Question

During a plant visit, it was noticed that a 12-m-long section of a \(10-\mathrm{cm}\)-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 \(10 \mathrm{~km} / \mathrm{h}\). 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 work day. 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

The Nusselt number is significant in calculating heat loss through convection because it represents the ratio of convective to conductive heat transfer. This value is crucial in determining the convective heat transfer coefficient, which in turn is needed to calculate the heat loss due to convection.

Step-by-step solution

Find the heat loss through convection

To start, we need to find the convective heat transfer coefficient, which we'll then use to find the heat loss due to convection. To find the convective heat transfer coefficient, we can use the formula: $$h_c = Nu \cdot \frac{k}{D}$$ Where: - \(Nu\) is the Nusselt number, - \(k\) is the thermal conductivity of the fluid (air), - \(D\) is the diameter of the pipe. First, let's find the Nusselt number using the formula: $$Nu = 0.3 + \frac{0.62 \cdot Re^{1/2} \cdot Pr^{1/3}}{(1+(\frac{0.4}{Pr})^{2/3})^{1/4}}$$ Before calculating the Nusselt number, we will need to determine the Reynolds number (\(Re\)) and Prandtl number (\(Pr\)) from given data.

Find the Reynolds and Prandtl number

To find the Reynolds number (\(Re\)), we use the formula: $$Re = \frac{v \cdot D}{ν}$$ Where: - \(v\) is the velocity of the fluid (velocity of wind) (convert from 10 km/h to m/s), - \(D\) is the diameter of the pipe (convert 10 cm to m), - \(ν\) is the kinematic viscosity of air at the film temperature (approximately, \(μ_{air} = 1.9 × 10^{-5} \mathrm{m^2/s}\)). Next, we will find the Prandtl number (\(Pr\)) using the formula: $$Pr = \frac{c_p \cdot μ}{k}$$ Where: - \(c_p\) is the specific heat capacity of air (approximated using \(c_{p_{air}} = 1006 \mathrm{J/(kg·K)}\)), - \(μ\) is the dynamic viscosity of air (approximated using \(μ_{air} = 1.7 × 10^{-5} \mathrm{kg/(m·s)}\)), - \(k\) is the thermal conductivity of air (approximated using \(k_{air} = 0.026 \mathrm{W/(m·K)}\)). Now, we have all the needed values for \(Re\) and \(Pr\). Calculate them and then calculate the Nusselt number.

Calculate the heat loss through convection

With the computed Nusselt number, we can now calculate the convective heat transfer coefficient, \(h_c\). Next, we will calculate the heat loss through convection using the formula: $$Q_{conv} = h_c \cdot A_s \cdot (T_s - T_{∞})$$ Where: - \(A_s\) is the surface area of the pipe, - \(T_s\) is the surface temperature (75°C), - \(T_{∞}\) is the ambient temperature (5°C).

Calculate the heat loss through radiation

To calculate the heat loss through radiation, we need to use the following equation: $$Q_{rad} = ε \cdot σ \cdot A_s \cdot (T_s^4 - T_{surround}^4)$$ Where: - \(ε\) is the emissivity (0.8), - \(σ\) is the Stefan-Boltzmann constant \((5.67 × 10^{-8} \mathrm{W/(m^2·K^4)})\), - \(A_s\) is the surface area of the pipe, - \(T_s\) is the surface temperature in Kelvin (convert 75°C to K), - \(T_{surround}\) is the average temperature of surrounding surfaces in Kelvin (convert 0°C to K). Next, find the total heat loss by summing up the heat loss due to convection and radiation.

Calculate the heat loss during a 10-hour workday

Multiply the total heat loss by the 12-meter length of the pipe, and then by the number of working hours per day (10 hours), to obtain the heat loss during a 10-hour workday.

Calculate the amount of money saved by insulating the steam pipes

First, calculate the reduction in heat loss by finding 90% of the original heat loss. Assume the plant operates every day of the year for 10 hours. Then, multiply this reduced heat loss by the number of operating days per year (365 days). Next, divide this by the efficiency of the gas-fired steam generator (80%) and then multiply by the cost per therm of natural gas (\(\$1.05 / \mathrm{therm}\)). The result will give us the amount of money saved per year as a result of insulating the steam pipes.

Key concepts

Convective Heat Transfer

Convective heat transfer is an essential concept when it comes to understanding how heat moves between a surface and a fluid, like air, flowing across it. This process involves the transfer of heat due to the motion of the fluid, which can be either natural or forced. In the case of forced convection, like the scenario with the steam pipe exposed to wind, the movement of the fluid is externally prompted, such as by the wind itself.

• The convective heat transfer coefficient (\( h_c \)) quantifies this transfer rate and is influenced by factors such as fluid speed and surface characteristics.
• Using the Nusselt number (\( Nu \)), which relates the convective and conductive heat transfers, can help calculate this coefficient accurately.

Understanding these basics allows one to determine how efficiently a surface loses heat to its environment, as seen in the exercise involving the steam pipe.

Radiation Heat Transfer

Radiation heat transfer is the process of heat transfer via electromagnetic waves, and it does not require any medium to occur. It becomes significant at high temperature differences. In the provided problem, the steam pipe emits thermal radiation due to its elevated temperature compared to its surroundings.

• Emissivity (\( \varepsilon \)), a measure of a material's ability to emit energy as thermal radiation, plays a crucial role. For a given surface area, the rate of radiation emission increases with higher emissivity values.
• The Stefan-Boltzmann Law gives the energy radiated per unit area of a black body and is described by the equation \( Q_{rad} = \varepsilon \cdot \sigma \cdot A_s \cdot (T_s^4 - T_{surround}^4) \).

This understanding is crucial to calculate how much heat the steam pipe loses to its surroundings due to radiation.

Thermal Insulation

Thermal insulation is a method used to reduce heat transfer between objects in thermal contact or in range of radiative influence. In the case of the steam pipe, insulating the pipe would drastically lower the heat lost to the surroundings, making it more energy-efficient.

• Insulation works by minimizing conduction, convection, and radiation, contributing to significant energy savings in the long run.
• By trapping heat within the pipe, insulation ensures that less energy is required to maintain the steam temperature, leading to reduced operational costs.

In this exercise, achieving a 90% reduction in heat loss through insulation reflects efficient use of insulation techniques.

Energy Efficiency

Energy efficiency involves utilizing energy in such a way that the desired outcomes are achieved with less waste. When applied to the steam pipe scenario, energy efficiency measures how effectively the pipe retains its heat.

• The efficiency of the steam generator (80% in this scenario) shows that not all energy input is converted into useful steam energy, highlighting the importance of minimizing losses, such as through convective and radiative heat transfers.
• Improving insulation directly enhances energy efficiency by conserving the generated steam and reducing the need for constant energy input.

Efficient energy usage results in cost savings and reduced environmental impacts, making it an important consideration for industrial operations.

Nusselt Number

The Nusselt number (\( Nu \)) is a dimensionless number that represents the ratio of convective to conductive heat transfer across a boundary. It aids in analyzing the efficiency of convective heat transfer compared to simple conduction.

• The formula used for the Nusselt number includes contributions from the Reynolds and Prandtl numbers, which account for fluid dynamics and conductive transfer specifics, respectively.
• In our steam pipe scenario, calculating the Nusselt number helps determine the convective heat transfer coefficient, providing insights into how effectively heat is removed from the pipe surface by the moving air.

You're able to learn about how materials and design influence thermal efficiency through understanding the Nusselt number.

Reynolds Number

The Reynolds number (\( Re \)) is another dimensionless number used to predict flow patterns in different fluid flow situations. It provides insight into whether a flow will be laminar or turbulent, influenced by factors such as fluid velocity, viscosity, and characteristic length.

• The formula for Reynolds number is \( Re = \frac{v \cdot D}{u} \), indicating the relationship between inertial and viscous forces within the fluid.
• For the steam pipe, determining \( Re \) clarifies if the air flow around the pipe induces turbulent conditions, which typically enhance the heat transfer rate.

By understanding and calculating the Reynolds number, engineers can optimize systems for better thermal performance.

Stefan-Boltzmann Law

The Stefan-Boltzmann Law is a principle that describes how the thermal energy radiated per unit area by a black body is proportional to the fourth power of its absolute temperature. This is crucial in calculating radiation heat transfer in any setting.

• The constant \( \sigma \) represents the Stefan-Boltzmann constant \(5.67 \times 10^{-8} \mathrm{W/(m^2 \cdot K^4)} \).
• Used in the formula \( Q_{rad} = \varepsilon \cdot \sigma \cdot A_s \cdot (T_s^4 - T_{surround}^4) \), it helps find the total heat lost via radiation based on temperature differences.

Understanding this law enables the accurate calculation of heat loss in systems where radiation plays a prominent role, such as in the steam pipe problem.