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HEAT PROCESSES
Heat transfer in ducts,
fouling
Noncircular profiles and equivalent diameter of pipe. Compact and plate heat
exchangers. Hydraulic and thermal analysis of chevron type heat exchanger (H.Martin).
Heat transfer enhancement (static mixers, centrifugal forces, Deans vortices). Flow
invertors. Performance criteria (PEC). Fouling (example: crude oil fouling - Polley model
Rudolf Žitný, Ústav procesní a
and diagrams).
zpracovatelské
techniky ČVUT FS 2010
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Noncircular ducts
Eliptical, rectangular
ducts, channels with
longitudinal fins
Multiply connected
regions (annular,
tube bundle in shell
and tube exchanger)
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Noncircular ducts
General cross section of a channel can be characterized by equivalent hydraulic
diameter Dh, that is used in definition of Reynolds and Nusselt numbers.
Cross section
surface
Volume of channel
4 A 4V
Dh 

P
S
Perimeter of cross
section
Surface of wall
At turbulent flows the same correlations for pressure drop (friction factor) and
heat transfer (Nusselt number) can be used. Correlations for circular pipe are usually used,
however the cross sections with sharp corners (triangles, cusped ducts) lead to error up to 35% .
Equivalent diameter is used also in laminar flows, but different correlations for
different cross sections must be used (from this point of view the laminar regime is more complicated).
Modified definitions of equivalent diameter exist for specific classes of cross sections (e.g. average distance from the point
of maximum velocity in triangles, or square root of the cross section area, see next slides).
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Noncircular ducts-examples
Laminar flow – pressure drop and asymptotic Nusselt number (extremes)
Increased Nu and f
when compared with
circular pipe
b
4ab
2b

2(a  b) 1  b / a
a
a 2
a 3
 f Re  96(1  1.355  1.9467( )  1.701( )  ...)
b
b
b
Dh 
The value 96 corresponds to laminar
flow and very thin gap. Compare with
fRe=64 for circular pipe
a
a
Nu  7.541(1  2.61  ....)
b
Decreased Nu and f
when compared with
circular pipe
 f Re  28
Asymptotic Nu. The value 7.54
corresponds to a narrow flat channel
with the constant wall temperature.
Compare with the limiting value 3.66 for
circular pipe.
Shah R.K., London A.I. “Laminar flow forced convection in ducts”, Supplement 1 to Advances in
Heat transfer eds. Irvine, Hartnett, Academic Press, N.Y. 1978,
Referred by Rohsenow “Handbook of heat transfer”, McGraw Hill, Boston, 1998
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Friction factor f = f/4
Warning: There exist two different friction factors for pressure drop calculation,
be careful whether you are using the correct one
1
L
p   f  u 2
2
D
1
L
p  f  u 2
8
D
f-Fanning
friction factor
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Parallel plate heat exchangers
Simultaneous development of temperature and velocity profiles (laminar).
Both plates at constant temperature
0.024Gz1.14
Nu  7.55 
1  0.0358Gz 0.64 Pr 0.17
Tw
Stephan K. Chem.Ing.Techn. 31 (1959),p773-787
Tw
WHAT IS CORRECT???
One plate is insulated
Wrong
behaviour
at
There are two different correlations in two very respected books used by
thousands
profesionals
You can found this
correlation in VDI
Warmeatlas
Tw
One plate is at constant temperature
Similar but
different
correlation in
Rohsenow’s book
Gz. Thermal
boundary layer
increases with Gz
0.061Gz1.2
Nu  4.86 
1  0.091Gz 0.17
Mercer W.E., et al: J.Heat Transfer 89 (1967),p.251-67
0.061Gz1.2
Nu  4.86 
1  0.091Gz 0.7 Pr 0.17
Shah R.K., London A.I. “Laminar flow forced convection in ducts”, Supplement 1
to Advances in Heat transfer eds. Irvine, Hartnett, Academic Press, N.Y. 1978,
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Parallel plate heat exchangers
Have you noticed the basic difference between
correlations for circular tube and parallel channel?
The difference is in the exponent of Gz (1/3 for
tube, 1/2 for planar channel)
ax
ux
 ( x) plate 
 ( x)tube 
3
ax
ux

x

x
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Corrugated plates Heat exchangers
How to calculate pressure drop and heat transfer coefficient in plate heat
exchangers with corrugated heat transfer walls?
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Corrugated plates paper Martin Holger
Applications of CFD is rather demanding and not very accurate.
According to my opinion the best way how to calculate pressure drop and heat
transfer in heat exchangers with corrugated plates is the semiempirical method
described by Martin Holger in Chemical Engineering and Processing, 1996.
Pay attention to the following features:
How to blend results for friction factors corresponding to
different flow patterns (longitudinal and furrow flows)
How to apply analogy between momentum and heat transfer
(how to predict heat transfer from friction factors). Quite unique
feature is the Leveque analogy.
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Corrugated plates paper Martin Holger
The first problem: how to define
equivalent hydraulic diameter?
Dh 
4V
S
D plate distance
 wavelength
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Corrugated plates paper Martin Holger
Friction factor correlation
Heat transfer (generalised Leveque)
Brave idea to apply Leveque
concept also at turbulent flows!
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Corrugated plates paper Martin Holger
Few more details about Friction factor correlation
Functions 0 1 are defined separately for laminar and turbulent regime
Flow along walleys (like in a
straight pipe)
Flow in a wavy channel,
characterized by separation of
vortices at down and up-hills
Few more details on Heat transfer (generalised Leveque)
L is the distance between two crossings (and not the
length of plate). This distance is quite small so that the
thermal boundary layer is thin enough to fulfill the
Leveque’s assumption (it is assumed that the boundary
layer is restored at each crossing)
Is it really Leveque? Yes,
because at laminar flow
Re=constant
Leveque analogy is discussed in paper Martin H.:The generalized Leveque equation and
its practical use for the prediction of heat and mass transfer rates from pressure drop,
Chem.Eng.Science, 57 (2002), pp.3217-3223
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HT enhancement
Dalí
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HT enhancement
How to increase heat transfer coefficient at internal channel flows (in pipes)?
1. Artificial wall roughness, porous wall
2. Fins, grooves, dimples
3. Inserts (static mixers, twisted tape, wire mesh, invertors)
4. Centrifugal forces (coiled tubes, bends)
5. Vibration, ultrasound, nanoparticles…
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HT enhancement
Heat transfer augmentation (Nu increase – desirable effect) is usually
accompanied by pressure drop increase (undesirable effect). There exist
many different PEC (Performance Evaluation Criteria) characterising
efficiency of considered modification (only those giving PEC>1 should be
used) .
The most frequently used PEC 
Nu
Nu0
3
f0
f
This PEC follows from comparison of the two identical pipes (the same diameter and length), one pipe
is empty (Fanning friction factor f0) the second one is modified by inserts, fins,… (higher f). So that the
pumping power will be the same the flowrate in the augmented pipe (f>f0) must be decreased
Assuming the same temperature approach T in the both
pipes, the thermal power is proportional to the Nu and the
PEC can be interpreted as
Comparison of thermal
powers for the same
pumping power, the same flow
rates but different lengths
Q
Nu f 0
PEC 

Q0 Nu0 f
Proof!
QV
Nu

Q0V0 Nu0
3
f0
f
V

V0
3
f0
f
Proof!
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HT enhancement –wall
Heat transfer can be increased by a modification of wall such that the heat
transfer surface is extended (fins, dimples), and the thermal boundary layer is
disrupted (for example by vortices generated at protrusions or dimples).
Only a little bit controversial enhancement by dimples will be presented in next
slides.
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HT enhancement – dimpled wall
H. Lienhart et al. / Int. J. Heat and Fluid Flow 29 (2008) 783–791
Drag reduction by dimples? – A complementary experimental/numerical investigation
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HT enhancement – inserts
Inserts (static mixers, twisted tape) extend heat transfer surface (as far
as a good thermal contact with pipe wall is ensured) and generate
secondary flows diminishing thermal boundary layer. Inserts are
effective first of all in laminar flow regime (PEC is highest at low Re),
but heat transfer enhancement in turbulent regime is also significant.
Wire coils disrupt thermal boundary layer (suitable for laminar flows),
wire mesh affects the main flow and is effective in turbulent flows.
Advantage: Tiny wire at wall has only small effect upon pressure drop.
What is surprising: inserts usually suppress fouling!
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HT enhancement SM Kenics
Static mixers (Kenics, Sulzer,Helax,…) serve for mixing of liquids but
also for the heat transfer intensification.
Standard solution consists in filling
the whole tube by SM elements (tight
arrangement).
For Kenics SM the heat transfer at
laminar flows is increased as (see
Joshi, Nigam, Cibulski)
Nuln  3.66  3.89Gz1/3
Compare with empty pipe (Leveque)
Nuln  3.66  1.618Gz1/3
On the other hand, the tube filled by SM elements exhibits higher
pressure drop (friction factor)
Question: f, f0 represent
Fanning or the Darcy
Weissbach friction factor?
f  110 / Re tube filled by Kenics SM
f 0  16 / Re empty tube
Answer : Fanning (see empty tube)
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HT enhancement twisted tape
International Communications in Heat and Mass Transfer 38 (2011) 348–352
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HT twisted tape & wire coil
Lieke Wang, Bengt Sundén Performance comparison of some tube inserts International
Communications in Heat and Mass Transfer, Volume 29, Issue 1, January 2002, Pages 45-56
Twisted tape (laminar/transition/turbulent)
Swirl number
Nu 
Sw  Re
D

 0.1Sw0.677 Pr 0.265 ( )0.14

w
300  Re  30000
Wire coil (laminar/transition/turbulent)
D
H
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HT– centrifugal forces
Centrifugal forces in coiled pipes (spirals, helically coiled pipes) create
secondary flows similar to vortices induced e.g. by a twisted tape. Local
effect of centrifugal forces and secondary flow appear also in bends (for
example U-tube acts as a partial flow inverter).
Advantage: Increased Nu is not accompanied by too large pressure
drop increase. Positive effect is significant reduction of fouling (spiral
heat exchangers are suitable for dirty fluids, fibrous pulps,…).
Residence time characteristics are improved (residence times of fluid
particles moving at axis of pipe and in vicinity of wall are not so different
as in a straight pipe).
Disadvantage: Effect of centrifugal forces disappears at creeping flow,
therefore this technique cannot be applied for highly viscous liquids
(Re<10)
Dean, W.R., Note on the motion of fluid in a curved pipe, Phil.Mag.Ser.7, vol.4, no.4, pp.208, 1927.
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HT– centrifugal forces
Some trivial facts:
u
Centrifugal force acting on
particle of mass m
Fc=2mu2/Dc
Dc
m
Force acting on plate with cross
section D x 1
Fi=ur2D
ur
D
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HT enhancement coiled pipe
Centrifugal forces generate two counter-rotating vortices
(secondary flow). Characteristic velocity of circulation ur
(transversal velocity) can be estimated from balance of
equilibrium force Fc and inertial force Fd
2
D
 ur2 D  k  u 2
Dc
Intertial force related to unit
length of pipe (dynamic
pressure acts on area D)
centrifugal force on unit length
of pipe (acting on volume in
the whole cross section)
From the force equilibrium follows the ratio between radial and axial velocity
Thermal boundary layer and penetration depth
D
 2   at   a

ur
Nu 
Mori a Nakayama (1965)
D D

 
D
ur
D
k
u
Dc
ur
Du D
D

 Pr Re
 Pr De
aD
a Dc
Dc
Nu  
4.75 De
1 1
See also M.M. Mandal et al. / Chemical Engineering Science 65 (2010) 999–1007
This is only brief
derivation showing
principles
77
4 Pr 2
De  Re
D
 12
Dc
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HT enhancement coiled pipe
Have you noticed similarity between Dean’s and
Swirl number?
D
De  Re
Dc
D
Sw  Re
H
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Flow inversion
Flow inversion
transfers overheated
fluid from wall to axis
Partial flow
inversion in bends
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Flow inversion in a bend paper Zitny
Zitny R, Luong TCT, Strasak P, et al.: Heat Transfer Enhancement and RTD in Pipes
with Flow Inversion. Heat Transfer Engineering, Vol. 25 (2004), pp. 67-79
R
Centrifugal forces in a bend
generate secondary flows and the
flow inversion (counter-rotating
vortices transfer fluid particles
from pipe axis toward wall)
L
Rc
L
Secondary vortex
Centrifugal force

Flow inversion in a bend
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Optimum flow inversion causes half-rotation of the secondary vortex and this situation
is achieved at about Re.F=100 (laminar flow)
Gz=50
F  /2 Rc/R=2
F  /2 Rc/R=4
F  /2 Rc/R=16
F  /2 Rc/R=50 S
F  /4 Rc/R=50 S
F  Rc/R=2
1.4
1.3
Nu/Nus
1+0.37[1-exp(-0.01ReF )]
1.2
R
L
R
1.1
c
L

1
10
100
Re.F
1000
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Flow inverter paper Zitny
Zitny R, Thi C.T.L, Sestak J : Heat Transfer Enhancement in a Pipe Using a Flow Inverter.
Heat Transfer Engineering, Vol. 30 (2009), pp. 952-960
inverter
Hot center,
cold wall
Incoming stream is
mechanically subdivided into
the central and the wall
region and mutually
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Flow inverter
In case of Re<10 centrifugal forces are not strong enough to generate secondary
flows and flow inversion. “Mechanical” subdivision operates also at Re<<1.
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Flow inverter
Performance Evaluation Criterion
PEC 
Nu
Nu0
3
f0
f
1.200
A Re=50
A Re=0.1
1.150
S Re=50
S Re=0.1
PEC
1.100
1.050
1.000
0.950
0.900
0.001
0.010
1/Gz
0.100
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Extended surfaces
Previous analysis was concentrated upon the heat transfer enhancement
by increasing heat transfer coefficient. Inserts or modifications of pipe
walls increases at the same time the heat transfer surface, however this
additional surface can be fully accounted for only if the thermal resistance
of inserts or fins is negligible.
Dalí
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Extended surfaces (fins)
Compact heat exchanger
T2
H
T1
T2
T2
Q12
Tw1
1
T1
B
Plate and fin
T2
T2

H/2
T1
Tw
T1
b
x
b T2
In the case that the thermal resistance of walls is zero (infinitely large thermal
conductivity of fins) the surface of fins can be added to the heat transfer surface and
Q12  1 (T1  Tw1 )( B  H )
In the case that thermal resistance of fins cannot be neglected the heat transfer
surface must be reduced
Q   (T  T )( B  H )
12
1
1
w1
fin
Efficiency of fin fin can be calculated from temperature profile T(x) in a fin, determined
by Fourier equation
2-because the fin is
d T 2
0 2 
(T1  T )
dx
b
2
heated from both sides
completed by boundary conditions
T  Tw
at x  0
dT
0
dx
at x  H / 2
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Extended surfaces (fins)
Solution of previous equation yields temperature profile along the height of fin
T ( x)  T1  (Tw  T1 )(
e
x
1 e
2
b
2
H
b

e
x
1 e
2
b
2
H
b
)
Efficiency of fin is calculated from temperature gradient at the heel of fin (the
gradient determines heat flux at the heel)
 fin 
Q
Q 
dT
|x  0
2b
H2
dx


tanh
2
 H (Tw  T1 )
H
2b
1
tgh Bi
Bi
b
0.5
0
0
1
In a similar way the efficiency of circular fin can be derived
K    I     I1  1  K1   0 
2
circular  2 0 2 1 1 1 0
.
1  0 K1  1  I 0  0   I1  1  K 0  0 
where  are dimensionless radii
2
3
I1,K1 are
modified Bessel
functions
r
  2 Bi ,
b
Bi 
b
,
2
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Extended surfaces (fins)
Example:
Calculate efficiency of rectangular fin of constant thickness
1mm, height H=20mm made from stainless steel for heat
transfer coefficient 3000 W/m2/K
Result =0.16
If the same fin will be from aluminium, the efficiency
increases to =0.54
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Fouling
Formation of deposits on heat transfer surface increases
thermal resistance (and pressure drop)
Photographs from paper
Precipitation
Corrosion
Khalil Ranjbar:Effect of flow induced corrosion
and erosion on failure of a tubular heat
exchanger. Materials and Design 31 (2010)
613–619
Chemical deposits
Biochemical deposits
Solidification
There are many ways how to mitigate fouling: addition of tiny particles (nano, pulps), sonication,
pulsating electrical field, turbulisation of flow (e.g. wire mesh usually mitigates fouling):
S.N. Kazi, G.G. Duffy, X.D. Chen: Fouling mitigation of heat exchangers with natural fibres. Applied Thermal Engineering 50 (2013) 1142-1148
Y.I. Cho, B.G. Choi: Validation of an electronic anti-fouling technology in a single-tube HE. Int. J. Heat and Mass Transfer. 42 (1999), 1491-1499
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Fouling
Fouling evolution
1. Induction period
2. Negative fouling (e.g. promoted nucleate boiling, heat transfer increased)
3. Linear fouling (constant rate of deposits formation, thermal resistance increases)
4. Falling fouling (decreasing rate of fouling formation)
5. Asymptotic fouling (zero rate)
Rf=h/
t
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Fouling fouling rate models
Chemical fouling of oil products (Ebert Panchal model)
dR f

E
 0.8 1/3 exp(
)   w
dt
Re Pr
RT film
Deposits as a product
of chemical reaction
with activation energy
E
Rate of deposits
removal proportional to
wall shear stress
The production rate is proportional to the volume of reaction zone –
overheated thermal boundary layer of thickness 

Nu   c Re0.8 Pr1/3

dR f
dt

c
E
 exp(
)   w

RT film
B.L. Yeap, D.I. Wilson, G.T. Polley, S.J. Pugh: Mitigation of Crude Oil Refinery Heat Exchanger Fouling Through
Retrofits Based on Thermo-Hydraulic Fouling Models Chemical Engineering Research and Design, Volume 82, Issue
1, January 2004, Pages 53-71
G. T. Polley, D. I. Wilson, B. L. Yeap, S. J. Pugh Evaluation of laboratory crude oil threshold fouling data for application
to refinery pre-heat trains Applied Thermal Engineering, Volume 22, Issue 7, May 2002, Pages 777-788
W.A. Ebert, C.B. Panchal, Analysis of Exxon crude slipstream coking data, in: C.B. Panchal, et al. (Eds.), Fouling
Mitigation of Industrial Heat-Exchange Equipment, Begell House, 1997, pp. 451–460.
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Fouling fouling rate models
Asymptotic fouling is characterized by
dR f
dt
0
and from the Ebert Panchal fouling model follows the value of critical wall
shear stress ensuring zero fouling rate
w 

E
exp(

)
0.8
1/3
 Re Pr
RT film
This criterion is used for heat exchanger design by Poddar diagrams.
See next slide
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Fouling fouling rate models
G.T. Polley et al. Use of crude oil fouling threshold data in heat
exchanger design. Applied Thermal Engineering 22 (2002) 763–776
T.K. Poddar, G.T. Polley, Optimising the design of shell-and-tube heat exchangers, Chemical Engineering Progress
(September) (2000).
Problem specification:
Calculate number of tubes and
length of S&T HE for given thermal
duty (power), flowrates in shell and
tubes, maximum pressure drops in
shell and tubes.
Poddar diagram
Region of design
parameters (L,n) satisfying
constraints on duty and
pressure drop (in this case is
limiting the shell side)
Optimum design:
600 tubes in
bundle, length 3.2
m
Length
Unpleasant situation-for the
optimum design parameters a
fouling in tubes can be expected
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HP5
EXAM
Noncircular channels
Concept of equivalent diameter (Dh
is used in definition Nu and Re)
4 Across sec tion
Dh 
Pwetted perimeter
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What is important (at least for exam)
Heat transfer and thermal boundary layer
thermal boundary layer grows faster at a plate than at the wall of pipe
 fluid


 ( x)tube 
 ( x) plate 
3
ax
ux
ax
ux
Parallel plate channel
tube (Léveque)

x

x
plate channel

constant temperature of both plates
x
0.024Gz1.14
Nu  7.55 
 0.67 Gz
0.64
0.17
1  0.0358Gz Pr
for short plate
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What is important (at least for exam)
Corrugated plates (chevron HE)
Fanning friction factor f (denoted as  in original
paper by H.Martin) is calculated from correlation
as a function of chevron angle and Reynolds
number.
Generalised Léveque correlation is based upon
analogy between momentum and heat transfer.
Nu is calculated from friction factor (take into
account that fRe=const at laminar flow regime)
Nu  0.4
3
f  Re 2 Pr
d
L
This correlation holds at laminar and turbulent
flow regime!
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What is important (at least for exam)
Inserts in pipes and centrifugal forces (heat transfer enhancement)
-static mixers (enhanced Leveque
Nuln  Gz1/3
)
-twisted tape (Nu depends upon swirl number
Sw  Re
D
H
)
-helical coils (Nu depends upon Dean number
De  Re
D
Dc
)
H
Extended heat transfer surface (fins)
tgh
Bi
b
The effective heat transfer surface of fins must be reduced by  fin 
H
where H is height, b is thickness of fin and Biot number is
Bi
b
b
Bi 
2
b
H
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What is important (at least for exam)
Fouling in pipes (Ebert Panchal 3 parametric model ,,E-activation energy, the
model assumes that rate of deposits formation is proportional to the volume of
overheated fluid inside turbulent thermal boundary layer, see Dittus Boelter
correlation Nu~Re0.8Pr1/3)
dR f

E
 0.8 1/3 exp(
)   w
dt
Re Pr
RT film