Clarke's equation

In combustion, Clarke's equation is a third-order nonlinear partial differential equation, first derived by John Frederick Clarke in 1978. ^{[1] [2] [3] [4]} The equation describes the thermal explosion process, including both effects of constant-volume and constant-pressure processes, as well as the effects of adiabatic and isothermal sound speeds. ^[5] The equation reads as ^[6]

${\displaystyle {\frac {\partial ^{2}}{\partial t^{2}}}\left({\frac {\partial \theta }{\partial t}}-\gamma \delta e^{\theta }\right)=\nabla ^{2}\left({\frac {\partial \theta }{\partial t}}-\delta e^{\theta }\right)}$

or, alternatively ^[7]

${\displaystyle \left({\frac {\partial ^{2}}{\partial t^{2}}}-\nabla ^{2}\right){\frac {\partial \theta }{\partial t}}=\left(\gamma {\frac {\partial ^{2}}{\partial t^{2}}}-\nabla ^{2}\right)\delta e^{\theta }}$

where ${\displaystyle \theta }$ is the non-dimensional temperature perturbation, ${\displaystyle \gamma >1}$ is the specific heat ratio and ${\displaystyle \delta }$ is the relevant Damköhler number. The term ${\displaystyle \partial \theta /\partial t-e^{\theta }}$ describes the thermal explosion at constant pressure and the term ${\displaystyle \partial \theta /\partial t-\gamma e^{\theta }}$ describes the thermal explosion at constant volume. Similarly, the term ${\displaystyle \partial ^{2}/\partial t^{2}-\nabla ^{2}}$ describes the wave propagation at adiabatic sound speed and the term ${\displaystyle \gamma \partial ^{2}/\partial t^{2}-\nabla ^{2}}$ describes the wave propagation at isothermal sound speed. Molecular transports are neglected in the derivation.

It may appear that the parameter ${\displaystyle \delta }$ can be removed from the equation by the transformation ${\displaystyle (x,t)\to (\delta x,\delta t)}$, it is, however, retained here since ${\displaystyle \delta }$ may also appear in the initial and boundary conditions.

Example: Fast, non-diffusive ignition by deposition of a radially symmetric hot source

Suppose a radially symmetric hot source is deposited instantaneously in a reacting mixture. When the chemical time is comparable to the acoustic time, diffusion is neglected so that ignition is characterised by heat release by the chemical energy and cooling by the expansion waves. This problem is governed by the Clarke's equation with ${\displaystyle \theta =(T_{m}-T)/\varepsilon T_{m}}$, where ${\displaystyle T_{m}}$ is the maximum initial temperature, ${\displaystyle T}$ is the temperature and ${\displaystyle \varepsilon T_{m}\equiv RT_{m}^{2}/E\ll T_{m}}$ is the Frank-Kamenetskii temperature (${\displaystyle R}$ is the gas constant and ${\displaystyle E}$ is the activation energy). Furthermore, let ${\displaystyle r}$ denote the distance from the center, measured in units of initial hot core size and ${\displaystyle t}$ be the time, measured in units of acoustic time. In this case, the initial and boundary conditions are given by ^[6]

${\displaystyle t=0:\,-\theta =r^{2},\quad r=0:\,{\frac {\partial \theta }{\partial r}}=0,\quad r\gg 1:\,-\theta =r^{2}+(j+1){\frac {\gamma -1}{\gamma }}t^{2},}$

where ${\displaystyle j=(0,1,2)}$, respectively, corresponds to the planar, cylindrical and spherical problems. Let us define a new variable

${\displaystyle \varphi (r,t)=\theta +r^{2}+(j+1){\frac {\gamma -1}{\gamma }}t^{2}}$

which is the increment of ${\displaystyle \theta (r,t)}$ from its distant values. Then, at small times, the asymptotic solution is given by

${\displaystyle \varphi =\gamma \delta te^{-r^{2}}+{\frac {1}{2}}(\gamma \delta t)^{2}e^{-2r^{2}}+\cdots }$

As time progresses, a steady state is approached when ${\displaystyle \delta \leq \delta _{c}}$ and a thermal explosion is found to occur when ${\displaystyle \delta >\delta _{c}}$, where ${\displaystyle \delta _{c}}$ is the Frank-Kamenetskii parameter; if ${\displaystyle \gamma =1.4}$, then ${\displaystyle \delta _{c}=0.50340}$ in the planar case, ${\displaystyle \delta _{c}=0.73583}$ in the cylindrical case and ${\displaystyle \delta _{c}=0.91448}$ in the spherical case. For ${\displaystyle \delta \gg \delta _{c}}$, the solution in the first approximation is given by

${\displaystyle \varphi =-\ln(1-\gamma \delta te^{-r^{2}})}$

which shows that thermal explosion occurs at ${\displaystyle t=t_{i}\equiv 1/(\gamma \delta )}$, where ${\displaystyle t_{i}}$ is the ignition time.

Generalised form

For generalised form for the reaction term, one may write

${\displaystyle \left({\frac {\partial ^{2}}{\partial t^{2}}}-\nabla ^{2}\right){\frac {\partial \theta }{\partial t}}=\left(\gamma {\frac {\partial ^{2}}{\partial t^{2}}}-\nabla ^{2}\right)\delta \omega (\theta )}$

where ${\displaystyle \omega (\theta )}$ is arbitrary function representing the reaction term.

See also

• Frank-Kamenetskii theory

References

1. Clarke, J. F. (1978). "A progress report on the theoretical analysis of the interaction between a shock wave and an explosive gas mixture", College of Aeronautics report. 7801, Cranfield Inst. of Tech.
2. Clarke, J. F. (1978). Small amplitude gasdynamic disturbances in an exploding atmosphere. Journal of Fluid Mechanics, 89(2), 343–355.
3. Clarke, J. F. (1981), "Propagation of Gasdynamic Disturbances in an Explosive Atmosphere", in Combustion in Reactive Systems, J.R. Bowen, R.I. Soloukhin, N. Manson, and A.K. Oppenheim (Eds), Progress in Astronautics and Aeronautics, pp. 383-402.
4. Clarke, J. F. (1982). "Non-steady Gas Dynamic Effects in the Induction Domain Behind a Strong Shock Wave", College of Aeronautics report. 8229, Cranfield Inst. of Tech. https://repository.tudelft.nl/view/aereports/uuid%3A9c064b5f-97b4-4527-a97e-a805d5e1abd7
5. Bray, K. N. C.; Riley, N. (2014). "John Frederick Clarke 1 May 1927 – 11 June 2013". Biographical Memoirs of Fellows of the Royal Society . 60: 87–106. doi: 10.1098/rsbm.2014.0012 .
6. Vázquez-Espí, C., & Liñán, A. (2001). Fast, non-diffusive ignition of a gaseous reacting mixture subject to a point energy source. Combustion Theory and Modelling, 5(3), 485.
7. Kapila, A. K., and J. W. Dold. "Evolution to detonation in a nonuniformly heated reactive medium." Asymptotic Analysis and the Numerical Solution of Partial Differential Equations 130 (1991).