Chapter 8, Problem 8.99P

A concentric lube arrangement, for which the inner and outer diameters are $\text{8}0\text{ mm}$ and $100\text{ mm}$ . respectively, is used to remove heat from a biochemical reaction occurring in a $\text{1}-\text{m}$ -long settling tank. Heat is generated uniformly within the tank at a rate of ${10}^5{\text{W/m}}^3$ , and water is supplied to the annular region at a rate of $0.\text{2 kg}/\text{s}$ .

(a) Determine the inlet temperature of the supply water that will maintain an average tank surface temperature of $\text{37}°\text{C}$ . Assume fully developed flow and thermal conditions. Is this assumption reasonable?

(b) It is desired to have a slight, axial temperature gradient on the tank surface, since the rate of the biochemical reaction is highly temperature dependent. Sketch the axial variation of the water and surface temperatures along the flow direction for the following two cases: (i) the fully developed conditions of part (a), and (ii) conditions for which entrance effects are important. Comment on features of the temperature distributions. What change to the system or operating conditions would you make to reduce the surface temperature gradient?