Gas absorption with rapid reaction, The following problem arises in gas absorption with fast reaction

Gas absorption with rapid reaction, The following problem arises in gas absorption with fast reaction. The notations have been slightly changed from Eq. (10.5) earlier: 2 d2Ca/ dx2 = cA[1 + q∗cA − q∗px=0(x − 1)] (14.58) where q∗ is defined as the reciprocal of q in Eq. (10.44). The parameter is the reciprocal of Ha. The boundary conditions for the diffusing gas cA are c(x = 0) = 1 and c(x = 1) = 0. The problem has two interesting characteristics. (i) The solution is of a boundary-layer type for Ha → ∞. (ii) An undetermined parameter px=0, the slope of the concentration gradient, appears. This parameter is not known at the start and can only be estimated a posteriori. Here we wish to examine the solution for large Ha. For fast reaction Ha is large and hence is a small parameter, leading to a singular perturbation problem. Show that the solution for equal to zero leads to an instantaneous reaction profile and develop the next approximation, which is approximate to O()2. Find the enhancement factor. Then use the results and find the next-higher-order solution.