Pulsating linear in situ combustion: why do we often observe oscillatory behavior?

Abstract

We have studied simplified, pulsating, one-dimensional, in situ combustion processes. For two cases, with different reaction stoichiometry, oscillations in temperature, flue gas rate, and flue gas composition are demonstrated and the parameter space resulting in oscillatory behavior is identified. To understand the role of different parameters, linear stability of the problem is studied. Because linear stability analysis requires the solution of uniform front propagation, we investigated an asymptotic analytical solution of the problem. We found an original formula for the front propagation velocity. The analytical solution enabled us to define four dimensionless parameters including Zeldovich (Ze) number, Damkohler (Da) number, a specialized air-fuel ratio (B), and a ratio incorporating air and rock heat capacities (Δ₁). Using linear stability analysis, we show that the stability of the problem is also governed by these four parameters. Because Δ₁ ≈ 1 for typical laboratory conditions, the set of (Ze, Da, B) is used to construct the stability plane; consequently, several important design considerations are suggested. Both larger air injection rate and air enriched in oxygen increase the front propagation speed but push the system toward oscillatory behavior. Conversely, the introduction of catalysts and metal additives, that decrease the activation energy of reactions, increases the front speed and stability. Similarly, increasing the amount of fuel available for the combustion makes the design more stable and drives the combustion front to propagate more quickly.

Appendix

1.1 7.1 Front Velocity Calculation

We define the non dimensional parameter

$${\Psi} =\frac{{\int}_{T_{x}}^{T_{F}}{y\ C\ dT}} {(T_{F}-T_{x}){(y\ C)}_{x}} \\ $$

By plotting Ψ vs. T (Fig. 14), we observe empirically that at T = T_R, Ψ ≈ 1.

Fig. 14

Ψ vs. T used for front velocity analysis

So we have the following

$${\int}_{T_{R}}^{T_{F}}{y\ C} dT = y_{0} C_{0} (T_{F}-T_{R}) $$

From Eq. 8 (with $\frac {\partial }{\partial \theta }= 0$), we have

$$\frac{\partial C}{\partial T} \frac{\partial T}{\partial \xi} V =R_{0}\ y\ C\ {\Theta}(T-T_{R})\\ $$

By assuming that $\frac {\partial T}{\partial \xi }$ is constant between T_R and T_F and integration we write:

$$-C_{0}\frac{\partial T}{\partial \xi} V = R_{0} y_{0} C_{0} (T_{F}-T_{R})\\ $$

Additionally, from Eq. 13 and knowing that at ξ = 0, C = C₀,and T = T_R, we have:

$$\frac {\partial T}{\partial \xi} =\ \frac{-(T_{R}-T_{i})\ V c_{s} {\Delta}_{2}}{\lambda}\\ $$

Hence, we reach (22).

1.2 7.2 Non-dimensional Form of the Energy Balance Equation

Starting from the energy balance equation

$$\lambda\ A \frac{\partial^{2} T}{\partial \xi^{2}} +Q\ \dot{R}\ A=\dot{m} c_{p} -A\ V c_{s} \frac{\partial T}{\partial \xi} +A\ c_{s} \frac{\partial T}{\partial \theta} \\ $$

and using Eq. ?? and dimensionless temperature, we have:

$$\begin{array}{@{}rcl@{}} &&\frac{\lambda\ A (T_{f}-T_{i})}{L^{2}}\frac{\partial^{2} \overset{\frown}{T}} {\partial \overset{\frown}{\xi}^{2}} +\\ &&(T_{F}-T_{i}) \frac {(A\ Vc_{s} - {\dot m_{inj}} )}{V\ C_{0}} \ddot R \frac{\overset{\frown}{Y}\overset{\frown}{C}} {\overset{\frown}{m}}\ e^{Ze(1-\frac{1}{\overset{\frown}{T}} )}\\ &&= (\frac{\dot{m}_{inj}\ c_{p}\ (T_{F}-T_{i})} {L} )\overset{\frown}{m} \frac{\partial \overset{\frown}{T}} {\partial \overset{\frown}{\xi} } - (\frac{A\ Vc_{s}(T_{F}-T_{i})}{L} )\frac{\partial \overset{\frown}{T}} {\partial \overset{\frown}{\xi} } \\ &&+ (\frac{A\ c_{s}(T_{F}-T_{i})}{L} )\frac{\partial \overset{\frown}{T}} {\partial \overset{\frown}{\theta} } \end{array} $$

Dividing both sides by T_F − T_i and introducing Δ₁ we have:

$$\begin{array}{@{}rcl@{}} &&\frac{\lambda\ A}{L^{2}}\frac{\partial^{2} \overset{\frown}{T}} {\partial \overset{\frown}{\xi}^{2}} + {\Delta}_{1}\frac {Ac_{s}}{C_{0}} \ddot R \frac{\overset{\frown}{Y}\overset{\frown}{C}} {\overset{\frown}{m}}\ e^{Ze(1-\frac{1}{\overset{\frown}{T}} )}\\ &&= -{\Delta}_{1} (\frac{A\ V\ c_{s}}{L} )\overset{\frown}{m} \frac{\partial \overset{\frown}{T}} {\partial \overset{\frown}{\xi} } + (\frac{A\ Vc_{s}}{L} )(\overset{\frown}{m}-1)\frac{\partial \overset{\frown}{T}} {\partial \overset{\frown}{\xi} } \\ &&+ (\frac{A\ c_{s}}{\theta} )\frac{\partial \overset{\frown}{T}} {\partial \overset{\frown}{\theta} } \end{array} $$

in which we have added and subtracted the $\overset {\frown }{m} (\frac {A\ Vc_{s}}{L} )\frac {\partial \overset {\frown }{T}} {\partial \overset {\frown }{\xi } }$ term. By using Eq. 22, $\frac {\lambda \ A}{L^{2}}$ in the first term of the LHS, is written as:

$$\frac{\lambda\ A}{L^{2}}=\frac{\lambda\ A\ \ddot{R}^{2}}{V^{2} {C_{0}}^{2}}=\frac{A\ c_{s} \ddot{R}}{C_{0}} (\frac{T_{F}}{T_{R}} ) Ze {\Delta}_{2} \approx \frac{A\ c_{s} \ddot{R}}{C_{0}} Ze {\Delta}_{2} $$

By inserting L and 𝜃 in the RHS of the equation, we see that all the terms contain $\frac {A\ c_{s} \ddot {R}}{C_{0}}$. By dividing the equation by $\frac {A\ c_{s} \ddot {R}}{C_{0}}$ and rearranging we reach Eq. 29.

1.3 7.3 Convergence and Stability

We show the convergence of our simulation by temporal and spatial refinement of the simulation. We also show the stability of our numerical simulation to both infinitesimal and finite perturbations.

In general, for σ < 1, we found that the fully resolved system is achieved by having Δx = 0.3 mm and Δt = 0.01 min. Figures 15 and 16 show the flue gas rates and oxygen composition for the case of Δx = 0.3 mm and Δx = 0.15 mm. We see that convergence is achieved.

Fig. 15

Flue gases rate for case 1 with Δx = 0.3 mm and Δx = 0.15 mm and Δt = 0.01 min. Note convergence as Δx decreases

Fig. 16

Oxygen composition in flue gases for case 1 with Δx = 0.3 mm and Δx = 0.15 mm and Δt = 0.01 min. Note convergence as Δx decreases

For σ > 1, we found that the convergence is slower both in temporal and spatial domain. Figure 17 shows the flue gas rate for a time step size of 0.01 min having Δx = 0.3, 0.15,and 0.075 mm. Figure 18 shows the flue gas rates for Δx = 0.3 mm having time step sizes of 0.001, 0.0005, and 0.0001 min.

Fig. 17

Flue gases rate for case 2 with Δx = 0.3 mm, Δx = 0.15, and Δx = 0.075 mm and Δt = 0.01 min. Note convergence as Δx decreases

Fig. 18

Flue gases rate of case 2 for Δx = 0.3 mm with time step sizes of 0.001, 0.0005, and 0.0001 min. Note convergence as Δx decreases

We also have tested the stability of our numerical simulation against infinitesimal and finite perturbations. The results show that that an infinitesimal change in the parameters does not cause deviation from the original results. Also a finite change in the parameters, causes the results to be in finite difference with the original results. Figure 19 shows the stability of the original numerical simulation (with with Ea= 120000 J/mol) toward toward infinitesimal change (Ea= 120000.001 J/mol) and finite change (Ea= 120100 J/mole). As we can see from Fig. 19, the infinitesimal change of activation energy does not change the final result. Also, a finite change in activation energy makes the result to be in finite difference with the original solution.

Fig. 19

Stability of numerical simulation (with Ea= 120000 J/mol) toward infinitesimal change (Ea= 120000.001 J/mol) and finite change Ea= 120100 J/mol

1.4 7.4 Results

It is interesting to see how the oscillation develops with time for different scenarios. Here, we show the results of flue gas compositions, temperature profile, and flue gases rate for cases 1.2, 1.3, 1.4, 1.5, 2.2, 2.3, 2.4, and 2.5 in Figs. 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, and 34.

Fig. 20

Flue gases composition profiles for case 1.2

Fig. 21

Flue gases composition profiles for case 1.3

Fig. 22

Flue gases composition profiles for case 1.4

Fig. 23

Flue gases composition profiles for case 1.5

Fig. 24

Temperature profiles at different times for case 1.2

Fig. 25

Temperature profiles at different times for case 1.3

Fig. 26

Temperature profiles at different times for case 1.4

Fig. 27

Temperature profiles at different times for case 1.5

Fig. 28

Flue gases composition profiles for case 2.2

Fig. 29

Flue gases composition profiles for case 2.3

Fig. 30

Flue gases composition profiles for case 2.4

Fig. 31

Flue gases composition profiles for case 2.5

Fig. 32

Temperature profiles at different times for case 2.2

Fig. 33

Temperature profiles at different times for case 2.3

Fig. 34

Temperature profiles at different times for case 2.4

Fig. 35

Temperature profiles at different times for case 2.5

Fig. 36

Flue gases rate for case 1.1

Fig. 37

Flue gases rate for case 1.2

Fig. 38

Flue gases rate for case 1.3

Fig. 39

Flue gases rate for case 1.4

Fig. 40

Flue gases rate for case 1.5

Fig. 41

Flue gases rate for case 2.1

Fig. 42

Flue gases rate for case 2.2

Fig. 43

Flue gases rate for case 2.3

Fig. 44

Flue gases rate for case 2.4

Fig. 45

Flue gases rate for case 2.5