1

Spring 2003 ECE309: Chapters 9,10,11 1

Lumped System Analysis

bt

i

eTT

TtT −

∞

∞ =−−)(

PCV

Ahb

ρ=

Spring 2003 ECE309: Chapters 9,10,11 2

Criteria for Lumped System Analysis

LTk

Th

/body the withinconduction

body theof surface at the ConvectionBiNumberBiot

∆∆===

AVL / LengthsticCharacteri ==

1.0≤=kA

hVBi

2

Spring 2003 ECE309: Chapters 9,10,11 3

Convection Heat TransferConvection heat transfer strongly depends on the:

fluid properties,dynamic viscosity µ,thermal conductivity k,density ρ,specific heat Cp,fluid velocity V,geometry of the solid surface,type of fluid flow

0=∂∂−==

y

fluidCondConv y

Tkqq &&

∞

=

−∂∂−

=TT

yTkh

s

yfluid 0)/(

Spring 2003 ECE309: Chapters 9,10,11 4

Nusselt Number, Nu

The larger the Nusselt number, the more effective is the convection. A Nusselt number Nu = 1 for a fluid layer represents heat transfer by pure conduction.

Thqconv ∆=&

δT

kqcond

∆=&

Nu==k

h

q

q

cond

conv δ&

&

3

Spring 2003 ECE309: Chapters 9,10,11 5

Velocity Boundary Layer

The region of the flow above the plate bounded by δV in which the effects of the viscous shearing forces caused by fluid viscosity are felt is called the velocity boundary layer.

The thickness of the boundary layer, δv, is arbitrarily defined as the distance from the surface at which V = 0.99 V∞.

Spring 2003 ECE309: Chapters 9,10,11 6

Laminar & Turbulent FlowsLaminar flow is characterized by smooth streamlines and highly ordered motion,Turbulent Flow is characterized by velocity fluctuations and highly disordered motion.

νδ∞== V

fources viscouse

forcesinertia Re

∞

]/[m ),/( viscosityKinematic

[m] length, sticCharacteri

[m/s] velocity stream- Free

2 s

V

ρµνδ

5plateflat critical, 10 5 Re ×≈

Critical Reynolds number is the Reynolds number at which the flow becomes turbulent.

4

Spring 2003 ECE309: Chapters 9,10,11 7

Thermal Boundary LayerA thermal boundary layer develops when a fluid at a specified temperature flows over a surface which is at a different temperature.

The thickness of the thermal boundary layer δt at any location is defined as the distance from the surface at which the temperature difference (T-Ts) equals 0.99(T∞-Ts).

The thickness of the thermal boundary layer increases in the flow direction.

The relative thickness of the velocity and the thermal boundary layers is described by Prandtl Number.

k

CPµαν ==Pr

>><<≈

Oils. 1Pr

Metals,d Liqui1Pr

Gases, Most1Pr

Spring 2003 ECE309: Chapters 9,10,11 8

Flow Over Flat PlatesLaminar:

Turbulent:

Combined:

3/12/1 PrRe332.0 xx

x k

xhNu ==

3/12/1 PrRe664.0 Lk

hLNu ==

3/15/4 PrRe0296.0 xx

x k

xhNu ==

3/15/4 PrRe037.0 Lk

hLNu ==

)6.0(Pr ≥

≤≤×≤≤

75 10Re105

60Pr6.0

x

≤≤×≤≤

75 10Re105

60Pr6.0

L

+= ∫∫

L

x

turbulentx

x

arlax

cr

cr

dxhdxhL

h ,

0

min,

1

All properties are evaluated at film temperature as: 2∞+= TT

T sf

5

Spring 2003 ECE309: Chapters 9,10,11 9

Natural Convection, Physical MechanismThe magnitude of the natural convection heat transfer between a surface and a fluid is directly related to the mass flow rate of the fluid.

The higher the mass flow rate, the higher is the heat transfer rate.

The mass flow is established by the dynamic balance of buoyancy and friction.

Spring 2003 ECE309: Chapters 9,10,11 10

Grashof Number, GrGrashof Number, Gr, is a measure of the relative magnitudes of the buoyancy force and the opposing friction force acting on the fluid.

°−∆ ∞

/s][m ,/ , viscositykinematic

[m] length, sticCharacteri

C][ ,

[1/K] ,1 expansion, volumeof Coeff.

][m/s onaccelerati nalGravitatio

2

2

ρµνδ

βTTT

/T

g

s

The Grashof number provides the main criteria in determining whether the fluid flow is laminar or turbulent in natural convection.

Turbulent 10Gr

ar Lamin10Gr9

9

>≤

2

3

2

)(forces viscousforcesbuoyancy

Grν

δβνρ

βνρρ ∞−=∆=∆== TTgVTgVg s

6

Spring 2003 ECE309: Chapters 9,10,11 11

Natural Convection Over Surfaces

The values of the constants C and n depend on the geometry of the surface and the flow regime,

The value of n is usually ¼ for laminar flow and 1/3 for turbulent flow.

The value of the constant C is normally less than 1. (Table 11-1, P 587)

nn

k

hNu RaCPr)C(Gr === δ

Pr)(

Pr2

3

νδβ ∞−== TTg

GrRa s

Where Rayleigh number, Ra, is

Spring 2003 ECE309: Chapters 9,10,11 12

Natural Convection Inside Enclosures

Pr)(

2

321

νδβ TTg

Ra−=

°C][ surfaces cold &hot of esTemperatur ,

[m] plates cold &hot thebetween stance Di

21 TT

δ