Question

Consider steady one-dimensional heat transfer through a plane wall exposed to convection from both sides to environments at known temperatures \(T_{\infty 1}\) and \(T_{\infty 2}\) with known heat transfer coefficients \(h_{1}\) and \(h_{2}\). Once the rate of heat transfer \(\dot{Q}\) has been evaluated, explain how you would determine the temperature of each surface.

Short answer

Answer: The key step in solving this type of problem is to understand and calculate the thermal resistance of each part of the system (conduction resistance and convection resistance), and use it to determine the heat transfer rate and surface temperatures.

Step-by-step solution

Identify the thermal resistances

Each part of the system has a thermal resistance associated with it: the conduction resistance through the wall and the convection resistances for the fluid on both sides. The conduction resistance can be denoted as \(R_{cond}=\frac{L}{kA}\), and the convection resistances can be denoted as \(R_{conv1} = \frac{1}{h_1 A}\) and \(R_{conv2} = \frac{1}{h_2 A}\), where \(L\) is the wall thickness, \(k\) is the thermal conductivity of the wall material and \(A\) is the area.

Calculate the total resistance

Now we need to find the total resistance in the system, which will be the sum of the three resistances mentioned above: \(R_{total} = R_{cond} + R_{conv1} + R_{conv2}\). This is because in a steady state and one-dimensional situation, the thermal resistances are in series.

Determine the heat transfer rate, \(\dot{Q}\)

Using the conservation of energy principle and the total resistance, we can determine the heat transfer rate, \(\dot{Q}\): \(\dot{Q} = \frac{T_{\infty 1} - T_{\infty 2}}{R_{total}}\). This formula relates the difference in environmental temperatures, the rate of heat transfer, and the total resistance in the system.

Calculate the surface temperatures

To find the temperature of each surface (\(T_1\) and \(T_2\)), we can use the formula \(\dot{Q} = h_1 A (T_{\infty 1} - T_1)\) and \(\dot{Q}= h_2 A (T_2 - T_{\infty 2})\). These equations express the heat transfer rate in terms of the heat transfer coefficients and surface temperatures.

Solve for surface temperatures \(T_1\) and \(T_2\)

Plugging the found value of \(\dot{Q}\) from Step 3 into the two equations from Step 4, we get two linear equations for \(T_1\) and \(T_2\), which can be solved analytically or numerically. In summary, we determined the rate of heat transfer \(\dot{Q}\) by evaluating the total resistance in the system and used the conservation of energy principle. Using this heat transfer rate, we were able to solve for the surface temperatures \(T_1\) and \(T_2\) on each side of the plane wall. The key step is to understand and calculate the thermal resistance of each part of the system and use it to determine the heat transfer rate and surface temperatures.

Key concepts

Thermal Resistance

Thermal resistance is a concept that helps us understand how well materials resist the flow of heat. Think of it like insulation, stopping heat from moving too easily. In any system where heat is transferring, there are various components contributing to the overall thermal resistance. In a plane wall exposed to convection, we have both conduction through the wall and convection on either side. Each of these has its own resistance.

**Components of Thermal Resistance:**

• Conduction Resistance (\(R_{cond}\)): This is associated with the wall and calculated using the formula:\[R_{cond} = \frac{L}{kA}\]where\(L\)is the thickness of the wall,\(k\)is the thermal conductivity, and\(A\)is the surface area.
• Convection Resistance (\(R_{conv}\)): Represented by\(R_{conv1}\)and\(R_{conv2}\)for each side, and calculated as:\[R_{conv1} = \frac{1}{h_1 A}, \quad R_{conv2} = \frac{1}{h_2 A}\]where\(h_1\)and\(h_2\)are the heat transfer coefficients for the respective sides.

All of these resistances are added together to get the total thermal resistance of the system. Understanding thermal resistance is essential in predicting how efficiently or inefficiently heat will transfer through materials.

Convection

Convection is a mode of heat transfer where heat is carried to or from a surface by a moving fluid, such as air or water. It occurs in two primary types: natural convection, driven by buoyancy forces due to temperature differences in the fluid, and forced convection, where a fan or pump moves the fluid. In building science, convection often involves air passing over surfaces, impacting heat transfer.

**Key Points about Convection:**

• Heat Transfer Coefficient (\(h\)): The measure of the convective heat transfer between a surface and a fluid in contact with it. A higher\(h\)value indicates a higher capacity for heat transfer, typical in forced convection scenarios.
• Impact on Temperature Profiles: On both sides of a wall, convection alters the surface temperature based on:\[\dot{Q} = h \times A \times (T_{\infty} - T_s)\]where\(\dot{Q}\)is the heat transfer rate,\(A\)is the area,\(T_{\infty}\)is the fluid's bulk temperature, and\(T_s\)is the surface temperature. This equation is rearranged to solve for the unknown surface temperatures once\(\dot{Q}\)is known.

Being mindful of convection allows you to better gauge how heat loss or gain occurs across surfaces and adjust designs or materials appropriately to optimize thermal performance.

Conduction

Conduction is the process of heat transfer through a material without any movement of the material itself. It relies on the interaction between molecules. When one part of a material is heated, those molecules vibrate and pass the energy along to neighboring molecules.

**Conduction in Solids:**

• Material Dependency: The rate of conduction is highly dependent on the material's properties, specifically its thermal conductivity (\(k\)). Metals, for example, have high thermal conductivity and are excellent conductors of heat.
• Formula for Conduction: The conduction through a wall is described through Fourier’s Law, often presented as:\[Q = \frac{k \times A \times (T_1 - T_2)}{L}\]where\(Q\)is the heat transferred,\(A\)is the cross-sectional area,\(T_1\)and\(T_2\)are temperatures across the material, and\(L\)is its thickness.

Understanding conduction helps in evaluating how much heat is lost or gained in buildings or products, helping design effective insulation systems.

Surface Temperature Calculation

To find the surface temperatures in scenarios where heat transfer is involved, knowing both the heat transfer rate and the resistances is vital. Calculating these temperatures will determine how the heat conducts and convects through materials, vital for design and analysis of thermal systems.

**Steps for Surface Temperature Calculation:**

• Start with Known Quantities: We already have\(\dot{Q}\)from previous calculations. Knowing this is crucial.
• Use Convective Heat Transfer Rate: For each surface, rearrange the convective heat transfer formula:\[\dot{Q} = h_1 \times A \times (T_{\infty 1} - T_1)\]and\[\dot{Q} = h_2 \times A \times (T_2 - T_{\infty 2})\]where\(T_1\)and\(T_2\)are the unknown surface temperatures to solve for.
• Solve the Equations: Insert the previously determined\(\dot{Q}\)value into these equations. You’ll have two linear equations and can solve them to find\(T_1\)and\(T_2\).These calculations reveal how much the temperature drops across each resistance, verifying the thermal performance of the system.

By analyzing surface temperatures, engineers ensure that materials and designs meet safety and efficiency standards.