Question 5 / 33:  A countercurrent heat exchanger operates with the following temperatures: cold fluid inlet 20°C, hot fluid inlet 90°C. The heat capacity of the hot fluid is 2.2 kJ/kg K. The heat capacity of the cold fluid is 1 kJ/kg K. The flow rate of the hot fluid is 5 kg/s. The flow rate of the cold fluid is 66 kg/s. The area for heat transfer is 10 m2. The overall heat transfer coefficient is 2418 W/m2 K. What are the outlet temperatures?
A  Thoto = 29.6° C, Tcoldo = 30.1°C
B  Thoto = 29.6° C, Tcoldo = 40.1°C
C  Thoto = 29.6 °C, Tcoldo = 50.1°C
D  Thoto = 19.6° C, Tcoldo = 20.1°C
E  Thoto = 39.6 °C, Tcoldo = 30.1°C
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Explanation:

This is an interesting calculation. We know the inlet temperatures of both fluids (Tcoldi = 20 C, and Thoti = 90 C); we know their flow rates and heat capacities (66 kg/s, 1 kJ/kg K) and (5 kg/s, 2.2 kJ/kg K); we know the area for heat exchange (10 m2); and we know the overall heat transfer coefficient (2418 W/m2 K). We wish to determine the outlet temperatures of each fluid. We have three unknowns: the two outlet temperatures and the heat transferred. We can write three expressions: (1) Q = (66 kg/s)(1 kJ/kg K) (Tcoldo - 20) (2) Q = (5 kg/s) (2.2 kJ/kg K) (90-Thoto) (3) Q = (2418 W/m2 K) (10 m2) ΔTlm The first two of these can be rearranged to give Tcoldo = 35 - 0.166Thoto We can then expand 3 to read Q = 24.18ΔT1 - ΔT2 / ln (ΔT1/ΔT2) where Δ T1= 90 - Tcoldo and ΔT2=Thoto-20. We may then solve the system iteratively by guessing a value of Thoto, calculating Tcoldo from equation 4, and then computing Q from equations 2 and equations 5. We continue to guess Thoto until we find one for which the two values of Q (from equations 2 and 5) agree. A spreadsheet is an easy way to implement such a calculation. Note that the outlet temperature of the hot fluid does not change by much for this situation.

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Thermal-Fluid Systems ME303

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Attribution:  Dr. Steve Gibbs. Thermal-Fluid Systems. The Saylor Academy 2014, http://www.saylor.org/courses/me303/
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