Transcript File

CONVECTION

Free (Natural)
&

 Forced
Convection
Dear Diary,
 Last year I replaced all the windows in my house
with those expensive double pane energy efficient
kind, but this week, I got a call from the contractor
who installed them. He was complaining that the
work had been completed a whole year ago and I
hadn't paid for them.
 Now just because I'm blonde doesn't mean that I am
automatically stupid. So, I told him just what his
fast talking sales guy had told ME last year....namely,
that in ONE YEAR these windows would pay for
 themselves! Hellooooo? It's been a year! (I told him.)
There was only silence at the other end of the line, so
I finally just hung up... He didn't
 call back. Guess I won that stupid argument.

FREE CONVECTION


When a solid body is heated by a hotter fluid
surrounding it (potato in an oven) heat is first
transferred to the body by convection then
subsequently conducted through it.
Convection

Heat transfer due to fluid movement on a macro scale

Density change, velocity, thermal conductivity, Cp,
important variables
3
NATURAL OR FREE CONVECTION



Laminar boundary layer on a flat plate where
temperature gradients are present in the flow
For small Ts-T∞ & μ, H/T problems are not acute
For high μ fluids (petroleum oils) where Tw-T∞ is
very large the fluid property variations (μ, ,ρ) heat
transfer rate calculations are difficult.
4
CONVECTION

This explains why heat transfer to fluids without phase change
is complex and treated as a set of special cases rather than as a
general theory.

Thermal boundary layer or prandtl boundary layer
-Surface of arbitrary shape
Vt
T∞
q
As
-v, T∞ , Ts ≠ T∞
q = h(Ts-T∞ ) ---------------- (1)
As
Ts
-v varies from point to point as
do h & q
CONVECTION

Total heat transfer
Q
 qdA
s
2
As
from 1 : Q  Ts  T   hdAs        3
As

Let h = average convection coefficient for the entire surface.
Then the total heat transfer rate is:
Q  hAs Ts  T           (4)
putting eq (3)  eq (4) : hAs Ts  T   Ts  T   hdAs      (5)
As
h
1
hdAs          6

As As
FLAT PLATE

Uoo Too
q
h varies with distance, x
As Ts
x
dx
L
1 L
from eq (6) h   hdx        (7)
L 0
Compare with mass transfer phenomenon where :
N A  h m As C A, S  C A, 
n A  h m As  A, s   A, 
EXAMPLE

Experimental results for the local heat transfer
coefficient, hx for flow over a flat plate with an
extremely rough surface were found to fit the relation:


hx(x) = ax-0.1
Where a (w/m1.9 K) is a coefficient and x(m) the distance from the
leading edge of the plate
(1)
(2)
Develop an expression for the ratio of the average heat transfer
coefficient for a plate of length, x to the total heat transfer
coefficient at x
Show in a quantitative manner the variation of hx and hx as a
function of x
CONVECTION BOUNDARY LAYERS (THERMAL & VELOCITY)

Fluid flow over non porous surface
u∞
T∞
Free stream
Thermal
Boundary layer
The fluid at the surface retards the
motion of the layers above
In the boundary layer temperature
gradient decrease with distance from
the leading edge
qconv = qcondt = -kfluid dt/dy -------(8)
Heat is then conducted away from the surface due to fluid motion
if q conv  hT and

h
hLc

 Nu
k/L
k
Ratio of
 
q cond  k
k
C P
T
L
then
q conv
hT

q cond
kT / L
Nusselt number
heat transfer to conduction over a surface of thickness, L 
Pr 
C 

 P

k
Pr andtl number thickness of thermal boundary layer 
DIMENSIONLESS NUMBERS

Reynolds number (ratio of inertia forces to viscous forces)
Re 

uLC


ud


Graetz number
.
.


.
m


 where G  A  u & m  uA 


Gd

.
m C P  u C P d 2
Gz 

kL
4kL

.
. 

d 2
 where m 
 u 
4


Prandtl number (ratio of molecular diffusivity of momentum to
molecular diffusivity of heat)
C P  C P


Pr  


 k / C P
k
k
DIMENSIONLESS NUMBERS
LuC P
Pe  Re* Pr 
k
Nu
h
St 

Re* Pr C P 

Peclet number

Stanton number

Grashoff number(ratio of buoyancy to viscous forces)
Gr 

Releigh number
gTL 
3

2
2

gTL
3

2
g Ts T  L 
Ra  Gr * Pr 
Pr
2

3
2
THE VOLUME EXPANSION COEFFICIENT
A measure of the variation of the density of a fluid with
temperature at constant pressure
 From thermodynamics we know that


1  v 
1   






v  T  P
  T  P
 
1 
1   

 T
 T  T
      T  T 
K 
1
at cons tan t
at cons tan t
For ideal gases where P  RT :  
1
T
P
P
K 
1
CONVECTION

For free convection u=0 so no need for Re, use Ra

For forced convection no need for Gr, use Re

For many gases Cpμ/k = constant for a wide range of
temperatures so Pr may be disregarded

Fluid properties are referenced at the film temperature
(Ts+T∞)/2
VERTICAL PLATES
AND CYLINDERS
-
FREE CONVECTION
Characteristic length is the height
 Kato et al conducted experiments to show that:


Nu  0.138Gr 0.36 Pr 0.175  0.55
Nu  0.683Gr

0.25
Pr
0.25

Pr




 0.861  Pr 
for 1  Pr  40
0.25
for Gr  10 9
Churchill& Chu
Nu  0.59 Ra
Nu  0.1Ra
1
1
3
4
for 10 4  Ra  10 9
for 10 9  Ra  1013
VERTICAL PLATES
AND CYLINDERS
A more complicated relationship is:
Nu  0.68 
0.670 Ra
1
4
1  0.492 / Pr  
9
4
16
forRa  109
9

0.387 Ra 6

Nu  0.825 
9

1  0.492 / Pr  16
1



8 
27


2
for the __ entire range
EXTERNAL FORCED CONVECTION


Flow fields and geometry too complicated for
analytical treatment hence emphasis on
correlations of experimental data
Flat plate (isothermal surface) in laminar fluid
flow

Critical flow at Re = 5.0 x 105
Nu  0.664 Re
1
2
Pr
1
3
For turbulent flow:
Nu  0.037 Re Pr
0.8
1
3
for 5 x10 5  Re  10 7
and 0.6  Pr  60
COMBINED
FLOW
 If
the plate is not long enough to disregard
laminar flow region resulting in combined
laminar and turbulent flow:
Nu  0.037 Re  871Pr
0.8
1
3
for 0.6  Pr  60
and 5.0 x 10 5  Re  10 7
EXAMPLE
 A 60.0 oC stream of engine oil flows at 2.0
ms-1over the upper surface of a 5.0 m long
flat plate that is kept at 20.0 oC. What is
the rate of heat transfer per unit width of
the entire plate?
TURBULENT FLOW THROUGH TUBES


Dittus & Boelter correlated works by a variety of
researchers using gases (air, CO2, H2O) and liquids
(H2O, acetone, kerosene, benzene) in smoth tubes.

For heating of the fluid

For cooling of the fluid
Nu  0.0241 Re
Nu  0.264 Re
0.8
0.8
Pr
Pr
0.4
0.3
McAdams reevaluated the Dittus/Boelter correlations
for both heating and cooling fluids:
0.8
0.4
Nu  0.023 Re
Pr
TURBULENT FLOW THROUGH TUBES

Winterton & Colburn found the literature of the
Dittus/Boelter correlation to be confusing and
introduced the j factor for heat transfer:
j H  St * Pr
0.67
 hd

C P 
 k
h
St  Nu * Re
 0.023 Re
 

 vd

1
Pr
1
0.2
 k


C 
 P




TURBULENT FLOW THROUGH TUBES

Multiplying jH by Re*Pr0.33 gives:
Nu  0.023 Re


0.8
Similar to the McAdams equation
Sieder & Tate looked at very viscous fluids where
there is marked difference between the fluid at
the wall and that in the bulk .
0.14
Nu  0.27 Re

Pr
0.33
0.8
Pr
0.33
 

 S



Knudsen & Katz found that this relationship was
d
valid for:
Re Pr
L
 10
Re Pr  Pe
BULK TEMPERATURE
For gases where Pr=0.74 (usually)
Nu  0.020 Re
0.8
Fluid properties calculated at the bulk temperature
No clear free stream in tubes hence the need for a
representative temperature (bulk temperature)
Sometimes referred to as the mixing cup temperature, the bulk
temperature is the average fluid temperature of the fluid in the
tube.
TUBES

**For fully developed turbulent flows in a tube, Nusselt
recommended:
0.055
Nu  0.036 Re

0.8
d 
Pr  
L
1
3
L
for 10   400
d
h varies with distance from the tube entrance
For low h/d ratios:
haverage
h
d 
 1  
l
0.7

Tube roughness also affects heat transfer capabilities

COPE found out that for a variety of tubes friction loss was six
times greater than for smooth tubes but the heat transfer
improved by ony100-120%.
SMOOTH TUBES


These equations, though simple to compute give
rather inaccurate results
A more accurate, though complicated, expression
was recommended by Petukov for fully developed
turbulent flow in smooth tubes:
 b

f / 8 Re* Pr

Nu 
2
1
1.07  12.7 f / 8 2  Pr 3  1   w


n  0.11
n0
for Tw  Tb
for gases



n
n  0.25 for Tw  Tb
SMOOTH TUBES


Properties are evaluated at the film temperature,
Tf=(Tw + Tb)/2 except for μb & μw.
The friction factor is given by:
f  1.82 log 10 Re  1.64
2

**For fully developed laminar flow in tubes, Hausen
proposed:
Nu  3.66 

0.0668d / L  Re* Pr
1  0.04(d / L) Re* Pr 
2
3
Note that for sufficiently long tubes Nu=3.66
FORCED

CONVECTION OUTSIDE SMOOTH TUBES
&
SPHERES
Common practice: Shell & tube H/E (Internal & external)
flow
Critical Re=2.0 x 105
 Reiher, Hilpert & Griffiths studied flow of gases past
various cylindrical shapes ranging from thin wires to
tubes of 150.0 mm diameter and temperatures reaching
1073K with gas velocity up to 30.0 ms-1 and 103<Re<105

Nu  0.26 Re
0.6
Pr
0.3
SMOOTH CYLINDER & SPHERE

Flow pattern complicated and have great impact on heat
transfer

Churchill & Burnstein offered a comprehensive relation :
Nu  0.3 
0.62 Re
1
2
Pr
3
1  0.4 / Pr  
2

1
1
3
4
  Re 
1  

  282,000 

5
7




4
5
For spheres Whitake recommends:
0.4    

3
2
Nu  2  0.4 Re  0.06 Re Pr  


 S 
for 3.5  Re  80,000 , 0.7  Pr  380
1
2
1
4
SMOOTH CYLINDERS & SPHERES

McAdams recommends the following relationship
for heat transfer to spheres:
 u d 
hd

 0.37
 
kf
 f 
0.6
for 17  Re  70,000

Properties are evaluated at the bulk temperature except
for μS which is evaluated at the wall temperature, Tw

Eg.

A long 10.0 cm diameter steam pipe with external
temperature of 110.0 oC passes through an open area that is
not protected against the winds. What is the rate of heat loss
from the pipe per unit length if the air is at 1.0 atm and 10.0
oC and the wind blows across the pipe at 8.0 ms-1?
LIQUID METALS


Mercury & bismuth have high thermal conductivities
but very small Prandtl numbers (0.01) so their
thermal boundary layers develop much faster than the
velocity boundary layer
Prandtl numbers for liquid metals
Metal
Temperature, K
Prandtl Mumber, Pr
Potassium
975
0.003
Sodium
975
0.004
Na/K alloy (56:44)
975
0.06
Mercury
575
0.008
Lithium
475
0.065
LIQUID

METALS
Assuming constant velocity
Nu  0.565Re* Pr 

2
for Pr  0.5
In order to find a correlation for all fluids Churchill
& Ozoe proposed:
Nu 
0.3387 Pr
1
3
Re
1
2
1  0.0468 / Pr  
2

1
1
3
for all Pr andtl numbers
4
For smooth, turbulent free, isothermal surfaces.
LIQUID

METALS
For fully developed turbulent flow of liquid metals in
smooth tubes, Lubarsky and Kaufman posited that:
Nu  0.625(Re* Pr) 0.4
for 10 2  Pr  10 4 and L / d  60

More recent data were correlated by Skupinshi, Tortel
and Vautrey with sodium-potassium mixtures:
Nu  4.82  0.0185Pe 0.827
for 3.6 x 10 3  Re  9.05 x 10 5

& 10 2  Pe  10 4
Witte measured heat transfer from a sphere to liquid
sodium and correlated:
Nu  2  0.386Re* Pr 
0.5
for 3.56 x 10  Re  1.55 x10
4
5
SUMMARY

General procedure

Evaluate the fluid properties, usually at the film
temperature

Establish the flow regime via the Reynolds number

Select the appropriate equation

Calculate Nu, h or Q
THANK YOU
