convective heat transfer

convective heat transfer

momentum diffusivity :  

 

themal diffusivity :

p

k

 c

 

Pr v c^p k



  

   

0

s s

y

h T T k T T

 y



    



Nu= tan

tan hL conductivethermal resis ce

k  convectivethermal resis ce

 

 

/ /

s y

s

T T y

hL

k T T L





  

 

D 2

Dt P

 v      

v g

2 :

p

c DT k T q

 Dt      τ v

dimensional analysis

forced convection in a closed conduit

# of variables; 7

# of dimensions; 5

# of ranks; 4

# of independent dimensionless groups; 3

1

a b c d

D k v

   

2

e f g h

D k v cp

  

3

i j k l

D k v h

  

1 Dv Re

 ^  ^{ 2 Pr}

cp

k

    ₃ hD Nu

  k  ^{Nu  f}₁





natural convection

 

0 1 T

   

 

buoyant 0

F    g

buoyant 0

F   g T

# of variables; 9

# of dimensions; 5

# of ranks; 5

# of independent dimensionless groups; 4

1

a b c d

L k cp

   

2

f g h j

L k g

   

3

k l m n o

L k g T

    

4

p q r s t

L k g h

   

1 c^p Pr

k

    ^{3 2} ₃   T

2 2

L g

 ^ 

2 3

Gr  g 2L T



  ^{Nu  f}₃





4 hL Nu

  k 

Laminar boundary layer

No matter how turbulent the flow is far from the surface

0



 





y u x

u_{x y}















 



 









 







 





2 2 2

2

y u x

u y

p y

u u x

u_x u^y _y ^y  ^{y y}





 











 









 







 





2 2 2

2

y u x

u x

p y

u u x

u_x u^x _y ^x  ^{x x}



layer boundary

the within 0

;

0 



 

y u_y p

Negligible viscous effect outside the boundary layer

0









y u x

ux y

2 2

y u y

u u x

u_x u^x _y ^{x x}



 



 







 



 

 ^{2 2} ₂ ³ ₃

y y

x y

x

y 

 







 









     

Boundary conditions

0 for

, 0 on

0  



 u y x

u_{x y}

0 for 

U x

u_x





U y

u_x for

0 for

, 0 on

0  

 

 

 

 y x

x y











 

 U x y

y for 0, and

 Blasius’ similarity transformation

( ) 1/ 2

( )

f U x

 

 

1/ 2

2 y U

 x



 

  

 

f '''  ff ''  0

' 0 at 0

' 2 as f f

f





  

  

U_x becomes nearly U

along the boundary layer 5

2 / 1

 



 



 x U

 

) ( x y  

Boundary layer thickness

Friction coeff.

1/ 2 0

2 1/ 2

1 2

( ) 0.664

0.664

(Re )

x y

f

x

u y

C U xU

 



     

    

Shear force



 



 







 

 ^L

y

x dx

y W u

F 0

0

s  ₁^s _{2 1/ 2 1/ 2}

L 2

/ 1.328 1.328

( / ) (Re )

f

F LW

U C UL

 ^{ }  ^

Re_x xU







Local Reynolds number

1/2 0

0.332

x Re

x

y y x

 _



   

laminar boundary layer; Blasius

2 2

2 2

x y

T T T T T

v v

t x y ^ x y ^

         

2

x y 2

T T T

v v

x y  y

    

  

2 2

x x x

x y

v v v

v v

x y  y

    

  

0 at 0

x vy

v y

v_  v_   1 at

vx

v_  y  

0 at 0

s s

T T T_ T y

  



1 at

s s

T T T_ T y

   



1/2 0

0.332

x Re

x

y y x

 _



   

 

0

0.332

s Rex

y

T T T

y _ ^ x

    

Nu 0.332 Re1/2 x

x x

h x

k  

for Pr=1

 

0 y

x s

y

q T

h T T k

A ^ y _

    



effect of Pr; Pohlhausen _Pr^1/3

t



 ^

 

0

0.332

s Rex

y

T T T

y _ ^ x

    

 

0

0.332

Re Pr

s x

y

T T T

y _ ^ x

    

Nu 0.332 Re1/2 x

x x

h x

k   h x^x Nu_x 0.332 Re^1/2_x Pr^1/3

k  

1/2 1/3

Nu_L 0.664 Re_L Pr hL

k  

2

s f

T T T   ^ fluid properties are evaluated at the film temperature

; laminar boundary layer over a flat plate

1/2 1/3

Nu_x  0.36 Re_x Pr (von Karman)

energy and momentum transfer analogy Reynolds analogy ; for Pr=1

0 0

x s

y s y

v T T

d d

dy v_{ } dy T_ T _

  

    0

p x

y

h c dv

v dy



 



0

2 2

0

2 / 2

x f

y

C dv

v v dy

 

 _  _{ }

 

coefficient of skin friction

 

2

f

p

h  C v c_ St

2

f p

h C

v c_ ^{ }

Colburn analogy ; 0.5<Pr<50, no form drag

St Pr2/3

2 Cf

 2

f H

j  C j_H St Pr^2/3

Colburn j-factor

   

0

s s

y

h T T k T T

 y



    



turbulent flow

x

x x

y L y

Ld dy

  

    _x d ^x

L dy

    _yx^turb   _x^{ }_y

y L y

y

T T L dt

    dy dT

T L

   dy

 

turb y

p y

q c v T

A   ^{ }

Prandtl analogy

 

St / 2

1 5 / 2 Pr 1

f f

C

 C

 

 

St / 2

1 5 / 2 Pr 1 ln 1 5 Pr 1 6

f

f

C C

         von Karman analogy

convective heat transfer correlations natural convection;

vertical plates, vertical cylinders,

horizontal plates, horizontal cylinders, spheres, rectangular enclosures

forced convection for internal flow;

laminar flow, turbulent flow

forced convection for external flow;

flow parallel to plane surfaces, cylinders in crossflow, single spheres, tube banks in crossflow,

special considerations;

whether to evaluate fluid properties at bulk or film temperature what significant length is used

what is the allowable Pr, Re range for a given set of data

natural convection; vertical plates

12.5cm high Ts=65C

Tinf=15C

2 3

Gr  g 2L T



  Pr c^p

k

 

 

2 1/6

9/16 8/27

0.387Ra Nu= 0.825

1 0.492 / Pr

L

 

  

 

 

    

 

Pr Ra  Gr

forced convection for internal flow laminar flow; Graetz solution

2

2 avg 1

x

v v r

R

   

     

1

x

T T

v r

x  ^r r r ^

      

2 avg

2 1 r T 1 T

v r

R x  r r r

          

          

 

 

at 0 for 0

T Te x   r R at 0,

T Ts x  r  R 0 at 0, 0

T x r

r

   



2

0 avg

e exp

n n

n

s e

T T r x

T T c f R Rv R

 





 

 



4 4 /

Re Pr / Pe

x D

D x  Gz Pe

4 D

x

 

Nu_x 3.658 for Twall constant Nu_x 4.364 for /q Awall  constant

1/3 0.14

Nu_D 1.86 Pe ^b

w

D L





 

 

    

2

0 avg

e exp

n n

n

s e

T T r x

T T c f R Rv R

 





 

 



Sieder and Tate

turbulent flow; Dittus and Boelter Nu_D  0.023Re0.8_D Prⁿ

1. n=0.4 if the fluid is being heated, n=3 if the fluid is being cooled

2. all fluid properties are evaluated at the arithmetic-mean bulk temperature

3. Re>10⁴

4. 0.7<Pr<100 5. L/D>60

Colburn;

Sieder and Tate ;

0.2 2/3

St=0.023Re_D^ Pr^

0.14 0.2 2/3

St=0.023Re_D Pr ^b

w





   

 

 

forced convection for external flow

flow parallel to plane surfaces laminar;

turbulent;

1/2 1/3

Nu_x 0.332 Re_x Pr

1/2 1/3

Nu_L 0.664 Re_L Pr

4/5 1/3

Nu_x  0.0288Re_x Pr

4/5 1/3

Nu_L 0.036 Re_L Pr St Pr2/3

2

fx x

 C (Colburn analogy)

1/5

0.0576

fx Re

x

C  (von Karman)

cylinders in crossflow

 

1/2 1/3 2/3 1/4

0.62 Re Pr Re

Nu 0.3 1

282,000 1 0.4 / Pr

D D

D

  

        