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Chapter 3: Problem 6

How is the combined heat transfer coefficient defined? What convenience does it offer in heat transfer calculations?

Short Answer

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**Answer:** The combined heat transfer coefficient, denoted as \(U\), represents the overall efficiency of heat transfer through a series of different materials or processes, accounting for all modes of heat transfer, such as conduction, convection, and radiation. It is defined as the reciprocal of the sum of the individual thermal resistances in a heat transfer system. The combined heat transfer coefficient offers convenience in heat transfer calculations by allowing complex systems to be analyzed using a single coefficient, simplifying the analysis process significantly. It also helps streamline the calculation of the overall heat transfer rate in a system, making it easier to determine the heat transfer rate across multiple layers or processes.

Step by step solution

01

Definition of Combined Heat Transfer Coefficient

The combined heat transfer coefficient, often denoted as \(U\), represents the overall efficiency of heat transfer through a series of different materials or processes. It accounts for all modes of heat transfer, such as conduction, convection, and radiation. The combined heat transfer coefficient is defined as the reciprocal of the sum of the individual thermal resistances \((R)\) in a heat transfer system: $$ U = \frac{1}{\sum R_i} $$ where \(R_i\) represents the thermal resistance of each layer or process.
02

Convenience in Heat Transfer Calculations

The combined heat transfer coefficient offers a significant convenience in heat transfer calculations because it allows complex systems to be analyzed using a single coefficient. Instead of having to individually calculate the resistances for each mode of heat transfer and for each layer or process in the system, the use of the combined heat transfer coefficient simplifies the analysis process significantly. Additionally, applying the combined heat transfer coefficient helps simplify the calculation of the overall heat transfer rate, \(Q\), in a system using the following equation: $$ Q = U \cdot A \cdot \Delta T $$ where \(A\) is the surface area through which heat transfer occurs and \(\Delta T\) is the temperature difference across the system. In summary, the combined heat transfer coefficient streamlines heat transfer calculations by condensing complex systems with multiple modes of heat transfer and layers into one parameter, making it easier to calculate the overall heat transfer rate.

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Key Concepts

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

Thermal Resistance
Imagine wearing layers of clothing on a cold day. Each layer you add increases your resistance to the cold air, trapping heat closer to your body. Similarly, in the context of heat transfer, thermal resistance is a concept that quantifies how much a material opposes the flow of heat.

Thermal resistance, often denoted by the symbol \( R \), is mathematically given by the equation: \[ R = \frac{\Delta T}{Q} \] where \( \Delta T \) is the temperature difference across the material and \( Q \) is the heat transfer rate. A higher thermal resistance indicates that the material is a better insulator, slowing down the heat flow.

In a multi-layer system, each layer contributes its own resistance, and these can be added up to find the total resistance the heat must overcome. This is why snug homes have insulation layers in their walls; they increase the thermal resistance, thereby keeping the indoor temperature regulated.
Heat Transfer Calculations
Working with heat transfer phenomena means playing with energy movement from a hotter place to a cooler one. Heat transfer calculations are crucial in various engineering tasks to ensure the safety and efficiency of equipment and buildings.

To perform these calculations, engineers use different equations for conduction, convection, and radiation, which represent the three primary modes of heat transfer. However, when dealing with real-world problems, where multiple layers and transfer modes are present, the calculations can get complex. This complexity can be reduced by using combined heat transfer coefficients, which we will discuss in more detail later.
Heat Transfer Modes
Heat moves in mysterious ways, but science helps us understand them as three distinct modes of heat transfer: conduction, convection, and radiation.

Conduction is like passing a secret from one person to another by whisper; it's the heat transfer through solids when they are in direct contact. Metals, for instance, are chatty and pass heat quickly due to their molecular structure. This is measured by a material’s thermal conductivity.

Convection is more like a crowd's movement at a concert—it's the heat transfer by the motion of fluids (liquids or gases). Warm fluid rises while cool fluid descends, creating a circulating current. A fan-forced oven uses this mode to cook food evenly.

Radiation is the enigmatic heat transfer through electromagnetic waves, similar to sunlight warming your face. It doesn't require a medium to travel through, so it works even in the vacuum of space.
Overall Heat Transfer Rate
When you're trying to heat up your dinner, the thing you really care about is how quickly it happens. Similarly, in heat transfer, we're often interested in the overall heat transfer rate, denoted by \( Q \). This tells us the amount of heat energy moving through a system over time.

To determine \( Q \), one can use the formula: \[ Q = U \cdot A \cdot \Delta T \] where \( U \) is the combined heat transfer coefficient, \( A \) is the area through which heat is being transferred, and \( \Delta T \) is the temperature difference across the system.

The overall heat transfer rate is influenced by the materials involved and the conditions under which heat transfer occurs. By knowing \( Q \) and the specific context, engineers can design systems that heat up or cool down as required.

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Most popular questions from this chapter

Two 3-m-long and \(0.4-\mathrm{cm}\)-thick cast iron \((k=\) \(52 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) steam pipes of outer diameter \(10 \mathrm{~cm}\) are connected to each other through two 1 -cm-thick flanges of outer diameter \(20 \mathrm{~cm}\). The steam flows inside the pipe at an average temperature of \(200^{\circ} \mathrm{C}\) with a heat transfer coefficient of \(180 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The outer surface of the pipe is exposed to an ambient at \(12^{\circ} \mathrm{C}\), with a heat transfer coefficient of \(25 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). (a) Disregarding the flanges, determine the average outer surface temperature of the pipe. (b) Using this temperature for the base of the flange and treating the flanges as the fins, determine the fin efficiency and the rate of heat transfer from the flanges. (c) What length of pipe is the flange section equivalent to for heat transfer purposes?

Explain how the fins enhance heat transfer from a surface. Also, explain how the addition of fins may actually decrease heat transfer from a surface.

A pipe is insulated to reduce the heat loss from it. However, measurements indicate that the rate of heat loss has increased instead of decreasing. Can the measurements be right?

Using cylindrical samples of the same material, devise an experiment to determine the thermal contact resistance. Cylindrical samples are available at any length, and the thermal conductivity of the material is known.

Exposure to high concentration of gaseous ammonia can cause lung damage. To prevent gaseous ammonia from leaking out, ammonia is transported in its liquid state through a pipe \(\left(k=25 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, D_{i}=2.5 \mathrm{~cm}\right.\), \(D_{o}=4 \mathrm{~cm}\), and \(L=10 \mathrm{~m}\) ). Since liquid ammonia has a normal boiling point of \(-33.3^{\circ} \mathrm{C}\), the pipe needs to be properly insulated to prevent the surrounding heat from causing the ammonia to boil. The pipe is situated in a laboratory, where the average ambient air temperature is \(20^{\circ} \mathrm{C}\). The convection heat transfer coefficients of the liquid hydrogen and the ambient air are \(100 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) and \(20 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), respectively. Determine the insulation thickness for the pipe using a material with \(k=\) \(0.75 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) to keep the liquid ammonia flowing at an average temperature of \(-35^{\circ} \mathrm{C}\), while maintaining the insulated pipe outer surface temperature at \(10^{\circ} \mathrm{C}\).
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