 Basics of Heat Transfer

COMPUTATIONAL HEAT

TRANSFER AND FLUID FLOW TRANSFER AND FLUID FLOW

PRADIP DUTTA

DEPARTMENT OF MECHANICAL ENGINEERING DEPARTMENT OF MECHANICAL ENGINEERING

INDIAN INSTITUTE OF SCIENCE BANGALORE

Outline Outline

 B i f H t T f

 Basics of Heat Transfer

 M h i l d i i f fl id fl d h

 Mathematical description of fluid flow and heat transfer: conservation equations for mass, momentum energy and chemical species momentum, energy and chemical species, classification of partial differential equations generalized control volume approach in Eulerian

g pp

frame

HEAT TRANSFER HEAT TRANSFER

BASICS

What is Heat Transfer?

What is Heat Transfer?

“Energy in transit due to temperature difference.”gy p Thermodynamics tells us:

 How much heat is transferred (Q)

H h k i d (W)

 How much work is done (W)

 Final state of the system Heat transfereat t ansfe tells us:te s us:

 How (by which mechanism) Q is transferred

 At what rate Q is transferred

 Temperature distribution within the body

Heat transfer complementary Thermodynamics Heat transfer complementary Thermodynamics

MODES MODES

 Conduction

- needs matter

- molecular phenomenon (diffusion process) i h b lk i

- without bulk motion of matter

 Convection

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

 Radiation d 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)

T >T T

.. . .

T

₂

T

₁

T

₁

>T

₂

. . . . . .

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

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…)

x Rate equations (1D conduction):

T₁

T

A  Differential Form

q = - k A dT/dx, W

k = Thermal Conductivity W/mK

q ^T2 k = Thermal Conductivity, W/mK

A = Cross-sectional Area, m² T = Temperature, K or ^oC

k 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 fluid T

_s

>T

_∞

moving fluid

T

_∞

q” T

_ss

 Energy transferred by diffusion + bulk motion of fluid

Rate equation (convection)

y U

_

q” y T

_

T(y) U

_

u(y)

Heat transfer rate q = hA( T T ) W

q T(y)

T

_s

u(y)

Heat transfer rate q = hA( T_s-T _ ) W Heat flux q” = h( T_s-T _ ) W / m²

h=heat transfer co-efficient (W /m²K)

not a fluid property alone, but also depends on flow p p y , p geometry, 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/myp ( ²K))

Free convection gases: 2 - 25

liquid: 50 - 100 Forced convection gases: 25 - 250

liquid: 50 - 20,000q , Boiling/Condensation 2500 -100,000

T

₁

q

₁

”

T

₂

q

₂

”

RATE:

q(W) or (J/s) Heat flow per unit time q(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=T

_s⁴

(W/ m

²

)

= emissivity (property)……..

=Stefan Boltzmann constant

=Stefan-Boltzmann constant

Rate equations (Contd )

(Contd….)

q” _d ”

T_{s u r}

q”_{co n v.}

q _{r a d.}

Area = A

T Area = A

T_s

R di ti h b t l f d

Radiation exchange between a large surface and surrounding

Q” = (T ⁴ T ⁴) W/ m² Q _{r a d} = (T_s⁴-T_sur⁴) W/ m²

Substantial derivative Substantial derivative

 ^x

₁

^{, x}

₂

^{, x}

₃

^{, t} 



 

d dx d

dx d

dx

D

_{1 2 3}



 



 



 



     



dt x

dt x

dt x

t

Dt  

₁



₂



₃

dx D











dt dx x

t Dt

D _i

 i

 



 

 



D











  

local ti l

substantia

i

i x

t u Dt

D



 



 

 



component convective component

oca derivative

substa t a

Conservation principle(s) Conservation principle(s)

• mass is conserved

• momentum is conserved

• energy is conserved gy

• chemical species are conserved

•• …



 ^x

₁

^{, x}

₂

^{, x}

₃

^{, t} 



 

D D D







V

dV



 





source V

V V

change

dV Dt dV

dV D Dt

D Dt

D^{ }



^{ }



^{ }



^

Conservation principle p p

for an elementary volume

dV

^::

 ^dV  ^dV

Dt

D  ^ 

    ^dV

Dt D Dt

dV D Dt dV

D     

Dt Dt

Dt  











 dV

Dt dV D Dt

dV D  





dV Dt

u dV D _i 





 x dV

Dt _i

i 

 

  



 

Conservation principle

  ^{D u}

D         ^dV x

u Dt

dV D Dt

D

i i





 







 

  



x dV u u x

t

_i

i i

i





 







 



 



    

  _ ^{ dV}

    

i

i





  ^{u dV}

x

t

_i ⁱ



 

  

   

Conservation principle Conservation principle

for an elementary volume

dV

^::

 D

   





  _



 

i i

x u Dt

D

i d f fi l l V

u

_i

V

  ^{ }

 _



 

 



i i

x u t

integrated for a final volume V

bounded by a surface

A

^::

u

_i

n

_i

dA

dV V A

      A





 

 



 

V V

i V i

dV x u

t dV Dt dV

D



 

 called as “Reynolds

Transport Equation





^_ ^



^



V

V A

i

iu dA dV

n

t dV  

 (RTE)” - a relation

between a system and control volume.

Conservation of mass Conservation of mass

Total mass remains constant. For a control volume,

the balance of mass flux gives the mass accumulation rate the balance of mass flux gives the mass accumulation rate.

 ^dV  ⁰

D  ^dV  ^{ 0}

Dt 

set :

-

^{density, then}

 



-

^mass

  ^



V

dV

-

0

^{no source}

 

Conservation of mass Conservation of mass

continuity equation

for an elementary volume

dV

^::

  V

 



i t t d f fi l l

V

^{b d d b}

u

_i

n

_i

dA

dV V A

 

_i

i

x u

t 





 

 



integrated for a final volume

V

bounded by a surface

A

^::

n

_i

A

dA u

n t dV

V A

i

 ^ _ ^{  }  ^

i ^{Here (ni ui} ) is a scalar quantity:

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

V -for inward flow- negative

Incompressible flow Incompressible flow

for an incompressible fluid:

 =const

^:

f p f



written for an elementary volume

dV

^::



V

 0





i i

x u

or a final volume

V

bounded by a surface

A

^::

u

_i

n

_i

dA

dV V A A

 0



A

i

i

u dA

n

A

Conservation of momentum Conservation of momentum

Rate of change of momentum will cause fluid acceleration.

For a control volume we consider the balance of momentum fluxes For a control volume, we consider the balance of momentum fluxes.



^dVu_j



^F_j

Dt

D  

j

j F

dV Du 





_j



_j

Dt

th l it

d it



u



  force j on

accelerati

mass Dt



set :

momentum

then velocity,

density

- -

j j

dV u

u

















e unit volum per

force

-

j V

 F



Conservation of momentum Conservation of momentum

momentum equation

for an elementary volume

dV

^:

 

^

 



 

 

  

_{j i}



_j

i

j u u F

u x

t 



 





 

   

i ij j

i j i

j u u f x

u x

t 

 

 

 



 







d

p  ij

 

 

i

j x

x  



Conservation of momentum

V u

_i

dV V

integrated for a final volume

V

bounded by a surface

A

^::

u

_i

n

_i

dA

dV

A

 

^{    }





^{momentum change outletconvectioninlet}





^



 

^{( . )}











_^{  }

d A

i i j

V

j

dA dA

dV f

dA u

n u

dV

t



u













 



body forces spherical stress deformation stress







^{ }



A

i d ij A

j V

jdV pn dA n dA

f





Stress - strain relationship p

xx

₂₂ ^{u1  u1}

^{ }

^x2

^

²¹

xx

₂₂

xx

₁₁

p ^d 

  

 

x dV x

dV p dF x

i ij i

ij i

ij

j 













 



 

 

  

 

3 3 2

2 3

2 0 0

u

x u x

u u u



 

 



 





1 1  u u  u

 

₂

1 1

1

1 0 u u x

x

u   



 

2 1 2

1 1

2

12 2

1 2

1

x u x

u x

u



 



 







 



 



Stress-strain relationship Stress strain relationship

for Newtonian fluids :

1

2

u

generalized :

12 2

1

12

 2 

 

 

x

d ij d

ij



  2

deformation stress = = 2 X viscosity X deformation strain rate

d ij kk

ij

ij   

  

3 1

j j

j 3



  

 

 _j  _{i k}

d u u u

1 2 1



 

   

 



  

 

k k ij

j i i

d j

ij  x x  x

 2 2 3

Navier-Stokes equations Navier Stokes equations

 

^

 

^{ p  d}



    

i ij j

j i

j i

j x x

f p u

x u

t u 

 



 

 

 













 



 











 





i j i

i d ij

x u x

x 





 











 





















k k ij

i j

i

i x

u x

x u

x  

3 2

 

  

 

 

  

 

^d

u ₁ u

 

 











 

 

 











 





i i j

i j i

i ij

x u x

x u x

x  



3

1

Navier-Stokes equations Navier Stokes equations

  ^  

    

 

  

 

 

  



 

 

j i j i

j

u u p

u x u

t  u 

1  

 











 

 

 











 



 

i i j

i j i

j

j

x

u x

x u x

x

f p  

 3

1

for an incompressible fluid

 =const

with a constant viscosity

 =const

with a constant viscosity

 const

 

^{1 2 j}

j p u

f u

u u 

 



 

 

  

₂

i j

j i

j

i u u f x x

x

t  

 

 



 

Surface force decomposition Surface force decomposition

p dV dV

dF

d ij

ij

  

 

 

 



 

 

 

x dV dV x

dF x

i i

ij i

j

 

 



  

 

  

since:

j i

ij

x

p x

p



 



 

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

dV p dV

dF

d ij

j 

 



 





x

x_{j i}

j  

Work decomposition Work decomposition

dx dx

₃₃



dV dV

dx dx

₂₂

u

u

₂₂ ¹

1 2

2 dx

x u u



 

12 dx

 





dx dx

₁₁

xx

₁₁

xx

₂₂



^{12 12  x1 dx1}

total work done on an elementary volume:

dx dx

₁₁

xx

₁₁

xx

₃₃

total work done on an elementary volume:



 







 



 







  ¹² _{1 2 3 2} ² ₁

12 dx

x u u

dx dx x dx

 







 



 

2 3 2 12

1 1

u dx dx

x x



Work decomposition Work decomposition



^u



^dV

dx x dx x dx

u

u x _{2 12}

1 3

2 1 1

2 12

1 12

2

  



 



 







 



 

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

 

  





ⁱ

j ij

j j ij

j

i

x

F u u

x u 

 

 

  













work n deformatio work

kinetic work

total

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"



dVuj



Fj

Dt

D



 ^uj

Dt

ij j

ju f dV dV u

dV u

D 

 

 

 













 





 

 

 



 

kinetic energy change work by external forces

2 _i ^j

j j

j

j dV u

dV x f

Dt dV 

 

  

 

 



 

Conservation of thermal (or internal) energy ( ) gy

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

of change of internal energy

 ^dV  ^q

ⁱ _{ij ij}

Dt

D    



 



q

_i

dV V

j j

x

i

Dt 

k  T

n

_i

dA

dV

i

A

i

k x

q 

 



 



 ⁱ

j ij i

i x

u x

k T dV x

Dt D



 



 











  







 



 





sfer heat tran

n work deformatio

sfer heat tran

conduction change

energy

internal





Conservation of thermal energy

set :

e temperatur heat

specific

-

 cT



 

gy

energy

internal

-

V

T u dV cT

 

 

  





^



fer heat trans

-

i j ij

i

i x

u x

k T

x 

 



 











  



thermal energy equation for an elementary volume

dV

^:

  

_i



i

T x cTu

t cT

 

 



 

 



  

i j ij

i

i x

u x

k T

x 

 



 











  

Conservation of thermal energy

integrated for a final volume

V u

_i

dV V

gy

bounded by a surface

A

^{: i}

n

_i

dA

dV

A

 ^{ }  ^{       }



 



^{energy change internalinlet energy outletconvection}

internal



 ^



dA u

cTn dV

cT 

  









 



 

_A ^{i i}

V

T u

dA u

cTn dV

t  cT 













d ti d f i k



 _ ^{  } _



V i

j ij

A

i i

x dV dA u

x n

k T 

sfer heat tran

n work deformatio

sfer heat tran

conduction

Conservation of chemical species

Net species mass transfer across the control volume faces + chemical reaction within the control volume results in a change of concentration of chemicalg f f Species in the control volume



^{ms dV}



^{qis Rs}

D  

 

q

_i^s

dV V

 

i

x R dV

Dt m 

 

 

 s

n

_i

dA

dV

A

i s s

s

i x

q m



 

 D

 

_

chemical to

due

k source/sin

s i

s s

i

s R

x m dV x

Dt m

D  



 











 



 





 D

reaction chemical to

due transfer

mass

diffusion change

ion

concentrat    

Conservation of chemical species

set :

ion concentrat

mass

pecies

-

ms ^s

 

mass

species -

s V

s dV

m









^



reaction

chemical

and diffusion

s -

i s s

i

x R m

x  



 











  D



mass transfer equation for an elementary volume

for an elementary volume

dV

^:

  

i



s i

s m u

m x t





 

 



  

s i

s s

i

x R m

x 

 











  D

Conservation of chemical species

integrated for a final volume

V q

_i^s

dV V

bounded by a surface

A

^:

n

_i

dA

A

ti i

 

^{    }





^{concentration change speciesinlet convectionoutlet}

species





_^{  }



A

i i s V

s dV m n u dA

t m 





^_ ^





V s A

i i

s s

A V

dV R

dA x n

m t

D









 



 



reaction chemical

to due k source/sin transfer

mass

diffusion

species

V

A 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 l

H Gu

Fu Eu

Cu Bu

Au

_xx

 2

_xy



_yy



_x



_y

 

where A - H are functions of x and y only

• Elliptic PDEs: B

²

- AC < 0

( d h i i h h )

(steady state heat equations without heat source)

• Parabolic PDEs: B

²

- AC = 0

(t i t h t t f ti )

(transient heat transfer equations)

• Hyperbolic PDEs: B

²

- 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 transfer coefficient, forced convection and boiling?

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

• 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 energy required will be greater? Explain why?

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

MODULE 1: Review Questions (contd )

(contd...)

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

heat 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 t

normal to each other. Will this condition hold for convection heat transfer?

• Distinguish between Eulerian and Lagrangian approach in fluid

h 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?