After traveling around the world making wine for several years, you decide to settle in California and open up a winery that specializes in Syrah. From your experience at crush positions in Australia, you decide to purchase several $)=(802f{t}^3$ rotary fermentors with $4500$ gal working volumes. Because you don't have much money to start, you go with a low$-$cost stainless$-$steel tank fabricator that has not made one of these tanks before, but is interested in starting. Therefore, you need to give him a lot of information on the design. The basic geometry is shown in the figure below.

$($a$)$ You plan to ferment your Syrah at a maximum temperature of $86F.$ To maintain this temperature, you are hoping to use a municipal water source that is typically at $72F$ during the late September harvest. The temperature at the outlet of the jacket should not exceed $78F.$ At your planned fermentation temperature, your experience tells you that you should have a maximum sugar utilization rate of about $5\mathrm{.}5$ Brix per day. Assuming an overall heat transfer coefficient of $15Bt\frac{u}{h}rFft,$ will you have enough heat transfer surface area if only the bottom half of the fermentor is jacketed? $(12\mathrm{.}9$ Btu$/$hr gal$=1$ W$/$L$).$

$($b$)$ What is the necessary volumetric flow rate of cooling water through the jacket? Would it be reasonable to use a dimpled jacket with this type of flow rate? The heat capacity and density of the cooling water are $1Bt\frac{u}{l}{b}_{m}F$ and $62\mathrm{.}4lb\frac{m}{f}{t}^3,$ respectively.