . '.
i
.% _
i
.
-. ' !
1
j
.!,,
- .: .,. , si.5
i
'
j
' ~
,
.$a
?*%'
.. :
+*j!
EXAMPLE OF CONSTRUCTED REACTOR The simplest type of isothermal flow reactor, bored on the calculations of this paper, has been constructed. This is a laminar premixed reactor system which i s designed to operate at 3oooo K. Details of this reactor, along with a specially designed induction furnace that maintains the reoctor at the desired temperature are shown above. The furnace i s powered by a Yl-kw., 10,000-cycle per second source. The tungsten sheath encloses the reactor which was fabricated from zirconium diboride. The latter material was chosen for kinetic experiments with systems containing the elements boron, nitrogen, and hydrogen. Zirconium . . iboride powder was hot-pressed (by the Carborundum
Co., Niagara Fails, N. Y.1 into solid cylinders, each 6 inches long and 2 inches in diameter. The pactor was then fabricated by drilling these solid cylinders into tubes of appropriate diameter by an electrical discharge technique; the latter was necessitated by the extreme hardness of this material. Four such tubes placed end to end within the tungsten sheath comprise a complete kinetic circuit. For the present reactor, designed to operate near a mass flow of gram per second, the heating section and the quench chamber are single zirconium diboride sections drilled to a tube diameter of 1.4 mm. Two additional sections.. ,Dlaced end on end., comorise the reaction . oath: ,~~ the diameters of these sections ore 0.9 cm.
IO-' gram per second are less satisfactory than higher mass flows. At loJ atm., the mass flow of lo-' per second is not practical because of the long path length and large radius imposed'by heat exchange and flow restrictions for an unpadred heater. Because of the greater velocities at low pressures, and the accompanying pressure gradients, packed heaters are not feasible. An analysis of the conditions at a mass flow of 10" gram per second, where the heating problem is considerably reduced, showed that only one or two orifices are possible for each orifice plate. It can be concluded that higher pressures are better suited for laminar postmixed systems. In the special case where one or both reactants have low vapor pressures at 3000' K., the above conclusion demonstrates the need for a carrier gas. While for the higher mass flows associated with turbulence, the m a s s flow of the model, boron, is a considerable quantity, at a mass flow of inert gas of lo-' gram per second the associated boron mass flow is only 5 X 10-5 gram per second (for 1 mm. of boron diffusing into a total pressure. of 1 atm.). This would result in the accumulation of exceedingly s m a l l quantities of product in the quenching
chamber, even for an extended experiment. F'roduct trapping is therefore a poor analytical technique for such a reaction system. Spectroscopy does not offer a reasonable alternative because stray radiation from the reactor wall is likely to mask the results of the chemical reaction under study. However the use of mass spectrometry in conjunction with the creation of a molecular beam instead of quenching is an interesting possibility. The diffusion barrier at a total mass flow of 10-1 gram per second is sufficiently large for any reasonably small surface area, so that a separate vaporizing chamber is not possible. The conditions under which the diffusion barrier would be small are not applicable to either laminar or turbulent kinetic flow systems.
40
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
~.
~
~~~
.
When Other Gases Are Used The conclusions reached by considering nitrogen are strictly valid for gases of very low emissivity; nitrogen has an emissivity of 6 X lod for a tube of 1 cm. diameter at atmospheric pressure and 3000' K. (9). For such gases, radiant heat transfer is negligible compared to
TABLE I.
Half Lines” of Bimolecular Reactions of Varying Activation Energies at 3000’ K .
ACKNOWLEDGMENT The authors are indebtedto A . Wheeler of this laboratory for his contribution to the diyusion calculations. P. Johansen and M . Sekala of the materials laboratory designed the induction furnace.
Ac tivation Energy, Kcal./ Steric R a t e Constant, Mole Factor CC./Mole Sec.
10 io 50 50 100 100
1 10-3 1 lop3 1
10-3
1 . 9 x 1013 1 . 9 x 1010 2 . 3 4 X 1O’O 2.34 X loi 5.75 x i o 6 5 . 7 5 x 103
NOMENCLATURE tljz(Sec.) a t 1 Atm.
2 . 1 x 10-9 2 . 1 x 10-6 1 . 6 8 X 10-6 1.68 X 6.95 x 10-3 6.95
t l j z (Sec.) Atrn.
at
2 . 1 x 10-6 2 . 1 x 10-3 1.68 X 1.68 6.95 6.95 X lo3
a Computed / o r biniolerulor renclions, using a nzodijed Arrhenius equation and assurnin$ an overage collision number of 7014 cc. per mole per sec.
convective heat transfer, and Equations 4 through 6 are all that would be necessary to calculate the minimum length of heat exchanger. However, for gases of appreciably higher emissivities, such as COz (which has an emissivity of 0.0035 under the same conditions) ( 4 ) , radiant heat transfer should be considered. At the temperatures under consideration, emissivity of a gas is directly related to its ability to absorb infrared radiation. Emissivities tend to increase with increasing degrees of freedom and increasing electronegativity differences between the constituent atoms. Symmetrical diatomic molecules would, therefore, have very low emissivities compared to asymmetric diatomic molecules and polyatomic molecules. By applying the Stephen-Boltzmann radiation equation, it was shown that the effect of radiation is still small, compared to convective heat transfer, for COa heated by unpacked heat exchangers. However, the greater radiative surface of packed heat exchangers might alter the conclusions. In making such calculations the degree of dissociation and enthalpy of dissociation of the specific gas under consideration must be known since dissociation could well negate any radiation effect. Exothermic Reactions
Under certain conditions of high exothermicity, large reactant pressures, high mass flow, and large reaction extent, it is possible that the energy liberated by the reaction would not be transferred rapidly to the container wall and could thus lead to considerable higher gas temperatures. If calculation of this effect by combined radiant and convective heat transfer to the wall should indicate that the reactant pressures were too high, the pressure and mass flow requirements could be met by dilution with an inert gas. From the data in Table I, it can be seen that dilution will tend to elevate the kinetic time range to practical values for bimolecular reactions of low activation energy while for reactions of high activation energy, dilution could take the kinetic time range out of the region of fast reactjon kinetics.
A A, C,
=
D f
= =
area of evaporating surface
= area of each hole in packed tube = heat capacity
diffusion constant frictional resistance of tube wall f’ = frictional resistance ( . 3 ) k = Boltzmann constant X = thermal conductivity L0.9 = length at which pressure falls to @ / l o of its initial value LD = distance between evaporating surface and carrier gas stream L,”in, = minimum length of heat exchanger rerluired to achieve desired temperature 7k = mass flow A4 = molecular weight N = number of holes in cross section of packed tube p = static pressure p ‘, = equilibrium vapor pressure of diffusing vapor b ,; = partial vapor pressure of diffusing vapor in the mixed stream P = perimeter of each hole in packed tube r,,,,, = maximum radius for turbulence rs = radius of packing sphere rt = radius of packed tube R = gas constant R!io,, = flow rate in molecules per second tjj? = half life of reaction T = absolute temperature T , = wall temperature ii = volume flow = fraction of free space e p = viscosity
LITERATURE CITED (1) Bockris, J. O’M., White, J. L., Mackensie, J. D., “Physicochemical Measurements at High Temperatures,” Academic Press, New York, 1959. (2) Campbell, I. E., “High Temperature Technology,” Wiley, New York, 1956. (3) Ergun, S., Chem. En
