Unsteady State Conduction

UNSTEADY-STATE HEAT CONDUCTION Applications where rate/duration of heating/cooling is a Design Parameter

General Conduction EquationFor Rectangular Geometry: The general form of these equations in multidimensions is:

Transient-conductionTemperature will vary with location within a system and with time.Temperature and rate heat transfer variation of a system are dependent on its internal resistance and surface resistance. If we have a ball with initial temperature of T0 and it is left in fluid at T. Heat is transferred by convection at the surface. As the surface temperature decreases, heat is transferred from the center of the ball to the surface, then to the fluid.

Thermal Analysis of Transient SystemRate of Change in energy of system = Rate of Heat transferRate of Heat Transfer = Rate of Convection /radiation at surface =Rate of Conduction transfer in the system

At any instant:

For Convection heat transfer across surface.For Radiation heat transfer across surface.

Characterization of System & EnvironmentIf the system conductivity is high or the system volume is small, the temperature response is very fast.The response has to do with what is called the internal resistance of the material. If the convection/radiation coefficient is very high, then the surface temperature of the system almost becomes identical to the fluid temperature quickly. Alternatively, for a low convection/radiation coefficient a large temperature difference exists between the surface and the fluid. The value of the coefficient controls what is known as the surface resistance to heat transfer.The instantaneous temperature variation within the system is dependent on the internal and surface resistances. The larger internal resistance or the smaller surface resistance, the larger temperature variation within the system, and vice versa.

Total thermal resistance of the system: RtotT0T

A Biot number is defined as:

Thermal Characterization of System & Environment using Biot number

System with negligible internal resistance For this case Bi 1.0.and the temperature profile within the body is quite uniform. The rate of change in internal energy of the body is equal to the rate of heat taken away from the surface by convection:

Rearranging:Integrating to any time interval:

Define, Thermal Time Constant

The total energy transferred in time t

The Fourier number (Fo) or Fourier modulus, named after Joseph Fourier, is a dimensionless number that characterizes transient behavior of a system. Conceptually, it is the ratio of the heat conduction rate to the rate of thermal energy storage. It is defined as:

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