Problem #$1$

In Inclass activity #$10$a$,$ we have derived the $-$NTU method for a parallel flow double$-$pipe heat exchanger. Here you are asked to follow the same method to derive the following $-$NTU equation for a counter flow double$-$pipe heat exchanger:

$=\frac{1-e^{-\text{N}\text{T}\text{U}(1-C_r)}}{1-C_r{e}^{-\text{N}\text{T}\text{U}(1-C_r)}}$

where $$ is the effectiveness of the heat exchanger, $C_r$ the ratio of heat capacitances of two fluids, and NTU the number of transfer unit; all three defined in the following table,

$=\frac{q}{q_{\text{m}\text{a}\text{x}}}=\frac{C_w(T_1-T_2)}{C_w(T_1-t_1)}=\frac{T_1-T_2}{T_1-t_1}if{C}_w{C}_c,C_r=\frac{C_{\text{m}\text{i}\text{n}}}{C_{\text{m}\text{a}\text{x}}}=\frac{C_c}{C_w}if{C}_w>C_c,$NTU$=\frac{U_o-A_o}{C_c}if{C}_w>C_c$