FREE CONVECTION
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7.2 Features and Parameters of Free Convection
1) Driving Force
In general, two conditions are required for fluids to be
set in motion in free convection.
- acceleration field (gravity)
- density gradient (temperature gradient)
2) Governing Parameters.
Two parameters play a key role in the determination of
the Nusselt number in free convection:
- the Grashof number
- the Prandtl number
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7.2 Features and Parameters of Free Convection
Grashof number
Coefficient of thermal expansion or Compressibility factor
For ideal gases it is given by
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7.2 Features and Parameters of Free Convection
Rayleigh number
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7.2 Features and Parameters of Free Convection
3) Boundary Layer
Boundary layer approximations for free convection are
valid for
Ra x  10
4
4) Transition from Laminar to Turbulent Flow
by
For vertical plates the transition Rayleigh number is given
Ra xt  10
9
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7.2 Features and Parameters of Free Convection
5) External vs. Enclosure Free Convection.
In external free convection a surface is immersed in a
fluid of infinite extent.
Enclosure free convection takes place inside closed
volumetric regions.
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7.2 Features and Parameters of Free Convection
6) Analytic Solutions.
Analytic solutions require the simultaneous integration of
the continuity, momentum and energy equations.
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7.3 Governing Equations
Analysis of free convection is usually based on following
approxi-mations:
(1)Density is assumed constant except in evaluating gravity
forces.
(2) The Boussinesq approximation which relates density change
to temperature change is used in formulating buoyancy force
in the momentum equation.
(3) Dissipation effect is neglected in the energy equation.
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7.3 Governing Equations
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7.3 Governing Equations
7.3.1 Boundary Layer Equations
From Eq.(7.6) reduce to
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7.3 Governing Equations
Applied (c) for Eq.(7.5), thus the x-component of the
Navier-Stokes equations simplifies to
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7.4 Laminar Free Convection over a Vertical Plate: Uniform Surface
Temperature
7.4.1 Assumptions.
(1) Continuum
(2) Newtonian
(3) Steady state
(4) Laminar flow
(5) Two-dimensional
(6) Constant properties
(7) Boussinesq approximation
(8) Uniform surface
(9) Uniform ambient temperature
(10) Vertical plate
(11) Negligible dissipation
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7.4 Laminar Free Convection over a Vertical Plate: Uniform Surface
Temperature
7.4.2 Governing Equations.
Based on the above assumptions we get the governing
equations
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7.4 Laminar Free Convection over a Vertical Plate: Uniform Surface
Temperature
7.4.3 Boundary Conditions
No slip
No dissipation
No density change
No density change
T= Ts
T= T
T= T
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7.4 Laminar Free Convection over a Vertical Plate: Uniform Surface
Temperature
7.4.4 Similarity Transformation
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7.4 Laminar Free Convection over a Vertical Plate: Uniform Surface
Temperature
By stream function y
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7.4 Laminar Free Convection over a Vertical Plate: Uniform Surface
Temperature
Using stream function y of Blasius solution
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7.4 Laminar Free Convection over a Vertical Plate: Uniform Surface
Temperature
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7.4 Laminar Free Convection over a Vertical Plate: Uniform Surface
Temperature
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7.4 Laminar Free Convection over a Vertical Plate: Uniform Surface
Temperature
7.4.5 Solution
Solution by numerical
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7.4 Laminar Free Convection over a Vertical Plate: Uniform Surface
Temperature
7.4.6 Heat Transfer Coefficient and Nusselt Number.
Based on Fourier’s law and Newton’s law
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7.4 Laminar Free Convection over a Vertical Plate: Uniform Surface
Temperature
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7.4 Laminar Free Convection over a Vertical Plate: Uniform Surface
Temperature
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7.4 Laminar Free Convection over a Vertical Plate: Uniform Surface
Temperature
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Example 7.1 Vertical at Uniform Surface Temperature
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Example 7.1 Vertical at Uniform Surface Temperature
Solution.
Assumptions
- Continuum
- Newtonian fluid
- Steady state
- Boussinesq approximations
- Two – dimensional
- Laminar flow
- Flat plate
- Uniform surface temperature
- No dissipation
- No radiation
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Example 7.1 Vertical at Uniform Surface Temperature
The properties are evaluated at the temperature
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Example 7.1 Vertical at Uniform Surface Temperature
Check laminar or turbulent by Rayleigh number
Thus the flow is laminar.
Axial velocity u is given by (7.20).
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Example 7.1 Vertical at Uniform Surface Temperature
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Example 7.1 Vertical at Uniform Surface Temperature
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Example 7.1 Vertical at Uniform Surface Temperature
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Example 7.1 Vertical at Uniform Surface Temperature
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Example 7.1 Vertical at Uniform Surface Temperature
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Example 7.1 Vertical at Uniform Surface Temperature
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Example 7.1 Vertical at Uniform Surface Temperature
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