HEAT CONVECTION FROM A SPHERE IN AN OSCILLATING STREAM

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b
y
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APPLICATIONS

• Heat and mass transfer rates are enhanced by the
  oscillation of the surrounding fluid. Useful in
  combustion, drying and the passage of sound waves
  through particulate systems.
• Particle-laden flows, Brownian motion, suspension
  rheometry, colloidal suspension, and particle
  motion in filters.

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MODES OF HEAT TRANSFER

• RADIATION
• CONDUCTION
• CONVECTION (forced free)

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GOVERNING EQUATIONS
Conservation of momentum
where
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Conservation of mass
First law of thermodynamics
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VECTOR RELATIONS
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Define
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then,
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and,
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SPHERICAL COORDINATES
r
a
SCALE FACTORS
g
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VARIABLES
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DIMENSIONLESS NUMBERS
kinematic viscosity
thermal diffusivity
volumetric expansion coef.
frequency of oscillation
surface temperature
far-field temperature
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USING THE SPHERICAL COORDINATE SYSTEM, THE
EQUATIONS REDUCE TO
(1)
(2)
(3)
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BOUNDARY CONDITIONS
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• Equations are two dimensional, time-dependent,
  nonlinear, coupled, and of infinite domain.
• No explicit boundary conditions for vorticity
  on the surface.
• Difficulties with finite- differences such as,
  indeterminate forms, etc..

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THE METHOD OF SOLUTION
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HOW ?!
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INTEGRALS NEEDED !!
and in general,
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After Mavromatis Alassar
where
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where
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Saalschutzs Theorem
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3-j SYMBOLS
Represent the probability amplitude that three
angular momentum j1, j2, and j3 with projections
m1, m2, and m3 are coupled to yield zero angular
momentum. They are related to Clebsch-Gordan
coefficients (C) by
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Clebsch-Gordan Coefficients
where
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(1)
(2)
(3)
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where,
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MODES BOUNDARY CONDITIONS
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NUMERICAL METHOD
CRANK-NICOLSON F.D. SCHEME
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SPECIALIZED STEP-BY-STEP METHOD
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PHYSICAL PARAMETERS
Nusselt Number
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Drag Coefficient
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Pressure Coefficient
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VALIDATION
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SOME RESULTS
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0.2500
0.0000
0.2300
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0.2625
0.2725
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0.2750
0.3125
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0.5550
0.5000
0.5650
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0.7500
0.6000
0.7250
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FUTURE RESEARCH

• Heat transfer from a sphere in a spinning
  infinite fluid.
• Heat transfer from other geometries such as
  oblate and prolate spheroids.

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SPHEROIDS
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Stream Function
Vorticity
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Energy
Boundary Conditions
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A NOTE ON THE PROBLEM OF SPINNING STREAM
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