Unsteady State Heat Conduction. Scope One dimensional conduction Transient condition (Unsteady...
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Transcript of Unsteady State Heat Conduction. Scope One dimensional conduction Transient condition (Unsteady...
![Page 1: Unsteady State Heat Conduction. Scope One dimensional conduction Transient condition (Unsteady state) –Temperature as a function of time and position.](https://reader030.fdocuments.in/reader030/viewer/2022012823/5697bfc61a28abf838ca7043/html5/thumbnails/1.jpg)
Unsteady State Heat Conduction
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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
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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)
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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
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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
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• s is the thermal time constant– A product of the resistance to convection heat
transfer and lumped thermal capacitance
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Heat transferred
0
[ ]t
q t hA T t T
Q q t dt
Derive Q for body which lump system analysis is valid…
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Biot number and Characteristic length
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Compute the Biot Number
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Compute the Biot Number
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Derive the following equation:
Fourier Number (dimensionless “time”)
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Fourier Number (Fo)
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Time of death analysis
Ans. 12.2 hr
What if the BIOT number is greater than 0.1?
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Average Temperature Analysis
(McCabe and Smith, 5th ed)
slab
Sphere
Infinite long cylinder
Sphere
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Heat transferred using the average temperature
T p p ave aQ mc T mc T T
Slab
Infinite long cylinder
Sphere
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Temperature (position,time)Local Temperature Analysis
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Geometric and thermal symmetry
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Transient Temperature ChartsLocal 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 …
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ChartSample Heisler charts
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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
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Complementary error function
( ) 1 ( )erf x erfc x
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(McCabe and Smith, 5th ed)
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Penetration distance
• The distance where the temperature change has reached about 1% of initial change in surface temp.
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Heat transferred
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Minimum burial depth of water pipes to avoid freezing
Answer: about 0.80 m
Determine the penetration distance for this condition…. (about 4 m?)
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Transient heat conduction in multi-dimensional system
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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.
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Total transient heat transfer
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Exercises
Answer: about 15 minutes
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Exercises