Unsteady State Heat Conduction

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Presentation transcript:

Unsteady State Heat Conduction

Scope One dimensional conduction Transient condition (Unsteady state) Temperature as a function of time and position Analysis Lumped Systems Average temperature analysis Transient heat conduction in infinite and semi-infinite solids, Convective boundary conditions

One dimensional transient heat conduction Thermal diffusivity T(x,t), 2 BC and 1 IC Analytical method using Method of Separation of Variables Numerically by Finite difference or Finite element method By Charts and Graphs Lumped analysis: T(t)

Lumped System Analysis Bodies behave like a “lump” whose interior temp. remains uniform at all time during the heat transfer process Example A hot copper ball coming from an oven

Validity of Lumped Analysis Assumed to cases where temperature gradient within the solid is small Occurs if the thermal resistance within the solid is very small compared to the external thermal resistance Solids with very high thermal diffusivity Applicable if Biot Number is less than 0.1

s is the thermal time constant A product of the resistance to convection heat transfer and lumped thermal capacitance

Heat transferred Derive Q for body which lump system analysis is valid…

Biot number and Characteristic length

Compute the Biot Number

Compute the Biot Number

Derive the following equation: Fourier Number (dimensionless “time”)

Fourier Number (Fo)

Time of death analysis Ans. 12.2 hr What if the BIOT number is greater than 0.1?

Average Temperature Analysis slab Infinite long cylinder Sphere Sphere (McCabe and Smith, 5th ed)

Heat transferred using the average temperature Slab Infinite long cylinder Sphere

Temperature (position,time) Local Temperature Analysis Temperature (position,time)

Geometric and thermal symmetry

Transient Temperature Charts Local temperature is a function of position and time (Heisler and Grober charts). Check for such charts in ChE Handbook/Unit Ops book… Always check the parameters to be used in the chart …

Sample Heisler charts Chart

Analysis for semi-infinite solids When solids are heated and cooled such that the temperature changes in the solids are found in the region near one surface only e.g., thick wall, earth surface

Complementary error function

(McCabe and Smith, 5th ed)

Penetration distance The distance where the temperature change has reached about 1% of initial change in surface temp.

Heat transferred

Minimum burial depth of water pipes to avoid freezing Answer: about 0.80 m Determine the penetration distance for this condition…. (about 4 m?)

Transient heat conduction in multi-dimensional system

Superposition approach (Product solution) The solution for a multi-dimensional geometry is the product of the solutions of the one dimensional geometries whose intersection is the multi-dimensional body.

Total transient heat transfer

Exercises Answer: about 15 minutes

Exercises