Immediately prior to crush this year, your foothills winery receives a new stainless steel jacketed tank to use for white wine fermentations. Unfortunately, you find out too late that your glycol cooling system is already at its limits and cannot be used for the new tank. Luckily, the municipal water source $($i$.$e$.$ city water$)$ is relatively cool in this region at $55F,$ and you think you can just about control the fermentation at the desired temperature of $65F$ using this water $($assuming that the water leaves the jacket at $($:$62F\}.$ The tank is $)=(1337f{t}^3$ total volume with a working volume of $)=(1070f{t}^3.$

$($a$)$ For $2\mathrm{.}5$Bri$\frac{x}{?}$ day maximum fermentation rate, the fermentation will generate heat at a maximum rate of $2\mathrm{.}5Bt\frac{u}{h}r$ gal. If the tank has a diameter of $9\mathrm{.}5ft,$ what is the height of jacket necessary to control the temperature at the desired level? The tank manufacturer tells you that the overall heat transfer coefficient for this fermentor should be $U=15Bt\frac{u}{h}rf{t}^{2}F.$

$($b$)$ What is the necessary flow rate for the municipal water to maintain temperature? Assume that the heat capacity of the water is $Cp=1Bt\frac{u}{b}{b}_{m}F$ and the density of the water is $62\mathrm{.}4l\frac{b_m}{f}{t}^3.$

$($c$)$ For the flow rate calculated, find the best size for the pipe entering and leaving the jacket and the circulation pump. Be sure to verify that you have turbulent flow and a "reasonable" pressure drop. The pipe can be assumed to be "smooth tube." The viscosity of the water is $)=(0\mathrm{.}000672l\frac{b_m}{f}ts.$

$($d$)$ Now that the tank and pipe are in place, you need to choose the size of the centrifugal circulation pump. The tank manufacturer tells you that the jacket will have a pressure drop of $10.$ The centrifugal pump used for circulation is at a height equal to the bottom of the fermentor, and the jacket begins $5ft$ above this. The pipe connecting the municipal water to the tank is a total of $30ft$ in length with three "standard" elbows. The return line exits the top of the jacket and goes back to a holding tank $30ft$ away $($at the same height$-$no elbows$).$ The municipal water source is supplied at $3$ psi pressure. A pump curve is attached.