Fractional Conversion and Concentration Profiles

Since the material-balance equations, 16.2-1 and -2, derived above refer to a particular radial position, we must integrate radially to obtain an average concentration cA at any axial position x, including at the outlet, where x = L. The latter is the observed outlet concentration that corresponds to the outlet fractional conversion. We develop the expression in terms of cA, 

At any position x, the rate of flow of A in terms of total and average quantities is 

On integrating over the entire cross-section, we obtain for the rate of flow of A  

To obtain the concentration profile cA(x) as well as the outlet concentration cA(L), we express equation 16.2-4 in terms of t rather than r. To achieve this, we use equations 2.5-1 and -2 to eliminate u(r) and ii, respectively, together with 

From this equation and equations 2.5-1 and -2, on elimination of u0, u(r) and ii, we obtain 

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