FORMULAE SHEET

CONVECTION

All symbols have their usual meaning.

Constants

Gravitational acceleration: g = 9.81 m/s2

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

Definitions

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

VAm

RTpv

TThq

p

c

s

sectiona at flux energy Thermal

rate,flow Mass

:law gas Ideal

Cooling, of Law sNewton'

gas. ideal anfor 11

t,coefficien expansion thermalVolumetric

y,diffusivit Thermal

/ , viscosityKinetic

TT

c/k

p

p

2D Continuity Equation:

2D x-Momentum Equation:

2D Energy Equation:

where viscous heat dissipation,

2D Boundary Layer Equations:

x-Momentum Equation:

Energy Equation:

Integral Momentum Equation:

Integral Energy Equation:

Forced Convection Over External Surfaces

Generally, nm PrReCNu

Unless otherwise stated, fluid properties are to be evaluated at the film temperature.

Forced Convection Over a Flat Plate:

For constant ,

.

0

y

v

x

u

Xy

u

x

u

x

p

y

uv

x

uu

2

2

2

2

qy

T

x

Tk

y

Tv

x

Tucp

2

2

2

2

222

2 y

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

Constant Surface Temperature:

plate.r rectangulaa for 11

t,coefficienfer heat trans Mean0

L

x

A

x dxhL

dAhA

h

For laminar flow ( 5Re 5 10x ):

; 5 3121 PrRex tx

For turbulent flow ( 5Re 5 10x ):

Pr02960 ; 05920 ; 370 31

51 5451

xxxx,fxturb Re.NuRe.CRex.

For mixed boundary layer conditions ( 5105LRe ):

Constant Surface Heat Flux:

For laminar flow ( 5Re 5 10x ):

For turbulent flow ( 5Re 5 10x ): 31

Pr03080 54

xx Re.Nu

For Unheated Starting Length, xo :

Forced Convection Across Long Cylinders:

where C and m are given by

ReD 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 0.618

40,000-400,000 0.027 0.805

31

21

31

21

6640 ;3320 PrRe.k

LhNuPrRe.Nu LLxx

)8710370( ; 17420740 80151 31

.

LLLLL,f Re.Prk

LhNuReRe.C

31

21

4530 PrRe.Nu xx

21

21

3281 ; 664022

xL,fx

x,s

x,f Re.CRe./u

C

3143

21

31

13320

x

xRePr.Nu o

xx

31PrReCk

DhNu m

DD

Forced Convection Across Spheres:

where all properties are evaluated at the free-stream temperature, except μs , which is evaluated

at the surface temperature of the sphere.

Forced Convection Across Non-Circular Cylinders

where C and m are given by

Forced Convection Across Tube Banks

where all properties, except Prs, are evaluated at the mean of the fluid inlet and outlet

temperatures, ReD,max is based on the maximum fluid velocity, and C1 and m are given in the

table below for number of tube rows for various alined and staggered arrangement of

tubes.

41

403221 060402

s

.

DDDμ

μPrRe.Re.

k

DhNu

31PrReCk

DhNu m

DD

41

360

1

s

.m

max,DDPr

PrPrReCNu

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

For :

where C2 for various is given in the table below:

20220

LL NDND NuCNu

Forced Convection in Tubes and Ducts

Unless otherwise stated, fluid properties should be evaluated at the mean or bulk temperature.

Forced Convection in Tubes

Friction factor,

Mean temperature, p

cA

p

mcm

dAuTcT c

PerimeterWetted

Area sectional-Cross4 Diameter,Hydraulic

hD

For thermally fully developed condition:

Log Mean Temperature Difference, io

iolm

T/Tln

TTT

Laminar Flow (ReD 2300):

Fully developed velocity profile:

where mean fluid velocity,

Friction factor, f = 64/ReD

dx

dpr

r

mum

8

2

0

2

0

2

0

2

12)(

r

r

u

ru

m

22 /u

Ddx/dpf

m

0)()(

)()(

xTxT

x,rTxT

x ms

s

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:

n

DD PrRe.Nuhh

540230

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

832 /fPr.St

Free Convection

Generally,

flow.ent for turbul 31 and flow,laminar for 41 with mmRaCPrGrCNum

L

m

LL

Unless otherwise stated, fluid properties are to be evaluated at the film temperature.

LaminarFree 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

Free Convection from Isothermal Planes and Cylinders

m

L

m

LL RaCPrGrCNu where C and m are given in the table below:

Constants for Use with Isothermal Surfaces

Geometry GrL Pr C m Characteristic

Length

Vertical plane and cylinder 10

4 – 10

9 0.59 1/4

Height 10

9 – 10

13 0.10 1/3

Horizontal cylinder

10-10

– 10

-2 0.68 0.058

Diameter

10-2

– 10

2 1.02 0.148

102 – 10

4 0.85 0.188

104 – 10

9 0.53 1/4

109 – 10

12 0.13 1/3

Hot surface facing up or

cold surface facing down

104 – 10

7 0.54 1/4

Area/Perimeter 10

7 – 10

11 0.15 1/3

Hot surface facing down or

cold surface facing up 10

5 – 10

11 0.27 1/4 Area/Perimeter

2

2

y

uTTg

y

uv

x

uu

00

2 dyTTgy

udyu

dx

d

s

414121 9520933 xGrPr.Prx.

541101for 4302 D

/

DD GrPrGr.k

DhNu

Free Convection from a Vertical Plate with Constant Surface Heat Flux

where

161341

11551

10102for 170 :Turbulent

1010for 600 :Laminar

Pr*GrPr*Gr.Nu

Pr*GrPr*.Gr.k

xhNu

xxx

xxx

x

2

4

kν

xqg.NuGr*Gr s

xxx