Problem 6 - Basic Heat And Mass Transfer

Question

An \(8 \mathrm{~cm}\)-long flat copper plate is part of a structure undergoing tests in a high-speed wind tunnel. The test conditions are an air velocity of \(1025 \mathrm{~m} / \mathrm{s}\), a pressure of \(50 \mathrm{kPa}\), and a temperature of \(280 \mathrm{~K}\). Determine the equilibrium temperature of the plate if its surface emittance is \(0.26\) and the tunnel walls are at \(355 \mathrm{~K}\). Take \(\operatorname{Re}_{t r}=0\).

Step-by-Step Solution

Answer

The equilibrium temperature of the plate is 355 K.

Step 1: Understand the Problem

We're asked to find the equilibrium temperature of a copper plate in a wind tunnel with specific conditions. Important factors include air velocity, pressure, temperature, plate emittance, and tunnel wall temperature.

Step 2: Define the Relevant Equation

To find the equilibrium temperature, use an energy balance at the surface of the plate. Since we're given the tunnel wall temperature, we will use the radiation heat transfer equation to compute the emitted and absorbed radiation by the plate:\[q_{emission} = \epsilon \sigma (T_s^4 - T_w^4)\]where \(\epsilon\) is the emittance, \(\sigma\) is the Stefan-Boltzmann constant \(5.67 \times 10^{-8} \ \text{W/m}^2\cdot\text{K}^4\), \(T_s\) is the surface temperature of the plate, and \(T_w\) is the temperature of the tunnel walls.

Step 3: Evaluate the Dynamic Conditions

Given \(\operatorname{Re}_{t r}=0\), it implies negligible convective heat transfer compared to radiation. Thus, only radiation heat exchange is considered between the plate and the walls.

Step 4: Rearrange and Solve the Equation

Using the radiation balance equation:\[0 = 0.26 \times 5.67 \times 10^{-8} (T_s^4 - 355^4)\]Solve for \(T_s\), the surface temperature.

Step 5: Simplify the Math

Simplify the equation:- Distribute and expand the terms inside the parenthesis.- Isolate \(T_s^4\):\[ T_s^4 = 355^4\]- Take the fourth root to solve for \(T_s\):\[ T_s = 355 \ \text{K} \]

Step 6: Conclusion

The equilibrium temperature of the plate, influenced by radiation from the tunnel walls and negligible convective heat transfer, is \(355 \ \text{K}\).

Key Concepts

Equilibrium Temperature

Equilibrium temperature is the temperature at which a system, such as the copper plate in the wind tunnel, reaches a stable state where the net heat transfer is zero. At this point, the heat emitted by the plate equals the heat absorbed, leading to thermal balance. In our exercise, the equilibrium temperature is determined by considering only radiation heat transfer, since convective heat transfer is negligible given \(\operatorname{Re}_{tr}=0\). By calculating the balance of energy emitted and absorbed via radiation, we find that the equilibrium temperature is reached when the copper plate and the surrounding tunnel walls have a similar thermal emission, which stabilizes to \(355 \, \text{K}\). Understanding equilibrium temperature is important in applications where heat exchange dynamics play a role, and achieving this equilibrium can ensure system stability.

Radiation Heat Transfer

Radiation heat transfer is a mode through which heat moves from one surface to another through electromagnetic waves. This process does not require any medium like air or water to occur, and it is based entirely on the thermal emission from surface to surface. In the case of the copper plate, the primary exchange of heat is through radiation with the tunnel walls. We use the radiation heat transfer equation to calculate the rate of heat radiation from the plate's surface: \(q_{emission} = \epsilon \sigma (T_s^4 - T_w^4)\). Where \(\epsilon\) is the emittance of the surface, \(\sigma\) is the Stefan-Boltzmann constant, \(T_s\) is the surface temperature of the plate, and \(T_w\) is the temperature of the tunnel walls.This illustrates how crucial radiation heat transfer is when looking at systems where direct material contact and convection are not significant modes of heat transfer due to high flow velocity or other considered factors.

Energy Balance

Energy balance is a fundamental concept ensuring that the total energy entering a system equals the energy leaving it, leading to a steady state situation. In terms of our exercise, the energy balance is solely analyzed in terms of radiation since convection is negligible. We can write the energy balance equation specifically in terms of radiation as \(0 = \epsilon \sigma (T_s^4 - T_w^4)\). Here, we set the net heat transfer across the system to zero, indicating that both the energy lost by radiation from the plate and the energy gained from the tunnel walls' radiation are equivalent. Achieving energy balance means that the temperature will no longer change, pointing to the condition of equilibrium in temperature, vital in maintaining system stability in thermal environments.

Stefan-Boltzmann Law

The Stefan-Boltzmann Law is a critical physical law for understanding radiation heat transfer, stating that the radiant energy emitted by a black body is proportional to the fourth power of its absolute temperature. The amount of radiative heat transfer can be calculated with the formula: \(E_b = \sigma T^4\), where \(E_b\) is the emitted energy per unit area, \(\sigma\) is the Stefan-Boltzmann constant \(5.67 \times 10^{-8} \, \text{W/m}^2\cdot\text{K}^4\), and \(T\) is the temperature in Kelvin.In the context of the copper plate system, the Stefan-Boltzmann Law helps describe how much energy is emitted or absorbed as radiant heat. Since the plate isn’t a perfect black body, we consider its emittance \(\epsilon\), adjusting the formula for practical use: \(q = \epsilon \sigma T^4\). This equation helps compute radiation between the copper plate and tunnel walls, facilitating the calculation of the equilibrium temperature based on radiative energy exchange. Understanding this principle is essential for analyzing systems where radiation serves as the dominant form of heat transfer.