These notes for GATE Mechanical Engineering describe all the important fundamentals for heat conduction, from the subject of Heat Transfer. 

Heat transfer: It is defined as the transmission of heat energy from one region to another because of a temperature gradient. There are three modes of heat transfer, (i) Conduction, (ii) Convection and (iii) Radiation. Heat conduction is majorly due to direct physical contact of a higher temperature medium with a lower temperature medium of a same or a different body. Heat convection is due to the property of moving matter (can be natural or forced) to carry heat from a higher temperature region to a lower temperature region. Heat radiation is due to the property of the matter to emit and absorb electromagnetic radiation, even without a carrying medium.

There are two very important terms one should understand while studying heat transfer, and the same can be applied to the above mentioned three modes :

1) Steady state heat transfer: In such a heat transfer, the temperature of the body does not vary with respect to time, but it does vary with respect to the position, i.e.  and .

2) Unsteady state heat transfer: In such a heat transfer, the temperature of the body varies with respect to both time and position, i.e.  and .

Thermal conductivity:

It is a property specific to the material which is carrying heat. It is defined as the amount of energy that can be conducted through a body or medium of unit thickness and unit area in unit time for a temperature difference of .

  • Thermal conductivity of metals and alloys decrease with rise in temperature.
  • Thermal conductivity of gases increase with rise in temperature.

Fourier's law of heat conduction:

This is the most fundamental laws for heat conduction. This law states that the rate of heat conduction is directly proportional to the surface area (A) in a direction perpendicular to the surface area and temperature gradient (dT/dx). Refer the figure given below.

or 

In the above expression, is called the thermal conductivity of the material through which heat is flowing. Units of thermal conductivity are kJ/s-mK or kW/mK.

Thermal resistance:

Heat flow can be considered analogous to electric current flow. As per Ohm's law for electricity, the current I flowing through a wall of resistance R, as shown in the figure, is proportional to the potential difference V across it. 

 

Recognize the following terms and then find the analogous terms in heat conduction:

1) Flow quantity = current (I)  heat (Q)

2) Potential difference or driving force = V  temperature difference (dT)

Where  = Thermal resistance to the heat flow offered by the material of the wall. 

Thermal resistance combinations:

(a) Series combination: Refer the figure below. 

Let  = temperature of interface and assuming that the rate of heat flow is equal through both the walls in series.

When there are multiple thermal resistances in series, we can write, 

  and 

(b) Parallel combination: Refer the figure below for a parallel combination.

Where,  and 

Also, 

(c) Thermal contact resistance: 

Due to surface roughness, the contact surfaces touch only at discrete locations. Due to this, the heat flow area at the interface becomes very small as compared to the geometric area of the face which causes large resistance to the heat flow at the interface. This resistance is termed as thermal contact resistance. The result of thermal contact resistance is that there is a drop in the temperature at the interface. 

The value of thermal contact resistance depends on the metals involved, the roughness of the surfaces, the contact pressure and temperature and the matter occupying the void spaces. 

This brings us to the end of the notes on Heat Conduction basics. In the next article, we will look at the Heat conduction through a cylinder. 

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Please share some practice questions also. 

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