Problem #$3(5$ pts $+1$ pt extra credit$)$

The throttling valve is theoretically a source of inefficiency in a refrigeration cycle, as it has a substantial decrease in temperature and pressure $($and increase in entropy$)$ with no power out.

Suppose you are an engineer who invents a special turbine that can take in R$-134$a as a saturated liquid, and the R$-134$a leaves as a saturated liquid $+$ vapor mixture. The isentropic efficiency of your special turbine is $55\%,$ but the power generated from this can help power the compressor, reducing the net work into the system.

Your boss asks you to evaluate how the performance of the cycle in problem $2$ would change if you replaced the throttling valve from state $3$ to state $4$ with this special turbine. Keep in mind that $h_1,h_2($and ${}_c),$ and $h_3$ will all be the same as problem $2($you don't need to recalculate $h_2),$ and $Q_L^{}=20kW$ still.

Determine the following:

The for your special turbine; report as a positive value

The mass flow rate for this system $($it will be close to the mass flow rate from problem $2,$ but not exactly the same$).$

The electrical power saved using this system compared with the system $(W_{\text{n}\text{e}\text{t}\text{,}\text{p}\text{r}\text{o}\text{b}2}^{}-W_{\text{n}\text{e}\text{t}\text{,}\text{p}\text{r}\text{o}\text{b}3}^{})$; remember that the mass flow rate has changed

$(1$ pt extra credit$)$ Adding this turbine will make the freezer cost $$4000$ more. If electrical power costs $$\frac{0\mathrm{.}13}{k}W-$hr $($i$.$e$.,$ using $1$ kW of power for one hour costs $$0\mathrm{.}13),$ how long would it take to save $$4000$ in energy costs $($to make it worth the extra cost up front$)?$ Report in years as a decimal $($e$.$g$.,17\mathrm{.}34$ years$)$