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# optimization - PROCESS OPTIMIZATION Optimization Theproce...

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Optimization. The process of improving an existing situation, device, or system such as a chemical process. Improving what? For a chemical process/system, the factor to be optimized is usually a cost or a profit item. For example: total cost per unit of production (\$/ton), total profit, amount of product per time (ton/month), NPW, or % conversion. Objective function. A relationship that connects the value to be optimized with the variables that affect it. For example: Cost=f(T, P, x 1 , x 2 , ….x n ) PROCESS OPTIMIZATION ChE 4253 - Design I

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Example: Finding the insulation thickness that gives the least total cost for a pipe line Single Variable Optimization ChE 4253 - Design I 0 500 1000 1500 2000 0 1 2 3 4 5 6 Cost per year, \$ Insulation thickness, in Total cost Fixed cost Cost of heat loss
Example: In a mathematical form (x is the insulation thickness) Fixed charges =ax + b Cost of heat loss =c/x + d Total Cost = C T (x) = ax +c/x +(b+d) Graphical method: Shown in previous figure Analytical method: Single Variable Optimization ChE 4253 - Design I 0 2 = - = x c a dx dC T 2 1 = a c x

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-1000 -500 0 500 1000 1500 2000 0 1 2 3 4 5 6 C T , dC /dx, d 2 /dx Insulation thickness, x Total cost First Derivative Second derivative Example: 1st derivative = 0 means min or max. Need to look at 2 nd derivative as well Single Variable Optimization ChE 4253 - Design I 3 2 2 x c dx C d T =
Example: Most probably you have a discrete option instead of a continuous, like this one, due to standard sizes: Single Variable Optimization ChE 4253 - Design I 0 500 1000 1500 2000 0 1 2 3 4 5 6 C T Insulation thickness, x Total cost

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Important: 1) The point of intersection is not the minimum/maximum. 2) There might be multiple minima/maxima (e.g., A, B, C). 3) Very likely to have optimum along at least one constraint. Process Optimization ChE 4253 - Design I Annualized Cost Independent variable, x A B C Objective function Minimum allowable x Maximum
Important: 4) The optimum is not found when it has merely been bracketed. Example: Optimization of a flash unit with temperature. Process Optimization ChE 4253 - Design I Annualized Cost Independent variable, T A B C Objective Function ?

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Now there are two or more variables that affect the cost. The general approach is similar. Example: Consider the case where the total cost is a function of two independent variables x and y. C T (x,y) = ax +b/(xy) +cy+d Graphical method: Need to show a three-dimensional plot (iso-surface plot) with x, y as the two axes and C T as the third axis. Alternative is to set different values of y constant and plot C T as a function of x for these different values of constant y. Optimization with Two or More Variables ChE 4253 - Design I
Graphical Method: Example: C T (x,y) = ax +b/(xy) +cy+d Optimization with Two or More Variables ChE 4253 - Design I C T Variable, x A B C D E F y=y 1 2 3 4 5 6

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Analytical Method: Example: C T (x,y) = ax +b/(xy) +cy+d Optimization with Two or More Variables ChE 4253 - Design I y x b a x C T 2 - = 3 1 2 3 1 2 = = c ab y a cb x 2 xy b c y C T - = If more than 2 variables, then the same procedure would be followed with the number of equations being equal to the number of independent variables.
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optimization - PROCESS OPTIMIZATION Optimization Theproce...

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