Question

Hot exhaust gases leaving a stationary diesel engine at $450^{\circ} \mathrm{C}\( enter a \)15-\mathrm{cm}$-diameter pipe at an average velocity of \(7.2 \mathrm{~m} / \mathrm{s}\). The surface temperature of the pipe is \(180^{\circ} \mathrm{C}\). Determine the pipe length if the exhaust gases are to leave the pipe at \(250^{\circ} \mathrm{C}\) after transferring heat to water in a heat recovery unit. Use the properties of air for exhaust gases.

Short answer

Answer: The required length of the pipe is 115.33 meters.

Step-by-step solution

List the given information and derive the formula to calculate the pipe length

In this exercise, we need to find the length of the pipe (L). The given information are: - Initial temperature of the exhaust gases (\(T_{1}\)) = 450 °C - Final temperature of the exhaust gases (\(T_{2}\)) = 250 °C - Diameter of the pipe (D) = 15 cm = 0.15 m (converted to meters) - Average velocity of the exhaust gases (V) = 7.2 m/s - Surface temperature of the pipe (\(T_s\)) = 180 °C - Use properties of air for exhaust gases We will use the convective heat transfer formula to determine the required pipe length. \(q = hA(T_{1} - T_{2})\) Where: - q is the heat transfer rate - h is the convective heat transfer coefficient (for air) - A is the surface area of the pipe - \(T_{1}\) and \(T_{2}\) are the initial and final temperatures of the exhaust gases, respectively To calculate the pipe length (L), we will use the relationship between the surface area (A) and the length: \(A = 2\pi rL\) Where r is the radius of the pipe.

Calculate the heat transfer rate (q)

To calculate the heat transfer rate (q), we can use the formula for convective heat transfer: \(q = hA(T_{1} - T_{2})\) However, we need to find the convective heat transfer coefficient (h) first. For air flowing through a pipe, the h value can be found using the Dittus-Boelter equation: \(h = k\left(\frac{RePr}{D}\right)^{0.4}\) Where: - k is the thermal conductivity of the air - Re is the Reynolds number - Pr is the Prandtl number - D is the diameter of the pipe We can now calculate the Reynolds number (Re) and the Prandtl number (Pr) using the given data: \(Re = \frac{VD}{\nu}\) (where ν is the kinematic viscosity of the air) For air, \(\nu = 1.58 × 10^{-5} \frac{\text{m}^2}{\text{s}}\) and \(k = 0.026 \frac{\text{W}}{\text{m}\cdot\text{K}}\). To calculate Pr, we can use the formula: \(Pr = \frac{\nu}{\alpha}\) (where α is the thermal diffusivity of the air) For air, \(\alpha = 2.22 × 10^{-5} \frac{\text{m}^2}{\text{s}}\). Therefore, \(Pr = \frac{1.58 × 10^{-5}}{2.22 × 10^{-5}} = 0.71\). Now, we can calculate Re and h: \(Re = \frac{(7.2)(0.15)}{1.58 × 10^{-5}} = 68500\) \(h = (0.026)\left(\frac{(68500)(0.71)}{0.15}\right)^{0.4} = 68.31 \frac{\text{W}}{\text{m}^2\cdot\text{K}}\) With the h value, we can now calculate the heat transfer rate (q): \(q = (68.31)(2\pi(0.075)L)(450 - 250)\) \(q = 256L\)

Calculate the pipe length (L)

We can now use the heat transfer rate formula to calculate the pipe length (L): \(q = 256L\) However, we need to find the value of q using the given data. To do that, we can use the formula for the mass flow rate (m_dot): \(m_\text{dot} = \rho AV\) Where ρ is the density of the air, A is the cross-sectional area of the pipe, and V is the velocity of the exhaust gases. For air, ρ = 1.164 kg/m³. Now, we can calculate A and m_dot: \(A = \pi r^2 = \pi (0.075)^2 = 0.0177 \text{ m}^2\) \(m_\text{dot} = (1.164)(0.0177)(7.2) = 0.1475 \frac{\text{kg}}{\text{s}}\) We can now use the specific heat capacity of air (Cp) to find the heat transfer rate (q): \(q = m_\text{dot} \times C_\text{p} \times (T_{1} - T_{2})\) For air, \(C_\text{p} = 1005 \frac{\text{J}}{\text{kg}\cdot\text{K}}\). Now, we can calculate q: \(q = (0.1475)(1005)(450 - 250) = 29525 \text{ W}\) Now, we can plug the value of q back into the equation for q in the Step 2: \(29525 = 256L\) \(L = \frac{29525}{256} = 115.33 \text{ m}\)

Final answer

The required length of the pipe is 115.33 meters.