International Communications in Heat and Mass Transfer 118 (2020) 104891
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International Communications in Heat and Mass Transfer
journal homepage: www.elsevier.com/locate/ichmt
Natural convection cooling of an infrared suppression device (IRS) with
conical funnels- a computational approach
T
Aurovinda Mohantya, , Santosh Kumar Senapatib, Sukanta K. Dashb
⁎
a
b
Department of Mechanical Engineering, Veer Surendra Sai University of Technology (VSSUT), Burla, Odisha 768018, India
Department of Mechanical Engineering, Indian Institute of Technology Kharagpur, West Bengal 721302, India
ARTICLE INFO
ABSTRACT
Keywords:
IRS device
Conical funnels
Natural convection
Heat transfer
Nusselt number
Cooling time
The present study involves the numerical computations of the free convection heat transfer from an Infrared
suppression device with the conical funnels, which is used in the exhaust of the marine gas turbines and helicopters. The computations have been performed in the range of 1010 ≤ Ra ≤1012. The study also involves
predicting the effect of the diameter ratio and the percentage overlap. Besides, our study proposes a correlation
for the Nusselt number and the induced mass flow rate in terms of Rayleigh number, diameter ratio, and percentage overlap. The study also computes the cooling time of the IRS device. The Nusselt number initially
increases with the diameter ratio, attains a maximum, and finally reduces to a constant value. Highest values of
Nu are observed with the positive overlap of the funnels, whereas the lowest values are obtained for the negative
overlap. Moreover, the present study also compares the cooling characteristics of the IRS having cylindrical and
the conical funnels. The comparisons show that higher values of the Nu are obtained for the cylindrical funnels
compared to the conical ones. The findings of the present study can be significant to both the engineers of the
ship industries and the academicians.
1. Introduction
An Infrared suppressor (IRS) plays a vital role in the operation of a
marine gas turbine. Usually, the temperature of the exhaust gases
leaving these turbines is high enough to emit a significant amount of
infrared radiation. Typically, in the conflicting regions in the ocean,
these radiations can be easily detected by the enemy IR guided appliances, which will not only cause a threat to the naval ships but also the
merchant ships. Such situations are highly undesirable. An IRS device
equipped at the turbine exit essentially provides a continuous cooling of
the hot exhaust gases released from the turbine and thereby suppresses
the emission of the infrared radiations. An IRS device consists of a series
of funnels arranged one above the other, as shown in Fig. 1 (a) with
different types of overlap, which has been discussed later in the
manuscript.
The hot gases from the turbine first enter into the nozzle (s) located
centrally at the turbine exit and then ejected into the IRS device in the
form of a high-speed jet. As a consequence, a low-pressure zone forms
inside the funnels of the IRS device. The pressure difference thus created, causes the suction of the ambient air into the IRS device through
the space between the successive funnels. Thus, the entrainment and
the interaction of the cold stream of air with the existing hot stream of
gas result in an appreciable reduction in the temperature of the hot
exhaust gas inside the IRS. A reduction in the exhaust gas temperature
substantially suppresses the infrared emissions. Thus, improving the
performance (cooling the hot exhaust from the turbines) of an IRS device in mitigating the infrared radiations has been an exciting subject of
research over the years and a few such studies worth discussing here
briefly.
One of the early studies on the IRS device by Birk and Davis [1]
discusses the need for incorporation of an Infrared suppression device
and its effect on the design of the naval ship in terms of the additional
weight, noise and vibration and engine back pressure. The article of
Birk and VanDam [2] reports different types of IRS devices and emphasizes on the technical criterion to be used for choosing the right IRS
as per the situation. The same study also emphasizes on the nozzle
design and reports the significant role played by the same in the mixing
of the hot exhaust and the fresh air. The study of Thompson and Vaitekunas [3] discusses different factors that significantly affect the ship's
IR signatures and proposes various ways of suppressing them. The existing literature also contains some studies on the lobed type infrared
suppressors [4–9]. The combined experimental and computational
Corresponding author.
E-mail addresses: aurovindam@gmail.com, amohanty_me@vssut.ac.in (A. Mohanty), senapatis431@gmail.com, sks90@iitkgp.ac.in (S.K. Senapati),
sdash@mech.iitkgp.ac.in (S.K. Dash).
⁎
https://doi.org/10.1016/j.icheatmasstransfer.2020.104891
Available online 30 September 2020
0735-1933/ © 2020 Elsevier Ltd. All rights reserved.
International Communications in Heat and Mass Transfer 118 (2020) 104891
A. Mohanty, et al.
p
pressure, N/m2
Symbols Description
atmospheric pressure, N/m2
Patm
Prt
Turbulent Prandtl number
R
Specific gas constant, J/kg-K
Ra
Rayleigh number
u, v, w
velocity components in x, y, z directions respectively, m/s
x, y, z
Cartesian coordinates
Greek symbols Description
α
thermal diffusivity, m/s2
β
thermal expansion coefficient, 1/K
∆T
base-to-ambient temperature difference, K
ε
rate of dissipation of turbulent kinetic energy, m2 s−3
μ
molecular viscosity, kg/m-s
μeff
effective viscosity, kg/m-s
μt
turbulent viscosity, kg/m-s
v
kinematic viscosity, m/s2
ρ
density, kg/m3
σk
Prandtl number for k
Prandtl number for ε
σε
τw
wall shear stress, kg/m-s2
Nomenclature
Symbols
A
D
L
DR
OL
g
h
Nu
Nux
Rax
Ra
Q
T∞
Tw
Tfilm
M
K
k
Nu
Description
Area of Convection, m2
Funnel diameter
Total Length of IRS device
Diameter ratio
Percentage of overlap
Acceleration due to gravity, m/s2
Average heat transfer coefficient, W/m2-K
Average Nusselt number
Local Nusselt number
Local Rayleigh number
Rayleigh number based on total length of IRS device
Convection heat transfer, W
Ambient temperature, K
Funnel wall temperature, K
Average Film temperature, K
Non-dimensional induced mass
Thermal conductivity of air
turbulent kinetic energy, m2s−2
Average Nusselt number
time (the temperature-time history) of such devices is to perform a
three-dimensional natural convection simulation and to predict the
cooling time through the computation of the heat loss and Nusselt
number.
A visit into the exiting studies on the natural convection (both experimental and computational studies) reveals that one set of the studies deal with the estimation of the Nusselt number of the cylinders
(both solid and hollow), and spheres which are often encountered in the
steel industries [19–22]. Another set of studies focuses on the natural
convection in the enclosures [23–26]. In these studies, the prime objective of focus is to estimate the Nusselt number as a function of various geometrical and operating parameters under different orientations
and to propose suitable correlations for the same. Likewise, a bunch of
studies focuses on enhancing the heat transfer from various surfaces
using the different types of extended surfaces or fins [22,27–32]. Recently, Gupta et al. [33] have studied the performance of a porous fin
with prescribed fin temperature. Another set of studies centers on the
natural or mixed convection from the chimneys, and electronic chips.
However, none of the existing studies on natural convection focuses
on the cooling characteristics of the IRS devices, despite its significant
practical relevance except for the recent study by the present authors
[34]. The funnels of an IRS device are usually of cylindrical shape and
recently performance in terms of air entrainment to a conical shape IRS
has been reported by Ganguly and Dash [16–18]. The previous study of
the present authors [34] estimated the cooling time of an IRS with
cylindrical funnels. Therefore, the main idea of the present study is to
evaluate the Nusselt number as a function of several parameters and to
compute the cooling time of an IRS device with the conical-shaped
funnels. Moreover, the study also provides a comparison of the performance of the cylindrical and the conical funnels in terms of the
average Nusselt number and the cooling time.
study of Mishra and Dash [10] presents the effects of various geometrical and operating parameters on the ratio of mass suction to the mass
injected for an IRS with louvered cylindrical funnels. The study reveals
that the acceleration length of flow in such pipes is much smaller than
that in the simple pipes for the same Reynolds number. Another set of
computational studies from the same authors [10,11] reveal that the
mass suction rate for the IRS with louvered cylindrical funnels is a
strong function of the louvered opening area, but is independent of the
number of Louvers per row. The same studies also report that placing
the louvers at the bottom portion of the funnels helps in improving the
suction of air. The studies of Barik et al. [12–15] reveal that the louvered opening area and the protruding nozzle lengths are the two most
significant parameters affecting the air suction into the funnels. Moreover, the entrainment rate of the hot exhaust gas into the funnels is
more with the non-circular jets. Besides, the aspect ratio of the noncircular nozzles also affects the entrainment process significantly. The
recent studies of Ganguly and Dash [16–18] compare the performances
of cylindrical, conical, and the louvered conical IRS devise in terms of
the air entrainment and reports that beyond a certain geometric ratio
the louvered conical IRS device performs significantly better than the
other two arrangements. The combined experimental and computational study of Ganguly and Dash [16–18] also predicts the role of the
pertinent parameters on the performance of these devices. Moreover,
most of these studies also provide correlation(s) for the mass suction
into the funnels as a function of the pertinent parameters under different operating conditions.
Improving the performance of an IRS device, as described in the
above-mentioned studies, is just one aspect of the research on such
devices. However, there is another aspect related to these devices,
which is equally significant, typically when a ship reaches a port for
maintenance. Once a ship arrives at a port, its engine is pulled through
the funnels of the IRS for servicing. Therefore, during that time, the
funnels should be nearly at the ambient temperature so that the
maintenance work can start. Hence, on the arrival of a ship to the port,
first, the IRS device is allowed to cool. The natural convection, owing to
its inherent advantages such as no external power consumption and
cheap, becomes the preferred choice for cooling. Therefore, it takes a
finite time to cool the IRS from a high temperature to the ambient, and
the corresponding time is referred to as the cooling time of the IRS
device. The service engineers at the maintenance site often need the
information on cooling time. One of the ways to estimate the cooling
2. Problem description
In the present work, the natural convection heat transfer from a fullscale IRS device with conical funnels has been studied computationally.
The corresponding schematic has been depicted in Fig. 1 (b). The 3D
image of a lab scale conical IRS set up [16] has been depicted in Fig. 1
(c). The details of the geometrical parameters of base funnel of the IRS
device considered in the present case have been presented in Table 1.
As per the design of conical IRS described in Ganguly and Dash [17],
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A. Mohanty, et al.
the length of other funnels have been taken as same as that of the base
funnel i.e., L1 = L2 = L3 = L4.
The sole effects of the three pertinent parameters, namely, the
Rayleigh number (Ra), the diameter ratio, and the percentage overlap
of the funnels have been predicted on the Nusselt number and the heat
transfer rate. Based on almost 190 simulation results, a correlation for
the Nusselt number has been developed that shows its functional dependency on the Rayleigh number (Ra), the diameter ratio, and the
percentage overlap of the funnels. Using the correlation and considering the lumped parameter analysis, finally, the cooling time of the
IRS device has been computed. Moreover, the effects of each of these
parameters (as mentioned above) on the cooling time have been discussed.
In the present study, the Rayleigh number (Ra), which is an essential parameter in any natural convection situation, has been taken in the
range of 1010 to 1012. The range of the Rayleigh number (Ra) corresponds to the turbulent regime of the natural convection, as verified
experimentally [35] . The diameter ratio (DR) is one of the significant
parameters when it comes to the analysis of an IRS device. In the present study, the ratio of the diameters of the two consecutive funnels has
been
considered
as
the
diameter
ratio,
i.e.
D
D
D
DR = Db2 = Db3 = Db4 = Dt 2 = Dt3 = Dt 4 . Where, Dband Dtrepresent the
t1
t2
t3
b1
b2
b3
bottom and top diameter of a conical funnel respectively. The diameter
ratio has been considered in the range of 1.03 to 1.3 to predict its effect.
The overlap of the funnels has been shown in Fig. 1(a). There can be
three possible types of overlap in any IRS device, such as (a) the positive overlap, (b) the negative overlap, and (c) the zero overlap. In the
case of the positive overlap, the funnels protrude into each other. In the
present case, the overlap between two consecutive funnels has always
been kept fixed, i.e., the percentage length of the first funnel protruding
into the second one is the same as the percentage length of the second
funnel protruding into the third one and so on. In the case of the negative overlap, the funnels are maintained at some fixed distance from
each other. In the present case, the distance between any two consecutive funnels of the IRS has been kept fixed for the negative overlap.
In the case of the zero overlap, the funnels are flushed with one another
without any overlap. In the present study, the effects of all these three
kinds of overlap (positive, zero, and negative) have been predicted. For
this purpose, the overlap has been considered in the range of -20 to 20
%.
It is to be noted that the objective of the present study is to compute
the Nusslet number and the rate of heat loss from the IRS device with
the conical funnel. Therefore, unlike the conventional studies on the
IRS devices, in the present study no central jet has been considered that
ejects the hot gases into the funnel system. In other words, here, a situation has been considered where the gas turbine is in the idle state,
and the funnels (conical shaped) of the IRS devices have been allowed
to cool naturally.
A 2D axisymmetric domain, as shown in Fig. 1 (b), has been considered in the present study. For the present computational study, both
the inner and outer walls of the funnels have been maintained at the
same temperature (Tw). The average heat transfer coefficient (or the
average Nusselt number) has been obtained from the overall heal loss
taking place from the funnels as described in the Eq. (13) presented in
sec-3 of the current article.
D
D
D
3. Mathematical modeling
The mathematical equations and the boundary conditions governing
the physics of the present problem, have been described very briefly in
this section for the sake of brevity. A more detailed description may be
obtained from the previous articles [19–22,29–32,34,36]. The air with
constant thermo-physical properties has been considered as the fluid
medium. The flow field has been assumed to incompressible, turbulent,
and steady. Moreover, the effects of viscous dissipation have been assumed to be negligible. The standard k-ε model has been incorporated
to resolve the closure issues. Likewise, the near-wall turbulence has
been taken care of using the enhanced wall treatment approach. To
ensure that the near-wall turbulence is resolved correctly, the Y+ has
been kept between 1 and 5 [37], which has been achieved by adjusting
the cell size adjacent to the wall.
The Cartesian form of the governing equations for the flow and heat
transfer have been presented in this section briefly.
The continuity equation representing the conservation of the mass
is.
vi
=0
xi
(1)
The momentum equation is expressed as
( vi vj )
xj
= gi
(p + 2/3k )
+
(µ + µt )
xi
xj
vj
vi
+
xj
xi
(2)
The energy equation neglecting the viscous dissipation term is
Fig. 1. (a) Schematic of an IRS with the conical funnels (Not to scale) (b)
Computational domain (two-dimensional axisymmetric geometry), (Not to
scale) (c) 3D image of a lab scale IRS with conical funnel [16].
( vj T )
xj
3
=
xj
µ
µ
+ t
Pr
Prt
T
xj
(3)
International Communications in Heat and Mass Transfer 118 (2020) 104891
A. Mohanty, et al.
Table 1
Geometrical parameters of the IRS device.
Material
Bottom diameter of the base funnel
Top diameter of the base funnel
Length of base funnel
Thickness of the funnels
Mild steel
1m
0.8 m
1m
3 mm
The transport equation for the turbulent kinetic energy (k) is
xj
( kvj ) =
xj
µ+
µt
k
k
xj
+ Gk + Gb
All these boundary conditions have also been depicted pictorially in
Fig. 1 (b).
The fluid properties have been evaluated at the film temperature
(Tfilm). The film temperature is defined as the mean of the ambient and
the wall temperature. Mathematically,
(4)
The transport equation for the turbulent dissipation rate is (ε)
xj
( vj ) =
xj
µ+
µt
xj
+
k
(C1 Gk + C1 C3 Gb
C2
)
Tfilm =
(5)
3.3. The pertinent dimensionless numbers
The two pertinent dimensionless numbers that govern any natural
convection situation includes the Rayleigh number and the Nusselt
number, which have been described briefly in this section.
The Rayleigh number (Ra) is defined as follows.
3.1. The equation of state
The modeling of the Buoyancy term necessitates the use of a suitable
equation of state that can dictate the relationship between density,
temperature, and pressure. There exist two such approximations that
can help in expressing the density as a function of the temperature,
namely, the Boussinesq approximation and the ideal gas approximaT
tion. The former one is most suitable when T < < 1 is satisfied. The
temperature difference between the funnels and the surrounding is very
high (almost of the order of 100 K or higher) for the IRS devices. Thus,
this criterion is not at all satisfied. Hence, the ideal gas approximation is
more suitable for the current situation, and the same has been implemented in the numerical procedure. The expression for the equation
of state is
Ra =
h=
(12)
QL
A (Tw T )
(13)
Apart from the above two dimensionless numbers, it is also essential
to define another dimensionless parameter that can represent the induced mass flow rate entering into the funnels, which is also an essential parameter in the case of IRS devices.
The non-dimensional form of the induced mass flow rate is
.
M=
m
(14)
0 0 D1
Where ρ0 and υ0 represent the reference density and the kinematic
viscosity respectively. D1 refers to the diameter of the base funnel.
Eq. (15) has been solved to obtain the cooling time of the funnels,
i.e., the time required for the hot funnels to reach the ambient temperature. The heat transfer coefficient involved in the expression has
been obtained from the correlation of the Nu developed in the present
study.
(7)
Except for the left face of the computational domain, all other faces
have been subjected to a pressure outlet boundary condition which
means at these faces the pressure is set to atmospheric pressure and the
temperature is set to the ambient value, i.e.,
(8)
mfunnel C
The turbulent intensity of 5% has been considered for accounting
any backflow at the pressure outlet.
The left face of the domain has been subjected to an axis boundary
condition.
Mathematically,
= 0, Vy = 0
Q
A (Tw T )
Nu =
It is necessary to provide suitable mathematical expressions at the
boundaries to obtain the approximate solutions for the pertinent
equations using numerical techniques. These have been described
briefly in this section.
The no-slip, no penetration, and a constant wall temperature have
been prescribed on the surface of the funnel walls, i.e.
()
(11)
Where, the total heat transfer rate (Q) has been obtained from the
numerical computation.
Based on the characteristic length, i.e. the entire length of the
funnel, the mean value of Nusselt number for the conical- shaped IRS
has been defined mathematically as:
(6)
3.2. Boundary conditions
P = Patm and T = T
g TL3
Where, ΔT = Tw − T∞ represents the temperature difference between the IRS funnel wall and the ambient fluid. L is the characteristic
length scale, which has been taken as the total funnel length.
The average Nusselt number is essentially a way of representing the
average heat transfer coefficient.
The average heat transfer coefficient has been defined as follows.
Where, Rc represents the specific gas constant of air, i.e., the
working fluid for the present problem. Pop is the operating pressure,
which is specified to a value of 101.325 kPa during the simulation.
Therefore, one can notice from Eq. (6) that the density of air is a
function of the temperature and not the local pressure.
T = Tw and vx = vy = 0
(10)
Where, T∞is the ambient temperature.
A more elaborative discussion on the parameters involved in the
above equations, along with their suitable mathematical expressions
and the corresponding modeling procedure has been provided in our
previous article [34].
pop = R c T
1
(Twall + T )
2
dT
= hA (T
dt
T )
(15)
4. Numerical procedure
In the present computational study, a cell centroid based finite volume technique has been used. Hence, the governing partial differential
equations have been integrated over each control volume (including the
(9)
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A. Mohanty, et al.
control volumes associated with the boundaries) of the computational
domain to convert them into a set of linear algebraic equations by
writing them in flux form using the finite difference method. The resulting set of algebraic equations has been solved using the Gauss-Seidal
point by point iteration technique. Besides, the convergence behavior
has been improved by incorporating the Algebraic multi-grid (AMG)
solver. The central differencing scheme has been adopted to discretize
the diffusive terms in all equations. Likewise, the first-order upwind
scheme has been used for all the convective terms at the beginning of
the simulation. The first-order upwind scheme is stable by nature, and
stability is desirable at the beginning. On the attainment of the convergence with the first-order upwind scheme, the simulations have
further been carried out with the second-order upwind scheme until
convergence is attained to obtain more accurate solutions. The discretization of the pressure term in the momentum equation has been
carried out using the body force weighted scheme. The coupling between the pressure and the velocity fields have been established using
the SIMPLE algorithm. A predefined scaled residual of 10−4 has been
taken as one of the criteria for the convergence. The same for the energy
equation has been set to 10−6. To further ensure the convergence, the
variation of different quantities of interest, such as the temperature,
Nusselt number, etc. has been monitored at various locations within the
domain. As the solution marches towards the convergence, the difference in these quantities between two successive iterations reduces. On
the attainment of the convergence, the relative difference between the
two successive iterations falls below 0.001 %. Moreover, for the converged solution, the net imbalance of the mass and energy has been
taken to be less than 0.5 % of the smallest calculated flux.
method with the experimental findings of Eckert and Diaguila [35] for
the variation of the local Nusselt number with the Rayleigh number.
The comparisons, as shown in Fig. 2, reveal a reasonably good agreement between the experimental data and the present grid-independent
computations. Thus, the same numerical method has been adopted in
the present study to predict the cooling time of the IRS device with
conical funnels. The procedure adopted for the grid-independent study
has been described below.
It is to be noted that the present problem involves a turbulent natural convection situation, owing to which it is necessary to use a turbulence model that can suitably relate the mean velocity with the
fluctuating velocity of the induced flow field to yield physically realistic
results. In the literature there exist four turbulent models namely, the
standard k-ε model, the RNG k-ε model, the Realizable k-ε model, and
the SST k-ω model which have been frequently used for modeling
various turbulent flow situations. In the present study, the experimental
case of Eckert and Diaguila [35] has been simulated with all these
turbulent models taking one at a time to predict the local Nusselt
number. The comparisons of the model predictions have been shown in
Fig. 2(b). It is observed that with the SST k-ω model, there is a significant over prediction of the computed results. On the other side,
under-predictions in the computed results are observed with the RNG kε model and the Realizable k-ε model. It can be observed that the underpredictions in the computed results have been found to be appreciable
at higher Rayleigh numbers. However, there is hardly any difference in
the computed results obtained with any of these two models. The closest agreement between the numerical predictions and the experimental
findings have been achieved with the standard k-ε model with a maximum deviation of 7.02 %. Thus, in the present work, the standard k-ε
model has been chosen for the rest of the simulations. Such an approach
for choosing appropriate numerical model is very common in literature
and is applied for a wide range of problems [38–42].
Both the 2D axisymmetric domain and the 3D domain have been
used for validation purposes. It can be seen from Fig. 2 (c) that there are
no differences in the computed results with both these domains.
Therefore, the choice of the 2D axisymmetric domain, which has been
made in the present study to save the computational time, is further
4.1. Validation of the present numerical methodology
In the existing literature, the experimental studies on natural convection cooling of the IRS device appear to be missing despite its significant practical importance. Therefore, to validate the present numerical procedure, the experimental case for the turbulent natural
convection from vertical tubes has been taken as the benchmark. Fig. 2
depicts the comparison of the results obtained from our numerical
Fig. 2. Comparison between the present computations and the experimental findings of Eckert [33] for different (a) mesh size (b) turbulent models (c) domain.
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A. Mohanty, et al.
justified.
increases, the average velocity with which the fresh air enters into the
funnels also increases.
As a result, the strength of the induced flow field inside the funnels
increases. The entrained air moves upward inside the funnels while
carrying the heat from the inner surfaces and finally exits the IRS device. Thus, at higher Ra, relatively thinner plumes are observed signifying higher heat loss, whereas the plumes become thicker as the Ra
reduces; signifying smaller heat loss as depicted in Fig. 7.
Fig. 8 depicts the effect of the percentage of overlap (OL) on the
heat transfer rate at two different diameter ratios. As observed from
Fig. 8, the heat transfer rate is slightly more for the positive overlap of
the funnels and increases as the percentage overlap increases. As
mentioned earlier in the case of positive overlap, the top portion of one
funnel is placed into the bottom end of its neighborhood funnel present
above it. Thus, an annular space is formed for any two consecutive pairs
of funnels. The presence of an annular space creates an additional
suction effect. The suction effect tends to draw the air into the funnel
with a higher velocity. As a result, heat loss increases. An increase in the
percentage overlap intensifies the suction effect, which enhances the
suction velocity of the fresh air inside the device, causing the heat loss
to increase. In the case of the negative overlap, the funnels are at a
certain distance from each other. Therefore, in this case, for a consecutive pair of funnels, the bottom funnel always tends to create
gliding motion of the fresh air and direct it to the interior of the top
funnel. The gliding motion of air thus creates some sort of suction effect
in the case of the negative overlap also. However, this suction effect is
relatively weak as compared to the suction effect produced in the case
of positive overlap. Therefore, due to a weaker suction effect in the case
of negative overlap, the velocity with which the air drives into the
funnels is very less, causing a smaller heat loss.
The same can also be visualized from the vector plot shown in Fig. 9
(a, b). It is observed that the backflow flow tendency at the entrance to
the second funnel is relatively high in case of the negative overlap. As a
consequence, a relatively higher suction effect is observed with the
positive overlap of the funnels. Besides, the thermal plume at the entrance to the second funnel is relatively thicker in the case of negative
overlap (Fig. 9(c, d)), which further confirms to the present finding that
the heat transfer is higher for the positive overlap of the funnels.
Besides, as the extent of negative overlap increases, the distance
between the consecutive funnels also increases. Hence, the suction effect produced by the negative overlap becomes even weaker. Therefore,
the heat transfer rate gets minimized with the increase in the negative
overlap. Thus, the heat transfer rate is highest with positive overlap of
the funnels followed by the zero-overlap case, and the lowest values are
obtained in case of the negative overlap of the funnels. A comparison of
Fig. 8 (a) and (b) reveals that as the diameter ratio increases, the effect
of funnel overlap becomes less prominent. It appears that at a higher
4.2. Mesh and domain size sensitivity tests
The distribution of the grid in the computational domain has been
shown in Fig. 3. For the near-wall region in the radial direction and the
region from the inlet of the funnel to the exit of the fourth funnel in the
axial direction, sufficiently fine cells have been used to capture the
near-wall turbulence, the flow pattern and the thermal plumes more
accurately. On the other side, the cells in the outer domain have been
kept relatively coarse.
Seven grids of different sizes have been generated using the ANSYS
meshing tool for the mesh refinement test. The average Nusselt number
has been calculated with each of these grids under identical conditions.
It can be seen from Fig. 4(a) and Table 2 that the average Nusselt
number varies with grid size up to Grid-5. However, beyond Grid-5, it is
found to be independent of the size of the grid. Therefore, Grid-6 with
4.5 × 105 cells has been taken as the optimum grid size. The same
procedure has been followed to obtain the optimum grid size for each
geometry of the IRS device considered in the present analysis.
Likewise, a domain-independent test has been performed to obtain
an optimum domain size beyond which the numerical predictions no
longer remain a function of it. To do the same, 6 different domains have
been taken by varying the length in the axial direction. The simulations
have been carried out with each of the domains separately for the same
operating conditions. From Fig. 4(b) and Table 3 it can be observed that
the domain independence is achieved when the domain length is taken
almost eight times the length of the IRS device.
5. Results and discussion
In this section, the results obtained from the present computational
study have been presented sequentially. The section has been divided
into three subsections. In the first portion, a parametric study has been
performed to predict the sole effect of the three primary parameters,
namely, the diameter ratio, the percentage overlap, and the Rayleigh
number on the heat transfer rate, the Nusselt number, and the induced
mass flow rates through the funnels. More insight into the physical
phenomena has been provided with the aid of the velocity vectors and
the temperature contours. In the second portion, a correlation has been
proposed for the average Nusselt number as a function of the diameter
ratio, the percentage overlap, and the Rayleigh number. In the third
portion, the same correlation has been used to compute the cooling
time of the IRS device based on the lumped parameter analysis. Besides,
the effects of the pertinent parameters on the cooling time have also
been explained. Moreover, the cooling curves of the conical funnels
have been compared with that of the cylindrical ones.
5.1. Effect of diameter ratio and funnel overlap on the heat transfer
Fig. 5 depicts the role of the diameter ratio (DR) on the heat transfer
rate from the hot vertical funnels of the IRS device at different Rayleigh
number. The diameter ratio has been considered in the range of 1.02 to
1.3. Fig. 5 depicts an augmentation in the heat transfer rate with an
increase in the diameter ratio irrespective of the funnel overlap and the
Rayleigh numbers. An increase in the diameter ratio causes an increase
in the surface area of the funnels. An increase in the surface area of the
funnels leads to more heat loss.
Moreover, Fig. 5 also depicts an augmentation in the heat transfer
rate with an increase in the Rayleigh number. It is essential to analyze
the induced flow field to understand the effect of the Rayleigh number
on the heat transfer rate. Fig. 6 shows the velocity vectors at different
Rayleigh numbers. It can be observed, as the Rayleigh number increases, the funnels of the IRS device come in contact with more
amount of fresh air from the immediate neighborhood regions. Thus,
more amount of fresh air drives into the funnels. Therefore, as the Ra
Fig. 3. Arrangement of the cells over the funnel walls in the computational
domain.
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A. Mohanty, et al.
Fig. 4. Nusselt number variation with (a) grid size (b) domain size.
Table 2
Sensitivity of the average Nu on the grid size.
Number of cells
2 × 105 (Grid-1)
2.5 × 105
(Grid-2)
3 × 105
(Grid-3)
3.5 × 105
(Grid-4)
4 × 105
(Grid-5)
4.5 × 105
(Grid-6)
6.75 × 105
(Grid-7)
Nu
Relative differences (In Percentage)
564.3
–
531.3
5.85
503.2
5.30
493.1
2.0
489.2
0.79
488.6
0.12
488.4
0.04
diameter ratio, the heat transfer rate mainly takes place due to the
increase in the surface area of the funnels and so the effect of the
overlap is not felt significantly.
Table 3
Sensitivity of the average Nu on the domain size.
Domain size
(in L)
1.5
3.5
5.5
7.5
10.5
13.5
Nu
Relative differences (In
Percentage)
564.3
–
531.3
3.63
503.2
3.45
493.1
3.40
489.2
0.24
488.6
0.021
5.2. Effect of diameter ratio and funnel overlap on Nusselt number
Fig. 10 shows the effects of the diameter ratio on the Nusselt
number for different percentage overlap of the funnels. It is observed
that the Nusselt number increases with an increase in the diameter
ratio, attains a peak, and then reduces to attain a constant value for all
Fig. 5. Variation of heat transfer rate with diameter ratio at different Rayleigh number and percentage overlap.
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Fig. 6. Vector plots for the IRS device with conical funnels at different Rayleigh numbers for DR = 1.1.
Fig. 7. Temperature contours for the IRS device with conical funnels at different Rayleigh numbers for DR = 1.1.
Fig. 8. Variation of heat transfer rate with funnel overlap at different Rayleigh number and diameter ratio.
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A. Mohanty, et al.
Fig. 9. Velocity vectors and Temperature contours closer to the inlet of second funnel at DR = 1.05 and Ra = 2 × 1011.
Fig. 10. Nusselt number variation with DR for different Rayleigh number and different funnel overlap.
types of funnel overlap. At a smaller diameter ratio, the gap between
the successive funnels is too small and a relatively hot zone of air
prevails, which fails to create an adequate suction of the ambient air
into the funnels. As the diameter ratio increases, the gap (the annular
space) between the funnels becomes larger, which reduces the hindrance to the entrainment of the ambient air. Therefore, the suction
effect becomes stronger. Entrainment of air with higher velocity
strengthens the natural convection current, and the Nusselt number
starts rising. However, beyond a critical value of the diameter ratio, the
funnel gap becomes so large that the suction effect becomes weaker.
Therefore, the Nusselt number decreases and finally attains a nearly
constant value. In the present study, the critical diameter is about 1.1,
at which the gap is optimum to produce a maximum suction effect. In
the case of the positive overlap, the increase or the decrease of the
Nusselt number before and after the critical diameter is sharper (a
significant peak in the Nusselt number is observed) compared to the
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the effect of the percentage overlap on the dimensionless induced mass
flow rate. It is observed that as the percentage overlap increases, relatively lesser mass of air enters into the funnels. In other words, more
amount of air enters into the funnels with the negative overlap of the
funnels, followed by the zero overlap of the funnels and finally for the
positive overlap of the funnels. The observation is found to be true both
at smaller and larger diameter ratios. In the case of positive overlap of
the funnels, although the stronger suction effect causes the air velocity
to increase, at the same time, however, the flow area also reduces
significantly. As a consequence, relatively less amount of air gets inducted into the funnels for the positive overlap, which reduces further
as the percentage of overlap increases.
6. Correlations for different parameters of interest
6.1. The correlation for the Nusselt number
Fig. 11. A detail view of velocity vectors closer to the inlet of second funnel for
different DR at Ra = 2 × 1011.
Finally, a correlation has been developed for the Nu in terms of Ra,
DR, and OL using the results of 190 simulations. The same has been
represented in Eq. (16). A non-linear regression has been adopted for
this purpose. The value of R2 for the developed correlation is 0.9967. It
can be observed from Fig. 14 that the calculated values of the Nu from
the numerical simulation and the predicted values from the correlation
agree reasonably well with an accuracy of ± 8%.
The developed correlation is:
other overlap cases. This may be attributed to the presence of a relatively stronger suction effect in the case of the positive overlap of the
funnels. Fig. 11 depicts the velocity vectors of the induced flow field at
different diameter ratios. Fig. 11 (a) depicts the presence of a negligible
backflow at the entrance to the funnels for DR = 1.1 (critical value).
However, as the diameter ratio increases, an increase in the backflow at
the entrance is observed (Fig. 11(a) to (c)). Therefore, beyond the critical diameter, the ambient air gets more resistance at the entrance of
the funnel, which tends to reduce the suction velocity of the ambient
air. As a result, beyond the critical diameter, the Nusselt number reduces.
(16)
Nu = C0 + C1 (Ra)n1 (DR)n2 (C2 + (OL)n3)
Where,
C0 = 60.16, C1 = 0.0357, n1 = 0.346, n2 =
0.1081, C2 = 1.1032, n3
(17)
= 3.48
The above correlation is valid under the following range of parameters.
5.3. Evaluation of induced mass flow rate as a function of DR and OL at
different Rayleigh numbers
1010
The effects of the relevant parameters such as DR (diameter ratio),
OL (Percentage of funnel overlap), and Ra (Rayleigh number) on induced mass flow rate (due to suction caused by natural convection,
through the funnels of Conical IRS) have been described in the present
section. Fig. 12 shows the effect of the diameter ratio on the induced
mass flow rate (non-dimensional form) through the funnels at different
Rayleigh numbers. It is observed that as the DR increases, more air
enters into the funnels. As pointed out in the earlier section that an
increase in the diameter ratio causes a reduction in the suction velocity
of the ambient air. However, an increase in the DR, on the other hand,
causes an increase in the flow area.
As a consequence, more air enters into the funnels at a larger diameter ratio; however, it moves with a smaller velocity. Fig. 13 shows
Ra
1012; 1.02
DR
1.3;
20%
OL
20%
(18)
6.2. Correlation for induced mass flow rate
Based on the present simulation results, a correlation for the dimensionless induced mass flow rate (M) has also been developed, and
the corresponding expression is
M = C0 + C1 (Ra)n1 (DR)n2 (C2 + (OL)n3) + C3 (Ra)n4
c0 =
(19)
5812.6; c1 = 0.068; n1 = 0.3982; n2 = 3.1874; c2 = 9.875; n3
= 0.2376
c3 = 0.4212; n4 = 0.3948
(20)
The parameter range within which the above correlation is valid are
Fig. 12. Effect of Rayleigh number and percentage overlap on mass suction for different diameter ratios.
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Fig. 13. Effect of funnel overlap on dimensionless mass suction at different Rayleigh number and diameter ratio.
as follows.
1010
Ra
1012; 1.02
DR
1.3;
20%
OL
20%
(21)
For Eq. (19), the value of R2 is 0.9916. It can be seen from Fig. 15
that the predicted values and the computed values of the mass suction
rate have an excellent agreement within the accuracy of ± 8%.
6.3. Cooling time prediction for the IRS device with the conical funnels
To calculate the cooling time of the IRS, Eq. (15) has been solved.
For solving Eq. (15), one needs to know the heat transfer coefficient for
a typical situation beforehand.
The same can be calculated from the correlation for the Nusselt
number presented in Eq. (16). Fig. 16(a) shows the cooling time curve
for the IRS for different diameter ratios of the funnel, whereas, Fig. 16
(b) shows the same for different percentage of overlap of the funnels. It
can be observed From Fig. 16 that as the diameter ratio reduces and the
percentage overlap increases, a slight reduction in the cooling time of
the IRS is observed. As discussed earlier, an increase in the DR beyond
1.1 and a reduction in the percentage overlap reduces the strength of
the natural convection. The same is also reflected in the cooling time
curve.
Fig. 14. Comparison of the predicted and the computed values of Nusselt
number.
6.4. Comparisons of the performance of the cylindrical and conical shapes
of the funnels on the cooling characteristics
In the previous investigation [34], the present authors have estimated the cooling time of an IRS with cylindrical funnels. Therefore, in
the present manuscript it is worth comparing the performance of conical funnels with that of the cylindrical funnels in terms of the cooling
characteristics. Fig. 17 (a) shows the comparison of Nusselt number of
Fig. 15. Comparison of the predicted and the computed values of the mass
suction rate (M) into the conical funnel setup.
Fig. 16. Cooling time curve for IRS device with conical funnels at (a) different diameter ratios and (b) percentage overlap of the funnels.
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Fig. 17. Comparison of the performance of the cylindrical and conical funnels in terms of (a) Nusselt number (b) cooling time.
the cylindrical funnels with that of the conical funnels. It is observed
that at a particular Ra, the Nusselt number is higher for the IRS with the
cylindrical funnels compared to that of the conical funnels.
The above finding can be explained from the vector plots for both
the cylindrical and conical funnels, as shown in Fig. 18. For the sake of
brevity, the vector plots near to the entrance of the second funnels have
been shown. A close observation of Fig. 18 depicts a higher suction
velocity for the cylindrical funnels, compared to the conical arrangement of funnels, which leads to higher Nu for the cylindrical funnel set
up. As a consequence, the cylindrical funnels take relatively less time to
reach the ambient temperature as depicted in Fig. 17 (b).
Nusselt number, the heat transfer rate, and the induced mass flow rate.
Based on the computational study the following major conclusions are
made.
1. An increase in the diameter ratio augments the heat transfer from
the conical funnels; however, the Nusselt number initially increases
till a maximum value is obtained, then reduces and finally reaches a
constant value. Therefore, at the higher range of the diameter ratio
(almost DR ≥ 1.15), the Nusselt number ceases to be a function of
the diameter ratio.
2. An increase in the Rayleigh number and the diameter ratio causes
more air entrainment into the funnels. In other words, the induced
mass flow rate gets augmented for all ranges of percentages overlap.
3. As the extent of negative overlap increases the suction effect gets
suppressed, which further causes a reduction in the heat transfer
rate.
4. For the diameter ratio (DR) beyond 1.1, the Nusselt number remains
almost constant for all ranges of overlaps considered in the present
study. Because of this, the cooling rate of the IRS device at higher
diameter ratios does not change much.
5. In the present study, the correlations developed for the Nu and mass
7. Conclusions
In the present work, a computational study was undertaken to study
the natural convection cooling of an IRS suppression device with conical funnels. The numerical procedure was thoroughly validated with
the benchmark experimental data and satisfactory agreement was obtained. The effect of the three major parameters namely, the Rayleigh
number(1010 ≤ Ra ≤ 1012), the diameter ratio (1.02 ≤ DR ≤ 1.3) and
the percentage overlap (−20 % ≤ OL ≤ 20%)were studied on the
Fig. 18. Comparison of the vector plots closer to the inlet of second funnel for the IRS with (a) cylindrical funnel (b) Conical funnel at DR = 1.05, OL = 20% and
Ra = 2 × 1011.
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A. Mohanty, et al.
suction rate for the IRS device with conical funnels have an accuracy
of ± 8%. These correlations bear significant importance to both the
practicing engineers of the shipping industry and the academicians
for calculating the cooling time.
6. The cylindrical funnels show higher suction effect compared to the
conical funnels. Therefore, higher Nu is observed for the cylindrical
funnels. Moreover, an IRS with the cylindrical funnels cools faster
than an IRS with the conical funnels.
[19] M.K. Dash, S.K. Dash, 3D numerical study of natural convection heat transfer from a
hollow horizontal cylinder placed on the ground, Int. J. Therm. Sci. 140 (2019)
429–441, https://doi.org/10.1016/j.ijthermalsci.2019.03.015.
[20] S. Acharya, S.K. Dash, Turbulent natural convection heat transfer from a vertical
hollow cylinder suspended in air: a numerical approach, Therm. Sci. Eng. Prog. 15
(2020) 100449, https://doi.org/10.1016/j.tsep.2019.100449.
[21] S. Acharya, S.K. Dash, Natural convection heat transfer from perforated hollow
cylinder with inline and staggered holes, J. Heat Transf. 140 (2018), https://doi.
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[22] B. Singh, S.K. Dash, Natural convection heat transfer from a finned sphere, Int. J.
Heat Mass Transf. 81 (2015) 305–324, https://doi.org/10.1016/j.
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enclosures for engineering applications. the particular case of the parallelogrammic
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[25] H.F. Öztop, P. Estellé, W.M. Yan, K. Al-Salem, J. Orfi, O. Mahian, A brief review of
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[27] M. Ahmadi, G. Mostafavi, M. Bahrami, Natural convection from rectangular interrupted fins, Int. J. Therm. Sci. 82 (2014) 62–71, https://doi.org/10.1016/j.
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[28] K. Zhang, M.J. Li, F.L. Wang, Y.L. He, Experimental and numerical investigation of
natural convection heat transfer of W-type fin arrays, Int. J. Heat Mass Transf. 152
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[29] V. Karlapalem, S. Rath, S.K. Dash, Orientation effects on laminar natural convection
heat transfer from branching-fins, Int. J. Therm. Sci. 142 (2019) 89–105, https://
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[30] J.R. Senapati, S.K. Dash, S. Roy, Numerical investigation of natural convection heat
transfer over annular finned horizontal cylinder, Int. J. Heat Mass Transf. 96 (2016)
330–345, https://doi.org/10.1016/j.ijheatmasstransfer.2016.01.024.
[31] J.R. Senapati, S.K. Dash, S. Roy, Numerical investigation of natural convection heat
transfer from vertical cylinder with annular fins, Int. J. Therm. Sci. 111 (2017)
146–159, https://doi.org/10.1016/j.ijthermalsci.2016.08.019.
[32] S. Acharya, S.K. Dash, Natural convection heat transfer from a horizontal hollow
cylinder with internal longitudinal fins, Int. J. Therm. Sci. 134 (2018) 40–53,
https://doi.org/10.1016/j.ijthermalsci.2018.07.039.
[33] A. Gupta, S. Gautam, A. Sahoo, A. Mohanty, Performance evaluation of porous fin
with prescribed tip temperature: an analytical and numerical approach, Int. J. Heat
Mass Transf. 156 (2020) 1–16, https://doi.org/10.1016/j.ijheatmasstransfer.2020.
119736 119736.
[34] A. Mohanty, S.K. Dash, S. Roy, Natural convection cooling of an infrared suppression (IRS) device with cylindrical funnels, Int. J. Therm. Sci. 141 (2019) 103–113,
https://doi.org/10.1016/j.ijthermalsci.2019.03.032.
[35] E.R.G. Eckert, D. A.J, Investigation of Free-Convection Heat Transfer in Vertical,
(1952).
[36] J.R. Senapati, S.K. Dash, S. Roy, 3D numerical study of the effect of eccentricity on
heat transfer characteristics over horizontal cylinder fitted with annular fins, Int. J.
Therm. Sci. 108 (2016) 28–39, https://doi.org/10.1016/j.ijthermalsci.2016.04.
021.
[37] ANSYS Inc ( Release-15.0), ANSYS Fluent Theory Guide, (2015).
[38] S.K. Senapati, S.K. Dash, Computation of pressure drop and heat transfer in gassolid suspension with small sized particles in a horizontal pipe, Part. Sci. Technol.
(2019) 1–14, https://doi.org/10.1080/02726351.2019.1649335.
[39] S.K. Senapati, S.K. Dash, Pressure recovery and acceleration length of gas-solid
suspension in an abrupt expansion-an Eulerian-Eulerian approach, Chem. Eng. Sci.
226 (2020) 1–20, https://doi.org/10.1016/j.ces.2020.115820 115820.
[40] S.K. Senapati, S.K. Dash, Computation of pressure drop for dilute gas–solid suspension across thin and thick orifices, Particuology. (2020) 1–13, https://doi.org/
10.1016/j.partic.2020.07.003.
[41] S.K. Senapati, S.K. Dash, Pressure recovery characteristics for a dilute gas–particle
suspension flowing vertically downward in a pipe with a sudden expansion: a twofluid modeling approach, Particuology. 50 (2020) 100–111, https://doi.org/10.
1016/j.partic.2019.07.001.
[42] S.K. Senapati, S.K. Dewangan, Comparison of performance of different multiphase
models in predicting stratified flow, Comput. Therm. Sci. 9 (2017) 1–11, https://
doi.org/10.1615/ComputThermalScien.2017017248.
Declaration of Competing Interest
The authors declare that they have no known competing financial
interests or personal relationships that could have appeared to influence the work reported in this paper.
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