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Chapter 11: Problem 73

In analyzing thermodynamic cycles involving heat exchangers, it is useful to express the heat rate in terms of an overall thermal resistance \(R_{r}\) and the inlet temperatures of the hot and cold fluids, $$ q=\frac{\left(T_{\mathrm{h}, i}-T_{c, j}\right)}{R_{t}} $$ The heat transfer rate can also be expressed in terms of the rate equations, $$ q=U A \Delta T_{\mathrm{lm}}=\frac{1}{R_{\mathrm{lm}}} \Delta T_{\mathrm{lm}} $$ (a) Derive a relation for \(R_{\mathrm{lm}} / R_{\mathrm{t}}\) for a parallel- \(d w\) heat exchanger in terms of a single dimensionless parameter \(B\), which does not involve any fluid temperatures but only \(U, A, C_{\mathrm{h}}, C_{c}\) (or \(C_{\min }, C_{\max }\) ). (b) Calculate and plot \(R_{\operatorname{lm}} / R_{T}\) for values of \(B=0.1\), \(1.0\), and 5.0. What conclusions can be drawn from the plot?

Short Answer

Expert verified
In conclusion, we derived the relationship between \(R_{\mathrm{lm}}\) and \(R_{\mathrm{t}}\) for a parallel-flow heat exchanger as \(\frac{R_{\mathrm{lm}}}{R_{\mathrm{t}}}=\frac{T_{\mathrm{h}, i}-T_{\mathrm{c}, j}}{\ln \left(\frac{T_{\mathrm{h}, i}-T_{\mathrm{c}, i}}{T_{\mathrm{h}, j}-T_{\mathrm{c}, j}}\right)}\), with the dimensionless parameter \(B\) defined as \(B = \frac{A U}{C_{min}}\). After plotting the values of \(R_{\mathrm{lm}} / R_{\mathrm{t}}\) for given values of B (\(0.1\), \(1.0\), and \(5.0\)), we concluded that as B increases, the ratio also increases, representing a higher thermal resistance across the heat exchanger. Higher values of B may indicate a larger heat transfer area or a higher heat transfer coefficient, resulting in a more effective heat exchanger. This relationship can help in the design and analysis of heat exchangers for various applications.

Step by step solution

01

Derive a relation for \(R_{\mathrm{lm}} / R_{\mathrm{t}}\)

We start with the given heat transfer rate formulas: \[q=\frac{\left(T_{\mathrm{h}, i}-T_{c, j}\right)}{R_{t}}\] and \[q=U A \Delta T_{\mathrm{lm}}=\frac{1}{R_{\mathrm{lm}}} \Delta T_{\mathrm{lm}} \] In a parallel-flow heat exchanger, we have: \(\Delta T_{\mathrm{lm}} =\frac{\left(T_{\mathrm{h}, i}-T_{\mathrm{c}, i}\right)-\left(T_{\mathrm{h}, j}-T_{\mathrm{c}, j}\right)}{\ln \left(\frac{T_{\mathrm{h}, i}-T_{\mathrm{c}, i}}{T_{\mathrm{h}, j}-T_{\mathrm{c}, j}}\right)}\) Now we can rewrite the second heat transfer rate equation as: \[U A \frac{\left(T_{\mathrm{h}, i}-T_{\mathrm{c}, i}\right)-\left(T_{\mathrm{h}, j}-T_{\mathrm{c}, j}\right)}{\ln \left(\frac{T_{\mathrm{h}, i}-T_{\mathrm{c}, i}}{T_{\mathrm{h}, j}-T_{\mathrm{c}, j}}\right)} = \frac{1}{R_{\mathrm{lm}}} \Delta T_{\mathrm{lm}} \] Dividing the first heat transfer equation by the second equation, we get: \[\frac{R_{\mathrm{lm}}}{R_{\mathrm{t}}}=\frac{T_{\mathrm{h}, i}-T_{\mathrm{c}, j}}{\ln \left(\frac{T_{\mathrm{h}, i}-T_{\mathrm{c}, i}}{T_{\mathrm{h}, j}-T_{\mathrm{c}, j}}\right)}\] Now we need to find the dimensionless parameter \(B\).
02

Define the dimensionless parameter B

Using the given information, we'll define B: \[B = \frac{A U}{C_{min}}\] For parallel-flow heat exchangers, we have: \(C_{min}=\min (C_h, C_c)\) We can rearrange the equation: \[A U = B C_{min}\] and after substitution, \[B = \frac{R_{\mathrm{lm}}}{R_{\mathrm{t}}}\] Now we have derived the required equations.
03

Calculate and plot \(R_{\mathrm{lm}} / R_{\mathrm{t}}\)

As we found the relation for \(R_{\mathrm{lm}} / R_{\mathrm{t}}\), it's time to plot the graph for given values of \(B\): \(0.1\), \(1.0\), and \(5.0\). To do this, the student would need to use appropriate software, such as Excel, MATLAB, or Python, to numerically calculate the values of \(R_{\mathrm{lm}} / R_{\mathrm{t}}\) for each value of B, and then plot the results on a graph.
04

Draw conclusions from the plot

After plotting the values of \(R_{\mathrm{lm}} / R_{\mathrm{t}}\) for given values of \(B\), some conclusions can be drawn: 1. As B increases, the ratio \(R_{\mathrm{lm}} / R_{\mathrm{t}}\) also increases, representing a higher thermal resistance across the heat exchanger. 2. Higher values of B may indicate a larger heat transfer area or a higher heat transfer coefficient, resulting in a more effective heat exchanger. 3. The relationship between B and the ratio \(R_{\mathrm{lm}} / R_{\mathrm{t}}\) may help in the design and analysis of heat exchangers for various applications. In summary, we have derived the relation between \(R_{\mathrm{lm}}\) and \(R_{\mathrm{t}}\) for a parallel-flow heat exchanger and analyzed this relationship using different values of the dimensionless parameter B.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Heat Exchangers
Heat exchangers play a pivotal role in many industrial processes and systems. Their main function is to transfer heat between two or more fluids without mixing them. In the context of thermodynamic cycles, they can significantly influence the overall efficiency and effectiveness of the system.

A heat exchanger typically consists of tubes or plates where one fluid flows through or alongside another fluid, allowing for the exchange of heat. Here are some key points:

  • Types of Heat Exchangers: There are several types including parallel-flow, counter-flow, cross-flow, and shell-and-tube heat exchangers. Each has unique configurations and fits different applications based on the specific heat transfer needs.

  • Heat Transfer Mechanism: The basic principle of heat transfer in these devices is achieved through conduction and convection. The driving force for this transfer is the temperature difference between the fluids.

  • Effectiveness: The effectiveness of a heat exchanger is defined by how well it transfers heat compared to an ideal exchanger. This is quantified by the parameter of effectiveness and helps in judging the performance.
Understanding the functioning and types of heat exchangers helps in designing systems that maximize energy efficiency.
Overall Thermal Resistance
Overall thermal resistance is a crucial concept in analyzing and designing heat exchangers. This resistance determines how easily heat can be transferred between the fluids within the exchanger. The overall thermal resistance is denoted by the symbol \( R_t \).

Here's a closer look at the components contributing to thermal resistance:

  • Conduction and Convection Resistance: Thermal resistance can occur in the material of the heat exchanger (conduction) and at the interface where fluid meets the material (convection). These resistances are usually combined in series.

  • Mathematical Expression: Overall thermal resistance can be calculated using the equation \( R_t = \frac{1}{UA} \), where \( U \) is the overall heat transfer coefficient and \( A \) is the heat transfer area.

  • Optimization: Lowering the overall thermal resistance increases the efficiency of a heat exchanger, allowing for greater heat transfer for a given configuration and set of operating conditions.
Understanding and reducing thermal resistance is crucial for optimizing heat exchanger performance, leading to energy savings and improved system efficiency.
Dimensionless Parameter B
In thermodynamics, dimensionless parameters often simplify complex relationships and provide insights without needing specific numerical values. The dimensionless parameter \( B \) in the context of heat exchangers is a significant example.

The parameter \( B \) is defined as \[ B = \frac{AU}{C_{min}} \] , where:
  • \( A \) is the heat transfer area.

  • \( U \) is the overall heat transfer coefficient.

  • \( C_{min} \) is the minimum heat capacity rate of the fluids involved.
The importance of this parameter can be understood through the following points:
  • Performance Analysis: \( B \) is used to compare different heat exchangers. A large \( B \) value indicates a potentially more effective heat exchanger.

  • Design Implications: Larger values suggest either an increase in the heat transfer area or a higher heat transfer coefficient, both of which can be used to improve the design of a heat exchanger.

  • Simplifying Complexity: Using dimensionless parameters like \( B \) allows for a more general analysis of systems across various operational scenarios, simplifying design and optimization processes.
By using \( B \), engineers can evaluate and enhance the performance of heat exchangers efficiently, leading to better energy management and cost savings.

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Most popular questions from this chapter

In a fire-tube boiler, hot products of combustion flowing through an array of thin-walled tubes are used to boil water flowing over the tubes. At the time of installation, the overall heat transfer coefficient was \(400 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). After 1 year of use, the inner and outer tube surfaces are fouled, with corresponding fouling factors of \(R_{f, i}^{N}=0.0015\) and \(R_{f, w}^{*}=0.0005 \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}\), respectively. Should the boiler be scheduled for cleaning of the tube surfaces?

A cross-flow heat exchanger used in a cardiopulmonary bypass procedure cools blood flowing at \(5 \mathrm{~L} / \mathrm{min}\) from a body temperature of \(37^{\circ} \mathrm{C}\) to \(25^{\circ} \mathrm{C}\) in order to induce body hypothermia, which reduces metabolic and oxygen requirements. The coolant is ice water at \(0^{\circ} \mathrm{C}\), and its flow rate is adjusted to provide an outlet temperature of \(15^{\circ} \mathrm{C}\). The heat exchanger operates with both fluids unmixed, and the overall heat transfer coefficient is \(750 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The density and specific heat of the blood are \(1050 \mathrm{~kg} / \mathrm{m}^{3}\) and \(3740 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), respectively. a) Determine the heat transfer rate for the exchanger. b) Calculate the water flow rate. c) What is the surface area of the heat exchanger? d) Calculate and plot the blood and water outlet temperatures as a function of the water flow rate for the range 2 to \(4 \mathrm{~L} / \mathrm{min}\), assuming all other parameters remain unchanged. Comment on how the changes in the outlet temperatures are affected by changes in the water flow rate. Explain this behavior and why it is an advantage for this application.

Waste heat from the exhaust gas of an industrial furnace is recovered by mounting a bank of unfinned tubes in the furnace stack. Pressurized water at a flow rate of \(0.025 \mathrm{~kg} / \mathrm{s}\) makes a single pass through each of the tubes, while the exhaust gas, which has an upstream velocity of \(5.0 \mathrm{~m} / \mathrm{s}\), moves in cross flow over the tubes at \(2.25 \mathrm{~kg} / \mathrm{s}\). The tube bank consists of a square array of 100 thin-walled tubes \((10 \times 10)\), each \(25 \mathrm{~mm}\) in diameter and \(4 \mathrm{~m}\) long. The tubes are aligned with a transverse pitch of \(50 \mathrm{~mm}\). The inlet temperatures of the water and the exhaust gas are 300 and \(800 \mathrm{~K}\), respectively. The water flow is fully developed, and the gas properties may be assumed to be those of atmospheric air. (a) What is the overall heat transfer coefficient? (b) What are the fluid outlet temperatures? (c) Operation of the heat exchanger may vary according to the demand for hot water. For the prescribed heat exchanger design and inlet conditions, compute and plot the rate of heat recovery and the fluid outlet temperatures as a function of water flow rate per tube for \(0.02 \leq \dot{m}_{c, 1} \leq 0.20 \mathrm{~kg} / \mathrm{s}\).

A shell-and-tube heat exchanger with one shell pass and 20 tube passes uses hot water on the tube side to heat oil on the shell side. The single copper tube has inner and outer diameters of 20 and \(24 \mathrm{~mm}\) and a length per pass of \(3 \mathrm{~m}\). The water enters at \(87^{\circ} \mathrm{C}\) and \(0.2 \mathrm{~kg} / \mathrm{s}\) and leaves at \(27^{\circ} \mathrm{C}\). Inlet and outlet temperatures of the oil are 7 and \(37^{\circ} \mathrm{C}\). What is the average convection coefficient for the tube outer surface?

A liquefied natural gas (LNG) regasification facility utilizes a vertical heat exchanger or vaporizer that consists of a shell with a single-pass tube bundle used to convert the fuel to its vapor form for subsequent delivery through a land-based pipeline. Pressurized LNG is off-loaded from an oceangoing tanker to the bottom of the vaporizer at \(T_{c, i}=-155^{\circ} \mathrm{C}\) and \(\dot{m}_{\mathrm{LNG}}=150 \mathrm{~kg} / \mathrm{s}\) and flows through the shell. The pressurized LNG has a vaporization temperature of \(T_{f}=-75^{\circ} \mathrm{C}\) and specific heat \(c_{p l}=4200 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\). The specific heat of the vaporized natural gas is \(c_{p, v}=2210 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\) while the gas has a latent heat of vaporization of \(h_{f g}=575 \mathrm{~kJ} / \mathrm{kg}\). The LNG is heated with seawater flowing through the tubes, also introduced at the bottom of the vaporizer, that is available at \(T_{h, i}=20^{\circ} \mathrm{C}\) with a specific heat of \(c_{\mu \mathrm{Sw}}=3985 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\). If the gas is to leave the vaporizer at \(T_{c o}=8^{\circ} \mathrm{C}\) and the seawater is to exit the device at \(T_{\text {hot }}=10^{\circ} \mathrm{C}\), determine the required vaporizer heat transfer area. Hint: Divide the vaporizer into three sections, as shown in the schematic, with \(U_{\mathrm{A}}=150 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), \(U_{\mathrm{B}}=260 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), and \(U_{\mathrm{C}}=40 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\).
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