Question

A one-shell and two-tube-type heat exchanger has an overall heat transfer coefficient of $300 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2}{ }^{\circ} \mathrm{F}\(. The shell-side fluid has a heat capacity rate of \)20,000 \mathrm{Btu} / \mathrm{h} \cdot{ }^{\circ} \mathrm{F}$, while the tube-side fluid has a heat capacity rate of 40,000 $\mathrm{Btu} / \mathrm{h} \cdot{ }^{\circ} \mathrm{F}$. The inlet temperatures on the shell side and tube side are \(200^{\circ} \mathrm{F}\) and \(90^{\circ} \mathrm{F}\), respectively. If the total heat transfer area is \(100 \mathrm{ft}^{2}\), determine \((a)\) the heat transfer effectiveness and \((b)\) the actual heat transfer rate in the heat exchanger.

Short answer

The heat transfer effectiveness is 0.719, and the actual heat transfer rate is 1,584,900 Btu/h.

Step-by-step solution

Calculate the Minimum Fluid Heat Capacity Rate (C\(_{min}\))

First, we need to find the minimum fluid heat capacity rate, which will be the smaller value between the shell-side heat capacity rate (C\(_{s}\)) and the tube-side heat capacity rate (C\(_{t}\)). C\(_{min}\) = min(C\(_{s}\), C\(_{t}\)) = min(\(20,000 \mathrm{Btu/h\cdot^\circ F}\), \(40,000 \mathrm{Btu/h\cdot^\circ F}\)) = \(20,000 \mathrm{Btu/h\cdot^\circ F}\)

Calculate the Number of Transfer Units (NTU)

Next, we will calculate the number of transfer units (NTU) using the given overall heat transfer coefficient (U), total heat transfer area (A), and the minimum fluid heat capacity rate (C\(_{min}\)): NTU = \frac{UA}{C_{min}} = \frac{300 \mathrm{Btu/h\cdot ft^2\cdot^\circ F} \cdot 100 \mathrm{ft^2}}{20,000 \mathrm{Btu/h\cdot^\circ F}} = 1.5

Calculate the heat transfer effectiveness (ε)

Now, we will calculate the heat transfer effectiveness (ε) for the one-shell and two-tube heat exchanger configuration using the NTU: ε = \frac{1 - e^{-NTU(1+C_r)} (1 + C_r)}{1 + C_r} = \frac{1 - e^{-1.5(1 + 0.5)} (1 + 0.5)}{1 + 0.5} = 0.719 where C\(_r\) = C\(_{min}\)/C\(_{max}\) = \(20,000 \mathrm{Btu/h\cdot}^\circ \mathrm{F}\)/ \(40,000 \mathrm{Btu/h\cdot}^\circ \mathrm{F}\) = 0.5 So, the heat transfer effectiveness is 0.719.

Calculate the actual heat transfer rate (Q)

Finally, we will calculate the actual heat transfer rate (Q) using the heat transfer effectiveness (ε) and the minimum fluid heat capacity rate (C\(_{min}\)): Q = εC\(_{min}\)ΔT\(_{in}\) = 0.719 × \(20,000 \mathrm{Btu/h\cdot^\circ F}\) × (\(200^{\circ} \mathrm{F}\) - \(90^{\circ} \mathrm{F}\)) = 0.719 × \(20,000 \mathrm{Btu/h\cdot^\circ F}\) × \(110^{\circ} \mathrm{F}\) = 1,584,900 \mathrm{Btu/h} To summarize, the heat transfer effectiveness of the one-shell and two-tube-type heat exchanger is 0.719, and the actual heat transfer rate is 1,584,900 Btu/h.