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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

P M V Subbarao

Associate Professor

Mechanical Engineering Department

IIT Delhi

More Inventions through Mathematical Modeling…


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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:

T

The heat flux across the slab

x


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Local Heat flux in a slab:

Global heat transfer rate:


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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:


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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?


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Most General form of Fourier Law of Conduction

k can be a vector !!!!

We are at cross roads !!!!!


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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.



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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.






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Mass Transfer Conduction……

  • 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


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Various Mass Transfer Phenomenon Conduction……

Evaporation:

Drying

Baking

Frying

Boiling

Diffusion:

Salt through cheese.

Smoke through meat.

Curing solution through meat

Not mass transfer:

Moving a fluid from one place to another


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Osmosis Conduction……

  • 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.