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FREE CONVECTION

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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

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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 ConvectionRayleigh number

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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

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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.

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7.2 Features and Parameters of Free Convection6) 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 Equations7.3.1 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 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

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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

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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

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7.4 Laminar Free Convection over a Vertical Plate: Uniform Surface Temperature7.4.4 Similarity Transformation

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7.4 Laminar Free Convection over a Vertical Plate: Uniform Surface Temperature

By stream function

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7.4 Laminar Free Convection over a Vertical Plate: Uniform Surface Temperature

Using stream function 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 Temperature7.4.5 Solution

Solution by numerical

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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

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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 TemperatureSolution.

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 TemperatureThe properties are evaluated at the temperature

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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).

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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