Question

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

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

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.

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, \(h\).

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, \(R_{\text{eq}}\).

Key concepts

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.