Heat Transfer

Two systems isolated at different temperatures will exchange heat until their temperatures became the same (thermal equilibrium). These heat exchanges can occur in three different ways: Conduction, Convection and Irradiation (radiation).

The heat transfer rate \(\Phi \) between systems is the ratio of the amount of heat \(Q \) transferred through the area \(S \) in a time interval \(\Delta t)\). Mathematically it is: $$\Phi = \frac{Q}{\Delta t}.$$

Conduction

For heat conduction, the thermal energy is transmitted from particle to particle (atoms, molecules or electrons) by direct collision of them. In the vacuum, the heat cannot be transferred by conduction.

Heat transfer between solids is usually done by conduction.

In the case of thermal conduction, the following law is important:

Fourier Law

The thermal energy \(Q \) transmitted through a rectangular object in a certain time interval \(\Delta t \) is: $$\frac{\Delta Q}{\Delta t} = -\frac{k A(T_2 – T_1)}{\Delta x},$$ where \(\Delta x\) is the length of the object in the direction of heat flow, \(A \) is the cross-sectional area through which heat flows, and \(T_1\) and \(T_2 \) are cold and hot temperatures at boundaries, respectively, and \(K\) is the thermal conductivity constant.

Example 1

Example 2

Example 3

Example 4

The table shows the thermal conductivity of some materials.

Thermal conductivity
Materials	Condutivity (W/m.K)
Stainless steel	14
Copper	401
Dry air	0.026
Polyurethane foam	0.024
Mineral wool	0.043
Fiberglass	0.048
White pine	0.11
Glass	1.0

Convection

"During convection the energy is transmitted by the flow of gas or liquid".

During heat transfer by convection, the hottest molecules are moved from one place to another where the thermal energy is carried by them. For convection it is necessary that the molecules have mobility, that is, that the system is a fluid (liquid or gas).

The process of heat transfer by convection is very complex, so there is no simple and general equation to describe it.

All the winds in the atmosphere are large convective currents.

Radiation

"Even in vacuum, heat spreads."

Radiation is the process of transmitting energy through electromagnetic waves, such as lights, radio waves, TV, "x-ray" etc. This type of energy transfer can also occur in the vacuum.

All objects that are above absolute zero radiate energy.

A body, being in thermal equilibrium with its surroundings, radiates and absorbs energy at the same rate, and therefore its temperature remains constant.

Definitions and important laws:

Black Body

It is a body that absorbs all the radiation that falls on it and has an emissivity equal to 1.

Stefan-Boltzmann Law

The radiation power, \(P_r = \frac{\Delta Q}{\Delta t}\), of a surface area \(A \), with a temperature \(T \) and emissivity \(\varepsilon\), can be expressed by the relation: $$P_r = A \varepsilon \sigma T^4,$$ where \(\sigma \) is called the Stefan-Boltzmann constant \((\sigma=5,67.10^{-8} W/m^2 k^4)\) and \(\varepsilon\) is the emissivity of the surface. Generally, the emissivity of a light surface is greater than a dark one. Its values are obtained in laboratory and are tabulated.

Newton's Cooling Law

The rate of cooling (or heating) of a body is proportional to the temperature difference between the body and its surroundings. That is, given the body temperature \(T(t)\) and the temperature of the environment \(M(t)\) at time \(t\), we then have: $$ \frac{\Delta T}{\Delta t} = k [M(t) - T(t)], $$ where \(k\) is a proportionality constant.

Observations

• Microwave ovens heat and/or cook food by irradiation.
• The energy required for the photosynthesis of the plants is obtained by irradiation.