Please be clear with units, show all steps, and explain for 5 stars! Use Kelvin

Question

Please be clear with units, show all steps, and explain for 5 stars! Use Kelvin instead of Celsius too!

One of the major limitations in scale-up of very large fermentors is believed to be the removal of heat, more so than the transfer of oxygen within the reactor. The reasoning behind this hypothesis is that heat transfer area increases in proportion to the reactor radius squared while the volume (and hence the total heat generated by microorganisms in solution) increases with the radius cubed. The reactor vessel is surrounded by a cooling jacket. Water enters the jacket at Tc = 20degreeC and leaves at Th = 35degreeC. The rate of heat removed by the cooling water is given by Q = UoA Delta T = UoA(Th -Tc), where the overall heat transfer coefficient is Uo = 350 kcal m-2 h-1 degreeC-1 and A is the surface area of the sidewall of the cylindrical reactor. The reactor contents are constantly stirred, and this agitation generates heat at a rate of 1.5 kcalL-1h-1. In addition, the cells' metabolism generates heat at a rate of 30 kcal L-1h-1. You are asked to determine the largest possible reactor volume (in L) given the constraint that the reactor height to diameter ratio is fixed at 2:1.

Explanation / Answer

Total heat generation Q = 30 + 1.5 = 31.5 kcal/L/h

V = pi/4*D^2*L m^3 = pi/4*D^2*L *10^3 L

Total heat generation in volume V = 31.5*10^3 *pi/4*D^2*L kcal/hr

Also, Q = UA(Th - Tc) where A = pi*D*L

So, 31.5*10^3 *pi/4*D^2*L = 350*(pi*D*L)*(35 - 20)

Simplifying, D = 0.66667 m

Since H/D = 2

we get H = 1.33333 m

Volume V = pi/4*D^2*L

= 3.14/4*0.667^2 * 1.333

= 0.465185 m^3 = 465.185 L

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