Fourier's law of heat conduction

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Presentation transcript:

Fourier's law of heat conduction

Steady state heat conduction Steady state means that temperature in an object may vary by location but it does not change with time.

Fourier's law describes the rate of heat conduction in a solid Boundary Conditions x = x1, T = T1 x = x2 , T = T2 Separating the variables, Integrating from x1 to x (some interior location within the slab)

Where L= thickness of the slab Thermal Resistance concept: In the flow of electricity, the electrical resistance is determined from the ratio of electric potential divided by the electric current. Similarly, we may consider heat resistance as a ratio of the driving potential (temperature difference) divided by the rate of heat transfer. Thus,

HEAT CONDUCTION IN MULTILAYERED SYSTEMS COMPOSITE RECTANGULAR WALL (IN SERIES) In a single layer, the rate of heat transfer is:

Using the Resistance Concept:

Conduction in cylindrical objects Fourier's Law in cylindrical coordinates 𝑞 𝑟 =−𝑘(2𝜋𝑟𝐿) 𝑑𝑇 𝑑𝑟 Boundary Conditions T =Ti at r = ri T=To at r = r0

Separating variables: Integrating :

OVERALL HEAT TRANSFER COEFFICIENT Heat transfer to the internal wall Heat transfer through the wall Where Alm

Outside the wall By summing the three equation 1,2, and 3 the result is :

Critical thickness for insulation

COMPOSITE CYLINDRICAL TUBE for a single-layer pipe: and resistance due to conduction Thermal resistance diagram:

Heat Conduction in a Spherical Shell Fourier's Law in radial coordinates Substituting the area of a sphere Integrating, between r = r1 and r2, and T1 and T2

The thermal resistance is expressed as

Conduction with internal heat generation In some cases it energy may be produced in an identical shape inside the system like electrical heaters or in respiration in fresh food products such as fruits and vegetables.

And for a pure conductive material Rate of heat in + rate of heat generation = rate of heat out + rate of heat accumulation By taking the limit And for a pure conductive material

Note that if the heat generation zero then : By integration Then we get

At x=-L, T=Tw And at x=L, T=Tw And by solving 1 and 2 At x=0, To=center temperature:

Total heat at both sides =total heat generated at steady state case.

Home work should be submitted after 6 days from the lecture. Home work #2 Temperature change with change of distance from the center of generation Draw the relationship between temperature and distance x from the center of the wall knowing that: Ta q' L H k 25 5 1 10 2 °C w/m3 m w/m2.°k w/m.°k Home work should be submitted after 6 days from the lecture.