Question

During a visit to a plastic sheeting plant, it was observed that a 45 -m-long section of a 2 -in nominal \((6.03-\mathrm{cm}\)-outerdiameter) steam pipe extended from one end of the plant to the other with no insulation on it. The temperature measurements at several locations revealed that the average temperature of the exposed surfaces of the steam pipe was $170^{\circ} \mathrm{C}\(, while the temperature of the surrounding air was \)20^{\circ} \mathrm{C}$. The outer surface of the pipe appeared to be oxidized, and its emissivity can be taken to be 0.7. Taking the temperature of the surrounding surfaces to be \(20^{\circ} \mathrm{C}\) also, determine the rate of heat loss from the steam pipe. Steam is generated in a gas furnace that has an efficiency of 84 percent, and the plant pays \(\$ 1.10\) per therm (1 therm \(=105,500 \mathrm{~kJ}\) ) of natural gas. The plant operates \(24 \mathrm{~h}\) a day, 365 days a year, and thus \(8760 \mathrm{~h}\) a year. Determine the annual cost of the heat losses from the steam pipe for this facility.

Short answer

Based on the given dimensions and properties of the steam pipe, the total heat loss is found to be 13613.39 W. The annual energy loss is calculated as 1.194 x 10^11 J. As a result, the annual cost of the heat losses from the steam pipe for this facility is $14650.35.

Step-by-step solution

Calculate the convective heat transfer

We will use Newton's law of cooling to find the convective heat transfer: \(q_{conv} = hA(T_s - T_\infty)\), where \(q_{conv}\) is the convective heat transfer, \(h\) is the convective heat transfer coefficient (which we need to estimate), \(A\) is the surface area of the steam pipe, \(T_s\) is the temperature of the steam pipe surface, and \(T_\infty\) is the temperature of the surrounding air. The surface area of the steam pipe can be calculated as: \(A = 2\pi rL\), where \(r\) is the outer radius of the steam pipe and \(L\) is the length of the steam pipe. First, we will estimate the value of \(h\) by using the Nusselt number and the thermal conductivity of the air \((k = 0.0262 \mathrm{W/m \cdot K})\): \(Nu = \frac{hD}{k}\), where \(Nu\) is the Nusselt number and \(D\) is the diameter of the steam pipe. Assuming a Nusselt number of 4 (typical for natural convection around a cylinder), we can calculate the value of \(h\): \(h = \frac{Nu \cdot k}{D} = \frac{4 \cdot 0.0262 \mathrm{W/m \cdot K}}{0.0603 \mathrm{m}} = 1.738 \mathrm{W/m^2 \cdot K}\). Now we can calculate the convective heat transfer: \(q_{conv} = hA(T_s - T_\infty) = 1.738 \mathrm{W/m^2 \cdot K} \cdot 2\pi \cdot 0.03015 \mathrm{m} \cdot 45 \mathrm{m} \cdot (170 - 20) \mathrm{K} = 9297.61 \mathrm{W}\).

Calculate the radiative heat transfer

We will use the Stefan-Boltzmann law to find the radiative heat transfer: \(q_{rad} = \epsilon \sigma A(T_s^4 - T_{sur}^4)\), where \(q_{rad}\) is the radiative heat transfer, \(\epsilon\) is the emissivity of the steam pipe, \(\sigma\) is the Stefan-Boltzmann constant \((5.67 \times 10^{-8} \mathrm{W/m^2 \cdot K^4})\), \(T_s\) is the temperature of the steam pipe surface in Kelvin, and \(T_{sur}\) is the temperature of the surrounding surfaces in Kelvin. First, we need to convert the temperatures to Kelvin: \(T_s = 170^{\circ}\mathrm{C} + 273.15 = 443.15 \mathrm{K}\) \(T_{sur} = 20^{\circ}\mathrm{C} + 273.15 = 293.15 \mathrm{K}\). Now we can calculate the radiative heat transfer: \(q_{rad} = 0.7 \cdot (5.67 \times 10^{-8} \mathrm{W/m^2 \cdot K^4}) \cdot 2\pi \cdot 0.03015 \mathrm{m} \cdot 45 \mathrm{m} \cdot (443.15^4 - 293.15^4) \mathrm{K^4} = 4315.78 \mathrm{W}\).

Calculate the total heat loss

The total heat loss is the sum of the convective and radiative heat transfers: \(q_{total} = q_{conv} + q_{rad} = 9297.61 \mathrm{W} + 4315.78 \mathrm{W} = 13613.39 \mathrm{W}\).

Calculate the annual energy loss

The annual energy loss can be calculated as: \(E_{loss} = q_{total} \cdot t_{year} = 13613.39 \mathrm{W} \cdot 8760 \mathrm{h} \cdot 3600 \mathrm{s/h} = 1.194 \times 10^{11} \mathrm{J}\).

Calculate the annual cost of the heat losses

The annual cost of the heat losses can be calculated by considering the efficiency of the gas furnace, the cost of natural gas, and the annual energy loss: \(C_{loss} = \frac{E_{loss}}{\eta \cdot E_{therm} \cdot 10^{3}} \cdot C_{therm} = \frac{1.194 \times 10^{11} \mathrm{J} }{0.84 \cdot 105,500 \mathrm{kJ/therm} \cdot 10^{3}} \cdot 1.10 \mathrm{\$/therm} = \$ 14650.35\). The annual cost of the heat losses from the steam pipe for this facility is $14650.35.