Heat Conduction: Conditions and Mechanism | Thermal Engineering

In this article we will discuss about the conditions and mechanism of heat conduction.

Conditions of Heat Conduction:

The general heat conduction equation is of first order in time co-ordinates, and of second order in spatial co-ordinates. For the solution of single integration one constant is required, and for the solution of each double integration two constants are required. Obviously for the complete solution of general conduction equation, seven integration constants are needed. These constants are evaluated through a set of initial and boundary conditions.

1. Initial Conditions:

The initial conditions describe the temperature distribution in a medium at the initial moment of time, and these are needed only for the transient (time-dependent) problems.

The initial conditions can be expressed as:

At τ = 0; t = t (x, y, z) … (2.31 a)

For a uniform initial temperature distribution, a simple but typical form of the above identity can be recast as-

At τ = 0; t = t₀ = constant … (2.31 b)

2. Boundary Conditions:

The boundary conditions refer to physical conditions existing at the boundaries of the medium, and specify the temperature or the heat flow at the surface of the body.

The Typical Boundary Conditions are:

(i) Prescribed Surface Temperature:

The temperature distribution t_s is prescribed at a bounding surface for each moment of time-

t_s = t(x, y, z, τ) …(2.32)

For a boundary condition of this type pertaining to slab, we have-

 

(ii) Prescribed Heat Flux:

The heat flux is prescribed at the boundary surface, and is expressed as:

Here ∂t/∂x = 0 at x = 0 describes an insulated or adiabatic boundary. Such a condition can also exist at the plane of symmetry.

(iii) Convective Condition:

This condition is encountered at a solid boundary when there is equality between heat transfer to the surface by conduction and that leaving the surface convection.

 

The convective boundary conditions described above can be expressed in the compact from as-

where h_i is the specific convection coefficient, F_i is the prescribed function and ∂t/∂n_i is the temperature derivative at the bounding surface in the direction of outward normal.

It is rather difficult to achieve solution of general heat conduction in closed form due to complex nature of the problem. Finite difference and finite element types of numerical techniques are available and are used for the approximate solution of heat conduction problems.

Mechanism of Heat Conduction:

In insulators (glass, wood, asbestos) conduction of heat takes place due to vibration of atoms about their mean positions. When heat is given to one part of an insulating substance, atoms belonging to that part are put in a violent state of agitation and start vibrating with greater amplitudes.

Consequently these more active particles collide with less active atoms lying next to them. During collision, the less active atoms also get excited i.e., thermal energy is imparted to them. The process is repeated layer after layer of molecules/atoms until the other part of the insulator is reached.

In metals besides atomic vibrations, there are large number of free electrons which also participate in the process of heat conduction. When a temperature difference exists between the different parts of the metal, a general drift of these free electrons occurs in the direction of decreasing temperature. It is this drift of free electrons which makes the metals so much better as conductors than other solids. These free electrons account for the observed proportionality between the thermal and electrical conductivities of pure metals.

The electrons do not contribute to heat conductivity in insulators. Electrons in an insulators are not free but fixed in the valence band. According to band structure of solids, the energy gap between valence band and conduction band is quite large and electrons cannot move to conduction band and contribute towards heat as well as electrical conductivity.

Insulators have low value of thermal conductivity due to their porosity, which may contain air.

Many factors are known to influence the thermal conductivity of a material such as:

i. Chemical composition of the substance or substances of which it is composed,

ii. Gaseous, liquid and solid phase in which the substance exists,

iii. Crystalline, amorphous and porous structure of the substance,

iv. Temperature and pressure to which the substance is subjected,

v. Homogeneous or non-homogeneous character of the material.

The factors with the greatest influence are the chemical composition, phase change and temperature. For a particular material, only the temperature effect has to be accounted for.

Generally a liquid is a better conductor than a gas, and a solid is a better conductor than a liquid. This aspect can be best illustrated by considering the three phases of mercury.

a. Mercury is solid at 193°C and has a thermal conductivity of 48 W/m-deg.

b. At 0°C, mercury becomes liquid and its thermal conductivity drops to 8.0 W/m-deg.

c. Mercury acquires the gaseous phase at 200°C and then has a thermal conductivity as low as 0.034 W/m-deg.

A partial explanation to this aspect stems from the fact that while in a gaseous state, the molecules of a substance are spaced relatively far apart and their motion is random. Obviously then the energy transfer by molecular impact is much more slow than in the case of a liquid where the motion is still random but in which the molecules are more closely packed.

The same is true concerning the difference between the thermal conductivity of the liquid and solid phases. However, many other factors become important when the substance transforms to solid state.

Listed below are the applications where poor conductivity of air restricts the heat transmission by conduction:

(i) Eskimos make double walled glass houses; air is enclosed between the walls and that reduces the outflow of heat from the inside of houses.

(ii) Woollen fibres are rough and hence have fine pores filled with air. Both wool and air are bad conductors of heat and do not allow the body heat to flow to the atmosphere.

(iii) Two thin blankets are warmer than a single blanket of double the thickness because the two blankets enclose between them a layer of air. A simple blanket of double the thickness does not have air entrapped in it, and so it does not provide as good an insulation as the two thin blankets,

(iv) Birds often swell their feathers to enclose air and thus prevent the outflow of body heat.