heat transfer

# 1

Heat Transfer Su Yongkang

School of Mechanical Engineering

HEAT TRANSFER

CHAPTER 6

Introduction to convection

# 2



Boundary Layer Similarity Parameters

• The boundary layer equations (velocity, mass, energy continuity) represent low speed, forced convection flow.

• Advection terms on the left side and diffusion terms on the right side of each equation, such as: Advection Diffusion

• Non-dimensionalize the equations by setting:

2*s* / and T

T-T

and

)( velocity freestream theis V where

surface theoflength sticcharacteri is L where

VpPT

T

UV

vvand

V

uu

L

yyand

L

xx

s

2

2

y

T

y

Tv

x

Tu

# 3



Boundary Layer Similarity Parameters (Cont’d)

• The boundary layer equations can be rewritten in terms of the non-dimensional variablesContinuity

x-momentum

energy

• With boundary conditions

0*

*

*

*

y

v

x

u

2*

*2

*

*

*

**

*

**

y

u

VLx

P

y

uv

x

uu

2*

*2

*

**

*

**

y

T

VLy

Tv

x

Tu

1 ),(

] if 1[ )(

),( :Freestream

0 )0,( ; 0 )0,( ; 0 )0,( :Wall

***

****

***

******

yxT

UVV

xUyxu

yxTyxvyxu

# 4



Boundary Layer Similarity Parameters (Cont’d)

• From the non-dimensionalized boundary layer equations, dimensionless groups can be seen

Reynolds #

Prandtl #

Substituting gives the boundary layer equations:

VLLRe

0*

*

*

*

y

v

x

u

2*

*2

*

*

*

**

*

**

Re

1

y

u

x

P

y

uv

x

uu

L

PrRe

1

2*

*2

*

**

*

**

y

T

y

Tv

x

Tu

L

Continuity:

x-momentum:

Energy:

rP

# 5



Back to the convection heat transfer problem…

• Solutions to the boundary layer equations are of the form:

• Rewrite the convective heat transfer coefficient

• Define the Nusselt number as:

*

****

*

*

*

****

Pr,,Re,,

plateflat for 0 : where,Re,,

dx

dPyxfT

dx

dP

dx

dPyxfu

L

L

L

TT

Ly

TTTT

TT

k

TT

yT

k

TT

qh s

y

s

s

s

f

s

y

f

s

xx

0

0

0*

*

*

y

fx

y

T

L

kh

*

**

0*

*Pr,,Re,

* dx

dPxf

y

TL

yNu

# 6



Nusselt number for a prescribed geometry

(For a prescribed geometry, is known)

Many convection problems are solved using Nusselt number correlations incorporating Reynolds and Prandtl numbers

• The Nusselt number is to the thermal boundary layer what the friction coefficient is to the velocity boundary layer.

Average Pr,Re k

Lh

Local Pr,Re, k

hL

f

*

f

L

L

fNu

xfNu

*

*

dx

dP

*

**

0

*

*

2 ,Re, Re

2

2dx

dPxf

y

u

VC L

yL

sf

# 7



Heat transfer coefficient, simple example

• Given:Air at 20ºC flowing over heated flat plate at 100ºC. Experimental measurements of temperatures at various distances from the surface are as shown

• Find: convective heat transfer coefficient, h

tsmeasuremen alExperiment

# 8



Heat transfer coefficient, simple example

• Solution:Recall that h is computed by

• From Table A-4 in Appendix, at a mean fluid temperature

(average of free-stream and surface temperatures)

the air conductivity, k is 0.028 W/m-K

• Temperature gradient at the plate surface from experimental data is -66.7 K/mm = -66,700 K/m

• So, convective heat transfer coefficient is:

0

TT

y

Tk

hs

yf

x

Km

W 345.23

80

)66700(0.028-

2

h

2msTT

T

CT 602)10020(m

# 9



Example: Experimental results for heat transfer over a flat plate with an extremely rough surface were found to be correlated by an expression of the form

where is the local value of the Nusselt number at a position x measured from the leading edge of the plate. Obtain an expression for the ratio of the average heat transfer coefficient to the local coefficient.

3/19.0 PrRe04.0 xxNu

xNu

# 10



# 11



• General boundary layer equations

• Nusselt number for heat transfer coefficient in the thermal boundary layer

• Many convection problems are solved using Nusselt number correlations incorporating Reynolds and Prandtl numbers.

SUMMARY

0

y

v

x

u

2

2

1

y

u

x

P

y

uv

x

uu

0

y

P

2

2

2

y

u

cy

T

y

Tv

x

Tu

p

Average Pr,Re

Local Pr,Re,

*

Lf

Lf

fk

LhNu

xfk

LhNu

# 12



Momentum and Heat Transfer Analogy

# 13




Where we’ve been ……

• Development of convective transport of heat transfer equations.

Where we’re going:

• Momentum and heat transfer (Reynolds) analogy.

# 14




KEY POINTS THIS SECTION• Physical significance of the dimensionless

parameters (Reynolds, Prandtl numbers)

• Describe how the convective heat transfer equations can be related (Heat Transfer Analogy)

• Review of the general convection equations, prepare for application to external and internal flows.

# 15



Review boundary layer equations for heat transfer

• For heat transfer (conservation of energy)

usually small, except for high speed or highly viscous flow

Begin to see where the momentum and heat transfer analogy is going …..

2

2

2

y

u

cy

T

y

Tv

x

Tu

p

2*

*2

*

*

*

**

*

**

Re

1

y

u

x

P

y

uv

x

uu

L

PrRe

1

2*

*2

*

**

*

**

y

T

y

Tv

x

Tu

L

2

2

1

y

u

x

P

y

uv

x

uu

Momentum equation

# 16



• If and , we obtain:

These two equations are of precisely the same form.

We know that if , .

The boundary conditions for these two equations are:

The boundary conditions are equivalent. Therefore, the boundary layer velocity and temperature profiles must be of the same functional form.

0/* dxdp 1Pr

Re

1 2*

*2

*

**

*

**

y

T

y

Tv

x

Tu

L

2*

*2

*

**

*

**

Re

1

y

u

y

uv

x

uu

L

0/* dxdp Vu

1 ),(

1; )(

),( :Freestream

0 )0,(

; 0 )0,( :Wall

***

****

***

***

yxT

V

xUyxu

yxT

yxu

# 17



Momentum and Heat Transfer Analogy(continued)

For the solution, the function f must be the same.

As we know,

We conclude that

Pr,Re,,T

Re,,***

***

L

L

yxf

yxfu

LLyL

sf xf

y

u

VC Re,

Re

2

Re

2

2

*

0

*

*

2

Pr,Re, Nu *

0

*

*

*

L

y

xfy

T

NuC Lf

2

Re

# 18



Momentum and Heat Transfer Analogy(continued)

• Define the Stanton number St,

• The analogy takes the form

• The restrictions: the validity of the boundary layer approximations, and .

The modified Reynolds, or Chilton-Colburn, analogy has the form

PrRe

Nu

Vc

hSt

p

StC f 2

Reynolds analogy

0/* dxdp 1Pr

2/ 3Pr (0. 6 Pr 60)2fc St j

Colburn j factor

For laminar flow, it’s only appropriate when 0/* dxdp

# 19



Boundary Layer Similarity Parameters

• Recall the non-dimensional parameters

Reynolds # - Ratio of the inertia to the viscous forces of a fluid flow

• Prandtl #Ratio of the momentum to the thermal diffusivity in a fluid flow.

For laminar boundary layers,

Where n is a positive exponent.

Ls

l VL

LV

LV

F

FRe

/

/2

2

k

cpPr

n

th

Pr

# 20



Example: Forced air at and is used to ℃cool electronic elements on a circuit board. One such element is a chip, 4 mm by 4 mm, located 120 mm from the leading edge of the board. Experiments have revealed that flow over the board is disturbed by the elements and that convection heat transfer is correlated by an expression of the form

Estimate the surface temperature of the chip if it is dissipating 30 mW.

3/185.0 PrRe04.0 xxNu

25T smV /10

# 21



# 22



• General boundary layer equations

• Nusselt number for heat transfer coefficient in the thermal boundary layer

• Empirical evaluation of Nusselt number involves correlations incorporating Re and Pr

Convective Transport Equations Summary

0

y

v

x

u

2

2

1

y

u

x

P

y

uv

x

uu

0

y

P

2

2

2

y

u

cy

T

y

Tv

x

Tu

p

Average Pr,Re

Local Pr,Re,

*

Lf

Lf

fk

LhNu

xfk

LhNu

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

coefficient )( TThq s

# 23



Convective Transport Equations Summary (Cont’d)

VLVL

L Re

th

Pr Pr

np

k

μ c

Reynolds #

Prandtl #

Reynolds analogy

StC f 2

# 24



Nothing Is Impossible To A Willing Heart

heat transfer

Documents