Question

Consider a person standing in a room maintained at \(20^{\circ} \mathrm{C}\) at all times. The inner surfaces of the walls, floors, and ceiling of the house are observed to be at an average temperature of \(12^{\circ} \mathrm{C}\) in winter and \(23^{\circ} \mathrm{C}\) in summer. Determine the rates of radiation heat transfer between this person and the surrounding surfaces in both summer and winter if the exposed surface area, emissivity, and the average outer surface temperature of the person are \(1.6 \mathrm{~m}^{2}, 0.95\), and \(32^{\circ} \mathrm{C}\), respectively.

Short answer

Answer: The rate of radiation heat transfer between the person and the surrounding surfaces in Winter is 170.1 W, and in Summer is 55.6 W.

Step-by-step solution

Convert temperatures to Kelvin

First, we need to convert the given temperatures from Celsius to Kelvin. To do this, add 273.15 to each temperature: 1. Room temperature: \(20 + 273.15 = 293.15\mathrm{~K}\) 2. Inner surfaces temperature (Winter): \(12 + 273.15 = 285.15\mathrm{~K}\) 3. Inner surfaces temperature (Summer): \(23 + 273.15 = 296.15\mathrm{~K}\) 4. Average outer surface temperature of the person: \(32 + 273.15 = 305.15\mathrm{~K}\)

Use the Stefan-Boltzmann law

The Stefan-Boltzmann law for radiation heat transfer is given by: $$q = \epsilon\sigma A (T_{1}^{4} - T_{2}^{4})$$ Where \(q\) - rate of radiation heat transfer \(\epsilon\) - emissivity \(\sigma\) - Stefan-Boltzmann constant (\(5.67 \times 10^{-8} \mathrm{W/m^2K^4}\)) \(A\) - exposed surface area \(T_{1}, T_{2}\) - temperatures of the surfaces For our problem, \(\epsilon = 0.95\), \(A = 1.6\mathrm{~m}^2\), \(T_{1} = 305.15\mathrm{~K}\) (person's outer surface temperature) and \(T_{2}\) is the inner surfaces temperature of the house (which varies depending on the season).

Calculate radiation heat transfer for Winter season

Substitute the known values for the Winter season into the Stefan-Boltzmann law: $$q_{winter} = 0.95 \times 5.67 \times 10^{-8} \times 1.6 \times (305.15^{4} - 285.15^{4})$$ $$q_{winter} = 170.1\mathrm{~W}$$ The rate of radiation heat transfer between the person and the surrounding surfaces in Winter is \(170.1\mathrm{~W}\).

Calculate radiation heat transfer for Summer season

Substitute the known values for the Summer season into the Stefan-Boltzmann law: $$q_{summer} = 0.95 \times 5.67 \times 10^{-8} \times 1.6 \times (305.15^{4} - 296.15^{4})$$ $$q_{summer} = 55.6\mathrm{~W}$$ The rate of radiation heat transfer between the person and the surrounding surfaces in Summer is \(55.6\mathrm{~W}\). In conclusion, the rate of radiation heat transfer between the person and the surrounding surfaces in Winter and Summer are \(170.1\mathrm{~W}\) and \(55.6\mathrm{~W}\), respectively.

Key concepts

Stefan-Boltzmann Law

The Stefan-Boltzmann Law is a fundamental principle of thermal physics. It describes how the power radiated by a body is related to its temperature in absolute scale (Kelvin). This law states that the total energy radiated per unit surface area is proportional to the fourth power of the black body’s temperature. Mathematically, it is expressed as:
\[ q = \epsilon \sigma A (T_{1}^{4} - T_{2}^{4}) \]
where:

• \(q\) is the rate of radiation heat transfer,
• \(\epsilon\) is the emissivity of the surface,
• \(\sigma\) is the Stefan-Boltzmann constant \( (5.67 \times 10^{-8} \mathrm{W/m^2K^4}) \),
• \(A\) is the surface area,
• \(T_{1}\) and \(T_{2}\) are the absolute temperatures of the object and surroundings respectively.

This law helps us calculate how much energy a body loses or gains through radiation. It is important in understanding how objects radiate energy based on their temperatures in various environments.

Emissivity

Emissivity is a measure of how effectively a real surface emits thermal radiation compared to a perfect black body. A black body, which is an idealized physical body, perfectly absorbs all incident radiation and also radiates energy at the maximum possible rate for its temperature. Emissivity is a dimensionless number between 0 and 1 that indicates the efficiency of radiation.

• An emissivity of 1 means the surface is a perfect emitter, like a black body.
• An emissivity of 0 means the surface doesn't emit any thermal radiation.

In practical applications, knowing the emissivity of surfaces helps predict the actual heat transfer between objects and their environments. Most materials have emissivities less than 1, and in this exercise, the person has an emissivity of 0.95, making them a very efficient emitter of thermal radiation.

Thermal Radiation

Thermal radiation is a mode of heat transfer which involves the emission of electromagnetic waves from the surface of an object. This process does not require any medium to travel through, meaning it can even occur in a vacuum. All objects emit thermal radiation as long as their temperature is above absolute zero, with the intensity and character of the radiation depending on the object's temperature.
The thermal radiation spectrum includes infrared, followed by visible and ultraviolet radiation as temperatures increase. In everyday life, thermal radiation is a key player in heat transfer processes. For instance, when a person stands near a warm wall, they directly gain heat via thermal radiation from the wall's surface.
Understanding thermal radiation is vital for calculating heat dynamics in various scenarios, such as in this exercise where we determine heat changes during seasonal shifts.

Surface Temperature

Surface temperature refers to the measure of thermal energy of an object's surface at a particular point in time. It plays a crucial role in determining how much thermal radiation an object will emit. Higher surface temperature results in higher radiation emissions. In terms of the Stefan-Boltzmann law, the surface temperature influences the rate of heat transfer due to the \(T^{4}\) dependency.
In the given exercise, we compute how the surface temperature of the person and the room affect heat transfer during winter and summer. The temperature is converted to Kelvin for accurate calculations. For example, the person's surface temperature is converted from 32°C to 305.15K, which we then use in the Stefan-Boltzmann equation. This conversion is crucial because it quantifies the energy flow from the person to the surrounding surfaces effectively.