91Ó°ÊÓ

We have a long, thin-walled pipe enclosed in a square concrete casing. To analyze this problem, we will consider the system as two concentric cylinders. One represents the pipe (inner cylinder) and the other represents the concrete casing (outer cylinder). The effective thermal conductivity of the system can be found using the formula for the overall heat transfer coefficient: \[U = \frac{1}{\frac{1}{h_i} + \frac{\ln{(r_o / r_i)}}{2\pi k_c} + \frac{1}{h_o}}\] where: - \(U\) is the overall heat transfer coefficient - \(h_i\) is the convective heat transfer coefficient inside the pipe - \(h_o\) is the convective heat transfer coefficient outside the casing - \(r_i\) is the inner radius of the pipe - \(r_o\) is the outer radius of the concrete casing - \(k_c\) is the thermal conductivity of the concrete In this problem, we are not given the values for \(h_i\), \(h_o\), and \(k_c\). Therefore, we will be looking for the heat loss per unit length for any given heat transfer coefficients.

Given the temperature difference between the steam and the outer surfaces of the concrete casing, and the overall heat transfer coefficient, we can calculate the heat transfer rate per unit area of the pipe as: \[q = U \Delta T\] where: - \(q\) is the heat transfer rate per unit area - \(\Delta T = T_{steam} - T_{outer}\) is the temperature difference between the steam and the outer surfaces of the concrete casing

Now that we have the heat transfer rate per unit area, we can calculate the heat loss per unit length of the pipe. To do this, we multiply the heat transfer rate per unit area by the surface area of the pipe over a given length. \[Q_L = q \times A_\text{pipe}\] where \(Q_L\) is the heat loss per unit length of the pipe, and \(A_\text{pipe}\) is the surface area per unit length. In this case, the pipe is cylindrical, so its surface area per unit length can be calculated as: \[A_\text{pipe} = 2\pi r_\text{pipe} \times L\] Substituting the values given in the problem statement, we have: \[Q_L = q \times 2\pi \frac{0.5}{2} \times L\] The heat loss per unit length of the pipe will be provided in terms of the overall heat transfer coefficient, so the final expression for the heat loss per unit length of the pipe can be written as: \[Q_L = U \Delta T \times \frac{\pi}{2} \times L\] Since we do not know the values of the heat transfer coefficients and the thermal conductivity of the system, we cannot find an exact value for the heat loss per unit length of the pipe. However, we have a general expression that we can use to determine the heat loss for any given overall heat transfer coefficient and temperature difference.