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The thermal theory based equation for correlation between temperature and flammability limits of hydrocarbons
Fuel 214 (2018) 55–62
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Full Length Article
The thermal theory based equation for correlation between temperature and flammability limits of hydrocarbons
MARK
⁎
Mingqiang Wu, Gequn Shu, Hua Tian , Xueying Wang, Yuewei Liu State Key Laboratory of Engines, Tianjin University, People’s Republic of China
A R T I C L E I N F O
A B S T R A C T
Keywords: Temperature dependence Flammability limits Thermal theory Hydrocarbon
Theoretical equation to evaluate the effect of temperature on the flammability limits of pure hydrocarbons was proposed in present study and compared with other methods available in the literatures. Theoretical equation was based on the thermal theory. Verification of the linear equation has been implemented on the cases of hydrogen, methane, propane, propylene, butylene, pentane and isobutane. The equation in this paper shows an average absolute relative error of 0.43% and 0.93% for lower and upper flammability limits, respectively. The equation possesses better prediction accuracy than other available methods at UFL. The reason for better accuracy is that the simple chemical kinetics about oxygen consumption was used to calculate reaction heat at upper flammability limit. Finally, the temperature range of application of theoretical equation was discussed in detail.
1. Introduction Studies have shown that hydrocarbons possess the better thermodynamic (a high decomposition temperature) and environmental properties (zero ozone depletion potential) than hydrofluorocarbon in medium-high temperature Organic Rankine Cycle (ORC) system [1,2]. Thus, hydrocarbons are often used for waste heat recovery in engine. However, flammability of such compounds imposes restrictions on their practical application. Thus, it is an important task to investigate flammability characteristics of such compounds for ensuring the safety in practical application. The lower and upper flammability limits are the flammability properties regularly used to evaluate the flammability hazards of gases [3–5]. Many industrial processes such as ORC generally operate at the temperature range of 81–222.5 °C [6,7]. In order to guarantee the safety of operation, it is necessary to get the flammability limits of hydrocarbons at elevated temperature. However, most experimental data for flammability limits was generally measured at 25 °C and few data measured at non-ambient temperature was available for many hydrocarbons. As we all know, the flammable zone of hydrocarbons will broaden if the temperature rises; that is to say, the lower flammability limit (LFL) becomes lower and the upper flammability limit (UFL) becomes higher [8–10]. Thus, safety instructions using the flammability limits at room temperature may result in a severe explosion hazard when temperature rises. Therefore, it is necessary for researchers to investigate the effect of temperature on flammability limits.
⁎
Several researchers [8,11–13]have deeply studied the flammability of fuel using different methods, which can be approximately divided into two principal categories: a) the method based on CAFT; and b) the group contribution method based on molecular structure. The latter method requires a large amount of experimental data measured at different temperatures to build the model. Thus, The GC method may be unfit particularly for predicting the temperature dependence of the flammability limits. The latter method is frequently applied to predict flammability limits at ambient temperature. Most forecasting methods based on the calculated adiabatic flame temperature were proposed to account for the temperature dependence of lower flammability limit. This method can determine the LFL of the hydrocarbon when the approximation of the threshold temperature is provided. White [14] deemed that the limit flame temperature remains the same no matter how the initial temperature changes. The modified Burgess-Wheeler law suggested by Zabetakis [15] was a very useful tool in solving the effect of temperature on the LFLs of hydrocarbons:
L = L25°−
0.75 (T −25°) ΔHC
(1)
where T is the temperature in °C, L25° is the LFL at 25 °C, and ΔHC is the heat of combustion in kilocalories per mole. Assuming adiabatic flame temperature was independent of initial temperature, Britton and Frurip [16] considered that the lower flammability limits of the hydrocarbon was linear between the CAFT and initial temperature. Another empirical formula was present by Britton and Frurip:
Corresponding author at: State Key Laboratory of Engines, Tianjin University, No. 92, Weijin Road, Nankai Region, Tianjin 300072, People’s Republic of China. E-mail address: [email protected] (H. Tian).
http://dx.doi.org/10.1016/j.fuel.2017.10.127 Received 3 September 2017; Received in revised form 4 October 2017; Accepted 29 October 2017 0016-2361/ © 2017 Elsevier Ltd. All rights reserved.
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Nomenclature AARE CAFT LFL UFL GC n m T0
L = LT °
T L(U) L0(U0) LT(UT) Q CP HT H ad ΔHc∗ ΔH c
average absolute relative error calculated adiabatic flame temperature lower flammability limit upper flammability limit group contribution number of carbon atom in hydrocarbon number of hydrogen atom in hydrocarbon initial temperature (known)
Tad−T Tad−T0
predict the temperature dependence of UFL than the Eq. (4). Mendiburu [19,20] tried to use chemical equilibrium to calculate product and reaction heat at UFL and got acceptable accuracy. The above methods indicated that reaction heat at UFL is not equal to U ∗ΔHC.. Chemical kinetics of combustion of hydrocarbons play an important part at UFL [21,22] in air. Thus, it is difficult to precisely calculate the reaction heat without involving chemical kinetics or chemical equilibrium in the modeling process. As described above, there is not an effective and theoretical method to easily estimate reaction heat at UFL. Thus, a simple chemical kinetics about oxygen is introduced to calculate reaction heat. It is a special interest to know whether the UFLs at different initial temperature of low hydrocarbon could be explained well by the equation using simple chemical kinetics.
(2)
where Tad is the CAFT at LFL. Kondo [17] measured the temperature dependence of upper and lower flammability limits of methane, propane, isobutane, ethylene and propylene and used the modified Burgess-Wheeler law to estimate lower flammability limits with respect to different temperatures. Mendiburu [8,18] researched temperature dependence of lower flammability limits of CeHeO and CeH compounds at atmospheric pressure in air. If the variable K in equation proposed by Mendiburu were equal to 1, then the equation proposed by Britton and Frurip would be same as the equation suggested by Mendiburu. For upper flammability limit, methods to explain temperature dependence is relatively scarce compared with lower flammability limit. Zabetakis [15] also interpreted the effect of temperature using the modified Burgess-Wheeler law:
UT = U25° +
0.75 (T −25°) ΔHC
2. Method (3)
Establishing and validation of the equation contain a procedure consisting of four steps:
where ΔHC is the heat of combustion in kilocalories per mole, U25° is the UFL at temperature 25 °C, and UT is the estimated UFL at temperature T. The prediction accuracy of the modified Burgess-Wheeler law is poor. Kondo [17] presented the linear equation based on assumption of a constant heat of combustion per mole of oxygen to explain the dependence of UFL on initial temperature:
UT = U25° +
100Cp,L Q
(T −25°)
(1) Build the equation related to T, U(or L); (2) Calculate the CAFTs of hydrocarbons with help of the flammability limits at T0; (3) Re-formulate equations related to T, U(or L) and getting correlation between U (or L) and T; (4) Use the developed equations to estimate the lower and upper flammability limits of hydrocarbon-air mixtures at different initial temperatures and validate the reliability of the equations by comparing the estimated values with observed values available in the references;
(4)
where Q is heat of combustion per mole of oxygen and Cp,L is the heat capacity of unburnt gas at UFL at 25 °C. Kondo proposed the linear equation easily only replacing heat of combustion with constant heat of combustion per mole of oxygen in Eq. (3), however, the prediction accuracy of the method is poor. In order to improve the prediction accuracy, another empirical equation was put forward based on the geometric mean G.
100Cp,L (T −25°)⎫ UT = U25° ⎧1 + ⎬ ⎨ L 25° ΔHc ⎭ ⎩
temperature lower(upper) flammability limit lower(upper) flammability limit at T0 (known parameter) lower(upper) flammability limit at T heat of combustion per mole of oxygen specific heat at constant pressure enthalpy at the temperature of T enthalpy at the calculated adiabatic flame temperature reaction heat at upper flammability limit heat of the combustion
Values of CAFT for different pure hydrocarbon generally were in the range of 1000–1600 K according to literatures [22,23]. In order to precisely get the values of CAFT, the adiabatic flame temperature was calculated by the CHEMKIN software based on chemical equilibrium and minimization of Gibb’s free energy for hydrocarbon-air mixture at fixed enthalpy and pressure when the values of flammability limits of hydrocarbon-air mixture were known. The values of CAFT of hydrocarbons at LFL and UFL are obtained and compared with the values in Refs. [12,23]. As shown in Table 1, A good consistency can be achieved between the present values and those in Refs. [12,23], which demonstrates that the results of the CAFT obtained from CHEMKIN software are valid.
(4a)
Mendiburu [19,20] also developed semi-empirical method using chemical equilibrium to estimate the UFL at different initial temperatures for CeH compounds and CeHeO compounds. The prediction accuracy is acceptable; however, the derivation process of the formulas is a bit complex. For lower flammability limits, most methods to evaluate the temperature dependence are empirical formulas except the equation suggested by Mendiburu [8] and Liaw [4]. The prediction accuracy for those methods is acceptable. This is because using value of L∗ΔHC can better take the place of value of reaction heat considering complete combustion at LFL. However, for the upper flammability limits, U ∗ΔHC cannot well explain reaction heat due to incomplete combustion. Kondo proposed U ∗Q rather than U ∗ΔHC to represent reaction heat. The predicted results show that neither the Eq. (3) proposed by Zabetakis nor the Eq. (4) presented by Kondo can accurately represents reaction heat at UFL. The empirical Eq. (4a) proposed by the Kondo can better
Table 1 Comparison between the present CAFTs and those in Refs. [12,23].
56
Compounds
UFL[12] (vol%)
CAFT at UFL (K)
Ref.[12] (K)
LFL[23] (vol%)
CAFT at LFL (K)
Ref.[23] (K)
Methane Ethane Propylene Propane
15 12.5 11 10
1769.2 1392.7 1440.8 1249.1
1763 1387 1452 1247
5 3 2.4 2.1
1482.5 1535.7 1624 1530.4
1482 1534 1621 1530
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hydrocarbon-air; ΔHcCn Hm represents heat of the combustion per mole fuel gas. Based on burning one mole of the mixture at LFL, the amount of components is shown in Table 2 before and after combustion. Supposing combustion of Eq. (5) takes place under adiabatic environment, Eq. (6) can be derived from the energy balance:
Table 2 Quantities of the components in the mixtures at LFL before and after burning. Compounds
No. of moles before reaction
No. of moles after reaction
Hydrocarbon Oxygen Nitrogen Carbon dioxide Water
L (1-L)/4.773 3.773(1-L)/4.773 0 0
0 (1-L)/4.773-nL-(mL)/4 3.773(1-L)/4.773 nL (mL)/2
T ad α1 HCO + α2 HHT2O + α3 HOT2 + α4 HNT2 + LΔHcCn Hm → α1 HCO + α2 HHad2O 2 2
+ α3 HOad2 + α4 HNad2
The symbol ad and T stand for the CAFT and the temperature, respectively. Inserting values of α1−α4 into Eq. (6) and getting Eq. (7):
Table 3 Quantities of the components in the mixtures at UFL before and after burning. Compounds
No. of moles before reaction
No. of moles after reaction
Hydrocarbon Oxygen Nitrogen Carbon monoxide Carbon dioxide Water Hydrogen
U (1-U)/4.773 3.773(1-U)/4.773 0 0 0 0
0 0 3.773(1-U)/4.773 Un 0 2(1-U)/4.773-Un U(n + m/2)-2(1-U)/4.773
ad T ad T ad T LΔHCnHm = L∗n (HCO 2−HCO2 ) + m ∗ L /2(HH 2O−HH2 O ) + (1−L)(Hair −Hair )
−L (n + m /4)(HOad2 −HOT2).
Exp.LFL(vol %)
CAFT at LFL (K)
Exp.UFL(vol %)
CAFT at UFL (K)
Methane Propane Isobutane Propylene Methane
5 2.07 1.689 2.21 5.05
1482 1500 1561 1519 1486
15.5 9.7 7.8 10.1 15.62
1717 1295 1248 1452 1709
(7)
The L, lower flammability limit, is gained by re-arranging Eq. (7):
L=
100CP,air (Tad−T ) . T T nHCO + m /2 ∗ HHT 2O−Hair −(n + m /4) HOT2 + A 2
(8)
ad ad ad ad in which A = ΔHCnHm−nHCO 2−m /2 ∗ HH 2O + Hair + (n + m /4) ∗ HO2 . T T T T Value of nHCO 2 + m /2 ∗ HH 2O−Hair −(n + m /4) HO2 is so small comparing with that of A that the former can be omitted. Thus, a brief linear relation between lower flammability limit and temperature is obtained. As shown in Eq. (9)
Table 4 Calculated adiabatic flame temperatures at experimental LFL and UFL in Refs. [10,17]. Compounds
(6)
L=
100CP,air (Tad−T ) . A
(9)
2.2. Estimation of the UFL at different initial temperatures Combustion products of most hydrocarbons at LFL always are H2O and CO2 for sufficient oxygen to support complete combustion. But the prediction at UFL is totally different from that at LFL due to oxygen deficiency. Thus, it is difficult to precisely calculate combustion products. As pointed out in Refs.[21,23], the chemical kinetics, in terms of rate constants, has important influence on prediction at UFL. Westbrook [24] proposed a two-reaction model to calculate combustion products at UFL. According to him, reaction rates between radical species such as O, H, OH and hydrocarbon molecules such as CH4, C2H6 are generally much greater than between the same radical species and CO or H2. Therefore, the radical species won’t react with CO or H2 while fuel or other hydrocarbon species still remain. Thus, it is difficult to precisely calculate combustion products without involving the chemical kinetics at UFL because there is not enough oxygen to oxidize H2 and CO. When establishing the equation to estimate UFL at the different initial temperatures, the hypotheses hereinafter are adopted:
Table 5 Thermochemical data used to obtain the predicted values of flammability limits. Compounds
Heat capacity CP,J/(K mol)
Heat of combustion (kJ/mol)
CH4 C3H6 C3H8 iC4H10 H2 H2O CO CO2 N2 O2
34.92 64.61 73.93 97.15 28.85 75.35 29.14 37.21 29.12 29.36
890.3 2057 2220 2869 285.8 – 283 – – –
2.1. Determination of the LFL at different initial temperatures (1) The combustion reaction at UFL of hydrocarbon takes place at atmospheric pressure; (2) The calculated adiabatic flame temperatures stay the same at UFL with the initial temperature increasing or reducing; (3) The Cp is constant with the temperature range from 5 to 150 °C. (4) The chemical kinetics about oxygen consumption. Oxygen is used to calculate the combustion products and reaction heat. The details about the chemical kinetics about oxygen consumption Oxygen are as follows:
Some of the major hypotheses are shown in order to build the theoretical equation for estimating the LFL: 1) The combustion reaction at LFL of hydrocarbon takes place at atmospheric pressure; 2) The calculated adiabatic flame temperatures remain unchanged at LFL with the initial temperature increasing or reducing; 3) The hydrocarbon reacts completely at LFL; 4) The Cp of is constant with the temperature range from 5 to 150 °C.
Firstly, the CO and H2 generate from combustion of hydrocarbon if oxygen is deficient; secondly if there remains residual oxygen when CO and H2 have already generated, H2 should be oxidized to H2O; and finally CO will be oxidized to CO2 if there is still residual oxygen. Lewis et al. [25] and Bartok and Sarofim [26] primarily put forward the simple chemical kinetics about oxygen consumption oxygen. This paper tries to calculate the effect of temperature on flammability limits by adopting the simple chemical kinetics. Combustion products and
Assuming one mole mixture of hydrocarbon and air burns at LFL, the overall reaction of mixture can be obtained easily.
LCn Hm +
1 (1−L)(O2 + 3.773N2) → α1 CO2 + α2 H2 O + α3 O2 + α4 N2 4.773 + LΔHcCn Hm
(5)
where L is the LFL volume fraction of hydrocarbon in mixture of 57
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Fig. 1. Comparison of between predicted results estimated by different methods and observed results for methane, propylene, propane and isobutane, respectively.
air burns at UFL, the overall reaction of mixture can be obtained easily.
reaction heat of hydrocarbons at UFL are calculated by using the hypothesis (4) and experimental data in Ref. [17]. The computed results using the priority hypothesis show that combustion products are CO, H2O and H2 excluding CO2. Based on burning one mole of the mixture at UFL and the simple chemical kinetics, the amount of components is shown in Table 3 before and after combustion. As we all know, the amount of oxygen in combustion at UFL is consumed completely. Assuming one mole mixture of hydrocarbon and
UCn Hm +
1 (1−U )(O2 + 3.773N2) → α1 CO + α2 CO2 + α3 H2 O 4.773 + α4 H2 + α5 N2 + ΔHc∗
(10)
With the same derivation process used at LFL in this paper and some equations used in Ref. [12], the Eq. (11) can be obtained: 58
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flammability limits estimated by different methods. The method of Britton and Frurip, the modified Burgess–Wheeler law and the equation in this study all can highly predict the variation of lower flammability limit. The AARE is applied to analysis which method is better. Table 6 shows AARE results of those methods for predicting the LFL at different initial temperatures. The method built in the present paper shows slightly better results than other two methods in terms of AARE in Table 6. The AARE of other two methods is close to each other and is less than 0.7%. This phenomenon can be explained easily. The detailed explanation is that all methods are based on the assumption of constant heat of combustion per mole of fuel gas at the lower flammability limit; however, other two methods are oversimplified empirical formulas so that they ignore enthalpy value of products at Tad that are included in A in Eq. (9). On the whole, the temperature dependence of lower flammability limit of hydrocarbon can be estimated accurately by all three methods. This is because that liberated heat which determines value of slope in the equation at LFL is same for all three methods. The Eq. (9) can precisely estimate the temperature dependence of lower flammability limit from 5 °C to 100 °C. The data of flammability limits at high temperature is rare. A way is put forward without using experimental data to evaluate the applicability of the equation to high temperature at LFL. The Eq. (9) is built based on the assumption that the CAFT of the mixture of hydrocarbon-air is independent of initial temperature at LFL. According to this hypothesis, if LFL of the mixture is close to 0 with the initial temperature increasing, the initial temperature of the flammable mixture containing little fuel should be slightly larger than the adiabatic flame temperature at 278 K. Assuming value of L in Eq. (9) is 0, the initial temperature can be calculated using Eq. (9). As shown in Table 7, the initial temperature at LFL = 0 is slightly larger than the CAFT at 278 K. Thus, the equation proposed at LFL can be applied to high temperature.
Table 6 Comparison of the different methods for determining the LFL and UFL at different initial temperatures. Methods
AARE at LFL (%)
AARE at UFL (%)
The modified Burgess–Wheeler law Britton and Frurip The method proposed by Kondo This work
0.68 0.58
2.18 1.44 0.93
0.43
Table 7 Comparison of between CAFT at 278 K and the initial temperature at LFL = 0. Compounds
CAFT(K)
TLFL=0 (K)
RDa(%)
CH4 C3H6 C3H8 iC4H10
1482 1519 1500 1561
1588 1648 1523 1689
7.15 8.49 1.53 8.20
a
RD is relative difference.
U=
(42CP,H 2O−42CP,H 2 + 79CP,N 2) T −C K −B
(11)
In which, ad K = nΔHCO−ΔHCnHm + (m /2 + n + 0.42)ΔHH 2 + nHCO −(n + 0.42) HHad2O
+ (m /2 + n + 0.42) HHad2−0.79HNad2
(12)
T B = nHCO −(n + 0.42) HHT 2O + (m /2 + n + 0.42) HHT 2−0.79HNT 2
(13)
C = −42ΔHH 2 + 42CP,H 2O Tad−42CP,H 2 Tad + 79CP,N 2 Tad
(14)
Value of B is so small compared with that of K that the former can be omitted. The linear relation expression (Eq. (15)) between upper flammability limit and temperature can be deduced.
U=
(42CP,H 2O−42CP,H 2 + 79CP,N 2) T −C K
3.1.2. UFL at different initial temperatures The CAFT results of hydrocarbon-air mixtures at upper flammability limits were calculated and presented in Table 4. The temperatures calculated in Table 4 were used to predict the upper flammability limits of the hydrocarbon-air mixtures at different initial temperatures. Thermochemical data adopted to gain the estimated values is listed in Table 5. Using the above data, temperature dependence of upper flammability limit can be calculated. Thus, the effect of temperature on upper flammability limit could be estimated by using the Eq. (15). Comparison of predicted results of upper flammability limit estimated by different methods with the observed values available in literature is shown in Fig. 2. As shown in Fig. 2, An excellent agreement between the observed and estimated values is observed for the Eq. (15). The result estimated by Eq. (15) is better than the result predicted by the Eq. (4a) proposed by Kondo. The performance of the modified Burgess-Wheeler law is the worst among three methods. The deviation between observed and predicted values can be analyzed quantitatively through the AARE. Table 6 shows AARE results of those methods at different initial temperatures. The above situation can be explained excellently by analyzing slope of equations. As pointed out at LFL, the heat liberated through combustion is a main reason determining value of slope in the equation. At the upper flammability limit zone, the reaction heat per mole of hydrocarbon is totally different from that at lower flammability limit zone. Furthermore, when the concentration of hydrocarbon increases in UFL zone with initial temperature increasing, the reaction heat per mole of hydrocarbon in combustion reduces sharply. If we still adopted constant reaction heat per mole of hydrocarbon at UFL, the slope would be smaller than actual value. Comparison of the Eq. (4a) with Eq. (3) show that the reaction heat in denominator in the Eq. (3) is U25/L25 times larger than that of the Eq. (4). Thus, the slope in Eq. (4) is bigger that of Eq. (3). As we all know, the chemical kinetics about oxygen consumption has little influence on prediction at LFL for enough oxygen [23]. However for UFL evaluation, the chemical kinetics has an
(15)
3. Results and discussion The linear equation to estimate the change of flammability limits at different initial temperatures is presented, based on the constant calculated adiabatic flame temperature and energy conservation, and will be validated with experimental data from the literature [4,10,14,17,27].The linear equations proposed in the present paper also are compared with methods available in the reference. The average absolute relative error (AARE) is used as a tool to judge which one is better.
AARE =
1 n
n
∑ i=1
|refi −esti | refi
× 100% (16)
in which ref is short for the observed value in literature, est is short for the predicted value. 3.1. Comparing the predicted results with experimental data [10,17] 3.1.1. LFL at different initial temperatures The CAFT results of hydrocarbon-air mixtures at LFL were calculated and presented in Table 4. The CAFT results in Table 4 were used to predict the lower flammability limits of the hydrocarbon-air mixtures at different initial temperatures. Thermochemical data used to gain the estimated values is listed in Table 5. The data of the thermodynamic properties comes from the NIST Chemistry WebBook [28]. Using the above data, temperature dependence of flammability limits can be calculated using the Eq. (9). Fig. 1 clearly shows comparison of predicted results of lower 59
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Fig. 2. Temperature dependence of upper flammability limit for methane, propylene, propane and isobutane, respectively.
products may transformed from CO, H2O and H2 to CO, C or unburnt hydrocarbon and H2 with temperature raising continually. Thus, the relation between temperature and upper flammability limit may not always be linear; the graph of temperature and upper flammability limit may be broken line from room temperature to adiabatic flame temperature. This is because that reaction heat reduces rapidly, which results in sharp slope when combustion products transform from CO, H2O
important influence on prediction accuracy and it is difficult to accurately calculate the reaction heat for lacking of oxygen. Thus, using the priority basis about oxygen can better determine liberated heat and composition of combustion products. For the application of the equation to high temperature at upper flammability limit, it may be different from LFL. According to the priority basis about oxygen, the composition of the combustion 60
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Fig. 3. Comparing the predicted results with the observed results from different Refs. [4,14,27].
and H2 to CO, C or unburnt hydrocarbon and H2. The Eq. (15) may be unfit for extraordinary high temperature.
shown in Fig. 3, a good agreement between the observed and predicted values is obtained. The predicted results of five fuels in Fig. 3 can be divided into two cases: ones are hydrogen and n-pentane, and others are propylene, butylene, and methane. For the former, the theoretical correlation presented can predict the outcome pretty well both at LFL and UFL. For the latter, the theoretical correlation underestimates the observed values of UFL at the last point. That may means the theoretical
3.2. Comparing the results with experiment data from Refs. [4,14,27] For temperature less than 100 °C, the prediction accuracy is guaranteed on the basis of experimental data measured by [4,14,27]. As 61
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correlation may be not suitable for a high temperature. Just as mentioned in Section 3.1.2, the application of the equation to high temperature at upper flammability limit is limited due transformation of combustion products. Counting the components indicates that the transformation of combustion products have taken place at the last point. Thus, the lower reaction heat leads to the larger slope. For the LFL of five fuels, there is a better consistency. In general, The theoretical correlation is effective way to solve the temperature dependence of flammability limits for hydrocarbons under 100 °C.
[3] [4] [5]
[6]
[7]
4. Conclusion [8]
Most of the existing methods used to predict the effect of temperature on flammability limits of hydrocarbon are empirical methods and most methods at UFL are unable to accurately calculate reaction heat. Aiming at the problems existing in those methods, this paper has developed theoretical method to estimate the temperature dependence of flammability limits and introduced the simple chemical kinetics about oxygen consumption to calculate reaction heat. Based on the research, some significant conclusions are summarized as follow:
[9] [10]
[11]
[12]
(1) For the lower flammability limit, value of heat of combustion times L is taken as the value of reaction heat in all methods, so deviation is found to be very small. However, the correlation in this paper investigated the effect of enthalpy value of products at Tad that was simplified in empirical formula. The accuracy of the prediction is raised by 21% in this method, compared with two empirical methods. (2) For the upper flammability limit, the simple chemical kinetics about oxygen consumption was investigated to precisely determine reaction heat. Mathematical analysis of the correlation shows reaction heat is the dominant factor which determines the prediction accuracy. The accuracy of the prediction for the correlation containing the priority hypothesis is improved by 35%, compared with the modified expression of Kondo. (3) Finally, the temperature range of application of theoretical correlation was discussed in detail. The results showed that the range of application of theoretical equation for LFL was wider than that of UFL.
[13]
[14] [15] [16] [17] [18]
[19]
[20] [21]
[22]
Acknowledgements
[23]
This work was supported by a grant from the National Natural Science Foundation of China (No. 51676133).
[24] [25]
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(Organic Rankine Cycle) for engine waste heat recovery. Energy 2014;74(5):428–38. Shen X, Zhang B, Zhang X, et al. Explosion behaviors of mixtures of methane and air with saturated water vapor. Fuel 2016;177:15–8. Liaw HJ, Chen KY. A model for predicting temperature effect on flammability limits. Fuel 2016;178:179–87. Gharagheizi F, Ilani-Kashkouli P, Mohammadi AH. Estimation of lower flammability limit temperature of chemical compounds using a corresponding state method. Fuel 2013;103(1):899–904. Shu G, Zhao M, Tian H, et al. Experimental comparison of R123 and R245fa as working fluids for waste heat recovery from heavy-duty diesel engine. Energy 2016;115(1):756–69. Shu G, Zhao M, Tian H, et al. Experimental investigation on thermal OS/ORC (Oil Storage/Organic Rankine Cycle) system for waste heat recovery from diesel engine. Energy 2016;107:693–706. Mendiburu AZ, Carvalho JAD, Coronado CR. Estimation of lower flammability limits of C–H compounds in air at atmospheric pressure, evaluation of temperature dependence and diluent effect. J Hazard Mater 2015;285:409–18. Cui G, Li Z, Yang C. Experimental study of flammability limits of methane/air mixtures at low temperatures and elevated pressures. Fuel 2016;181:1074–80. Cui G, Yang C, Li ZL, et al. Experimental study and theoretical calculation of flammability limits of methane/air mixture at elevated temperatures and pressures. J Loss Prev Process Ind 2016;41:252–8. Frutiger J, Marcarie C, Abildskov J, et al. Group-contribution based property estimation and uncertainty analysis for flammability-related properties. J Hazard Mater 2016;318:783. Shu G, Long B, Tian H, et al. Flame temperature theory-based model for evaluation of the flammable zones of hydrocarbon-air-CO2 mixtures. J Hazard Mater 2015;294:137–44. Shu G, Long B, Hua T, et al. Evaluating upper flammability limit of low hydrocarbon diluted with an inert gas using threshold temperature. Chem Eng Sci 2015;138:810–3. White AG. Limits for the propagation of flame in inflammable gas–air mixtures.III.The effect of temperature on the limits. J Chem Soc 1925;127:672–84. Zabetakis MG. Flammability characteristics of combustible gases and vapors. US Dept Inter Bur Mines; 1965. 1–129. Britton LG, Frurip DJ. Further uses of the heat of oxidation in chemical hazard assessment. Process Saf Prog 2003;22(1):1–19. Kondo S, Takizawa K, Takahashi A, et al. On the temperature dependence of flammability limits of gases. J Hazard Mater 2011;187(1–3):585. Mendiburu AZ, Carvalho JAD, Coronado CR, et al. Determination of lower flammability limits of C–H–O compounds in air and study of initial temperature dependence. Chem Eng Sci 2016;144:188–200. Mendiburu AZ, Jr DCJ, Coronado CR. Estimation of upper flammability limits of C–H compounds in air at standard atmospheric pressure and evaluation of temperature dependence. J Hazard Mater 2016;304(4):512. Mendiburu AZ. Determination of upper flammability limits of C–H–O compounds in air at reference temperature and atmospheric pressure. Fuel 2017;188:212–22. Shebeko YN, Fan W, Bolodian IA, et al. An analytical evaluation of flammability limits of gaseous mixtures of combustible–oxidizer–diluent. Fire Saf J 2002;37(6):549–68. Melhem GA. A detailed method for estimating mixture flammability limits using chemical equilibrium. Process Saf Prog 1997;16(4):203–18. Vidal M, Wong W, Rogers WJ, et al. Evaluation of lower flammability limits of fuel–air–diluent mixtures using calculated adiabatic flame temperatures. J Hazard Mater 2006;130(1–2):21. Westbrook CK. Simplified reaction mechanisms for the oxidation of hydrocarbon fuels in flames. Combust Sci Technol 1981;27(1–2):31–43. Lewis B, von Elbe G. Combustion, flames and explosions of gases. third ed. New York: Academic Press Inc; 1987. Bartok W, Sarofim AF. Fossil fuel combustion. New York: J. Wiley & Sons; 1991. Wierzba I, Wang Q. The flammability limits of H2–CO–CH4, mixtures in air at elevated temperatures. Int J Hydrogen Energy 2006;31(4):485–9. NIST Chemistry WebBook. NIST Standard Reference Database SRD Number 69, http://webbook.nist.gov/chemistry/ 2017 [accessed October 1, 2017].
Contents lists available at ScienceDirect
Fuel journal homepage: www.elsevier.com/locate/fuel
Full Length Article
The thermal theory based equation for correlation between temperature and flammability limits of hydrocarbons
MARK
⁎
Mingqiang Wu, Gequn Shu, Hua Tian , Xueying Wang, Yuewei Liu State Key Laboratory of Engines, Tianjin University, People’s Republic of China
A R T I C L E I N F O
A B S T R A C T
Keywords: Temperature dependence Flammability limits Thermal theory Hydrocarbon
Theoretical equation to evaluate the effect of temperature on the flammability limits of pure hydrocarbons was proposed in present study and compared with other methods available in the literatures. Theoretical equation was based on the thermal theory. Verification of the linear equation has been implemented on the cases of hydrogen, methane, propane, propylene, butylene, pentane and isobutane. The equation in this paper shows an average absolute relative error of 0.43% and 0.93% for lower and upper flammability limits, respectively. The equation possesses better prediction accuracy than other available methods at UFL. The reason for better accuracy is that the simple chemical kinetics about oxygen consumption was used to calculate reaction heat at upper flammability limit. Finally, the temperature range of application of theoretical equation was discussed in detail.
1. Introduction Studies have shown that hydrocarbons possess the better thermodynamic (a high decomposition temperature) and environmental properties (zero ozone depletion potential) than hydrofluorocarbon in medium-high temperature Organic Rankine Cycle (ORC) system [1,2]. Thus, hydrocarbons are often used for waste heat recovery in engine. However, flammability of such compounds imposes restrictions on their practical application. Thus, it is an important task to investigate flammability characteristics of such compounds for ensuring the safety in practical application. The lower and upper flammability limits are the flammability properties regularly used to evaluate the flammability hazards of gases [3–5]. Many industrial processes such as ORC generally operate at the temperature range of 81–222.5 °C [6,7]. In order to guarantee the safety of operation, it is necessary to get the flammability limits of hydrocarbons at elevated temperature. However, most experimental data for flammability limits was generally measured at 25 °C and few data measured at non-ambient temperature was available for many hydrocarbons. As we all know, the flammable zone of hydrocarbons will broaden if the temperature rises; that is to say, the lower flammability limit (LFL) becomes lower and the upper flammability limit (UFL) becomes higher [8–10]. Thus, safety instructions using the flammability limits at room temperature may result in a severe explosion hazard when temperature rises. Therefore, it is necessary for researchers to investigate the effect of temperature on flammability limits.
⁎
Several researchers [8,11–13]have deeply studied the flammability of fuel using different methods, which can be approximately divided into two principal categories: a) the method based on CAFT; and b) the group contribution method based on molecular structure. The latter method requires a large amount of experimental data measured at different temperatures to build the model. Thus, The GC method may be unfit particularly for predicting the temperature dependence of the flammability limits. The latter method is frequently applied to predict flammability limits at ambient temperature. Most forecasting methods based on the calculated adiabatic flame temperature were proposed to account for the temperature dependence of lower flammability limit. This method can determine the LFL of the hydrocarbon when the approximation of the threshold temperature is provided. White [14] deemed that the limit flame temperature remains the same no matter how the initial temperature changes. The modified Burgess-Wheeler law suggested by Zabetakis [15] was a very useful tool in solving the effect of temperature on the LFLs of hydrocarbons:
L = L25°−
0.75 (T −25°) ΔHC
(1)
where T is the temperature in °C, L25° is the LFL at 25 °C, and ΔHC is the heat of combustion in kilocalories per mole. Assuming adiabatic flame temperature was independent of initial temperature, Britton and Frurip [16] considered that the lower flammability limits of the hydrocarbon was linear between the CAFT and initial temperature. Another empirical formula was present by Britton and Frurip:
Corresponding author at: State Key Laboratory of Engines, Tianjin University, No. 92, Weijin Road, Nankai Region, Tianjin 300072, People’s Republic of China. E-mail address: [email protected] (H. Tian).
http://dx.doi.org/10.1016/j.fuel.2017.10.127 Received 3 September 2017; Received in revised form 4 October 2017; Accepted 29 October 2017 0016-2361/ © 2017 Elsevier Ltd. All rights reserved.
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Nomenclature AARE CAFT LFL UFL GC n m T0
L = LT °
T L(U) L0(U0) LT(UT) Q CP HT H ad ΔHc∗ ΔH c
average absolute relative error calculated adiabatic flame temperature lower flammability limit upper flammability limit group contribution number of carbon atom in hydrocarbon number of hydrogen atom in hydrocarbon initial temperature (known)
Tad−T Tad−T0
predict the temperature dependence of UFL than the Eq. (4). Mendiburu [19,20] tried to use chemical equilibrium to calculate product and reaction heat at UFL and got acceptable accuracy. The above methods indicated that reaction heat at UFL is not equal to U ∗ΔHC.. Chemical kinetics of combustion of hydrocarbons play an important part at UFL [21,22] in air. Thus, it is difficult to precisely calculate the reaction heat without involving chemical kinetics or chemical equilibrium in the modeling process. As described above, there is not an effective and theoretical method to easily estimate reaction heat at UFL. Thus, a simple chemical kinetics about oxygen is introduced to calculate reaction heat. It is a special interest to know whether the UFLs at different initial temperature of low hydrocarbon could be explained well by the equation using simple chemical kinetics.
(2)
where Tad is the CAFT at LFL. Kondo [17] measured the temperature dependence of upper and lower flammability limits of methane, propane, isobutane, ethylene and propylene and used the modified Burgess-Wheeler law to estimate lower flammability limits with respect to different temperatures. Mendiburu [8,18] researched temperature dependence of lower flammability limits of CeHeO and CeH compounds at atmospheric pressure in air. If the variable K in equation proposed by Mendiburu were equal to 1, then the equation proposed by Britton and Frurip would be same as the equation suggested by Mendiburu. For upper flammability limit, methods to explain temperature dependence is relatively scarce compared with lower flammability limit. Zabetakis [15] also interpreted the effect of temperature using the modified Burgess-Wheeler law:
UT = U25° +
0.75 (T −25°) ΔHC
2. Method (3)
Establishing and validation of the equation contain a procedure consisting of four steps:
where ΔHC is the heat of combustion in kilocalories per mole, U25° is the UFL at temperature 25 °C, and UT is the estimated UFL at temperature T. The prediction accuracy of the modified Burgess-Wheeler law is poor. Kondo [17] presented the linear equation based on assumption of a constant heat of combustion per mole of oxygen to explain the dependence of UFL on initial temperature:
UT = U25° +
100Cp,L Q
(T −25°)
(1) Build the equation related to T, U(or L); (2) Calculate the CAFTs of hydrocarbons with help of the flammability limits at T0; (3) Re-formulate equations related to T, U(or L) and getting correlation between U (or L) and T; (4) Use the developed equations to estimate the lower and upper flammability limits of hydrocarbon-air mixtures at different initial temperatures and validate the reliability of the equations by comparing the estimated values with observed values available in the references;
(4)
where Q is heat of combustion per mole of oxygen and Cp,L is the heat capacity of unburnt gas at UFL at 25 °C. Kondo proposed the linear equation easily only replacing heat of combustion with constant heat of combustion per mole of oxygen in Eq. (3), however, the prediction accuracy of the method is poor. In order to improve the prediction accuracy, another empirical equation was put forward based on the geometric mean G.
100Cp,L (T −25°)⎫ UT = U25° ⎧1 + ⎬ ⎨ L 25° ΔHc ⎭ ⎩
temperature lower(upper) flammability limit lower(upper) flammability limit at T0 (known parameter) lower(upper) flammability limit at T heat of combustion per mole of oxygen specific heat at constant pressure enthalpy at the temperature of T enthalpy at the calculated adiabatic flame temperature reaction heat at upper flammability limit heat of the combustion
Values of CAFT for different pure hydrocarbon generally were in the range of 1000–1600 K according to literatures [22,23]. In order to precisely get the values of CAFT, the adiabatic flame temperature was calculated by the CHEMKIN software based on chemical equilibrium and minimization of Gibb’s free energy for hydrocarbon-air mixture at fixed enthalpy and pressure when the values of flammability limits of hydrocarbon-air mixture were known. The values of CAFT of hydrocarbons at LFL and UFL are obtained and compared with the values in Refs. [12,23]. As shown in Table 1, A good consistency can be achieved between the present values and those in Refs. [12,23], which demonstrates that the results of the CAFT obtained from CHEMKIN software are valid.
(4a)
Mendiburu [19,20] also developed semi-empirical method using chemical equilibrium to estimate the UFL at different initial temperatures for CeH compounds and CeHeO compounds. The prediction accuracy is acceptable; however, the derivation process of the formulas is a bit complex. For lower flammability limits, most methods to evaluate the temperature dependence are empirical formulas except the equation suggested by Mendiburu [8] and Liaw [4]. The prediction accuracy for those methods is acceptable. This is because using value of L∗ΔHC can better take the place of value of reaction heat considering complete combustion at LFL. However, for the upper flammability limits, U ∗ΔHC cannot well explain reaction heat due to incomplete combustion. Kondo proposed U ∗Q rather than U ∗ΔHC to represent reaction heat. The predicted results show that neither the Eq. (3) proposed by Zabetakis nor the Eq. (4) presented by Kondo can accurately represents reaction heat at UFL. The empirical Eq. (4a) proposed by the Kondo can better
Table 1 Comparison between the present CAFTs and those in Refs. [12,23].
56
Compounds
UFL[12] (vol%)
CAFT at UFL (K)
Ref.[12] (K)
LFL[23] (vol%)
CAFT at LFL (K)
Ref.[23] (K)
Methane Ethane Propylene Propane
15 12.5 11 10
1769.2 1392.7 1440.8 1249.1
1763 1387 1452 1247
5 3 2.4 2.1
1482.5 1535.7 1624 1530.4
1482 1534 1621 1530
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hydrocarbon-air; ΔHcCn Hm represents heat of the combustion per mole fuel gas. Based on burning one mole of the mixture at LFL, the amount of components is shown in Table 2 before and after combustion. Supposing combustion of Eq. (5) takes place under adiabatic environment, Eq. (6) can be derived from the energy balance:
Table 2 Quantities of the components in the mixtures at LFL before and after burning. Compounds
No. of moles before reaction
No. of moles after reaction
Hydrocarbon Oxygen Nitrogen Carbon dioxide Water
L (1-L)/4.773 3.773(1-L)/4.773 0 0
0 (1-L)/4.773-nL-(mL)/4 3.773(1-L)/4.773 nL (mL)/2
T ad α1 HCO + α2 HHT2O + α3 HOT2 + α4 HNT2 + LΔHcCn Hm → α1 HCO + α2 HHad2O 2 2
+ α3 HOad2 + α4 HNad2
The symbol ad and T stand for the CAFT and the temperature, respectively. Inserting values of α1−α4 into Eq. (6) and getting Eq. (7):
Table 3 Quantities of the components in the mixtures at UFL before and after burning. Compounds
No. of moles before reaction
No. of moles after reaction
Hydrocarbon Oxygen Nitrogen Carbon monoxide Carbon dioxide Water Hydrogen
U (1-U)/4.773 3.773(1-U)/4.773 0 0 0 0
0 0 3.773(1-U)/4.773 Un 0 2(1-U)/4.773-Un U(n + m/2)-2(1-U)/4.773
ad T ad T ad T LΔHCnHm = L∗n (HCO 2−HCO2 ) + m ∗ L /2(HH 2O−HH2 O ) + (1−L)(Hair −Hair )
−L (n + m /4)(HOad2 −HOT2).
Exp.LFL(vol %)
CAFT at LFL (K)
Exp.UFL(vol %)
CAFT at UFL (K)
Methane Propane Isobutane Propylene Methane
5 2.07 1.689 2.21 5.05
1482 1500 1561 1519 1486
15.5 9.7 7.8 10.1 15.62
1717 1295 1248 1452 1709
(7)
The L, lower flammability limit, is gained by re-arranging Eq. (7):
L=
100CP,air (Tad−T ) . T T nHCO + m /2 ∗ HHT 2O−Hair −(n + m /4) HOT2 + A 2
(8)
ad ad ad ad in which A = ΔHCnHm−nHCO 2−m /2 ∗ HH 2O + Hair + (n + m /4) ∗ HO2 . T T T T Value of nHCO 2 + m /2 ∗ HH 2O−Hair −(n + m /4) HO2 is so small comparing with that of A that the former can be omitted. Thus, a brief linear relation between lower flammability limit and temperature is obtained. As shown in Eq. (9)
Table 4 Calculated adiabatic flame temperatures at experimental LFL and UFL in Refs. [10,17]. Compounds
(6)
L=
100CP,air (Tad−T ) . A
(9)
2.2. Estimation of the UFL at different initial temperatures Combustion products of most hydrocarbons at LFL always are H2O and CO2 for sufficient oxygen to support complete combustion. But the prediction at UFL is totally different from that at LFL due to oxygen deficiency. Thus, it is difficult to precisely calculate combustion products. As pointed out in Refs.[21,23], the chemical kinetics, in terms of rate constants, has important influence on prediction at UFL. Westbrook [24] proposed a two-reaction model to calculate combustion products at UFL. According to him, reaction rates between radical species such as O, H, OH and hydrocarbon molecules such as CH4, C2H6 are generally much greater than between the same radical species and CO or H2. Therefore, the radical species won’t react with CO or H2 while fuel or other hydrocarbon species still remain. Thus, it is difficult to precisely calculate combustion products without involving the chemical kinetics at UFL because there is not enough oxygen to oxidize H2 and CO. When establishing the equation to estimate UFL at the different initial temperatures, the hypotheses hereinafter are adopted:
Table 5 Thermochemical data used to obtain the predicted values of flammability limits. Compounds
Heat capacity CP,J/(K mol)
Heat of combustion (kJ/mol)
CH4 C3H6 C3H8 iC4H10 H2 H2O CO CO2 N2 O2
34.92 64.61 73.93 97.15 28.85 75.35 29.14 37.21 29.12 29.36
890.3 2057 2220 2869 285.8 – 283 – – –
2.1. Determination of the LFL at different initial temperatures (1) The combustion reaction at UFL of hydrocarbon takes place at atmospheric pressure; (2) The calculated adiabatic flame temperatures stay the same at UFL with the initial temperature increasing or reducing; (3) The Cp is constant with the temperature range from 5 to 150 °C. (4) The chemical kinetics about oxygen consumption. Oxygen is used to calculate the combustion products and reaction heat. The details about the chemical kinetics about oxygen consumption Oxygen are as follows:
Some of the major hypotheses are shown in order to build the theoretical equation for estimating the LFL: 1) The combustion reaction at LFL of hydrocarbon takes place at atmospheric pressure; 2) The calculated adiabatic flame temperatures remain unchanged at LFL with the initial temperature increasing or reducing; 3) The hydrocarbon reacts completely at LFL; 4) The Cp of is constant with the temperature range from 5 to 150 °C.
Firstly, the CO and H2 generate from combustion of hydrocarbon if oxygen is deficient; secondly if there remains residual oxygen when CO and H2 have already generated, H2 should be oxidized to H2O; and finally CO will be oxidized to CO2 if there is still residual oxygen. Lewis et al. [25] and Bartok and Sarofim [26] primarily put forward the simple chemical kinetics about oxygen consumption oxygen. This paper tries to calculate the effect of temperature on flammability limits by adopting the simple chemical kinetics. Combustion products and
Assuming one mole mixture of hydrocarbon and air burns at LFL, the overall reaction of mixture can be obtained easily.
LCn Hm +
1 (1−L)(O2 + 3.773N2) → α1 CO2 + α2 H2 O + α3 O2 + α4 N2 4.773 + LΔHcCn Hm
(5)
where L is the LFL volume fraction of hydrocarbon in mixture of 57
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Fig. 1. Comparison of between predicted results estimated by different methods and observed results for methane, propylene, propane and isobutane, respectively.
air burns at UFL, the overall reaction of mixture can be obtained easily.
reaction heat of hydrocarbons at UFL are calculated by using the hypothesis (4) and experimental data in Ref. [17]. The computed results using the priority hypothesis show that combustion products are CO, H2O and H2 excluding CO2. Based on burning one mole of the mixture at UFL and the simple chemical kinetics, the amount of components is shown in Table 3 before and after combustion. As we all know, the amount of oxygen in combustion at UFL is consumed completely. Assuming one mole mixture of hydrocarbon and
UCn Hm +
1 (1−U )(O2 + 3.773N2) → α1 CO + α2 CO2 + α3 H2 O 4.773 + α4 H2 + α5 N2 + ΔHc∗
(10)
With the same derivation process used at LFL in this paper and some equations used in Ref. [12], the Eq. (11) can be obtained: 58
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flammability limits estimated by different methods. The method of Britton and Frurip, the modified Burgess–Wheeler law and the equation in this study all can highly predict the variation of lower flammability limit. The AARE is applied to analysis which method is better. Table 6 shows AARE results of those methods for predicting the LFL at different initial temperatures. The method built in the present paper shows slightly better results than other two methods in terms of AARE in Table 6. The AARE of other two methods is close to each other and is less than 0.7%. This phenomenon can be explained easily. The detailed explanation is that all methods are based on the assumption of constant heat of combustion per mole of fuel gas at the lower flammability limit; however, other two methods are oversimplified empirical formulas so that they ignore enthalpy value of products at Tad that are included in A in Eq. (9). On the whole, the temperature dependence of lower flammability limit of hydrocarbon can be estimated accurately by all three methods. This is because that liberated heat which determines value of slope in the equation at LFL is same for all three methods. The Eq. (9) can precisely estimate the temperature dependence of lower flammability limit from 5 °C to 100 °C. The data of flammability limits at high temperature is rare. A way is put forward without using experimental data to evaluate the applicability of the equation to high temperature at LFL. The Eq. (9) is built based on the assumption that the CAFT of the mixture of hydrocarbon-air is independent of initial temperature at LFL. According to this hypothesis, if LFL of the mixture is close to 0 with the initial temperature increasing, the initial temperature of the flammable mixture containing little fuel should be slightly larger than the adiabatic flame temperature at 278 K. Assuming value of L in Eq. (9) is 0, the initial temperature can be calculated using Eq. (9). As shown in Table 7, the initial temperature at LFL = 0 is slightly larger than the CAFT at 278 K. Thus, the equation proposed at LFL can be applied to high temperature.
Table 6 Comparison of the different methods for determining the LFL and UFL at different initial temperatures. Methods
AARE at LFL (%)
AARE at UFL (%)
The modified Burgess–Wheeler law Britton and Frurip The method proposed by Kondo This work
0.68 0.58
2.18 1.44 0.93
0.43
Table 7 Comparison of between CAFT at 278 K and the initial temperature at LFL = 0. Compounds
CAFT(K)
TLFL=0 (K)
RDa(%)
CH4 C3H6 C3H8 iC4H10
1482 1519 1500 1561
1588 1648 1523 1689
7.15 8.49 1.53 8.20
a
RD is relative difference.
U=
(42CP,H 2O−42CP,H 2 + 79CP,N 2) T −C K −B
(11)
In which, ad K = nΔHCO−ΔHCnHm + (m /2 + n + 0.42)ΔHH 2 + nHCO −(n + 0.42) HHad2O
+ (m /2 + n + 0.42) HHad2−0.79HNad2
(12)
T B = nHCO −(n + 0.42) HHT 2O + (m /2 + n + 0.42) HHT 2−0.79HNT 2
(13)
C = −42ΔHH 2 + 42CP,H 2O Tad−42CP,H 2 Tad + 79CP,N 2 Tad
(14)
Value of B is so small compared with that of K that the former can be omitted. The linear relation expression (Eq. (15)) between upper flammability limit and temperature can be deduced.
U=
(42CP,H 2O−42CP,H 2 + 79CP,N 2) T −C K
3.1.2. UFL at different initial temperatures The CAFT results of hydrocarbon-air mixtures at upper flammability limits were calculated and presented in Table 4. The temperatures calculated in Table 4 were used to predict the upper flammability limits of the hydrocarbon-air mixtures at different initial temperatures. Thermochemical data adopted to gain the estimated values is listed in Table 5. Using the above data, temperature dependence of upper flammability limit can be calculated. Thus, the effect of temperature on upper flammability limit could be estimated by using the Eq. (15). Comparison of predicted results of upper flammability limit estimated by different methods with the observed values available in literature is shown in Fig. 2. As shown in Fig. 2, An excellent agreement between the observed and estimated values is observed for the Eq. (15). The result estimated by Eq. (15) is better than the result predicted by the Eq. (4a) proposed by Kondo. The performance of the modified Burgess-Wheeler law is the worst among three methods. The deviation between observed and predicted values can be analyzed quantitatively through the AARE. Table 6 shows AARE results of those methods at different initial temperatures. The above situation can be explained excellently by analyzing slope of equations. As pointed out at LFL, the heat liberated through combustion is a main reason determining value of slope in the equation. At the upper flammability limit zone, the reaction heat per mole of hydrocarbon is totally different from that at lower flammability limit zone. Furthermore, when the concentration of hydrocarbon increases in UFL zone with initial temperature increasing, the reaction heat per mole of hydrocarbon in combustion reduces sharply. If we still adopted constant reaction heat per mole of hydrocarbon at UFL, the slope would be smaller than actual value. Comparison of the Eq. (4a) with Eq. (3) show that the reaction heat in denominator in the Eq. (3) is U25/L25 times larger than that of the Eq. (4). Thus, the slope in Eq. (4) is bigger that of Eq. (3). As we all know, the chemical kinetics about oxygen consumption has little influence on prediction at LFL for enough oxygen [23]. However for UFL evaluation, the chemical kinetics has an
(15)
3. Results and discussion The linear equation to estimate the change of flammability limits at different initial temperatures is presented, based on the constant calculated adiabatic flame temperature and energy conservation, and will be validated with experimental data from the literature [4,10,14,17,27].The linear equations proposed in the present paper also are compared with methods available in the reference. The average absolute relative error (AARE) is used as a tool to judge which one is better.
AARE =
1 n
n
∑ i=1
|refi −esti | refi
× 100% (16)
in which ref is short for the observed value in literature, est is short for the predicted value. 3.1. Comparing the predicted results with experimental data [10,17] 3.1.1. LFL at different initial temperatures The CAFT results of hydrocarbon-air mixtures at LFL were calculated and presented in Table 4. The CAFT results in Table 4 were used to predict the lower flammability limits of the hydrocarbon-air mixtures at different initial temperatures. Thermochemical data used to gain the estimated values is listed in Table 5. The data of the thermodynamic properties comes from the NIST Chemistry WebBook [28]. Using the above data, temperature dependence of flammability limits can be calculated using the Eq. (9). Fig. 1 clearly shows comparison of predicted results of lower 59
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Fig. 2. Temperature dependence of upper flammability limit for methane, propylene, propane and isobutane, respectively.
products may transformed from CO, H2O and H2 to CO, C or unburnt hydrocarbon and H2 with temperature raising continually. Thus, the relation between temperature and upper flammability limit may not always be linear; the graph of temperature and upper flammability limit may be broken line from room temperature to adiabatic flame temperature. This is because that reaction heat reduces rapidly, which results in sharp slope when combustion products transform from CO, H2O
important influence on prediction accuracy and it is difficult to accurately calculate the reaction heat for lacking of oxygen. Thus, using the priority basis about oxygen can better determine liberated heat and composition of combustion products. For the application of the equation to high temperature at upper flammability limit, it may be different from LFL. According to the priority basis about oxygen, the composition of the combustion 60
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Fig. 3. Comparing the predicted results with the observed results from different Refs. [4,14,27].
and H2 to CO, C or unburnt hydrocarbon and H2. The Eq. (15) may be unfit for extraordinary high temperature.
shown in Fig. 3, a good agreement between the observed and predicted values is obtained. The predicted results of five fuels in Fig. 3 can be divided into two cases: ones are hydrogen and n-pentane, and others are propylene, butylene, and methane. For the former, the theoretical correlation presented can predict the outcome pretty well both at LFL and UFL. For the latter, the theoretical correlation underestimates the observed values of UFL at the last point. That may means the theoretical
3.2. Comparing the results with experiment data from Refs. [4,14,27] For temperature less than 100 °C, the prediction accuracy is guaranteed on the basis of experimental data measured by [4,14,27]. As 61
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correlation may be not suitable for a high temperature. Just as mentioned in Section 3.1.2, the application of the equation to high temperature at upper flammability limit is limited due transformation of combustion products. Counting the components indicates that the transformation of combustion products have taken place at the last point. Thus, the lower reaction heat leads to the larger slope. For the LFL of five fuels, there is a better consistency. In general, The theoretical correlation is effective way to solve the temperature dependence of flammability limits for hydrocarbons under 100 °C.
[3] [4] [5]
[6]
[7]
4. Conclusion [8]
Most of the existing methods used to predict the effect of temperature on flammability limits of hydrocarbon are empirical methods and most methods at UFL are unable to accurately calculate reaction heat. Aiming at the problems existing in those methods, this paper has developed theoretical method to estimate the temperature dependence of flammability limits and introduced the simple chemical kinetics about oxygen consumption to calculate reaction heat. Based on the research, some significant conclusions are summarized as follow:
[9] [10]
[11]
[12]
(1) For the lower flammability limit, value of heat of combustion times L is taken as the value of reaction heat in all methods, so deviation is found to be very small. However, the correlation in this paper investigated the effect of enthalpy value of products at Tad that was simplified in empirical formula. The accuracy of the prediction is raised by 21% in this method, compared with two empirical methods. (2) For the upper flammability limit, the simple chemical kinetics about oxygen consumption was investigated to precisely determine reaction heat. Mathematical analysis of the correlation shows reaction heat is the dominant factor which determines the prediction accuracy. The accuracy of the prediction for the correlation containing the priority hypothesis is improved by 35%, compared with the modified expression of Kondo. (3) Finally, the temperature range of application of theoretical correlation was discussed in detail. The results showed that the range of application of theoretical equation for LFL was wider than that of UFL.
[13]
[14] [15] [16] [17] [18]
[19]
[20] [21]
[22]
Acknowledgements
[23]
This work was supported by a grant from the National Natural Science Foundation of China (No. 51676133).
[24] [25]
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