Unizor - Physics4Teens - Energy - Heat Transfer - Conduction

Notes to a video lecture on http://www.unizor.com

Heat Transfer - Conduction

As we know, heat is a form of internal energy that is related to molecular movement.

For solids the molecular movement is usually restricted to molecules' oscillation around some neutral positions.

For liquids the freedom of molecular motion is greater, but still restricted by external forces, like gravity, and surface tension. The average distance between molecules of liquids is relatively constant.

Gas molecules are usually taking all the space available for them. Such forces as gravity also restrict their movement (otherwise, the air molecules would fly away from our planet), but still allow substantial freedom. The average distance between molecules of gas mostly depends on a reservoir the gas is in, the larger the reservoir - the larger average distance between molecules.

Transfer of heat is transfer of molecular movement from one object or part of an object to another object or part of an object, from an object or part of an object with more intense molecular movement (relatively warmer) to an object or part of an object with less intense movement (relatively cooler).

There are three major ways to transfer heat from a hot object to a cold one:
Conduction,
Convection,
Radiation.
This lecture explains a concept of conduction.

Conduction

Conduction of heat energy is a transfer of molecular movement mostly applicable to solid objects - between two solid objects that touch each other, having an area of a contact, or within one object, one part of it having different temperature than another.
The conductivity is present in heat transfer in liquids and gases, but there it's usually combined with another form of heat transfer - convection, while in solids it's not the case, and we can study conductivity by itself.

An example is a building wall, one side of which towards the outside having temperature of the air outside, while inner surface of the wall having room temperature. The heat energy constantly flows from a warmer surface of the wall to the opposite cooler one with some rate of flow that depends on the thermal conductivity of the wall material. The wall material with higher level of thermal conductivity will transfer more heat energy to a cooler side within unit of time per unit of area, which is usually not a desirable property of the building walls.

The mechanism of heat transfer through conductivity can be explained as follows.
Imagine two objects (or two parts of the same object), a hot one with higher intensity of molecular movement and a cold one with lower intensity level of molecular movement, that touch each other along some surface, while completely insulated from heat around them. For example, you put a cold metal spoon into a styrofoam cup with hot tea.

Molecules of a hot object are hitting the molecules of a cold one, thus forcing the molecules of a cold object to move faster. These faster molecules of a cold object, in turn, hit their neighbors, forcing them to move faster. This process of transferring heat energy through contacting surfaces continues until the intensity of molecular movement gradually equalizes on average. A hot object will lose some energy of molecular movement, while a cold one will gain it. As a result, the temperatures of both will equalize.

Since the heat energy is a kinetic energy of molecular movement, that is a sort of mechanical energy, we expect that the total amount of energy for an isolated system will remain constant, whatever a hot object loses in its kinetic energy of molecular movement will be gained by a cold object. The total amount of heat energy will remain the same.

If you put a silver spoon into a cup of hot tea, it will heat up faster than a spoon made of steel, which, in turn, will heat up much faster then a spoon made of wood. The reason for this is that the thermal conductivity of different materials is different.

We can experimentally measure the thermal conductivity of different solids by having a standard rod of any solid material at certain starting temperature and heating its one end by bringing it to contact with some hot object. Measuring the temperature on the other end after different time intervals will give us a picture of growing temperature.

Some materials with higher thermal conductivity will have the temperature at the opposite end of a rod growing faster than in case of other materials.
Metals have much higher heat conductivity then plastic or wood, for example. That's why the handle of a tea kettle is usually made of plastic or wood. Diamonds have one of the highest thermal conductivity, even higher than silver.

More precise definition of thermal conductivity is related to a concept of heat flux(sometimes, called heat flow density or thermal flux, or thermal flow density). Heat flux is an amount of heat energy flowing through a unit of area during a unit of time.

Let's examine how heat flows through a building wall made of some uniform material from a warm room to cold air outside the building.
Assume, the room temperature is T_room and the cold air outside the building has temperature T_air . If the thickness of a wall is L, the temperature inside the wall T(x), as a function of the distance x from the surface facing outside, gradually changes from T(0)=T_air to T(L)=T_room.

It is intuitively understandable and experimentally confirmed that amount of heat energy flowing through a unit of area of such a wall during a unit of time (heat conductivity of a wall) is proportional to a difference between temperatures T_roomand T_air and inversely proportional to a thickness of a wall L:
q = −k·(T_room − T_air ) / L
(negative sign is used because the flow of heat is opposite to a direction of temperature growth).

The situation with heat fluxmight be compared with a water flow down a river between two points A and B. The difference in levels above the sea level of these points is similar to a difference in temperature between the inside and outside walls of a building. The distance between points A and B is similar to a thickness of a wall. It's reasonable to assume that amount of water flowing through a unit of area in a unit of time will be proportional to a difference between the levels of points A and B above the sea level and inversely proportional to a distance between these two points.

To make this definition of the heat flux more precise and independent of the way how the heat flows inside the wall, let's consider a thin slice of wall parallel to both sides from a point at distance x from the outdoor cold side to a point at distance x+Δx.
The heat flow through this thin slice of a wall, as a function of distance x, can be expressed similarly to the above:
q(x)=−k·[T(x+Δx)−T(x)] /Δx

Next step is, obviously, to reduce the thickness of the slice by making Δx infinitesimal, that is Δx→0, which leads to the following definition of the heat flux:
q(x) = −k·dT(x)/dx
This definition was formulated by Fourier in 1822 and is called Fourier's law of thermal conduction.
The coefficient k is called thermal conductivity.

To find the amount of heat Q(x,A) going through an area Aat distance x from the outside wall during a unit of time, we have to multiply the heat flux by an area:
Q(x,A) = −k·A·dT(x)/dx
As in a case of water flow along the river, if the temperature is linearly dependent on the distance from the outside wall, the derivative is constant and the flow of heat is constant. But, if the wall material is uneven, like in case of a river bed not being a straight line down, the heat flow rate will change.