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The Stefan-Boltzmann law is a key principle in understanding how objects radiate thermal energy. This law states that the power radiated per unit area of an object is proportional to the fourth power of its temperature in Kelvin. In mathematical terms, it is expressed as:

• \( P = \varepsilon \sigma A T^{4} \)

Where \( P \) is the power radiated, \( \varepsilon \) is the emissivity, \( \sigma \) the Stefan-Boltzmann constant \((5.67 × 10^{-8} \text{ W/m}^{2} \text{K}^{4})\), \( A \) is the surface area, and \( T \) is the absolute temperature.
In our example, the human body is radiating heat due to the difference in temperature between body surface (37°C) and the surrounding environment (24°C). By converting these temperatures into Kelvin and using the given emissivity and surface area, the rate at which thermal energy is dissipated by radiation can be calculated. This calculation is crucial for understanding how much thermal energy needs to be removed from the body to maintain a stable internal environment.

Sweating is an essential mechanism the body uses to regulate temperature. When you sweat, the water on the skin surface absorbs heat from the body and evaporates, thus cooling you down. The energy required for this evaporation is known as latent heat, and for sweat, it takes about 2430 kJ to evaporate 1 kg of water.
To find out how much thermal energy is removed from the body through the evaporation of sweat, we use the equation:

• \( Q = mL \)

Where \( m \) is the mass of sweat and \( L \) is the latent heat of evaporation per unit mass. In the gym example, if a person sweats out 0.40 kg of water in an hour, this energy can be calculated to yield the rate of thermal energy dissipation. This process effectively helps in maintaining the body’s thermal balance.

The metabolic rate is the rate at which your body consumes energy to maintain vital functions and perform activities. In the context of thermal energy dissipation, the metabolic rate represents the total energy produced by the body. In the given scenario, the person has a metabolic rate of 2500 kJ/h.
Since 80% of this energy is released as heat, it needs to be dissipated to maintain a consistent internal temperature. This heat is eliminated through various mechanisms such as radiation, evaporation, conduction, and convection. Understanding the metabolic rate is essential for calculating how much energy needs to be managed by these processes. It provides a basis to determine how the body efficiently removes excess heat.

Emissivity is a measure of how effectively a surface emits thermal radiation compared to an ideal black body. It ranges from 0 to 1, where 1 represents a perfect emitter. The concept of emissivity is important in calculating the radiative heat loss from the human body using the Stefan-Boltzmann law.
In our example, the person's skin has an emissivity of 0.97, meaning it is very close to being a perfect emitter. This high emissivity factor signifies that the skin is efficient at radiating away excess heat. By knowing the emissivity, one can determine how much heat is lost through radiation, which is a critical component in managing the body's energy balance during physical activities such as working out in a gym.