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Application of exergy analysis to the hydrodynamic theory of detonation in gases
Fuel Processing Technology 67 Ž2000. 131–145 www.elsevier.comrlocaterfuproc
Application of exergy analysis to the hydrodynamic theory of detonation in gases Richard Petela Technology Scientific Ltd., 152 Ranch Estates DriÕe N.W., Calgary, Alberta, Canada T3G 1K4 Received 1 October 1999; accepted 1 April 2000
Abstract The exergy analysis was applied to the phenomenon of detonation in gases. Separately were considered the component processes of the pressure shock and the chemical reaction of combustion. The considerations were illustrated by the examples of detonation of stoichiometric mixtures of dry air and hydrogen or carbon monoxide or methane. A measure of the thermodynamic imperfectness of process is the exergy loss in the process. It was found that exergy loss of the shock is larger than the exergy loss of the chemical reaction of combustion. A steady detonation characterized by the Chapman–Jouguet point occurs at the extremum of these losses. The exergy loss of combustion during steady deflagration, considered for comparison, is larger than the total exergy loss of detonation. It suggests that from thermodynamic viewpoint, the detonation is a better method of fuel utilization than the deflagration. Therefore, the consideration of any new industrial burner device based on the detonation process would be motivated. q 2000 Elsevier Science B.V. All rights reserved. Keywords: Exergy; Hydrodynamic; Detonation
1. Introduction The present paper contributes with the exergy analysis applied to detonation in gases. Combustion of premixed combustible gases can develop at either the relatively low propagation velocity Ždeflagration., e.g., not exceeding about 200 cmrs, or at the significantly larger velocity Ždetonation.. The detonation can occur in a long tube filled up with a premixed gaseous mixture of fuel and oxygen, within the flammability limit, when an ignition is applied at the closed end and at open other end. The hydrodynamic theory of the combustion wave propagation is well-described, e.g., by Glassman w1x. A 0378-3820r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 2 0 Ž 0 0 . 0 0 0 9 9 - 0
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planar, one-dimensional flame front, as shown in Fig. 1, traveling down the combustible mixture, is considered. The subscript 1 is used for the unburned gas conditions and subscript 2 for the burned gas conditions. The wave phenomena can be analytically described by the following independent equations for: Flow continuity: u1 u2 s Ž 1. Õ2 Õ2 Momentum conservation: P1 q
u12 Õ1
s P2 q
u 22
Ž 2.
Õ2
Energy conservation: h1 q
u12 2
q qex s h 2 q
u 22 2
Ž 3.
State 2: P2 Õ 2 s RT2
Ž 4.
where: Õ s gas specific volume; u s gas velocity; P s gas pressure; T s absolute temperature of the gas; h s total enthalpy of gas; qex s heat delivered to the gas from external source; R s individual gas constant. The initial state of the mixture is determined and its parameters are related according to the state equation: P1Õ1 s RT1
Ž 5.
The total enthalpies take into account the sensible and chemical parts: h1 s c p Ž T1 y T0 . q Ý g i CVi
Ž 6.
i
h 2 s c p Ž T2 y T0 .
Ž 7.
where: c p s specific heat of the gas; CVi s calorific value of ith component of the initial mixture; g i s gram fraction of ith component of the initial mixture; T0 s reference temperature, e.g., the absolute environment temperature.
Fig. 1. Propagation of combustion wave.
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Specific heat c p and isentropic exponens g , are assumed to be constant and the following relation is used: g cp s R Ž 8. gy1 Using Eqs. Ž1. and Ž2., the difference of squares velocities can be expressed as: u12 y u 22 s Ž P2 y P1 . Ž Õ 1 q Õ 2 .
Ž 9.
Introducing Eqs. Ž6. – Ž9. into Eq. Ž3., the well-known Hugoniot equation can be derived:
g gy1
1
Ž P2 Õ 2 y P1Õ1 . y Ž P2 y P1 . Ž Õ1 q Õ 2 . s q
Ž 10 .
2
where q s qex q Ý g i CVi
Ž 11 .
i
There are only four available equations, Eqs. Ž1. – Ž4., to solve the problem containing five unknowns, u1 , u 2 , Õ 2 , T2 and P2 . Therefore, the theoretical solution is not given by one determined point but is represented by the points of curve Ž10. from which, however, only a part corresponds to the realistic solutions. A unique solution of the problems requires some additional considerations of the mechanism of the process. For the determined value of q, which is a parameter of the family of curves, the representation of Eq. Ž10. are the curves in a coordinate system Ž P2 , Õ 2 ., as shown in Fig. 2. The curve at q ) 0 represents the states of gas after combustion, if the initial state of gas, entering the wave, is represented by point 1. This curve relates to the process which is composed of the two interacting phenomena considered separately.
X
X
Fig. 2. Representation of the two interacting detonations phenomena: shock 1–1 and combustion 1 –2, in P, Õ system of coordinates.
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X
X
Fig. 3. Representation of the two interacting detonations phenomena: shock 1–1 and combustion 1 –2, in T, s system of coordinates.
These phenomena are: the compression in the shock wave, represented by the Section 1–1X of the curve at q s 0, and the combustion, represented by the Section 1X –2 of the straight combustion line determined by Eq. Ž2.. The detonation phenomena can also be presented in the temperature–entropy diagram as shown in Fig. 3. Usually, the detonation wave is considered as composed consecutively of shock, induction period, and chemical reactions of combustion. It could be assumed that during induction period the temperature, pressure and velocity of gas remain unchanged. Therefore, neglecting the processes occurring during induction period, the temperature of gas from initial value T1 increases during shock to value T1X , and then increases again to T2 due to exothermic chemical reactions of combustion. Pressure of gas from initial value P1 increases during shock to the value P1X , and then drops during combustion to the value P2 larger than P1. Gas velocity from initial value u1 decreases during shock to u1X , and then increases during combustion to the value u 2 smaller than u1. When including point 1X into considerations, additional four unknowns T1X , P1X , u1X and Õ 1X have to be calculated using the four additional equations. For the adiabatic combustion process Ž1X –2., the energy conservation equation, with taking into account Eq. Ž8., can be applied:
g R
gy1
Ž T1X y T2 . q g F CVF q
u12X y u 22 2
s0
Ž 12 .
The momentum conservation Eq. Ž2. can be applied for the states 1X and 2: P1X q
u12X Õ 1X
s P2 q
u 22 Õ2
Ž 13 .
Continuity equation is: u1X Õ1
X
s
u2 Õ2
Ž 14 .
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State equation: P1X Õ1X s RT1X Ž 15 . Eqs. Ž3. and Ž12. represent energy conservation equations based on the first law of thermodynamics. The objective of the present paper is to involve also the second law of thermodynamics and, therefore, the corresponding equations of exergy balance will be included. 2. Exergy considerations Thermodynamic mechanism of various processes can be analyzed with use of the exergy w2–4x. Exergy analysis allows us to develop additional explanation, evaluation or interpretation of the substances values and processes performances which often differs dramatically from the energy analysis results. The basic equation in the exergy analysis, analogically to the energy analysis, is the exergy balance equation. Whereas, in steady state, the energy input is equal to the energy output of the system, the exergy input and exergy output for a real system are different and has to be compensated by the exergy loss due to the irreversibility of system processes. For example, the exergy balance equation for the detonation wave changing from state 1 to state 2, analogically to energy balance Eq. Ž3., takes a form: u12
u 22
q bq ex s b 2 q q db Ž 16 . 2 2 where: db s the total exergy loss due to thermodynamic imperfectness of the analyzed processes; bq ex s the exergy of heat delivered to the gas from external heat source; b 1 , b 2 s so-called thermal exergy of gas entering and exiting the considered system, respectively. The velocities u1 and u 2 in Eq. Ž16. are absolute which means relative to the earth. The thermal exergy includes the so-called physical and chemical parts. A physical exergy, results from the temperature and pressure discrepancies in relation to the temperature and pressure of the environment. A chemical exergy results from a difference in partial pressures of mixture components at environment temperature and the partial pressures of these components in the environment air. The thermal exergy b 1 of the combustible mixture, composed of different components numbered successively by i, can be calculated from the following formula derived for gaseous fuels w2–4x: b1 q
ž
b 1 s c p Ž T1 y T0 . y T0 c p ln
T1 T0
y Rln
P1 P0
/
q g i b F i q RT0Ý g i ln z i
Ž 17 .
i
where: b F i s the chemical exergy of ith components Žcorresponding to its calorific value.; g i , z i s the gram and mole Žrespectively. fraction of the ith mixture component. For simplification, formula Ž17. is written for the assumption that the standard temperature, for which the chemical enthalpy and exergy are tabulated, is equal to the environment temperature. It was also assumed that the partial pressure of water vapor in actual and standard environments are equal. These assumptions cause negligible error.
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The thermal exergy b 2 of the combustion products of complete combustion, composed of different components numbered successively by j, can be calculated from the formula derived for gaseous mixture composed of the air components w2–4x:
ž
b 2 s c p Ž T2 y T0 . y T0 c p ln
T2 T0
y Rln
P2 P0
/
q RT0Ý g j ln j
z j P2 Pj0
Ž 18 .
where: g j , z j s the gram and mole Žrespectively. fractions of jth products component, Pj0 s the partial pressures of jth products components in the environment air. According to the Gouy–Stodola law, the exergy loss caused by thermodynamic irreversibility of a process is calculated as follows: db s T0Ý Ž D s . k
Ž 19 .
k
where: D s s the entropy increase of k th body Žincluding, e.g., all substances and heat sources. taking part in the process. The exergy values of each particular input component or output component is a measure of its practical values, whereas the exergy loss is a measure of the quality of the process in the system. The larger the exergy of the component, the larger the practical value of this component. However, the smaller the exergy loss, the better the system performance. The exergy loss for real process is always positive. For an ideal Žreversible. process, the exergy loss is zero. Exergetic optimization of the process is based on the minimization of the exergy loss. However, when the exergy loss is negative then it means that the occurring of the considered process is impossible. In the case of adiabatic detonation process, the sum of the entropy increases consists only of the entropy increase D s of the gases: db s T0 D s
Ž 20 .
Assumptions for exergy considerations follow the assumptions for energy considerations w1x. In the simplified model of detonation, the kind of gas does not change during processes of shock and combustion Ž c p s const, g s const.. Both processes are replaced by the physical processes; the first process, compression with significant change of flow velocity, and the second process, preheating by the internal heat source Žcalorific value. with further flow velocity change. It is assumed that the residual chemical exergy of gases, resulting from discrepancy between its composition and the composition of environment, can be neglected. Consequently, the sum terms in Eqs. Ž17. and Ž18. can be neglected and the chemical exergy of mixture can be assumed equal to its calorific value, g i b F i s g i CVi . The above assumptions allow us to simplify Eqs. Ž17. and Ž18. as follows:
ž ž
b 1 s c p Ž T1 y T0 . y T0 c p ln b 2 s c p Ž T2 y T0 . y T0 c p ln
T1 T0 T2 T0
y Rln
y Rln
P1 P0 P2 P0
/ /
q g i CVi
Ž 21 . Ž 22 .
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It is possible to consider the shock and combustion processes as the separate successive steps. Therefore, according to Eq. Ž20., the exergy loss dbs due to irreversible shock can be calculated as: dbs s T0 Ž s1X y s1 .
Ž 23 .
and the exergy loss dbc due to irreversible combustion as: dbc s T0 Ž s2 y s1X .
Ž 24 .
The total exergy loss db comprise both losses: db s dbs q dbc
Ž 25 .
The exergy losses can be also calculated from the respective exergy balance equations. For example, the exergy loss for the shock can be calculated from the exergy balance equation for the shock process: b1 q
u12 2
s b 1X q
u12X 2
q dbs
Ž 26 .
where u1X is the velocity of gases during induction period and the specific exergy b 1X of this gases is:
ž
b 1X s c p Ž T1X y T0 . y T0 c p ln
T1X T0
y Rln
P1X P0
/
Ž 27 .
The extreme values of the above exergy losses can be determined from the relation obtained by equating of the partial derivative to zero. For example, the minimum total exergy loss would result from the following condition: E Ž db . EÕ 2
s0
Ž 28 .
The illustration of considerations is further carried out with some assumptions. The adiabatic process is considered, which means that the heat delivered to the gas from external source is zero, qex s 0 and thus also bq ex s 0. These assumptions are quite realistic as the detonation process occurs very quickly and there is a little time for exchanging heat with external source. The initial mixture is composed of a gaseous fuel and dry air. The initial temperature T1 of the mixture is equal to the standard temperature T0 of environment air ŽT1 s T0 s 258C., as assumed for the tabulated values of calorific values and chemical exergy of various combustible substances. The initial pressure of the mixture is assumed equal to the standard pressure Ž P1 s P0 s 100 kPa. of environment air. From theoretical considerations w1x, it results that the only steady-state solution for the detonation region of the Hugoniot curve, is given by point J, also called the Chapman–Jouguet detonation point. This is the point determined by tangency of the combustion line with the Hugoniot curve at a given parameter q, as shown, for example,
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Fig. 4. Exergy losses at detonation of hydrogenrair stoichiometric mixture.
Fig. 5. Steady deflagration at point K.
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Table 1 Some assumed data and calculation results on the detonation combustion of hydrogen, carbon monoxide or methane Data
Stoichiometric mixture of dry air and: H2
CO
CH 4
Component of combustible mixture, % by weight
H 2 s 2.85, N2 s 74.53, O 2 s 22.62 1.403 396.4 H 2 Os 25.4, N2 s 74.6
COs 28.98, N2 s 54.48, O 2 s16.54 1.400 290.0 CO 2 s 45.52, N2 s 54.48
119.96 118.23
10.1 9.83
CH 4 s 5.52, N2 s 72.47, O 2 s 22.01 1.388 299.9 CO 2 s15.14, H 2 Os12.39, N2 s 72.47 50.02 52.15
0.70141 2.4897
0.5116 2.8667
0.5281 2.5542
21.302 0.0430
17.942 0.0429
16.696 0.0432
13.20 8.09 21.29
12.26 7.10 19.36
12.91 8.03 20.94
27.55
25.09
26.90
Isentropic exponens of mixture, g Gas constant for mixture, R, JrŽkg K. Components of combustion products % by weight Calorific value of fuel, CV, MJrkg Chemical exergy of fuel, b F , MJrkg Parameters at point J: Õ 2 J , m3 rkg P2 J , MPa Parameters at point K : Õ 2 K , m3 rkg P2 K , MPa Exergy losses, Ž% of fuel exergy.: Detonation at point J Pressure shock Combustion reaction Total Deflagration at point K Total
for combustion of the hydrogenrair mixture, in Fig. 4, which shows also the exergy losses discussed in the next paragraph. For comparison to detonation, the deflagration is also considered. From theoretical considerations w1x, it results that the only steady-state solution for the deflagration region of the Hugoniot curve, is given by point K, also called the Chapman–Jouguet deflagration point. This point, shown in Fig. 5, is determined by tangency of combustion line with the Hugoniot curve at given parameter q, for the pressure range P2 - P1. More details on assumed data are discussed for the three following examples which are the combustion of hydrogen, carbon oxide or methane with the stoichiometric amount of dry air. Some data assumed for calculations and obtained results are shown in Table 1.
3. Combustion of hydrogen To calculate the unknown parameters at points 1X and 2, the system containing Eqs. Ž1. –8. and Eqs. Ž12. – Ž15., was used. The exergy losses were calculated form Eqs. Ž23.
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and Ž24.. The obtained results are shown in Fig. 4. All the three exergy losses reach an extremum at the Õ 2 J . The exergy loss of the shock reaches a minimum whereas the
Fig. 6. Energy and exergy balances of detonation wave of stoichiometric hydrogenrair mixture at point J.
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exergy loss of the combustion approaches a maximum. The total exergy loss, which is the sum of these two losses, reaches a flat minimum. It should be mentioned that the solutions represented on the right-hand side from point Õ 2 J , Ždashed lines., have no real significance. It results that the real steady-state detonation develops towards the minimum of the total exergy loss. The typical diagrams of the energy and exergy balances of detonation wave, including separately shock and combustion, are shown in Fig. 6. Data were calculated for the point J representing steady detonation. A mixture energy h1 , or exergy b 1 , is assumed as 100% Ž h1 s b 1 s 3.4189 MJrkg.. The energy balance shows a successive conversion of kinetic energy and total enthalpy, however does not indicate any losses. The kinetic energy of initial mixture and combustion products is high and significantly comparable to the respective total enthalpy. The exergy balance indicates irreversible losses during the shock and combustion phenomena. Obviously, if the velocities relative to the earth are used, the values of kinetic energy and kinetic exergy are the same. The values of the thermal exergy of gases are a little smaller than the respective values of enthalpy. The evolution of the components of the exergy balance of detonation wave for the Hugoniot curve points in the vicinity of point J is plotted in Fig. 7. The thermal exergy of initial mixture was assumed as 100%. It results that the exergy loss of the shock varies slightly around 13%, whereas the exergy loss of the combustion, also varying slightly, is smaller and varies around 8%. The kinetic energy, equal to the kinetic exergy, was denoted by the symbol e with respective subscript, e.g., e1 s u12r2.
Fig. 7. Components of the exergy balances of the hydrogenrair detonation wave as a function of specific volume Õ 2 .
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Fig. 8. Components of the exergy balances of the hydrogenrair deflagration wave as a function of specific volume Õ 2 .
Regarding deflagration, Fig. 8 presents the components of exergy balance varying with specific volume Õ 2 in the vicinity of value Õ 2 K . The thermal exergy b 1 of initial mixture was assumed as 100% Ž b 1 s 3.4189 MJrkg.. Kinetic energy of mixture was always very small Ž e1 - 0.06%. and is not shown in the diagram. With increasing Õ 2 the exergy b 2 of products decreases at the cost of kinetic exergy e 2 , which increases. The total exergy loss db changes very little and the flat maximum appears for Õ 2 s Õ 2 K . As the deflagration does not include a shock, this exergy loss, at the value of about 27.55%, is for combustion only, although a certain small portion of the exergy loss could be assigned to the conversion of chemical exergy of fuel into the kinetic exergy of the combustion products. Total exergy loss of deflagration Ž27.55%. is larger than the exergy loss of combustion in the detonation wave Ž8.09%. and is even larger than the total exergy loss of detonation Ž21.29%.. Based on the above values of the total exergy loss for Õ 2 J and Õ 2 K for the same fuel, which was hydrogen, a hypothetical conclusion can be withdrawn, that the combustion in a detonation wave is more efficient than in the deflagration. This conclusion was also confirmed for other fuels considered in the next paragraphs.
4. Combustion of carbon monoxide Some data on the considered mixture of CO and air and combustion products, are given in Table 1. The calculation results were obtained in the same way like for
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Fig. 9. Components of the exergy balances of the COrair detonation wave as a function of specific volume Õ 2 .
previously considered combustion of hydrogen. The evolution of the components of the exergy balance of the COrair detonation wave for the Hugoniot curve points in the vicinity of point J is plotted in Fig. 9. The thermal exergy b 1 of initial mixture was assumed as 100% Ž b 1 s 2.926 MJrkg.. It results that the exergy loss of shock varies slightly around 14%, whereas the exergy loss of combustion, also varying slightly, is smaller and varies around 6%. The characteristic extremum of e1 , bs , es , dbs and dbc appear at the specific volume Õ 2 J . Comparison of detonation and deflagration, similar to that for hydrogen, for combustion of COrair stoichiometric mixture, shows similar results. For example, with varying Õ 2 , the total exergy loss db changes insignificantly in the vicinity of Õ 2 K and achieves a flat maximum at Õ 2 s Õ 2 K . The total exergy loss of deflagration Ž25.09%. was found to be larger than the exergy loss of combustion Ž7.1%. in the detonation wave and was even larger than the total exergy loss of detonation Ž19.36%..
5. Combustion of methane Applying the same calculation procedure as for the previous fuels, the components of the exergy balance of the methanerair detonation wave for the Hugoniot curve points in the vicinity of point J were plotted in Fig. 10. The thermal exergy b 1 of the initial mixture was assumed as 100% Ž b 1 s 2.761 MJrkg.. The exergy loss of the shock varies slightly around 15%, whereas the exergy loss of the combustion, also varying slightly, is smaller and varies around 7%. The characteristic extremum of e1 , bs , es , dbs and dbc appear at the specific volume Õ 2 J .
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Fig. 10. Components of the exergy balances of the methanerair detonation wave as a function of specific volume Õ 2 .
Comparison of detonation and deflagration for combustion of methanerair stoichiometric mixture showed similar results to that obtained for previous fuels Žhydrogen and carbon monoxide.. With varying Õ 2 , the total exergy loss db changes very little in the vicinity of Õ 2 K and the flat maximum appears for Õ 2 s Õ 2 K . The total exergy loss of deflagration Ž26.9%. is larger than the exergy loss of combustion in the detonation wave Ž8.03%. and is even larger than the total exergy loss of detonation Ž20.94%..
6. Conclusion A simplified model of adiabatic detonation process in gases was considered using the concept of exergy. The energy analysis of detonation, as well as deflagration, shows no loss of energy in these processes. However, with use of the exergy analysis of detonation, the exergy loss due to the pressure shock and the combustion reactions, can be disclosed. It was found that the exergy loss of the shock is larger than that for the combustion. The deflagration shows only the exergy loss due to combustion. The total exergy loss for deflagration is larger than that for detonation. The difference in exergy losses for detonation and deflagration results from the high temperature and pressure of the final combustion products of detonation in comparison to these parameters during deflagration. It means that during detonation, the degradation of chemical energy of fuel is smaller. This advantage of detonation over deflagration can be intuitively deduced, but the exergy analysis exposes this fact numerically.
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The suggestion also results that to pursue for a better thermodynamic utilization of gaseous fuel, the consideration of possible design of any burner device system based on the detonation process, would be motivated. Nomenclature b specific exergy, Jrkg bq ex exergy of heat delivered from external source, Jrkg cp specific heat at constant pressure, JrŽkg K. CV calorific value, Jrkg e specific kinetic energy, Jrkg g gram fraction of component h total specific enthalpy, Jrkg P absolute pressure, Pa q heat delivered, Jrkg qex heat delivered from external source, Jrkg R individual gas constant, JrŽkg K. s specific entropy, JrŽkg K. T absolute temperature, K u flow velocity, mrs Õ specific volume, m3rkg z mole fraction of component Greek g db Ds
isentropic exponens exergy loss, Jrkg increase of entropy, JrŽkg K.
Subscripts c combustion i successive number j successive number k successive number F fuel s shock 0 environment 1 mixture 1X mixture after shock 2 combustion products
References w1x w2x w3x w4x
I. Glassman, Combustion, 3rd edn., Academic Press, San Diego, 1996. J. Szargut, R. Petela, Exergy, WNT, Warsaw, 1965, Žin Polish.. J. Szargut, R. Petela, Exergy, Energija, Moscow, 1968, Žin Russian.. J. Szargut, D.R. Morris, F.R. Steward, Exergy Analysis Of Thermal, Chemical, And Metallurgical Processes, Hemisphere Publishing, New York, 1988.
Application of exergy analysis to the hydrodynamic theory of detonation in gases Richard Petela Technology Scientific Ltd., 152 Ranch Estates DriÕe N.W., Calgary, Alberta, Canada T3G 1K4 Received 1 October 1999; accepted 1 April 2000
Abstract The exergy analysis was applied to the phenomenon of detonation in gases. Separately were considered the component processes of the pressure shock and the chemical reaction of combustion. The considerations were illustrated by the examples of detonation of stoichiometric mixtures of dry air and hydrogen or carbon monoxide or methane. A measure of the thermodynamic imperfectness of process is the exergy loss in the process. It was found that exergy loss of the shock is larger than the exergy loss of the chemical reaction of combustion. A steady detonation characterized by the Chapman–Jouguet point occurs at the extremum of these losses. The exergy loss of combustion during steady deflagration, considered for comparison, is larger than the total exergy loss of detonation. It suggests that from thermodynamic viewpoint, the detonation is a better method of fuel utilization than the deflagration. Therefore, the consideration of any new industrial burner device based on the detonation process would be motivated. q 2000 Elsevier Science B.V. All rights reserved. Keywords: Exergy; Hydrodynamic; Detonation
1. Introduction The present paper contributes with the exergy analysis applied to detonation in gases. Combustion of premixed combustible gases can develop at either the relatively low propagation velocity Ždeflagration., e.g., not exceeding about 200 cmrs, or at the significantly larger velocity Ždetonation.. The detonation can occur in a long tube filled up with a premixed gaseous mixture of fuel and oxygen, within the flammability limit, when an ignition is applied at the closed end and at open other end. The hydrodynamic theory of the combustion wave propagation is well-described, e.g., by Glassman w1x. A 0378-3820r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 2 0 Ž 0 0 . 0 0 0 9 9 - 0
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planar, one-dimensional flame front, as shown in Fig. 1, traveling down the combustible mixture, is considered. The subscript 1 is used for the unburned gas conditions and subscript 2 for the burned gas conditions. The wave phenomena can be analytically described by the following independent equations for: Flow continuity: u1 u2 s Ž 1. Õ2 Õ2 Momentum conservation: P1 q
u12 Õ1
s P2 q
u 22
Ž 2.
Õ2
Energy conservation: h1 q
u12 2
q qex s h 2 q
u 22 2
Ž 3.
State 2: P2 Õ 2 s RT2
Ž 4.
where: Õ s gas specific volume; u s gas velocity; P s gas pressure; T s absolute temperature of the gas; h s total enthalpy of gas; qex s heat delivered to the gas from external source; R s individual gas constant. The initial state of the mixture is determined and its parameters are related according to the state equation: P1Õ1 s RT1
Ž 5.
The total enthalpies take into account the sensible and chemical parts: h1 s c p Ž T1 y T0 . q Ý g i CVi
Ž 6.
i
h 2 s c p Ž T2 y T0 .
Ž 7.
where: c p s specific heat of the gas; CVi s calorific value of ith component of the initial mixture; g i s gram fraction of ith component of the initial mixture; T0 s reference temperature, e.g., the absolute environment temperature.
Fig. 1. Propagation of combustion wave.
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Specific heat c p and isentropic exponens g , are assumed to be constant and the following relation is used: g cp s R Ž 8. gy1 Using Eqs. Ž1. and Ž2., the difference of squares velocities can be expressed as: u12 y u 22 s Ž P2 y P1 . Ž Õ 1 q Õ 2 .
Ž 9.
Introducing Eqs. Ž6. – Ž9. into Eq. Ž3., the well-known Hugoniot equation can be derived:
g gy1
1
Ž P2 Õ 2 y P1Õ1 . y Ž P2 y P1 . Ž Õ1 q Õ 2 . s q
Ž 10 .
2
where q s qex q Ý g i CVi
Ž 11 .
i
There are only four available equations, Eqs. Ž1. – Ž4., to solve the problem containing five unknowns, u1 , u 2 , Õ 2 , T2 and P2 . Therefore, the theoretical solution is not given by one determined point but is represented by the points of curve Ž10. from which, however, only a part corresponds to the realistic solutions. A unique solution of the problems requires some additional considerations of the mechanism of the process. For the determined value of q, which is a parameter of the family of curves, the representation of Eq. Ž10. are the curves in a coordinate system Ž P2 , Õ 2 ., as shown in Fig. 2. The curve at q ) 0 represents the states of gas after combustion, if the initial state of gas, entering the wave, is represented by point 1. This curve relates to the process which is composed of the two interacting phenomena considered separately.
X
X
Fig. 2. Representation of the two interacting detonations phenomena: shock 1–1 and combustion 1 –2, in P, Õ system of coordinates.
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X
X
Fig. 3. Representation of the two interacting detonations phenomena: shock 1–1 and combustion 1 –2, in T, s system of coordinates.
These phenomena are: the compression in the shock wave, represented by the Section 1–1X of the curve at q s 0, and the combustion, represented by the Section 1X –2 of the straight combustion line determined by Eq. Ž2.. The detonation phenomena can also be presented in the temperature–entropy diagram as shown in Fig. 3. Usually, the detonation wave is considered as composed consecutively of shock, induction period, and chemical reactions of combustion. It could be assumed that during induction period the temperature, pressure and velocity of gas remain unchanged. Therefore, neglecting the processes occurring during induction period, the temperature of gas from initial value T1 increases during shock to value T1X , and then increases again to T2 due to exothermic chemical reactions of combustion. Pressure of gas from initial value P1 increases during shock to the value P1X , and then drops during combustion to the value P2 larger than P1. Gas velocity from initial value u1 decreases during shock to u1X , and then increases during combustion to the value u 2 smaller than u1. When including point 1X into considerations, additional four unknowns T1X , P1X , u1X and Õ 1X have to be calculated using the four additional equations. For the adiabatic combustion process Ž1X –2., the energy conservation equation, with taking into account Eq. Ž8., can be applied:
g R
gy1
Ž T1X y T2 . q g F CVF q
u12X y u 22 2
s0
Ž 12 .
The momentum conservation Eq. Ž2. can be applied for the states 1X and 2: P1X q
u12X Õ 1X
s P2 q
u 22 Õ2
Ž 13 .
Continuity equation is: u1X Õ1
X
s
u2 Õ2
Ž 14 .
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State equation: P1X Õ1X s RT1X Ž 15 . Eqs. Ž3. and Ž12. represent energy conservation equations based on the first law of thermodynamics. The objective of the present paper is to involve also the second law of thermodynamics and, therefore, the corresponding equations of exergy balance will be included. 2. Exergy considerations Thermodynamic mechanism of various processes can be analyzed with use of the exergy w2–4x. Exergy analysis allows us to develop additional explanation, evaluation or interpretation of the substances values and processes performances which often differs dramatically from the energy analysis results. The basic equation in the exergy analysis, analogically to the energy analysis, is the exergy balance equation. Whereas, in steady state, the energy input is equal to the energy output of the system, the exergy input and exergy output for a real system are different and has to be compensated by the exergy loss due to the irreversibility of system processes. For example, the exergy balance equation for the detonation wave changing from state 1 to state 2, analogically to energy balance Eq. Ž3., takes a form: u12
u 22
q bq ex s b 2 q q db Ž 16 . 2 2 where: db s the total exergy loss due to thermodynamic imperfectness of the analyzed processes; bq ex s the exergy of heat delivered to the gas from external heat source; b 1 , b 2 s so-called thermal exergy of gas entering and exiting the considered system, respectively. The velocities u1 and u 2 in Eq. Ž16. are absolute which means relative to the earth. The thermal exergy includes the so-called physical and chemical parts. A physical exergy, results from the temperature and pressure discrepancies in relation to the temperature and pressure of the environment. A chemical exergy results from a difference in partial pressures of mixture components at environment temperature and the partial pressures of these components in the environment air. The thermal exergy b 1 of the combustible mixture, composed of different components numbered successively by i, can be calculated from the following formula derived for gaseous fuels w2–4x: b1 q
ž
b 1 s c p Ž T1 y T0 . y T0 c p ln
T1 T0
y Rln
P1 P0
/
q g i b F i q RT0Ý g i ln z i
Ž 17 .
i
where: b F i s the chemical exergy of ith components Žcorresponding to its calorific value.; g i , z i s the gram and mole Žrespectively. fraction of the ith mixture component. For simplification, formula Ž17. is written for the assumption that the standard temperature, for which the chemical enthalpy and exergy are tabulated, is equal to the environment temperature. It was also assumed that the partial pressure of water vapor in actual and standard environments are equal. These assumptions cause negligible error.
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The thermal exergy b 2 of the combustion products of complete combustion, composed of different components numbered successively by j, can be calculated from the formula derived for gaseous mixture composed of the air components w2–4x:
ž
b 2 s c p Ž T2 y T0 . y T0 c p ln
T2 T0
y Rln
P2 P0
/
q RT0Ý g j ln j
z j P2 Pj0
Ž 18 .
where: g j , z j s the gram and mole Žrespectively. fractions of jth products component, Pj0 s the partial pressures of jth products components in the environment air. According to the Gouy–Stodola law, the exergy loss caused by thermodynamic irreversibility of a process is calculated as follows: db s T0Ý Ž D s . k
Ž 19 .
k
where: D s s the entropy increase of k th body Žincluding, e.g., all substances and heat sources. taking part in the process. The exergy values of each particular input component or output component is a measure of its practical values, whereas the exergy loss is a measure of the quality of the process in the system. The larger the exergy of the component, the larger the practical value of this component. However, the smaller the exergy loss, the better the system performance. The exergy loss for real process is always positive. For an ideal Žreversible. process, the exergy loss is zero. Exergetic optimization of the process is based on the minimization of the exergy loss. However, when the exergy loss is negative then it means that the occurring of the considered process is impossible. In the case of adiabatic detonation process, the sum of the entropy increases consists only of the entropy increase D s of the gases: db s T0 D s
Ž 20 .
Assumptions for exergy considerations follow the assumptions for energy considerations w1x. In the simplified model of detonation, the kind of gas does not change during processes of shock and combustion Ž c p s const, g s const.. Both processes are replaced by the physical processes; the first process, compression with significant change of flow velocity, and the second process, preheating by the internal heat source Žcalorific value. with further flow velocity change. It is assumed that the residual chemical exergy of gases, resulting from discrepancy between its composition and the composition of environment, can be neglected. Consequently, the sum terms in Eqs. Ž17. and Ž18. can be neglected and the chemical exergy of mixture can be assumed equal to its calorific value, g i b F i s g i CVi . The above assumptions allow us to simplify Eqs. Ž17. and Ž18. as follows:
ž ž
b 1 s c p Ž T1 y T0 . y T0 c p ln b 2 s c p Ž T2 y T0 . y T0 c p ln
T1 T0 T2 T0
y Rln
y Rln
P1 P0 P2 P0
/ /
q g i CVi
Ž 21 . Ž 22 .
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It is possible to consider the shock and combustion processes as the separate successive steps. Therefore, according to Eq. Ž20., the exergy loss dbs due to irreversible shock can be calculated as: dbs s T0 Ž s1X y s1 .
Ž 23 .
and the exergy loss dbc due to irreversible combustion as: dbc s T0 Ž s2 y s1X .
Ž 24 .
The total exergy loss db comprise both losses: db s dbs q dbc
Ž 25 .
The exergy losses can be also calculated from the respective exergy balance equations. For example, the exergy loss for the shock can be calculated from the exergy balance equation for the shock process: b1 q
u12 2
s b 1X q
u12X 2
q dbs
Ž 26 .
where u1X is the velocity of gases during induction period and the specific exergy b 1X of this gases is:
ž
b 1X s c p Ž T1X y T0 . y T0 c p ln
T1X T0
y Rln
P1X P0
/
Ž 27 .
The extreme values of the above exergy losses can be determined from the relation obtained by equating of the partial derivative to zero. For example, the minimum total exergy loss would result from the following condition: E Ž db . EÕ 2
s0
Ž 28 .
The illustration of considerations is further carried out with some assumptions. The adiabatic process is considered, which means that the heat delivered to the gas from external source is zero, qex s 0 and thus also bq ex s 0. These assumptions are quite realistic as the detonation process occurs very quickly and there is a little time for exchanging heat with external source. The initial mixture is composed of a gaseous fuel and dry air. The initial temperature T1 of the mixture is equal to the standard temperature T0 of environment air ŽT1 s T0 s 258C., as assumed for the tabulated values of calorific values and chemical exergy of various combustible substances. The initial pressure of the mixture is assumed equal to the standard pressure Ž P1 s P0 s 100 kPa. of environment air. From theoretical considerations w1x, it results that the only steady-state solution for the detonation region of the Hugoniot curve, is given by point J, also called the Chapman–Jouguet detonation point. This is the point determined by tangency of the combustion line with the Hugoniot curve at a given parameter q, as shown, for example,
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Fig. 4. Exergy losses at detonation of hydrogenrair stoichiometric mixture.
Fig. 5. Steady deflagration at point K.
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Table 1 Some assumed data and calculation results on the detonation combustion of hydrogen, carbon monoxide or methane Data
Stoichiometric mixture of dry air and: H2
CO
CH 4
Component of combustible mixture, % by weight
H 2 s 2.85, N2 s 74.53, O 2 s 22.62 1.403 396.4 H 2 Os 25.4, N2 s 74.6
COs 28.98, N2 s 54.48, O 2 s16.54 1.400 290.0 CO 2 s 45.52, N2 s 54.48
119.96 118.23
10.1 9.83
CH 4 s 5.52, N2 s 72.47, O 2 s 22.01 1.388 299.9 CO 2 s15.14, H 2 Os12.39, N2 s 72.47 50.02 52.15
0.70141 2.4897
0.5116 2.8667
0.5281 2.5542
21.302 0.0430
17.942 0.0429
16.696 0.0432
13.20 8.09 21.29
12.26 7.10 19.36
12.91 8.03 20.94
27.55
25.09
26.90
Isentropic exponens of mixture, g Gas constant for mixture, R, JrŽkg K. Components of combustion products % by weight Calorific value of fuel, CV, MJrkg Chemical exergy of fuel, b F , MJrkg Parameters at point J: Õ 2 J , m3 rkg P2 J , MPa Parameters at point K : Õ 2 K , m3 rkg P2 K , MPa Exergy losses, Ž% of fuel exergy.: Detonation at point J Pressure shock Combustion reaction Total Deflagration at point K Total
for combustion of the hydrogenrair mixture, in Fig. 4, which shows also the exergy losses discussed in the next paragraph. For comparison to detonation, the deflagration is also considered. From theoretical considerations w1x, it results that the only steady-state solution for the deflagration region of the Hugoniot curve, is given by point K, also called the Chapman–Jouguet deflagration point. This point, shown in Fig. 5, is determined by tangency of combustion line with the Hugoniot curve at given parameter q, for the pressure range P2 - P1. More details on assumed data are discussed for the three following examples which are the combustion of hydrogen, carbon oxide or methane with the stoichiometric amount of dry air. Some data assumed for calculations and obtained results are shown in Table 1.
3. Combustion of hydrogen To calculate the unknown parameters at points 1X and 2, the system containing Eqs. Ž1. –8. and Eqs. Ž12. – Ž15., was used. The exergy losses were calculated form Eqs. Ž23.
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and Ž24.. The obtained results are shown in Fig. 4. All the three exergy losses reach an extremum at the Õ 2 J . The exergy loss of the shock reaches a minimum whereas the
Fig. 6. Energy and exergy balances of detonation wave of stoichiometric hydrogenrair mixture at point J.
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exergy loss of the combustion approaches a maximum. The total exergy loss, which is the sum of these two losses, reaches a flat minimum. It should be mentioned that the solutions represented on the right-hand side from point Õ 2 J , Ždashed lines., have no real significance. It results that the real steady-state detonation develops towards the minimum of the total exergy loss. The typical diagrams of the energy and exergy balances of detonation wave, including separately shock and combustion, are shown in Fig. 6. Data were calculated for the point J representing steady detonation. A mixture energy h1 , or exergy b 1 , is assumed as 100% Ž h1 s b 1 s 3.4189 MJrkg.. The energy balance shows a successive conversion of kinetic energy and total enthalpy, however does not indicate any losses. The kinetic energy of initial mixture and combustion products is high and significantly comparable to the respective total enthalpy. The exergy balance indicates irreversible losses during the shock and combustion phenomena. Obviously, if the velocities relative to the earth are used, the values of kinetic energy and kinetic exergy are the same. The values of the thermal exergy of gases are a little smaller than the respective values of enthalpy. The evolution of the components of the exergy balance of detonation wave for the Hugoniot curve points in the vicinity of point J is plotted in Fig. 7. The thermal exergy of initial mixture was assumed as 100%. It results that the exergy loss of the shock varies slightly around 13%, whereas the exergy loss of the combustion, also varying slightly, is smaller and varies around 8%. The kinetic energy, equal to the kinetic exergy, was denoted by the symbol e with respective subscript, e.g., e1 s u12r2.
Fig. 7. Components of the exergy balances of the hydrogenrair detonation wave as a function of specific volume Õ 2 .
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Fig. 8. Components of the exergy balances of the hydrogenrair deflagration wave as a function of specific volume Õ 2 .
Regarding deflagration, Fig. 8 presents the components of exergy balance varying with specific volume Õ 2 in the vicinity of value Õ 2 K . The thermal exergy b 1 of initial mixture was assumed as 100% Ž b 1 s 3.4189 MJrkg.. Kinetic energy of mixture was always very small Ž e1 - 0.06%. and is not shown in the diagram. With increasing Õ 2 the exergy b 2 of products decreases at the cost of kinetic exergy e 2 , which increases. The total exergy loss db changes very little and the flat maximum appears for Õ 2 s Õ 2 K . As the deflagration does not include a shock, this exergy loss, at the value of about 27.55%, is for combustion only, although a certain small portion of the exergy loss could be assigned to the conversion of chemical exergy of fuel into the kinetic exergy of the combustion products. Total exergy loss of deflagration Ž27.55%. is larger than the exergy loss of combustion in the detonation wave Ž8.09%. and is even larger than the total exergy loss of detonation Ž21.29%.. Based on the above values of the total exergy loss for Õ 2 J and Õ 2 K for the same fuel, which was hydrogen, a hypothetical conclusion can be withdrawn, that the combustion in a detonation wave is more efficient than in the deflagration. This conclusion was also confirmed for other fuels considered in the next paragraphs.
4. Combustion of carbon monoxide Some data on the considered mixture of CO and air and combustion products, are given in Table 1. The calculation results were obtained in the same way like for
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Fig. 9. Components of the exergy balances of the COrair detonation wave as a function of specific volume Õ 2 .
previously considered combustion of hydrogen. The evolution of the components of the exergy balance of the COrair detonation wave for the Hugoniot curve points in the vicinity of point J is plotted in Fig. 9. The thermal exergy b 1 of initial mixture was assumed as 100% Ž b 1 s 2.926 MJrkg.. It results that the exergy loss of shock varies slightly around 14%, whereas the exergy loss of combustion, also varying slightly, is smaller and varies around 6%. The characteristic extremum of e1 , bs , es , dbs and dbc appear at the specific volume Õ 2 J . Comparison of detonation and deflagration, similar to that for hydrogen, for combustion of COrair stoichiometric mixture, shows similar results. For example, with varying Õ 2 , the total exergy loss db changes insignificantly in the vicinity of Õ 2 K and achieves a flat maximum at Õ 2 s Õ 2 K . The total exergy loss of deflagration Ž25.09%. was found to be larger than the exergy loss of combustion Ž7.1%. in the detonation wave and was even larger than the total exergy loss of detonation Ž19.36%..
5. Combustion of methane Applying the same calculation procedure as for the previous fuels, the components of the exergy balance of the methanerair detonation wave for the Hugoniot curve points in the vicinity of point J were plotted in Fig. 10. The thermal exergy b 1 of the initial mixture was assumed as 100% Ž b 1 s 2.761 MJrkg.. The exergy loss of the shock varies slightly around 15%, whereas the exergy loss of the combustion, also varying slightly, is smaller and varies around 7%. The characteristic extremum of e1 , bs , es , dbs and dbc appear at the specific volume Õ 2 J .
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Fig. 10. Components of the exergy balances of the methanerair detonation wave as a function of specific volume Õ 2 .
Comparison of detonation and deflagration for combustion of methanerair stoichiometric mixture showed similar results to that obtained for previous fuels Žhydrogen and carbon monoxide.. With varying Õ 2 , the total exergy loss db changes very little in the vicinity of Õ 2 K and the flat maximum appears for Õ 2 s Õ 2 K . The total exergy loss of deflagration Ž26.9%. is larger than the exergy loss of combustion in the detonation wave Ž8.03%. and is even larger than the total exergy loss of detonation Ž20.94%..
6. Conclusion A simplified model of adiabatic detonation process in gases was considered using the concept of exergy. The energy analysis of detonation, as well as deflagration, shows no loss of energy in these processes. However, with use of the exergy analysis of detonation, the exergy loss due to the pressure shock and the combustion reactions, can be disclosed. It was found that the exergy loss of the shock is larger than that for the combustion. The deflagration shows only the exergy loss due to combustion. The total exergy loss for deflagration is larger than that for detonation. The difference in exergy losses for detonation and deflagration results from the high temperature and pressure of the final combustion products of detonation in comparison to these parameters during deflagration. It means that during detonation, the degradation of chemical energy of fuel is smaller. This advantage of detonation over deflagration can be intuitively deduced, but the exergy analysis exposes this fact numerically.
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The suggestion also results that to pursue for a better thermodynamic utilization of gaseous fuel, the consideration of possible design of any burner device system based on the detonation process, would be motivated. Nomenclature b specific exergy, Jrkg bq ex exergy of heat delivered from external source, Jrkg cp specific heat at constant pressure, JrŽkg K. CV calorific value, Jrkg e specific kinetic energy, Jrkg g gram fraction of component h total specific enthalpy, Jrkg P absolute pressure, Pa q heat delivered, Jrkg qex heat delivered from external source, Jrkg R individual gas constant, JrŽkg K. s specific entropy, JrŽkg K. T absolute temperature, K u flow velocity, mrs Õ specific volume, m3rkg z mole fraction of component Greek g db Ds
isentropic exponens exergy loss, Jrkg increase of entropy, JrŽkg K.
Subscripts c combustion i successive number j successive number k successive number F fuel s shock 0 environment 1 mixture 1X mixture after shock 2 combustion products
References w1x w2x w3x w4x
I. Glassman, Combustion, 3rd edn., Academic Press, San Diego, 1996. J. Szargut, R. Petela, Exergy, WNT, Warsaw, 1965, Žin Polish.. J. Szargut, R. Petela, Exergy, Energija, Moscow, 1968, Žin Russian.. J. Szargut, D.R. Morris, F.R. Steward, Exergy Analysis Of Thermal, Chemical, And Metallurgical Processes, Hemisphere Publishing, New York, 1988.