This problem examines the sensitivity of a given chemical substance as a functio

Question

This problem examines the sensitivity of a given chemical substance as a function of the amount of reaction time given the amount of catalyst. This exercise is concerned with an industrial process in which the reaction time of a chemical substance depends on the temperature and the amount of catalyst. We are interested in minimizing the reaction time and its sensitivity to the temperature. Currently we are not satisfied with the specified reaction time of a catalyst we are using. We have to determine the amount of catalyst that will give us the least reaction time. The difference in the reaction time (from the original reaction time) is given by the following equation: y(x, alpha) = 3/2 x^2 + 3/2 x (1 - alpha)- 2/3 alpha^2 Solve the problem using the Uncertainty Sensitivity Index Method (USIM). A negative value of y indicates a reduction in the original reaction time, while a positive value indicates an increase. In short, the greater the negative value we obtain the better, because we are reducing our original reaction time. y(x, a) denotes the difference in reaction time x denotes the difference from the original amount of catalyst. (A negative value denotes a reduction of the original amount, while a positive value denotes an increase.) alpha denotes the model's parameter (temperature). Assume a nominal value of alpha = 2.

Explanation / Answer

y=3/2 x2+3/2(1-alpha)x-2/3 alpha2

y is function of x and alpha

so Take differentiation with respect to x.

so

dy/dx=3x+3/2(1-alpha)

For lest reaction time

dy=0

alpha=2

0=3x+3/2(1-2)

x=0.5

Now taking second differntiantion with respect to x

d2y/dx2 =3 >0 means we find x at dy/dx=0 that give us y minimum.

so amount of catalyst is 0.5 that gives least reaction time.

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