Question

A vertical \(1.5\)-m-high, 2.8-m-wide double-pane window consists of two layers of glass separated by a \(2.0-\mathrm{cm}\) air gap at atmospheric pressure. The room temperature is \(26^{\circ} \mathrm{C}\) while the inner glass temperature is \(18^{\circ} \mathrm{C}\). Disregarding radiation heat transfer, determine the temperature of the outer glass layer and the rate of heat loss through the window by natural convection.

Short answer

Based on the step-by-step solution provided, answer the following question: Question: Calculate the temperature of the outer glass layer and the rate of heat loss through a double-pane window by natural convection, given its dimensions and the temperatures of the room and inner glass layer. Assume air as an ideal gas and use the properties of air at the film temperature. Answer: To calculate the temperature of the outer glass layer and the rate of heat loss, follow these steps: 1. Find the film temperature by taking the average of the inner and outer glass layer temperatures (with the initial assumption that the outer glass temperature is equal to the room temperature). 2. Calculate the Grashof number using the film temperature, which analyzes the ratio of buoyancy force to viscous force. 3. Determine the Nusselt number by using the Grashof number, which analyzes the ratio of convective heat transfer to conductive heat transfer. 4. Calculate the heat transfer coefficient (h) using the Nusselt number. 5. Utilize the heat transfer equation to compute the temperature of the outer glass layer and the rate of heat loss. Remember to update the film temperature and other parameters once the outer glass layer temperature is found for a more accurate result.

Step-by-step solution

Film Temperature and Air Properties

Find the film temperature, which is the average temperature of the inner and outer glass layers. Since we don't know the temperature of the outer glass layer, we'll assume it to be the room temperature for the initial calculation. We'll use this average temperature to find the relevant properties of air. The film temperature is given by: $$ T_f = \frac{T1 + T2}{2} $$

Grashof Number

Calculate the Grashof number, which is a dimensionless quantity used to describe the ratio of buoyancy force to viscous force. It's used to determine the flow regime of natural convection. The Grashof number is defined as: $$ Gr = \frac{g \beta (T1 - T2) L^3}{\nu^2} $$

Nusselt Number

Find the Nusselt number, which is a dimensionless quantity used in analyzing heat transfer by natural convection. It's the ratio of convective heat transfer to conductive heat transfer. For a vertical surface, the Nusselt number correlation in natural convection is given by: $$ Nu = 0.59 (Gr \cdot Pr)^{1/4} $$

Heat Transfer Coefficient

Calculate the heat transfer coefficient (h), which relates the heat transfer rate to the temperature difference between the solid surface and the surrounding fluid. The heat transfer coefficient is given by: $$ h = \frac{Nu \cdot k}{L} $$

Heat Transfer Equation

Use the heat transfer equation to find the temperature of the outer glass layer and the rate of heat loss through the window. The heat transfer equation for natural convection is given by: $$ Q = hA(T1 - T2) $$ Rearrange this equation to find the temperature difference \((T1-T2)\): $$ T1 - T2 = \frac{Q}{hA} $$ Now, we can find the temperature of outer glass layer (T2). After finding the outer glass layer temperature, recalculate the film temperature and other parameters using the updated outer glass layer temperature for a more accurate result. Finally, use the updated parameters to calculate the rate of heat loss through the window.