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Fundamental Concepts of Convection :Fundamental Concepts of Convection :Flow and Thermal ConsiderationsFlow and Thermal Considerations

Chapter Six and Appendix DChapter Six and Appendix DSections 6.1 through 6.8 Sections 6.1 through 6.8

and D.1 through D.3and D.1 through D.3

6.1 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 thickness increases in the flow direction.δ

( ) 0.99u yu

δ∞

→ =

– Why does increase in the flow direction? δ

– Manifested by a surface shear stressthat provides a drag force, .

sτDF 0s y

uy

τ µ =∂

=∂ s

D s sA

F dAτ= ∫

– How does vary in the flowdirection? Why?

sτ

• 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 thicknessincreases in the flow direction.

tδ

– Why does increase in theflow direction?

tδ

– Manifested by a surface heat flux and a convection heat transfer coefficient h .

sq′′

( ) 0.99st

s

T T yT T

δ∞

−→ =

−

0s f yTq ky =

∂′′ = −∂

0/f y

s

k T yh

T T=

∞

− ∂ ∂≡

−– If is constant, how do and

h vary in the flow direction? ( )sT T∞− sq′′

6.2 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:

( )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 h d xL

= ∫

6.3 Boundary Layer Transition

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

• Why is the Reynolds number an appropriate parameter for quantifying transitionfrom 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< <

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 the convection coefficient decay in the turbulent region? Why does it increase significantly with transition to turbulence, despite the increase in the boundary layer thickness?

6.4 The Boundary Layer Equations

• Consider concurrent velocity and thermal boundary layer development for steady, two-dimensional, incompressible flow with constant fluid propertiesand negligible body forces.

( ),, pc kµ

• Apply conservation of mass, Newton’s 2nd Law of Motion and conservation of energy to a differential control volume and invoke the boundary layer approximations.

Velocity Boundary Layer:2 2

2 2 , u u p dpx y x dx

∞∂ ∂ ∂≈

∂ ∂ ∂

Thermal Boundary Layer:2 2

2 2∂ ∂∂ ∂

T Tx y

• Conservation of Mass:

0∂ ∂+ =

∂ ∂u vx y

In the context of flow through a differential control volume, what is the physicalsignificance of the foregoing terms, if each is multiplied by the mass density of the fluid?

• Newton’s Second Law of Motion:2

2x-directi :on 1u u dp uu 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

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 conditionsmay it be neglected?

• Conservation of Energy:

22

2

⎛ ⎞∂ ∂ ∂ ∂+ = + ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠p

T T T uu vx y y c y

να

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

determining similarity parameters that facilitate application of results obtainedfor a surface experiencing one set of conditions to geometrically similar surfacesexperiencing different conditions.

• 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

=

=

( ) ( )and , , , , , , , ; , , , , , ,p pT f x y L V c k h f x L V c kρ µ ρ µ= =

• Key similarity parameters may be inferred by non-dimensionalizing the momentumand energy equations.

• Recast the boundary layer equations by introducing dimensionless forms of theindependent and dependent variables.

* * * * * s

s

T Tx y u vx y u v TL L V V T T∞

−≡ ≡ ≡ ≡ ≡

−

• Neglecting viscous dissipation, the following normalized forms of the x-momentum and energy equations are obtained:

* * * 2 ** *

* * * *2

* * 2 ** *

* * *2

1Re

1Re Pr

L

L

u u dp uu vx y dx y

T T Tu vx y y

∂ ∂ ∂+ = − +

∂ ∂ ∂

∂ ∂ ∂+ =

∂ ∂ ∂

Re the

Pr

Reynolds Numb

the

er

Prandtl Number

L

p

VL VLv

c vk

ρµµ

α

≡ = →

≡ = →

• For a prescribed geometry, ( )* * *, ,ReLu f x y=

*

*

*0 0

sy y

u V uy L y

µτ µ= =

∂ ∂⎛ ⎞= = ⎜ ⎟∂ ∂⎝ ⎠

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

*

*

2 *0

2/ 2 Re

sf

L y

uCV yτ

ρ=

∂≡ =

∂( )

*

**

*0

,ReLy

u f xy

=

∂=

∂

( )*2 ,ReRef L

L

C f x=

What is the functional dependence of the average friction coefficient?

How may the Reynolds and Prandtl numbers be interpreted physically?

• 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 ThT T L T T y L y

= ∞

∞ ∞ ==

− ∂ ∂ − ∂ ∂= = − = +

− − ∂ ∂

The dimensionless local convection coefficient is then

( )*

**

*0

,Re ,PrLf y

hL TNu f xk 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?

6.6 Physical Significance of the Dimensionless Parameters6.6 Physical Significance of the Dimensionless Parameters

2 inertial forceRe/ viscous forceL

VL VV L

ρ ρµ µ

= = =

t

t

t

Pr

for a gas: for a liquid metal: for an oil:

n

t

δδ

δ δ

δ δ

δ δ

≈

≈

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

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

Advection terms Diffusion

* * 2 ** *

* * *21

R eT T Tu vx y y

∂ ∂ ∂+ =

∂ ∂ ∂

* * 2 ** *

* * *21

R eu u uu vx y y

∂ ∂ ∂+ =

∂ ∂ ∂

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

* *

* ** *

* *0 0

Re 2

y y

f

u Tu Ty y

C Nu

= =

∂ ∂= =

∂ ∂

=

With Pr = 1, the Reynolds analogy, which relates important parameters of the velocity and thermal boundary layers, is

2fC

St=

or, with the defined as,

Stanton nu mber

p

h NuStVc Re Prρ

≡ =

• Modified Reynolds (Chilton-Colburn) Analogy:– An empirical result that extends applicability of the Reynolds analogy:

23Pr 0.6 Pr 60

2f

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*.

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,

( )LhLNu f Re ,Pr .k

= =

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 2

L,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,2

2 1s,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. Howeverfor 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.