combustion Combustion, or burning, is a high-temperature exothermic redox chemical reaction between a fuel (the reductant) and an oxidant, usually atmospheric oxygen, that produces oxidized, often gaseous products, in a mixture termed as smoke. Combusti ...

, Clarke's equation is a third-order nonlinear

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...

, first derived by

John Frederick Clarke John Frederick Clarke FRS (1 May 1927 – 11 June 2013) was a professor, an aeronautical engineer, and a pilot. Biography After his schooling, he got training from Fleet Air Arm as a Navy Pilot and then from Royal Air force at Lossiemouth. ...

in 1978.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 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 speed The speed of sound is the distance travelled per unit of time by a sound wave as it propagates through an elastic medium. At , the speed of sound in air is about , or one kilometre in or one mile in . It depends strongly on temperature as wel ...

s. The equation reads as :

(\theta_t-\gamma e^)_=(\theta_t-e^\theta)_

where

\theta

is the non-dimensional temperature perturbation and

\gamma

is the

specific heat ratio In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volu ...

. The term

\theta_t-e^\theta

describes the explosion at constant pressure and the term

\theta_t-\gamma e^\theta

describes the explosion at constant volume. Similarly, the term

(\ )_-(\ )_

describes the wave propagation at adiabatic sound speed and the term

\gamma(\ )_-(\ )_

describes the wave propagation at isothermal sound speed. Molecular transports are neglected in the derivation.

References

{{reflist, 30em Partial differential equations Fluid dynamics Combustion

See also

References