Response to an ideal pulse input of tracer

Mathematically, the solution to the partial differential equation 2.13 for a pulse input of M moles of tracer into a pipe of cross-sectional area A is 2)  

That this equation is indeed a solution of equation 2.13. may be verified by partial differentation with respect to t and z and substituting. It also satisfies the initial condition corresponding to an ideal pulse, i.e. when t - 0+, z = 0, C - but, at z 0, C= 0. In addition, at any time t 0, the total moles of tracer anywhere in the pipe must be equal to M. 

Now let us see what happens to the tracer concentration at a fixed position z = L equation 2.15 then becomes  

Mathematically, the skewness of the CL versus t curve can be identified with the presence of t in the equation 2.17 in the positions arrowed  

In many instances, however, the change in shape of the concentration wave in passing the observation point is negligibly small. Mathematically, this corresponds to a value DjuL 0.01, approximately. In this case the arrowed t in the equation above may be replaced by t = L/u, the mean residence time of the pulse in the section of pipe between z = 0 and z = L. (Since the shape of the concentration wave is now considered not to change in passing the observation point, this is also the time at which the peak of the wave passes z = L) 

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