Derive \epsilon - NTU Expression for Double-Pipe Counter Flow Heat Exchanger

[CRich]

Homework Statement

I'm trying to derive the \epsilon - NTU Expression for a double-pipe counter flow heat exchanger. I know what I need to do the only problem I am having is:

I don't know how to algebraically go from

ln( \frac{\Delta T2}{\Delta T1} ) = -UA ( \frac{1}{Ch} + \frac{1}{Cc} )

to

ln( \frac{Th,o-Tc,i}{Th,i-Tc,o} ) = -\frac{UA}{Cmin}(1-\frac{Cmin}{Cmax})

2. Homework Equations & attempt at problem

I said (1/Ch + 1/Cc) = (\frac{Th,i-Th,o}{q} + \frac{Tc,o-Tc,i}{q})

Then I used the relationship: \epsilon = \frac{q}{qmax}

where qmax = Cmin(Thi-Tci)

...so q = \epsilon * qmax

substituted these equations in ...

I have a giant mess of Cmin and Cmax

Any help is greatly appreciated!

The only other equations that may be beneficial are:

q = mh*Cph*(Th,i - Th,o)
and
q = mc*Cpc*(Tc,i - Tc,o)
and
\frac{Cmin}{Cmax} = \frac{mh Cph}{mc Cpc} = \frac{Tc,o - Tc,i}{Th,i - Th,o}

 

[CRich]

Anyone?