08. Basic Concepts in Convection_Bounday Layers

7/28/2019 08. Basic Concepts in Convection_Bounday Layers

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8. BASIC CONCEPTS IN CONVECTION

Introduction

Characteristic of convection: Fluid motion Focal point: Determination of heat transfer coefficient

- souinitel pestupu tepla Determination of:

Temperature distribution in the fluid


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General ObservationsThe Role of Fluid Motion

For the electric bulb:

wq& = surface flux

wT = surface temperature

T = free stream temperatureu = free stream velocity

For a fixed input power how to lower surface temperature?

? Raise or lower u Change the cooling fluid?

VT

+

sTsq

1.6Fig.

u

wq&wT


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water lowers surface temperature toair

from

fluid

the

changingorIncreasing u

Surface temperature depends on

Conclusion:

Fluid motion and fluid nature

play important roles in convection

Solve for wT

qTT ww

&+=

fixed

Newtons law:

( )= TTq ww& (8.1)[W/m2]


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For =f(x) = L dxxL

0

)(1

(8.3)

Local heat transfer coefficient, = f(S), f(x)

Average heat transfer coefficient,

is not uniform over a surfaceIn general,

( )= TTq ww& [W/m2]

For =f(S) = SdSS1

(8.2)=

TTSQ ww& [W]

wq&

x

y


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Similarly for Mass Transfer (as for Heat Transfer)

Consider a lake or a pond and

its surface from which water

evaporates (or an evaporatingdroplet)

Two substances, one labeledA (water vapor)

is transferred intoB (dry or humid air).

A,w

mass concentration of substanceA density [kg/m

3

]at surface temperature and assumed in saturated state

A, mass concentration of substanceA at free stream

conditions (temperature, humidity, pressure)


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For the entire surfaceS: [kg/s]= A,A,wA Sm&

is an average mass transfercoefficient [m/s]

=S

dSS

1

(8.5)

Mass transfer is proportional to concentration difference

( )= TTq ww&

Heat transfer is proportional to temperature difference

is mass transfer coefficient [m/s] sou. pestupu hmoty

(8.4)=A,A,wA

m [kg/s.m2]thenIf A,A,w


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Conclusion:

naturefluidandmotionfluidondependscoefficienttransferheat

geometry,givenaFor

What is the objective of this chapter?

Examine thermal interaction between a surface and a

moving fluid and determine:

(1) The heat transfer coefficient

(3) Surface temperature Tw(2) Surface heat flux wq

&


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introduced ?isWhy(2) analytically?determinedisHow)1(

Heat transfer coefficient


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Combine Newtons and Fouriers laws

(8.8)

=

=

=

TT

yT

TT

q

w

0yf

w

w&

Apply Fourier's law to thefluidat surface

0y

fwy

Tq

=

=& (8.6) Heat conducted across a thin sticky

layer on the surface

Balance between conducted heatand heat convected downstream

the surface - consider Newton's

law( )= TTq ww&

VT x

sT

6.2Fig.

),( yxT

wT

u

wq&(8.7)


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To get from (8.8), we must determine temperature

distribution in the fluid and to obtain temperature

gradient in the fluid

),(xT

0/ =yyT

(8.8)

=

=

=TT

yT

TT

q

w

0yf

w

w&


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Governing Equations for Convection Heat

Transfer

Focal point in convection:

a

temperatureofDeterminationfluidmovingindistribution

Basic laws governing temperature distribution:

(3) Conservation of energy

(1) Conservation of mass(2) Conservation of momentum


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Assumptions:

(1) Two-dimensional u(x,y), v(x,y)(2) Single phase flow (water, air, etc)

Conservation of Mass: The ContinuityEquation

(b)(a) 6.4Fig.

x

y

ym&

dyy

mm

yy

+ &&

xm&dx

x

mm xx + &&

dxdx

dydy


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Apply conservation of mass to an element dxdy:

removedRate-addedmassofRatewithinchangemassofRate

(a)

Apply (a) Using previous Figure:

= mass flow rate entering element in thex-direction

= mass flow rate entering element in the y-direction

m = mass within element

Express (b) in terms of fluid density and velocity:

(b)

)()(t

mdy

ydx

x =

+

+


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m = mass of element:

dxdym (f)

= flow area

V= velocity normal to

= density

(c)VA

u and v are the velocity components in thexand y

directions

Apply (c) to the element

(d)udy

(e)vdx


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(d)(f) into (b)

(8.9)0)()( =++ yx

u

t

v

Incompressible fluid: constant

(8.10)0+

yx

u v

is the continuity equation


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Conservation of Momentum:

The Navier-Stokes Equations of Motion

Assume: 2-D

Newton's law of motion: Apply to element dxdyx-direction

in

(a)xx maF =xa = acceleration in thex-direction

m = mass of the element

dxdym (b) Acceleration :xa

we need: u = u(x,y,t)


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The total change in u is

dtt

udy

y

udx

x

udu

++

=Divide by dtand note that dx/dt= u and dy/dt= v

(c)t

u

y

u

x

uu

dt

duax

++

= v

y

u

x

uu

+

v = convective acceleration

tu = local acceleration

xF : Two types of external forces:


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Surface forces, sF :Normal: pressurep and normal stress xx

Tangential: shearing stress xy

Body forces, bF : GravitydxdygFb (d)

(e)+ sbx FTotal external forces:


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dxyx

dxdy

y

yxyx )(

+ dyxx dydx

x

xxxx )(

+

pdy dydxx

p

p )(

+dxdy

6.5Fig.

(e) and (f) into (d)

dxdy

yxx

pgF

yxxxx )(

++

(g)

(f)dxdyyxx

pF xyxxs )( ++

Surface forces:


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By analogy:y-direction

xyy

p

yx

u

t

xyyy +

+

++

)( vvvv (8.12)Too many unknowns!

Important assumption: The variables ,xx

,yy

,xy

yxand are eliminated using empirical relations. For

incompressible fluids:

(b), (c) and (g) into (a)

yxx

pg

y

u

x

uu

t

u yxxx +

+

++

)( v (8.11)


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Fluids that obey these relations, such as water, air and

oil, are referred to asNewtonian fluids

Polymers, honey, etc. do not follow these relationsand are known as non-Newtonian fluids

(8.13)-(8.15) into (8.11) and (8.12), assume constant

viscosity

(8.13)x

u

xx

= 2

y

yy

= v2 (8.14)

(8.15)

+

=x

v

y

uyxxy


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(8.16))()( 2

2

2y

u

x

u

x

pgy

u

x

uut

u

+

+

++

2vand

)()(2

2

2

2

yxy

p

yxu

t ++++vvv

vvv (8.17)

(8.16) and (8.17) are the equations of motion inrectangular coordinates. They are also known as the

Navier-Stokes equations of motion.


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(3) Constant viscosity (4) Two-dimensional flow

(5) Gravity pointing in the positivex-direction

Limitations on (8.16) and (8.17):

(1) Newtonian fluids

(2) Constant density


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Energy by conduction and convection

Apply conservation of energy, to element dxdy :

dtdEEEE akoutgin ==+ &&&& [W]


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(a), (b), (c) and (d) into conservation of energy and using

the continuity equation

(8.18)p

zdr

2

2

2

2

c

Q

y

T

x

T

y

Tv

x

Tu

t

T)(

&

+

+

=

+

+

a = thermal diffusivity (souinitel tepeln vodivosti):

pc

a =

Equation (8.18) is the energy equation in rectangularcoordinates for 2-D constant property fluids


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(1) First term: Local rate of energy change

(2) Second term: Net energy convected with fluid

(3) Third: Net energy conducted in thexandy

directions

(4) Fourth term: Energy generation

Physical significance of each term in (8.18):

(8.18)p

zdr2

2

2

2

c

Q

y

T

x

T

y

Tv

x

Tu

t

T)(

&

+

+

=

+

+

(1) (2) (3) (4)


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Summary of the Governing Equations for

Convection Heat Transfer: Mathematical

Implications

Assumptions:(1) Newtonian fluid

(2) Two-dimensional

(3) Negligible changes in kinetic and potential energy

(4) Constant properties (except in buoyancy)

(5) Gravity is in the positivex-direction

Continuity:

0+

yx

u v


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x-momentum:

)()(2

2

2

2

y

u

x

ux

p

y

uvx

uut

u+

+

=

+

+

y-momentum:

)()(2

2

2

2

yxy

p

yxu

t +

+

++

vvvv

vv

Energy:

p

zdr2

2

2

2

cQ

yT

xT

yTv

xTu

tT

&

+

+

=

+

+


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Boundary layer features:

(1) Velocity at the surface vanishes. This is the no-slip

condition

(2) Velocity changes rapidly across the boundary layerthickness . At the edge uu

(3) Viscosity plays no role outside the velocity boundary

layer

Conditions for the existence of the velocity boundary layer:

(2) High Reynolds number (Re > 100)

(1) Streamlined body without flow separation

6.10Fig.

R

y

xy


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heating

Thermal boundary layer:

Under certain conditions the

effect of thermal interaction

between a fluid and a surface will

be confined to a thin region near

the surface called the thermal

boundary layer

The edge of this region is defined by the thickness

whereT

( )= TT0,99TT ww

( )=

=

= TT

y

Tq w

0y

fw&Heat flux transferred at the wall:


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heat flux qw0yy

T=

tangent

For (Tw-T) = const,whats the behavior of?

x

( )

=

=TT

yT

w

0yf

Heat Transfer Coefficient

As the boundary layer increases,

the temperature gradient decreases.

Why?

The same temperature difference (Tw-T)applies to a larger distance

decreases and so does.

xT

xT


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(1) Streamlined body without flow separation

(2) High product of the Reynolds and Prandtl numbers

(Re Pr>100)

Luc

c

LuPe

pp ))( === (NumberPeclet

(4) Temperature changes rapidly across the thermal

boundary layer thickness t . At the edge TT(5) In general, both velocity and thermal boundary layer

are thin

Conditions for the existence of the thermal boundary layer:


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Transition Reynolds number Re


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Transition Reynolds number, tRe

Used to check if the flow is laminar or turbulent Ret is determined experimentally

Its value depends on geometry, surface roughness,

pressure gradient,

Magnitude of transRe can be changed by manipulating

surface roughness, pressure gradient,

For uniform flow over a semi-infinite plate:

500000

xu

Retrans

x,trans =

ux

y

For flow through smooth tubes:

2300

DuRetrans = u

L i T b l t Fl


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Laminar vs. Turbulent Flow

xLaminar Transition Turbulent


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x-momentum:

)()(2

2

2

2

y

u

x

uxp

yuv

xuu

tu

+

+

=

+

+

y-momentum:

)()(2

2

2

2

yxy

p

yxu

t +

+

++

vvvv

vv

Energy:

p

zdr2

2

2

2

cQ

yT

xT

yTv

xTu

tT

&

+

+

=

+

+

M th ti l Si lifi ti f B d


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Mathematical Simplifications for Boundary

Layer Flows

Not always all terms in the momentum equations are

necessary to take into account

Often, incompressible flow, =const, often constant

physical quantities , , negligible mass forces

(gravitational etc.), no internal heat source.

Often u>>v,x

v,

y

v,

x

u

y

u

>>

x

T

y

T

>>

ux

y


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Summary of Boundary Layer Equations

for Steady Laminar Flow

Assumptions:

(1) Newtonian fluid

(2) Two-dimensional

(3) Negligible changes in kinetic and potential energy

(4) Constant properties

(5) Streamlined surface

(6) High Reynolds number (Re > 100)

(7) High Peclet number (Pe > 100).


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(8) Steady state

(9) Laminar flow

(12) No gravity

(10) No dissipation, 0

(11) No buoyancy, 0

(13) No energy generation,


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Classification of Convection Heat Transfer

1. Forced convection vs. free convection

2. External vs. internal flow

3. Boundary layer flow vs. low Reynolds number flow

4. Compressible vs. incompressible flow5. Laminar vs. turbulent flow

6. Newtonian vs. non-Newtonian fluid


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Fluid Properties

Fluid properties needed to solve convection problems:

Specific heat cpThermal conductivity

Prandtl numberPr

Thermal diffusivity a

Dynamic viscosity Kinematic viscosity Density

Heat Transfer Coefficient and Dimensionless


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Heat Transfer Coefficient and Dimensionless

Criteria

( )

=

=TT

yT

w

0yfFrom boundary layer

7 quantities

4 primary dimensions

J/K, kg, m, s

Buckingham theorem

3 dimensionless similarity

parameters - numbers

General functional dependence

for forced convection),,,,,( cLuf =

LNu = Nusselt number

uL

Re = Reynolds number

a

cPr == Prandtl number

Formula with dimensionless numbers correlation equations:


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( )PrRe,Nu =

Formula with dimensionless numbers correlation equations:

forced convection

How such equations can be obtained?

Mostly by experiments or by analytical solution

for simple situations or systems (e.g. flow over a flat plate)

Forced convection in a tube laminar or turbulent (entrance

length L/d, fully developed region)

Cross flow over a cylinder, tube bundle

Forced convection for external flow on a flat plate

Natural convection (another dimensionless number entersinto play Grashoff number)

Flow with viscous dissipation (another dimensionless number

enters into play Eckert number) etc.

Actual form of the equation depends on the system:

08. Basic Concepts in Convection_Bounday Layers

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