Calculating Heat Current

The heat current is the rate at which heat is transferred over time. Because it is a rate of heat energy over time, the SI unit of heat current is joule per second, or watt (W).

Heat flows through material objects through the conduction, with heated particles imparting their energy to neighboring particles. Scientists studied the flow of heat through materials well before they even knew that the materials were made up atoms, and heat current is one of the concepts that was helpful in this regard. Even today, though we understand heat transfer to be related to the movement of individual atoms, in most situations it is impractical and unhelpful to try to think of the situation in that way, and stepping back to treat the object on a larger scale is the most appropriate way to study or predict the movement of heat.

Mathematics of Heat Current

Because heat current represents the flow of heat energy over time, you can think about it as representing a tiny amount of heat energy, dQ (Q is the variable commonly used to represent heat energy), transmitted over a tiny amount of time, dt. Using the variable H to represent heat current, this gives you the equation:

H = dQ / dt

If you've taken pre-calculus or calculus, you might realize that a rate of change like this is a prime example of when you would want to take a limit as the time approaches zero. Experimentally, you can do that by measuring the heat change at smaller and smaller time intervals.

Experiments conducted to determine the heat current have identified the following mathematical relationship:

H = dQ / dt = kA (T_H - T_C) / L

That may seem like an intimidating array of variables, so let's break those down (some of which have already been explained):

• H: heat current
• dQ: small amount of heat transferred over a time dt
• dt: small amount of time over which dQ was transferred
• k: thermal conductivity of the material
• A: cross-sectional area of the object
• T_H - T_C: the temperature difference between the warmest and coolest temperatures in the material
• L: the length across which the heat is being transferred 

There's one element of the equation that should be considered independently:

(T_H - T_C) / L

This is the temperature difference per unit length, known as the temperature gradient.

Thermal Resistance

In engineering, they often use the concept of thermal resistance, R, to describe how well a thermal insulator prevents heat from transferring across the material. For a slab of material of thickness L, the relationship for a given material is R = L / k, resulting in this relationship:

H = A(T_H - T_C) / R