Upload
nelson-ross
View
220
Download
0
Embed Size (px)
DESCRIPTION
Sarthit Toolthaisong 7.2 Features and Parameters of Free Convection Grashof number Coefficient of thermal expansion or Compressibility factor For ideal gases it is given by
Citation preview
Sarthit Toolthaisong
FREE CONVECTION
Sarthit Toolthaisong
7.2 Features and Parameters of Free Convection1) 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
Sarthit Toolthaisong
7.2 Features and Parameters of Free Convection Grashof number
Coefficient of thermal expansion or Compressibility factor
For ideal gases it is given by
Sarthit Toolthaisong
7.2 Features and Parameters of Free ConvectionRayleigh number
Sarthit Toolthaisong
7.2 Features and Parameters of Free Convection3) Boundary Layer
Boundary layer approximations for free convection are valid for
4) Transition from Laminar to Turbulent Flow
For vertical plates the transition Rayleigh number is given by
410xRa
910xtRa
Sarthit Toolthaisong
7.2 Features and Parameters of Free Convection5) 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.
Sarthit Toolthaisong
7.2 Features and Parameters of Free Convection6) Analytic Solutions.
Analytic solutions require the simultaneous integration of the continuity, momentum and energy equations.
Sarthit Toolthaisong
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.
Sarthit Toolthaisong
7.3 Governing Equations
Sarthit Toolthaisong
7.3 Governing Equations7.3.1 Boundary Layer Equations
From Eq.(7.6) reduce to
Sarthit Toolthaisong
7.3 Governing Equations
Applied (c) for Eq.(7.5), thus the x-component of the Navier-Stokes equations simplifies to
Sarthit Toolthaisong
7.4 Laminar Free Convection over a Vertical Plate: Uniform Surface Temperature7.4.1 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
Sarthit Toolthaisong
7.4 Laminar Free Convection over a Vertical Plate: Uniform Surface Temperature7.4.2 Governing Equations.
Based on the above assumptions we get the governing equations
Sarthit Toolthaisong
7.4 Laminar Free Convection over a Vertical Plate: Uniform Surface Temperature7.4.3 Boundary Conditions No slip
No dissipation
No density change
No density change
T= Ts
T= T
T= T
Sarthit Toolthaisong
7.4 Laminar Free Convection over a Vertical Plate: Uniform Surface Temperature7.4.4 Similarity Transformation
Sarthit Toolthaisong
7.4 Laminar Free Convection over a Vertical Plate: Uniform Surface Temperature
By stream function
Sarthit Toolthaisong
7.4 Laminar Free Convection over a Vertical Plate: Uniform Surface Temperature
Using stream function of Blasius solution
Sarthit Toolthaisong
7.4 Laminar Free Convection over a Vertical Plate: Uniform Surface Temperature
Sarthit Toolthaisong
7.4 Laminar Free Convection over a Vertical Plate: Uniform Surface Temperature
Sarthit Toolthaisong
7.4 Laminar Free Convection over a Vertical Plate: Uniform Surface Temperature7.4.5 Solution
Solution by numerical
Sarthit Toolthaisong
7.4 Laminar Free Convection over a Vertical Plate: Uniform Surface Temperature7.4.6 Heat Transfer Coefficient and Nusselt Number.
Based on Fourier’s law and Newton’s law
Sarthit Toolthaisong
7.4 Laminar Free Convection over a Vertical Plate: Uniform Surface Temperature
Sarthit Toolthaisong
7.4 Laminar Free Convection over a Vertical Plate: Uniform Surface Temperature
Sarthit Toolthaisong
7.4 Laminar Free Convection over a Vertical Plate: Uniform Surface Temperature
Sarthit Toolthaisong
Example 7.1 Vertical at Uniform Surface Temperature
Sarthit Toolthaisong
Example 7.1 Vertical at Uniform Surface TemperatureSolution.
Assumptions - Continuum
- Newtonian fluid- Steady state- Boussinesq approximations- Two – dimensional- Laminar flow- Flat plate- Uniform surface temperature- No dissipation- No radiation
Sarthit Toolthaisong
Example 7.1 Vertical at Uniform Surface TemperatureThe properties are evaluated at the temperature
Sarthit Toolthaisong
Example 7.1 Vertical at Uniform Surface TemperatureCheck laminar or turbulent by Rayleigh number
Thus the flow is laminar.Axial velocity u is given by (7.20).
Sarthit Toolthaisong
Example 7.1 Vertical at Uniform Surface Temperature
Sarthit Toolthaisong
Example 7.1 Vertical at Uniform Surface Temperature
Sarthit Toolthaisong
Example 7.1 Vertical at Uniform Surface Temperature
Sarthit Toolthaisong
Example 7.1 Vertical at Uniform Surface Temperature
Sarthit Toolthaisong
Example 7.1 Vertical at Uniform Surface Temperature
Sarthit Toolthaisong
Example 7.1 Vertical at Uniform Surface Temperature
Sarthit Toolthaisong
Example 7.1 Vertical at Uniform Surface Temperature