heat transfer

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# 1 Heat Transfer Su Yongkang School of Mechanical Engineering HEAT TRANSFER CHAPTER 6 Introduction to convection

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heat transfer

Transcript of heat transfer

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Heat Transfer Su Yongkang

School of Mechanical Engineering

HEAT TRANSFER

CHAPTER 6

Introduction to convection

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Heat Transfer Su Yongkang

School of Mechanical Engineering

CH6 – INTRODUCTION

Where we’ve been ……Basic Conduction Heat Transfer Finished

Fourier’s law:

Where we’re going:Begin study of convective heat transfer.

Newton’s law of cooling:

dx

dtkq

tkq

)( TThq s

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Heat Transfer Su Yongkang

School of Mechanical Engineering

Convective transfer problem

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Heat Transfer Su Yongkang

School of Mechanical Engineering

CH6 INTRODUCTION

KEY POINTS THIS CHAPTER• What are the key variables when analyzing

convection heat transfer?

• Review boundary layer concept and significance

• General idea of relationship between velocity and thermal profiles in a boundary layer.

• Effect of laminar versus turbulent flow on heat transfer potential

• Boundary layer similarity

• This chapter will be taught in two lectures: the first includes text book sections §6.1 to 6.4the other includes text book sections §6.5 to 6.10

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Heat Transfer Su Yongkang

School of Mechanical Engineering

Convection overview

• Consider a flat plate of length L, in air flow with velocity u and temperature T

• Local heat flux is: where h is the local heat transfer coefficient

• Total heat transfer rate:

haverage heat transfer coefficient

Determination of ‘h’ will rely on analytical as well as empirical data

)( TThq s

ss A ssA s hdATTdAqq )(

)( TTAhq ss

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Heat Transfer Su Yongkang

School of Mechanical Engineering

Convection overview (Cont’d)

• Same principal applies to any arbitrary shape, not just a flat plate

• Average convection heat transfer coefficient:

So, we need to know how h varies with x, the distance from the leading edge……..

What do you think key parameters that might influence h?

q sdA

ss TA ,

or, for unit width: sA s

s

hdAA

h1

Lhdx

Lh

0

1

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Heat Transfer Su Yongkang

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Key parameters

• Transfer potential: forced flow or free flow

• Phase change: boiling and condensation

• Flow conditions: laminar or turbulent flow

• Geometries: shape, size, position and roughness.

• Properties: density, viscosity, thermal conductivity, specific heat, and so on.

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Heat Transfer Su Yongkang

School of Mechanical Engineering

Example

Given• Experimental results for measured local heat

transfer coefficient h for flow over a flat plate with a rough surface

• where: a = coefficient x = distance from leading edge

– Find expression for average heat transfer coefficient, and the relation of average heat transfer coefficient to the local coefficient

)( 3.0 axxhx

7.0

0

3.0

0

3.0

0.7

1

1

xx

ah

dxxx

adxxa

xh

x

xx

x

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Heat Transfer Su Yongkang

School of Mechanical Engineering

The Convection Boundary Layers

Velocity Boundary Layer

• For fluid flow over a flat plate, which disturbs the fluid flow:– As y: where u is velocity in

x-direction– As y0: (no-slip condition)– The boundary layer thickness is defined as

the value at which:– The boundary layer thickness varies with

x• Shear Stress

• Local friction coefficient

0

y

s y

u

2

2u

C sf

uu

0u

uyu 99.0)(

Dynamic viscosity

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Heat Transfer Su Yongkang

School of Mechanical Engineering

The Convection Boundary Layers

Thermal Boundary Layer

• A hot or cold plate alters the temperature distribution in the air– As y:– As y0:– The thermal boundary layer thickness is

defined as the value at which:

– The thermal boundary layer thickness, t

also varies (increases) with x

99.0)(

TT

yTT

s

airs

TTs

TyT )(

sTyT )(

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Heat Transfer Su Yongkang

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The Convection Boundary Layers

Thermal Boundary Layer (Cont’d)

Heat Flux• Heat flux analogous to shear stress in velocity

boundary layer• Heat flux proportional to the temperature

gradient at the surface,ANDsince u(y=0) =0, energy transfer to/from fluid occurs by conduction only!

• Since thermal boundary layer gets larger along x direction, the temperature gradient changes with x, and therefore

TTs

fluid thermal conductivity

wall temperature gradient

____________________

0

y

fs y

Tkq

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Heat Transfer Su Yongkang

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The Convection Boundary Layers

Thermal Boundary Layer (Cont’d)Heat Flux (Cont’d)• Using Newton’s law of cooling:

We obtain

• While increases with increasing x, temperature gradients in the boundary layer must decrease with increasing x.

• Accordingly, and h decrease with increasing x.

)( TThq s

TT

yTkh

s

yf 0/

sq

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The Convection Boundary Layers

Laminar Versus Turbulent Flow

Characterization of laminar flow• Low amount of “mixing” of fluid within the

boundary layer (smooth flow)

Characterization of turbulent flow• High amount of mixing of fluid within the

boundary layer (irregular flow)• High amount of mixing means increased surface

friction as well as convection transfer rates (heat and mass)

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Heat Transfer Su Yongkang

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Convection Heat Transfer Variations Along Flat Plate

• Consider flat plate with:and all laminar flow

• Thermal boundary layer defined by

Compare temperature gradients at points 1 and 2 to evaluate the heat flux rate (and hence the heat transfer coefficient)

C0 and C100 TTx

21

C0T

C100sT1 2

99.0

TT

TT

s

s

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Heat Transfer Su Yongkang

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Convection Heat Transfer Variations Along Flat Plate (Cont’d)

• Consider flat plate with: C0 and C100 TTx

2

1

C100C1C100C1

at x1 at x2

To determine the location that transition begins, we define the Reynolds number,

And the critical Reynolds number,

xu

xRe

5, 105Re

c

cx

xu

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Heat Transfer Su Yongkang

School of Mechanical Engineering

Convective Transfer Equations

Topic of the Day

Project Teams

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Heat Transfer Su Yongkang

School of Mechanical Engineering

Convective Transport Equations

Where we’ve been ……Last section:Overview of the topic of convective transport of heat.

Where we’re going:

• Convection transfer detailed equations

• Heat and mass transfer analogy

• Eventually get to applications in external and internal flow.

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Heat Transfer Su Yongkang

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Convective Transport Equations

KEY POINTS THIS SECTION

• Detailed development of boundary layer equations for velocity, temperature and species concentrations

• What approximations can be made in the boundary layers?

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Heat Transfer Su Yongkang

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Recall the convection overview

• Local heat flux is:

where h is the local heat transfer coefficient

)( TThq s

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Heat Transfer Su Yongkang

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Develop convection transfer equations

• Consider steady, 2-D flow of a viscous, incompressible fluid with constant properties.

• Key point to remember:At each point in the fluid, conservation of mass, energy and momentum must be satisfied.

• Appendix E contains detailed development of the full boundary layer equations, for example:

Conservation of mass (continuity)

0

y

v

x

u

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Heat Transfer Su Yongkang

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The magnitude of variables in the thermal boundary layer

variables x(main flow direction)

y u v t

magnitude 1 1 1

x

T

y

T

Thermal boundary layer

y

v

y

u ,

x

v

y

u ,

x

u

y

u and ;

vu

Velocity boundary layer

Boundary Layer Approximations

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Heat Transfer Su Yongkang

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Continue convection transfer equations

• Quick overview of fluid equationsConservation of mass (continuity):

OR

x-momentum equation:

y-momentum equation:

0 0

y

v

x

u

y

v

x

u

0

y

v

x

u

forces"Body " 2

2

2

2

y

u

x

u

x

P

y

uv

x

uu

x

u

y

u 0

0

2

2

1

y

u

x

P

y

uv

x

uu

:where

forces"Body " 2

2

2

2

y

v

x

v

y

P

y

vv

x

vu

0 0 0 0 0

0

y

PSo:

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Heat Transfer Su Yongkang

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Continue convection transfer equations

Energy equation:

qy

v

x

u

x

v

y

u

y

T

x

Tk

y

Tv

x

Tuc p

222

2

2

2

2

2

0 0 0 0

0

2

y

u

c p

NOTE: is usually small unless u is high

(as in sonic flows) or is high (such as flow of oils).

Result is 4 equations and 4 unknowns:

Unknowns are: u, v, P, and T

Since:

P(x) can be obtained from free stream flow.

dx

dP

x

P and only, )x(fP then 0

y

P

x

T

y

T Why?

x

v

y

u Why?

2

2

2

y

u

cy

T

y

Tv

x

Tu

p

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Heat Transfer Su Yongkang

School of Mechanical Engineering

• Review boundary layer concept.• General relationship between velocity and

thermal boundary layers.• Convective heat transfer is dependent on

the temperature gradient at the fluid/solid interface

• Boundary layer grows with distance from the leading edge, and this decreases the local heat transfer rates.

• Turbulent boundary layers have much greater potential for heat and mass transfer due to velocity fluctuations.

• Convection transfer equations (4).

Go back and review fluids course notes!

SUMMARY