International Communications in Heat and Mass Transfer 161 (2025) 108411
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International Communications in Heat and Mass Transfer
journal homepage: www.elsevier.com/locate/ichmt
A computational study for enhancement in thermal performance of plus
(+)-shaped ribbed solar air heater duct
Kashinath Dhamudia a,b , Jnana Ranjan Senapati b,*
a
b
Department of Mechanical Engineering, Parala Maharaja Engineering College, Berhampur 761003, India
Department of Mechanical Engineering, National Institute of Technology, Rourkela 769008, India
A R T I C L E I N F O
A B S T R A C T
Keywords:
SAHD
TGPP
SAHPI
Plus-shaped rib
Relative roughness pitch
This article uses the CFD approach to present heat transfer augmentation and flow characteristics in solar air
heater duct (SAHD) plus(+)-shaped rib turbulators. In this study, a two-dimensional domain is considered for
simulation, and the collector plate of SAHD is made from aluminum, which is exposed to a consistent heat flux of
1000 W/m2. The numerical exercise uses ANSYS-Fluent R22 code to analyze the consequence of different
dimensionless parameters crucial in characterizing fluid flow and heat transfer dynamics. The parameters of
interest include Reynolds number (Re), relative roughness pitch (P/e), and relative roughness height (e/Dh). The
investigation gives rise to noteworthy results that put up in terms of average Nusselt number enhancement factor,
average enhancement friction factor, thermo-geometric performance parameter (TGPP), solar air heater per­
formance index (SAHPI), pumping power, and visualization of different fluid properties for the SAHD. The study
was done by varying P/e and Re from 7.14 to 17.85 and 4000 to 22,000, respectively, at a constant e/Dh of 0.042.
The analysis reveals that a solar air heater performance index (SAHPI) of 2.54 is achieved as heat removal is
augmented for the SAHD. The RNG k − ε model with enhanced wall treatment is chosen as the best model to
simulate the governing equations. The average Nusselt number and the friction factor correlations, based on Re
and P/e, have been determined through a non-linear regression analysis.
SAHD:
1. Introduction
The importance of diverse energy sources has grown significantly
over recent decades, as they are crucial for driving industrialization and
promoting global economic growth. Solar energy, in particular, is
viewed as a critical solution for addressing climate change while
meeting rising energy demands sustainably. Solar radiation, an abun­
dant and clean energy source, is freely available and has immense po­
tential for global utilization. In the 1970s, sharp increases in oil prices
and the environmental impact of non-renewable energy sources pushed
researchers to prioritize sustainable energy solutions. While transition­
ing to alternative energy sources like solar power has presented financial
and technical challenges, the benefits to society make this shift essential.
Solar energy, for example, has a range of applications, one of which is
heating air through a solar air heater duct (SAHD). As illustrated in
Fig. 1, SAHD are widely used in process industries, drying applications,
and room heating. However, one fundamental limitation of SAHD sys­
tems is the relatively low heat transfer rate from the collector plate.
Indeed, there are two primary means to maximizing heat removal in
i) Heat transfer could be enhanced by increasing the convective surface
area of the collector plate by adding ribs or grooves.
ii) Creating swirl or turbulence within the fluid flow domain promotes
heat transfer.
A plethora of studies have been reported on the thermal character­
istics of the SAHD. However, to have a systematic review of the litera­
ture, the entire study is divided into two parts: experimental studies and
numerical/computational studies. In the following paragraph, the
experimental studies are discussed.
Sudo et al. [1] experimentally studied forced-convection features
between upflow and downflow in narrow rectangular channels to
differentiate Dittus-Boelter, Sieder-Tate, and Colburn correlations. They
found that any correlations are applicable considering the equivalent
hydraulic diameter for rectangular channels. They also concluded that
the Dittus-Boelter correlation establishes noble extrapolation within the
error of ±20 % for Re larger than 4000. Prasad and Sani [2] researched
to explore the impact of ribbed surfaces on SADH. Their findings
* Corresponding author.
E-mail address: senapatijr@nitrkl.ac.in (J.R. Senapati).
https://doi.org/10.1016/j.icheatmasstransfer.2024.108411
Available online 9 December 2024
0735-1933/© 2024 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
K. Dhamudia and J.R. Senapati
International Communications in Heat and Mass Transfer 161 (2025) 108411
Nomenclature
Aar
Aas
Dh
e
H
h
L1
L3
L2
P
Ti
Ta
Tf
To
W
Pr
P/e
Re
Absorber plate area with ribs (m2)
Absorber plate area without ribs (m2)
Hydraulic diameter of duct (mm)
Rib height (mm)
Depth of duct (mm)
Convective Heat transfer coefficient (W/m2K)
Entry length(mm)
Exit section length (mm)
Test section length (mm)
Pitch distance (mm)
Inlet fluid temperature (K)
Mean absorber plate temperature (K)
Mean fluid temperature (K)
Outlet fluid temperature (K)
Width of duct (mm)
Prandtl number
Relative roughness pitch.
Reynolds number
Abbreviations
SAHPI
Solar air heater performance index
TGPP
Thermo-geometric performance parameter
SAHD
Solar air heater duct
CFD
Computational Fluid Dynamics
Greek letters
α
Thermal diffusivity
αt
Turbulent diffusivity
ΔP
Pressure drop at absorber plate
k
Thermal conductivity (W/m K)
μeff
Effective viscosity
ρ
Air density (Kg /m3)
τ1r
Shear forces of the roughed surface.
τ3s
Total shear forces of smooth surfaces.
Non-dimensional numbers
e/Dh
Relative roughness height
fav
New average friction factor
fr
Average friction factor of the duct with a rough surface
fs
Average friction factor of smooth duct.
Average Nusselt number of rough duct
Nur
Nus
Average Nusselt number of non ribbed duct
Subscripts
a
Absorber plate
r
Rib (Roughen) duct
s
Smooth duct
t
Turbulent
compared to a smooth duct. Kumar et al. [8] explore numerous V-shaped
ribs with gap roughness on the absorber plate, revealing the optimal
thermo-hydraulic performance with a relative gap (Gd/Lv) of 0.69, a
relative gap width (g/e) of 1, e/Dh of 0.043, and a P/e of 10. Sharma and
Kalamkar [9] investigate SAHD with thin ribs by maintaining a blockage
effect (e/H = 0.1). They found a maximum thermo-hydraulic perfor­
mance in the range of 8 ≤ P/e ≤ 10 due to the low reattachment-free
shear layer leading to flow separation and causing low heat removal.
Gawande et al. [10] executed an experimentation and CFD examination
using the RNG k − ε model to study L-shaped ribs. The study involved an
e/Dh of 0.042 and a P/e of 1.14, revealing a maximum thermo-hydraulic
performance of 1.90 at a Reynolds number of 15,000. Sani and Verma
[11] researched dimple-shaped artificial ribs, and resolute maximum
Nusselt number and friction factor values were achieved at e/Dh of
0.0379 and 0.0289, respectively, while maintaining a constant P/e of 10.
Alsaiari et al. [12] explored heat transfer augmentation in conical ribs
through experiments, and the highest thermo-hydraulic efficiency of
82.33 % with (P/e) of 20 at Re = 3000 was achieved. Azad et al. [13]
experimentally worked on discrete arc geometry on SAHD and
concluded that Re = 14,000 has a maximum thermo-hydraulic perfor­
mance of 1.68. Alam et al. [14] experimented on V-shaped perforated
blockages in SAHD. They reported about a 50 % rise in thermo-hydraulic
performance for (P/e) of 8 over a solid blockage. Arunkumar et al. [15]
experimented on the SAHD employing rectangular perforated duct in­
serts. The results revealed that a height ratio of 0.667 corresponds to the
highest thermo-hydraulic efficiency of 87.06 %. Sharma et al. [16]
performed an experiment on the thermal management of SAHD with
roughened sine wave baffles. Their findings revealed a significant
enhancement, with a maximum increase of 3.07 observed in the thermal
enhancement factor at a Re of 9000. Skullong et al. [17] an experimental
analysis has been performed over quadruple perforated-delta-winglet
ribed on SAHD and they identified a maximum thermo-hydraulic per­
formance of 1.902. Additionally, they developed correlations for friction
factor and Nusselt number, with deviations of ±10%and ±7.5%,
respectively. Kumar et al. [18] employed discrete W-shaped roughness
over the heated wall of SAHD to calculate heat transfer and friction
properties experimentally. They noted that at an angle of attack of 600
Fig. 1. Outline diagram of SAHD.
indicated that both Nusselt number and friction factor decrease when
(P/e) drops below 8 to 10 due to deficiency of a reattachment-free shear
layer, resulting in flow separation. Ahn [3] studied heat removal and
friction factors in rectangular ducts with ribbed surfaces. His findings
indicated that triangular ribs raise heat removal more effectively than
square ribs. Karwa et al. [4] explored their studies using transverse,
rectangular, chamfered ribs. They found an enhancement in thermal
efficiency from 10 to 40 % due to a roughened surface on the absorber
plate, whereas the Nusselt numbers increased by 50–120 % at an e/Dh of
0.0441. Alfarawi et al. [5] worked on rectangular SAHD with hybrid ribs
and established a correlation for the Nusselt number with a deviation of
±10 %. Sahu and Bhagoria [6] conducted an experimental analysis of
SAHD featuring broken 90◦ transverse ribs. Their research demonstrated
that these ribs enhance heat transfer coefficient by 1.25 to 1.4 times
compared to non-ribed ducts under similar conditions. Hans et al. [7]
inspected V-ribs over the collector plate. They discovered that this
artificial roughness resulted in an enhancement of the Nusselt number
and friction factor by approximately 6 and 5 times, respectively,
2
K. Dhamudia and J.R. Senapati
International Communications in Heat and Mass Transfer 161 (2025) 108411
and e/Dh = 0.0338, the ribs resulted in an optimal value of Nusselt
number and friction factor, which were 2.16 and 2.75 times higher,
respectively, compared to the duct with no rib. Agrawal et al. [19] led an
experimental survey on SAHD featuring discrete roughened ribs. The
maximum thermo-hydraulic performance was 2.84 at a P/e ratio of 8.33
and a Reynolds number of 13,935. Sethi et al. [20] carried out an
experiment on the SAHD with dimple-shaped artificial turbulators. They
proposed empirical relations for the Nusselt number and friction factor
as functions of the Reynolds number and roughness geometry parame­
ters that are valid within ±8%. Sivakumar et al. [21] did an experi­
mental comparison of a pin-fin absorber plate with a flat plate.They
found thermos-hydraulic efficiency for pin-fin absorber plates in the
range of 10 to 30 % compared to the flat plate range (1 to 24 %). Mund
et al. [22] did an experimental investigation on a rectangular duct
impinging jet solar air heater by varying the length of the perforated jet
plate and found a jet diameter of 3 mm with a length of perforated of
1140 mm has a maximum thermo-hydraulic performance of 1.96 at Re
= 4913. Sharma et al. [23] investigated a computational and experi­
mental study on six different baffle designs (transverse, inclined trans­
verse, dimple, inclined dimple, arc, and sine wave) in SAHD. Their
research found that sine wave baffles attained the peak thermohydraulic performance of 2.05 at a Re = 15,000. Korpale et al. [24]
carried out experimental and computational analysis with rectangular
section fins. They found a maximum thermo-hydraulic performance of
2.77 at Re of 20,000. Patel et al. [25] conducted an experimental and
numerical analysis of SAHD with discrete reverse NACA profile ribs and
obtained a maximum thermo-hydraulic performance of 2.65 at Reynolds
of 6000 with a maximum Nusselt number enhancement of 102.08 at Re
=18,000. Pandey et al. [26] carried out an experimental investigation
using multiple arcs with a gap on the collector plate, and they found a
maximum Nusset number of 5.85 at Re = 21,000.
Computational Fluid Dynamics (CFD) has increasingly gained trac­
tion among researchers for evaluating the performance metrics of Solar
Air Heater Ducts (SAHD). This interest is fueled by the rapid advance­
ments in computational power, allowing for more complex simulations
and in-depth analysis of SAHD designs. However, due to the faster
processing times and reduced computational demands, 2D simulations
are generally preferred over 3D simulations in many studies. Mahanand
and Senapati [27] analyzed the effects of transverse inverted T-shaped
ribs on the thermo-fluid behavior of SAHD. They obtained a maximum
thermo-hydraulic performance of 1.86 with P/e of 7.14, e/Dh of 0.042,
and a Re = 15,000. Barik et al. [28] designed a SAHD with different
types of ribs to enhance its thermo-fluid performance. Their observa­
tions revealed that by adjusting the Reynolds number between 5000 and
18,000 for T-shaped ribs, the thermo-hydraulic performance ranged
from 1.58 to 1.7. Using CFD techniques, Ngo et al. [29] deployed conic
constants in artificial roughness profiles to investigate heat flow and
pressure changes in SAHD. The Nusselt number increases through
decreasing friction factor, as the conic constant decreases and maximum
thermo-hydraulic performance of 1.56 are reached at Reynolds number
of 8000.
Kumar and Sani [30] conducted a CFD analysis of SAHD using arcshaped ribs and found a optimal overall enhancement ratio of 1.7 with
e/Dh = 0.0426.Yadav et al. [31] implemented circular transverse wire
ribs over heated plate of SAHD, with a P/e = 7.14 and e/Dh of 0.042.
This setup led to a considerable improvement, as evidenced by the
highest average Nusselt number of 2.31 times that of a smooth surface at
a Re of 18,000. Moreover, the highest thermo-hydraulic performance of
1.65 was attained at P/e = 10.71. Kumar et al. [32] investigated carried
on SAH over triangular duct by attaching forward-facing rectangular
ribs. Their research disclosed that a maximal thermo-hydraulic perfor­
mance of 2.15 is noticed at P/e = 10 and an e/Dh of 0.043 and formu­
lated correlations for Nusselt number and friction factor with percentage
deviation of ±12 % and ±13 %, respectively. In review, Yadav and
Bhagoria [33] examined the CFD approach, specifically focusing on 2D
analysis models; the results closely align with experimental findings
compared to 3D simulations. Sing et al. [34] carried a CFD exploration of
SAHD with non-irregular cross-section transverse ribs and found a
maximum Nusselt number enhancement of 1.78 for irregular crosssection saw-tooth ribs. Mahanand and Senapati [35] conducted a
comparative study on an SAHD equipped with quarter-circular ribs.
They observed that a P/e of 7.14 resulted in the maximum thermohydraulic performance of 1.88 at a Reynolds number of 15,000.
Kumar and Verma [36] investigated sinusoidal protrusion ribs and
continuous transverse sinusoidal ribs in SAHD and found that both show
the optimum value of thermo-hydraulic performance at P of 10 mm by
changing the Re from 4000 to 15,000. Yadav and Bhagoria [37] inves­
tigated the SAHD featuring square section transverse ribs, examining a
range of Re from 3800 to 18,000 and P/e ratios in the range of 7.14 to
17.86. Their study revealed that the highest thermo-hydraulic perfor­
mance of 1.82 was reached at a Re of 18,000 and a P/e ratio of 7.14. Sign
et al. [38] did a comparative study with multiple broken transverse and
square-waved ribs. Their findings reveal a maximum thermo-hydraulic
performance of 2.10 and 1.62 at Re of 15,000. Thakur et al. [39] per­
formed CFD analysis with a novel hyperbolic rib and found a maximum
thermo-hydraulic performance of 2.16 at Re = 6000 for a Pitch of 10
mm. Promthaisond and Eiamsa-ard [40] did a numerical investigation
on absorber plates with wavy-triangular ribs and found a maximum
thermo-hydraulic performance of 2.62 at Re = 3000. Haldar et al. [41]
carried a CFD analysis of a solar air heater with wavy roughness and
found a maximum thermo-hydraulic performance of 1.96 at Re =
12,000. Vyas et al. [42] did a parametric investigation on aerofoil and
bio-inspired fins. They found that bio-inspired fins achieved maximum
thermal efficiency of 92.69 % at a Re of 21,000 and a thermohydraulic
efficiency of 84.34 % at a Re of 15,000. Besides this literature, several
experimental and computational studies on different rib turbulators in
SAHD can be found by reviewing various research works by (Varun et al.
[43], Bhushan and Singh [44], Alam et al. [45], Mund et al. [46] and
Mahanand and Senapati [47]) have been performed to assess the per­
formance of SAHD featuring ribs on the collector plate.
This literature review indicates that numerous experimental and
computational investigations have been conducted on SAHD employing
various rib roughnesses. These studies include rib shapes ranging from
rectangular, circular, semi-circular, and trapezoidal to irregular shapes
such as NACA profiles and dimples, etc. Even minor alterations in
roughness geometry can significantly impact the thermal performance
of SAHD, as these ribs enhance heat transfer by generating turbulence
near the heated surface (absorber plate) of the flow domain. Although
abundant studies have been performed by considering different rough­
ness geometries on collector plates, there remains considerable oppor­
tunity for analyzing the thermal characteristics of the SAHD by using
novel types of ribs. Augmentation in the heat removal from the absorber
plate is possible with increasing convective surface area and intensity of
turbulence. With increasing height of ribs (e), the convective area can be
increased, but that leads to more pressure drop and, subsequently, a rise
in friction factor. Thus, instead of increasing rib height and sacrificing
pumping power, the heat transfer rate can be enhanced by adopting a rib
geometry that promotes greater turbulence near the absorber plate. This
inspires us to use a complex rib configuration, i.e., plus(+)-shaped ribs
on the collector plate, which has not been reported in the open litera­
ture. The motivation behind using this unique shape of the rib is that it
creates more turbulence because of its extra protruding parts (horizontal
and vertical), as displayed in Fig. 2. Therefore, in this study, the heat
transfer augmentation and flow characteristics in solar air heater ducts
(SAHD) with plus (+)-shaped rib turbulators are investigated numeri­
cally using ANSYS Fluent R22. This exercise aims to evaluate the impact
of roughness parameters (P/e) on the thermal characteristics of SAHD,
while keeping (e/Dh) fixed with different Reynolds numbers.
2. Problem statement
The present simulation of the SAHD rectangular duct is carried out
3
K. Dhamudia and J.R. Senapati
International Communications in Heat and Mass Transfer 161 (2025) 108411
Fig. 2. Configuration of (+) shaped rib with extra protrusion.
by dividing the flow domain into three sections, viz., entry section (L1),
test section (L2), and exit section (L3). The dimensions for the analysis
domain are chosen as per ASHRAE 93–2010 [48] standards, and it is
√̅̅̅̅̅̅̅̅
recommended that the entry and exit length have to be > 5 WH and
√̅̅̅̅̅̅̅̅
> 2.5 WH, respectively, where (W) and (H) are width and depth of the
duct. The current study is performed by choosing ‘W’ and ‘H’ values to
be 100 mm and 20 mm with L1, L2, and L3 of 225 mm, 275 mm, and 121
mm, respectively, as displayed in Fig. 3. The hydraulic diameter (Dh) is
calculated to be 33.334 mm by using Eq. 13, with considering duct di­
mensions, at constant aspect ratio (W/H) of 5. The test section features a
collector plate made of aluminum, with plus-shaped artificial ribs
attached, also made of aluminum, as its thermal conductivity is high.
Throughout the work, the collector plate is consistently subjected to a
heat flux of 1000 W/m2. A rib height of 1.4 mm was selected to reduce
the influence of flow passage blockage. The aim of this exercise is to
determine the SAH Performance Index (SAHPI) and the heat transfer
enhancement of the SAHD. Additionally, a numerical simulation is run
to analyze fluid behavior in the SAHD, including velocity, pressure, and
turbulence intensity, within the fluid domain as artificial ribs are
attached. The ongoing analysis is carried out by keeping constant e/Dh of
0.042 and varied P/e from 7.14 to 17.85 as shown in Fig. 4. The nu­
merical exploration is done by considering ten Reynolds numbers
(4000,6000,8000, 10000, 12000,14000,16000, 18000,20000, and
22000) to get optimum (P/e) for plus-shaped turbulators.
Primary goals of this work are outlined as follows:
Fig. 4. Schematic of absorber plate with plus-shaped ribs.
ii. Air is employed as the working fluid and its properties are pre­
sumed to be uniform, and measured at mean bulk temperature in
a specified cross-section.
iii. No slip conditions are imposed at interface of fluid and solid.
iv. The radiation effect is neglected.
v. Thermal conductivity of collector plate and ribs are independent
of temperature change.
3.1. Governing equations
With the above mentioned assumptions, the governing differential
equations (Biswas and Eswaran [49], Yadav and Bhagoria [50]) are
given as follows.
Continuity equation
∂
(ρui ) = 0
∂xi
(1)
Where ρ and ui are density and flow velocity in the x-direction, used
to find mass balance.
Momentum equation
[ (
)]
)
)
∂ (
∂p
∂
∂ui ∂uj
∂ (
ρui uj = −
+
μ
+
− ρu,i uij
(2)
+
∂xi
∂xj
∂xi ∂xj
∂xj ∂xi
I. Analyze the impact of (P/e) on SAHD with varying Reynolds
number.
II. Predict the flow field, temperature, pressure, and turbulence in
pictorial form to analyze the thermo-fluid behavior in SAHD.
III. To get optimal (P/e) of plus-shaped rib with maximal heat
removal enhancement in terms of SAHPI.
IV. Develop the correlations for the average Nusselt number and
friction factor for current study.
Energy balance equation
[
]
∂
∂
∂T
(ρui T) =
(α + αt )
∂xi
∂xj
∂xj
(3)
Where α and αt are thermal diffusivity and turbulent thermal diffu­
sivity, which is represented as follows:
α=
3. Mathematical formulation
αt =
This section of ongoing analysis of fluid domain includes, governing
equations and boundary conditions with following assumptions.
k
=
k
=
ρCp
ρCp
μ
ν
=
ρPr Pr
(4)
μt
νt
=
ρPr Prt
(5)
The standard k-ε turbulence model includes transport equations for
the turbulent kinetic energy (TKE) and turbulent dissipation rate. In
contrast, the renormalization group (RNG) k-ε turbulence model applies
to low and high Reynolds numbers and accounts for swirling flows, such
i. Incompressible and steady- state turbulent flow.
Fig. 3. Schematic of 2-D computational flow area.
4
K. Dhamudia and J.R. Senapati
International Communications in Heat and Mass Transfer 161 (2025) 108411
as those around weir structures. In the RNG k − ε turbulence model, the
Navier-Stokes equations are solved utilizing a mathematical technique
referred to as the “renormalization group” (RNG). The constants in the
RNG k-ε model, derived through analytical methods, differ from those in
the standard k-ε model. Launder and Spalding [51] provided standard
empirical values for the dependent constants in these models, as shown
in Table 1, by solving the following governing equations.
For the standard k − ε turbulence model.
Turbulent kinetic energy
(
)
μeff ∂k
∂
ρ
×
(6)
uj k −
= G − ρε
∂xj
σ k ∂xj
Dissipation rate of k
(
)
∂u ε
∂ μeff ∂k
(C1 Gε − C2 Gε2 )
ρ j =
×
+
∂xj σk ∂xj
∂xj
k
(
G = μt
∂ ∂uj ∂ui
+
∂xj ∂xi ∂xj
conductivity of working fluid taken from Table 2).
The ‘h’ of air at test section calculate by applying energy principle.
The energy supplied to the ribbed collector plate is equal to energy
received in the air of SAHD.
/
)
(
Q
1000
= 1000 W m2 = h Ta − Tf , h =
(15)
Aar
Ta − Tf
Here Tf = Mean fluid temperature =
(7)
SAHPI =
ε
μeff = μ 1 +
√̅̅̅̅̅̅
Cμ
μ
fr =
(10)
ρ × V × Dh
μ
(11)
Here, Re is varied from 4000 to 18,000, from this the inlet velocity to
the flow domain calculated by taking properties of air from Table 1.
Re × μ
ρ × Dh
((W + 2H)τ3s + W × τ1r )L = τ × 2 × (W + H) × L
4 × Area
4×W×H
=
Wighted perimeter 2 × (W + H)
hDh
k
(13)
Here, τ3s = total shear force of smooth surface, τ1r = shear force for
roughed surface and τ = average shear force.
The friction factor can also calculate by basic Fanning relation.
fs =
Table 1
The constants are recommended for standard and RNG k–ε turbulence models.
Models Standard RNG
C1
1.44
C2
1.92
Cμ
0.09
σk
1.00
σc
1.30
C1
1.42
C2
1.68
τs
1
×
2
ρ × Vs
2
(friction factor of smooth surface)
(23)
Table 2
Thermo-physical properties of air and aluminum taken from the FLUENT
database.
(14)
Here, h = convective heat transfer coefficient. And ‘k ‘is the thermal
Constants
(22)
τ=
For this study height (H) and width (W) of duct are taken 20 mm,100
mm, respectively [27,31].
Average Nusselt number of ribbed SAHD evaluated by
Nur =
((W + 2H)τ3s + W × τ1r )L
2 × (W + H) × L
(21)
(12)
Where, the Dh = hydraulic diameter is calculated by.
Dh = Hydraullic diameter =
(20)
Again, Yadav and Bhagoria [50], and Yadav et al. [54] have pro­
posed that the results obtained from the 2-dimensional model are close
to actual results as analyzed with the 3-D flow. While in practical case
calculating average friction factor for test section of SAHD affected by
one ribbed and three smooth surface as shown in Fig. 5.
Here, surface 1 represent ribbed absorber plate, surface 3 represent
smooth bottom wall and surface 2,4 represent smooth side wall of test
section.
The average friction factor considering four surface can be evaluated
with the correlation given by Prasad and Sani [2] The empirical formula
develops by resolving the forces of the surfaces.
The complete numerical exercise mainly focuses on finding the heat
removal rate, pressure loss, SAHPI, and flow behavior of duct using plusshaped turbulators in a collector plate. The results are shown in terms of
performance parameters such as average friction factor enhancement
factor, average Nusselt number enhancement factor, SAHPI, and
thermo-geometric performance parameter (TGPP). The evaluation of
these parameters involves the following equations.
V=
(19)
fs = 0.085 × Re− 0.25
3.2. Important mathematical relations
Re =
(ΔP)Dh
2 × ρ × L × V2
Here, L = test section length, ΔP = pressure drop, V = inlet velocity
and Dh = hydraulic diameter.
The mean friction factor for smooth duct evaluated by the correlation
given by modified Blasius equation ([10,23])
)2
ε
(18)
The average friction factor of ribbed surface is calculated by the
relation
(9)
+μ
k
√̅̅̅
(17)
1/3
Nus = 0.023 Re0.8 Pr0.4
The RNG k-ε turbulence model is represented similarly to the stan­
dard k-ε model. The primary difference lies in the effective viscosity, as
described by Hou and Zou [52] and given by,
(
Nur /Nus
(fr /fs )
Where, Nus average Nusselt number of smooth duct and the corre­
lation given by Dittus-Bolter.
(8)
The effective viscosity μeff
μeff = μt + μ = ρCμ
(16)
To, Ti = outlet and inlet temperature of the fluid.
Ta = Mean absorber plate temperature.
The SAHPI of ribbed absorber plate is evaluated as a relation given by
Webb and Eckert [53].
)
k2
To + Ti
2
Cμ
0.0845
5
Properties (at 300 K)
Air
Aluminum (Absorber plate and
ribs)
Density, ρ (kg/m3)
Specific heat,Cp (J/kg-K)
Thermal conductivity, k (W/mK)
Dynamic viscosity,μ (kg/m-s)
1.225
1006.43
0.0242
2719
871
202.4
1.7894e− 5
–
K. Dhamudia and J.R. Senapati
International Communications in Heat and Mass Transfer 161 (2025) 108411
exercise was done by taking eight different inlet velocities ranging from
1.749 m/s to 9.62 m/s, and these velocities were calculated by using an
Eq. (12), that corresponds to the Reynolds number range 4000 to
22,000. The boundary conditions for the computational domain are
mathematically represented as follows:
At the inlet boundary:
u = uinlet , v = w = 0, Tinlet = 300K and 4000 ≤ Re ≤ 22000
(29)
At the outlet boundary:
P = Patm , and
fr =
1
×
2
fav =
ρ × Vr
q̇ = 1000
2
(friction factor of rib surface)
(24)
(Average friction factor)
(25)
τ
ρ × Vav
1
×
2
2
2
2
Considering the assumption, 12 × ρ × Vs = 12 × ρ × Vr = 12 × ρ × Vav
The new average friction factor as follows
fav =
((W + 2H)fs + W × fr )
2 × (W + H)
Nur /Nus
(fav /fs )1/3
No − slip condition( u = v = w = 0) and q̇ = 0 (no heat transfer)
Nur /Nus
Aar /Aas
(31)
(32)
The thermophysical properties, as mentioned in Table 2, are assumed
to be constant, and measured at the bulk mean temperature of the fluid.
It is worthwhile to mention that the temperature of the was taken as 300
K from the beginning.
2
4. Numerical methodology
(26)
The current investigation employs ANSYS 22 R2 and focuses on a
two-dimensional rectangular duct as the analysis domain, featuring
plus-shaped turbulators affixed to a hot aluminum collector plate, as
depicted in Fig. 3. The losses originating from conduction, convection,
radiation, and diffusion are not considered in this simulation. This sec­
tion encompasses information on the numerical approach, mesh gen­
eration, pertinent mathematical equations, grid independence
assessment, validation of turbulence models, and validation of the nu­
merical methodology.
(27)
The above relation is used by Alsaiari et al. [12] and Prasad et al.
[55], and they found it has more realistic results. The attachment of plus
ribs in the collector plate enhances the heat removal rate with increasing
surface area (absorber plate + ribs). So, it is essential to see the per­
formance evaluation of ribbed ducts. The performance parameter is
defined as the (TGPP) represent the ratio of average Nusselt number
enhancement factor (Nur /Nus ) to the area enhancement factor (Aar /Aas ).
This configuration is used by Zheng et al. [56],where (Aar is the area of
the collector plate with ribs and Aas represents area without ribs.
TGPP =
W
, with no − slip condition ( u = v = w = 0)
m2
At other walls (other than the absorber plate):
By considering above relation Modified (SAHPI) given as follows
SAHPI =
(30)
At the absorber plate:
Fig. 5. Line diagram of test section with four surface of flow domain.
τ1r
dϕ
= 0, where ϕ = u, v, w, T, k and ε
dx
4.1. Numerical scheme
The governing equations underwent discretization employing a
second-order upwind scheme. The Semi-Implicit Method for Pressure
Linked Equation (SIMPLE) algorithm is used to solved velocity and
pressure coupling. Convergence criteria were set to 10− 6 for the mo­
mentum and turbulence equations, and 10− 9 for the energy equation.
(28)
3.3. Boundary conditions
4.2. Mesh generation
In the present analysis, the rectangle domain comprises an inlet,
outlet, and two walls (top and bottom walls), as shown in Fig. 6. The
inlet boundary maintains a uniform velocity at 300 K.The outlet
boundary is considered a pressure outlet with a fixed value. As explained
earlier, the rectangular domain is divided into three sections such as the
entry section, the exit section, and the test section. The top wall of the
test section is the absorber plate and is employed with uniform heat flux
with no-slip conditions. The other walls of the three sections are
considered to be adiabatic with the no-slip condition. The numerical
The domain modeling and meshing were conducted using ANSYS 22
R2. A uniform grid with high resolution was utilized, particularly near
the ribs of the SAHD, as illustrated in Fig. 7. This strategy was employed
to prevent formation of a laminar sub-layer near the collector plate.
4.3. Grid independent test
With a Reynolds number of 12,000, a roughness height (e) of 1.4
mm, and a pitch (P) of 10 mm, a grid independence test was conducted
Fig. 6. Diagram representation of flow domain with boundary conditions.
6
K. Dhamudia and J.R. Senapati
International Communications in Heat and Mass Transfer 161 (2025) 108411
Fig. 7. Meshing of flow domain with plus-shaped ribs.
by adjusting edge sizing the ribs with face sizing of the domain and face
meshing. The number of nodes was varied beyond 81,325 to determine
the optimal node count. It was found that a node count of 141,641
provided the best results, with a percentage difference in the Nusselt
number of 0.065 %, as displayed in Fig. 8.
4.4. Validation of turbulence models
To achieve accurate results in this study, selecting an appropriate
turbulence model is essential. The Nusselt number was computed using
five turbulence models in relation to the Reynolds number and
compared to the Dittus-Boelter equation, as stated in Eq. (13). This
comparison, depicted in Fig. 9, reveals the RNG k-ε model with EWT
produces the most accurate results, with a variation of ±6.7 %. Previous
studies have also shown that the RNG k-ε with EWT has minimal devi­
ation from experimental data. Therefore, this model is used in current
study to evaluate performance parameters of the SAHD.
4.5. Validation of numerical scheme
Validation of a numerical scheme entails ensuring that the compu­
tational algorithms produce results that accurately represent the inten­
ded real-world phenomena. This typically involves comparing the
numerical outputs to established analytical solutions or experimental
data to confirm the scheme’s accuracy and reliability. Here, RNG k-ε
model with EWT has been validated against the experimental results of
square rib tubulators from Skullong et al. [57]. The validation indicates
a maximum deviation of ±5 %, as displayed in Fig. 10.
Fig. 9. Comparison of Nusselt number with Reynolds number for different
turbulent models.
5. Results and discussion
In this computational analysis, heat transfer and flow field study of
SAHD equipped with transverse plus- shaped ribs are conducted keeping
Fig. 10. Comparison of experimental and present result with RNG k − ε
with EWT.
constant e/Dh of 0.042. The key findings from this analysis are obtain­
able in both quantitatively and qualitatively. The investigation yields
significant results, as the effect of Re and P/e on the average Nusselt
number, average friction factor, SAHPI, Thermo-Geometric Perfor­
mance Parameter (TGPP) and visualization different fluid properties are
deliberated in sub-sections as follows.
Fig. 8. Variation of nodes with Nusselt number.
7
K. Dhamudia and J.R. Senapati
International Communications in Heat and Mass Transfer 161 (2025) 108411
5.1. Assessment of heat transfer
removal rate. Similarly, with increasing pitch length, the number of ribs
decreases. This leads to smoother flow with less disruption of boundary
layer in the viscous sublayer, leading to minor localized turbulence,
resulting in decrease in the heat removal rate.
A comprehensive numerical research of plus-shaped rib-turbulators
on absorber plate, led a noteworthy enhancement in heat removal rate,
with enhancing the convective surface area of SAHD. Moreover, pres­
ence of ribs heightened the turbulence effect, leading to the disruption of
the boundary layer within flow domain, thereby further augmenting the
heat removal rate in SAHD. Fig. 11 shows the static temperature con­
tours around the first four plus-shaped transverse ribs near the inlet
section for three different Re, viz., 4000, 14,000, and 18,000. This in­
dicates that as the incoming velocity increases, the thermal boundary
layer becomes thinner. Since, the development of boundary layer op­
poses heat removal, a stripper boundary layer results maximizing rate of
heat removal. The contour plots, evidently shows a significant
improvement of heat transfer at higher Reynolds numbers.
5.1.2. Dispersal of local heat transfer coefficient along the absorber plate
Fig. 13 shows how the heat transfer coefficient varies along collector
plate with plus-shaped ribs within the inter-rib spaces. Each inter-rib
space contains a peak value of convective heat transfer coefficient (h)
and gradually decreasing along the flow direction, eventually stabilizing
after 8–9 inter-rib spaces. These local peaks arise from the striking of
incoming cold fluid on the ribs and the resulting turbulence near col­
lector plate around the ribs. Consequently, the ribbed SAHD exhibits
superior heat transfer compared to a smooth one.
5.1.3. Effect of velocity
Fig. 14 offers a visual depiction of velocity magnitude contours for a
of e/Dh = 0.042 across Reynolds numbers of 18,000 for different relative
pitch (P/e = 7.14,10.71,14.28 and 17.85). The introduction of ribs onto
the collector plate within rectangular duct instigates a phenomenon akin
to a nozzle effect, effectively accelerating the fluid as it traverses the
passage between the ribs. This acceleration prompts a notable
enhancement in fluid velocity downstream, thereby inducing turbulence
within the flow field.The heightened turbulence, in turn, engenders
vortices around the tips of the turbulators, thereby intensifying removal
of heat from collector plate to cold air within SAHD. Inside flow domain
detachment and subsequent reattachment of the fluid occurs along
interface of ribs. Notably, with increasing Reynolds number, the reat­
tachment point shifts towards the downstream region, indicative of
more dynamic fluid behavior.
On the opposing side of the ribs, a distinct recirculation zone
emerges, characterized by poor fluid attachment to the ribs, conse­
quently leading to diminished heat transfer. However, as fluid velocity
escalates the kinetic energy of the fluid led in reduction of boundary
layer thickness. This diminishment in thickness renders the separation
region more compact, thereby utilizing it to maintain a uniform velocity
profile among neighboring fluid particles.
5.1.1. Influence of Reynolds number on average Nusselt number
enhancement factor for different relative pitch
By elevating the Reynolds number, the fluid velocity experiences an
increase, causing high-energy fluid particles to collide with the plusshaped ribs of SAHD. The collision induces turbulence, enhancing heat
flow between the collector plate and fluid by facilitating the movement
of kinetic energy between fluid layers. A smaller pitch results in a
number of ribs on the collector plate, which enhances the exposed sur­
face area and, consequently, removal of heat from absorber plate.
Additionally, greater number of ribs enhance in mingling of fluid at
inter-rib regions, leading to a elevated ‘h’. Therefore, the Nusselt num­
ber rises with decrease of pitch for a given Reynolds number, as
described in Eq. (14). Fig. 12 (a) illustrates,the average Nusselt number
enhancement factor rises with an rise in Re for various (P/e) and reaches
its peak value of 3.09 at Re = 18,000 with a P/e of 7.14. Then it de­
creases with an increment of Re. The decrease of Nur /Nus occurs due to a
decrease in turbulence intensity, causing less fluid attachment to the
ribs, led a to low heat transmission. In the same way, Fig. 12 (b) shows
the Nusselt number with (P/e) for different Reynolds numbers. It tells
with increasing P/e the Nusselt number decreases. This because a
addition of ribs enhances heat transfer coefficient which improves heat
Fig. 11. Temperature contours of P/e = 7.14 for (a) Re = 4000, (b) Re = 14,000 and (c) Re = 18,000.
8
K. Dhamudia and J.R. Senapati
International Communications in Heat and Mass Transfer 161 (2025) 108411
Fig. 12. (a) Average Nusselt number enhancement factor(Nur/Nus) vs Re and (b) averageNusselt number (Nur) vs (p/e)
Fig. 13. Spreading of surface heat transfer coefficient along the collector plate
at P/e = 7.14 and Re of18000
5.1.4. Effect of turbulence parameters
The presence of ribs interrupts boundary layers, leading to an in­
crease in turbulence kinetic energy (TKE) SAHD. Fig. 15 illustrate the
behavior of turbulent kinetic energy for plus-designed ribs over absorber
plate having P/e = 7.14 for Re of 4000, 14,000, 18,000 and smooth duct
at Re = 18,000. The observed trends show that turbulence kinetic energy
peak at the first and second ribs of SAHD and progressively decrease in
downstream.The observed trends show that turbulence kinetic energy
peak at the first and second ribs of SAHD and progressively decrease in
downstream.The TKE of ribed surface are almost 10 time higher than
smooth duct at Re of 18,000. Fig. 16 shows the change in TKE along a
horizontal axis in test section, positioned 5 × 10–3 m from the collector
plate at three different Reynolds numbers (Re = 4000, 14,000, and
18,000).It reveals increase of Reynolds number, the TKE becomes more
pronounced. This increase because of the disruption of the viscous
sublayer in the regions between the ribs and higher Reynolds numbers
correspond to greater turbulent kinetic energy. Higher turbulence pa­
rameters signify the presence of more energetic fluid particles. Near the
tips of the ribs, stronger shear forces are present, requiring more ener­
getic particles to sustain downstream flow. Fig. 17 illuminated presence
of turbulators on the collector plate the behavior of turbulent intensity
(TI) for plus-designed ribs over collector plate at P/e = 7.14 for Re of
4000, 14,000, 18,000 and 22,000. These ribs create more (TI) in the flow
domain cause diminishes of shear force. This reduction in shear force
Fig. 14. Velocity contours of P/e = 7.14, 10.71, 14.28,17.85 and smooth tube
at Re =18,000.
leads formation of eddies in spaces between the ribs. These eddies
facilitate momentum transfer between fluid.
5.2. Influence of Reynolds number on average enhancement friction
factor
The incorporation of plus-shaped ribs in the SAHD improves heat
transfer furthermore, enhancing pressure drop in flow field. This addi­
tional pressure drop causes an average friction loss within the flow
domain, as calculated by Eq. (26), indicating that a smooth duct has less
friction loss compared to a ribbed duct. Therefore, it is essential to un­
derstand the relationship between friction loss, Reynolds number, and
rib geometry. Fig. 18 (a) reveals with enhancement of Re, average
friction factor enhancement ratio decrease. The rise of Re causes
enhancement of fluid velocity, which helps to decrease viscous-sublayer,
makes decrease of average frictiona factor. Therefore,the interplay be­
tween rib-induced turbulence and friction loss is key to optimizing
design and performance of SAHD. Furthermore, the average friction
factor drops as the (P/e) rises at specified Reynolds number and cause
9
K. Dhamudia and J.R. Senapati
International Communications in Heat and Mass Transfer 161 (2025) 108411
Fig. 15. Contours of TKE for P/e = 7.14 for Reynolds number (a) 4000 (b) 14,000, (c) 18,000 and (d) smooth tube at Re of 18,000.
due to vortex shedding, causing a higher average friction factor on the
collector plate due to the presence of more ribs. For a given P/e, with
Reynolds number, the viscous sub-layer becomes thinner and leads
decrease of friction factor values.
5.3. Influence of pressure drop and pumping power
The flow behavior in SAHD can be examined by looking at the
pressure contours for different (P/e = 7.14,10.71,14.28 and 17.85) at a
Re = 18,000 as shown in Fig. 19. Plus-shaped turbulators obstruct the
fluid flow, causing a pressure drop within SAHD. The contours clearly
demonstrate that increasing the P/e lowers static pressure and reduces
turbulence. When high-energy fluid particles hit the vertical surfaces of
the ribs, kinetic energy is lost, decreasing the fluid velocity. According to
the conservation of energy principle, this lost kinetic energy converts to
potential energy, raising the pressure near the ribs and causing flow
separation. This effect is visible in the contours, which show that higher
ribs result in increased static pressure and also enhance turbulence.
Moreover, a recirculation zone is observed between the ribs, where
the flow reattaches, leading to a rise in pressure in that area. Reduction
of pressure takes place from the entry to the exit section of SAHD.
Although ribs on the collector plate enhance heat transfer, they also
demand significant pumping power to counteract the frictional resis­
tance induced by the turbulators. The current study is focused on
identifying the optimal geometric configuration that maximizes heat
removal at minimum pumping power. Fig. 20 shows pumping power
requirements for various rib heights (P/e = 7.14,10.71,14.28 and
17.85). The findings reveal that for a lesser mass flow, the pumping
power needed remains relatively constant across the considered range of
P/e. However, an upper mass flow rate varies significantly with changes
in relative roughness pitch. This is because, at lower flow velocities, the
pressure drop shows minimal variation regardless of the relative
roughness pitch.
Fig. 16. Deviation of TKE along a horizontal axis (at a vertical distance of 5 ×
10− 3 m from the collector plate) in the test section.
reduction at quantity of ribs on collector plate. The fewer ribs in flow
domain cause less hinderance to the flow led to lesser pressure drop, as
shown in Fig. 19, and a decrease in the average friction factor.
Fig. 18(b) explores a comparision of fav versus P/e for various Rey­
nolds numbers. It is clear, reduction of friction factor gradually happens
as increase of P/e. This indicates that the airflow experiences less
disturbance while passing through the SAHD. Furthermore, the friction
factor decreases when there are fewer ribs as a reduction of pressure
drop occurs. Additionally, the tips of plus ribs lead to greater energy loss
10
K. Dhamudia and J.R. Senapati
International Communications in Heat and Mass Transfer 161 (2025) 108411
Fig. 17. Contours of TI for P/e = 7.14 for Reynolds number (a) 4000 (b) 14,000, (c) 18,000 and (d) 22,000.
Fig. 18. (a) Average enhancement friction factor (fav/fs) vs Reand (b) average friction factor (fav) vs (p/e).
5.4. Effect thermo-geometric performance parameter
optimizes the balance between enhancing heat transfer and managing
additional pressure drib induced due to turbulators. The optimal TGPP
value signifies the efficiency of this configuration in maximizing heat
transfer while keeping frictional losses within acceptable limits, making
it an effective design for the SAH system.
Inserting plus-designed turbulators on the collector plate of an SAHD
boosts heat removal rate by increasing surface area. Therefore,
analyzing the heat removal from the ribbed collector plate with fluid
interaction is essential. The heat transfer aumentation largely depends
on the rib geometry. To assess the thermo-geometric performance
parameter (TGPP), Eq. (28) is used, which is a ratio of average Nusselt
number enhancement factor to area enhancement factor. Fig. 21 shows
that a rib height of P/e = 7.14 achieves the highest TGPP of 2.16 at a Re
= 18,000. This indicates that this specific rib geometry spacing
5.5. Exploration of solar air heater performance index (SAHPI)
The performance SAHD with plus-shaped turbulators demonstrates
that while heat transfer is significantly enhanced, but it makes a draw­
back of increasing pumping power due to friction loss. The turbulators
11
K. Dhamudia and J.R. Senapati
International Communications in Heat and Mass Transfer 161 (2025) 108411
Fig. 19. Static pressure contours of P/e = 7.14, 10.71, 14.28,17.85and smooth tube at Re =18,000.
Fig. 21. Thermo-geometric performance parameter (TGPP) vs Reynolds.
Fig. 20. Pumping power vs Mass flowrate.
12
K. Dhamudia and J.R. Senapati
International Communications in Heat and Mass Transfer 161 (2025) 108411
on collector plate induce flow detachment, recirculation, vortex for­
mation, and reattachment, all of which contribute to the frictional
penalty. The effectiveness of the turbulator design lies in maximizing
heat transfer with minimizing additional pumping power requirement.
Fig. 20, shows how pumping power varies with the P/e of the turbula­
tors, indicating the trade-off between turbulence-induced heat transfer
and frictional losses. Fig. 23 reveals that the Solar Air Heater Perfor­
mance Index (SAHPI) which is ratio of average Nusselt number
enhancement factor to one-third power of the average enhancement
friction factor, reaches its topmost value of 2.54 at Re of 18,000 and P/e
= 7.14.
The Nusselt number is influenced by the convective heat transfer
coefficient; greater fluid attachment to the active area facilitates faster
heat removal from heated surface to cooler fluid, resulting in a higher
‘h’. In this computational analysis, greater fluid attachment occurs at P/
e = 7.14 at Re of 18,000. Fig. 22, indicates path line for Re of
4000,14,000,18,000 and 22,000 where separation and reattachment of
fluid occurs in between of the ribs. With enhancement of Reynolds
number there is decrement of boundary layer thickness makes more heat
removal,and at Re of 18,000 the sepation region is less due to more
turlence intensity as shown in Fig. 17. Additionaly Fig. 21 also shows
same trend at Re of 18,000.
Fig. 23. Solar air heater performance index (SAHPI)vs Reynolds number(Re).
5.7. Nusselt number and friction factor correlations
5.6. Compairasion of previous numerical work with present study
The correlations of Nur and fav are derived by adopting previously
published research works [17,20,59,60] with regression analysis of
numerical data in non-linear way to determine optimal values of vari­
able parameters, ensuring maximum SAHPI of SAHD. In this analysis,
Nur and fav are reliant on dimensionaless parameters such as ‘P/e’ and
Re, with a constant e/Dh. Plus-shaped turbulators are simulated across a
Table 3 compares the computational results of ribbed SAHD with
various structures. It highlights that the plus-shaped absorber plate
achieves a SAHPI of 2.54, outperforming designs with square, invertedT,sinusoidal protrusion, multiple broken transverse and square waved,
discrete reverse NACA profile ribs,and triangular ribs.
Fig. 22. Path line contours of P/e = 7.14 for (a) Re = 4000, (b) Re = 14,000, (c) Re = 18,000 and (d) Re = 22,000.
13
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International Communications in Heat and Mass Transfer 161 (2025) 108411
Fig. 24. Plot of predict versus simulated result (a) Nusselt number (b) friction factor correlation.
tors, analyzing both qualitatively and quantitatively. The main objective
is to pinpoint the ideal geometric configuration based on maximum
SAHPI. To achieve this, several non-dimensional parameters are
considered, such as P/e, e/Dh, and Reynolds numbers. The key conclu­
sions can be summarized as follows:
Table 3
Comparsion of previous numerical work with present study.
Authors
Rib geometry
Optimal
Parameters
SAHPI
Mahanand and
Senapati [27]
Inverted-T
1.86
Kumar et al. [32]
Chamfered rectangular
ribbed
Mahanand and
Senapati [35]
Quarter-circular
Kumar and Verma
[36]
Sinusoidal protrusion
Yadav and
Bhagoria [37]
Square
Singh et.al [38]
Multiple broken
transverse and square
waved
Discrete reverse NACA
profile
Re:15000
P/e:7.14,e/Dh =
0.042
Re:17000
P/e = 10, e/ Dh =
0.043
Re:15000
P/e:7.14,e/Dh =
0.042
Re:4000
P/e = 10, e/Dh =
0.03
Re:18000
P/e = 7.14,e/Dh
= 0.042
Re:15000
P/e = 10, e/ Dh =
0.043
Re:6000
P/e = 5,e/ Dh =
0.065
Re = 18,000
P/e = 7.14, e/Dh
= 0.042
Re:18000
P/e = 7.14, e/ Dh
= 0.042
Patel et.al [25]
Yadav and
Bhagoria [58]
Triangular
Present study
Plus-shaped
I. The RNG k − ε with the EWT model is found best among other
models as it agrees well with the Dituss-Bolter equation for a
smooth duct.
II. The highest average Nusselt number enhancement factor of 3.09
is attained at P/e = 7.14 and Re of 18,000, while the highest
average enhancement friction factor of 1.96 occurs at Reynolds
number of 4000 at P/e = 7.14.
III. The highest heat augmentation is observed with a TGPP of 2.17,
and the highest SAHPI of 2.54 is attained at P/e = 7.14 and Re of
18,000.
IV. Visualizing fluid flow, along with velocity and pressure contours
and turbulence characteristics, offers a clear insight into thermofluid behavior.
V. The Nur and fav relationships are established, it is highly benefi­
cial for both industrial and academic applications.
2.15
1.88
2.02
1.82
2.10 and
1.62
2.65
2.11
CRediT authorship contribution statement
2.54
Kashinath Dhamudia: Investigation. Jnana Ranjan Senapati: Su­
pervision, Conceptualization.
Declaration of competing interest
range of relative roughness pitch-to-height ratios (P/e) from 7.14 to
17.85 and Re from 4000 to 22,000.
These simulations aim to derive correlations that capture relation­
ship between these parameters and heat transfer augmentation. The
developed correlations are as follows:
Nur = 0.062 × Re0.732 × (P/e)− 0.103
(33)
fav = 0.289 × Re− 0.345 × (P/e)− 0.120
(34)
The authors declare the following financial interests/personal re­
lationships which may be considered as potential competing interests:
Jnana Ranjan Senapati reports financial support was provided by
National Institute of Technology Rourkela. JNANA RANJAN SENAPATI
reports a relationship with National Institute of Technology Rourkela
that includes: employment. JNANA RANJAN SENAPATI has patent NA
pending to NA. No conflict of interest If there are other authors, they
declare that they have no known competing financial interests or per­
sonal relationships that could have appeared to influence the work re­
ported in this paper.
The proposed correlations for Nur and fav (both for predicted and
simulated) are displayed in Fig. 24.The results indicate that the re­
lationships for Nur and fav are usable, with maximum deviations of
±6.5%and ±4.5%, respectively.
Data availability
Data will be made available on request.
6. Conclusion
This paper presents a computational study of plus-shaped turbula­
14
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International Communications in Heat and Mass Transfer 161 (2025) 108411
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