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Convection 3

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Energy equation

First law of thermodynamics

dWdQdE dE increment in the (kinetic plus thermal energy) of the system

dQ heat transfer to the system

dW work done on the system

Internal energy per unit mass of the fluid consists of the sum of thekinetic energy ( ) and thermal internal energy e = cv T

Writing the above equation in the substantial derivative form, so that

it applies to transport ofEby a moving system

2

22vu

Dt

DW

Dt

DQ

Dt

DE x1

y1

z1

dx

dy

dz

x

y

z

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Rate of increaseof energy E offluid element

Rate of heat transferto fluid element Rate of work doneon the fluid elementby surface and bodyforces

Rate of increaseof E in CV

Rate at which Eenters throughsurface of CV

Rate at which Eleaves throughsurface of CV

Rate of heat transferinto CV by conduction Rate of surface and body forces dowork on CV

Rate of increaseof E in CV

dxdydzt

E

Rate at which Eenters throughsurface of CV

Rate at which Eleaves throughsurface of CV

dxdydzy

Ev

x

uE

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dxdydzx

uEuEdydz

xy

z

ji

KdydzEu

dxdzEv

dydxdz

y

EvvEdxdz

Rate of increaseof E in CV

Rate at which Eenters throughsurface of CV

Rate at which Eleaves throughsurface of CV

dxdydzy

Ev

x

uE

t

E

CV showing rate of transport of

energy through the facesnormal to the x and y axes

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Rate of increaseof E in CV

Rate at which Eenters throughsurface of CV

Rate at which Eleaves throughsurface of CV

dxdydzy

Ev

x

uE

t

E

dxdydzy

Ev

y

vE

x

Eu

x

uE

tE

t

E

dxdydzyEv

xEu

tEdxdydz

yvE

xuE

tE

dxdydzy

Ev

x

Eu

t

Edxdydz

y

v

x

u

tE

dxdydzvu

e

tD

D

2

22

dxdydztD

DEdxdydz

y

Ev

x

Eu

t

E

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Rate of heat transferinto CV by conduction

x

y

z

ji

Kdydzqxdxdydz

x

qdydzq xx

dxdzqy

dxdydzy

qdxdzq

yy

dxdydzy

q

x

q yx

dxdydzy

T

kyx

T

kx

Negative sign arises because heat

transfer is counted as positive in the

positive coordinate direction

8/3/2019 Convection 3

7/26Rate of doing work = force velocity

x

y

z

ji

KudydzP dxdydzxuPdydzPu

vPdxdz

dxdydzy

vPdxdzPv

dxdydz

yvP

xuP

dxdydzy

vPdxdzPvPvdxdzdxdydz

x

uPdydzPudydzPu

Outward normal stresses are positive. Positive normal

stresses are tensile stresses; that is, they tend to stretchthe material. Compressive normal stress wil give positive

valuefor p

RATE OF WORK DONE BY PRESSURE FORCES

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xxxyxz

yy

yzyx

xyxzxx

First subscript denotes the direction of the normal to the

plane on which the stress actsSecond subscript denotes the direction of the stress

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x

y

z

ji

Kudydzxx

dxdydz

y

vdxdzv

yyyy

dxdydzy

v

x

u

dxdydzy

vdxdzvvdxdzdxdydz

x

udydzudydzu

yyxx

yyyyyy

xxxxxx

vdxdzyy

dxdyd

x

udydzu xxxx

RATE OF WORK DONE BY NORMAL STRESSES

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RATE OF WORK DONE BY SHEAR STRESSES

x

y

z

ji

Kvdydzyx

dxdydzy

u

x

v

dxdydzy

udxdzuudxdzdxdydz

x

vdydzvdydzv

xyyx

xyxyxy

xxxxyx

dxdzuxy

dydzdxx

vvdx

x

yxyx

dxdzdyy

uudy

y

xyxy

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Rate of increaseof E in CV

Rate at which Eenters throughsurface of CV

Rate at which Eleaves throughsurface of CV

Rate of heat transferinto CV by conduction Rate of surface and body forces dowork on CV

2

22vu

e

tD

D

y

Tk

yx

Tk

x

y

vP

x

uP

y

v

x

u yyxx yx

xyyxvfuf

y

u

x

v

xzxyxxx fzyxx

P

Dt

Du

X- momentum equation

xyxxx uf

y

u

x

u

x

Pu

Dt

Duu

A

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xyxxx ufyx

u

x

Pu

Dt

uD

Dt

Duu

2

2

xyxxx vfyx

v

x

Pv

Dt

vD

Dt

Dvv

2

2

yxyyyxyxxx vfufyx

v

yx

u

y

Pv

x

Pu

Dt

vuD

122

22

BA -B

y

v

x

v

y

u

x

u

y

v

x

uP

y

T

x

Tk

Dt

Deyyyxxyxx

1

2

2

2

2

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y

v

x

v

y

u

x

u

y

v

x

uP

y

T

x

Tk

Dt

Deyyyxxyxx

1

2

2

2

2

y

v

x

u

x

uxx

3

22

y

v

x

u

y

vyy

3

22

x

v

y

uyxxy

2222

3

222 xvyuyvxuyvxu

D

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P

eh Dt

PD

Dt

De

Dt

Dh

y

v

x

uP

Dt

DP1

y

v

x

uP

Dt

DP1

Dt

DP

Dt

DP1

Dt

P

D

22

y

v

x

v

y

u

x

u

y

v

x

uP

y

T

x

Tk

Dt

De

yyyxxyxx

1

2

2

2

2

y

v

x

uP

Dt

DP1

y

v

x

uP

y

T

x

Tk

Dt

Dh2

2

2

2

Dt

DP1

y

T

x

Tk

Dt

Dh2

2

2

2

0V.tD

D

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Dt

DP1

y

T

x

Tk

Dt

Dh2

2

2

2

Assuming a fluid of constant specific heat TCh P

Dt

DP

y

T

x

Tk

Dt

DTCP 2

2

2

2

ndissipatioviscousworkpressure

conductionconvection

P Dt

DP

y

T

x

T

ky

T

vx

T

ut

T

C

2

2

2

2

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Dt

DP

y

T

x

Tk

Dt

DTCP 2

2

2

2

2

* * * * * * s

s

T T x y u v P x , y , u v , P , and T

L L V V T T V

PRINCIPLE OF SIMILARITY

y

Pv

x

Pu

y

T

x

Tk

y

Tv

x

TuCP

2

2

2

2

For steady flows

2

23

2

2

2

2

2 L

V

y

Pv

x

Pu

L

V

y

T

x

T

L

TTk

y

Tv

x

Tu

L

TTVC

*

*

**

*

**

*

*

*

*s

*

**s

P

2222

2

22222

3

222

3

222

*

*

*

*

*

*

*

*

*

*

*

**

x

v

y

u

y

v

x

u

y

v

x

u

L

V

x

v

y

u

y

v

x

u

y

v

x

u

223

2

2

2

2

2L

V

TTVC

L

y

Pv

x

Pu

TT

L

VCL

V

y

T

x

T

TT

L

VCL

TTk

y

Tv

x

Tu

sP

*

*

**

*

**

sP*

*

*

*

sP

s*

**

sP*

*

**

*

**

sP*

*

*

*

P

*

**

TTLC

V

y

Pv

x

Pu

TTC

V

y

T

x

T

VCL

k

y

Tv

x

Tu

2

2

2

2

2

***** 222

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Re

Ec

y

Pv

x

PuEc

y

T

x

T

PrRey

Tv

x

Tu

*

*

**

*

**

*

*

*

*

*

**

2

2

2

21

PrReC

k

VLVLC

k

PP

1

sP*

*

**

*

**

sP*

*

*

*

P*

**

TTLC

V

y

Pv

x

Pu

TTC

V

y

T

x

T

VCL

k

y

Tv

x

Tu

2

2

2

2

2

sP TTCV

Ec

2

Eckert number is measure of the dissipation effects in the flow. Since this grows in

proportion to the square of the velocity, it can be neglected for small velocities.

In an air flow, V = 10 m/s, Cp = 1050 J/kg.K and a reference temperature difference of

10K,Ec 0.01.

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NONDIMENSIONALIZED CONVECTION AND SIMILARITYWhen viscous dissipation is negligible, the continuity, momentum,

and energy equations for steady incompressible, laminar flow of a

fluid with constant properties

0u v

x y

2

2

u u u Pu v

x y xy

2 2

2 2p

T T T T C u v k

x y x y

At 0 0 0

At 0 0 0 0 0 0

As

s

x u , y u , T , y T

y u x, , v x, ,T x, T

y u x, u , T x, T

With the boundary conditions

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2

* * * * * * s

s

T T x y u v P x , y , u v , P , and T

L L V V T T V

0

* *

* *

u v

x y

2

2

1* * * *

* *

* * * *L

u u u dPu v

Re x y y dx

2

2

1* * *

* *

* * *L

T T Tu v

Re Pr x y y

0 1 0 0 1 0 0

0 1 0 0 1

* * * * * * * *

* * * * * *

u , y , u x , , u x , , v x , ,

T , y , T x , , T x ,

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212f1f NuNuCC 2121 PrPrReRe

The number of parameters is reduced greatly by non-

dimensionalizing the convection equations

L1

Re1

L2

Re2

V1Water

V2Air

L,V ,T , ,

Re,Pr

Parameters before nondimensionalizing

Parameters after nondimensionalizing

F i t th l ti f * b d

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For a given geometry, the solution for u* can be expressed as

1* * * Lu f x , y ,Re

20 0

*

*

*s L*

y y

u V u V f x ,Re y L Ly

2 2 32 22

2 2

* * *s f ,x L l L

L

V LC f x ,Re f x ,Re f x ,Re

ReV V

Friction coefficient for a given geometry can be expressed in terms

of the Reynolds number Re and the dimensionless space variable x*alone (instead of being expressed in terms ofx, L, V, and ).This is a very significant finding, and shows the value of

nondimensionalized equations.

L

*

x,f Re,xC

*

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Dimensionless temperature T* for a given geometry

Note that the Nusselt number is equivalent to the dimensionlesstemperature gradient at the surface, and thus it is properly

referred to as the dimensionless heat transfer coefficient

* * * LT g x , y ,Re ,Pr

20

*

*

*x L*

y

hL T Nu g x ,Re ,Pr k y

0

0 0* *

* *y s

* *s s y y

k T yk T T T k Th

T T L T T Ly y

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Nusselt number is equivalent

to the dimensionless

temperature gradient at thesurface

m nL Nu C Re Pr

*y

0*

*

*

y

TNu

y

*

x

*T

Laminar

*

x L

x L

Local Nusselt number :

Nu f ( x , Re , Pr)

Average Nusselt number :

Nu f (Re , Pr)

A common form of Nusselt

number:

ANALOGIES BETWEEN MOMENTUM AND HEAT TRANSFER

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ANALOGIES BETWEEN MOMENTUM AND HEAT TRANSFER

Reynolds analogyand Chilton-Colbourn analogy

2

2

1* * *

* *

* * *L

u u u

u v Re x y y

2

2

1* * *

* *

* * *L

T T Tu v

Re x y y

1Pr

20 0

*

**

s L*y y

u V u V f x ,Re

y L Ly

2 2 32 22

2 2

* * *s f ,x L l L

L

V LC f x ,Re f x ,Re f x ,Re

ReV V

20

*

**

x L*

y

hL T Nu g x ,Re ,Pr

k y

2L

f ,x x

ReC Nu

REYNOLDS ANALOGY

h Nu

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p L

h NuSt

C V Re Pr

2

L f ,x x

Re

C Nu2

f ,xxC St

* *

* *

* *

* *

y y

L f , x x

Pr ofile : u T

u TGradients :

y y

Re Ana log y : C Nu

0 0

2

Laminar flow over a flat plate

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1 20 664 f ,x xC . Re

1 3 1 20 332 x Nu . Pr Re

2

f ,x

x

C

St p L

h Nu

St C V Re Pr

Laminar flow over a flat plate

1 3

2

L f ,x x

ReC Nu Pr

2 3

2

f ,x xH

p

C hPr j

C V

For . Here is called the Colburn j-factor. Although this

relation is developed using relations for laminar flow over a flat plate(for which = 0 ), experimental studies show that it is also

applicable approximately for turbulent flow over a surface, even in

the presence of pressure gradients. For laminar flow, however, the

analogy is not applicable unless = 0.

0 6 60. Pr Hj

**P x

**P x

Th f i d l l i fl i i