Question

A \(0.2-\mathrm{m}\)-long and \(25-\mathrm{mm}\)-thick vertical plate $(k=1.5 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})$ separates the hot water from the cold air at \(2^{\circ} \mathrm{C}\). The plate surface exposed to the hot water has a temperature of \(100^{\circ} \mathrm{C}\), and the surface exposed to the cold air has an emissivity of \(0.73\). Determine the temperature of the plate surface exposed to the cold air \(\left(T_{s, c}\right)\). Hint: The \(T_{s, c}\) has to be found iteratively. Start the iteration process with an initial guess of \(51^{\circ} \mathrm{C}\) for the \(T_{s, c^{*}}\)

Short answer

Question: A vertical plate with dimensions of \(20 \,\mathrm{cm}\times 25\,\mathrm{mm}\) and a thermal conductivity of 1.5 W/mK is exposed to hot water at \(100^{\circ} \mathrm{C}\) on one side and cold air on the other. The emissivity of the surface exposed to cold air is 0.73. The initial guess for the surface temperature exposed to cold air is \(51^{\circ} \mathrm{C}\). Calculate the temperature of the plate surface exposed to cold air using the energy balance equations for conduction, radiation, and convection. Assume a convection heat transfer coefficient of around \(15\,\mathrm{W}\,\mathrm{m}^{-2}\,\mathrm{K}^{-1}\) for the cold air. Answer: To calculate the temperature of the plate surface exposed to cold air, follow these steps: 1. Define the key equations and variables for the energy balance equations. 2. Initialize the variables and constants, and start the iteration. 3. Iterate the solution until convergence by updating the guess for the surface temperature exposed to cold air. 4. Report the final temperature value once the iterative process has converged. Using this approach, calculate the temperature of the plate surface exposed to the cold air by iterating through the energy balance equations until a stable temperature value is reached.

Step-by-step solution

Define equations and variables

We will use the following variables and equations to solve this problem: - \(T_{s,h}\): Temperature of the surface exposed to hot water, given as \(100^{\circ} \mathrm{C}\) - \(T_{s,c}\): Temperature of the surface exposed to cold air (The value we want to find) - \(T_{s,c^{*}}\): Initial guess for \(T_{s,c}\), given as \(51^{\circ} \mathrm{C}\) - \(k\): Heat transfer (thermal conductivity) of the plate, given as \(1.5 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) - \(L\): Length of plate, given as \(0.2 \mathrm{~m}\) - \(x\): Thickness of the plate, given as \(25 \mathrm{~mm} = 0.025 \mathrm{~m}\) - \(\epsilon\): Emissivity of the surface exposed to cold air, given as \(0.73\) - \(h_{c}\): Convection heat transfer coefficient for air, around \(15\,\mathrm{W}\,\mathrm{m}^{-2}\,\mathrm{K}^{-1}\) We will use the energy balance equations: 1. Conduction heat transfer: \(q_{cond} = k \cdot A \cdot \frac{T_{s,h} - T_{s,c}}{x}\) 2. Radiation heat transfer: \(q_{rad} = \epsilon \cdot \sigma \cdot A \cdot \left(T_{s,c}^4 - T_{\infty}^4\right)\) 3. Convection heat transfer: \(q_{conv} = h_{c} \cdot A \cdot \left(T_{s,c} - T_{\infty}\right)\) We will iterate through the solution using the following approach: - Calculate \(q_{cond}\), \(q_{rad}\), and \(q_{conv}\) for the given values - Update the \(T_{s,c^{*}}\) value to get a new estimate for \(T_{s,c}\)

Initialize variables and start iteration

Initialize the variables and constants with their given values. Calculate \(q_{cond}\) using the initial guess of \(T_{s,c^{*}}\). Start the iteration process: 1. Calculate \(q_{cond}\) using the current guess for \(T_{s,c^{*}}\) 2. Calculate \(q_{rad}\) using the current guess for \(T_{s,c^{*}}\) 3. Calculate \(q_{conv}\) using the current guess for \(T_{s,c^{*}}\) 4. Update \(T_{s,c^{*}}\) based on the energy balance equations To update \(T_{s,c^{*}}\), we use the energy balance equation: \(q_{cond} = q_{rad} + q_{conv}\). This can be rewritten as: \(T_{s,c^{*}}=\frac{k(T_{s,h} - T_{s,c})x - \sigma \epsilon ((T_{s,c^{*}})^4 - T_{\infty}^4)}{h_{c}}\) Calculate the new value of \(T_{s,c^{*}}\) using that expression.

Iterate until convergence

Continue the iterative process by repeatedly updating the \(T_{s,c^{*}}\) value and recalculating \(q_{cond}\), \(q_{rad}\), and \(q_{conv}\) until the temperature value converges to a stable value. Check for convergence by calculating the change in \(T_{s,c^{*}}\) in each iteration. If the change is below a predefined tolerance, then consider the solution converged and stop iterations.

Report the final temperature value

After the iterative process has converged, report the final value of \(T_{s,c^{*}}\) as the temperature of the plate surface exposed to cold air, \(T_{s,c}\). Remember to perform the calculations with care and make sure to iterate the solution enough times to get an accurate temperature value for the plate surface exposed to the cold air.