Question

A furnace is shaped like a long equilateral-triangular duct where the width of each side is \(2 \mathrm{~m}\). Heat is supplied from the base surface, whose emissivity is \(\varepsilon_{1}=0.8\), at a rate of \(800 \mathrm{~W} / \mathrm{m}^{2}\) while the side surfaces, whose emissivities are \(0.5\), are maintained at \(500 \mathrm{~K}\). Neglecting the end effects, determine the temperature of the base surface. Can you treat this geometry as a two-surface enclosure?

Short answer

Answer: The base surface temperature is approximately 612.7 K, and this geometry cannot be treated as a two-surface enclosure.

Step-by-step solution

Apply the Stefan-Boltzmann Law

The Stefan-Boltzmann Law relates the heat transfer (power emitted) by a surface to its temperature and emissivity: \(q = \varepsilon \sigma(T^{4} - T_{s}^4)\) where: - \(q\) is the heat transfer per unit area [W/m²] - \(\varepsilon\) is the emissivity of the surface - \(\sigma\) is the Stefan-Boltzmann constant, \(5.67 \times 10^{-8} \mathrm{W/m^{2}K^4}\) - \(T\) is the temperature of the surface [K] - \(T_s\) is the surrounding surface temperature [K] The base surface has an emissivity of \(\varepsilon_{1}=0.8\) and receives a heat transfer of \(800 \mathrm{~W}/\mathrm{m}^{2}\). We will call the base surface temperature \(T_{1}\). The side surfaces have an emissivity of \(\varepsilon_{2}=0.5\) and are maintained at \(500 \mathrm{~K}\) (i.e., \(T_{s} = 500 \mathrm{~K}\)).

Energy balance

We can set up an energy balance equation using the Stefan-Boltzmann Law for the base surface and side surfaces: \(q_{1} = \varepsilon_{1} \sigma (T_{1}^4 - T_{s}^4)\) Substitute the given values: \(800 = 0.8 \times 5.67 \times 10^{-8} \times (T_{1}^4 - 500^4)\)

Solve for the base surface temperature, \(T_1\)

Solve the energy balance equation for \(T_{1}\): \(800 = 4.54 \times 10^{-8} \times (T_{1}^4 - 500^4)\) \(800 = 4.54 \times 10^{-8} \times (T_{1}^4 - 62500000000)\) \(T_{1}^4 = \frac{800}{4.54 \times 10^{-8}} + 62500000000\) \(T_{1}^4 = 17624382989\) \(T_{1} = \sqrt[4]{17624382989}\) \(T_{1} \approx 612.7 \mathrm{~K}\)

Determine if we can treat this as a two-surface enclosure

Calculate the angle factor (view factor) between the base surface and the side surfaces, \(F_{12}\): \(F_{12} = \frac{1}{2} \times (1 - \cos(60^{\circ}))\) \(F_{12} = 0.25\) Since \(F_{12} = 0.25\) is not close to zero, we cannot treat this geometry as a two-surface enclosure. It means that radiant heat exchange between surfaces should be taken into account. To summarize, the base surface temperature is approximately \(612.7 \mathrm{~K}\), and this geometry cannot be treated as a two-surface enclosure.

Key concepts

Thermodynamics

Thermodynamics is a branch of physics dealing with heat, work, and energy. At the heart of thermodynamics lie the four fundamental laws which govern the behavior of thermal energy in systems. The first law, also known as the law of energy conservation, states that energy cannot be created or destroyed, only transformed from one form to another. The second law introduces the concept of entropy, indicating that systems naturally progress towards disorder. The third law addresses the impossibility of reaching absolute zero temperature. Lastly, there's the zeroth law which involves thermal equilibrium and introduces the concept of temperature.

These principles provide the foundation for understanding heat transfer mechanisms, including radiation, conduction, and convection, which are essential for solving problems related to the temperature of objects, such as the exercise given where the temperature of a furnace surface is determined.

Heat Transfer

Heat transfer is the movement of heat from a region of higher temperature to one of lower temperature and can occur via three primary mechanisms: conduction, convection, and radiation. Conduction is the transfer of heat through a solid or between solids in direct contact, driven by a temperature gradient. Convection is the transfer of heat through a fluid, which could be liquid or gas, and is influenced by the fluid's motion. Radiation, however, does not require a medium and transfers energy through electromagnetic waves.

In the described exercise, we focus on radiative heat transfer, particularly how the Stefan-Boltzmann Law describes the rate at which heat is emitted by the surfaces of the furnace. The application of thermodynamics principles, specifically concerning heat transfer, enables us to calculate the base surface temperature within the given conditions.

Radiative Heat Exchange

Radiative heat exchange refers to the transfer of heat energy through electromagnetic radiation. This process is fundamental to the behavior of the furnace in our problem. Objects emit radiation according to their temperature and emissivity—a measure of how effectively they radiate energy. The Stefan-Boltzmann Law quantitatively describes this relationship, showing that the power radiated per unit area by a surface is proportional to the fourth power of the surface’s temperature and the surface’s emissivity.

In the furnace problem, we see a practical application of radiative heat exchange. The heated base surface radiates heat energy outwards, and this transfer of heat is modeled by the equation provided by the Stefan-Boltzmann Law. The law takes into account the temperatures of both the emitting and surrounding surfaces, thereby allowing us to solve for the unknown temperature given the other parameters.

Emissivity

Emissivity is a dimensionless quantity that measures the efficiency with which a surface emits thermal radiation compared to a perfect black body. A black body is an idealized physical entity that absorbs all incident radiation, and it has an emissivity of 1. Real-world objects have emissivities ranging from 0 to 1; the closer the value is to 1, the more effectively the object emits radiation.

In the context of our exercise, different surfaces of the furnace have distinct emissivities (\( \boldsymbol{\frac{1}{0.8}} \) for the base and \( \boldsymbol{0.5} \) for the sides). This difference in emissivity is crucial when we apply the Stefan-Boltzmann Law to determine the rate of radiative heat exchange and, consequently, the temperature. By acknowledging that each surface has its own emissivity, we can more accurately predict the flow of heat between surfaces in the furnace.

Temperature Calculation

Temperature calculation, especially in the field of thermodynamics, involves determining an object's temperature based on various factors like heat transfer rate, material properties, and surrounding conditions. In the example provided, the calculation involves an iterative process. An initial equation derived from the Stefan-Boltzmann Law relates heat transfer rate, emissivity, and temperatures of the interacting surfaces.

The step-by-step solution demonstrates how to isolate the unknown temperature (the base surface temperature of the furnace) from the equation then solve for it using algebra and available data (emissivity and the rate of heat transfer). Temperature calculation often requires understanding and manipulation of these types of formulas to predict how a system will behave under given thermal conditions.