Using the results shown in Table 11.3, determine an expression for the ratio of the effectiveness of a counterflow heat exchanger to that of a parallel flow heat exchanger. Considering the same heat exchanger used in either counter- or parallel flow mode, what is the limiting value of the effectiveness ratio for extremely small area (low cost) heat exchangers? What is the limiting value of the effectiveness ratio for heat exchangers of infinite area (high cost)? Plot the effectiveness ratio over the range \(0 \leq \mathrm{NTU} \leq 10\) for \(C_{r}=\) \(0.25,0.5\), and 0.75. Under what conditions (high or low cost) does the counterflow mode most significantly outperform the parallel flow mode?

Data From Table 11.3:-

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TABLE 11.3 Heat exchanger effectiveness relations [5] Flow Arrangement Parallel flow Relation E = 1- exp[-NTU(1 + C,)] 1+C, (11.28a) 1-exp[-NTU(1 - C,)] Counterflow 8 = (C, <1) 1-C, exp[-NTU(1 - C,)] 8 = NTU 1+ NTU (C,= 1) (11.29a) Shell-and-tube One shell pass (2, 4, ... tube passes) = 2{1 + C, + (1 + C = 1+ exp[-(NTU), (1+ C)/]] 1- exp[-(NTU), (1+C)/2]] (11.30a) n shell passes (2n, 4n,... tube passes) Cross-flow (single pass) 8 = 1-EC 1-8 1-EC, 1-E C (11.31a) Both fluids unmixed E = 1-exp (NTU)022 (exp[-C, (NTU) 0.78]-1) 1}] (11.32) Cmax (mixed), C (unmixed) min Cmin (mixed), Cax (unmixed) All exchangers (C, = 0) (1-exp(-C,[1-exp(-NTU)])) = 1-exp(-C1- exp[-C, (NTU)]}) E = 1-exp(-NTU) = (11.33a) (11.34a) (11.35a)