A Kinematic Temperature Wave

To show the power of the proposed method we consider the following problem. A cylindrical bed of particles is thermally insulated on its curved surface and a fluid flows uniformly through it. The particles may be of different shapes (denoted by a suffix m) and of different sizes of any one shape (denoted by suffix ), but they are sufficiently small and well mixed that in any sample from the bed the same fraction fm of the heat capacity is due to particles of 

Assuming the particles to be sufficiently small for the surrounding temperature to be sensibly constant, the following equations describe the variation of temperature. 

Let hs = heat capacity of the particles per unit volume of the bed, fmn = fraction of this heat capacity contributed by the m, nth kind of particle, 

K = thermal conductivity of the fluid, a = voidage of the bed, v = linear velocity of fluid. 

Then Hf = h/T = heat content of fluid per unit volume of bed, 

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The theory of kinematic waves, initiated by Lighthill Whitham, is taken up for the case when the concentration k and flow q are related by a series of linear equations. If the initial disturbance is hump-like it is shown that the resulting kinematic wave can be usually described by the growth of its mean and variance, the former moving with the kinematic wave velocity and the latter increasing proportionally to the distance travelled. Conditions for these moments to be calculated from the Laplace transform of the solution, without the need of inversion, are obtained and it is shown that for a large class of waves, the ultimate wave form is Gaussian. The power of the method is shown in the analysis of a kinematic temperature wave, where the Laplace transform of the solution cannot be inverted. 

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