89
ORIGINAL RESEARCH PAPER .
Multi-Objective Optimization of Tio2-Water Nanofluid Flow in Tubes
Fitted with Multiple Twisted Tape Inserts in Different Arrangement
H. Safikhani1,*, S. Eiamsa-ard2
1Department of Mechanical Engineering, Faculty of Engineering, Arak University, Arak 38156-88349,Iran
2Department of Mechanical Engineering, Faculty of Engineering, Mahanakorn University of Technology, Bangkok 10530,
Thailand
1. Introduction
Several heat transfer enhancement (HTE) techniques have been used in many engineering applications such as nuclear reactor, chemical reactor, chemical process, automotive cooling, refrigeration, and heat exchanger, etc. HTE techniques are powerful tools to increase heat transfer rate and thermal performance as well as to reduce of the size of Transfer system in installing and operating costs. The
techniques can be classified into 2 categories; (1) active method: by supplying external power source to the fluid or the equipment; (2) passive method: by turbulence promoter (such as special surface geometries, twisted tape, propeller, tangential inlet nozzle, snail entry, axial/radial guide vane, spiral fin) or fluid additives (such as nanofluid), without using any direct external power source. Due to its easy installation/operation and cost saving passive method has drawn great attention.
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Corresponding author
Email address:[email protected]
ARTICLE INFO.
Article history
Received 20 April 2015 Accepted 12 June 2015
Keywords
Dual/triple/quadruple twisted tapes Heat transfer enhancement. Multi-objective optimization NSGA II
TiO2/water nanofluid
Abstract
In this paper, experimentally derived correlations of heat transfer and pressure drop are used in a Pareto based Multi-Objective Optimization (MOO) approach to find the best possible combinations of heat transfer and pressure drop of TiO2-water nanofluid flow in tubes fitted
with multiple twisted tape inserts in different arrangement. In this study there are four independent design variables: the number and arrangement of twisted tape inserts (N), TiO2 volume fraction (φ),
Reynolds number (Re) and Prandtl number (Pr). Seven twisted tape arrangement in three different categories are investigated. The objectives are maximizing the non-dimensional heat transfer coefficient (Nu) and minimizing the non-dimensional pressure drop (f Re).
Safikhani & Eiamsa-ard/ TPNMS 3 (2015) 89-99
90
One important group of devices used in passive method is swirl flow devices which produce secondary recirculation on the axial flow leading to an increase of tangential and radial turbulent fluctuation. This allows a greater mixing of fluid inside a heat exchanger tube and subsequently reduces the thickness of the boundary layer [1-6]. Among the swirl generators of tube inserts, twisted tapes (TTs) have gained great attention and widely used for producing compact heat exchangers and upgrading the heat transfer rate of the existing heat exchanger due to its low cost, acceptable thermal performance and ease of manufacture installation [2]. In the past investigations, the heat transfer enhancement by twisted tape insert has been considered on both experimental and numerical works. Saha et al. [7] studied the heat transfer and pressure drop behaviors in a tube fitted with regularly-spaced twisted tape elements. Ray and Date [8] predicted the heat transfer in a square sectioned duct fitted with twisted tape in both laminar and turbulent flows. Akhavan-Behabadi et al. [9] reported the influence of the twisted tape on heat transfer and pressure drop characteristics in horizontal evaporators for the flow using R-134a. Eiamsa-ard et al. [10] investigated the heat transfer and pressure drop behaviors in a double pipe heat exchanger fitted with regularly-spaced twisted tape elements at several space ratios. Effect of the combined conical-ring and twisted tape on the heat transfer, friction factor and thermal performance factor characteristics were also studied by Promvonge
and Eiamsa-ard [11]. Influence of the tube equipped with the short-length twisted tape on the mean Nusselt number, friction factor and thermal performance factor characteristics for several tape length ratios was examined by ard et al. [12]. Again, Eiamsa-ard et al. [13-15] reported the effect of the dual twisted tape elements in tandem, twin twisted tape with counter/co-swirling flow and delta-winglet twisted tape on the heat transfer enhancement, friction factor and thermal performance factor. They found that all tape arrangements provide better heat transfer rate than those of the typical twisted tape while the friction factors are also increasing. Eiamsa-ard and Promvonge [16] also studied the effect of helical tape with/without core-rod and regularly-spaced helical tape swirl generators on the heat transfer and pressure drop characteristics in a heat exchanger tube. Chang et al. [17] examined the heat transfer and pressure drop characteristics in a tube fitted with single, twin and triple twisted tapes in a heat exchanger tube. Among the tested tapes, the triple twisted tapes offered the highest heat transfer rate and thermal performance factor. Recently, Eiamsa-ard and Kiatkittipong [18] performed an experimental and numerical study to investigate the effect of the number and arrangement of twisted tape inserts on the thermal and pressure drop performance of TiO2-water nanofluid flow. They
investigated seven different arrangements of twisted tapes (ST, Co-DTs, C-DTs, Co-TTs, Co-QTs, PC-QTs, and CC-QTs) and finally they presented three pair of correlation for Nusselt number and friction Nomenclature
y pitch length of twisted tape (m)
D Tube diameter(m) Greek symbol
f Friction factor
Density(kg m-3)h Heat transfer coefficient (Wm-2K-1)
Dynamic viscosity (N s m-2)k Thermal conductivity (W m-1 K-1) φ Volume fraction of nanofluid
L Length of tubes (m) Abbreviations
N Number of twisted tapes NSGA Non-dominated Sorting Genetic Algorithms Nu Nusselt Number (hDh /k) ST single tape as single swirl flow generator
P Pressure (Pa) Co-DTs dual co-tapes as co-dual swirl flow generators Po Poiseuille number (= f Re) C-DTs dual counter tapes as counter-dual swirl flow
generators
Pr Prandtl number (
/
) Co-TTs triple co-tapes as co-triple swirl flow generatorsRe Reynolds number (
UDh/
) Co-QTs quadruple co-tapes as co-quadruple swirl flow generatorsU Velocity (m s-1) CC-QTs quadruple counter tapes as counter-quadruple
swirl flow generators
K i n t T D a t g b t r p n e a h t p t
factor. The em Kiatkittipong investigate nanofluid flo tapes in differ
Table 1
Details of twi
Nanofluid sized particle and CuO are transfer syste greater therm base fluids. transfer enh researchers [ particle load friction loss nanofluid enhancement addition, nan heat transfer tapes [24-29] possessed b techniques, p
Twisted tape Numbof tap
ST 1 Co-DTs 2 C-DTs 2 Co-TTs 3 Co-QTs 4 PC-QTs 4 CC-QTs 4 mpirically co g are used
the multi-o ow in tubes
rent arrangem
sted tape insert
d which is the es or nanopa
e currently a em as alterna mal conductiv
The influen hancement w
19-23]. In ge ding gave hig
. With prop yielded ap t with insig nofluid was u
enhancement ]. Apparently better perfo particularly at ber pe Tape Width (W) T p le
19 mm 5 8 mm 22
Same as Co-DTs S Co Same as Co-DTs S Co Same as Co-DTs S Co Same as Co-DTs S Co Same as Co-DTs S Co rrelations of in the pre objective op
fitted with m ment. ts. e suspensions articles such attractive for ative medium vities as comp nces of nano were studied eneral, nanoflu gher heat tra per particle ppreciable gnificant fric
utilized toget t devices, esp y, the compo
ormance th high Reynold Tape pitch ength (y) Twist ratio (y/W) T
7 mm 3.0 25 mm 3.0
Same As o-DTs Same as Co-DTs Same as o-DTs Same as Co-DTs Same as o-DTs Same as Co-DTs Same as o-DTs Same as Co-DTs Same as o-DTs Same as Co-DTs Eiamsa-ard a esent study ptimization multiple twist
s of nanomete as TiO2, Al2
r using in he ms due to th
pared to that ofluid on he d by seve uid with high ansfer rate a
loading, mo heat transf ction loss. ther with oth pecially twist ound techniqu han individu ds number [1
Tape Thickness
(δ)
Swir type
0.8 mm S-Same as
ST Co D
Same as ST
Coun D-S
Same as ST Co T
Same as ST Co Q
Same as ST Coun Q-Ss P-A Same as ST Coun Q-Ss
C-91
and to of ted er-O3 eat eir of eat ral her and ost fer In her ted ues ual 8].Fig. 1
arrange Acc that th tape to Howev optima pressur explore and na transfe objecti be iden multi-o this pa propos applied optimiz [31-34
In t of Nus in a approa heat t
rl e S D-Ss nter Ss T-Ss Q-Ss nter s in A nter s in A
1. Multiple ement which ar
cording to the he heat transf ogether with n
ver, the influe al arrangemen re drop perf ed. Simultane anofluids wi er and pressu ive optimizat ntified. One o objective opt
aper is NSG sed by Deb
d abundant zation of en 4].
this paper, ex sselt number (
Pareto bas ach to find t transfer and
twisted tapes re investigated
e above revie fer enhancem nanofluid is a ences of opti nt of twisted
formance of eous using o ill result in ure drop; henc tion, optimal of the most c timization alg GA II algori
[30] for the tly for t ngineering is
xperimentally (Nu) and frict sed multi-ob the best poss d pressure
s inserts in in this paper
ew, it has been ment by using
a promising a imal number tapes on ther f nanofluids of twisted tap
increasing t ce, by doing design point complete and gorithms also ithm. This a first time h the multi-o sues in rece
y derived cor tion factor (f) bjective opti sible combin
drop of Ti
differen n proven g twisted approach. and also rmal and are not pe inserts the heat a multi-ts should the best o used in algorithm has been objective ent years
rrelations f) are used imization nations of
n t r i i i b
2
d t R t q T c q C t d c t I a o o a nanofluid flo twisted tape shown that relationships involved in th in round tube in different a based multi-o
Fig. 2. Expe performance o
nserts [18]
2. Defining
In the pr design varia twisted tape Reynolds nu this study, s quadruple tw The schemat shown in figu
The seven 1 can be cla category 1, a same directio quadruple tw Co-TTs and tapes have c designed for case of dual t twisted in op In the case o aligned to be opposite to th In addition of two pai arrangements
ow in round inserts in d t some in as useful he thermal pe es fitted with arrangement c objective opti
erimental set of tubes fitted
the design v
resent study ables: the nu inserts (N), mber (Re) an seven differe wisted tapes ar
ic definition ure 1 and tabl n twisted tape assified in th all tapes are on. In this ca wisted tapes a
Co-QTs, resp counter-swirl
dual and qua twisted tapes pposite directi of quadruple e twisted in t hat of other tw
n, the quadru irs of tape s, to produce
Safik
tubes fitted different arra nteresting a optimal des erformance of
multiple twis can be discov mization appr
up for inves d with multip
variables
there are fo umber and a
TiO2 volum
nd Prandtl n ent single, d rrangement a and details o le 1.
es which are s hree different
aligned to be ase, single, d are assigned a pectively. In flow. This adruple twiste
, two tapes ar ions and assig
twisted tapes the same dire wo tapes. uple counter ta
es are in e (1) parallel
khani & Eiams
d with multip angement. It and importa sign principl f nanofluid flo sted tape inse vered by Pare
roach.
stigating therm ple twisted tap
ur independe arrangement me fraction (φ
number (Pr). dual, triple a are investigate
of each one a
shown in figu t categories. e twisted in t dual, triple a as ST, Co-DT
category 2, t arrangement ed tapes. In t re aligned to gned as C-DT s, two tapes a
ection which apes consistin two differe l counter-swi sa-ard/ TPNMS
92
ple is ant les ow erts eto ma pes ent of φ), In and ed. are ure In the and Ts, the is the be Ts. are is ng ent irlflow a counte swirl f quadru cross produc quadru which mentio study i Fig. 3. b) SEM Des shown Ac that th variatio nanoflu
3. Def
In h and p minim Kiatkit numeri and arr and p nanoflu The appara
S 3 (2015) 89-9
and (2) cross er-swirl flow, flow in the uple counter t counter-swir ced swirl flow uple counter
are classified oned notes, t
is performed f
Details of TiO M image, c) XR
sign variables n in table 2.
ccording to R he flow reg
on range o uid as workin
fining the ob
heat exchange pressure dro mized respect
ttipong [18] ical study to i rangement of pressure dro
uid flow. e schematic atus of them is
99
s counter-swi , the tapes in same directi tapes are assi rl flow, the
w in the opp tapes are a d in category the MOO pr for three diffe
O2 nanoparticle
RD pattern.
s and their ra
Re values of gime is turbu of Pr is rel
ng fluid.
bjective fun
ers tubes, he op should tively. Recen performed investigate th f twisted tape op performa
c arrangeme s shown in fig
irl flow. For n each pair p
ion; in this igned as PC-Q
tapes in ea posite directio assigned as y 3. Accordin rocess in the erent categori
es: a) macro ph
ange of variat
table 2, it is ulent. Moreo lated to TiO
nctions
eat transfer co be maximiz ntly, Eiamsa
an experime he effect of th
inserts on th ance of Ti
ent of exp gure 2. r parallel produced case the QTs. For ach pair ons. The CC-QTs ng to the e present ies. hotograph, tions are obvious over the
O2-water
oefficient zed and -ard and ental and e number e thermal iO2-water
T D i e d C w N e K i n t c C F r Table 2
Design variab
Moreover, information experimental different arra C-DTs, Co-which were c Finally the Nusselt num empirically Kiatkittipong investigate nanofluid flo tapes in d correlation o
k hD Nu and
DTs, Co-4 . 0 618 . 0
Pr
Re
(
103
.
0
Nu
233 . 0Re
1
(
437
.
0
f
Fig. 4. Multi-related to twist
Design Variables C S Co-Fro
N 1
φ (%) 0 Re 54
Pr 3
bles and their ra
figure 3 about TiO2 n
conditions. angements of TTs, Co-QT classified in th ey presented mber (Nu) an
correlations g are used the multi-o ow in tubes different arr f Eiamsa-ard d f=
TTs and Co-Q
768 . 0
(
)
N
1
(
25 . 1(
1
)
N
1
-objective Par ted tapes of cat
Category 1: ST, Co-DTs, -TTs, Co-QTs om To 1 4 0 0.21 00 15200 3 6
ange of variati
shows s nanoparticles They inve f twisted tape Ts, PC-QTs
hree different three pair of nd friction f s of Eia in the pre objective op
fitted with m angement. T d and Kiatkit
for category
QTs tapes are
438 . 0
)
1
(
268 . 0)
eto results for tegory 1 Category 2: ST, C-DTs, PC-QTs From To
1, 2 and 4 0 0.21 5400 15200
3 6
ons.
some detail s and the oth estigated sev
s (ST, Co-DT and CC-QT categories. f correlation f
factor (f). T amsa-ard a esent study
ptimization multiple twist
The empiric ttipong [18] f y 1 included S
e as follows:
(
(
r Nu and f R
Category 3: CC-QTs From To 4 0 0.21 5400 15200 3 6
93
led her ven Ts, Ts) for The and to of ted cal for ST, (1) (2) Re Nu PC-QT 6 . 0Re
Nu
0Re
f
Fin
QTs ta
61 . 0Re
Nu
2 . 0Re
0
f
Fig. 5. related It sh respect examp leading instead Re f same b should 36]. Mul and 6 y
0
and f for cat Ts tapes are as
4 . 0 618
Pr
N
1
(
104
.
0
234 . 0N
1
(
468
.
0
nally
Nu
and
apes are as f
4 . 0 18
Pr
N
1
(
096
.
0
234N
1
(
416
.
0
Multi-objecti d to twisted taphould be no t to changing ple increasing g to increase i d of f,
2 D / U L / P
(P
behavior with d be investig
ltiplying a Re yield the f Re
tegory 2 incl s follows:
789 . 0
(
1
)
N
217 .
1
(
1
)
)
N
d
f
for
categfollows:
834 . 0
(
1
)
N
221 .
1
(
1
)
)
ive Pareto res pes of category
oted that the g Re is not sim
g Re leading in Δp. Hence
another Poiseuille num
h Δp with res ated in optim
e on both sid number as fo
luded ST, C-D
439 . 0
)
266 . 0)
ory
3 includ
468 . 0
)
262 . 0)
sults for Nu a 2
e behavior o milar to that o to decrease to solve this
parameter mber), which
spect to chan mization pro
des of equati ollows: DTs and (3) (4)
ded
(5) (6)and f Re
of f with of Δp, for
in f but problem,
namely h has the
nging Re, cess [35,
C C C p t d e d F r
4
n
t
p m t p o a b m Category 1: 733 . 0Re
43
.
0
Re
f
Category 2: 766 . 0Re
46
.
0
Re
f
Category 3: 766 . 0Re
41
.
0
Re
f
Therefore paper is m transfer coe dimensional equations 1, different arraFig. 6. Multi-related to twist
4. Multi-ob
nanofluid f
twisted tape
In order performance multiple twis the experime presented in objective op algorithms [3 been chosen mutation prob 2 . 1)
N
1
(
37
21 . 1)
N
1
(
68
22 . 1)
N
1
(
16
the objectiv maximizing Nfficient) and pressure dro 3,5,7, 8 and ngements of t
-objective Par ted tapes of cat
bjective opti
flow in tub
es in differe
to investig of round tu ted tape inser entally derive
section 3 are ptimization p 30]. In all run n with cross bability (Pm)
Safik
268 . 0 5
(
1
)
266 . 0 17
(
1
)
262 . 0 21
(
1
)
ve functions Nu (non-dim d minimizing op) which ar
d 9 respecti twisted tapes
reto results fo tegory 3
imization of
bes fitted w
ent arrangem
gate the op ubes which rts in differen ed correlation e now emplo procedure us
s a population sover probab
as 0.7 and 0.0
khani & Eiams
(
6
(
2
(
in the prese mensional he g f Re (no re presented ively for thr
.
or Nu and f R
f TiO2-wate
with multip
ment
ptimal therm are fitted w nt arrangemen ns which we yed in a mul sing NSGA
n size of 60 h bility (Pc) a
07 respective sa-ard/ TPNMS
94
(7) (8) (9) ent eat on-in ree Reer
le
mal ith nts, ere lti-II has and ly. The tw (non-d (non-d optimiz variabl objecti the foll Catego
Subjec
Minim
Maxim
Fig. 7. 3 Catego Subj Min Max Figu objecti twistedS 3 (2015) 89-9
wo conflictin dimensional h dimensional
zed simultan les N, φ, Re ive optimizat lowing form: ory 1:
3
54
0
1
:
to
ct
f
mize
N
mize
Overlap graph ory 2: 3 54 0 N : to ject f nimize N ximizeures 4-6 show ive functions d tapes. 99 ng objectives heat transfer pressure dro neously with
e and Pr (T tion problem
6
Pr
1
Re
400
0021
.
0
4
N
,
N
(
f
Re
,
N
(
f
Nu
2 1
h of Pareto resu
6 Pr 15 Re 400 0021 . 0 and 2 , 1 N , N ( f Re f , , N ( f Nu 4 3
w the Pareto f s for three d
in this study coefficient) op) that sh
respect to th Table 1). Th can be form
5200
1
Re)
,
Pr)
Re,
,
ults of category
5200 4 Re) , Pr) Re,
front of the m different categ
y are Nu and f Re hould be he design he multi-mulated in
(10)
y 1, 2 and
(11)
C F w a o w t e B H c t Category 3: to Subject Minimize Maximize
Fig. 8. Overlap with the relate and 3
It is clear one another, where one of the other one from one po function gets each category B and C (cate H and I (c correspondin table 3. Pr 3 R 5400 0 4 N : Re f f Nu
p graph of the ed experimenta
that the poin meaning that f their objectiv e is different. oint to anothe s better and t
y, three optim egory 1), D, E category 3) g design vari
6 15200 Re 0021 . 0 Re) , , N ( f Pr Re, , , N ( f 2 1
e obtained optim al data [18] fo
nts have no d t no two poin ve functions In other wor er, definitely the other one mal points, de
E, and F (cate can be ob ables have be r)
(1
mal Pareto fro or category 1,
dominancy ov nts can be fou
is the same a ds, as we mo , one objecti gets worse. esignated by egory 2) and bserved, who een presented
95
12) ont 2 ver und and ove ive In A, G, ose d in The feature drop) a Nu (he figures insert l which tubes. In g which satisfie used [3 both o calcula the hig point. P obtaine they ad and f RIt differe Figu Pareto tapes. A them, categor moreov with re interes data o extract shows categor figure has rec experim highest optimiz The the Pa achiev Figure the inp three d It is all cat obviou linearly
e points illus es. Points A,D
and similarly eat transfer) in s 4-6 that incr
leading to inc is due to mo
general, it w both objec ed. To find s 33]. For this objective func ate the norm ghest norm v Points B, E a ed from this dequately sati Re.
would be i ent Pareto fron
ure 7 shows t fronts relate As shown the however it ry 3 is ab ver this Paret espect to the t sting and use of Eiamsa-ard
ted Pareto f the overlap ries and the indicates that cognized acc mental data w
t Nu. This p zation proces
changes of th areto front
ing the suit 9 illustrates put design va different categ
obvious from tegories and us in this fig
y.
trated in figu D, G exhibit t y points A,D, n each catego reasing the n crease in Nu ore increasing
would be idea ctive functio such a point,
purpose, we ctions to be b of these func value constitu and H are the
approach, an isfy both obje
interesting to nts and comp the overlap gr
ed to three ere is not mu seems that le to reach to front has cl two other Par eful to comp
d and Prom front of pres
p of the P e related exp
t in each cate curately the with respect t point verifie ss performed he design var
can be effe table therma the changes o ariables from
gories. m this figure
is equal to gure, in each
ures 4-6 hav the least f Re
G exhibit th ory. It is obvi number of twi but increasin g in mixing o
al to find a ons are ad , mapping m assume the v between 0 an
ctions; the po utes the idea
points that ha nd it can be ective functio
o overlay t are them in o raph of three
categories o uch difference the Pareto
higher Nu loser range of reto fronts. It pare the expe mvonge [18]
sent study. areto front perimental da egory, the Par
best boundar to the lowest s the validit in the present riables associ ective and u al design co of Nu and f R point A to p
that Pr is co 6. Similarly, h N, Re varie
ve unique (pressure he highest ious from isted tape ng in f Re of fluid in
point at dequately method is
values of nd 1, and oint with al design ave been said that ons of Nu
the three one curve. different f twisted e between front of number f Nu- f Re would be erimental with the Figure 8 of three ata. This reto front ry of the
f Re and ty of the t study.
ated with useful in onditions. Re,versus
oint I for
onstant in , as it is es almost
Fig. 9. Optimmal variations o
Safikhani
of Nu and f Re
& Eiamsa-ard
e with respect to
96
d/ TPNMS 3
(a)
(b)
(c) o design variab
(2015) 89-99
bles: (a) catego 9
97
Table 3
The values of objective functions and their associated design variables of the optimum points.
5. Conclusion
In this paper, multi-objective optimization of TiO2
-water nanofluid flow in round tubes fitted with multiple twisted tape inserts in different arrangement has been successfully implemented using the combination of experimentally derived correlations and NSGAII algorithm. The design variables were the number and arrangement of twisted tape inserts (N), TiO2 volume fraction (φ), Reynolds number (Re) and
Prandtl number (Pr). Seven twisted tape arrangement in three different categories were investigated and the ultimate goal was to simultaneously increase the non-dimensional heat transfer coefficient (Nu) and reduce the non-dimensional pressure drop (f Re). It was shown that some interesting and important relationships as useful optimal design principles were discovered by Pareto based multi-objective optimization approach, which could not be obtained except by this method.
Finally the optimal Pareto front of the present study were overlaid with the available experimental data and it was seen that the Pareto front is able to recognize the best boundary of experimental data with respect to the lowest f Re and highest Nu, which verifies the validity of the optimization process performed in the present study.
References
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Webb, Performance evaluation criteria
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use of enhanced heat transfer surfaces in heat exchanger design, Int. J. Heat Mass Transf 24 (1981) 715-726.[3] K. Wongcharee, S. Eiamsa-ard, Friction and heat transfer characteristics of laminar swirl flow through the round tubes inserted with alternate clockwise and counter-clockwise twisted-tapes, Int. Commun. Heat Mass Transf 38 (2011) 348– 352.
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[7] S. K. Saha, A. Dutta, S. K. Dhal, Friction and heat transfer characteristics of laminar swirl flow through a circular tube fitted with regularly spaced twisted-tape elements. Int. J. Heat Mass Transfer 44 (22) (2001) 4211-4223.
[8] S. Ray, A. W. Date, Friction and heat transfer characteristics of flow through square duct with twisted tape insert. Int. J. Heat Mass Transfer 46 (5) (2003) 889-902.
[9] M. A. Akhavan-Behabadi, Ravi Kumar, A. Mohammadpour, M. Jamali-Asthiani, Effect of twisted tape insert on heat transfer and pressure drop in horizontal evaporators for the flow of R-134a. Int. J. Refrig 32 (5) (2009) 922-930.
[10] S. Eiamsa-ard, C. Thianpong, P. Promvonge, Experimental investigation of heat transfer and flow friction in a circular tube fitted with regularly spaced twisted tape elements. Int. Commun. Heat Mass Transfer 33 (10) (2006) 1225-1233.
[11] P. Promvonge, S.Eiamsa-ard, Heat transfer behaviors in a tube with combined conical-ring and twisted-tape insert. Int. Commun. Heat Mass Transfer 34 (7) (2007) 849-859.
[12] S. Eiamsa-ard, C. Thianpong, P. Eiamsa-ard, P. Promvonge, Convective heat transfer in a circular tube with short-length twisted tape
Point N φ Re Pr Nu f Re
Category 1 A 1 0.00 5400 6 72.76 757.74 B 2 0.0013 13488 6 175.02 2539.37 C 4 0.0021 15200 6 279.01 5270.80
Category 2 D 1 0.00 5400 6 56.49 786.35 E 2 0.0014 13022 6 176.96 2528.79 F 4 0.0021 15200 6 291.32 5300.52
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