A Cylindrical Heat Exchanger
� 2008 Waterloo Maple Inc.
Introduction
This worksheet considers a common design of heat exchanger consisting of a double walled cylinder. A cooling liquid in the outer cylinder flows in the opposite direction to the hot gas inside the inner pipe.
Description of the Problem
We consider a heat exchanger as shown above with water used for the cooling fluid, and the inner pipe containing air to be cooled. The inlet temperature of the air is =100�C, and its outlet temperature, , should be less than 80�C. The water has an inlet temperature, , of 20�C and should have an outlet temperature, , of not more than 40�C. The heat transfer coefficient of the heat exchanger is 600 W/K and is the length of the heat exchanger. The heat capacitance flow is 1000 J/s. The heat capacitance flow has to be calculated.
Solution
The differential equations describing the heat exchanger are:

(3.1) 

(3.2) 
The initial conditions needed to solve this differential equations are the inlet temperatures at and at .

(3.3) 
Inserting the values x=L, =1000, kA=600 into these equations, we get functions and as a function of :

(3.4) 

(3.5) 
We'll solve these for

(3.6) 

(3.7) 
Since both conditions must be fufilled, we choose C corresponding to the higher value for .
Graphical Display of the Solution
Graphical display of the air temperature at the outlet with respect to .
#Plot Air Temp
Graphical display of the water temperature at the outlet with respect to .
#Plot Water Temp
Plot of the Temperature as a function of distance along the tube.

(4.1) 

(4.2) 
#Plot Temp as Fct of Distance
Conclusion
Using Maple for mathematical modeling, it is possible to calculate the temperature of the water and the air as a function of distance along a cylindrical heat exchanger.
Reference
Frank P. Incropera and David P. De Witt, Fundamentals of Heat and Mass Transfer, John Wiley & Sons.
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