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Fourier Law of Conduction . P M V Subbarao Associate Professor Mechanical Engineering Department IIT Delhi. More Inventions through Mathematical Modeling…. Statement of Fourier’s Law.

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fourier law of conduction

Fourier Law of Conduction

P M V Subbarao

Associate Professor

Mechanical Engineering Department

IIT Delhi

More Inventions through Mathematical Modeling…

statement of fourier s law
Statement of Fourier’s Law

The (mod of) heat flux, q’’, (the flow of heat per unit area and per unit time), at a point in a medium is directly proportional to the temperature gradient at the point.

Temperature gradient across the slab of thickness Dx:


The heat flux across the slab



Local Heat flux in a slab:

Global heat transfer rate:

mathematical description
Mathematical Description
  • Temperature is a scalar quantity.
  • Gradient of temperature (▼T) will be a vector.
  • Heat flux is defined with direction and Magnitude : A Vector.
  • Mathematically it is possible to represent heat flux as:

Using the principles of vector calculus:

mathematical fantasy to physical existence
Mathematical Fantasy to Physical Existence
  • Will k be same in all directions?
  • Why k cannot be different each direction?
  • Why k cannot be a vector?

Will mathematics approve this ?

What could be the most general acceptable behavior of k, approved by both physics and mathematics?

most general form of fourier law of conduction
Most General form of Fourier Law of Conduction

k can be a vector !!!!

We are at cross roads !!!!!

physically mathematically feasible model
Physically -mathematically Feasible Model
  • Taking both physics and mathematics into consideration, the most feasible model for Fourier’s Law of conduction is:

Thermal conductivity of a general material is a tensor.

fire resistant wood
Fire Resistant Wood
  • Among the assessment properties of wood composite of structural members in building construction, fire performance is important and getting more attention nowadays.
  • A new composite called molded carbon phenolic spheres (CPS),
  • A mixture of sugi wood charcoal powders and phenol formaldehyde resin molded with a hot press is developed by a research group in Japan.
  • The heat due to a fire accident should be thrown out fast outside the building.

Mass Transfer

  • Mass transfer: The transfer of mass (Chemical species) into or out of a substance.
  • The transfer of a chemical compound from one phase to another
  • Examples:
  • Evaporation: liquid → gas
  • Diffusion: high concentration → low concentration

Various Mass Transfer Phenomenon







Salt through cheese.

Smoke through meat.

Curing solution through meat

Not mass transfer:

Moving a fluid from one place to another

  • Osmosis is the net movement of water across a partially permeable membrane from a region of high solvent potential to an area of low solvent potential, up a solute concentration gradient.
  • Osmosis is responsible for the ability of plant roots to suck up water from the soil.
  • There are many fine roots, which have a large surface area, water enters the roots by osmosis, and
  • Generates the pressure required for the water to travel up the plant.
  • Osmosis can also be seen very effectively when potato slices are added to a high concentration of salt solution.
  • The water from inside the potato moves to the salt solution, causing the potato to shrink and to lose its 'turgor pressure'.
  • The more concentrated the salt solution, the bigger the difference in size and weight of the potato slice.
  • For example, freshwater and saltwater aquarium fish placed in water of a different salinity than that they are adapted to will die quickly, and in the case of saltwater fish, rather dramatically.