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?