4. Effectiveness - NTU Method for Cocurrent Exchanger

From energy balance between the cold and hot fluid, we get \[\begin{align*} C_h(T_{hi}-T_{ho}) &= C_c(T_{co}-T_{ci}) \\ T_{ho} &= T_{hi} - \frac{C_c}{C_h}(T_{co}-T_{ci}) \tag*{(6)} \end{align*}\] Using Eqn.(6) in Eqn.(5), we get \[\begin{aligned} \frac{T_{hi} - \dfrac{C_c}{C_h}(T_{co}-T_{ci}) - T_{co}}{T_{hi}-T_{ci}} &= e^{-BAU} \\ \frac{(T_{hi}-T_{ci}) - \dfrac{C_c}{C_h}(T_{co}-T_{ci}) - T_{co} + T_{ci}}{T_{hi}-T_{ci}} &= e^{-BAU} \end{aligned}\]

\[\begin{aligned} \frac{(T_{hi}-T_{ci}) - \dfrac{C_c}{C_h}(T_{co}-T_{ci}) - (T_{co} - T_{ci})}{T_{hi}-T_{ci}} &= e^{-BAU} \\ 1-\frac{T_{co}-T_{ci}}{T_{hi}-T_{ci}}\left(1+\frac{C_c}{C_h}\right) &= e^{-BAU} \end{aligned}\] \[1 - e^{-BAU} = \frac{T_{co}-T_{ci}}{T_{hi}-T_{ci}}\left(1+\frac{C_c}{C_h}\right) \tag*{(7)}\]

From the definition of \(\varepsilon\), we have \[\varepsilon = \frac{Q}{C_{min}(T_{hi}-T_{ci})} \qquad \Longrightarrow \quad T_{hi}-T_{ci} = \frac{Q}{\varepsilon C_{min}} \tag*{(8)}\] For the cold fluid, \[Q = C_c(T_{co}-T_{ci}) \qquad \Longrightarrow \quad T_{co}-T_{ci} = \frac{Q}{C_c} \tag*{(9)}\]

Using Eqns.(8) and (9) in Eqn.(7), we get \[\begin{aligned} 1 - e^{-BAU} &= \varepsilon \frac{C_{\text{min}}}{C_c}\left(1+\frac{C_c}{C_h} \right) \end{aligned}\] Rearranging, we get \[\varepsilon = \frac{1-e^{-BAU}}{\dfrac{C_{\text{min}}}{C_c}+ \dfrac{C_{\text{min}}}{C_h}} \tag*{(10)}\] From Eqn.(4) we have, \[B = \frac{1}{C_h} + \frac{1}{C_c}\] And, from the definition of NTU = \(N\) we have \[N = \frac{AU}{C_{\text{min}}}\] Using these in Eqn.(10), we get

\[\begin{aligned} \varepsilon &= \frac{1-\exp\left[-\left(\dfrac{1}{C_h} + \dfrac{1}{C_c} \right)NC_{\text{min}} \right]}{\dfrac{C_{\text{min}}}{C_c} + \dfrac{C_{\text{min}}}{C_h}} \\ &= \frac{1-\exp\left[-N\left(\dfrac{C_{\text{min}}}{C_h} + \dfrac{C_{\text{min}}}{C_c} \right) \right]}{\dfrac{C_{\text{min}}}{C_c} + \dfrac{C_{\text{min}}}{C_h}} \end{aligned}\] From the definition of \(C\). we have \[C = \frac{C_{\text{min}}}{C_{\text{max}}}\] Let \(C_h=C_{\text{min}}\). Then, \[\varepsilon = \frac{1-\exp[(-N(1+C)]}{1+C}\]

If \(C_c=C_{\text{min}}\). Then also, we get \[\varepsilon = \frac{1-\exp[(-N(1+C)]}{1+C}\] Hence, for the cocurrent heat exchanger, the relation between effectiveness (\(\varepsilon\)) and NTU (\(N\)) is given by \[\boxed{\varepsilon = \frac{1-\exp[(-N(1+C)]}{1+C}}\]