Question

A cylindrical container whose height and diameter are \(8 \mathrm{~m}\) is filled with a mixture of \(\mathrm{CO}_{2}\) and \(\mathrm{N}_{2}\) gases at \(600 \mathrm{~K}\) and \(1 \mathrm{~atm}\). The partial pressure of \(\mathrm{CO}_{2}\) in the mixture is \(0.15 \mathrm{~atm}\). If the walls are black at a temperature of \(450 \mathrm{~K}\), determine the rate of radiation heat transfer between the gas and the container walls.

Short answer

Answer: The rate of radiation heat transfer between the CO2 gas and the container walls is approximately 832 W.

Step-by-step solution

Gather information and establish formulas

From the given information, we have the following variables: Height (H) = 8 m Diameter (D) = 8 m Temperature of the gas (T1) = 600 K Pressure of the gas (P) = 1 atm Partial pressure of CO2 (P_CO2) = 0.15 atm Temperature of the container walls (T2) = 450 K Stefan-Boltzmann constant (σ) = 5.67 x 10^-8 W/m²K We will use the Stefan-Boltzmann Law for radiation heat transfer: Q = σ * A * (T1^4 - T2^4) In which 'Q' is the heat transfer rate, 'A' is the surface area of the container, 'T1' is the gas temperature, 'T2' is the container wall temperature, and 'σ' is the Stefan-Boltzmann constant.

Calculate the surface area of the container

To calculate the surface area of the cylindrical container, we can use the following formula: A = 2 * π * r * (r + H) In which 'A' is the surface area, 'r' is the radius of the cylinder, and 'H' is the height of the cylinder. In our case, the diameter and the height of the cylinder are both 8 m. So, the radius 'r' is 4 m. A = 2 * π * 4 * (4 + 8) A ≈ 301.6 m²

Calculate the net heat transfer rate

We have the surface area 'A', gas temperature 'T1', and container walls temperature 'T2'. We can plug these values into the Stefan-Boltzmann Law: Q = σ * A * (T1^4 - T2^4) Q = (5.67 x 10^-8 W/m²K) * (301.6 m²) * (600^4 - 450^4) Q ≈ 5546 W

Determine the rate of radiation heat transfer

Now, that we have the net heat transfer rate, we must take into account the partial pressures of both gases to determine the corresponding heat transfer rate of the CO2 gas: Partial pressure of N2 (P_N2) = P - P_CO2 = 1 - 0.15 = 0.85 atm Ratio between partial pressure of CO2 and total pressure: Valpha= P_CO2/P = 0.15 / 1 = 0.15 Finally, we can calculate the rate of radiation heat transfer between the CO2 gas and the walls: Q_CO2 = Q * Valpha Q_CO2 = 5546 W * 0.15 Q_CO2 ≈ 832 W The rate of radiation heat transfer between the CO2 gas and the container walls is approximately 832 W.

Key concepts

Stefan-Boltzmann Law

The Stefan-Boltzmann Law is crucial when discussing radiation heat transfer. It describes how the heat radiated by a black body (an idealized object that perfectly absorbs all radiation) changes with its temperature. The formula is expressed as \( Q = \sigma \cdot A \cdot (T_1^4 - T_2^4) \), where \( Q \) represents the heat transfer rate, \( \sigma \) is the Stefan-Boltzmann constant \( (5.67 \times 10^{-8} \, \text{W/m}^2\text{K}^4) \), \( A \) is the surface area, and \( T_1 \) and \( T_2 \) are the temperatures of the respective surfaces involved.
This helps us predict how much energy is radiated between different bodies. The power of the Stefan-Boltzmann Law lies in its ability to provide insight into the impact of temperature differences on heat transfer, which scales with the fourth power of temperature. Meaning, even small temperature differences can lead to significant heat flow.

partial pressure

Partial pressure is a key concept in understanding gas mixtures and their behavior under various conditions. In a gaseous mixture, the total pressure is the sum of the pressures exerted by each individual gas. The individual pressure contributed by a single gas is known as its partial pressure.
For example, in our problem, the partial pressure of \( \text{CO}_2 \) is given as \( 0.15 \, \text{atm} \), which is part of a total pressure of \( 1 \, \text{atm} \). You calculate the partial pressure by multiplying the mole fraction of the gas in the mixture by the total pressure. This relationship is crucial when analyzing the effects of each component in a gas mixture.
Understanding partial pressures allows us to apply the correct weighting to each gas component when determining the total heat transfer or reaction rates. In radiation heat transfer problems, partial pressures inform us of the effective amount of radiation each gas contributes.

cylindrical container

A cylindrical container serves as a practical example to apply concepts of geometry in calculating physical properties such as surface area or volume that are needed in heat transfer calculations. In the context of radiation heat transfer, the surface area \( A \) of a cylindrical container is vital for determining the amount of heat that can be transferred through radiation.
To find the surface area of a cylinder, the formula used is \( A = 2 \pi r (r + H) \), where \( r \) is the radius, and \( H \) is the height of the cylinder. In the provided exercise, both the diameter and height of the cylinder are \( 8 \, \text{m} \), giving a radius of \( 4 \, \text{m} \), and thus a surface area of approximately \( 301.6 \, \text{m}^2 \).
This area is fundamental in the Stefan-Boltzmann Law to calculate how much radiation is transferred to or from the gas within, helping assess heat transfer efficiencies.

net heat transfer rate

The net heat transfer rate is what determines the actual amount of energy being transferred from one body to another, taking into account all influencing factors. In radiation heat transfer problems, it determines how effectively heat moves between surfaces.
Using the Stefan-Boltzmann Law, the net heat transfer rate \( Q \) is first calculated using both the surface area \( A \) of the container and the temperatures \( T_1 \) and \( T_2 \) of the contents and the container walls, respectively. In our problem, this resulted in a preliminary \( Q \) value of approximately \( 5546 \, \text{W} \).
To find the specific heat transfer rate due to \( \text{CO}_2 \), one must consider the partial pressure ratio, leading to a reduced rate of \( 832 \, \text{W} \). This differentiation helps pinpoint the contribution of different gas components to the overall net heat flow.