COMPUTATIONAL HEAT TRANSFER AND FLUID FLOWTRANSFER AND FLUID FLOW

PRADIP DUTTADEPARTMENT OF MECHANICAL ENGINEERINGDEPARTMENT OF MECHANICAL ENGINEERING

INDIAN INSTITUTE OF SCIENCEBANGALORE

OutlineOutline

B i f H t T fBasics of Heat Transfer

M h i l d i i f fl id fl d hMathematical description of fluid flow and heattransfer: conservation equations for mass,momentum energy and chemical speciesmomentum, energy and chemical species,classification of partial differential equationsgeneralized control volume approach in Euleriang ppframe

HEAT TRANSFERHEAT TRANSFER BASICS

What is Heat Transfer?What is Heat Transfer?

“Energy in transit due to temperature difference.”gy pThermodynamics tells us: How much heat is transferred (Q)

H h k i d (W) How much work is done (W) Final state of the systemHeat transfer tells us:eat t ansfe te s us: How (by which mechanism) Q is transferred At what rate Q is transferred Temperature distribution within the body

Heat transfer ThermodynamicscomplementaryHeat transfer Thermodynamicscomplementary

MODESMODES

Conduction- needs matter- molecular phenomenon (diffusion process)

i h b lk i- without bulk motion of matterConvection

heat carried away by bulk motion of fluid- heat carried away by bulk motion of fluid- needs fluid matter

Radiationd o- does not need matter- transmission of energy by electromagnetic waves

APPLICATIONS OF HEAT TRANSFER

Energy production and conversion Energy production and conversion- steam power plant, solar energy conversion etc.

Refrigeration and air-conditioning D ti li ti Domestic applications

- ovens, stoves, toaster Cooling of electronic equipment Manufacturing / materials processing

- welding, casting, forming, heat treatment, laser machining Automobiles / aircraft design Automobiles / aircraft design Nature (weather, climate etc)

(Needs medium, Temperature gradient)TT >T

.. . . . . . . . . T2T1

T1>T2. . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

q’’

. . . . .

. . . . . .. . . . . . . .. . . . . .

. . . . . . . . . . . q

. . . . .solid or stationary fluid………….. . ..

RATE:

. . . . . . .solid or stationary fluid

RATE:

q(W) or (J/s) (heat flow per unit time)

Conduction (contd )(contd…)

xRate equations (1D conduction):

T1

T

A Differential Formq = - k A dT/dx, Wk = Thermal Conductivity W/mKq T2 k = Thermal Conductivity, W/mKA = Cross-sectional Area, m2

T = Temperature, K or oCk x = Heat flow path, m

Difference Form q = - k A (T2 – T1) / (x2 – x1)q k A (T2 T1) / (x2 x1)

Heat flux: q” = q / A = - kdT/dx (W/m2)

(negative sign denotes heat transfer in the direction of decreasing temperature)

moving fluidTs>T∞moving fluid

T∞

q” Tss

Energy transferred by diffusion + bulk motion of fluid

Rate equation(convection)

q”y y

U TT(y)u(y)

U

Heat transfer rate q = hA( T T ) W

q T(y)Ts

u(y)

Heat transfer rate q = hA( Ts-T ) W

Heat flux q” = h( Ts-T ) W / m2

h=heat transfer co-efficient (W /m2K)

not a fluid property alone, but also depends on flow p p y , pgeometry, nature of flow (laminar/turbulent), thermodynamics properties etc.

Convection (contd…)

Free or natural convection (induced by buoyancy forces) May occur with

phase change (boiling,

Forced convection (induced by external

(boiling, condensation)

(induced by external means)

Convection (contd…)( )

Typical values of h (W/m2K)yp ( )

Free convection gases: 2 - 25

liquid: 50 - 100

Forced convection gases: 25 - 250

liquid: 50 - 20,000q ,

Boiling/Condensation 2500 -100,000

T1q1”

T2q2”

RATE:q(W) or (J/s) Heat flow per unit timeq(W) or (J/s) Heat flow per unit time.

Flux : q” (W/m2)

Rate equations (Radiation)

RADIATION:

Heat Transfer by electro-magnetic waves or photons( no medium required. )

Emissive power of a surface (energy released per unit area):Emissive power of a surface (energy released per unit area):

E=Ts4 (W/ m2)

= emissivity (property)……..

=Stefan Boltzmann constant=Stefan-Boltzmann constant

Rate equations (Contd )(Contd….)

”q” d

Ts u r

q”co n v.q r a d.

Area = AT Area = ATs

R di ti h b t l f dRadiation exchange between a large surface and surrounding

Q” = (T 4 T 4) W/ m2Q r a d = (Ts4-Tsur

4) W/ m2

Substantial derivativeSubstantial derivative txxx ,,, 321

ddx

ddx

ddxD 321

dtxdtxdtxtDt 321

dxD dtdx

xtDtD i

i

D

tilocallsubstantia

ii x

utDt

D

componentconvectivecomponent

ocaderivativesubsta t a

Conservation principle(s)Conservation principle(s)• mass is conserved• momentum is conserved• energy is conservedgy• chemical species are conserved•• …

txxx ,,, 321

DDD

V

dV

source

VVVchange

dVdVDtDdV

DtD

DtD

Conservation principlep pfor an elementary volume dV ::

dVdVDtD

dVDtD

DtDdVdV

DtD

DtDtDt

dV

dVDtD

DtDdV

dVDt

dVuD i

dV

xDt i

i

Conservation principle

uDD

dV

xu

DtDdV

DtD

i

i

dVxu

xu

t i

i

ii

dV

ii

dVuxt i

i

Conservation principleConservation principlefor an elementary volume dV ::

D

i

i

xu

DtD

i d f fi l l V ui V

ii

uxt

integrated for a final volume V bounded by a surface A ::

ui

ni dAdV V

AA

V V

iiV

dVux

dVt

dVDtD

called as “Reynolds Transport Equation

VV A

ii dVdAundVt

(RTE)” - a relation between a system and control volume.

Conservation of massConservation of massTotal mass remains constant. For a control volume, the balance of mass flux gives the mass accumulation ratethe balance of mass flux gives the mass accumulation rate.

0dVD 0dVDt

set : - thendensity,

- mass V

dV

- 0 source no

Conservation of massConservation of masscontinuity equation

for an elementary volume dV::

V

i t t d f fi l l V b d d b

ui

ni dAdV V

A

ii

uxt

integrated for a final volume V bounded by

a surface A ::

ni A

dAundVtV A

ii Here (ni ui ) is a scalar quantity:

-for outward flow (away from the surface)- positive V

-for inward flow- negative

Incompressible flowIncompressible flowfor an incompressible fluid: =const :f p f written for an elementary volume dV ::

V0

i

i

xu

or a final volume V bounded by a surface A ::

ui

ni dAdV V

AA0

Aii dAun

A

Conservation of momentumConservation of momentumRate of change of momentum will cause fluid acceleration. For a control volume we consider the balance of momentum fluxesFor a control volume, we consider the balance of momentum fluxes.

jj FdVuDtD

jj F

DudV

jjDt

thl itd itu

force

j

onacceleratimass Dt

set :

momentum

thenvelocity,density

-

-

j

j

dVu

u

eunit volumper force - j

V

F

Conservation of momentumConservation of momentummomentum equation

for an elementary volume dV :

jiji

j Fuux

ut

i

ijjij

ij x

fuux

ut

dijp

ij xx

Conservation of momentum

V ui dV Vintegrated for a final volume Vbounded by a surface A ::

ui

ni dAdV

A

inletoutlet

convectionchange momentum

).(

d

Aiij

Vj

dAdAdVf

dAunudVut

stressn deformatiostress sphericalforcesbody

A

idij

Aj

Vj dAndApndVf

Stress - strain relationshipp

xx22 211 xuu 21

xx22

xx11

p d

dVxx

pdVx

dFi

ij

iij

i

ijj

3

3

2

232 00

uxu

xuuu

11 uuu 2111

1 0 xuuxu

2

1

2

1

1

212 2

121

xu

xu

xu

Stress-strain relationshipStress strain relationshipfor Newtonian fluids :

1 2u

generalized :

122

112 2

x

dij

dij 2

deformation stress = = 2 X viscosity X deformationstrain rate

dijkkijij

31

jjj 3

kijd uuu

112

k

kij

j

i

i

jdij xxx

32

2

Navier-Stokes equationsNavier Stokes equations

dp i

ij

jjij

ij xx

pfuux

ut

i

j

ii

dij

xu

xx

k

kij

ij

i

i xu

xxu

x

32

d uu 1

i

i

ji

j

ii

ij

xu

xxu

xx

31

Navier-Stokes equationsNavier Stokes equations

j

iji

j

uup

uux

ut

1

i

i

ji

j

ijj x

uxx

uxx

pf 31

for an incompressible fluid =constwith a constant viscosity=constwith a constant viscosity const

21 jj upfuu

u

2ij

jiji xx

fuuxt

Surface force decompositionSurface force decomposition

dVpdVdFdijij

dVxx

dVx

dFii

iji

j

since:

jiij x

pxp

the total stress Fj can be separated into the spherical and deformation components:

dVdVpdFdij

j

xx ij

j

Work decompositionWork decompositiondxdx33

dVdVdxdx22

uu22 11

22 dx

xuu

12 dx

dxdx11xx11

xx221

112 dx

x12

total work done on an elementary volume:

dxdx11xx11xx33

total work done on an elementary volume:

12

232112

12 dxxuudxdxdx

x

23212

11

udxdxxx

Work decompositionWork decomposition

dVux

dxdxdxxu

xu 122

1321

1

212

1

122

generalized total work done on an elementary volume can be separatedgeneralized - total work done on an elementary volume can be separatedinto the kinetic and deformation components:

i

jijjjijj

i xu

Fuux

workndeformatio

workkinetic

worktotal

Conservation of mechanical energy

"change of kinetic energy is a result of work d b t l f "

start with the momentum equation :

done by external forces"

jj FdVuDtD

juDt

ijjj udVdVfuu

dVD

forces externalby work changeenergy kinetic

2 ji

jj

jj udVx

dVfdVDt

Conservation of thermal (or internal) energy( ) gy

Net heat flux entering a control volume results in rate of change of internal energyof change of internal energy

ijijiqdV

DtD

qi dV V

jjixDt

Tk ni dA

dV

Aii x

kq

i

jij

ii xu

xTk

xdV

DtD

sferheat trann workdeformatio

sferheat tranconductionchangeenergy internal

Conservation of thermal energyset :

etemperaturheat specific - cT

gy

energy internal - V

uT

dVcT

ferheat trans - i

jij

ii xu

xTk

x

thermal energy equation

for an elementary volume dV :

ii

T

cTux

cTt

i

jij

ii xu

xTk

x

Conservation of thermal energy

integrated for a final volume V ui dV V

gy

bounded by a surface A :i

ni dAdV

A

outlet inlet

convectionenergy internalchangeenergy internal

dAucTndVcT

A

iiV

uT

dAucTndVcTt

kd f id ti

V i

jij

Ai

i

dVxu

dAnxTk

sferheat trann work deformatio

sferheat tranconduction

Conservation of chemical species

Net species mass transfer across the control volume faces + chemical reactionwithin the control volume results in a change of concentration of chemicalg f fSpecies in the control volume

ssis RqdVmD

qis dV V

i

Rx

dVmDt

sni dA

dV

Ai

sss

i xmq

D

chemicaltoduek source/sin

s

i

ss

i

s Rxm

xdVm

DtD

D

reactionchemical todue

transfermassdiffusionchangeion concentrat

Conservation of chemical species

set : ionconcentrat mass pecies - sm s

mass species -

s

V

s dVm

reaction chemical

and diffusion -s

i

ss

i

Rxm

x

D

mass transfer equation

for an elementary volume for an elementary volume dV :

is

i

s umx

mt

s

i

ss

i

Rxm

x

D

Conservation of chemical species

integrated for a final volume V qis dV V

bounded by a surface A : ni dAA

tii

outlet inlet

convection specieschangeion concentrat species

Aii

s

V

s dAunmdVmt

V

s

Ai

i

ss

AV

dVRdAnxm

t

D

reaction chemical toduek source/sin

transfermassdiffusion species

VA i

Class of Linear Second-order PDEs

• Linear second order PDEs are of the form• Linear second-order PDEs are of the form

h A H f ti f d lHGuFuEuCuBuAu yxyyxyxx 2

where A - H are functions of x and y only• Elliptic PDEs: B2 - AC < 0

( d h i i h h )(steady state heat equations without heat source)

• Parabolic PDEs: B2 - AC = 0(t i t h t t f ti )(transient heat transfer equations)

• Hyperbolic PDEs: B2 - AC > 0 (wave equations)(wave equations)

MODULE 1: Review Questions• What is the driving force for (a) heat transfer (b) electric current flow

and (c) fluid flow?Whi h f th f ll i i t t i l t ?• Which one of the following is not a material property?(a) thermal conductivity (b) heat transfer coefficient (c)emissivity

• What is the order of magnitude of thermal conductivity for (a) metals(b) solid insulating materials (c) liquids (d) gases?

• What are the orders of magnitude for free convection heat transfercoefficient, forced convection and boiling?

• Under what circumstances can one expect radiation heat transfer to besignificant?

• An ideal gas is heated from 40ºC to 60ºC (a) at constant volume andg ( )(b) at constant pressure. For which case do you think the energyrequired will be greater? Explain why?

• A person claims that heat cannot be transferred in a vacuum. Evaluatepthis claim.

MODULE 1: Review Questions (contd )(contd...)

• In which of the three states (solid/liquid/gas) of a matter, conductionheat transfer is high/low? Explain your claimheat transfer is high/low? Explain your claim.

• Name some good and some poor conductors of heat.• Show that heat flow lines and isotherms in conduction heat transfer are

l t h th Will thi diti h ld f ti h tnormal to each other. Will this condition hold for convection heattransfer?

• Distinguish between Eulerian and Lagrangian approach in fluidh i Wh i th E l i h ll d?mechanics. Why is the Eulerian approach normally used?

• Derive the Reynolds transport equation.• Derive the continuity (mass conservation) equation in differential form

for incompressible flow.• Define substantial derivative. What is the physical significance of the

substantial derivative in an Eulerian framework?