Question

Consider a 25-mm-diameter and 15-m-long smooth tube that is maintained at a constant surface temperature. Fluids enter the tube at \(50^{\circ} \mathrm{C}\) with a mass flow rate of \(0.01 \mathrm{~kg} / \mathrm{s}\). Determine the tube surface temperatures necessary to heat water, engine oil, and liquid mercury to the desired outlet temperature of \(150^{\circ} \mathrm{C}\).

Short answer

Question: Determine the tube surface temperatures necessary to heat water, engine oil, and liquid mercury to the desired outlet temperature of 150°C. Answer: To determine the tube surface temperatures necessary to heat water, engine oil, and liquid mercury to the desired outlet temperature of 150°C, follow these steps: 1. Calculate the heat transfer rates using the energy balance equation for each fluid. 2. Calculate the heat transfer rate using the heat transfer formula (conduction and convection) for each fluid. 3. Determine the tube surface temperatures for each fluid by substituting the calculated values into the rearranged heat transfer equation.

Step-by-step solution

Find heat transfer rates using energy balance for each fluid.

First, find the specific heat capacities of water, engine oil, and liquid mercury. The energy balance equation is: \(Q = m\cdot c_p (\Delta T)\) where \(Q\) is the heat transfer rate, \(m\) is the mass flow rate, \(c_p\) is the specific heat capacity, and \(\Delta T\) is the change in temperature. The mass flow rate is given as \(m = 0.01 \mathrm{~kg/s}\). The change in temperature for all fluids is the same as the difference between the desired outlet temperature and the inlet temperature: \(\Delta T = T_{outlet} - T_{inlet} = 150-50 = 100^{\circ}C\) Find the specific heat capacities of water (\(c_{p,w}\)), engine oil (\(c_{p,eo}\)), and liquid mercury (\(c_{p,lm}\)). Then, calculate the heat transfer rates for each fluid using the energy balance equation.

Calculate the heat transfer rate using the heat transfer formula for each fluid.

Next, we'll use the heat transfer equation for conduction and convection in a tube: \(Q = hA_s(T_s - T_m)\) where \(Q\) is the heat transfer rate, \(h\) is the heat transfer coefficient, \(A_s\) is the surface area of the tube, \(T_s\) is the tube surface temperature, and \(T_m\) is the mean temperature between the inlet and outlet temperatures. To find the surface area of the tube, use the formula: \(A_s = 2 \pi r L\) where \(r\) is the radius of the tube and \(L\) is its length. Calculate the mean temperature for each fluid: \(T_m = \frac{T_{inlet}+T_{outlet}}{2}\) For each fluid, look up the heat transfer coefficient (\(h\)) in a reference. Then, rearrange the heat transfer equation to solve for the tube surface temperature (\(T_s\)) for each fluid: \(T_s = T_m + \frac{Q}{hA_s}\)

Determine the tube surface temperatures for each fluid.

Using the heat transfer rates found in Step 1, the surface area calculated in Step 2, and the heat transfer coefficients (from reference) for each fluid, compute the tube surface temperature (\(T_s\)) for water, engine oil, and liquid mercury. In conclusion, follow these steps to determine the tube surface temperatures necessary to heat water, engine oil, and liquid mercury to the desired outlet temperature of \(150^{\circ}C\): 1. Calculate the heat transfer rates using the energy balance equation for each fluid. 2. Calculate the heat transfer rate using the heat transfer formula (conduction and convection) for each fluid. 3. Determine the tube surface temperatures for each fluid by substituting the calculated values into the rearranged heat transfer equation.