A miniature heat sink heat exchanger is constructed with circular cross section channels (similar to that in Example 7.4), each of which has a diameter of 3.0 mm, drilled through a rectangular block. Cooling water at 20°C flows at the rate of 0.5 kg/h through each circular pipe, which has a constant surface temperature of 80°C. Assuming fully developed velocity and temperature distribution conditions, determine the length of the drilled pipe if the outlet water temperature is 60°C. Also, based on your calculations, verify the fully developed condition assumption.
GIVEN
• Heat exchanger constructed with circular cross section channels.
• Diameter of channel (Di)= 3 mm = 3*10-3 m
• Cooling water inlet temperature (Tin)= 200C
• Cooling water flow rate (m• )=0.5 kg/h = 1.39*10-4 kg/s
• Cooling water outlet temperature (Tout)= 600C
• Surface temperature(Ts)=800C
FIND
(a) Length of the drilled pipe.
(b) Verify fully developed condition assumption
PROPERTIES AND CONSTANTS
or Water at 60°C:
Density (?) = 982.8 kg/m3
Specific heat (cp) = 4182.8 J/(kg K)
Thermal conductivity (k) = 0.657 W/(m K)
Absolute viscosity (?) = 483.7 × 10–6 (Ns)/m2
Prandtl Number (Pr) = 3.02
The Reynold’s Number for given flow is
Thus the flowrate is in laminar region.
Thus, for a fully developed flow at constant wall temperature Nu=3.657
The rate of heat transfer to the cooling water is given by
For Ts=800C, finding LMTD given we have
Now, we have from equation
We have,
Since, the length of the drilled pipe is 10.7 cm and the velocity profile is fully developed at 1.8 cm,
and thermal profile is fully developed in 5.5 cm, the fully developed assumption is valid.
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