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Chapter 13: Problem 101

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

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

Step by step solution

01

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.
02

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
03

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
04

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.

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Most popular questions from this chapter

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