Heat Loss from a Sweaty Jersey

Question

A sports jersey is completely drenched with sweat. How much heat will be lost from the sportsman if the jersey dries out while the sportsman wears it? The jersey is made out of 1.5 meters of cloth. Assume unknown variables and provide a mathematical solution.

Solution

To calculate the heat lost from the sportsman as their jersey dries out, we will use the concept of latent heat of vaporization, which is the energy required to evaporate water (sweat) from the jersey.

Assumptions

The jersey is completely soaked, and we assume it holds \( m \, \text{kg} \) of water (sweat).
The temperature of the sweat is around body temperature (\( 37^\circ \text{C} \)), meaning evaporation occurs close to this temperature.
The latent heat of vaporization of water (\( L_v \)) is approximately \( 2260 \, \text{kJ/kg} \) at \( 37^\circ \text{C} \).

Key Formula

The heat loss \( Q \) due to the evaporation of sweat can be calculated using the formula:

\( Q = m \cdot L_v \)

where:

\( m \) is the mass of water (in kg).
\( L_v \) is the latent heat of vaporization of water.

Estimating the Mass of Water (\( m \))

The jersey is made out of 1.5 meters of cloth. The amount of water the cloth can hold depends on the type of fabric and its absorbency. For this calculation, we assume that the jersey can hold approximately \( 100\% \) of its own weight in water.

Let’s denote:

\( \rho \): Density of water (\( 1000 \, \text{kg/m}^3 \)).
\( A \): Total surface area of the jersey (\( 1.5 \, \text{m}^2 \)).
\( t \): Thickness of the jersey (assumed to be \( 0.0015 \, \text{m} \)).

The volume of water absorbed by the jersey \( V \) is given by:

\( V = A \cdot t \)

Substituting the values:

\( V = 1.5 \, \text{m}^2 \cdot 0.0015 \, \text{m} = 0.00225 \, \text{m}^3 \)

The mass of the water held by the jersey is:

\( m = \rho \cdot V = 1000 \, \text{kg/m}^3 \cdot 0.00225 \, \text{m}^3 = 2.25 \, \text{kg} \)

Heat Loss Calculation

Using the formula for heat loss:

\( Q = m \cdot L_v \)

Substituting \( m = 2.25 \, \text{kg} \) and \( L_v = 2260 \, \text{kJ/kg} \):

\( Q = 2.25 \, \text{kg} \cdot 2260 \, \text{kJ/kg} = 5085 \, \text{kJ} \)

Conclusion

The total heat lost from the sportsman as the jersey dries out would be approximately 5085 kJ. This significant amount of heat loss highlights why evaporation of sweat is such an effective cooling mechanism for the body.