SOLUTION ONE

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Considering that all three turbines are being used, we determine the flow $Q_T$ to each turbine that will give the maximum total energy production. Our limitations are that the flows much sum to the total incoming flow and the given domain restrictions must be observed. Consequently, we use Lagrange multipliers to find the values for the individual flows (as functions of $Q_T$) that maximize the total energy $KW_1 + KW_2 + KW_3$ subject to the constraints $Q_1 + Q_2 + Q_3 = Q_T$ and the domain restrictions on each $Q_i$.

Using Lagrange, we set the partial derivative of each $KW$ equation with respect to $Q_1$, $Q_2$, and $Q_3$ respectively, equal to the partial of $Q_1 + Q_2 + Q_3 = Q_T$. $$Q_1 = [0.1277 - (8.16 \cdot 10-5)Q_1](170 – 1.6 \cdot 10^{-6}Q_T^2) = \lambda$$ $$Q_2 = [0.1358 - (9.38 \cdot 10-5)Q_2](170 – 1.6 \cdot 10^{-6}Q_T^2) = \lambda$$ $$Q_3 = [0.1380 - (7.68 \cdot 10-5)Q_3](170 – 1.6 \cdot 10^{-6}Q_T^2) = \lambda$$

Because each partial is equal to lambda, it is possible to set each partial equal to each other. By doing this, we can solve $Q_2$ and $Q_3$ in terms of $Q_1$. $$Q_2 = 86.354 + 0.8699Q_1$$ $$Q_3 = 134.115 + 1.0625Q_1$$ Then we can substitute back into the original constraint equation in order to find $Q1$ in terms of $Q_T$. By plugging this value of $Q_1$ into the two equations above we can then determine $Q_2$ and $Q_3$ in terms of $Q_T$. $$Q_1 = 0.341Q_T - 75.134$$ $$Q_2 = 0.297Q_T + 20.995$$ $$Q_3 = 0.362Q_T + 54.285$$

SOLUTION TWO

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To determine which values of $Q_T$ will work, we must first look at the general domain restrictions. $170 - 1.6 \cdot 10^{-6}Q_T^2$ by default has to be greater than or equal to 0 because the power generated by the turbine cannot be zero. Therefore $Q_T$ must be less than 10,307.76 ft³/s. Looking at the individual minimums and maximums of each turbine then, and summing their maxes and minimums, we find that $Q_T$ must be limited by $750 \le Q_T \le 3445$.

This restriction is however not the correct restriction because the Lagrange equations give different values for lower and upper limits. To find the maximum value of $Q_T$ we use the the equations of $Q_1$, $Q_2$, and $Q_3$ in terms of $Q_T$ and set them to the individual maxes of 1110, 1110, and 1225 respectively. The lowest possible maximum value of $Q_T$ will be the overall maximum of water flow that the this model will maximize. $$Q_1 = .341Q_T - 75.134 \le 1110 \quad Q_1 \le 3475$$ $$Q_2 = .297Q_T + 20.995 \le 1110 \quad Q_2 \le 3671.628$$ $$Q_3 = .362Q_T + 54.285 \le 1225 \quad Q_3 \le 3231.34$$ In this scenario turbine 3 gives the lowest possible maximum of $Q_T$.

A similar process is used for the greatest possible minimums of the system. By setting the three equations equal to their respective minimums, we can determine the overall minimum of the solution by taking the greatest minimum given by these restrictions. $$Q_1 = .341Q_T - 75.134 \ge 250 \quad Q_1 \ge 953.5$$ $$Q_2 = .297Q_T + 20.995 \ge 250 \quad Q_2 \ge 772.1$$ $$Q_3 = .362Q_T + 54.285 \ge 250 \quad Q_3 \ge 540.2$$ Therefore we are able to determine that turbine 1 gives the greatest possible minimum of $Q_T$ which is the overall minimum of the system.

The overall restriction of values of $Q_T$ that are valid is therefore $954.47 \le Q_T \le 3231.34$.

SOLUTION THREE

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To determine the distribution for turbines with an incoming flow of 2500 ft³/s and verify that our result is indeed a maximum, we plug in $Q_T = 2500$ into our equations for $Q_1$, $Q_2$, $Q_3$ that are defined only in terms of $Q_T$. From this, we find that: $$Q_1 = 777.366, \quad Q_2 = 762.495, \quad Q_3 = 960.035$$ We then plug in these values into the original equations for $KW_1$, $KW_2$, and $KW_3$ to find the values for each individual output. Adding these values yields the total power output of the maximum value given by the 2500 ft₃/s $Q_T$ restriction.
Given this maximum, we find that $KW_{Total} = 28410.61$ kilowatts.

To verify that this value is indeed a maximum, we take other nearby distributions on both upper and lower sides of the spectrum. When using values of: $$Q_1^1 = 790, \quad Q_2^1 = 755, \quad Q_3^1 = 955$$ We get the total output as $KW_{Total}^1 = 28410.045$ kilowatts, which is less than the previously found maximum output value.

To further this relationship of a maximum found at the original turbine flows, we take a second distribution of values. When using values of: $$Q_1^2 = 760, \quad Q_2^2 = 770, \quad Q_3^2 = 970$$ We get the total output as $KW_{Total}^2 = 28352.32$ kilowatts, which is also less than the previously found maximum value.

Therefore, we ascertain the distribution of turbines for kilowatt output to be: $$Q_1 = 777.366, \quad Q_2 = 762.495, \quad Q_3 = 960.035$$

SOLUTION FOUR

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To determine whether using all three turbines or just one would be more beneficial for a flow of 1000 cubic feet per second, we first plug in the value of $Q_T = 1000$ into our equations found in solution one that define $Q_1$, $Q_2$, and $Q_3$ in terms of $Q_T$. We find that $Q_1 = 265.866$, $Q_2 = 317.505$, and $Q_3 = 416.585$, which yields a total power output of 8398.207 kilowatts.

We then compare this value to the power outputs produced when only one turbine is used, i.e. when $Q_1$, $Q_2$, and $Q_3$ are equal to $Q_T$. If $Q_1 = Q_T = 1000$, we find that 11,452.884 kilowatts are produced. For $Q_2 = 1000$, 10843.276 kilowatts are produced. For $Q_3 = 1000$, 12,222.472 kilowatts are produced.

Therefore, we determine that the amount of power produced by using only turbine 3 is much greater than if all turbines were used.

For a total flow of 600 ft³/s, only one turbine can be used. If the flow were to be distributed between all three turbines, the individual flows would be less than the minimums presented. Because no turbine can have less than 250 ft³/s flowing through, we find what each power output would be if each turbine had 600 ft³/s flowing through: $$Given \quad Q_1 = 600, \quad KW_1 = 7292$$ $$Given \quad Q_2 = 600, \quad KW_2 = 6837$$ $$Given \quad Q_3 = 600, \quad KW_3 = 7108$$

Since the greatest amount of power is produced by turbine 1 in this case, this should be the only turbine used for a flow of 600 ft³/s.

SOLUTION FIVE

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Considering the use of only two turbines with a total incoming flow of 1500 ft³/s, the max power output can be determined by finding the maximum power output of each combination and comparing accordingly. Through Lagrange multipliers, we are able to determine the following relationships for turbine flows:

•      For turbines 1 and 2:   $Q_2 = 86.354 + 0.870Q_1$
•      For turbines 2 and 3:   $Q_3 = 28.646 + 1.221Q_2$
•      For turbines 1 and 3:   $Q_3 = 134.115 + 1.063Q_1$

Knowing that the total flow between two turbines must be 1500 ft³/s, we are able to determine the following distributions of flow between turbines in ft³/s:

•      Turbines 1 and 2:   $Q_1 = 755.986, \quad Q_2 = 744.014$
•      Turbines 2 and 3:   $Q_2 = 662.368, \quad Q_3 = 837.632$
•      Turbines 1 and 3:   $Q_1 = 662.247, \quad Q_3 = 837.753$

Finally, we plug these values into the power output functions for two turbines, derived from the summation of the individual turbine output functions, to calculate the max total power outputs given two turbines in kilowatts:

•      Turbines 1 and 2: 17454.843
•      Turbines 2 and 3: 17720.568
•      Turbines 1 and 3: 18208.300

Accordingly, to generate max power with two turbines and a total flow of 1500 ft³/s, we determine that turbines 1 and 3 should be used with 662.247 ft³/s diverted to turbine 1 and 837.753 ft³/s diverted to turbine 3.

Using the relationships found with Lagrange multipliers in solution one regarding an individual turbine’s flow and total flow with three turbines, we can calculate the max power output at 1500 ft³/s with all three turbines in use. To do so, we derive a three turbine power output function which is the summation of all three individual power output functions and find that maximum power output is 16538.68111 ft³/s.

It is thus more efficient with a flow of 1500 ft³/s to only use the two turbines.

SOLUTION SIX

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Ordinarily, we would use our model found in solution one that maximizes power output by using all three turbines. However, a flow of 3400 ft³/s is above the maximum total flow for which this model is relevant.

If we look back at the restraints found in solution two however: $$Q_1 = 0.341Q_T - 75.134 \le 1110, \quad Q_1 \le 3475$$ $$Q_2 = 0.297Q_T - 20.995 \le 1110, \quad Q_2 \le 3672$$ $$Q_3 = 0.362Q_T - 54.285 \le 1225, \quad Q_3 \le 3231$$ We can see that $Q_3$ is limited by the lowest total flow. Therefore, if we assume that $Q_3$ is equal to its maximum value at 1225, then we can use Lagrange multipliers to determine what the distribution should be to turbines 1 and 2 to maximize output.