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Homework 2 2013

17mn 15si 135cr 43ni in this formula the uts that is

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Unformatted text preview: Mn +15Si +13.5Cr + 4.3Ni In this formula the UTS that is calculated is in thousands of pounds per square inch (ksi) and the chemical symbols represent the composition of that element in the steel. We will refer to the coefficients as strengthening factors. An increase in the concentration of an element with a large strengthening factor will produce a much greater change in the UTS than the same increase for an element with a low strengthening factor. A typical composition for such a steel could be: 0.022%C, 1%Mn, 3.51%Mo, 3.4%Ni, 0.06%V, 0.23%Cr, 0.2%Si, 0.01%N, 91.5% Fe. The percentages are in weight percent. That is, for this composition, if you had 100 pounds of the material, it would contain 1 pound of Mn, 3.5 pounds of Mo, 3.4 pounds of Ni, etc. To use the first formula, you simply put the percent value for a given element into the formula where the chemical symbol is given. If these values are plugged into the formula for UTS you will find that the result is 158ksi. Your goal is to modify this steel to increase the UTS to 250 ksi by increasing the composition of the elements V, Ni, Cr, Si while minimizing the cost of the additions. First we must consider the cost of these elements. Values that we can use for this calculation are as follows: Si $0.96 per pound, Ni $6.40 per pound, Cr $1.2 per pound, and V $12.00 per pound. (Note, these cost values are not necessarily the true cost today.) There are also limits on the amount that each of these elements can be added to avoid unwanted phases in the material. While the weight percent of none of the elements can be decreased, the upper limits are as follows: Si, 0.4%; Cr, 2%; Ni, 5%; and V, 1.3%. The questions are the following; the first three can be presented in an excel spread sheet. Be sure to include a screen shot for any work done in excel. a. Write down the objective function. b. Write down the constraints. c. Using Excel, determine the optimum new composition that achieves the strength level and minimizes cost. d. Explain your results in terms of the optimization of strength and cost. e. Assume that the price of Ni drops to $4.00 per pound and the cost of V begins to increase. At what price of V per pound does it become economically feasible to increase Ni to its limit of 5% and decrease the amount of V added. This will best be addressed by using your spread sheet and changing the input numbers iteratively. Problem 3: Optical loss minimization You are designing a filter box for optical telecommunications. Inside the box is a specially treated optical fiber, through which the light must travel in order to remove certain spurious signals. The light enters the box vertically at (x,y) = (-1,0) (All dimensions are in centimeters), and leaves the box vertically in the opposite direction at (x,y) = (+1,0) (See figure). The fiber inside the box is 25 cm in length. Your objective is to lay out the fiber path such that optical losses in the box are minimized. Losses arise when an optical fiber is ben...
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