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Consider a surface of area \(A\) at which the convection and radiation heat transfer coefficients are \(h_{\text {conv }}\) and \(h_{\mathrm{rad}}\), respectively. Explain how you would determine \((a)\) the single equivalent heat transfer coefficient, and \((b)\) the equivalent thermal resistance. Assume the medium and the surrounding surfaces are at the same temperature.

Short Answer

Expert verified
Question: Calculate the single equivalent heat transfer coefficient and the equivalent thermal resistance for a surface with a convection heat transfer coefficient of \(h_{\text {conv }} = 10 \, W/m^2K\) and a radiation heat transfer coefficient of \(h_{\mathrm{rad}} = 5 \, W/m^2K\). The surface area is \(A = 2 \, m^2\). Answer: The single equivalent heat transfer coefficient, \(h\), is the sum of the convection and radiation heat transfer coefficients, so \(h = h_{\text {conv }} + h_{\mathrm{rad}} = 10 + 5 = 15 \, W/m^2K\). To find the equivalent thermal resistance, \(R_{\text{eq}}\), use the formula \(R_{\text{eq}} = \frac{1}{h \cdot A} = \frac{1}{15 \cdot 2} = 0.0333 \, K/W\). Therefore, the single equivalent heat transfer coefficient is \(15 \, W/m^2K\) and the equivalent thermal resistance is \(0.0333 \, K/W\).

Step by step solution

01

Understand the heat transfer coefficients

Heat transfer coefficients represent the efficiency of heat transfer in convection and radiation. In this case, \(h_{\text {conv }}\) represents the convection heat transfer coefficient, and \(h_{\mathrm{rad}}\) represents the radiation heat transfer coefficient.
02

Calculate the combined heat transfer coefficient

The combined heat transfer coefficient, \(h\), is obtained by adding the individual heat transfer coefficients. The equation for the combined heat transfer coefficient is: \[h = h_{\text {conv }} + h_{\mathrm{rad}}\] Substitute the given values of \(h_{\text {conv }}\) and \(h_{\mathrm{rad}}\) in the equation to get the equivalent heat transfer coefficient, \(h\).
03

Calculate the equivalent thermal resistance

The thermal resistance (\(R_{\text{eq}}\)) is defined as the ratio of the temperature difference between the surface and the environment to the heat transferred through the surface. The equation for the thermal resistance is: \[R_{\text{eq}} = \frac{1}{h \cdot A} \] Substitute the calculated value of \(h\) from Step 2 and given area \(A\) in the equation to obtain the equivalent thermal resistance, \(R_{\text{eq}}\).

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Key Concepts

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

Convection Heat Transfer
Convection heat transfer involves the movement of heat due to the flow of fluid, which could be liquid or gas, over a surface. This is a common mechanism observed in everyday activities, like boiling water or feeling the breeze on your skin. Convection occurs in two forms:
  • Natural Convection: This happens when fluid movement is caused by buoyancy forces that occur due to temperature differences in the fluid, without any external influence like fans or pumps. For example, the natural circulation of air in a room with a heater.
  • Forced Convection: This is achieved when an external force, such as a fan or a pump, moves the fluid over the surface to enhance heat transfer. An example would be the cooling system in a car engine.

In convection, the heat transfer coefficient, denoted as \(h_{\text{conv}}\), quantifies the rate of heat exchange per unit area and temperature difference between the surface and the fluid. A higher coefficient means more efficient heat transfer.
To optimize convection heat transfer, it is crucial to understand the properties of the fluid, as well as the geometry and orientation of the surface involved.
Radiation Heat Transfer
Radiation heat transfer occurs through electromagnetic waves, primarily in the infrared spectrum. This method does not require a medium, meaning it can occur in a vacuum. It is the way the Sun heats the Earth through space. All objects emit radiation energy, and the rate is influenced by their temperature and surface characteristics.
The efficiency of radiation heat transfer is represented by the radiation heat transfer coefficient, \(h_{\mathrm{rad}}\), which depends on the emissivity of the surface, its surface area, and the temperatures of the surface and surroundings.
  • Emissivity: This is a measure of a material's ability to emit thermal radiation compared to a perfect black body. A surface with high emissivity is more efficient in radiating heat.
  • Surface Area and Temperature: Larger areas and higher temperatures generally increase the rate of radiation heat transfer.
Understanding these properties helps in managing heat transfer in systems where radiation plays a significant role, such as in satellite thermal controls or in architectural designs promoting energy efficiency.
Thermal Resistance
Thermal resistance is a measure of a material's ability to resist the flow of heat. It is an essential concept in engineering, especially in thermal management of mechanical and electrical systems. The thermal resistance \(R_{\text{eq}}\) is similar to electrical resistance but deals with heat instead of electricity.
The thermal resistance for a heat transfer process is calculated using the formula:\[ R_{\text{eq}} = \frac{1}{h \cdot A} \]where \(h\) is the combined heat transfer coefficient (sum of convection and radiation coefficients) and \(A\) is the surface area. A smaller \(R_{\text{eq}}\) signifies a system that readily allows the flow of thermal energy.
  • Importance in Design: In designing systems like heat exchangers or insulating materials, minimizing thermal resistance helps to ensure efficient heat transfer.
  • Balance in Systems: Proper thermal management involves balancing thermal resistance to maintain optimal temperatures, avoiding overheating or excessive cooling.
By carefully calculating and managing thermal resistance, engineers can design systems that maintain energy efficiency while meeting functional requirements.

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Most popular questions from this chapter

The outer surface of an engine is situated in a place where oil leakage can occur. When leaked oil comes in contact with a hot surface that has a temperature above its autoignition temperature, the oil can ignite spontaneously. Consider an engine cover that is made of a stainless steel plate with a thickness of \(1 \mathrm{~cm}\) and a thermal conductivity of \(14 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). The inner surface of the engine cover is exposed to hot air with a convection heat transfer coefficient of \(7 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) at \(333^{\circ} \mathrm{C}\). The outer surface is exposed to an environment where the ambient air is \(69^{\circ} \mathrm{C}\) with a convection heat transfer coefficient of \(7 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). To prevent fire hazard in the event of oil leak on the engine cover, a layer of thermal barrier coating (TBC) with a thermal conductivity of \(1.1 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) is applied on the engine cover outer surface. Would a TBC layer of \(4 \mathrm{~mm}\) in thickness be sufficient to keep the engine cover surface below autoignition temperature of \(200^{\circ} \mathrm{C}\) to prevent fire hazard?

We are interested in steady state heat transfer analysis from a human forearm subjected to certain environmental conditions. For this purpose consider the forearm to be made up of muscle with thickness \(r_{m}\) with a skin/fat layer of thickness \(t_{s f}\) over it, as shown in the Figure P3-138. For simplicity approximate the forearm as a one-dimensional cylinder and ignore the presence of bones. The metabolic heat generation rate \(\left(\dot{e}_{m}\right)\) and perfusion rate \((\dot{p})\) are both constant throughout the muscle. The blood density and specific heat are \(\rho_{b}\) and \(c_{b}\), respectively. The core body temperate \(\left(T_{c}\right)\) and the arterial blood temperature \(\left(T_{a}\right)\) are both assumed to be the same and constant. The muscle and the skin/fat layer thermal conductivities are \(k_{m}\) and \(k_{s f}\), respectively. The skin has an emissivity of \(\varepsilon\) and the forearm is subjected to an air environment with a temperature of \(T_{\infty}\), a convection heat transfer coefficient of \(h_{\text {conv }}\), and a radiation heat transfer coefficient of \(h_{\mathrm{rad}}\). Assuming blood properties and thermal conductivities are all constant, \((a)\) write the bioheat transfer equation in radial coordinates. The boundary conditions for the forearm are specified constant temperature at the outer surface of the muscle \(\left(T_{i}\right)\) and temperature symmetry at the centerline of the forearm. \((b)\) Solve the differential equation and apply the boundary conditions to develop an expression for the temperature distribution in the forearm. (c) Determine the temperature at the outer surface of the muscle \(\left(T_{i}\right)\) and the maximum temperature in the forearm \(\left(T_{\max }\right)\) for the following conditions: $$ \begin{aligned} &r_{m}=0.05 \mathrm{~m}, t_{s f}=0.003 \mathrm{~m}, \dot{e}_{m}=700 \mathrm{~W} / \mathrm{m}^{3}, \dot{p}=0.00051 / \mathrm{s} \\ &T_{a}=37^{\circ} \mathrm{C}, T_{\text {co }}=T_{\text {surr }}=24^{\circ} \mathrm{C}, \varepsilon=0.95 \\ &\rho_{b}=1000 \mathrm{~kg} / \mathrm{m}^{3}, c_{b}=3600 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, k_{m}=0.5 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K} \\ &k_{s f}=0.3 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, h_{\text {conv }}=2 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}, h_{\mathrm{rad}}=5.9 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K} \end{aligned} $$

Two finned surfaces with long fins are identical, except that the convection heat transfer coefficient for the first finned surface is twice that of the second one. What statement below is accurate for the efficiency and effectiveness of the first finned surface relative to the second one? (a) Higher efficiency and higher effectiveness (b) Higher efficiency but lower effectiveness (c) Lower efficiency but higher effectiveness (d) Lower efficiency and lower effectiveness (e) Equal efficiency and equal effectiveness

Inconel \(^{\circledast}\) refers to a class of nickel-chromium-based superalloys that are used in high-temperature applications, such as gas turbine blades. For further improvement in the performance of gas turbine engine, the outer blade surface is coated with ceramic-based thermal barrier coating (TBC). Consider a flat Inconel \({ }^{\circledR}\) plate, with a thickness of \(12 \mathrm{~mm}\), is coated with a layer of TBC, with a thickness of \(300 \mu \mathrm{m}\), on its surface. At the interface between the Inconel \({ }^{\circledR}\) and the TBC, the thermal contact conductance is \(10,500 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The thermal conductivities of the Inconel \({ }^{\circledast}\) and the TBC are \(25 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) and \(1.5 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), respectively. The plate is in a surrounding of hot combustion gasses at \(1500^{\circ} \mathrm{C}\), and the convection heat transfer coefficient is \(750 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Determine the temperature at the mid-plane of the Inconel \({ }^{\oplus}\) plate, if the outer surface temperature is \(1200^{\circ} \mathrm{C}\).

Steam in a heating system flows through tubes whose outer diameter is \(5 \mathrm{~cm}\) and whose walls are maintained at a temperature of \(180^{\circ} \mathrm{C}\). Circular aluminum alloy 2024-T6 fins \((k=186 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) of outer diameter \(6 \mathrm{~cm}\) and constant thickness \(1 \mathrm{~mm}\) are attached to the tube. The space between the fins is \(3 \mathrm{~mm}\), and thus there are 250 fins per meter length of the tube. Heat is transferred to the surrounding air at \(T_{\infty}=25^{\circ} \mathrm{C}\), with a heat transfer coefficient of \(40 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Determine the increase in heat transfer from the tube per meter of its length as a result of adding fins.

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