ConvectionConvectionHeat HeatHeatTransfer Transfer...

ConvectionConvectionConvectionConvectionConvectionConvectionConvectionConvection HeatHeatHeatHeatHeatHeatHeatHeat TransferTransferTransferTransferTransferTransferTransferTransfer

-------- IntroductionIntroductionIntroductionIntroductionIntroductionIntroductionIntroductionIntroduction ---------------- IntroductionIntroductionIntroductionIntroductionIntroductionIntroductionIntroductionIntroduction --------

Profa. Flávia Zinani

PPGEM

[email protected]@unisinos.br –– 6A2346A234

What is convectionWhat is convection

Convection

IntroductionIntroduction

Introduction


Convection = Advection + Diffusion

Flow direction



Flow direction

Diffusion



Flow direction

Diffusion

Advection

ObjectiveObjective

Objective

To understand the physical

mechanisms that underlie the

processes of convection heat transfer.processes of convection heat transfer.

Boundary Layer FeaturesBoundary Layer Features

Boundary Layers: Physical Features

• Velocity Boundary Layer

– A consequence of viscous effects

associated with relative motion

between a fluid and a surface.

– A region of the flow characterized by

shear stresses and velocity gradients.

– A region between the surface

and the free stream whose ( )0.99

u yδ → =and the free stream whose

thickness increases in

the flow direction.

δ( )

0.99u y

uδ

∞

→ =

– Why does increase in the flow direction? δ

– Manifested by a surface shear

stress that provides a drag

force, .sτDF

0s y

u

yτ µ =

∂=

∂

s

D s sA

F dAτ= ∫

– How does vary in the flow

direction? Why?sτ

Boundary Layer Features (cont.)Boundary Layer Features (cont.)

• Thermal Boundary Layer

– A consequence of heat transfer

between the surface and fluid.

– A region of the flow characterized

by temperature gradients and heat

fluxes.

– A region between the surface and

the free stream whose thickness

increases in the flow direction.tδ

( )0.99

s

t

s

T T y

T Tδ

∞

−→ =

−increases in the flow direction.

– Why does increase in the

flow direction?tδ

– Manifested by a surface heat

flux and a convection heat

transfer coefficient h .sq′′

0s f y

Tq k

y=

∂′′ = −

∂

0/f y

s

k T yh

T T

=

∞

− ∂ ∂≡

−

– If is constant, how do and

h vary in the flow direction?

( )sT T∞−sq′′

( )s sq h T T∞′′ = −

Boundary Layer Features (cont.)Boundary Layer Features (cont.)

• Thermal Boundary Layer

– A consequence of heat transfer

between the surface and fluid.

– A region of the flow characterized

by temperature gradients and heat

fluxes.

– A region between the surface and

the free stream whose thickness

increases in the flow direction.tδ

( )0.99

s

t

s

T T y

T Tδ

∞

−→ =

−increases in the flow direction.

– Why does increase in the

flow direction?tδ

– Manifested by a surface heat

flux and a convection heat

transfer coefficient h .sq′′

0s f y

Tq k

y=

∂′′ = −

∂

0/f y

s

k T yh

T T

=

∞

− ∂ ∂≡

−

– If is constant, how do and

h vary in the flow direction?

( )sT T∞−sq′′

Boundary Layer FeaturesBoundary Layer Features

Boundary Layers: Physical Features

Local and Average CoefficientsLocal and Average Coefficients

Distinction between Local andAverage Heat Transfer Coefficients

• Local Heat Flux and Coefficient:

( )∞′′ = −s sq h T T

• Average Heat Flux and Coefficient for a Uniform Surface Temperature:• Average Heat Flux and Coefficient for a Uniform Surface Temperature:

( )s sq hA T T∞= −

s sAq q dA′′= ∫ ( )

ss sAT T hdA∞= − ∫

1s sA

s

h hdAA

= ∫

• For a flat plate in parallel flow:

1 L

oh hdx

L= ∫

ProblemProblem

Main problem

The main problem in convection heat

transfer is the determination of the

convection heat transfer coefficient, h.

h is a function of the temperature profile h is a function of the temperature profile

at the thermal boundary layer.

0/f y

s

k T yh

T T

=

∞

− ∂ ∂≡

−

TransitionTransition

Boundary Layer Transition

• How would you characterize conditions in the laminar region of boundary layer• How would you characterize conditions in the laminar region of boundary layer

development? In the turbulent region?

• What conditions are associated with transition from laminar to turbulent flow?

• Why is the Reynolds number an appropriate parameter for quantifying transition

from laminar to turbulent flow?

• Transition criterion for a flat plate in parallel flow:

, critical Rey nolds numberRe cx c

u xρµ∞≡ →

location at which transition to turbulence beginscx →5 6

,~ ~

10 Re 3 x 10x c< <

TurbulenceTurbulence

•• What is turbulence?What is turbulence?

•• Effect of turbulence on Effect of turbulence on

NavierNavier--Stokes equations.Stokes equations.


•• Reynolds averaging.Reynolds averaging.

•• Reynolds stresses.Reynolds stresses.

InstabilityInstability

•• All flows become unstable above a certain Reynolds number.All flows become unstable above a certain Reynolds number.

•• At low Reynolds numbers flows are laminar.At low Reynolds numbers flows are laminar.

•• For high Reynolds numbers flows are turbulent.For high Reynolds numbers flows are turbulent.

•• The transition occurs anywhere between 2000 and 1E6, The transition occurs anywhere between 2000 and 1E6,

depending on the flow.depending on the flow.

•• For laminar flow problems, flows can be solved using the For laminar flow problems, flows can be solved using the


•• For laminar flow problems, flows can be solved using the For laminar flow problems, flows can be solved using the

usual conservation equations.usual conservation equations.

•• For turbulent flows, the computational effort involved in For turbulent flows, the computational effort involved in

solving those for all time and length scales is prohibitive.solving those for all time and length scales is prohibitive.

•• An engineering approach to calculate timeAn engineering approach to calculate time--averaged flow averaged flow

fields for turbulent flows is a theme of research.fields for turbulent flows is a theme of research.

InstabilityInstability


•• Unsteady, Unsteady, aperiodicaperiodic motion in which all three velocity motion in which all three velocity

components fluctuate, mixing matter, momentum, and energy.components fluctuate, mixing matter, momentum, and energy.

•• Decompose velocity into mean and fluctuating parts:Decompose velocity into mean and fluctuating parts:

UUii(t) (t) ≡≡ UUii + + uuii(t).(t).

What is turbulence?What is turbulence?


•• Similar fluctuations for pressure, temperature, and species Similar fluctuations for pressure, temperature, and species

concentration concentration values.values.

Time

Examples of simple turbulent flowsExamples of simple turbulent flows

•• Some examples of simple turbulent flows are a jet entering a domain with Some examples of simple turbulent flows are a jet entering a domain with

stagnant fluid, a mixing layer, and the wake behind objects such as stagnant fluid, a mixing layer, and the wake behind objects such as

cylinders.cylinders.

•• Such flows are often used as test cases to validate the ability of Such flows are often used as test cases to validate the ability of

computational fluid dynamics software to accurately predict fluid flows.computational fluid dynamics software to accurately predict fluid flows.


jet mixing layer wake


•• The photographs show the flow in The photographs show the flow in

a boundary layer.a boundary layer.

•• Below ReBelow Recritcrit the flow is laminar the flow is laminar

and adjacent fluid layers slide past and adjacent fluid layers slide past

each other in an orderly fashion. each other in an orderly fashion.

•• The flow is stable. Viscous effects The flow is stable. Viscous effects


•• The flow is stable. Viscous effects The flow is stable. Viscous effects

lead to small disturbances being lead to small disturbances being

dissipated.dissipated.

•• Above the transition point ReAbove the transition point Recritcrit

small disturbances in the flow small disturbances in the flow

start to grow. start to grow.

•• A complicated series of events A complicated series of events

takes place that eventually leads takes place that eventually leads

to the flow becoming fully to the flow becoming fully

turbulent.turbulent.

Transition in boundary layer flow over flat plateTransition in boundary layer flow over flat plate


Transition in boundary layer flow over flat plateTransition in boundary layer flow over flat plate


Turbulent spots Fully turbulent flowT-S waves

Turbulent boundary layerTurbulent boundary layer

Top view


Merging of turbulent spots and transition to turbulence

in a natural flat plate boundary layer.

Side view

Turbulent boundary layerTurbulent boundary layer


Close-up view of the turbulent boundary layer.

Transition in a channel flowTransition in a channel flow

•• Instability and turbulence is also Instability and turbulence is also

seen in internal flows such as seen in internal flows such as

channels and ducts.channels and ducts.

•• The Reynolds number is constant The Reynolds number is constant

throughout the pipe and is a throughout the pipe and is a

function of flow rate, fluid function of flow rate, fluid

properties and diameter.properties and diameter.


properties and diameter.properties and diameter.

•• Three flow regimes are shown:Three flow regimes are shown:

–– Re < 2200 with laminar flow.Re < 2200 with laminar flow.

–– Re = 2200 with a flow that Re = 2200 with a flow that

alternates between turbulent alternates between turbulent

and laminar. This is called and laminar. This is called

transitional flow.transitional flow.

–– Re > 2200 with fully turbulent Re > 2200 with fully turbulent

flow.flow.

Transition in a jet flowTransition in a jet flow


LargeLarge--scale vs. smallscale vs. small--scale structurescale structure


Large Structure Small Structure

Weddell Sea off AntarcticaWeddell Sea off Antarctica


2929

Alaska's Aleutian IslandsAlaska's Aleutian Islands

•• As air flows over and As air flows over and

around objects in its around objects in its

path, spiraling eddies, path, spiraling eddies,

known as Von Karman known as Von Karman

vortices, may form.vortices, may form.

•• The vortices in this The vortices in this

image were created when image were created when


3030

image were created when image were created when

prevailing winds prevailing winds

sweeping east across the sweeping east across the

northern Pacific Ocean northern Pacific Ocean

encountered Alaska's encountered Alaska's

Aleutian Islands Aleutian Islands


3131Alexander Selkirk Island in the southern Pacific Ocean20 km


3232

Smoke ringSmoke ring


A smoke ring (green) impinges on a plate where it interacts with the slow moving

smoke in the boundary layer (pink). The vortex ring stretches and new rings form.

The size of the vortex structures decreases over time.

Homogeneous, decaying, gridHomogeneous, decaying, grid--generated turbulencegenerated turbulence


Turbulence is generated at the grid as a result of high stresses in the immediate vicinity of the

grid. The turbulence is made visible by injecting smoke into the flow at the grid. The eddies are

visible because they contain the smoke. Beyond this point, there is no source of turbulence as the

flow is uniform. The flow is dominated by convection and dissipation. For homogeneous

decaying turbulence, the turbulent kinetic energy decreases with distance from grid as x-1 and the

turbulent eddies grows in size as x1/2.

Flow transitions around a cylinderFlow transitions around a cylinder

•• For flow around a cylinder, the flow starts separating at Re = 5. For Re below 30, For flow around a cylinder, the flow starts separating at Re = 5. For Re below 30,

the flow is stable. Oscillations appear for higher Re.the flow is stable. Oscillations appear for higher Re.

•• The separation point moves upstream, increasing drag up to Re = 2000.The separation point moves upstream, increasing drag up to Re = 2000.


Re = 9.6 Re = 13.1

Re = 30.2 Re = 2000

Re = 26

Re = 10,000

Turbulence: high Reynolds numbersTurbulence: high Reynolds numbers

Turbulent flows always occur at Turbulent flows always occur at high Reynolds numbershigh Reynolds numbers. They are caused by the . They are caused by the

complex interaction between the viscous terms and the inertia terms in the complex interaction between the viscous terms and the inertia terms in the

momentum equations.momentum equations.


3636

Laminar, low Reynolds

number free stream flow

Turbulent, high Reynolds

number jet

Turbulent flows are chaoticTurbulent flows are chaotic


3737

One characteristic of turbulent flows is their One characteristic of turbulent flows is their irregularityirregularity or randomness. A full or randomness. A full

deterministic approach is very difficult. Turbulent flows are usually described deterministic approach is very difficult. Turbulent flows are usually described

statistically. Turbulent flows are always chaotic. But not all chaotic flows are statistically. Turbulent flows are always chaotic. But not all chaotic flows are

turbulent.turbulent.

Turbulence: diffusivityTurbulence: diffusivity


3838

The The diffusivity diffusivity of turbulence causes rapid mixing and increased rates of of turbulence causes rapid mixing and increased rates of

momentum, heat, and mass transfer. A flow that looks random but does not exhibit momentum, heat, and mass transfer. A flow that looks random but does not exhibit

the spreading of velocity fluctuations through the surrounding fluid is not turbulent. the spreading of velocity fluctuations through the surrounding fluid is not turbulent.

If a flow is chaotic, but not diffusive, it is not turbulent. If a flow is chaotic, but not diffusive, it is not turbulent.

Turbulence: dissipationTurbulence: dissipation


3939

Turbulent flows are Turbulent flows are dissipativedissipative. Kinetic energy gets converted into heat . Kinetic energy gets converted into heat

due to viscous shear stresses. Turbulent flows die out quickly when no due to viscous shear stresses. Turbulent flows die out quickly when no

energy is supplied. Random motions that have insignificant viscous losses, energy is supplied. Random motions that have insignificant viscous losses,

such as random sound waves, are not turbulent.such as random sound waves, are not turbulent.

Turbulence: rotation and vorticityTurbulence: rotation and vorticity

Turbulent flows are Turbulent flows are rotationalrotational; that is, they have non; that is, they have non--zero vorticity. Mechanisms zero vorticity. Mechanisms

such as the stretching of threesuch as the stretching of three--dimensional vortices play a key role in turbulence.dimensional vortices play a key role in turbulence.


4040

Vortices

What is turbulence?What is turbulence?

•• Turbulent flows have the following Turbulent flows have the following characteristics:characteristics:–– One characteristic of turbulent flows is their One characteristic of turbulent flows is their irregularityirregularity or randomness. A full or randomness. A full

deterministic approach is very difficult. Turbulent flows are usually described deterministic approach is very difficult. Turbulent flows are usually described statistically. Turbulent flows are always statistically. Turbulent flows are always chaoticchaotic. But not all chaotic flows are . But not all chaotic flows are turbulent. Waves in the ocean, for example, can be chaotic but are not turbulent. Waves in the ocean, for example, can be chaotic but are not necessarily turbulent.necessarily turbulent.

–– The The diffusivity diffusivity of turbulence causes rapid mixing and increased rates of of turbulence causes rapid mixing and increased rates of momentum, heat, and mass transfer. A flow that looks random but does not momentum, heat, and mass transfer. A flow that looks random but does not


momentum, heat, and mass transfer. A flow that looks random but does not momentum, heat, and mass transfer. A flow that looks random but does not exhibit the spreading of velocity fluctuations through the surrounding fluid is exhibit the spreading of velocity fluctuations through the surrounding fluid is not turbulent. If a flow is chaotic, but not diffusive, it is not turbulent. The trail not turbulent. If a flow is chaotic, but not diffusive, it is not turbulent. The trail left behind a jet plane that seems chaotic, but does not diffuse for miles is then left behind a jet plane that seems chaotic, but does not diffuse for miles is then not turbulent.not turbulent.

–– Turbulent flows always occur at Turbulent flows always occur at high Reynolds numbershigh Reynolds numbers. They are caused by . They are caused by the complex interaction between the viscous terms and the inertia terms in the the complex interaction between the viscous terms and the inertia terms in the momentum equations. momentum equations.

–– Turbulent flows are Turbulent flows are rotationalrotational; that is, they have non; that is, they have non--zero zero vorticityvorticity. . Mechanisms such as the stretching of threeMechanisms such as the stretching of three--dimensional vortices play a key dimensional vortices play a key role in turbulence.role in turbulence.

What is turbulence? What is turbulence? -- ContinuedContinued

–– Turbulent flows are Turbulent flows are dissipativedissipative. Kinetic energy gets converted into . Kinetic energy gets converted into

heat due to viscous shear stresses. Turbulent flows die out quickly heat due to viscous shear stresses. Turbulent flows die out quickly

when no energy is supplied. Random motions that have insignificant when no energy is supplied. Random motions that have insignificant

viscous losses, such as random sound waves, are not turbulent.viscous losses, such as random sound waves, are not turbulent.

–– Turbulence is a Turbulence is a continuum continuum phenomenon. Even the smallest eddies are phenomenon. Even the smallest eddies are

significantly larger than the molecular scales. Turbulence is therefore significantly larger than the molecular scales. Turbulence is therefore


significantly larger than the molecular scales. Turbulence is therefore significantly larger than the molecular scales. Turbulence is therefore

governed by the equations of fluid mechanics.governed by the equations of fluid mechanics.

–– Turbulent flows are flows. Turbulence is a Turbulent flows are flows. Turbulence is a feature of fluid flowfeature of fluid flow, not of , not of

the fluid. When the Reynolds number is high enough, most of the the fluid. When the Reynolds number is high enough, most of the

dynamics of turbulence are the same whether the fluid is an actual fluid dynamics of turbulence are the same whether the fluid is an actual fluid

or a gas. Most of the dynamics are then independent of the properties of or a gas. Most of the dynamics are then independent of the properties of

the fluid.the fluid.


Boundary Layer Transition

• How would you characterize conditions in the laminar region of boundary layer• How would you characterize conditions in the laminar region of boundary layer

development? In the turbulent region?

• What conditions are associated with transition from laminar to turbulent flow?

• Why is the Reynolds number an appropriate parameter for quantifying transition

from laminar to turbulent flow?

• Transition criterion for a flat plate in parallel flow:

, critical Rey nolds numberRe cx c

u xρµ∞≡ →

location at which transition to turbulence beginscx →5 6

,~ ~

10 Re 3 x 10x c< <

Transition (cont.)Transition (cont.)

What may be said about transition if ReL < Rex,c? If ReL > Rex,c?

• Effect of transition on boundary layer thickness and local convection coefficient:

Why does transition provide a significant increase in the boundary layer thickness?

Why does the convection coefficient decay in the laminar region? Why does it increase

significantly with transition to turbulence, despite the increase in the boundary layer

thickness? Why does the convection coefficient decay in the turbulent region?

Boundary Layer EquationsBoundary Layer Equations

The Boundary Layer Equations

• The equations in Cartesian coordinates, 2-D, steady, incompressible flow with

constant fluid properties. ( ),, pc kµ

• Apply conservation of mass, Newton’s 2nd Law of Motion and conservation of energy

to a differential control volume.



• Consider concurrent velocity and thermal boundary layer development for steady,

two-dimensional, incompressible flow with constant fluid properties and

negligible body forces.( ),, pc kµ

negligible body forces.


to a differential control volume and invoke the boundary layer approximations.

Velocity Boundary Layer:

2 2

2 2, ∞∂ ∂ ∂

≈∂ ∂ ∂u u p dp

x y x dx≪

Thermal Boundary Layer:2 2

2 2

∂ ∂∂ ∂T T

x y≪



• The equations in Cartesian coordinates, 2-D, steady, incompressible flow with

constant fluid properties. ( ),, pc kµ


to a differential control volume.

Boundary Layer Equations (cont.)Boundary Layer Equations (cont.)

• Conservation of Mass:

0∂ ∂

+ =∂ ∂u v

x yIn the context of flow through a differential control volume, what is the physical

significance of the foregoing terms, if each is multiplied by the mass density of

the fluid?

• Newton’s Second Law of Motion:

2

x-directi :on

1 ∞∂ ∂ ∂+ = − +

u u dp uu v ν

2

1 ∞∂ ∂ ∂+ = − +

∂ ∂ ∂u u dp u

u vx y dx y

νρ

What is the physical significance of each term in the foregoing equation?

Why can we express the pressure gradient as dp∞/dx instead of / ?∂ ∂p x

Boundary Layer Equations (cont.)Boundary Layer Equations (cont.)

What is the physical significance of each term in the foregoing equation?

What is the second term on the right-hand side called and under what conditions

may it be neglected?

• Conservation of Energy:

22

2

∂ ∂ ∂ ∂+ = + ∂ ∂ ∂ ∂ p

T T T uu vx y y c y

να

Similarity ConsiderationsSimilarity Considerations

Boundary Layer Similarity• As applied to the boundary layers, the principle of similarity is based on

determining similarity parameters that facilitate application of results obtained

for a surface experiencing one set of conditions to geometrically similar surfaces

experiencing different conditions. (Recall how introduction of the similarity

parameters Bi and Fo permitted generalization of results for transient, one-

dimensional condition).

• Dependent boundary layer variables of interest are:

and or s q hτ ′′

• For a prescribed geometry, the corresponding independent variables are:

Geometrical: Size (L), Location (x,y)

Hydrodynamic: Velocity (V)

Fluid Properties:

Hydrodynamic: ,

Thermal : ,pc k

ρ µ

( )( )

Hence,

, , , , ,

, , , ,s

u f x y L V

f x L V

ρ µ

τ ρ µ

=

=

SimilaritySimilarity Considerations (cont.)Considerations (cont.)

( )( )

and

, , , , , , ,

, , , , , ,

p

p

T f x y L V c k

h f x L V c k

ρ µ

ρ µ

=

=

• Key similarity parameters may be inferred by non-dimensionalizing the momentum

and energy equations.

• Recast the boundary layer equations by introducing dimensionless forms of the

independent and dependent variables.* *x yx y

L L

u v

≡ ≡

* *

* s

s

u vu v

V V

T TT

T T∞

≡ ≡

−≡

−

• Neglecting viscous dissipation, the following normalized forms of the x-momentum

and energy equations are obtained: * * * 2 *

* *

* * * *2

* * 2 ** *

* * *2

1

Re

1

Re Pr

L

L

u u dp uu v

x y dx y

T T Tu v

x y y

∂ ∂ ∂+ = − +

∂ ∂ ∂

∂ ∂ ∂+ =

∂ ∂ ∂

SimilaritySimilarity Considerations (cont.)Considerations (cont.)

• Parâmetros de similaridade são importantes, pois nos permitem a

utilização dos resultados, obtidos em uma superfície submetida a um utilização dos resultados, obtidos em uma superfície submetida a um

conjunto de condições convectivas, em superfícies geometricamente

similares submetidas a condições inteiramente diferentes. Essas

condições podem variar, por exemplo, com a natureza do fluido,

com a velocidade do fluido e/ou com o tamanho característico, L.

Contanto que os parâmetros de similaridade e as condições de

contorno adimensionais sejam os mesmos para os dois conjuntos de

condições, as soluções das equações diferenciais para a velocidade e

a temperatura adimensionais também serão as mesmas.

Similarity Considerations (cont.)Similarity Considerations (cont.)

Reynolds NumbeRe the

Pr

r

Prandtl Number the

L

p

VL VL

v

c v

k

ρµµ

α

≡ = →

≡ = →

• For a prescribed geometry,

( )* * *, ,ReLu f x y=

*

*

*

0 0

s

y y

u V u

y L y

µτ µ

= =

∂ ∂ = = ∂ ∂

How may the Reynolds and Prandtl numbers be interpreted physically?

*0 0y yy L y

= =∂ ∂

The dimensionless shear stress, or local friction coefficient, is then

*

*

2 *

0

2

/ 2 Re

sf

L y

uC

V y

τρ

=

∂≡ =

∂

( )*

**

*

0

,ReLy

uf x

y=

∂=

∂

( )*2,Re

Ref L

L

C f x=

What is the functional dependence of the average friction coefficient?

Similarity Considerations (cont.)Similarity Considerations (cont.)

• For a prescribed geometry,

( )* * *, ,Re ,PrLT f x y=

( )( ) **

* *0

* *

00

/f y f fs

s s yy

k T y k kT T T Th

T T L T T y L y

= ∞

∞ ∞ ==

− ∂ ∂ − ∂ ∂= = − = +

− − ∂ ∂

The dimensionless local convection coefficient is then

( )*

**

*

0

,Re ,PrL

f y

hL TNu f x

k y=

∂≡ = =

∂

local Nusselt nu mberNu→

What is the functional dependence of the average Nusselt number?

How does the Nusselt number differ from the Biot number?

Significado dos números adimensionaisSignificado dos números adimensionais

•• Número de Reynolds: razão entre forças de inércia e Número de Reynolds: razão entre forças de inércia e

forças viscosas:forças viscosas:


•• Número de Número de PrandtlPrandtl: razão entre a difusividade de : razão entre a difusividade de

quantidade de movimento e a difusividade térmica. quantidade de movimento e a difusividade térmica.

Está relacionado ao crescimento relativo entre as Está relacionado ao crescimento relativo entre as

camadascamadas--limite fluidodinâmica e térmica:limite fluidodinâmica e térmica:

•• PrPr<<1: metais líquidos, difusão térmica mais eficiente <<1: metais líquidos, difusão térmica mais eficiente

que difusão de momentum, que difusão de momentum, δδtt>>>>δδ..

•• PrPr≈≈1: gases, 1: gases, δδtt≈≈δδ..

•• PrPr>>1: óleos, difusão de momentum mais eficiente >>1: óleos, difusão de momentum mais eficiente

que difusão térmica, que difusão térmica, δδtt<<<<δδ..


•• Número de Número de NusseltNusselt: representa o : representa o gradiente de gradiente de

temperatura adimensional na superfícietemperatura adimensional na superfície, mede , mede

a transferência de calor por convecção que a transferência de calor por convecção que

ocorre nesta superfícieocorre nesta superfície..

MeaningMeaning ofof dimensionlessdimensionless numbersnumbers

Reynolds AnalogyReynolds Analogy

The Reynolds Analogy• Equivalence of dimensionless momentum and energy equations for

negligible pressure gradient (dp*/dx*~0) and Pr~1:

Advection terms Diffusion

* * 2 ** *

* * *2

1

Re

T T Tu v

x y y

∂ ∂ ∂+ =

∂ ∂ ∂

* * 2 ** *

* * *2

1

Re

u u uu v

x y y

∂ ∂ ∂+ =

∂ ∂ ∂

• Hence, for equivalent boundary conditions, the solutions are of the same form:

* *

* *

* *

* *

0 0

Re

2

y y

f

u T

u T

y y

C Nu

= =

=

∂ ∂=

∂ ∂

=

Reynolds Analogy (cont.)Reynolds Analogy (cont.)

With Pr = 1, the Reynolds analogy, which relates important parameters of the velocity

and thermal boundary layers, is

2

fCSt=

or, with the defined asStanton number ,

p

h NuSt

Vc RePrρ≡ =

• Modified Reynolds (Chilton-Colburn) Analogy:

– An empirical result that extends applicability of the Reynolds analogy:– An empirical result that extends applicability of the Reynolds analogy:

23Pr 0.6 Pr 60

2

f

H

CSt j= ≡ < <

Colburn j factor for heat transfer

– Applicable to laminar flow if dp*/dx* ~ 0.

– Generally applicable to turbulent flow without restriction on dp*/dx*.

Reynolds Analogy (cont.)Reynolds Analogy (cont.)

Problem: Turbine Blade ScalingProblem: Turbine Blade Scaling

Problem 6.19: Determination of heat transfer rate for prescribed

turbine blade operating conditions from wind tunnel data

obtained for a geometrically similar but smaller

blade. The blade surface area may be assumed to be

directly proportional to its characteristic length . ( )sA L∝

SCHEMATIC:

ASSUMPTIONS: (1) Steady-state conditions, (2) Constant properties, (3) Surface area A is

directly proportional to characteristic length L, (4) Negligible radiation, (5) Blade shapes are

geometrically similar.

ANALYSIS: For a prescribed geometry,

( )LhL

Nu f Re , Pr .k

= =

Problem: Turbine Blade Scaling (cont.)Problem: Turbine Blade Scaling (cont.)

Therefore,

2 1Nu Nu=

( ) ( )2 2 2 1 1 1h L / k h L / k=

( )1 1 1

2 12 2 1 s,1

L L qh h

L L A T T∞= =

−

The Reynolds numbers for the blades are

( ) ( )2 2L,1 1 1 1 1 L,2 2 2 2 2Re V L / 15m / s Re V L / 15m / s .ν ν ν ν= = = =

Hence, with constant properties ( )1 2v v= , L,1 L,2Re Re .= Also, 1 2Pr Pr=

The heat rate for the second blade is then

( ) ( )( )

s,21 22 2 2 s,2 1

2 1 s,1

T TL Aq h A T T q

L A T T

∞∞

∞

−= − =

−

( )( )

( )s,22 1

s,1

T T 400 35q q 1500 W

T T 300 35

∞

∞

− −= =

− −

2q 2066 W.=

COMMENTS: (i) The variation in ν from Case 1 to Case 2 would cause ReL,2 to differ from

ReL,1. However, for air and the prescribed temperatures, this non-constant property effect is

small. (ii) If the Reynolds numbers were not equal ( ),1 2Re Re ,L L≠ knowledge of the specific form of

( ),Re PrLf would be needed to determine h2.

Problem: Nusselt NumberProblem: Nusselt Number

Problem 6.26: Use of a local Nusselt number correlation to estimate the

surface temperature of a chip on a circuit board.

KNOWN: Expression for the local heat transfer coefficient of air at prescribed velocity and

temperature flowing over electronic elements on a circuit board and heat dissipation rate for a 4 × 4 mm

chip located 120mm from the leading edge. chip located 120mm from the leading edge.

FIND: Surface temperature of the chip surface, Ts.

SCHEMATIC:

Problem: Nusselt Number (cont.)Problem: Nusselt Number (cont.)

PROPERTIES: Table A-4, Air (Evaluate properties at the average temperature of air in the boundary

layer. Assuming Ts = 45°C, Tave = (45 + 25)/2 = 35°C = 308K. Also, p = 1atm): ν = 16.69 ×

10-6

m2/s, k = 26.9 × 10

-3 W/m⋅K, Pr = 0.703.

ANALYSIS: From an energy balance on the chip,

conv gq E 30mW.= =ɺ

Newton’s law of cooling for the upper chip surface can be written as

= +

ASSUMPTIONS: (1) Steady-state conditions, (2) Power dissipated within chip is lost by convection

across the upper surface only, (3) Chip surface is isothermal, (4) The average heat transfer coefficient

for the chip surface is equivalent to the local value at x = L, (5) Negligible radiation.

s conv chipT T q / h A∞= + (2)

where 2

chipA .= ℓ

Assuming that the average heat transfer coefficient ( )h over the chip surface is equivalent to the local

coefficient evaluated at x = L, that is, ( )chip xh h L≈ , the local coefficient can be evaluated by

applying the prescribed correlation at x = L.

0.851/ 3x

xh x Vx

Nu 0.04 Prk ν

= =

0.851/ 3

Lk VL

h 0.04 PrL ν

=

Problem: Nusselt Number (cont.)Problem: Nusselt Number (cont.)

From Eq. (2), the surface temperature of the chip is

( )2-3 2sT 25 C 30 10 W/107 W/m K 0.004m 42.5 C.= + × ⋅ × =� �

COMMENTS: (1) The estimated value of Tave used to evaluate the air properties is reasonable.

(2) How else could chiph have been evaluated? Is the assumption of Lh h= reasonable?

( )0.85

1/ 3 2L -6 2

0.0269 W/m K 10 m/s 0.120 mh 0.04 0.703 107 W/m K.

0.120 m 16.69 10 m / s

⋅ × = = ⋅ ×

SuggestedSuggested problemsproblems

((6th 6th Ed. Ed. IncroperaIncropera, , etet al.al., 2008, 2008

Fundamentos de Transferência de Calor e de Massa)Fundamentos de Transferência de Calor e de Massa)

•• 6.11 6.11 andand 6.126.12

•• 6.156.15•• 6.156.15

•• 6.166.16

•• 6.286.28

•• 6.38 6.38 oror 6.396.39

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