1056
INDUSTRIAL AND ENGINEERING CHEMISTRY
relations policies may require the elimination of such evesores before they become intolerable. Recovering mineral values from waste phosphate slime is of national concern, as it represents a large potential source of an essential fertilizer constituent -48a first step, fundamental studies should be made of the mineral composition of the slimes and distribution of the phosphate and other potential mineral values determined in the subsieve particle size fractions. The basic information thus obtained should serve as a guide in developing ore-dressing methods applicable to the treatment of this niateiial.
Voi. 46, No. 5
LITERATURE CITED
(1) Conley, J. E., et al., U. 9. Bur. Nines. Rspt. Intest. 4401 (Kovem-
ber
1948).
(2) Greene, E. 1%‘. (to Minerals Separation North American Corp.), U. S. Patent 2,571.866 (Oot. 16, 1951). (3) Haseman, J. F., Ibid.. 2,680.303 (1903). (4) TVaggaman, W. H., “Phosphoric Acid, Phosphates, and Phosphatic Fertilizers,” A M E R I C I N CHEMICAL SOCIETYMonograph 34, 2nd ed., S e w York, Reinhold Publishing Corp.. 1952. (5) Walthall, J. H., private communication. (6) Warren, 9. P., unpublished report.
RECEIVED for review July 16, 1953.
ACCEPTEDJanuary 25. 1934.
Equilibrium Temperatures and Compositions behind a Detonation Wave ALEXANDER WEIR, JR., ~ K RICHARD D B. MORRISON Uniuersity of Michigan, 4ircraft Propulsion Laboratory, I’psilnnti, Mich.
T
HE combustion of Hanimable gaseous mixtures may be
divided into two categories. deflagration or “subsonic” combustion and detonation or “supersonic” combustion. The temperatures in the wake of deflagrations approach the adiabatic flame temperatures, inasmuch as the flow terms are usually of negligible importance. On the other hand, detonation waves are accompanied by strong convective flows which account for temperatures in the wake that are greatlyin excess of the adiabatic flame temperature. Information concerning the chemical composition a t temperatures accompanying detonations is lacking. Ordinary methods of chemical analysis are precluded in determining the composition immediately behind detonation waves, inasmuch as these waves may have velocities in excebs of 10,000 feet per second and the temperatures behind them ale on the order of 4000” K. Emission apectroscopy would be of great value in identifying certain components that are present but ~ o u l dnot, to the witers’ knowledge, provide an instantaneous quantitative analysis of all the gases immediately behind the wave. Hence, somewhzt indirect methods must be used t o obtain knowledge of the chemical composition behind a detonation wave The velocities of these supersonic detonation waves in hydrocarbon-oxygen mixtures 15 ere determined by the utilization of shock tube techniques (9). Laffitte and Breton ( 7 ) , Breton ( 2 ) . and Manson (8) reported detonation velocities of acetyleneoxygen and propane-oxygen mixtures but not hexane-oxygen mixtures. For consistency, Morrison’s data (0) \\ere used for the three fuels in the calculations reported herein. These experimentally determined velocities may be used in conjunction with chemical equilibrium constants to determine the temperature (and hence, the chemical equilibrium composition) which must occur t o satisfy the conservation of mass and momentum. These chemical equilibrium compoqltlons are the limiting values which will occur. They will probably be a closer approximation of the true composition than the valueq obtained by stoichiometryi.e , by assuming complete reactions without dissociation. This paper presents the temperatures and equilibrium compositions for detonation in propane-oxygen, hexane-oxygen, and acetylene-ouygen miutures APPLICATION OF MOME\TUhI THEOREM TO DETOli iTIOR WAVES
A Chapman-Jouguet ( S , 6 } detonation wave may be described aa a supersonic wave with heat addition behind it, that is
propagated through a gaseous Hammable mixture with minimum possible velocity consistent with the conservation laws. This paper considers only the case for Chapman-Jouguet detonations. It has been shown that the gases behind a Chapman-Jouguet detonation wave relative to the wave front a t a speed just equal t o their local sonic velocity-Le., the burned gases in a ChapmanJouguet wave-are moving a t a Mach number of 1 relative t o the wave. This fact, together with the conservation of maps and momentum and an equation of state, may be used to describe the changes across a detonation wave (Figure I).
PRESSURE RISE ACROSS D E T O N A ~ W ION ~E T
+ piulz
(Conwrvation of momentum)
PI
Equation of state--i.e., gau Ian-
p =
ideal
Definition of Mach Jfa number for ideal gases
=
P2
+ p3u;
P/T(R,M)
(1) (2
__
=
u / d q )
Combining Equations 2 and 3 we have P I L ~=
PrMa2
(4)
and by substituting Equation 4 in Equation 1, we obtain
(5) However, for a Chapman-Jouguet Detonation, the Mach numhei behind the wave, M a 2 ,i b equal to 1 and the relationship involvipg the bIach number of detonation ( M a ,= Man) is
This relationship makes it possible to calculate the preswre rise across a detonation wave from the experimentally determined Mach numberq of detonation by assuming that the detona-
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
May 1954
tion is of the Chapman-Jouguet type and that the ideal gas law and momentum conservation prevail. SPECIFIC VOLUMECHANGE ACROSS A DETONATIOK WAVE. For a constant area duct, the conservation of mass may be written as PlUI =
PZU2
(7)
By combining the conservation of mass (Equation 7 ) with the conservation of momentum (Equation l), we obtain
By substituting Equation 4 (a combination of the ideal gas law and the definition of hIach number) into Equation 8 and rearranging, we have
and since
p1
is the reciprocal of VI by definition, we have
1057
ESTIBIATIOS OF AVERAGE MOLECUL4R WEIGHT OF GASES BEHIND DETOY4TION WAVE
Equation 16 indicated that the average molecular weight ratio,
iCfzliM1, was required to compute the temperature behind a detonation wave The initial average molecular weight of the unburned gases is, of course, easily computed from the composition of the premixed fuel-oxygen mixture which the detonation wave traverses. Immediately behind the wave, after the reactions have occurred, the average molecular weight is not so easily obtainable. If, as previously stated, the assumption is made that chemical equilibrium exists, then the chemical composition (and the average molecular weight) is determined by the temperature. Thus, we have two unkno~vns, the temperature and average molecular weight behind a detonation nave (or their ratios to the unburned gaseq), but we also have two equations. The first equation, Equation 16, is obtained from the momentum theorem and the experimental detonation velocities, while the chemical equilibrium relationships, involving temperature and composition, may be considered as the second equation
or
or by substituting the previously derived relationship for the pressure rise ( P 2 / P 1shown ) in Equation 6, we have
Rearranging
1
+ y, - 1 -
(13)
MOL
Figure 2.
Equation 14 indicates that the specific volume or velocity change across a detonation wave can be computed from the experimentally determined Mach numbers of detonation by applying the momentum theorem if variations in y~are assumed to be negligible. The ideal gas law, conservation of mass, and assumption of a Chapman-Jouguet detonation are also involved. TEMPERATURE RISE ACROSS A DETONATION WAVE. The temperature rise across the detonation wave may be written with the aid of the ideal gas law as follows (Figure 1):
TJTt =
(~~~,/~~i)(P2/Pi)(~~2/Vi)
(15)
or, substituting the previously derived relationships for the pressure ratio (Equation 6) and the specific volume ratio (Equation 14): we have
&E"'
FE.
v,
1 >wr,AL F U E t Q O X I C E H
The following nine components are assumed to be present at equilibrium:
COZ,CO, HzO, 0,) Hz, OH, H, 0, and C Although the presence of Cz and CH has been well established in deflagration ( 4 ) , these components were not considered in this treatment of detonation. HOZ and 03 were also disregarded. Following the scheme of Winternitz ( 1 1 ) and von Stein (IO), we may express the partial pressures of the first six components in terms of the partial pressures of H, 0, and C by means of equilibrium relationships and thus obtain six of the required nine equations as follows: Pa,o = Kii P H Po ~ poa
=
Kia Po PH
(17)
(18) (19)
Km PH*
(20)
PO, = K2i PO,
(21)
Pa, Equation 16 indicates that while the pressure and specific volume change across a detonation wave can be computed from the experimental detonation velocity data with the aid of the momentum theorem, the temperature rise across a detonation wave cannot be found from the equations of mass, momentum, and state alone. To compute the temperature behind the detonation wave requires that the change in average molecular weight between the burned and unburned gases be known.
0
Mach Numbers of Detonation Waves
Pco = K19 pc PO (16)
M,*i.,
PCO,
=
Kzz PCPO,
(22)
The equilibrium constants, K , are functions only of the temperature if ideal gases are assumed. The remaining three equations may be obtained from an atomic material balance:
Vol. 46, No. 5
INDUSTRIAL AND E N G I N E E R I N G C H E M I S T R Y
1058 PO
=
PO
f
pH20
f
pc =
POH PC
f
+
'&Os
PO0
f
pC0
f 2PC02
4- pco,
(24) (23
Quoting Winternitz, the terms "on the left side of shese equations are (in the terminology of von Stein) the 'fictitious pressures' of the basic constituents-that is, the pressures which they would exert if all the reaction products were converted into the basic constituents in the gaseous state." Hence, to solve for the amount of each of the nine component8 assumed to be present a t equilibrium, we have nine equations (Equations 17 through 25). Kumerical values for the terms PH,Po, and Pc in Equations 23, 24, and 25 may be obtained from the eomposition of the initial fuel-oxygen mixture ahead of the detonation wave. Values of the equilibrium constants, R's, in Equations 17 through 22 are determined by the temperature behind the detonation wave. Hence, Equations 17 through 25 may be used in conjunction with Equation 16 obtained from the momentum theorem to calculate values of the average niolwular weight and temperature behind a detonation wave. The toniputatiorial procedure used is described in a later section.
All values are in mole percentages. The calculations were of necessity made only over the initial mixture composition range for which the experimental detonation velocities were available. The average molecular weight ratios before and after a detonat'ion wave were computed from Figures 3, 4, and 5 and are pic'serited in Figure 6 for detonations of hexane-oxygen, propancoxygen, and acetylene-oxygen mixtures.
QssmipTIoIvs The results presented in the following graphs involve the following assumptions: 1. The laws of conservation of mass and obeyed. 2. The ideal gas law ( P V = n R T ) is obeyed the heat capacities after a detonation wave is a although it is not necessarily equal to thp ratio pacities in the initial mixture. 3. The detonation wave is of the Chapman-Jouguet (1,s) type. 4. Chemical e q u i l i b r i u n i exists in the reaction products and only the follon-ing nine c o m p o n e n t s are present a t equilibrium: C02, CO, H20, 0 2 , €Iz, OH, H, 0 , and C.
momentum are and the ratio of constant value, of the heat ca-
I
I
I I
oCgHa
IN" lNITIAfo C 3 H 8 - i l
J MIXTU#
Figure 4. Computed Equilibrium Composition behind Detonation Tare
Assumptions involving the energy equation or a theoretical heat release or heat of combustion are not included. These results do not, of course, violate the law of conservation of energy.
The temperatures attained in the detonation of these thiee hydiocarbon-oxygen mixtures are shown in Figure 7 . T h e v values were computed by means of Equation 16 using the expetimental data presented in Figure 2 and the computd valuw ehomn in Figure 6.
RESULTS
COMPUT.AT1ONAL PROCEDURE
The Mach numbers of detonation waves traversing hexane-oxygen, propane-oxygen, and acetylene-oxygen mixtures are presented in Figure 2. These Mach numbers were computed from the detonation velocities obt'ained by Morrison (9) in shock tube experiments. The computed t'emp e r a t u r e s and compositions behind the detonation wave Figure 3. Computed are based on these values. Equilibrium ComposiThe computed equilibrium tion behind Detonation composition for detonation ocWave curring in gaseous hexaneoxwen mixtures a t atmospheric pressure are shown in Figure 3. Similar compositions behind the detonation wave for propane-oxygen mixtures are shown in Figure 4, while in Figure 5 the equilibrium composition for detonation of acetylene-oxygen mixtures is presented. "
Y
To solve the chemical equilibrium relationships, there are nine unknown constituents and nine equations. Equations 1; through 22 express the partial pressures (as a function of temperature) of six of the nine constituents in terms of the partial pressures of the other three, PH, PO, and p c . These values may he substitut>edinto the material balance equations ( 2 3 , 24, anti 25), so that three equations and three unknowns (the partial pressures of C, H, and 0) remain. These three simultaneous equations were solved for the partial pressure of O ( p o )to give
INDUSTRIAL AND ENGINEERING CHEMISTRY
May 1954
1059
+
i
Calculate A , A = 4(ZC1i po KZO) 8. Calculate B, B = K l s p o 1. - B fd B 2 ~APH.
+
9.
Calculate p ~p~,
=
+
A
10. Calculate p ~ 2 . 11. Calculate E from steps 5, 8, 9, E = 1 Kiipn’ Kispn Ki9 p c . 12. Calculate p o from steps 6 and 11, p o = -E i d E 2 2PoD. D 13. Compare the value of I)O obtained in step 12 with the value assumed in step 4. Repeat steps 4 through 12 until agreement is obtained. 14. Using the values of po (step 12), p c (step 3 ) , and pa (step 9) found to be satisfactory in step 13, calculate the partial pressure of the remainder of the constituents by means of Equations 17 through 22. 15. From the partial pressures of the nine constituents, calculate the mole fraction of each constituent, and determine the average molecular weight of this composition. 16. Using the average molecular weight behind the detonation wave obtained in step 15, compute a molecular weight ratio. Compare this value with the one assumed in step 1 and repeat steps 1 through 16 until agreement is obtained.
+
+
+
+
Figure 5.
Computed Equilibrium Composition behind Detonation Wave
The equilibrium constaiits ( K ’ s ) are determined by the temperature, and the values of Po, PB, and Pc can be determined from the initial mixture composition, so that p o is the only unknown in this equation. A5 the temperatures behind the detonation wave are obtained from the experimental detonation velocities by means of Equation 16 (in which the average molecular weight is also a variable), a double trial and error procedure was found to be expedient in solving Equations 16 and 26 simultaneously.
TRIAL a u n ERRORPROCEDURE used for a givcn initial mixture composition. 1. Assume a molecular weight ratio and compute the temperature behind the detonation wave with the use of Equation 16 and the Mach numbers of detonation presented in Figure 2. 2. Using this temperature, evaluate the six equilibrium constants. 3. Determine the values for Po, Pc, and PH from the initial mixture composition, the average molecular weight assumed, and the total pressure behind the detonation wave (from Equation 6 and Figure 2). PC = (mole fraction of fuel)(C atoms/mole fuel) (PZ)(MZ/MI) -1.
Assume a value of po.
+
j.
Calculate p c by the equation p c =
6.
Calculate D, D = 4(KZ1 R zp~~ ) .
+
1
Pc Kispo
+ Kzz pa
DISCUSSIOY OF RESULTS
The assumptions upon which these results were obtained have bpen stated. A questionable assumption is that of chemical equilibrium behind the detonation wave. The writers realize that chemical equilibrium is not achieved behind the detonation wave. For example, referring to the NO concentration in carbon monoxide-air explosions, Jost (6) states, “Bone ( 1 ) and his associates found yields in excess of equilibrium that were further increased by suddenly cooling the gases immediately upon reaching the maximum temperature.” (The “maximum temperatures” for which equilibrium conditions were computed were 1660”, 1740°, and 2270“ K.) However, unless this assumption of chemical equilibrium is made, the results reported here could not be obtained, since no knowledge of the average molecular weight behind the detonation wave is available. The writers believe that knowledge of the high temperatures behind these waves and the correspondingly high concentrations of 0, H, OH, and C are of sufficient interest to justify reporting these limiting values. The results presented are for both lean and rich mixtures. The stoichiometric ratios for the three fuels are: n-Hexane-oxygen mixtures = 9.52 mole % C ~ I I I ~ Propane-oxygen mixtures = 16.67 mole 7 0 C& Acetylene-oxygen mixtures = 28.57 mole % CzHz
_---
MOL PERCENT
Figure 6.
FUEL IN
INITIAL FLEL-OXYGEN
MlXTURE
Average Molecular Weight Ratios before and after Detonation Wave
MOL
PERCENT FUEL
N INTYL
FUEL-OXYGEA
MXTURE
Figure 7. Temperatures Attained in Detonation of Hydrocarbon-Oxygen Mixtures
1060
INDUSTFlIAL AND E N G I N E E R I N G CHEMISTRY
The equilibrium concentration behind the detonation with hexane-oxygen, shown in Figure 3, indicates that maximum concentrations of OH(10.5%) and 0(4.7%) occur a t the stoichiometric ratio, but' that the concentration of H continues to rise (t,o 8.3%)as the fuel concentration is increased. Kegligible amounts of C (0.0003%) are obtained a t the richer mixtures. I n Figure 4 for propane-oxygen detonations, tmhemaximum concentration of OH( 12.4%) occurs at the stoichiometric ratio. The concentration of 0 at, the stoichometric ratio (6.6%) is only slightly less than the value a t a leaner (10 mole % propane in initial mixture) mixture (7.0% 0).Significant amounts of H (6.6%) are present' a t t'he richer mixtures, while the concentration of C (0.0002%) is negligible. The equilibrium composition behind a detonation ivave traversing acetylene-oxygen mixtures, shown in Figure ,5, indicates that the maximum concentratious of OH (11.9%) and 0 (16.9%) occur a t the stoichiometric rat'io. Contrary tmothe previous curves, the concentration of H exhibits a maximum value (21,9%) at, a 50% mixture of acetylene and oxygen rather than as previously indicated increasing as the fuel composition increases. Also, it is seen that' t'he concentration of C is not negligible a t the richer mixtures, being 0.14% a t a 50% acetylene mixture and increasing to 30.6% C for a 65% acetylene mixture. I n general, these three graphs indicate that t h e concentrations of 0 and OH exhibit a maximum a t the stoichiometric fueloxygen mixture. HzO also exhibits such a maximum in the detonation of hexane-oxygen and possibly propane-oxygen mixtures, but not in the detonation of acetylene-oxygen mixtures. The remaining six components do not have a maximum value a t the stoichiometric fuel-oxygen mixture. The concentrations of 0, OH, H, and in the case of rich acetylene mixtures only, of C, are significant,lylarge a t these high temperat>ures. The molecular weight ratios shown in Figure 6 indicate 'that the simplifying assumption sometimes made, of no change in molecular weight across a detonation wave, can be in error by as much as a fact,or of 2. The temperat'ures behind the detonation x-ave, shown in Figure 7, exhibit a maximum (4550" K.) at' the etoichiometric ratio for acetylene-oxygen detonations, although additional calculations between 28 and 50% acetTlene may indicate that this is not true. The maximum temperature for propane-oxygen drt,onations (3970" K.) a t 20% propane is only 30" K. higher than the t,emperature (3940" K.I a t the stoichiometric ratio. It is doubtful ivhether this difference is significant. However, the data for hexane do not exhibit, a maximum temperature a t the stoichiometric ratio, but the temperat'ure increases as the concentration of hexane is increased. These temperatures in a11 case@are much greater than the adiabatic flame temperaturesfor example, betn-een 25 and 50% acetylene, it is about 1200" IC. greater than the adiabatic flame temperature. These high temperatures are, of course, due to the strong convective flows associat,ed with the detonation waves. NO31 EUC L.ATURE
Subscript 1 refers to conditions ahead of the detonation m v c Le., the unreacted gases. Subscript 2 refers to conditions behind the detonation w a v i.e., the reacted gasea.
A
Vol. 46. No. 5
term used in equilibrium calculations defined in computational procedure (Step 7 ) and Equation 27 B = term used in equilibrium calculations defined in computational procedure (Step 8) and Equation 28 D = term used in equilibrium calculations defined in computational procedure (Step 6) E = term used in equilibrium calculations defined in coniputational procedure (step 11) Iit: = equilibrium constant defined in Equation 17 Kl8 = equilibrium constant defined in Equation 18 K19 = equilibrium constant defined in Equation 19 K X = equilibrium constant defined in Equation 20 KP1 = equilibrium constant defined in Equation 21 Kz, = equilibrium constant defined in Equation 22 &' = average molecular weight velocity X u = Mach number = velocity of sound in same medium XOD= SIach number of detonation n-ave = ;Iln, P = pressure pa = defined in Equation 23 po = defined in Equation 23 pc = defined in Equation 23 DCO, = partial pressure of CO, in reacted gases = partial pressure of CO in reactrd gaser = partial pressure of H20 in reacted gaseq = paitial pressure of 0 2 in reacted gases = partial pressure of Hz in reacted gases = partial pressure of OH in reacted gases = partial pressure of H in reacted gases = partial pressure of 0 in reacted gases = Dartial pressure of C in reacted gase9 = universal gas constant = temperature = specific volume = velocity = ratio of specific heats = density =
~
-
LITERATURE CII'EU
Bone, IT. A, Xewitt, P. 11.. and Townsend, D. T. A , Pi.or: Roy. Soc. London,, A139, 67 (1933). Breton, 11. J., Ann. ofice natl. combustibles liqilides, 11, 487 (1936). Chapman, D. L., Phil. Mag.. (6j, 47, 90 (1899). Gaydon, A. G., "Spectroscopy and Combustion Theory," 2nd ed., London. England, Chapman and Hall, Ltd., 1948. Jouguet, E., J. Math., 1905, 347; 1906, 6; "Mecanique des explosifs," Paris, 1907. Jost, W., "Explosion and Combustion Processes in Gases," tr. hy Huber 0. Croft, P. 337, S e w York, lIcGraw-Hill Book Co., 1946. Laffitte. P., and Breton, 11. J., Compt. rend., 199, 146 (1931). Manson, N., and Ferrie, F., "Contribution to the Study of Spherical Detonation Waves," in "Fourth Symposium on Combustion," p. 486, Baltimore, N d . , Williams and TVilkins co., 1953. Morrison. R. B., Unirersity of Michigan, Rept. CLIIilI 97 (January 1952). Stein, bl. Y O U , Fomch. Gebiete Inge7iieuru., 14B,113-23 (1943) Tinternitz, P. F.. "Method for Calculating Simultaneous, Homogeneous Gas Equilibrium and Flame Temperatures," Third Symposium on Combustion, Flame, and Explosioii Phenomena, p. 623, Baltimore, Md., Williams and Wilkins Co., 1949. RECEIVED for review June 20, 1953.
ACCEPTEDFebruary 1 7 , 1 9 5 4 . Presented before t h e Division of Gas and Fuel Chemistry, Symposium on Properties and Reactions of Carbons. a t the 124th Meeting of the AMERICAX C H E m c A L SOCIETY. Chicago, Ill. A portion of thia work mas done under the sponsorship of the U. S. Air Force, Flight Research Laboratory.
