Question

Hot water is flowing at an average velocity of \(4 \mathrm{ft} / \mathrm{s}\) through a cast iron pipe $\left(k=30 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft} \cdot{ }^{\circ} \mathrm{F}\right)$ whose inner and outer diameters are \(1.0\) in and \(1.2\) in, respectively. The pipe passes through a 50 -ft-long section of a basement whose temperature is $60^{\circ} \mathrm{F}\(. The emissivity of the outer surface of the pipe is \)0.5$, and the walls of the basement are also at about \(60^{\circ} \mathrm{F}\). If the inlet temperature of the water is \(150^{\circ} \mathrm{F}\) and the heat transfer coefficient on the inner surface of the pipe is $30 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2} \cdot{ }^{\circ} \mathrm{F}$, determine the temperature drop of water as it passes through the basement. Evaluate air properties at a film temperature of \(105^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\) pressure. Is this a good assumption?

Short answer

Question: Determine the temperature drop of the water as it passes through a 50-ft-long cast iron pipe in a basement with given parameters. Answer: To determine the temperature drop of water as it passes through the 50-ft-long cast iron pipe in a basement, follow these steps: 1. Calculate the pipe's thermal resistance and convective heat transfer coefficient. 2. Calculate the heat lost by the water using conservation of energy. 3. Determine the temperature drop of the water with the heat lost and mass flow rate.

Step-by-step solution

Calculate the Pipe's Thermal Resistance and Convective Heat Transfer Coefficient

Let's first determine the pipe's thermal resistance. The given parameters are: Inner diameter, \(D_i = 1.0 \text{ in}\) Outer diameter, \(D_o = 1.2 \text{ in}\) Thermal conductivity of cast iron, \(k = 30 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft} \cdot { }^{\circ} \mathrm{F}\) For a cylindrical pipe, the thermal resistance (\(R_\text{pipe}\)) can be calculated as: \(R_\text{pipe} = \frac{\ln{(D_o / D_i)}}{2\pi L k}\) Where \(L\) is the length of the pipe, which is \(50 \mathrm{ft}\) in this case. Plug the values in and find the thermal resistance: \(R_\text{pipe} = \frac{\ln{(1.2/1.0)}}{2\pi (50)(30)}\) Next, we are given the convective heat transfer coefficient for the inside of the pipe, \(h_i = 30 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2}\cdot { }^{\circ} \mathrm{F}\). To calculate the overall heat transfer coefficient, \(U\), we will use the following formula: \(U = \frac{1}{R_\text{pipe} + \frac{1}{h_i}}\)

Calculate the Heat Lost by the Water

To find the heat lost by the water, we will use conservation of energy principles: \(Q_\text{lost} = UA(T_{in} - T_\text{basement})\) Where \(Q_\text{lost}\) is the heat lost by the water to the surroundings, \(A\) is the inside surface area of the pipe, \(T_{in}\) is the inlet temperature of the water, and \(T_\text{basement}\) is the basement temperature. For this problem, \(T_{in} = 150^{\circ} \mathrm{F}\) and \(T_\text{basement} = 60^{\circ} \mathrm{F}\). Inside surface area of the pipe can be calculated as: \(A = \pi D_i L\) Plug all the values and find the heat lost by the water.

Determine the Temperature Drop of the Water

To find the temperature drop, we will use the heat lost and the mass flow rate of water. In order to find the mass flow rate, we need to find the volumetric flow rate (\(Q_v\)) and the density of water (\(\rho\)). Given the average velocity (\(v\)) as \(4 \mathrm{ft} / \mathrm{s}\), we can calculate the volumetric flow rate as: \(Q_v = v \cdot A\) Now using the inlet temperature of the water, we can assume the density of the water is roughly 62.4 lb/ft³ (density of water at room temperature). So the mass flow rate (\(\dot{m}\)) can be calculated as: \(\dot{m} = Q_v \cdot \rho\) Finally, we can determine the temperature drop (\(\Delta T\)) using the heat lost and mass flow rate, along with specific heat capacity of water (\(c_p = 1 \mathrm{Btu} / \mathrm{lb} \cdot{ }^{\circ} \mathrm{F}\)): \(\Delta T = \frac{Q_\text{lost}}{\dot{m} c_p}\) Calculate the temperature drop of the water as it passes through the basement. In conclusion, follow these steps to determine the temperature drop of water as it passes through the cast iron pipe inside the basement: 1. Calculate the pipe's thermal resistance and convective heat transfer coefficient. 2. Calculate the heat lost by the water using conservation of energy. 3. Determine the temperature drop of the water with the heat lost and mass flow rate.