Question

During a plant visit, it was noticed that a \(12-\mathrm{m}\)-long section of a 12 -cm-diameter steam pipe is completely exposed to the ambient air. The temperature measurements indicate that the average temperature of the outer surface of the steam pipe is \(75^{\circ} \mathrm{C}\) when the ambient temperature is \(5^{\circ} \mathrm{C}\). There are also light winds in the area at \(25 \mathrm{~km} / \mathrm{h}\). The emissivity of the outer surface of the pipe is \(0.8\), and the average temperature of the surfaces surrounding the pipe, including the sky, is estimated to be \(0^{\circ} \mathrm{C}\). Determine the amount of heat lost from the steam during a 10 -h-long workday. Steam is supplied by a gas-fired steam generator that has an efficiency of 80 percent, and the plant pays \(\$ 1.05 /\) therm of natural gas. If the pipe is insulated and 90 percent of the heat loss is saved, determine the amount of money this facility will save a year as a result of insulating the steam pipes. Assume the plant operates every day of the year for \(10 \mathrm{~h}\). State your assumptions.

Short answer

Answer: To find the amount of money saved by insulating the steam pipes per year, follow these steps: 1. Calculate the area of the exposed pipe using the diameter and length. 2. Calculate the convective heat transfer coefficient using the wind speed. 3. Calculate the heat loss due to convection using the convective heat transfer coefficient, area, and temperature difference. 4. Calculate the heat loss due to radiation using the Stefan-Boltzmann constant, emissivity, and temperature difference. 5. Calculate the total heat loss by summing the convective and radiative heat losses. 6. Calculate the heat loss during a 10-hour workday by multiplying the total heat loss by the time. 7. Calculate the amount of money saved by insulating the steam pipes by reducing the heat loss, then calculating the fuel saved and the money saved per day, and finally scaling this value to the entire year. The amount of money saved per year will be the result from step 7.

Step-by-step solution

Calculate the area of the exposed pipe

To find the total heat loss, we first need to determine the area of the exposed pipe. We can do this using the given diameter and length of the pipe: Area (A) = π * pipe diameter * pipe length A = π * 0.12 m * 12 m

Calculate the convective heat transfer coefficient

Next, we need to find the convective heat transfer coefficient "h." Since the wind velocity is given, and assuming the Reynolds number is in the turbulent regime, we use the empirical correlation to calculate "h": Assuming the wind speed is uniform and parallel to the pipe, we have the following formula for convective heat transfer. h = 1.23 * wind_speed^0.54 wind_speed = (25 km/h) * (1000 m/km) / (3600 h/s) h = 1.23 * (wind_speed)^0.54

Calculate the heat loss due to convection

Now, we can calculate the heat loss due to convection using the following formula: Convective heat loss (Q_conv) = h * A * (T_surface - T_ambient) Q_conv = h * A * (75°C - 5°C)

Calculate the heat loss due to radiation

Now, we will calculate the heat loss due to radiation using the Stefan-Boltzmann constant, emissivity, and surrounding temperature: Radiative heat loss (Q_rad) = ε * σ * A * (T_surface^4 - T_surrounding^4) ε = 0.8 (emissivity) σ = 5.67 * 10^(-8) W/(m^2*K^4) (Stefan-Boltzmann constant) T_surface = 75°C + 273.15 T_surrounding = 0°C + 273.15 Q_rad = 0.8 * 5.67 * 10^(-8) W/(m^2*K^4) * A * (T_surface^4 - T_surrounding^4)

Calculate the total heat loss

We can now calculate the total heat loss from the pipe by summing the convective and radiative heat losses: Total heat loss (Q_total) = Q_conv + Q_rad

Calculate the heat loss during a 10-hour workday

To determine the amount of heat lost during a 10-hour workday, we multiply the total heat loss by the time: Heat loss during workday (Q_workday) = Q_total * 10 hours

Calculate the amount of money saved by insulating the steam pipes

First, we calculate the heat loss reduction by insulating the pipes: Reduced heat loss = 0.90 * Q_workday Then, we calculate the fuel saved by this reduced heat loss, considering the efficiency of the steam generator: Fuel saved = Reduced_heat_loss / (efficiency * natural_gas_energy_content) Finally, we calculate the money saved per day and scale this value to the entire year, assuming the plant operates every day: Money_saved_per_day = Fuel_saved * natural_gas_price_per_therm Money_saved_per_year = Money_saved_per_day * 365 days