Chapter 11, Problem 11.50P

In a supercomputer, signal propagation delays arereduced by resorting to high-density circuit arrangements which are cooled by immersing them in a specialdielectric liquid. The fluid is pumped in a closed loopthrough the computer and an adjoining shell-and-tubeheat exchanger having one shell and two tube passes.

During normal operation, heat generated within the computer is transferred to the dielectric fluid passing through the computer at a flow rate of ${\stackrel{\dot{}}{m}}_f=4.81\text{kg/s}$ . In turn, thefluid passes through the tubes of the heat exchanger andthe heat is transferred to water passing over the tubes. The
dielectric fluid may be assumed to have constant properties of $c_p=1040\text{J/kg}\cdot \text{K}$ , $\mu =7.65\times {10}^{-4}\text{kg/s}\cdot \text{m}$ , $k=0.058W\text{/m}\cdot \text{K}$ , and $Pr=14$ . During normal operation, chilled water at a flow rate of the ${\stackrel{\dot{}}{m}}_w=2.5\text{kg/s}$ and aninlet temperature of $T_{w,i}=5°C$ passes over the tubes. The water has a specific heat of $4200\text{J/kg}\cdot \text{K}$ and provides anaverage convection coefficient of $10,000{\text{W/m}}^2\cdot \text{K}$ overthe outer surface of the tubes.

(a) If the heat exchanger consists of 72 thin-walled tubes, each of 10-mm diameter, and fully developed how is assumed to exist within the tubes, what is the convection coefficient associated withflow through the tubes?
(b) If the dielectric fluid enters the heat exchanger at $T_{f,i}=25°C$ and is to leave at $T_{f,o}=15°C$ , what isthe required tube length per pass?
(c) For the exchanger with the tube length per passdetermined in part (b), plot the outlet temperature
of the dielectric fluid as a function of its flow ratefor $4\le {\stackrel{\dot{}}{m}}_f\le 6kg\text{/s}$ . Account for correspondingchanges in the overall heat transfer coefficient, butassume all other conditions to remain the same.
(d) The site specialist for the computer facilities isconcerned about changes in the performance of the water chiller supplying the cold water $\left({\stackrel{\dot{}}{m}}_w,T_{w,i}\right)$ and their effect on the outlet temperature $T_{f,o}$ of the dielectric fluid. With all other conditionsremaining the same, determine the effect of $a\pm 10\%$ change in the cold water flow rate on $T_{f,o}$ .
(e) Repeat the performance analysis of part (d) todetermine the effect of $a\pm 3°C$ change in the water inlet temperature on $T_{f,o}$ , with all other conditions remaining the same.