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DOUBLE PIPE FINNED HEAT EXCHANGER
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Proceedings of the 9th Biennial ASME Conference on Engineering Systems Design and Analysis
ESDA08
July 7-9, 2008, Haifa, Israel
ESDA 2008 59124
A SEMI EMPIRICAL METHODOLOGY FOR PERFORMANCE ESTIMATION OF A
DOUBLE PIPE FINNED HEAT EXCHANGER
G. Arvind Rao1, Yeshyahou Levy2
Turbo and Jet Engine Laboratory, Technion: - Israel Institute of Technology
Haifa 32000, Israel.
ABSTRACT
Finned tubes are one of the most widely used means of
passively enhancing the heat transfer in circular tube. Many
investigators have proposed different correlations for predicting
the performance of such heat exchangers based on their
experimental results. However, the practical usage of such
correlations is limited because of the variety of parameters that
can influence Nusselt number and friction factor. Most of the
correlations either have been developed with limited databases,
or are geometry specific. Using CFD for analyzing performance
of such heat exchangers is very computational intensive and
hence cannot be used for design optimization purposes. On the
other hand, empirical correlations have many limitations in terms
of their applicability.
The objective of the present article is to present a
physically based model for evaluating heat transfer and
frictional loss for an internally and / or externally finned double
pipe heat exchanger that can be applied in a wide range of
operating conditions of practical importance. This paper
describes a simple semi-empirical-numerical methodology to
evaluate heat transfer and pressure drop characteristics in a
finned tube heat exchanger with internal and/or external fins.
Conduction and turbulent forced convection are the prominent
modes of heat transfer. In order to resolve the operational
characteristics of double pipe finned heat exchangers, a
numerical methodology is presented which uses well known
existing correlations for flow in a smooth pipe and flow over a
flat plate. The method of successive substitution is used to
solve the problem numerically. The proposed methodology is
applied to some simple cases and the results compare well with
1
2
data and correlations available in the literature. It is found that
the addition of fins to such double pipe heat exchangers
reduces the Nusselt Number; however the corresponding heat
transfer rate increases owing to the increase in the overall heat
transfer per unit area.
Keywords: Double Pipe Heat Exchanger, Heat Transfer,
Internally Finned Tubes, Externally Finned Tubes
1. INTRODUCTION
Finned tubes are one of the most widely used means of
passively enhancing the heat transfer in circular tube (one of
the most widely used heat transfer surface in heat exchangers).
Finned surfaces are applicable where an additional area can be
provided for augmenting the heat transfer. An important
application for such air to air heat exchangers is in the open
regenerated Brayton-cycle [1] and alternative regenerator
configurations [2] of the land based gas-turbine power-plant,
wherein the compressor discharge air is preheated by the hightemperature exhaust gases. Similarly, such heat exchangers are
used for regeneration in industrial furnaces.
The
thermodynamic efficiency of the whole cycle is improved
because air enters the combustor at an elevated temperature,
with out the need for fuel burn and therefore the amount of heat
addition in the combustor is reduced.
Finned tubes are used to reduce the size of a heat
exchanger required for a specified heat duty, or to increase heat
transfer rate of an existing heat exchanger design. An internally
finned tube can provide a significant increase in the surface
area, and can offer an appreciable enhancement of heat transfer
Post Doctoral Fellow
Corresponding author. Tel.: +972-4-8293807; Fax: +972-4-8121604.
E-mail address: levyy@aerodyne.technion.ac.il (Y. Levy)
1
Copyright © 2008 by ASME
rate [3]. Among several available techniques for augmenting
heat transfer in heat exchanger tubes, the use of fins appear to
be the most promising, as described by Webb [4]. It is evident
that an efficient design of heat exchanger can substantially
increase performance of the entire system.
Finned tubes perform differently depending on whether the
flow is laminar or turbulent. Several studies have been
conducted to investigate the effect of fin characteristics on heat
transfer. For both laminar and turbulent flow regimes, the finned
tubes exhibit substantially higher heat transfer coefficients
when compared with corresponding smooth tubes. The
performance of finned pipe is mainly determined by the type of
flow, fin efficiency (which determines the average heat transfer
coefficient) and the friction factor, which is responsible for
pressure / pumping loss.
1.1 Motivation
Many investigators have proposed different correlations based
on their experimental results [5-9]. But the practical usage of
such correlations is limited because of the variety of parameters
that can influence Nusselt number and friction factor. Most of
the correlations have been developed with either limited
databases, or are geometry specific. Moreover since most of
the experimental test sections are small in length (typically less
than 1m), the temp erature variation within these test sections is
low and therefore the resulting correlations do not take into
account the variation of fluid properties along the test section.
However, when it comes to practical applications of such heat
exchangers, where a large amount of heat is to be extracted, and
length of the heat exchanger can be quite large. Hence, there
may be a substantial change in the fluid temperature along the
tube length and therefore variation of fluid properties can
significantly affect the performance predictions. An additional
motivation for carrying out the present research work is the fact
that in the literature there is information on either internally
finned tubes or externally finned tubes, but there is no
published work on double pipe heat exchangers with both
internal and external fins. Because of the above-mentioned
limitations, the objective of the current research is to develop a
physical model for evaluating heat transfer and frictional loss
for an internally and / or externally finned double pipe heat
exchanger, which can be applied in a wide range of operating
conditions of practical importance.
An additional motivation for present research is that most
of the numerical investigations that have been carried out for
finned pipe heat exchangers have used Computation Fluids
Dynamics with various turbulence models for studying the flow
pattern inside the heat exchanger [10, 11]. Even though such
techniques are quite accurate and give an in-depth physical
understanding of the problem, they are very time consuming,
and hence cannot be used to evaluate performance of the
complete heat exchanger, especially for industrial applications.
Also such techniques can not be used for optimization exercises
where a number of iterations are required before an optimum
design can be achieved. On the other hand, empirical
correlations have many limitations in terms of their applicable
operational range. Hence there is room for a relative quick semiempirical methodology to evaluate performance of such heat
exchangers that can also be used in design / optimization
exercises. As mentioned by many authors [12,13], there is
significant scope in optimizing the performance of finned tube
heat exchangers. Schematic of the various fin arrangements
investigated in the present paper is shown in Figure 1.
[a]
[b]
[c]
Figure 1. Fin arrangement in different types of heat
exchangers,
(a) Internally Finned Pipe
(b) Externally Finned Pipe
(c) Internally and Externally Finned Double Pipe
Heat Exchanger
2. LITERATURE SURVEY
As far as geometry of the internal fins is concerned, most
internal fins are strips with surfaces positioned longitudinally
along the tube axis. Many investigations, both experimental
and numerical, have been conducted on different kinds of
internally finned tubes using a variety of fluids (air, water, oil,
ethylene, etc) [5-15]. These studies examined the overall
performance in terms of circumferentially averaged friction
factors and heat transfer coefficients and examined the effects
of parameters like: number of fins, fin height, fin width, helix
angle, etc. Because of the wide range of fin geometries and
Reynolds numbers covered in their experiments, Jensen and
Vlankancic [9] suggested different governing processes
between ‘‘tall fin’’ and ‘‘micro-fin’’ tubes. While “tall fins” (as
investigated in this paper), are used in industrial furnace,
thermal power plants, etc, the ‘‘micro-fins’’ are generally used
for cooling in electronic equipments. Most of the investigators
obtained an increase in the Nusselt numbers from 15-180% as
compared to the smooth tube. However this increase in heat
transfer is counterbalanced by an increase in friction factor by
50-500 % [9].
Soliman et al. [18] analyzed fully developed laminar flow,
while accounting for conduction in the tube wall and fins but
keeping the outer surface of tube at a constant temperature. A
comprehensive experimental and numerical investigation of
laminar flow was performed by Shome and Jensen [19].
Rectangular fins were implicitly assumed in all these studies.
2
Copyright © 2008 by ASME
Patankar et al. [10] analyzed fully developed turbulent flow and
heat transfer characteristics for internally finned tubes and
annuli using the mixing length model. The local heat transfer
coefficients exhibited a substantial variation along the fin
height, lowest being at the fin root and highest being at the fin
tip.
Literature survey for heat transfer through pipes by
turbulent forced convection portrays that most of the
investigators have used finite difference equations of
conservation of energy, x-momentum and continuity equation
and have emphasized on the importance of modelling the
variation in transport properties for realistic results. Bankston
and McEligot [20] predicted wall temperature and pressure drop
characteristics for forced internal flows. They used power law
index approximation for modelling variation in fluid transport
properties. Malik & Pletcher [21] and Pletcher & Malik [22]
worked on prediction of turbulent flow heat transfer in annular
geometries with property variations in which they considered
the flow as axisymmetric, steady and fully turbulent.
2.1 Internal Fins
Masliyah and Nandakumar [12] have studied heat transfer
in internally finned tube. The fins were of triangular shape and
the number of fins was varied up to 24 and in the length up to
0.8 times of the tube radius. Finite element method was used to
analyze laminar flow in an internally finned circular tube with
uniform axial heat flux around the wall. They conclude that
Nusselt number based on inside diameter was higher than that
for a smooth tube without fins. They also found that for
maximum heat transfer rate, there exists an optimum number of
fins for a given configuration and application. Liu and Jensen
[23] used an unstructured finite-volume method with a two-layer
turbulence model to capture the near-wall turbulence in two
spirally finned tubes. The circumferentially averaged friction
factors and Nusselt numbers compared well with exp erimental
data of Jensen and Vlakancic [9].
One of the commonly used correlations for internally finned
tubes is that by Carnavos [7], who used 21 different types of
internally straight finned tubes. For straight longitudinal fins,
the correlation is given as,
A 
0 .4  fa 
Nu bh = 0.023⋅ Re 0bh.8 ⋅ Prbh
⋅
A 
 fc 
fh = 0.046
0 .10
 A fa 


A
 fn 
Re −bh0. 2⋅ 

A
⋅  n
 Aa




Afn = nominal flow area based on tube ID as if the fins were not
present, mm2
All flow quantities in the above equation are calculated at the
fluid bulk temperature.
The Carnavos correlation has been subsequently verified
by many investigators like Patankar et al. [10] and Trupp and
Haine [24]. Edwards et al. [25] experimentally investigated fully
developed turbulent flow in longitudinally finned tubes with
Laser Doppler Velocimeter (LDV) and found that the Carnavos
correlation agreed well with their data (within ±10% accuracy).
However they suggested that a Reynolds number exponent of 0.25 (instead of -0.2) would be more appropriate for evaluating
the friction factor.
2.2 External Fins
There is a dearth of literature when it comes to investigations in
externally finned annuli. One of the most important works in
this regard is by Braga and Saboya [5]. They performed
experiments to determine heat transfer coefficient and friction
factor for turbulent flow through annular ducts with 20
continuous longitudinal rectangular fins made of brass. They
proposed the following Reynolds number dependent heat
transfer coefficient (for air).
h ⋅ Dh
Nu =
= 0.00529 Re 0.8680·
(104 < Re < 5×104)
(3)
k
They also proposed the following correlation for evaluating the
friction factor
f = 2.88467·Re- 0.2863
(104 < Re < 5.2×104)
(4)
where the pressure drop is evaluated as
1 L
m& 2
∆P = f ⋅
⋅
2ρ Dh  π
2 
2
 4 ⋅ D o − ( Di + 2t1 ) 


(
)
(5)
It should be noted that the above mentioned relation
between pressure drop and friction factor as used by Braga and
Saboya is different than the conventionally used relation given
below
 ρ ⋅V 2  L
⋅
∆P = 4 ⋅ f ⋅ 
(6)
 2  D

 h
0 .5
(1)
0 .5
(2)
where Aa = actual heat transfer area (mm2/mm); An = nominal heat
transfer area (mm2/mm)
Afa = actual free flow area, mm2;
Afc = open core free flow
area at fin ID, mm2
3.
NUMERICAL MODELING
The heat transfer effectiveness of finned pipes depends on
factors like the fin conductance, fin dimensions and local heat
transfer coefficient. The present problem cannot be solved
analytically because of the system complexity involving
simultaneous conduction and convection processes with
variation in fluid properties, resulting in a non-linear system.
The schematic of the axial discretization of a double pipe finned
heat exchanger layout is shown in Figure 2. For better
understanding and clarity of the posed problem, a 3-dimensional
3
Copyright © 2008 by ASME
view of a small section of the heat exchanger is also shown in
Figure 3.
Annulus
duct wall
Figure 3
Annulus Flow (Fluid-2)
External F in
Core duct
wall
Axis
Core Flow (Fluid-1)
Annulus Flow (Fluid-2)
Internal Fin
External Fin
th
i elemental
strip
Figure 2 Schematic of the double pipe finned heat
exchanger and its discretization.
Annulus duct wall
Hence, the thermal model comprises of conduction through
walls and fins, and turbulent forced convection at the wall and
fin surfaces.
The governing equations for combined conduction and
convection model mainly include heat balance equations for the
discretized core and the annulus duct wall (refer figure 3).
Equations 7-13 are the heat balance equations for the ith element
of the discretized layout. Within an axially discretized element,
the fluid and wall temperatures are assumed constant. The
convection boundary conditions are air inlet temperatures and
pressures, convective heat transfer coefficient, and the wall
conductivity. Steady state forced convection is modelled on
surfaces using the relevant convective heat transfer coefficients
as described later.
For the core duct wall, heat input is due to convection from
the hot core flow and heat conducted from the inner fins. The
heat is then convected out to the cold annulus flow and
conducted out of the outer fins. Hence, the heat balance
equation for the k th element of core duct wall is written as,
q& cv , IW1 + q& cn, Ifin + q& cv ,OW1 + q& cn,Ofin = 0
(7)
where:
q& cv , IW
q& cv , OW
Outer Fin
Cold Annulus
flow
Core duct wall
(conducting)
Inner Fin
Hot Core
flow
Figure 3: Schematic of a segment in double pipe heat
exchanger with fins
3.1 Modeling Heat Transfer
As the Nusselt number for both annulus flow and core flow
is high due to the turbulent flow, temperature variation of the
core & annulus duct walls due to axial conduction of duct wall
is negligible as compared to the axial temperature change due to
forced convection.
Hence, for the present case, axial
conduction through duct walls is not considered and only radial
conduction of heat through tubes is taken into account. In
addition, due to relatively high turbulence level in the flow, the
radial variation of temperature within the fluid is negligible.
1
1
= h IW
1
= h OW
⋅ A IW
1
1
⋅ (T IW
1
⋅ A OW ⋅ (T OW
− T FI
1
)
− T FII
(8.1)
)
(8.2)
In the above equations, h IW1 and h OW1 are the convective
heat transfer coefficients at the core duct wall inner and outer
surfaces respectively. There are several correlations available in
literature to evaluate heat transfer coefficient for a flow through
a smooth tube. One of the most widely used such correlation is
the Dittus-Bolter Equation, and is given as [26]
Nu = 0.023 ⋅Re0.8 ⋅Pra
(9)
Where, a = 0.4 (if the fluid is being cooled); a = 0.3 (if the fluid
is being heated) and all the variables are evaluated at the bulk
temperature.
D 
Hence, hIW 1 = 0. 023 ⋅ Re 0F.81 ⋅ PrF0.14⋅  1  ,
(9.1)
 k F1 
For evaluating heat transfer coefficient in the annulus, (refer to
Eqn.8 & Figure 3) the hydraulic diameter is used for evaluating
the flow Reynolds Number.
D 
hIWO = 0. 023 ⋅ Re 0h.,8F 2⋅ Pr F0.24 ⋅  h  ,
(9.2)
 kF2 
ρ ⋅ V ( D − D1 − 2 × tW1 )
where Reh,F2 = F 2 F 2 2
(9.2.1)
µF 2
heat transfer by fins (used in Eqn. 7) can be calculated as,
1
1


q& cn,Ifin = − k Ifin ⋅ AIfin ⋅ (TFI − T Iw1 ) ⋅ ξ ⋅ 
−
 (10.1)
−2ξ L
+2ξ L
1 + e
1+e

1
1


q& cn, Ofin = − kOfin ⋅ AOfin ⋅ (TFII − TOw1 ) ⋅ ξ ⋅ 
−

− 2ξ L
1 + e
1 + e +2ξL 
(10.2)
4
Copyright © 2008 by ASME
ξ=
where
h fin ⋅ Pfin
k fin ⋅ Ac / s , fin
,
Pfin
2 × L fin,cr ,
=
Ac / s , fin = t fin × L fin, cr
(10.2.1)
The Harper and Brown [27] approximation as given by
Holman [28], is used for evaluating the corrected fin height
(Lfin,cr) for evaluating the heat transfer through longitudinal fins
with uninsulated ends
Lfin,cr = Lfin + tfin/2
(10.2.2)
To evaluate heat transfer coefficient over surface of these
fins (both inner and outer), the Nusselt number correlation for
flow over a flat plate is used [29] as given below,
Nu x,fin = 0.332 Re1/2⋅Pr1/3
for 100 < Re < 3×105
(10.2.3)
Nu x,fin = 0.0296 Re0.8⋅Pr1/3
for 3×105 < Re < 107
(10.2.4)
where all quantities are evaluated at film temperature. The
transition Reynolds number (3×105) indicates the point when
flow turns from laminar into turbulent.
Finally, heat conducted by the core duct wall is given by
L
q&cn ,W 1 = q& cv ,IW 1 + q&cn , Ifin = q& cv ,OW 1 + q& cn ,Ofin = 2π ⋅ k W 1 ⋅   ⋅ (TIW 1 − TOW 1 )
N
(11)
If the annulus duct wall is not insulated, additional
equations accounting for heat balance across annulus duct wall
have to be solved,
q& cv , IW 2 + q& cv ,OW 2 = 0
(12)
where,
q& cv
, IW 2
q& cv
, OW
= h IW
2
2
= h OW
⋅A
2
IW 2
⋅ A OW
⋅ (T IW
2
2
⋅ (T OW
− T FII
2
)
− T F∞
(12.1)
)
(12.2)
where h IW2 and h OW2 are the convective heat transfer coefficient
at inner and outer surfaces of the annulus duct wall
respectively, and can be evaluated by using Dittus-Bolter
correlation given by eqn 9.1 & 9.2. In Eqn. (12.2), TF∞ is the
ambient air temperature surrounding the heat exchanger. The
heat transferred from annulus wall due to conduction can be
calculated as,
L
q& cn,W 2 = q& cv ,IW 2 = − q& cv ,IW 2 = 2π ⋅ kW 2 ⋅   ⋅ (TIW 2 − TOW 2 )
N
(13)
To improve accuracy of the convective heat transfer
mechanism, the fluid thermo -physical and transport properties
are evaluated according to the local temperature and pressure,
as described in the subsequent sections.
3.2 Modelling Pressure Loss
The total pressure loss in the core flow is attributed to frictional
loss at inner pipe wall surface and frictional losses at fin
surfaces.
∆ PF1 = ∆ PW 1 + ∆ PIfin
(14)
where
V 2 ⋅ ρ   L 

∆ PW 1 = 4 × fW 1 ⋅  F 1 F1  ⋅ 

 N⋅D 
2
1

 
(14.1)
V 2 ⋅ ρ 
∆ PIfin = f Ifin ⋅  F1 F1  ⋅ L Ifin H Ifin + t fin ⋅ N fin
(14.2)


2


To evaluate friction losses due to the pipe surface, the
Darcy- Weisbach friction factor correlation for evaluating
friction factor for flow through a smooth pipe, given by Eqn. 15,
is used. All variables are evaluated at the film temperature [29],
(
((
(
f = 1.82 ⋅ Log ? ⋅ M ⋅
)
(? ⋅ R ⋅ T ) ) ⋅ Dh
)
µ -1. 64
)
−2
4
(15)
For computing frictional losses factor at the fin surface, the
correlation given by Schlichting [29] for evaluating friction
coefficient for flow over flat plate is used.
f x , fin = 0 .646 (Re x, fin ) −0.5
100 < Re < 5×104
(16)
f x , fin = 0. 0592 (Re x, fin) − 0. 2
5×104 < Re < 107
Here the transition Reynolds number has been reduced due to
the accelerated transition from laminar to turbulent flow because
of the pipe flow.
Similar to the core flow, pressure losses in annulus flow can
be attributed to frictional loss at the duct walls and the external
fin surfaces. Hence
∆ PF 2 = ( ∆POW1 + ∆PIW 2 ) + ∆POfin
(17)
The Reynolds Number based correlations given by Eq (18)
is used for evaluating the friction factor and its associated
pressure loss in the annulus [30].
−2
 ε 


 

Dh 
2. 5 


f = −2 log
+
(18)
 3. 7
Re F 2 f 




where hydraulic diameter is given by , Dh = Dh
= ( D2 − D1 − 2 × tW1 ) , and Reynolds number is given by Eqn
(9.2.1). The above equation is solved iteratively for the friction
factor (f). Friction due to external fins is evaluated in a similar
manner to that by internal fins given by Eq. (16).
3.3 Fluid Property Variation
It is well know that if fluid temperature variation is
significant within a system, the fluid transport and
thermodynamic properties vary to a large extent, especially for
5
Copyright © 2008 by ASME
3.4 Numerical Solution Method
The present problem cannot be solved analytically because
of the complex nature of system involving both conduction and
forced convection processes, resulting in a non-linear system.
Therefore zonal analysis method is used, wherein the layout is
discretized into certain number of elements and governing
equations are solved for each discretized axial element (refer to
Figure 2). Various parameters like wall temperature, fluid
temperature, fluid properties, heat transfer coefficient, friction
factor coefficient, etc, are assumed constant within a discretized
element.
The Method of Successive Substitution [32] is used to
solve the governing equations, which include heat balance
equations for all elements and expressions for evaluating
various terms involved in these equations. In this methodology,
variables like the wall temperatures, fluid temperatures, etc, are
assigned with an initial values and then proceeding through the
system of equations (Eqn. 1-10), all variables are recalculated
and successively substituted and iterated until satisfactory
convergence is achieved.
For the present exercise, a
convergence value of 10-3 is used.
there are no external fins in the heat exchanger and only the
internal fins are used for enhancing the heat transfer.
The comparison between Nusselt number and heat transfer
coefficient obtained from present methodology with
experimental results of Carnavos for the core flow is shown in
Figure 4 and 5 respectively. As seen from the figures, the
comparison between computed results and Carnavos correlation
is very good, the error is less than 5%. Hence validity of the
proposed methodology in evaluating the heat transfer and
pressure drop characteristics for an internally finned pipe is
established. It should be noted that Carnavos correlation is
evaluated at fluid bulk temperature where as the film temperature
is used for evaluating heat transfer in the semi-empirical
methodology.
450
Simulation
Series1
400
Carnavos
Series2
350
300
Nuh
high temperature flows. Therefore, fluid temperature variation in
a practical heat exchanger is expected to be large and hence it is
necessary that this variation be taken into account. Zografos
et.al. [31] have provided the equations to compute the thermo
physical and transport properties of seven commonly used
fluids (air, liquid water, water vapour, carbon dioxide, Freon-12,
engine oil, and mercury). They used the curve fitting process to
fit the variation in fluid properties as, dynamic viscosity,
constant pressure specific heat, thermal conductivity, and more
with temperature, in the form of polynomials. In the present
analysis, air is used as the working medium in both core and
annulus flow.
250
200
150
100
50
0
1000
10000
100000
1000000
Re h
Figure 4: Comparison of Nusselt number obtained from the
present simulations with the Carnavos correlation
500
Simulation
Series1
450
Carnavos
Series2
400
4.0 RESULTS AND DISCUSSIONS
H (Wm- 2K- 1)
350
The methodology described in the earlier section is used to
study heat transfer and pressure loss characteristics of heat
exchangers. As stated earlier, there are no results available in
the literature for double pipe heat exchangers with both internal
and external fins, therefore the results obtained from the
proposed semi-empirical numerical methodology are compared
with experimental results available in literature for heat
exchangers with only internal fins or heat exchangers with only
external fins.
300
250
200
150
100
50
0
1000
10000
100000
1000000
Reh
4.1 Heat Exchanger with Internal Fins only
As stated earlier, many investigators have looked into the
performance of internally finned tube with respect to its heat
enhancement capability and increase in pressure drop due to
the addition of fins. Both straight longitudinal fins and spiral
fins have been investigated. The Carnavos correlation is used
as a benchmark for comparison. In this case it is assumed that
Figure 5. Comparison of the heat transfer coefficient obtained
from the simulations with that obtained from the Carnavos
correlation
The comparison between friction factor and total pressure
loss for internally finned tube calculated by the proposed
methodology and that obtained by using Carnavos correlation
6
Copyright © 2008 by ASME
is shown in Figure 6 and Figure 7 respectively. It can be seen
that the discrepancy between the friction factor decreases with
increase in the flow Reynolds number and the error never
exceeds 6 %
the predicted friction factor, in case for external fins, is more
than that obtained in experiments. However it should be noted
that the difference between the computed and experimentally
obtained results diminishes with increase in Reynolds Number.
The difference is mainly attributed in prediction of frictional
drag at the fin surface. Since there is no other correlation /
experimental data available for predicting frictional losses in an
annular duct with fins, the above simulations can not be
compared further. However it can be seen from Figure 9 that the
total pressure loss predicted by the proposed methodology is in
good agreement with that obtained experimentally by Braga &
Saboya [30], the difference being less than 5%
0.008
Series1
Simulation
0.007
Carnavo
Series2
s
0.006
f
0.005
0.004
0.003
0.25
0.002
Simulation
FF1O_AVG,
Braga&Sboya
FF1O_BR,
0.001
0.2
0
1000
10000
100000
1000000
fh
ReF1
0.15
Figure 6. Comparison of friction factor obtained from the
simulations with the Carnavos correlation
0.1
4000
Simulation
Series1
0.05
3500
Carnavos
Series2
Ploss (Pa)
3000
0
0
10000
20000
30000
40000
50000
60000
70000
2500
Reh
2000
Figure 8. Comparison of friction factor obtained from
simulations with experimental correlation provided by Braga
& Saboya
1500
1000
500
2500
Simulation
PRLOSS,
0
1000
10000
100000
Braga
& Sboya
PRLF_BR,
1000000
ReF1
-∆P2
Figure 7. Comparison of the pressure loss obtained from
the simulations with that obtained from the Carnavos
correlation for unit length of the pipe
2000
1500
1000
4.2 Heat Exchanger with External Fins only
In this case, only external fins within the annular passage
are considered for enhancement of the heat transfer (the internal
fins within the core pipe are neglected in this analysis).
Performance of heat exchanger with similar conditions as used
by Braga and Saboya [30], the total pressure drop and heat
transfer coefficient across the annulus, evaluated by the earlier
described methodology, is shown in Figure 8 and Figure 9
respectively. It is seen that unlike in the case for internal fins,
500
0
0
10000
20000
30000
40000
50000
60000
70000
Reh
Figure 9. Comparison between the numerically obtained loss
in heat exchanger with experimental correlations of Braga &
Saboya
7
Copyright © 2008 by ASME
500
F1(K)
No Fins
Inner Fins
Inner(TF1)
Outer Fins
Outer(TF1)
Both Fins
Both(TF1)
495
490
TF1 (K)
Figure 10 shows comparison between numerically obtained
Nusselt number (by the proposed methodology) with that
obtained experimentally by Braga & Saboya. It is seen that the
two curves cross each other at low Reynolds Number, thus
indicating that dependency of Nusselt Number on the Reynolds
Number exponent is more than that given by Eqn (3). However,
the overall comparison is good with the maximum error being
less than 5 %
485
100
480
80
475
0
60
5
0.25
10
0.5
15
0.75
1.020
Nuh
Length (m)
40
NU1O,
Braga
& Saboya
NU_BR,
Simularions
20
0
0
10000
20000
30000
40000
50000
60000
70000
Reh
Figure 10: Comparison of the Nusselt Number from the
simulations with Braga & Saboya
Figure 11: Temperature variation of the core flow for different
cases
The variation of Nusselt Number for the core and the
annular fluid with increase in the number of fins (both external
and internal) is shown in Figure 12. It is observed that Nusselt
number decreases for both fluids, however since there is a
substantial increase in the over all heat transfer area, the
effective heat transfer rate increases due to the simultaneous
increase in the number of external and internal fins.
140
Core
flow
NU1O,
4.3 Internal and External Fins
120
Annular
NU1I,
100
80
Nu
From analysis of the results obtained in earlier sections, it is
apparent that the proposed semi-empirical-numerical
methodology can be used for evaluating performance of
internally or externally finned tubes. In this section, the double
pipe heat exchanger with both internal and external fins is
analyzed by using the proposed semi-empirical-numerical
methodology. Inlet conditions used for the test case are given
in Table 1. The variation of the core fluid temperature for
various cases (i.e. without fins, with only inner fins, with only
outer fins, with both inner and outer fins) is shown in Figure 11.
It is seen from Fig. 11 that the effectiveness of only inner fins or
only outer fins in transferring heat from the core fluid to the
annular fluid is almost the same. Therefore maximum heat is
transferred in the case when there are both external and internal
fins.
60
40
20
0
0
5
10
15
20
25
Number of Fins
Figure 12: Variation of Nusselt number with external and
internal fins
The increase in heat transfer due to fins also results in
enhanced pressure losses in the flow. The effect of fins (both
external and internal) on the pressure loss characteristic of the
double pipe heat exchanger is shown in Figure 13. Even though
the pressure loss in core flow is less as compared to that in
annular flow when there are no fins, the core fluid pressure
losses increase more rapidly with the increases in number of fins
8
Copyright © 2008 by ASME
200
Core
flow
PRLOSS,
Annular
FF1I_C,flow
Pressure loss (Pa)
160
120
80
40
0
0
5
10
15
20
25
Number of Fins
Figure 13: Pressure Loss Characteristics with fins for core and
annular flow
The effect of fin conductance and fin height on the flow
Nusselt number (for both core and annular flow) is shown in
Figure 14. The Nusselt number increases with increase in fin
conductance and reduces with increase in fin height, as
expected. However increase in the fin height results in an
increase in the overall heat transfer area and hence resulting in
enhancement of the overall heat transfer rate.
120
100
Inner Fin (k =100)
NU1I_100
Outer Fin (k = 100)
NU1O_100
Inner Fin (k =10)
NU1I,
Outer Fin (k = 10)
NU1O,
Nu
80
60
40
20
0
0
5
10
15
20
25
Fin Height (mm)
Figure 14: Variation of Nusselt number with fins conductance
and fin height
5. CONCLUSIONS
A new and fast calculation procedure / methodology is
described to evaluate performance of a double pipe heat
exchanger with internal and / or external fins. The methodology
uses well-known and established correlations for the flow in
smooth tubes and for flow over flat plates and hence is not
design specific. In addition, the variation in the fluid properties
due to variation in the temperature along the heat exchanger is
taken into account. Hence the methodology can be applied to
cases where the fluid undergoes large temperature changes.
The predicted performance matches well with experimental
results reported in literature for pipes with internal or external
fins, and therefore we can extrapolate the validity of the
developed model to the case of double pipe heat exchanger.
Since the proposed model is based upon analytical formulations
of semi-empirical correlations, the time required for numerical
computations is very small and hence this methodology can be
successfully used for optimization processes wherein a large
number of design iterations are required before arriving at an
optimal solution. Hence the present methodology conveniently
fills in the gap between costly CFD simulations and easily
available but inadequate empirical correlations for finned tube
heat exchangers, and gives the designer a tool to optimize the
design of such heat exchangers for practical applications.
6. ACKNOWLEDGMENTS
The authors are thankful to the Ministry of Higher Education,
Govt. of Israel and Israel science Foundation (ISF) for
supporting the research.
7. NOMENCLATURE
A
area [m2]
Bi
Biot number [-]
Cp
specific heat at constant pressure [J/kg-K]
dA
area of elemental strip [m2]
D
diameter of duct [m]
Dh
hydraulic diameter [m]
dx
width of elemental strip [m]
f
friction factor [-]
h
convective heat transfer coefficient [W/m2-K]
k
thermal conductivity
KT
thermal conductivity of fluid [W/m-K]
L
length [m]
M
Mach Number [-]
m&
mass flow rate [kg/s]
N
number of elements [-]
Nu
Nusselt Number [-]
P
pressure [Pa]
Pr
Prandtl number [-]
Q
total enthalpy [J]
q&
heat transfer rate [W]
R
gas constant of air [= 287 J/kg K]
Re
Reynolds Number [-]
t
thickness [m]
T
temperature [K]
V
velocity [m/s]
Greek scripts
µ
dynamic viscosity [Pa-s]
ρ
density [kg/m3]
9
Copyright © 2008 by ASME
γ
ratio of specific heats [-]
ξ
fin parameter [-]
Subscripts
b
bulk quantity
cv
convective heat transfer
cn
conduction heat transfer
cr
corrected
f
evaluated at film temperature
fin
related to fin
FI
core fluid
FII
annulus fluid
h
hydraulic quantity
I
inner surface
O
outer surface
w1
core duct wall
w2
annulus duct wall
x
distance
Superscripts
a
exponent in the Dittus-Bolter equation
8.
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
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11
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