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Chapter 11: Problem 41

When the outlet temperatures of the fluids in a heat exchanger are not known, is it still practical to use the LMTD method? Explain.

Short Answer

Expert verified
Answer: No, it is not practical to use the LMTD method directly when the outlet temperatures are unknown. However, alternative techniques can be employed to estimate the unknown outlet temperatures and subsequently apply the LMTD method for heat transfer calculations.

Step by step solution

01

Understand the LMTD method

The Log Mean Temperature Difference (LMTD) method is a widely used method to calculate the heat transfer in heat exchangers. It is based on the average temperature difference between the hot and cold fluids. The LMTD formula is given by: \[LMTD = \frac{(T_h1 - T_c1) - (T_h2 - T_c2)}{\ln \frac{T_h1 - T_c1}{T_h2 - T_c2}}\] where \(T_h1\) and \(T_h2\) are the hot fluid temperatures at the inlet and outlet, and \(T_c1\) and \(T_c2\) are the cold fluid temperatures at the inlet and outlet, respectively.
02

Identify the dependence on inlet and outlet temperatures

As seen from the LMTD formula, both inlet and outlet temperatures are required to calculate the LMTD value. If either the outlet temperatures of the hot fluid or cold fluid are unknown, it would not be possible to directly calculate the LMTD using this method.
03

Assess the practicality of using the LMTD method with unknown outlet temperatures

In situations where outlet temperatures are unknown and cannot be measured, it is impractical to use the LMTD method directly. However, engineers can use alternative approaches such as iterative methods or numerical techniques to estimate the outlet temperatures based on other known parameters (e.g., heat transfer coefficients, mass flow rates, and heat capacities).
04

Conclusion

In conclusion, it is not practical to use the LMTD method directly for a heat exchanger when the outlet temperatures of the fluids are unknown. However, alternative techniques can be employed to estimate the unknown outlet temperatures and subsequently apply the LMTD method for heat transfer calculations.

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Most popular questions from this chapter

In a textile manufacturing plant, the waste dyeing water $\left(c_{p}=4295 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)\( at \)80^{\circ} \mathrm{C}$ is to be used to preheat fresh water $\left(c_{p}=4180 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)\( at \)10^{\circ} \mathrm{C}$ at the same flow rate in a double-pipe counterflow heat exchanger. The heat transfer surface area of the heat exchanger is \(1.65 \mathrm{~m}^{2}\), and the overall heat transfer coefficient is $625 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\(. If the rate of heat transfer in the heat exchanger is \)35 \mathrm{~kW}$, determine the outlet temperature and the mass flow rate of each fluid stream.

A crossflow heat exchanger with both fluids unmixed has an overall heat transfer coefficient of \(200 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) and a heat transfer surface area of \(400 \mathrm{~m}^{2}\). The hot fluid has a heat capacity of \(40,000 \mathrm{~W} / \mathrm{K}\), while the cold fluid has a heat capacity of \(80,000 \mathrm{~W} / \mathrm{K}\). If the inlet temperatures of both hot and cold fluids are \(80^{\circ} \mathrm{C}\) and $20^{\circ} \mathrm{C}$, respectively, determine the exit temperature of the cold fluid.

A double-pipe counterflow heat exchanger is used to cool a hot fluid before it flows into a pipe system. The pipe system is mainly constructed with ASTM F2389 polypropylene pipes. According to the ASME Code for Process Piping, the recommended maximum temperature for polypropylene pipes is $99^{\circ} \mathrm{C}$ (ASME B31.3-2014, Table B-1). The heat exchanger's inner tube has negligible wall thickness. The convection heat transfer coefficients inside and outside of the heat exchanger inner tube are $1500 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\( and 1000 \)\mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}$, respectively. The fouling factor estimated for the heat exchanger is \(0.0001 \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}\). The hot fluid \(\left(c_{p}=3800\right.\) \(\mathrm{J} / \mathrm{kg} \cdot \mathrm{K})\) enters the heat exchanger at \(150^{\circ} \mathrm{C}\) with a flow rate of $0.5 \mathrm{~kg} / \mathrm{s}\(. In the cold side, cooling fluid \)\left(c_{p}=4200 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)$ enters the heat exchanger at \(10^{\circ} \mathrm{C}\) with a mass flow rate of $0.75 \mathrm{~kg} / \mathrm{s}$. Determine the heat transfer surface area that the heat exchanger needs to cool the hot fluid to \(99^{\circ} \mathrm{C}\) at the outlet so that it flows into the pipe system at a temperature not exceeding the recommended maximum temperature for polypropylene pipes.

A counterflow heat exchanger is used to cool oil $\left(c_{p}=2.20 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}\right)\( from \)110^{\circ} \mathrm{C}\( to \)85^{\circ} \mathrm{C}\( at a rate of \)0.75\( \)\mathrm{kg} / \mathrm{s}\( with cold water \)\left(c_{p}=4.18 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}\right)\( that enters the heat exchanger at \)20^{\circ} \mathrm{C}$ at a rate of \(0.6 \mathrm{~kg} / \mathrm{s}\). If the overall heat transfer coefficient is \(800 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), the heat transfer area of the heat exchanger is (a) \(0.745 \mathrm{~m}^{2}\) (b) \(0.760 \mathrm{~m}^{2}\) (c) \(0.775 \mathrm{~m}^{2}\) (d) \(0.790 \mathrm{~m}^{2}\) (e) \(0.805 \mathrm{~m}^{2}\)

A pipe system is mainly constructed with ASTM F441 CPVC pipes. The ASME Code for Process Piping (ASME B31.3-2014, Table B-1) recommends that the maximum temperature limit for CPVC pipes be \(93.3^{\circ} \mathrm{C}\). A double-pipe heat exchanger is located upstream of the pipe system to reduce the hot water temperature before it flows into the CPVC pipes. The inner tube of the heat exchanger has a negligible wall thickness, and its length and diameter are $5 \mathrm{~m}\( and \)25 \mathrm{~mm}$, respectively. The convection heat transfer coefficients inside and outside of the heat exchanger inner tube are $3600 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\( and \)4500 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, respectively. The hot fluid enters the heat exchanger at \(105^{\circ} \mathrm{C}\) with a flow rate of $0.75 \mathrm{~kg} / \mathrm{s}$. In the cold fluid stream, water enters the heat exchanger at \(10^{\circ} \mathrm{C}\) and exits at \(80^{\circ} \mathrm{C}\). Determine whether this double-pipe heat exchanger should employ the parallel flow or the counterflow configuration to ensure that the hot water exiting the heat exchanger is \(93.3^{\circ} \mathrm{C}\) or lower.
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