18.3 Extra Example - Fourier\'s Law of Heat Conduction: Fourier\'s Law of Heat C

Question

18.3 Extra Example - Fourier's Law of Heat Conduction:

Fourier's Law of Heat Conduction The heat (energy) flow down a temperature gradient through stationary matter is given by Fourier's law of heat conduction as J = -k T where J heat current density = heat flowing per unit time, per unit area T temperature gradient k thermal conductivity of the material We can think of - T as a thermal "driving force" which drives thermal energy. The total heat current I flowing across an area A is I = integral J dA where dA = n dA and n is a unit vector along the normal to the surface area. "Steady State" heat flow occurs when the heat flowing across an area is the same at all times, i.e., I = constant. Derive expressions for the steady state heat current I for the following geometries: heat flow in one-dimension across an area A, with no heat source radial heat flow in a cylinder, with no heat source radial heat flow in a sphere, with no heat source radial heat flow in a cylinder, with a heat source S = heat generated per unit time, per unit volume

Explanation / Answer

In thermodynamics, we considered the amount of heat transfer as a system undergoes a process from one equilibrium state to another. Thermodynamics gives no indication of how long the process takes. In heat transfer, we are more concerned about the rate of heat transfer. The basic requirement for heat transfer is the presence of a temperature difference. The temperature difference is the driving force for heat transfer, just as voltage difference for electrical current. The total amount of heat transfer Q during a time interval can be determined from

: Q Q dt kJ t 0 The rate of heat transfer per unit area is called heat flux, and the average heat flux on a surface is expressed as 2 W / m A Q

Steady Heat Conduction in Plane Walls Conduction is the transfer of energy from the more energetic particles of a substance to the adjacent less energetic ones as result of interactions between the particles.

Consider steady conduction through a large plane wall of thickness x = L and surface area A. The temperature difference across the wall is T = T2 – T1.   Note that heat transfer is the only energy interaction; the energy balance for the wall can be expressed: dt dE Q Q wall in out For steadystate operation,   Q Q const. in out It has been experimentally observed that the rate of heat conduction through a layer is proportional to the temperature difference across the layer and the heat transfer area, but it is inversely proportional to the thickness of the layer. (surface area)(temperature difference) rate of heat transfer

Q = KA dT /dx

Heat Conduction in Cylinders and Spheres Steady state heat transfer through pipes is in the normal direction to the wall surface (no significant heat transfer occurs in other directions). Therefore, the heat transfer can be modeled as steadystate and onedimensional, and the temperature of the pipe will depend only on the radial direction, T = T (r). Since, there is no heat generation in the layer and thermal conductivity is constant,

the Fourier law becomes: Q cylind = -KA dT /dr W   

where A = 2 Pixrxl

Q = T1 -T2 / R cyl

Rcyl = ln r2 /r1 / 2 pi k L

where Rcyl is the conduction resistance of the cylinder layer. Following the analysis above, the conduction resistance for the spherical layer can be found:

Q sph = T1 -T2 / R sph

Rsph = r2 - r1 / 4pir1r2 k

The convection resistance remains the same in both cylindrical and spherical coordinates, Rconv = 1/hA. However, note that the surface area A = 2rL (cylindrical) and A = 4r 2 (spherical) are functions of radius.

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