Energy Balance#

Heat Capacity#

Heat is the transfer (or flux) of thermal energy. Thermal energy and temperature are related by the heat capacity of the object. In particular, the heat capacity of an object is the amount of thermal energy \(\Delta E\) require to raise the temperature of the object by \(\Delta T\)

\[ C = \frac{\Delta E}{\Delta T} \]

Note that \(C\) has units J/K. In reality, the heat capacity depends on the temperature and pressure of the object but we will assume that \(C\) is constant for the objects that we consider.

We will construct models where temperature is changing over time and so we use the equation

\[ C \frac{dT}{dt} = \frac{dE}{dt} \]

to relate the rate of change of energy (ie. heat transfer) to the rate of change of temperature of an object over time.

Energy Balance#

The law of conservation of energy applied to an object that absorbs and emits heat yields the energy balance equation:

\[ C \frac{dT}{dt} = Q_{in} - Q_{out} \]

where:

  • \(T\) is the temperature of the object (K)

  • \(C\) is the heat capacity of the object (J/K)

  • \(Q_{in}\) is the heat rate of thermal energy (ie. heat) absorbed by the object (W = J/s)

  • \(Q_{out}\) is the rate of thermal energy (ie. heat) emitted by the object (W = J/s)

Dimensions and Units#

Quantity

Symbol

Dimensions

SI Units

thermal energy

\(E\)

M L2 T-2

J

temperature

\(T\)

\(\Theta\)

K

heat capacity

\(C\)

M L2 T-2 \(\Theta^{-1}\)

J/K

heat

\(Q\)

M L2 T-3

W