# Hydro Turbine Optimization

So I was doing the project related to Lagrange multiplier called hydroxyzine optimization recently and I have encountered problem on question4-6, which required me to plot a graph and find the optimum method for highest efficiency, does anyone know how to plot the graph? (Question1-3 is done)

• What's the graph expected to be? – CCola Dec 6 '18 at 18:10
• can you elaborate more? Since I was new to multivariable calculus... – CCola Dec 7 '18 at 1:14

## 1 Answer

Using the following MATHEMATICA script you can analyze and build the sought graphics

 KW1[Q_, QT_] := (-18.89 + 0.1277 Q - 4.08 10^-5 Q^2) (170 - 1.6 10^-6 QT^2) KW2[Q_, QT_] := (-24.51 + 0.1358 Q - 4.69 10^-5 Q^2) (170 - 1.6 10^-6 QT^2) KW3[Q_, QT_] := (-27.02 + 0.1380 Q - 3.84 10^-5 Q^2) (170 - 1.6 10^-6 QT^2)

The functions representing the turbines performance.

 NMaximize[ {KW1[Q1, QT] + KW2[Q2, QT] + KW3[Q3, QT], 250 - Q1 <= 0 && 1110 - Q1 >= 0 && 250 - Q2 <= 0 && 1110 - Q2 >= 0 && 250 - Q3 <= 0 && 1225 - Q3 >= 0 && Q1 + Q2 + Q3 - QT == 0}, {Q1, Q2, Q3, QT}]

Analyzing the three turbines together, with their restrictions.

 NMaximize[ {KW2[Q2, QT] + KW3[Q3, QT], 250 - Q2 <= 0 && 1110 - Q2 >= 0 && 250 - Q3 <= 0 && 1225 - Q3 >= 0 && Q2 + Q3 - QT == 0}, {Q2, Q3, QT}]

Analyzing only turbines KW2 and KW3

NMaximize[ {KW1[Q1, QT] + KW2[Q2, QT] + KW3[Q3, QT], Q1 + Q2 + Q3 - QT == 0}, {Q1, Q2, Q3, QT}]

Analyzing the three turbines without restrictions, etc.