In the Classroom
Rate Controlling Factors in a Bunsen Burner Flame Julio Andrade-Gamboa* and Hugo L. Corso Area de Química, Centro Regional Universitario Bariloche, Universidad Nacional del Comahue and Centro Atómico Bariloche, Comisión Nacional de Energía Atómica, Av. Bustillo Km 9.500, R8402AGP San Carlos de Bariloche, Río Negro, Argentina; *
[email protected] Fabiana C. Gennari Area de Química, Centro Regional Universitario Bariloche, Universidad Nacional del Comahue and Consejo Nacional de Investigaciones Científicas y Técnicas, Centro Atómico Bariloche, Comisión Nacional de Energía Atómica, Av. Bustillo Km 9.500, R8402AGP San Carlos de Bariloche, Río Negro, Argentina
Although the Bunsen burner is the most traditional heating device in a chemical laboratory, it has been infrequently studied by undergraduates. Several articles in this Journal show applications in which burners support flame tests (1), demonstrate stoichiometry (2), analyze heating efficiency (3), atomize solutions for atomic emission spectrometry (4), and stimulate critical (scientific) thinking (5). But a Bunsen burner, independently of any specific application, has educational importance per se because a flame is a very interesting chemical system. A Bunsen burner can be used to teach several chemical concepts at the undergraduate level. This observation was made about two decades ago by Craig et al., in a paper in this Journal (6) that starts, “Flames are an important class of chemical reactions to which little attention is ordinarily paid in undergraduate laboratories—other than as sources of instant heat.” In this manner Craig et al. introduced an experiment in which both the measurement and a detailed calculation of flame temperature, including a realistic gaseous composition, were done. Calculations of a mean flame temperature can also be made after several simplifications (7) and descriptions of flame zones are available (8) in a form suitable even for the high school level. Therefore, some aspects of the thermodynamics of a flame are covered. However, there is another important aspect related to flames that should be taught, namely the chemical–kinetic aspects. We have used a Bunsen burner to measure the burning velocity of methane combustion (9). Burning velocity depends on the overall combustion rate, which, in turn, results from the combination of several phenomena. Such an analysis is available in the advanced theoretical treatments of combustion and flame (10–15), but they are out of the range of undergraduate teaching. (Complete source of references, prior to year 1967, about theoretical and experimental principles of combustion and flame is found in the paper of Anderson, ref 16.) The possibility of doing a simple description of the chemical kinetics of a flame should be a useful complement, from the teaching point of view, to the thermodynamic description. In this paper, a simple model of combustion and flame that accounts for mass and energy transfer and kinetics of chemical reaction is presented. Based on experimental data,1 the model was used to establish the controlling regime of the Bunsen burner flame. Kinetics of a Process To develop the model proposed in this work it was useful to adopt a formulation common in chemical engineering that generalizes the rate, r, of a process as (17), 524
r =
F Ω
(1)
where F is the driving force and Ω is a resistance. The driving force depends on the kind of process. It can be a gradient of, for example, pressure, concentration, or temperature (some quantity proportional to free energy decreasing). The resistance can be a physical impediment, some quantity related to the activation energy, or a combination of both. For complex processes the resistance in eq 1 can be separated into different contributions, one for each single phenomenon (step). This procedure facilitates descriptions of complex processes and simplifies the chemical–kinetics analysis (18). The separation of a global resistance is easily done in cases in which the steps occur in parallel or in series, because the law of combination for Ω is the same as for electrical resistance. Then, for two steps working in series, with individual resistances Ω1 and Ω2, the total resistance Ω is given by Ω = Ω1 + Ω 2
(2)
and the overall rate, in series, is
rs =
F F = Ω Ω1 + Ω 2
(3)
On the other hand, if the steps occur in parallel, the total resistance is given by
1 1 1 = + Ω Ω1 Ω2
(4)
and the overall rate, in parallel, is rp =
F F F = + Ω Ω1 Ω2
(5)
These expressions allow us to analyze the influence of each step in the overall process. For instance, from eq 3, if one step has higher resistance (e.g., Ω1 > Ω2), its influence on the overall rate is greater. In particular, for Ω1 >> Ω2, then Ω ≅ Ω1, and the rate of the overall process reduces to the rate of step 1. In this situation step 1 is designated as the ratecontrolling step. For the case of steps in parallel, from eq 5 it can be seen that, if one step has a much higher resistance (e.g., Ω1 >> Ω2), the overall rate is essentially that of the step with the lower resistance (step 2). In this situation, although step 2 is not controlling, it represents the main contribution.
Journal of Chemical Education • Vol. 80 No. 5 May 2003 • JChemEd.chem.wisc.edu
In the Classroom
More general expressions, in which the driving forces for different steps are not necessarily the same, are obtained as follows. From eq 3 Ω 1 Ω1 Ω 1 1 = = + 2 = + rs F F F r1 r2
Therefore, in general, in series 1 1 1 = + rs r1 r2
(6)
From eq 5
rp =
F F F = + = r1 + r2 Ω Ω1 Ω2
extinguish, the adjacent unburned gas must be heated at temperatures higher than a minimum value, called the ignition temperature, which is lower than Tb. This heating is promoted by energy transfer, mass transfer, or both. The energy of combustion is transferred to “cold” zones, in particular to unburned gas (from right to left), at a rate that depends on the chemical reaction rate, the temperature gradient, and the thermal conductivity of the gas. Also, the consumption of combusting gas generates a concentration (or partial pressure) gradient that induces the molecular gas diffusion with respect to flame surface (from left to right in Figure 1). Other phenomena, such as diffusion of free radicals, although essential for the process, will be not considered in order to simplify the model. A Model for the Burning Velocity
Therefore, in general, in parallel
We will consider the methane combustion in air,
rp = r1 + r2
(7)
Flame Structure A flame is a thin region that separates the zone of unburned gas at temperature Ta from the gas products at the burning temperature, Tb. Figure 1 represents the longitudinal section of a portion of a flat flame. In the reference system in which the gas is at rest, a flame is a reaction front that moves (from right to left in Figure 1) at a linear velocity denominated the burning velocity, νB (14, p 84). If the unburned gas moves in an opposite direction to that of flame propagation, and at a rate equal to the burning velocity, the flame is fixed in space. This is known as flame stabilization and is attained in a Bunsen burner with the typical conical geometry, the burning velocity having a direction normal at each point of flame surface. In order for the flame not to
CH4 + 2O2 → CO2 + 2H2O which is the reaction that occurs in a Bunsen burner fed with natural gas, and assume the ignition temperature is equal to Tb. In order to develop a model, three steps will be considered: thermal conduction, molecular diffusion, and chemical reaction. The expressions corresponding to each step will be developed separately supposing limiting cases in which each step operates as the rate controlling step or makes the main contribution. The corresponding equations will be deduced considering the unburned gas as stationary. Modeling Each Step
Thermal Conduction as the Rate Controlling Step From Figure 1 the energy, Q, necessary to raise the temperature of the border of the overall flame, initially at Ta, to Tb is Q =
flame Tb
m δ
δ 0
c [T (x ) − Ta ] dx
where m is the total mass of unburned gas within the overall flame, c is its specific heat, δ is the flame thickness, and T(x) is the temperature profile in the flame, which is considered to be linear. Then T (x ) = Tb −
unburned gas
combustion products
(8)
Tb − Ta x δ
(9)
Assuming c as temperature independent1 and substitution of T(x) according to eq 9, integration of eq 8 results in
T − Ta Q = mc b 2
(10)
An energy flux controlled by thermal conduction is given by
1 dq dT T − Ta = K = K b A dt dx δ
Ta δ Figure 1. Flame structure and temperature gradient.
(11)
where A is the surface area of the flame and K is the thermal conductivity of CH4–air mixture. If the energy Q is transferred to the unburned gas at a rate equal to the thermal con-
JChemEd.chem.wisc.edu • Vol. 80 No. 5 May 2003 • Journal of Chemical Education
525
In the Classroom
duction rate, we obtain, from eq 11 and assuming K as temperature independent1, Q =
Q
dq =
0
∆t
KA 0
(Tb
− Ta ) (Tb − Ta ) dt = K A ∆t (12) δ δ
Combining eqs 10 and 12 we obtain KA
(Tb
− Ta ) mc ∆t = (Tb − Ta) δ 2
(13)
While the energy Q is transferred during a time ∆t, the flame front moves a length δ. Therefore, if the flame movement is controlled by thermal conduction only, the burning velocity is expressed as νB ≈ νQ = δ兾∆t. If gas expansion due to heating is neglected, we can replace m = ρδA, where ρ is the gas density at room temperature.1 Then, from eq 13 we obtain νB ⬇ νQ =
2K ρδ c
(14)
Molecular Diffusion as the Rate Controlling Step As mentioned above, in the flame there is a partial pressure or concentration gradient of gaseous reactants (CH4 and O2). We can suppose the reactant concentrations at flame borders of unburned and burned gas are C and zero, respectively, as shown in Figure 2. Therefore, the concentration gradient is (C − 0)兾δ. This gradient acts as the driving force for the molecular diffusion of each reactant. The molar flux from the unburned gas zone into the flame will be C dn = D δ A dt
where n and D are the number of moles and the diffusion coefficient of a gaseous reactant, respectively. If in eq 15, dn is replaced by CAdx, the following relation is obtained dx D = dt δ
(16)
Equation 16 gives the net average linear velocity of a molecule of a gaseous reactant. A representative value for D is the binary diffusion coefficient D12 for the CH4–air mixture. If the flame movement depends only on gaseous mass transfer by diffusion, the burning velocity will be given by νB ⬇ νD =
D12 δ
Chemical Reaction as the Rate Controlling Step In the case where the flame movement is controlled by the reaction rate, the energy from the chemical process is transferred to the unburned gas (this is equivalent to a high enough thermal conduction, i.e., K → ∞). The situation can be represented, as in Figure 3, with a flame at temperature Tb in contact with unburned gas at temperature Ta . Then, the energy transfer rate is dq = Rcomb δ A ∆H comb dt
(18)
where Rcomb and ∆Hcomb are the combustion rate (mol CH4 cm᎑3 s᎑1) and the combustion enthalpy (cal兾mol CH4), respectively. The energy dq associated with a flame displacement dx is the energy necessary to raise the temperature of a volume Adx of the unburned gas from Ta to Tb. Then, neglecting again the gas expansion,1 it can be expressed
(15)
flame
flame
C
Tb
unburned gas
combustion products
C=0 δ Figure 2. Gradient concentration of gaseous reactants.
526
(17)
unburned gas
combustion products
Ta δ Figure 3. Temperature profile for a flame chemically controlled.
Journal of Chemical Education • Vol. 80 No. 5 May 2003 • JChemEd.chem.wisc.edu
In the Classroom
dq = ρA c (Tb − Ta ) dx
(19)
Because this energy is transferred during the time dt, it follows from eq 19 that dq dx = ρA c (Tb − Ta ) dt dt
(20)
where νC = dx兾dt is the burning velocity controlled by chemical reaction. Therefore, from eqs 18 and 20 we obtain νB ⬇ νC =
Rcomb δ ∆Hcomb ρ c (Tb − Ta )
(21)
Combination of the Three Steps Equations 14, 17, and 21 represent asymptotic expressions for the burning velocity corresponding to conditions in which the flame moves controlled alternatively by one specific step. In a more general case, all phenomena contribute to the overall process, hence the three steps considered must be combined in a final expression for νB. The step of thermal conduction is in parallel with the step of gas diffusion, thus the contributions of both processes can be coupled based on eq 7 as νQ−D = νQ + νD
tion of the velocities of flame movement associated with each of the steps considered in the present model. Discussion We will use the present model to establish the rate-controlling regime in methane combustion. Table 1 summarizes the relevant experimental data used (9). Equation 24 is an expression of the burning velocity as a function of the velocities of all processes taken into account in the present model: thermal conduction, molecular diffusion, and chemical reaction. From this expression, the relative contribution of each phenomenon can be discussed based on some values of properties of the CH4–air mixture extracted or calculated from literature and shown in Table 2. From data of Table 2, taking the room temperature Ta = 293 K and assuming the CH4–air mixture as ideal (at room temperature,1 ρ = 3.3 × 10᎑4 g cm᎑3), the contributions of thermal conduction and molecular diffusion can be estimated using eqs 14 and 17, respectively. Thus
(22)
νQ =
1.5 cm2 s −1 δ
(25)
νD =
0.11 cm2 s −1 δ
(26)
Because the energy to be transferred to the unburned gas is produced in the combustion, the chemical reaction is a step in series with the other processes. Therefore, from eq 6, the burning velocity including the influence of the three steps is given by
Therefore, for any value of flame thickness, δ, the thermal conduction makes the main contribution to the burning velocity (see eq 22 or second term on denominator of eq 24). From data of Tables 1 and 2, the contribution of the combustion reaction (eq 21) can be estimated as
1 1 1 = + ν νC νB Q−D
νC = 3.4 × 106 cm 3 mol −1 δ Rcomb
(23)
Replacing in eq 23 νQ-D by eq 22 and rearranging, the final equation for νB is obtained
νB =
Replacing eqs 25, 26, and 27 in eq 24, for the burning velocity, we obtain νB =
1 1 1 + νC νQ + νD
(27)
(24)
1 −7
3 × 10 mol cm− 3 δ + Rcomb δ 2cm2 s −1
(28)
Equation 24 is an expression of the burning velocity as a func-
where for constants, only one significant figure is considered because of the approximations made.1
Table 1. Experimental Dataa
Table 2. Parameters for a CH4–Aira Mixture at Room Temperature
Parameter
Experimental Value
Burning velocity
7.7 cm s
᎑1
Flame temperature
1023 K
Atmospheric pressure
0.91 atmb
Methane mole fractionc
0.11
Oxygen mole fractionc
0.18
Air mole fractionc
0.89
Parameter
Value
cb
0.26 cal g᎑1 K᎑1
Kc
6.2 × 10᎑5 cal s ᎑1 cm᎑1 K᎑1
D12d
0.11 cm2 s ᎑1
∆Hcomb e
2.13 ×105 cal mol᎑1
a
Xair = 0.89.
a
b
b
c
c
d
Calculated from ref 19.
Ref 9.
Bariloche is at 780 m above sea level.
Corresponding to the operation of burner with the air inlet totally open.
Ref 20. Ref 21, pp 5–93.
e
Ref 21, pp 6–18.
JChemEd.chem.wisc.edu • Vol. 80 No. 5 May 2003 • Journal of Chemical Education
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In the Classroom
The rate controlling regime (chemical or transport) in a Bunsen flame can be deduced if the two terms in the denominator of eq 28 can be compared. If chemical reaction controls the overall process, from eq 27 and the measured burning velocity (νB = 7.7 cm兾s), the combustion rate (for δ a trial value of 0.1 cm) can be deduced to be Rcomb = 2.3 × 10᎑5 mol cm᎑3 s᎑1. Therefore, assuming a first-order reaction with respect to both CH4 and O2, an overall (bimolecular) constant rate of 7 × 106 mol᎑1 cm3 s-1 is calculated. This value is four or five orders of magnitude lower than the values of the rate constant of the main elementary steps of the methane combustion mechanism (14, pp 100–101). This means that flame propagation is controlled by transfer processes, as reported in the specialized literature (13, p 429; 14, p ii; 15, p 1). If we neglect the contribution of molecular diffusion, eq 14 will be the more representative expression for the burning velocity derived from the present model. From eq 14 the flame thickness can be expressed as,
δ =
2K ρ c νB
(29)
which only differs from the expression derived from advanced models (14, p 98) in the factor two. From eq 29, the calculated flame thickness is 0.2 cm. This is of the order of the trial value, used above, that allowed neglect of the chemical reaction as the controlling factor. Finally, the conditions determining the controlling regime in a flame can be qualitatively analyzed from eq 24 and used to introduce chemical–kinetics conceptions, which usually are related to heterogeneous systems (17). If the first term in the denominator of eq 24 is greater than the second one, chemical reaction controls the overall process. For any reaction system, the achievement of the chemical control condition is very important for a study of the intrinsic kinetics from direct measurements, because the derived parameters (rate constants, orders, activation energy, etc.) are the true ones. The present work shows that the overall process in a flame, at the experimental conditions imposed, is not chemically controlled because combustion is much faster than transport phenomena (the first term in the denominator of eq 24 is less than the second one). If the experimentation cannot be conducted under chemical control, the measured kinetic parameters are apparent, and the true ones could be assessed only by calculation if an appropriate model accounting for nonchemical processes applies (18). Equation 24 (or those improved expressions derived from it) is a simple example of a model that separates the contributions of the involved phenomena. Conclusion The model presented here is a chemical–kinetic description of the combustion process in a Bunsen burner flame, which can be used as a teaching tool to give, together with the thermodynamic analysis, a more complete physicochemical description of flame. In addition, the chemical–kinetic treatment is useful to introduce subjects such as mass and energy transfer in chemical reactions and the concepts of ratecontrolling regimes.
528
Note 1. In order to simplify the equations of the model, the temperature dependency of some quantities was neglected. Although these approximations introduce errors, only the order of magnitude of deduced quantities was used to extract conclusions.
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Journal of Chemical Education • Vol. 80 No. 5 May 2003 • JChemEd.chem.wisc.edu
