fundamentals of convection

7/27/2019 Fundamentals of Convection

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Chapter 6: Fundamentals of

Convection

Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.


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ObjectivesWhen you finish studying this chapter, you should be able to:

Understand the physical mechanism of convection, and itsclassification,

Visualize the development of velocity and thermal boundary layersduring flow over surfaces,

Gain a working knowledge of the dimensionless Reynolds, Prandtl,and Nusselt numbers,

Distinguish between laminar and turbulent flows, and gain anunderstanding of the mechanisms of momentum and heat transfer inturbulent flow,

Derive the differential equations that govern convection on thebasis of mass, momentum, and energy balances, and solve theseequations for some simple cases such as laminar flow over a flat

plate,

Nondimensionalize the convection equations and obtain thefunctional forms of friction and heat transfer coefficients, and

Use analogies between momentum and heat transfer, and determineheat transfer coefficient from knowledge of friction coefficient.


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Physical Mechanism of Convection

Conduction and convection are similar in that both

mechanisms require thepresence of a material medium.

But they are different in that convection requires the presence

offluid motion.

Heat transfer through a liquid or gas can be by conduction or

convection, depending on the presence of any bulk fluid

motion.

The fluid motion enhances heat transfer, since it brings

warmer and cooler chunks of fluid into contact, initiating

higher rates of conduction at a greater number of sites in a

fluid.


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Experience shows that convection heat transferstronglydepends on the fluid properties: dynamic viscositym, thermal conductivityk,

densityr, and specific heatcp, as well as the

fluid velocityV.

It also depends on thegeometryand the roughnessof the solid

surface. The rate of convection heat transfer is observed to be

proportional to the temperature difference and is expressed byNewtons law of coolingas

The convection heat transfer coefficient hdepends on theseveral of the mentioned variables, and thus is difficult todetermine.

2

(W/m )conv sq h T T (6-1)


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All experimental observations indicate that a fluid in

motion comes to a complete stop at the surface and

assumes a zero velocity relative to the surface (no-slip).

The no-slip condition is responsible for the development

of the velocity profile.

The flow region adjacent

to the wall in which the

viscous effects (and thus

the velocity gradients) aresignificant is called the boundary layer.


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An implication of the no-slip condition is that heat

transfer from the solid surface to the fluid layer

adjacent to the surface is bypure conduction, and canbe expressed as

Equating Eqs. 61 and 63 for the heat flux to obtain

The convection heat transfer coefficient, in general,

varies along the flow direction.

2

0

(W/m )conv cond fluid

y

Tq q k

y

(6-3)

0 2

(W/m C)fluid y

s

k T yh

T T

(6-4)


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The Nusselt Number It is common practice to nondimensionalize the heat transfer

coefficient hwith the Nusselt number

Heat flux through the fluid layer by convectionand byconductioncan be expressed as, respectively:

Taking their ratio gives

The Nusselt number represents the enhancement of heat transferthrough a fluid layer as a result of convection relative toconduction across the same fluid layer.

Nu=1pure conduction.

chLNuk

(6-5)

convq h T (6-6)

condTq k

L (6-7)

/

conv

cond

q h T hLNu

q k T L k

(6-8)


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Classification of Fluid Flows

Viscous versus inviscid regions of flow

Internal versus external flow

Compressible versus incompressible flow Laminarversus turbulent flow

Natural (or unforced) versus forced flow

Steady versus unsteady flow One-, two-, and three-dimensional flows


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Velocity Boundary Layer Consider the parallel flow of a fluid over aflat plate.

x-coordinate: along the plate surface y-coordinate:from the surface in the normal direction.

The fluid approaches the plate in thex-direction with a uniform

velocity V.

Because of the no-slip condition V(y=0)=0. The presence of the plate is felt up to d.

Beyond dthe free-stream velocity remains essentially unchanged.

The fluid velocity, u, varies from 0 aty=0 to nearly Vaty=d.


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Velocity Boundary Layer

The region of the flow above the plate bounded by d

is called the velocity boundary layer.

dis typically defined as

the distanceyfrom the

surface at which

u=0.99V.

The hypothetical line of

u=0.99Vdivides the flow over a plate into tworegions:

the boundary layer region, and

the irrotational flow region.


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Surface Shear Stress

Consider the flow of a fluid over the surface of a plate. The fluid layer in contact with the surface tries to drag theplate along via friction, exerting afriction forceon it.

Friction force per unit area is called shear stress, and isdenoted by t.

Experimental studies indicate that the shear stress for mostfluids is proportional to the velocity gradient.

The shear stress at the wall surface for these fluids isexpressed as

The fluids that that obey the linear relationship above arecalled Newtonian fluids.

The viscosity of a fluid is a measure of its resistance todeformation.

2

0

(N/m )sy

u

y

t m

(6-9)


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The viscosities ofliquidsdecreasewith temperature, whereas

the viscosities ofgasesincreasewith temperature.

In many cases the flow velocity profile isunknown and the surface shear stress ts

from Eq. 69 can not be obtained.

A more practical approach in external flow

is to relate tsto the upstream velocity Vas

Cfis the dimensionless friction coefficient(most cases is

determined experimentally).

The friction force over the entire surface is determined from

22

(N/m )2

s f

VC

rt (6-10)

2

(N)2

f f s

VF C A

r (6-11)


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Thermal Boundary Layer Like the velocity a thermal boundary layerdevelops when a

fluid at a specified temperature flows over a surface that is at

a different temperature.

Consider the flow of a fluid

at a uniform temperature of

T over an isothermal flatplate at temperature Ts.

The fluid particles in the

layer adjacent assume the surface temperature Ts.

A temperature profile develops that ranges from Tsat thesurface to T sufficiently far from the surface.

The thermal boundary layerthe flow region over the

surface in which the temperature variation in the direction

normal to the surface is significant.


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The thicknessof the thermal boundary layerdtat any

location along the surface is defined as the distancefrom the surface at which the temperature difference

T(y=dt)-Ts= 0.99(T-Ts).

The thickness of the thermal boundary layerincreases

in the flow direction.

The convection heat transfer rate anywhere along the

surface is directly related to the temperature gradient

at that location.


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Prandtl Number

The relative thickness of the velocity and thethermal boundary layers is best described by the

dimensionlessparameterPrandtl number, defined

as

Heat diffuses very quickly in liquid metals (Pr1)

and very slowly in oils (Pr1) relative to momentum. Consequently the thermal boundary layer is much

thicker for liquid metals and much thinner for oils

relative to the velocity boundary layer.

Molecular diffusivity of momentumPrMolecular diffusivity of heat

pck

m

(6-12)


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Laminar and Turbulent Flows

Laminar flow the flow is characterized by

smooth streamlines and highly-ordered

motion.

Turbulent flow the flow ischaracterized by velocity

fluctuations and

highly-disordered motion. The transitionfrom laminar

to turbulent flow does not

occur suddenly.


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The velocity profile in turbulent flow is much fuller than that in

laminar flow, with a sharp drop near the surface.

The turbulent boundary layer can be considered to consist of

four regions: Viscous sublayer

Buffer layer

Overlap layer

Turbulent layer

The intense mixingin turbulent flow enhances heat and

momentum transfer, which increases the friction force on the

surface and the convection heat transfer rate.


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Reynolds Number The transition from laminar to turbulent flow depends on the

surface geometry,surface roughness,flow velocity,surface

temperature, and type of fluid. The flow regimedepends mainly on the ratio of the inertia forces

to viscous forcesin the fluid.

This ratio is called the Reynolds number, which is expressed for

external flow as

At large Reynolds numbers (turbulent flow) the inertia forces arelarge relative to the viscous forces.

Atsmallormoderate Reynolds numbers (laminar flow), theviscous forces are large enough to suppress these fluctuationsand to keep the fluid inline.

Critical Reynolds numberthe Reynolds number at which the

flow becomes turbulent.

Inertia forcesRe

Viscous forces

c cVL VLr

m (6-13)


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Heat and Momentum Transfer in

Turbulent Flow

Turbulent flow is a complex mechanism dominated by

fluctuations, and despite tremendous amounts of research the

theory of turbulent flow remains largely undeveloped.

Knowledge is based primarily on experiments and the empirical

or semi-empirical correlations developed for various situations.

Turbulent flow is characterized by random and rapid fluctuations

of swirling regions of fluid, called eddies.

The velocity can be expressed as the sum

of an average value uand afluctuating

componentu

'u u u (6-14)


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It is convenient to think of the turbulent shear stress as

consisting of two parts:

the laminar component, and

the turbulent component.

The turbulent shear stresscan be expressed as

The rate of thermal energy transport by turbulent eddies is

The turbulent wall shear stress and turbulent heat transfer

mt turbulent (or eddy) viscosity.

kt turbulent (or eddy) thermal conductivity.

' 'turb pq c v T r

' 'turb u vt r

' ' ;turb t turb p t u T

u v q c vT k y yt r m r

(6-15)


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The total shear stress and total heat flux can be

expressed as

and

In the core region of a turbulent boundary layer

eddy motion (and eddy diffusivities) are much larger

than their molecular counterparts.

Close to the wall the eddy motion loses its intensity. At the wall the eddy motion diminishes because of

the no-slip condition.

turb t t u u

y yt m m r

(6-16)

(6-17) turb t p t T T

q k k cy y

r


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In the core region the velocity and temperature profiles

are very moderate.

In the thin layer adjacent to the wall the velocity and

temperature profiles are very steep.

Large velocity and temperature gradients at the

wall surface.The wall shear stress

and wall heat flux are much larger

in turbulent flow than they

are in laminarflow.


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Derivation of Differential Convection

Equations

Consider the parallel flow of a fluid over a surface. Assumptions:

steady two-dimensional flow,

Newtonian fluid,

constant properties, and laminar flow.

The fluid flows over the surface with a uniform free-stream velocity V, but the velocity within boundary

layer is two-dimensional (u=u(x,y), v=v(x,y)). Three fundamental laws:

conservation of masscontinuity equation

conservation of momentummomentum equation

conservation of energyenergy equation


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The Continuity Equation

Conservation of mass principle the mass cannot be created or destroyed during a process.

In steady flow:

The mass flow rate is equal to: ruA

Rate of mass flowinto the control volume

Rate of mass flowout of the control volume= (6-18)

ruA


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u

u+u/xdx

x,y

dx

dy

v+v/ydy

v

Repeating this for they direction

1u dyr The fluid leaves the control volume from the left surface at a rate of

1u

u dx dyx

r

the fluid leaves the control volume from the right surface at a rate of

(6-19)

The continuity equation

and substituting the results into Eq.

618, we obtain

1 1

1 1

u dy v dx

u vu dx dy v dy dxx y

r r

r r

(6-20)

Simplifying and dividing by dxdy

0u v

x y

(6-21)


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The Momentum Equation The differential forms of the equations of motion in

the velocity boundary layer are obtained by applyingNewtons second law of motion to a differential

control volume element in the boundary layer.

Two type of forces:

body forces,

surface forces.

Newtons second law of motion for the control

volume

or

(Mass)Acceleration

in a specified direction

Net force (body and surface)

acting in that direction=X

, ,x surface x body xm a F F d

(6-23)

(6-22)


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where the mass of the fluid element within the control

volume is

The flow is steady and two-dimensional and thus

u=u(x,y), the total differential ofuis

Then the acceleration of the fluid element in thex

direction becomes

1m dx dyd r (6-24)

u udu dx dyx y

(6-25)

x

du u dx u dy u ua u v

dt x dt y dt x y

(6-26)


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The forces acting on a surface are due topressure and

viscous effects.

Viscous stress can be resolved intotwo perpendicular components:

normal stress,

shear stress.

Normal stress should not be confused withpressure.

Neglecting the normal stresses the net surface force

acting in thex-direction is

,

2

2

1 1 1

1

surface xP PF dy dx dx dy dx dy

y x y x

u Pdx dy

y x

t t

m

(6-27)


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Substituting Eqs. 621, 623, and 624 into Eq. 620 and

dividing by dxdy1 gives

Boundary Layer Approximation

Assumptions:

1) Velocity components:

u>>v

2) Velocity gradients:

v/x0and v/y0

u/y >> u/x

3) Temperature gradients:

T/y >> T/x

When gravity effects and other body forces are negligible the

y-momentum equation

2

2

u u u P

u vx y y xr m

(6-28)

The x-momentum

equation

0P

y

(6-29)


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Conservation of Energy Equation The energy balance for any system undergoing any

process is expressed asEin-Eout=Esystem. During asteady-flow process Esystem=0.

Energy can be transferred by

heat,

work, and

mass.

The energy balance for a steady-flow control volumecan be written explicitly as

Energy is a scalar quantity, and thus energyinteractions in all directions can be combined in one

equation.

0in out in out in out by heat by work by mass

E E E E E E (6-30)


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Repeating this for they-direction and adding

the results, the net rate of energy transfer to thecontrol volume by mass is determined to be

Note thatu/x+v/y=0 from the continuityequation.

in out by mass

p p

p

E E

T u T vc u T dxdy c v T dxdy

x x y y

T Tc u v dxdy

x y

r r

r

(6-32)


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Energy Transfer by Heat Conduction

The net rate of heat conduction to the volume element

in thex-direction is

Repeating this for they-direction and adding the

results, the net rate of energy transfer to the control

volume by heat conduction becomes

,

2

2

1

xin out x x

by heat x

Q TE E Q Q dx k dy dx

x x x

T

k dxdyx

(6-33)

2 2 2 2

2 2 2 2in out

by heat

T T T T E E k dxdy k dxdy k dxdy

x y x y

(6-34)


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Energy Transfer by Work

The work done by abody force is determined bymultiplying this force by the velocity in the direction

of the force and the volume of the fluid element.

This work needs to be considered only in thepresence of significant gravitational, electric, ormagnetic effects.

The workdone bypressure (the flow work) is alreadyaccounted for in the analysis above by using enthalpyfor the microscopic energy of the fluid instead ofinternal energy.

The shear stresses that result from viscous effects areusually very small, and can be neglected in manycases.


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The Energy Equation The energy equation is obtained by substituting Eqs.

632 and 634 into 630 to be

When the viscous shear stresses are not negligible,

where the viscous dissipation function is obtained

after a lengthy analysis to be

Viscous dissipation may play a dominant role in high-

speed flows.

2 2

2 2p

T T T T c u v k

x y x yr

(6-35)

2 2

2 2p T T T T c u v k x y x yr m

(6-36)

2 22

2u v u v

x y y x

(6-37)


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Solution of Convection Equations for a

Flat Plate (Blasius Equation)

Consider laminar flow of a fluid overaflat plate.

Steady, incompressible, laminar flow

of a fluid with constant properties

Continuity equation

Momentum equation

Energy equation

2

2

u u uu v

x y y

(6-40)

u v

x y

(6-39)

2

2

T T Tu v

x y y

(6-41)

B d diti


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Boundary conditions

Atx=0

Aty=0

Asy

When fluid properties are assumed to be constant,the first two equations can be solved separately for

the velocity components uand v. knowing uand v, the temperature becomes the only

unknown in the last equation, and it can be solvedfor temperature distribution.

0, , 0,u y V T y T

,0 0, ,0 0, ,0 su x v x T x T

, , ,u x V T x T

(6-42)


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The continuity and momentum equations are solved

by transforming the two partial differential equations

into a single ordinary differential equation by

introducing a new independent variable (similarity

variable).

The argument the nondimensional velocity profile

u/Vshould remain unchanged when plotted againstthe nondimensional distancey/d.

dis proportional to (x/V)1/2, therefore defining

dimensionless similarity variable as

might enable a similarity solution.

Vyx

(6-43)

Introducing a stream function y(x y) as


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Introducing astream function y(x,y) as

The continuity equation (Eq. 639) is automatically

satisfied and thus eliminated.

Defining a functionf() as the dependent variable as

The velocity components become

;u vy x

y y

(6-44)

/

fV x V

y

(6-45)

1

2 2

x df V dfu V V

y y V d x d

x df V V dfv V f f

x V d Vx x d

y y

y

(6-46)

(6-47)

B diff i i h d l i h


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By differentiating these uand vrelations, the

derivatives of the velocity components can be shown

to be

Substituting these relations into the momentum

equation and simplifying

which is a third-order nonlinear differential equation.Therefore, the system of two partial differential

equations is transformed into a single ordinary

differential equation by the use of a similarity

variable.

2 2 2 2 3

2 2 2 3; ;

2

u V d f u V d f u V d f V

x x d y x d y x d

(6-48)

3 2

3 22 0

d f d f f

d d (6-49)


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The boundary conditions in terms of the similarity

variables

The transformed equation with its

associated boundary conditions

cannot be solved analytically, and

thus an alternative solution method

is necessary.

The results shown in Table 6-3 was

obtained using different numerical approach.

The value ofcorresponding to u/V=0.99 is =4.91.

00 0, 0, 1

df df

f d d

(6-50)

Substituting =4 91 and y=d into the definition of the


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Substituting 4.91 andy dinto the definition of the

similarity variable (Eq. 643) gives 4.91=d(V/x)1/2.

The velocity boundary layer thickness becomes

The shear stress on the wall can be determined from its

definition and the u/yrelation in Eq. 648:

4.91 4.91

Rex

x

V xd

(6-51)

2

2

0 0

w

y

u V d f Vy x d

t m m

(6-52)

2

0.3320.332Re

w

x

V VVx

rm rt (6-53)

Substituting the value of the second derivative offat h=0

from Table 63 gives

1/ 2

, 20.664Re

/ 2

wf x xC

V

t

r

(6-54)

Then the average local skin friction coefficient becomes


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The Energy Equation Introducing a dimensionless temperature qas

Noting that both Tsand Tare constant, substitution

into the energy equation Eq. 641 gives

Using the chain rule and substituting the uand v

expressions from Eqs. 646 and 647 into the energy

equation gives

,, ss

T x y T x yT T

q

(6-55)

2

2u v

x y y

q q q

(6-56)

22

2

1

2

df d d Vy df d d d V f

d d dx x d d dy d y

q q q

(6-57)

Simplifying and noting that Pr=/ gives


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Simplifying and noting that Pr=/gives

Boundary conditions:

Obtaining an equation forqas a function ofalone confirms thatthe temperature profiles are similar, and thus a similarity solution

exists. forPr=1, this equation reduces to Eq. 649 when qis replaced by

df/d.

Equation 658 is solved for numerous values of Prandtl numbers.

ForPr>0.6, the nondimensional temperature gradient at the surface isfound to be proportional to Pr1/3, and is expressed as

2

22 Pr 0

d df

d d

q q

(6-58)

0 0, 1q q

1/ 3

0

0.332Prd

d

q

(6-59)

Th t t di t t th f i


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The temperature gradient at the surface is

Then the local convection coefficient and Nusselt number become

and

Solving Eq. 658 numerically for the temperature profile fordifferent Prandtl numbers, and using the definition of the thermal

boundary layer, it is determined that

00 0 0

1/ 30.332Pr

s s

y y y

s

dT d d T T T T

dy y d dy

VT T

x

q q

(6-60)

0 1/ 3

0.332Prys

x

s s

k T yq Vh k

T T T T x

(6-61)

1/3 1/ 20.332 Pr Re Pr>0.6x

x

h xNu

k (6-62)

1/ 3

Prtd d


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Nondimensional Convection Equation

and Similarity

Continuity equation

x-momentum equation

Energy equation

Nondimensionalized variables

0u v

x y

(6-21)

2

2

u u u P u v

x y y xr m

(6-28)

2 2

2 2p

T T T T c u v k

x y x yr

(6-35)

* * * * * *

2; y ; u ; v ; P ; s

s

T Tx y u v Px T

L L V V V T Tr

Introd cing these ariables into Eqs 6 21 6 28 and


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Introducing these variables into Eqs. 621, 628, and

635 and simplifying give

Continuity equation

x-momentum equation

Energy equation

with the boundary conditions

* *

* *0u v

x y

(6-64)

* * 2 * ** *

* * *2 *

1

ReL

u u u P u v

x y y x

(6-65)

* * 2 ** *

* * *2

1

Re Pr L

T T Tu v

x y y

(6-66)

* * * * * * * *0, 1 ; ,0 0 ; , 1 ; ,0 0u y u x u x v x (6-67)

* * * * * *0, 1 ; ,0 0 ; , 1T y T x T x


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For a given type of geometry, the solutions of

problems with the same Re andNu numbers are

similar, and thus Re andNu numbers serve as

similarity parameters.

A major advantage of nondimensionalizing is the

significant reduction in the number of parameters.

The original problem involves 6 parameters (L, V, T,Ts, , ), but the nondimensionalized problem

involves just 2 parameters (ReLand Pr).

L, V, T, Ts,, Nondimensionalizing

6 parameters

ReL, Pr

2 parameters

F i l F f h F i i d


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Functional Forms of the Friction and

Convection Coefficient

From Eqs. 6-64 and 6-65 it can be inferred that

Then the shear stress at the surface becomes

Substituting into its definition gives the localfriction coefficient,

* * *1 , ,ReLu f x y (6-68)

*

**

2*

0 0

,Res Ly y

u V u V f x

y L y L

m mt m

(6-69)

* * *, 2 2 32 22

,Re ,Re ,Re2 2 Re

sf x L L L

L

V LC f x f x f x

V V

t m

r r

(6-70)


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Similarly the solution of Eq. 6-66

Using the definition ofT*, the convection heat

transfer coefficient becomes

Substituting this into the Nusselt Number relation

gives

* * *1 , ,Re ,Pr LT g x y (6-71)

* *

* *0

* *0 0

y s

s s y y

k T y k T T T k Th

T T L T T y L y

(6-72)

*

* *

2*

0

, Re , Pr x L

y

hL TNu g xk y

(6-73)


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It follows that the averageNu and Cfdepends on

These relations are extremely valuable:

The friction coefficient can be expressed as a function of

Reynolds number alone, and

TheNusselt numberas a function ofReynolds and Prandtl

numbers alone.

The experiment data for heat transfer is oftenrepresented by a simple power-law relation of the

form:

3 4Re , Pr ; ReL f LNu g C f (6-74)

Re Pr L

m nNu C (6-75)

A l i B t M t d


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Analogies Between Momentum and

Heat Transfer

Reynolds Analogy(ChiltonColburn Analogy) undersome conditions knowledge of the friction coefficient, Cf,can be used to obtainNu and vice versa.

Eqs. 665 and 666 (the nondimensionalized momentumand energy equations) forPr=1 and P*/x*=0:

x-momentum equation

Energy equation

which are exactly of the same form for the dimensionless

velocity u*

and temperature T*

.

* * 2 ** *

* * *2

1

ReL

u u uu v

x y y

(6-76)

* * 2 ** *

* * *2

1

ReL

T T Tu v

x y y

(6-77)

The boundary conditions for u* and T* are also identical


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The boundary conditions foru and T are also identical.

Therefore, the functions u* and T* must be identical.

The Reynolds analogy can be extended to a wide range of

Prby adding a Prandtl number correction.

* *

* *

* *0 0y y

u T

y y

(6-78)

,

Re(Pr=1)

2

Lf xC Nu

(6-79)Reynolds analogy

1/2

, 0.664Ref x xC 1/3 1/ 20.332Pr Re Pr>0.6

xNu (6-82)

1 3

,

RePr 0.6 Pr 60

2

f x xC Nu (6-83)


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6.39


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fundamentals of convection

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