Clarke's equation explained

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]

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

or, alternatively^[7]

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

where

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

\delta can be removed from the equation by the transformation (x,t)\to(\delta x,\delta t) , it is, however, retained here since \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 instantnaeously in a reacting mixture. When the chemical time is comparable to the acoustic time, diffusion is neglected so that igntion 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

\theta = (T _{m-T)/\varepsilon} T _{m} , where T _{m} is the maximum initial temperature, T is the temperature and \varepsilon T _{m} \equiv RT ^{2 /E} _{m} \ll T _{m} is the Frank-Kamenetskii temperature (R is the gas constant and E is the activation energy). Furthermore, let r denote the distance from the center, measured in units of initial hot core size and t be the time, measured in units of acoustic time. In this case, the initial and boundary conditions are given byt =0 : - \theta = r ^{2,} r =0 : (\partial \theta)/(\partial r) =0 , r\gg 1 : - \theta = r ^{2} + (j +1 ) (\gamma -1)/(\gamma) t ^{2,}

where

j = ( 0 , 1 , 2 ) , respectively, corresponds to the planar, cylindrical and spherical problems. Let us deifine a new variable\varphi(r,t) = \theta + r ^{2} + (j +1 ) (\gamma -1)/(\gamma) t ^{2}

which is the increment of

\theta(r,t) from its distant values. Then, at small times, the asymptotic solution is given by \varphi = \gamma\delta t e ^{- r ^{2}} + (1)/(2) (\gamma\delta t) ^{2e} ^{-2 r ^{2}} + …

As time progresses, a steady state is approached when

\delta\leq \delta _{c} and a thermal explosion is found to occur when \delta>\delta _{c} , where \delta _{c} is the Frank-Kamenetskii parameter; if \gamma =1 . 4 , then \delta _{c=0.50340} in the planar case, \delta _{c} = 0 . 73583 in the cylindrical case and \delta _{c=0.91448} in the spherical case. For \delta\gg \delta _{c} , the solution in the first approximation is given by \varphi =-ln ( 1- \gamma\delta t e ^{- r ^{2}} )

which shows that thermal explosion occurs at

t = t _{i\equiv} 1 /(\gamma\delta) , where t _{i} is the ignition time.

Generalised form

For generalised form for the reaction term, one may write

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

where

\omega(\theta) is arbitray function representing the reaction term.

See also

• Frank-Kamenetskii theory

Notes and 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. 10.1098/rsbm.2014.0012. John Frederick Clarke 1 May 1927 – 11 June 2013. Biographical Memoirs of Fellows of the Royal Society. 60. 87–106. 2014. Bray . K. N. C.. Kenneth Bray. Riley . N.. free.
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).