Dynamics of a Counter-current Heat Exchanger

For each fluid we could write an equation such as Eqn. (14.10). If the heat capacity of the wall would be ignored, we would get a set of two simultaneous equations that could be solved. The outlet temperature responses of the two fluids would consist of a summation of exponential functions, which are an indication of higher order models. An approximate solution of the set of two differential equations is discussed, among others, by Friedly (1972), Harriott (1%4) andMozley (1956). 

Note that one equation has a term +dT /dz and the other has a term -dT / 3z since the one fluid is warming up and the other is cooling down. 

An additional approximation is made by using the arithmetic mean of the temperature  

From a substitution of Eqm (14.39) into Eqm (14.38) the transfer function between one fluid outlet temperature and the other fluid Met temperature can then easily be derived  

Note that Tr is the residence tMe of the hquid in the heat exchanger. As can be seen, the approximation is second order. When the one fluid increases in temperature, the outlet temperature of the other flrrid will also start to increase. The approxirrration can be rrtade more accmate by incorporating the heat capacity of the wall in eqrral parts in the heat capacity of the fluids. 

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