Transient Heat Conduction.pdf
Jan 07, 2016
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Transient Heat Conduction
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Lumped System Analysis
Some bodies are essentially isothermal and can be treated as a lump system.
For an isothermal solid, the energy balance for the time interval dt can be expressed as:
Heat Transfer into the body
during dt
= The increase in
the energy of the body during dt
SOLID BODY m = mass
V = volume = density
Ti = initial temperature T = T(t)
h T
As
=
= >
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Integrating from time,t = 0 (T = Ti) to t = t (T = T(t))
Taking the exponential of both sides and rearranging:
b is a positive quantity whose dimension is (time)-1, and is called the time constant.
= = = =
=
=
=
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The temperature of body T(t) at time t can be determined, or alternatively the time required for the temperature to reach specified temperature (T(t)).
The temperature of a body approaches the ambient temperature, T exponentially.
The temperature of the body changes rapidly in the beginning and rather slowly later on.
A large value of b indicates that the body approaches T in a short time.
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Rate of Convection Heat Transfer The rate of convection heat transfer between the body
and the ambient can be determined from Newtons law of cooling.
The total heat transfer between the body and the ambient over the time internal 0 to t is simple the change on the energy content of the body.
The maximum heat transfer between the body and its surroundings (when the body reaches T)
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Criteria for Lumped System Analysis
Lumped system analysis is not always appropriate:
Biot number (Bi) =
Characteristic Length, =
Bi =
1 =
=
Lumped system analysis assumes a uniform temperature distribution throughout the body, which is true only when the thermal resistance of the body to heat conduction is zero.
The smaller the Bi, the more accurate the analysis.
It is generally accepted that lump system analysis is applicable if:
