me3122 handbook of heat transfer equations 2014

8/10/2019 ME3122 Handbook of Heat Transfer Equations 2014

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HANDBOOK OF EQUATIONS, TABLES

AND CHARTS FOR

ME3122/ME3122E HEAT TRANSFER

Department of Mechanical Engineering

National University of Singapore


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1

CONDUCTION HEAT TRANSFER

1stlaw of thermodynamics: WQdU

Conduction:

Convection:

Radiation: where -4-28 KWm10675 .

Control Volume:

Surface:

Heat Conduction Equation:

Cartesian:

Cylindrical:

Spherical:


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One-Dimensional Walls

Fin Equations:

kA/hPmmdx

d 22

2

2

where0

which has the general solution mxmx eCeC 21 .

Fin Efficiency:

Fin Effectiveness:

Overall Surface Efficiency:

ftf

t

t

max

t

o A

NA

hA

q

q

q

11

0

whereunfinnedft

ANAA .


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3

Lumped Capacitance Method:

,

, , ,

Other Equations (Thermal Properties):

Solids:

Free electrons:

Gases:

Joule heating: RIEg2

Interfaces:

Heat wave speed:

Two semi-infinite solids touch:


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4

CONVECTION HEAT TRANSFER

All symbols have their usual meaning.

ConstantsGravitational acceleration: g = 9.81 m/s2

Specific gas constant for air: R= 287 J/kgK

Definitions

Kinematic viscosity, /

Thermal diffusivity, / pck

Volumetric thermal expansion coefficient,TT p

11

for an ideal gas.

General

Dimensionless Groups

uc

h

PrRe

NuSt

PrGrRa

LTTgGr

k/hLNu

/Pr

/VL/VLRe

p

x

x

xx

LL

sL

L

L

Number,Stanton

Number,Rayleigh

Number,Grashof

Number,Nusselt

Number,Prandtl

Number,Reynolds

2

3

Tcm

y

u

VAm

RTpv

TThq

p

c

s

sectionaghflux throuenergyThermal

stress,Shear

rate,flowMass

:lawgasIdeal

Cooling,ofLawsNewton'


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2D Continuity Equation:

2D x-Momentum Equation:

2D Energy Equation:

where viscous dissipation,

2D Boundary Layer Equations:

x-Momentum Equation:

Energy Equation:

Integral Momentum Equation:

Integral Energy Equation:

Forced Convection Over External Surfaces

Generally,nm

PrReCNu

Forced Convection Over a Flat Plate:

For constant , .

0

y

v

x

u

Xy

u

x

u

x

p

y

u

vx

u

u

2

2

2

2

qy

T

x

Tk

y

Tv

x

Tucp

2

2

2

2

222

2y

v

x

u

x

v

y

u

2

2

y

u

y

uv

x

uu

2

2

y

T

y

Tv

x

Tu

00

)(

yy

udyuuu

dx

d

0

0

yy

TdyTTu

dx

d t


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6

Uniform Surface Temperature (Isothermal):

11

0

L

x

A

x dxhL

dAhA

h

For laminar flow (5Re 5 10

x ):

;5 3121 PrRex tx

For turbulent flow (5

Re 5 10x ):

Pr02960;05920;370 31

51 5451

xxxx,fxturb Re.NuRe.CRex.

For mixed boundary layer conditions ( 5105LRe ):

Uniform Surface Heat Flux (Isoflux):

For laminar flow (5Re 5 10

x ):

For turbulent flow ( 5Re 5 10x ):3

1

Pr03080 54xx Re.Nu

For Unheated Starting Length,xo, with laminar flow for both isothermal and isoflux

conditions:

Forced Convection Across Long Cylinders:

where Cand mare given byReD C m

0.4-4 0.989 0.330

4-40 0.911 0.385

40-4000 0.683 0.466

4000-40,000 0.193 0.61840,000-400,000 0.027 0.805

31

21

31

21

6640;3320 PrRe.k

LhNuPrRe.Nu LLxx

)8710370(;1742074080151 3

1

.LLLLL,f Re.Prk

LhNuReRe.C

31

21

4530 PrRe.Nu xx

21

21

3281;66402

2

xL,fx

x,s

x,f Re.CRe./u

C

31430

1

x/xNuNu oxxx o

31PrReC

k

DhNu

m

DD


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Forced Convection Across Spheres:

where all properties are evaluated at thefree-streamtemperature, except s, which isevaluated at the surface temperature of the sphere.

Forced Convection Across Non-Circular Cylinders

where Cand mare given by

Forced Convection Across Tube Banks

where all properties, exceptPrs, are evaluated at the average of the fluid inlet and outlet

temperatures,ReD,maxis based on the maximum fluid velocity, and C1and mare given in the

table below for number of tube rows for various aligned and staggered arrangements

of tubes.

41

403221 060402

s

.

DDD

PrRe.Re.

k

DhNu

31PrReC

k

DhNu

m

DD

41

360

1

s

.m

max,DDPr

PrPrReCNu


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(a)Aligned tube rows (b) Staggered tube rows

For : where C2for various is given in the table

below:

20220 LL NDND NuCNu


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Forced Convection in Tubes and Ducts

Friction factor,

PerimeterWettedAreasectional-Cross4Diameter,Hydraulic hD

For thermally fully-developed condition:

Laminar Flow (ReD2300):

Fully developed velocity profile:

where mean fluid velocity,

Friction factor, f= 64/ReD

Nuand ffor Fully-Developed Laminar Flow in Tubes of Various Cross-Sections

dx

dpr

r

mum

8

20

2

0

2

0

2

12)(

r

r

u

ru

m

2

or2

2

2

m

m

u

D

Lfp

/u

Ddx/dpf

0)()(

)()(

xTxT

x,rTxT

x ms

s


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Turbulent Flow (ReD> 2300):

For smooth tubes and ducts, the Dittus-Boelter equation:

with n= 0.4 for heating of fluid, and n= 0.3 for cooling of fluid

Friction factor for smooth tubes: 26417900 .Reln.f D

Friction factor for rough tubes of roughness e: 290745733251 .DRe/.D./eln.f

Reynolds-Colburn Analogy

For flow over a flat plate:

For flow in a tube or duct:

FREE CONVECTION

Generally,

flow.entfor turbul31andflow,laminarfor41with mmRaCPrGrCNu mLm

LL

Laminar Free Convection on an Isothermal Vertical Plate:

Boundary layer momentum equation:

Integral Momentum Equation for Free Convection BL:

Boundary layer thickness,

Critical Ra= 109.

Free Convection from an Isothermal Sphere

n

DD PrRe.Nu hh540230

2;2 3232 /CPr.St/CPr.St L,fLx,fx

832 /fPr.St

2

2

y

uTTg

y

uv

x

uu

00

2dyTTg

y

udyu

dx

d

s

414121 9520933 xGrPr.Prx.

541 101for4302 D/

DD GrPrGr.k

DhNu


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11

Free Convection from Isothermal Planes and Cylinders

Free Convection from a Vertical Plate with Constant Surface Heat Flux

where

mLm

LL RaCPrGrCNu

Geometry GrL Pr C mCharacteristic

Length

Vertical plane and cylinder104109 0.59 1/4

Height10 10 0.10 1/3

Horizontal cylinder

10-1010-2 0.68 0.058

Diameter

10- 10 1.02 0.148

102104 0.85 0.188

104109 0.53 1/4

1091012 0.13 1/3

Hot surface facing up or

cold surface facing down

104107 0.54 1/4Area/Perimeter

1071011 0.15 1/3

Hot surface facing down or

cold surface facing up1051011 0.27 1/4 Area/Perimeter

161341

11551

10102for170:Turbulent

1010for600:Laminar

Pr*GrPr*Gr.Nu

Pr*GrPr*.Gr.k

xhNu

xxx

xxx

x

2

4

k

xqg.NuGr*Gr sxxx


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12

RADIATION HEAT TRANSFER

Solid angle: 2/ rAn , ddsind

Radiation:where

-4-28

KWm10675

.

surssursr

sursr

"

rad

TTTTh

TThq

22

Spectral directional Intensity:

Diffuse emitter:

Blackbody: 4)( TTEb

Spectral black body emissive power

).)T/Cexp(

C)T,(E b, m(W/m

1

2

2

5

1

m.K104391and/mmW.107423where 42248

1 .C.C

Weins displacement law: m.K2898max T

Emissivity of real surfaces:

4)()()( TTETTE b

Absorptivity of surface:

GGabs

Semitransparent medium: 1


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Black Body Radiation Functions


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14

View factors:

32

133122132

AA

FAFAF ,

Radiation exchange between black-body surfaces:

Radiation network approach:

resistancespatial1re whe1

resistancesurface1where1

121

121

2112

FA/FA/

JJq

A/A/

JEq b

Radiation Exchange Network for a Two-Surface Enclosure

22

2

21111

1

4

2

4

112

111AFAA

TTq

,


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View factor for aligned parallel rectangles

View factor for coaxial parallel disks


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View factor for perpendicular rectangles with common edge

HEAT EXCHANGERS

Log Mean Temperature Difference, io

iolm

T/Tln

TTT

where fiR and fiR are fouling factors.

Capacity rate, pcmC , is infinite for a condensing or boiling fluid.

ooo

fo

w

i

fi

ii

BA

AhA

RR

A

R

Ah

TTq

11

exchangerheatindifferenceatureMax temper

fluid)(minimum

rateferheat transpossibleMax

rateferheat transActualess,Effectiven

T

max

min

max

minRatio,RateCapacityC

C

cm

cmCr

nits)Transfer Uof(NumberNTUC/UA min


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Correction Factor for Single Pass Cross-Flow Heat

Exchangers with the Shell Side Fluid Mixed, and the

Other Fluid Unmixed.

Correction Factor for a Single Pass Cross-Flow

Heat Exchanger withBoth Fluids Unmixed.


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-NTU Charts for Heat Exchangers

Effectiveness of parallel flow heat exchangers Effectiveness of counterflow heat exchangers

Effectiveness of Heat Exchangers with One Shell

Passand Two (or Multiples of Two)Tube Passes.

Effectiveness of Heat Exchangers with Two Shell

PassesandFour (or Multiples of Four)Tube Passes.


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Heat Exchanger Effectiveness Relations

Heat Exchanger NTU Relations

Use the above two equations with

me3122 handbook of heat transfer equations 2014

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