Chapter 4: Problem 4

Explain in what sense the set of equations Equation (4.11) provides a solution for the surface-to-surface radiative heat transfer problem. How would you solve the equations (a) in the case that the temperature of all surfaces is known and (b) in the case that for some surfaces the temperature and for other surfaces the heat flux are known?

Short Answer

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Question: Explain how Equation 4.11 helps in solving the surface-to-surface radiative heat transfer problem and describe the steps to solve this problem in cases (a) and (b). Answer: Equation 4.11 provides a solution to the surface-to-surface radiative heat transfer problem by relating radiative heat transfer rates between surfaces based on their physical and geometrical properties. Solving this system of equations allows determining temperatures or heat fluxes on each surface, ensuring energy balance in the system. In case (a), when all surface temperatures are known, the steps are: 1. Substitute the known temperature values into Equation 4.11. 2. Determine the emissivity and view factors of the surfaces. 3. Solve the system of equations to find the radiative heat fluxes between the surfaces. In case (b), when some surfaces have known temperatures and some have known heat fluxes, the steps are: 1. Substitute the known temperature and heat flux values into Equation 4.11. 2. Determine the emissivity and view factors of the surfaces. 3. Solve the system of equations to find unknown temperatures and heat fluxes using an iterative approach such as the Newton-Raphson method.

Step by step solution

1. Understanding the surface-to-surface radiative heat transfer problem and Equation 4.11

The surface-to-surface radiative heat transfer problem deals with heat transfer between surfaces via electromagnetic radiation. It is an important part of many engineering problems, particularly in systems involving high temperatures. Equation 4.11 represents a system of equations for radiative heat exchange between surfaces. Without the specific equation, it is difficult to give detailed information; however, in general, the radiative heat transfer equations involve parameters like surface temperatures, view factors, and emissivity of the surfaces.

2. Importance of Equation 4.11 in solving the problem

Equation 4.11 plays a crucial role in solving the surface-to-surface radiative heat transfer problem, as it relates the radiative heat transfer rate between surfaces based on their physical and geometrical properties. By solving this system of equations, one can determine either the temperatures or heat fluxes on each surface, ensuring the energy balance is maintained within the system.

3. Solving the equations with known surface temperatures (case a)

If the temperature of all surfaces is known, we can solve Equation 4.11 to find the radiative heat flux between surfaces. Steps to solve the equations are: 1. Substitute the known temperature values into Equation 4.11. 2. Determine the emissivity and view factors of the surfaces (if not provided) using the given temperature values. 3. Solve the system of equations to find the radiative heat fluxes between the surfaces. Methods to solve the system of equations include matrix inversion, Gaussian elimination, or iterative methods like Gauss-Seidel or the method of successive over-relaxation.

4. Solving the equations with known surface temperatures and heat fluxes (case b)

In this case, we have some surfaces with known temperatures and others with known heat fluxes. The goal is to solve Equation 4.11 to find the unknown temperatures and heat fluxes. Steps to solve the equations are: 1. Substitute the known temperature and heat flux values into Equation 4.11, creating a system of equations with the remaining unknown variables (unknown temperatures or heat fluxes). 2. Determine the emissivity and view factors of the surfaces (if not provided) using the given temperature values. 3. Solve the system of equations to find the unknown temperatures and heat fluxes. This case might require an iterative approach, such as the Newton-Raphson method, to solve the system of equations, particularly when there is strong coupling between the variables.

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Key Concepts

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

Surface-to-Surface Radiation

Surface-to-surface radiation is a critical aspect of thermal engineering, dealing primarily with the transfer of heat through electromagnetic waves between surfaces. Unlike conduction and convection, radiation does not require a medium to transfer heat, making it a predominant mode of heat transfer in vacuums and at high temperatures.
When analyzing such problems, engineers take into account not only the temperatures of the surfaces but also their emissivity and reflectivity, the geometry of the involved bodies, and spatial orientation, commonly described by view factors. These components play a substantial role in calculating the amount of energy radiated from one surface to another.
Through understanding this concept, students can tackle complex problems such as calculating the heat transfer in a furnace, between spacecraft components in outer space, or within electronic devices where cooling or heat dissipation is crucial.

Radiative Heat Exchange Equations

The radiative heat exchange equations are a set of mathematical expressions used to quantify the radiant energy transfer between different surfaces. These equations are derived from the fundamental laws of thermodynamics and electromagnetic theory, particularly the Stefan-Boltzmann law, which states that the thermal radiation from a surface is proportional to the fourth power of its absolute temperature.
In these equations, the heat transfer rate is influenced by factors such as surface area, temperature, emissivity, and the view factor. To completely describe the radiative interaction within a system, one must establish an energy balance that considers the radiative heat exchange with all other surfaces in view. Solving these equations provides engineers with the necessary information to predict surface temperatures or heat fluxes, crucial for designing thermally efficient systems.

Thermal Engineering

Thermal engineering is a subset of mechanical engineering that focuses on the movement of heat energy and its conversion to different forms of energy. It involves the analysis, design, and maintenance of systems that heat and cool, often requiring a deep understanding of heat transfer modes, including conduction, convection, and radiation.
Professionals in this field use radiative heat exchange equations to design heating, ventilation, and air conditioning (HVAC) systems, thermal insulation, energy systems in automobiles, and much more. By mastering the principles of thermal engineering, students can innovate solutions to control temperature and enhance the reliability and performance of a wide array of thermal systems.

Gauss-Seidel Method

The Gauss-Seidel method is an iterative technique used to solve systems of linear equations, commonly applied when dealing with large sets of data. It is particularly useful in solving the matrix equations that often result from discretizing the equations governing radiative heat transfer.
In this method, each equation in the system is rearranged to express one variable as a function of the remaining variables. Then, using initial guesses for the solution, these equations are solved sequentially, and the latest available value for each variable is used to calculate the next. This process is repeated until the changes in the variables between successive iterations fall below a predetermined threshold, indicating convergence to a solution.
The efficiency of this method, especially in terms of its simplicity and limited memory requirements, makes it a popular choice among engineers when dealing with heat transfer problems.

Newton-Raphson Method

The Newton-Raphson method is another powerful iterative technique, widely utilized to find successively better approximations to the zeros (or roots) of a real-valued function. Its application within the context of thermal engineering often pertains to non-linear systems of equations arising in complex heat transfer analyses.
Starting from an initial guess, the method uses the function's derivative to approximate the slope near the guess and uses the intercept with the x-axis to formulate a better estimate of the zero. The procedure is repeated until a sufficiently accurate value is obtained. This method can converge to a solution more rapidly than other iterative methods, especially when the initial guess is close to the actual root, and the function involved does not have flat or highly curved regions near the zero.
Given its robustness and precision, the Newton-Raphson method is often the preferred choice when precise control over temperature and heat flux is necessary, making it an indispensable tool in the arsenal of thermal engineers.

Short Answer

Step by step solution

1. Understanding the surface-to-surface radiative heat transfer problem and Equation 4.11

2. Importance of Equation 4.11 in solving the problem

3. Solving the equations with known surface temperatures (case a)

4. Solving the equations with known surface temperatures and heat fluxes (case b)

Key Concepts

Surface-to-Surface Radiation

Radiative Heat Exchange Equations

Thermal Engineering

Gauss-Seidel Method

Newton-Raphson Method

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