Question

Hot water at an average temperature of \(90^{\circ} \mathrm{C}\) passes through a row of eight parallel pipes that are \(4 \mathrm{~m}\) long and have an outer diameter of \(3 \mathrm{~cm}\), located vertically in the middle of a concrete wall \((k=0.75 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) that is $4 \mathrm{~m}\( high, \)8 \mathrm{~m}\( long, and \)15 \mathrm{~cm}$ thick. If the surfaces of the concrete walls are exposed to a medium at $32^{\circ} \mathrm{C}\(, with a heat transfer coefficient of \)12 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, determine the rate of heat loss from the hot water and the surface temperature of the wall.

Short answer

Question: Determine the rate of heat loss from the hot water and the surface temperature of the wall in a scenario where there are eight pipes embedded in a concrete wall, with a pipe diameter of 0.03 m, wall thickness of 0.15 m, thermal conductivity of 0.75 W/m·K, and a heat transfer coefficient of 12 W/m²·K. The temperature of the water inside the pipes is 90°C, and the surrounding medium temperature is 32°C. Answer: To determine the rate of heat loss from the hot water and the surface temperature of the wall, follow these steps: 1. Calculate the cross-sectional area of a single pipe and the total area of all eight pipes using the given pipe diameter. 2. Calculate the thermal resistance of the concrete wall and the thermal resistance of the combined convective film using the wall thickness, thermal conductivity, and heat transfer coefficient. 3. Use the overall thermal resistance to calculate the heat loss from the hot water, considering the temperatures of the water and the surrounding medium. 4. Calculate the surface temperature of the wall using the heat loss and the thermal resistance of the combined convective film.

Step-by-step solution

Calculate the Cross-sectional Area of a Single Pipe and the Total Area of All Eight Pipes

First, let's calculate the cross-sectional area of a single pipe: \(A_{pipe} = \pi \times (\frac{D_{pipe}}{2})^2\) Where \(D_{pipe} = 0.03 \mathrm{~m}\) is the diameter of the pipe. Then, calculate the total area of all eight pipes: \(A_{total} = 8 \times A_{pipe}\)

Calculate the Thermal Resistance of the Concrete Wall and the Thermal Resistance of the Combined Convective Film

Now, we need to calculate the thermal resistance of the concrete wall: \(R_{concrete}= \frac{L}{k \cdot A_{total}}\) Where \(L=0.15 \mathrm{~m}\) is the thickness of the wall, and \(k=0.75 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) is the thermal conductivity of the concrete. Next, we need to calculate the thermal resistance of the combined convective film: \(R_{conv} = \frac{1}{h \cdot A_{total}}\) Where \(h=12 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) is the heat transfer coefficient.

Use the Overall Thermal Resistance to Calculate the Heat Loss from the Hot Water

We can calculate the overall thermal resistance, \(R_{overall}\), by adding the thermal resistance of the concrete wall and the thermal resistance of the combined convective film: \(R_{overall} = R_{concrete} + R_{conv}\) Now, use the overall thermal resistance to calculate the heat loss from the hot water: \(q = \frac{T_{water} - T_{medium}}{R_{overall}}\) Where \(T_{water} = 90^{\circ} \mathrm{C}\) and \(T_{medium} = 32^{\circ} \mathrm{C}\).

Calculate the Surface Temperature of the Wall

Finally, we can calculate the surface temperature of the wall: \(T_{surface} = T_{medium} + q \times R_{conv}\)