MHMT11
Momentum Heat Mass Transfer
D
     source
Dt
Heat transferconvection
Thermal boundary layer. Forced convection (pipe,
plate, sphere). Natural convection.
Rohsenow W.M., Hartnett J.P., Cho Y.I.: Handbook of Heat Transfer. McGraw Hill, 3rd Edition, 1998
Rudolf Žitný, Ústav procesní a
zpracovatelské techniky ČVUT FS 2010
MHMT11
Heat transfer - convection
Heat flux from a surface to fluid is evaluated using heat transfer coefficient
, assuming linear relationship between heat flux and driving force, which
is the temperature difference between solid surface and temperature of
fluid far from surface (outside a thermal boundary layer)
q   (Tw  T f )
Heat transfer coefficient is related to thickness of thermal boundary layer


T
y
T
T
Tw
x
MHMT11
Thermal boundary layer
Integral boundary layer theorem can be derived in the same way like for
the momentum boundary layer
H
H 2
T
T
T
Upper bound H 
(
u

u
)
dy

a
0 x x y y
0 y 2 dy
H
(
0
H

0
H

0
H

0
u xT
u u yT
u
T
T x 
 T x )dy  a[ ] yy 0H
x
x
y
x
y
u xT
T
dy  [u yT ] yy 0H  a
| y 0
x
y
u xT
dy  T
x
H

0
u x
T
dy  a
| y 0
x
y

T
(u x (T  T ))dy  a
| y 0
x
y
d
dx
H
 ux (T  T )dy  ux (T  T ) | H
0
d H
T
 a
| y 0
dx
y
this is zero at boundary
MHMT11
Thermal boundary layer
Final form of integral boundary equation

u x (T  T )
qw
d
[(Tw ( x)  T )U ( x) 
dy ] 
dx
U ( x)(Tw ( x)  T )
c p
0
T thermal energy thickness
qw heat flux
This equation can be applied as soon as the momentum boundary layer
was identified (velocity profile inside the thermal boundary layer must be
known).
This equation is used for prediction of heat transfer at flows along plates,
pipes, cylinder, spheres…
However, in the following slides we shall analyze heat transfer in a
pipe and along a plate in a simpler way, using the penetration theory
(instead of integral theorem). The penetration theory predicts not so
accurate but still qualitatively correct results.
MHMT11
Heat transfer – pipe
Forced convection: Reynolds
number is given, Nusselt number is
to be calculated.
Nu  3 Re Pr D / L
Duchamp
MHMT11
Heat transfer – pipe (laminar)
Convective heat transfer in a pipe for the laminar flow and fully developed
velocity profile (therefore it is not necessary to solve the problem of
penetration
momentum boundary layer).
theory
Velocity profile
u( y)  u
8y
D
2 
4ax axD

u ( ) 2u 
axD
 3
2u
parabolic
velocity
profile
D
slope of the
velocity
profile
behind the boundary
layer is the inlet
temperature T0
Re
Pr
Re Pr  Pe
Leveque formula

Tw
x
uD 2
uD  D

ax
 a x
Gz Graetz number
T0
y
 D D 3 2uD 2
Nu 
 


ax
2
D
uD
Nux 
 c3
 c 3 Gz x

ax
Remark: This correlation predicts local value of (x), which
is stressed by lower index in Nusselt and Graetz number
MHMT11
Heat transfer – pipe (laminar)
Anyway, qualitatively the same result can be obtained using integral theorem

qw
Tw  T0
d T
u
(
T

T
)
dy


a
x
0
dx 0
cp
T
assuming linear temperature and velocity profile in the thermal boundary layer
y
8y
T  Tw  (Tw  T0 )
u( y)  u
T
D

Tw  T0
d T 8y
y
u
(
T

T
)(1

)
dy

a
w
0
dx 0 D
T
T
d 8 T2
aD
(
)
dx 6
T u
 T3 
9 aDx
8 u
…this results differs only by a constant (9/8) from the
previous result (1/2) but this constant is anyway usually
modified using experiments or more accurate
assumptions.
MHMT11
Heat transfer – pipe (laminar)
Local value (x) increases to infinity with decreasing distance from the tube
inlet. From practical point of view a more important is the average value of 
at pipe of the length L .
1
 c 3
D
 3c 3
D
3
 ln    dx 
Re
Pr
dx

Re
Pr

1.6
Gz
L0
D L 0
x
D 2
L
D
L
L
index ln is used because this ln is
related to the mean logarithmic
temperature difference used in the
heat exchanger design
Correlations used in practice for heat transfer prediction in laminar flows in pipes
 ln D
Léveque formula for Gz>50 (short pipes)
 1.67 3 Gz

 D
0.0668Gz
Hausen formula for arbitrary long pipes
Nu  ln  3.66 
2/3

1  0.04Gz
Nu 
Note the fact that at very long tubes the Nu (and ) is constant 3.66, and that the
Hausen correlation reduces to Leveque for Gz
MHMT11
Heat transfer – pipe (turbulent)
Turbulent flow is characterised by the energy transport by turbulent eddies which is
more intensive than the molecular transport in laminar flows. Heat transfer
coefficient and the Nusselt number is therefore greater in turbulent flows. Basic
differences between laminar and turbulent flows are:
Nu is proportional to
3
0.8
u in laminar flow, and u in turbulent flow.
Nu doesn’t depend upon the length of pipe in turbulent flows significantly (unlike
the case of laminar flows characterized by rapid decrease of Nu with the length L)
Nu doesn’t depend upon the shape of cross section in the turbulent flow regime (it
is possible to use the same correlations for elliptical, rectangular…cross sections
using the concept of equivalent diameter – this cannot be done in laminar flows)
The simplest correlation for hydraulically smooth pipe designed by Dittus Boelter
is frequently used (and should be memorized)
Nu  0.023Re0.8 Pr m
m=0.4 for heating
m=0.3 for cooling
MHMT11
Heat transfer - plate
Analysis of heat transfer at external flows
(around a plate, cylinder, sphere…)
differs from internal flows (for example
heat transfer in pipe) by the fact that
velocity profile at surface is not known in
advance and therefore not only the
thermal boundary layer but also the
momentum boundary must be solved
Nux  Re x 3 Pr
Duchamp
MHMT11
Heat transfer - plate
Heat transfer in parallel flow along a plate is characterised by simultaneous
development of thermal and momentum boundary layers. It will be assumed, that the
thickness of momentum boundary layer H is greater than thermal boundary layer T
y

 x (this is previously obtained
Linear velocity profile u ( y )  U 
u ( T )  U  T
result for thickness of the
 H  4.6
H
H
U  momentum boundary layer)
Thermal boundary layer thickness
H
T
 T2 
T
Tw
x
Nu 
4ax
4ax
4ax
x

H 
4.6
u (T )
U T
U T
U 
x
U 
 H  4.6
y
T 
x

loca value of
heat transfer
4ax
u ( T)

x
T  (18.4a)1/3 ( )1/6

U
Note the fact, that the ratio of thermal and
momentum boundary layer thickness is
independent of x and velocity U
( xU  )1/2  1/6


( ) 
 T (18.4a)1/3 
x
1
18.41/3
this constant should
be replaced by 0.332
in accurate solution
(
xU  

Re
)1/2 (
 1/3
)
a
Pr
MHMT11
Heat transfer - plate
Mean value of heat transfer is obtained from the previous formula by integration
along the length of plate L
H 
y
x
U 
T   H / Pr1/3
x
Laminar flow regime
L
Nu  0.664 Re L Pr1/3
Turbulent flow regime (Re> 500000)
1/3
Nu  0.0365Re0.8
Pr
L
Compare these results with the heat transfer in pipe
Pipe: Laminar flow NuRe1/3 turbulent flow NuRe0.8
Plate: Laminar flow NuRe1/2 turbulent flow NuRe0.8
MHMT11
Heat transfer – sphere…
Flow around a sphere (Whitaker)
Front side
Important for heat transfer from droplets…
Rear side
(wake)
See next
slide
Nu  2  (0.4 Re1/2  0.06 Re2/3 ) Pr 0,4
0,71Pr380 3,5Re7,6.104 .
Cross flow around a cylinder (Sparrow 2004)
Important for shell&tube and fin-tube heat exchangers
Nu  0.25  (0.4 Re1/2  0.06 Re2/3 ) Pr 0.37
0,67Pr300 1Re105 .
Cross flow around a plate (Sparrow 2004)
Nu front  0.6 Re
Nurear  0.17 Re 2/3
Heat transfer – sphere
MHMT11
Conduction outside a sphere, see 1D solution of temperature profile
Tw
D
1 d 2 dT
(r
)0
2
r dr
dr
T
r
T (r )  T 
D
(T  Tw )
2r
q   (T  Tw )  
T
2
|r  D /2 
(T  Tw )
r
D
D
Nu 
2

MHMT11
Natural convection
Velocity and Reynolds
number is not known in
advance. Flow is induced
only by buoyancy. Instead of
Reynolds it is necessary to
use the Rayleigh number.
Duchamp
MHMT11
Natural convection
Vertical wall. Wall temperature Tw>Tf. Heated fluid flows upward along
the plate due to buoyant forces.
Forces equilibrium (wall shear stress=buoyant force)
u
 m  g T 
 w  g

um
Penetration depth at distance x

x
4ax
 2  4a
 um  2
um

x
Substituting um into force balance
4ax
4 ax
 3  g T     4
y

g T
x x
g Tx3
g Tx3
Nu 
 4
 c4
 c 4 Ra
 
4 a
a
where Ra is Rayleigh number
(x represents a characteristic dimension in the direction of gravity, e.g. height of wall)
MHMT11
Natural convection
Turbulent flow regime occurs at very high Ra>109 and instead the 4th root of Ra
the 3rd root is used in correlations
Nu  cturb 3 Ra
Note the fact, that at turbulent flow regime the heat transfer coefficient is almost
independent of x (size of wall)
x
g Tx3
g T
3
3
 cturb
   cturb 

a
a
MHMT11
EXAM
Forced convection
Natural convection
MHMT11
What is important (at least for exam)
What is it heat transfer coefficient and how is related to thermal
boundary layer thickness
Forced convection


Nu(Re,Pr)
q   (Tw  T f )
T
 ln D
 1.6 3 Gz  1.6 3 Re Pr D / L

Heat transfer in a pipe (laminar)
Nu 
Heat transfer a plate (laminar)
Nu  0.664 Re L Pr1/3
Heat transfer turbulent
Nu
Re0.8 Pr1/3
MHMT11
What is important (at least for exam)
Free convection
Nu(Ra)
Laminar flow (Re<1010)
Turbulent flow
x
g Tx3
Nu 
 c4
 c 4 Ra

a
x
g Tx3
Nu 
 ct 3
 ct 3 Ra

a