Eko Siswanto, Denny Widhiyanuriyawan, Mochammad Agus Choiron, Djarot Bangun Darmadi, Yasuo Katoh

Corresponding email: eko_s112@ub.ac.id

Corresponding email: eko_s112@ub.ac.id

**Published at : ** 18 Jan 2023

**Volume :** **IJtech**
Vol 14, No 1 (2023)

**DOI :** https://doi.org/10.14716/ijtech.v14i1.3660

Siswanto, E., Widhiyanuriyawan, D., Choiron, M.A., Darmadi, D.B., Katoh, Y., 2023. The Effects of Effective Thermal Conductivity of Porous Materials Under Vapor Flow in Sudden Enlargement-Contraction Channel on Local Heat Transfer.

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Eko Siswanto | Mechanical Engineering Departmen,t Faculty of Engineering, Brawijaya University, Jalan MT. Haryono No. 167, Malang 65145, Indonesia |

Denny Widhiyanuriyawan | Mechanical Engineering Departmen,t Faculty of Engineering, Brawijaya University, Jalan MT. Haryono No. 167, Malang 65145, Indonesia |

Mochammad Agus Choiron | Mechanical Engineering Departmen,t Faculty of Engineering, Brawijaya University, Jalan MT. Haryono No. 167, Malang 65145, Indonesia |

Djarot Bangun Darmadi | |

Yasuo Katoh | Mechanical Engineering Department, System Design and Engineering, Yamaguchi University, Tokiwadai 2-16-1, Yamaguchi 7558611, Japan |

Abstract

This study aims to obtain the local heat
penetration and local heat convection characteristics in a system where a flow
of heated vapour cedes thermal energy to a horizontal heat-sink plate. The heat penetrates
a bed of porous materials with variations in the thermal conductivity inside a
sudden-enlargement-contraction-channel. The two non-dimensional parameters of
interest are the local Nusselt and Metais-Eckert numbers. The solid particles
used for the porous bed are copper, carbon steel, and ceramics. The study is
conducted numerically via the software package Ansys-Fluent to solve the
Navier-Stokes equation for the conservation of mass, momentum, and energy to
obtain the profiles of local temperature and velocity, both in the porous bed
and in the vapour stream. Results of this study show that the overall
effective-thermal-conductivity of porous materials filled with vapour mainly
affects the local Nusselt number. The local Nusselt number increases with
increasing overall effective thermal conductivity of porous materials. The
sudden enlargement-contraction of the channel affects the local Metais-Eckert
number. This finding is proven by the average Metais-Eckert number's variation
along the duct and by its non-proportionality to the
overall-effective-thermal-conductivity of each porous material. Together with
the local Reynolds number, the local Metais-Eckert number describes the flow
regimes that locally occur in the system, namely, transitional-combined-convection,
laminar-combined-convection, and laminar-forced-convection. Additionally, based
on the local Nusselts number, this study locates the critical point of change
from enlargement-affected-zone to contraction-affected-zone at 80 mm along the
duct's axis, which means that the switchover-point is not at the centre of the
channel.

Effective thermal conductivity; Metais-Eckert number; Porous materials bed; Sudden enlargement-contraction channel

Introduction

Because of their
importance in industrial applications, research and development projects on
porous materials are carried out. Among the most notable research works in the
literature is the paper by Zulkarnain, Sharudin, and Ohshima (2022). They show that polymer foams
have superior properties in low density by using, as
basic material, *et al.* (2018) researched porous
adsorbents used for natural gas adsorption in gas storage technologies.

In
line with the development of science and technology on heat transfer, the usage
of porous or permeable materials has replaced the extended solid slab method (Kundu *et al.*, 2012; Bassam
and Hijleh, 2003; Kiwan and Al-Nimr, 2001).
The
idea behind the slab method of increasing the heat transfer by expanding the
surface area available to the heat transfer through solid slabs finds its
limitation in space availability. In contrast, a porous material provides a
much bigger surface area for heat transfer.

A good
knowledge of a system's local heat transfer characteristics is required to
create a more accurate design and better control of heat release from a heat
source to a heat sink. This is the case, for example, in heat exchangers at the
inlet and outlet sections or in a heat releaser's regions where sudden changes
in shape occur. In a porous media system, heat penetration and heat convection
play a dominant role in heat transport (Siswanto, Katsurayama, and Katoh, 2011a). Therefore,
the present study focuses on determining the characteristics of local heat
penetration and local heat convection in a system where a heated vapour moves
inside a sudden-enlargement-contraction-channel and transfers its thermal
energy to a horizontal heat-sink plate penetrating a bed of porous materials.
The study also accounts for variations in the porous material's thermal
conductivity.

Porous materials have porous and capillary holes
scattered throughout a solid volume, resulting in cavities where flows of mass
and energy can occur. Slabs made from porous materials have a higher ratio of
surface area to volume than those built out of a continuum solid. Several
experimental and numerical studies on fluids used as heat sources flowing
tangentially over porous material bed layers exist in the literature. Among the
experimental studies are worth mentioning those by Nagata *et al.* (2013), Siswanto, Katsurayama, and Katoh (2011b),
Siswanto, Katsurayama, and Katoh (2010) where
the heat transfer on laminar condensation using a bed of porous materials was
indagated. More specifically, in the experiments carried out, the authors did
flow a saturated vapour stream over glass beads and alumina balls with
different thermal conductivity and wettability to obtain the characteristic
curves of condensate-propagation and heat-flow that penetrates the porous
materials bed in a channel. Results allowed the authors to understand how the
propagation characteristics of the condensate flow are affected by the
materials' wettability. Propagation can be linear dominant, nonlinear dominant,
or chaotic. However, this level of information about the heat flow
characteristics in the system is still inadequate. The heat flow
characterization made in terms of its Jacob or Kutaleladze numbers or via some
perturbation parameters like those determined by Kaviany (Siswanto, Katsurayama, and Katoh, 2011a)
is still based on some non-dimensional numbers generally
defined by differences between the average temperature values along the
channel.

The studies mentioned above on heat transfer with
saturated vapour (Siswanto *et al.*, 2016; Nagata *et al.*, 2013; Siswanto, Katsurayama, and Katoh, 2011a; Siswanto, Katsurayama, and Katoh, 2010) report
Jacob number values of less than 100. This finding means that, although the
propagation of condensate into the porous media is affected by the wettability
of materials, the condensation from saturated vapour to liquid is predominantly
controlled by heat transfer rather than by viscous inertia (Siswanto, Katsurayama, and Katoh, 2011a).
Those studies though, do not present the effect and contribution of the
effective-thermal-conductivity on the local heat transfer. Hence, this study
focuses on the effect of the materials' effective-thermal-conductivities
(instead of wettability) on the local heat transfer in a channel where a stream
of saturated vapour flows over a bed made from various porous materials

Yamaguchi, Katoh, and Kurima (2007) and Katoh *et al.* (2007) measured the
effective thermal conductivity of porous materials in the presence of two
different thermodynamic states of the fluid flowing through the voids inside
two thicknesses of a layer of porous material. Based on Yamaguchi's work, the
measurements of effective-thermal-conductivity of porous material in this study
are correlated as in equations (1), (2), and (3):

The term in expression (1) is the
effective thermal conductivity of the average thickness of the bed upper layer

To better explore the influence of the bed material thermal conductivity on results, we have selected for the present study three materials which cover a broad spectrum of thermal conductivity values, namely copper with k_{Cu}=385 (W/mK), carbon steel with k_{cs}=43 (W/mK), and ceramics with k_{cer}=1.298 (W/mK). The selected materials represent high, medium, and low thermal conductivities, respectively. The choice of these materials is made to obtain clear information on the effect of thermal conductivity and does not involve changes in wettability, as in already cited previous studies, which made use of materials with only slightly different values of thermal conductivity, (i.e., glass beads with k_{gb}=1.035 (W/mK) and alumina balls with k_{ab}=18.84 (W/mK)).

To
obtain details on the local heat transfer characteristics along the channel,
for the porous bed upper surface and the inside (i.e., void) of the porous
material bed the overall effective thermal conductivity k ef is used. In the
present study, the Navier-Stokes system of equations for the conservation of
mass, momentum, and energy in the saturated vapour flow is solved. With this
approach, an ideally infinite number of observation points can be obtained,
with detailed information such as local temperatures, local pressures, local
velocities, and some local properties of the saturated vapour for a variety of
pressure and temperature values.

Experimental Methods

*2.1. Materials,
Procedures, and Test section*

This investigation uses three
different porous beds made of copper (*Cu*), carbon steel (*cs*), and
ceramics (*cer*) particles with thermal conductivity _{Cu}= 385 (W/mK), _{cs}43 (W/mK) and k_{cer} = 1.298 (W/mK), respectively. The
thermal conductivity of the saturated vapour ^{-3} (m). The porosity

The temperature ^{-3} (m) in length *L*, 20x10^{-3} (m)
in width *W*, and 40x10^{-3} (m) in height *H*. The size of
the porous bed that occupies the chamber's volume is 240x10^{-3} (m) in
length *l*, 20x10^{-3} (m) in width *w*, and 20x10^{-3}
(m) in height *z*. The saturated vapour enters the chamber with an inlet
velocity ^{-3} (m). The vapour exits the chamber through a hole of
diameter ^{-3} (m).

**Figure 1** Test section

As
previously stated, the saturated vapour flow is subjected to a sudden expansion
as it enters the chamber.

Figure 1
depicts the installation of the porous bed, the channel of saturated vapour,
and the chamber of the test section. The same figure shows the glass-made test
section walls (with a thickness ^{-3} (m)) in
transparent colour. The light green colour refers to the duct channels of
saturated vapour, while the grey colour indicates the porous bed. The dark
green colour is the contact region between the vapour stream and the porous
bed's upper surface. The copper plate is in contact with the porous bed's
bottom face (not visible in the figure). Finally, Figure 2 reports the
flowchart of the model of this current study.

*2.2. Computation, Governing
Equations, and Flowchart of Model** *

The software ANSYS Workbench 18.1 (Academic Research license) is used to
compute the local temperature gradient, local property parameters, and heat
transfer both in the porous materials bed and the saturated vapour stream.
Brick-type 8-node-elements are used for meshing the porous materials bed, the
volume occupied by the saturated vapour, the chamber walls, and the bottom of
the copper plate (ANSYS
Inc., 2017). The CONTA174 (ANSYS
Inc., 2017) allows us to define the contact and sliding regions between the saturated
vapour stream and the porous bed upper surface, between walls and vapour,
between walls and porous material bed, and between the porous bed and copper
plate. This method allows for an analysis of the coupled field at the contact
region. Figure 1 also shows the contact region between the vapour and the upper
surface of the porous bed.

This ANSYS solves the Navier-Stokes conservation equation for mass,
momentum, and energy in the saturated vapour flow with heat transfer. The
following paragraphs give a brief review of those equations.

Under the hypothesis that no other mass is added to the system, the mass
conservation equation for the compressible saturated vapour writes (ANSYS Inc., 2017; Welty *et
al.*, 2008),

** **

**Figure 2** Flowchart of model

Under the assumption that no heat source exists in the test section
other than the heated saturated vapour, the energy conservation equation in
terms of sensible enthalpy

where

Finally, Figure 2 shows the model’s flowchart for the
current study.

Results and Discussion

*3.1.
**Local Nusselts
Numbers *

The local Nusselt
number Nu_{x}

Where _{x }are those valid for transitional
and laminar flows.

For the transitional flow, local Nusselt numbers Nu_{x }

The average
Nusselts number

The Nu_{x}

Where _{wx}_{x}

Finally, the average local Nusselts number

here,

Figure 3 shows the
computed local

Regarding the
values of

Furthermore,
one can drive an important piece of information from the local Nusselt numbers

We can further
validate the simulation results by comparing the average local Nusselt numbers

**Figure 3** The local Nusselt
number

*3.2.
**Metais-Eckert
Numbers *

An experiment from
Metais and Eckert provides the value of the Metais-Eckert numbers (*ME*)
by measuring the magnitude of the vapour's convection heat rate against the
length of the channel. The study also correlates the local Metais-Eckert number
values

The _{x} values along the channel is visible. To simplify the discussion on the

Analysis of Figure
4 shows that inside area A the porous material with the highest *ME*_{x }*ME*_{x }suggest that, in the convective heat transfer over the porous materials,
the role of the porous materials' overall effective thermal conductivity is negligible. This statement is
proven by the fact that the *ME*_{x }*ME*_{x }

**Figure
4 **Local
Metais-Eckert numbers

With reference to
equation (16) and under the assumption that *D _{H}*

Therefore, by assuming constant the gravity
acceleration *g _{x }*

*3.3.
**Convection and
Flow Regimes *

In
this part of the study, mapping of the convective flow regimes based on the
value of local Metais-Eckert numbers *ME _{x}*

Local convection exists along
the first 40 (mm) in the channel, i.e., evaluated at *x*=20 (mm) and *x*=40
(mm). For all types of porous materials, the local Reynolds number range is and the local Metais-Eckert number range is As for Metais's and Eckert's experiment (Holman, 2010), if

**Figure 5** Mapping of
laminar forced convection, laminar combined convection, and transition
combined convection regimes based on Metais and Eckert

Moving deeper
inside the channel and at the axial coordinate ranges of
local Reynolds and Metais-Eckert numbers are 437.22464.78 and 8,715.32 19,305.57 respectively. As per the references (Holman, 2010), if

Furthermore, a
non-uniform local convection regime occurs in the channel at x = 220 (mm). At
this point, the values of for porous copper and ceramic materials are (18,458.33; 440.04) and
(14,288.95; 465.61), respectively. This finding puts the local convection for
porous-copper and porous ceramics materials in the combined convection regime
with laminar flow due to

Finally, we can summarize the results of the present study. Table 1 presents a recapitulation of what has been found in the course of the study.

**Table 1** Recapitulation of results obtained in the
present study

Conclusion

Previous
results and discussion allow us to establish relations between the effective
thermal conductivity of porous materials under saturated vapour in a sudden
enlargement-contraction channel with the local Nusselts numbers, the local
Metais-Eckert numbers, and the local convection flows regimes. The local
Nusselt numbers in the channel are
affected by the porous material overall-effective-thermal-conductivity , where the term can also represent the heat penetrability of
the porous materials. Variations of the local Metais-Eckert numbers *ME _{x }*are mostly influenced by the local vapour
kinematic viscosity

Acknowledgement

The
authors are obliged to thank the Brawijaya University, Malang, Indonesia, for
providing the laboratory facilities, ANSYS Academic Research license software
and the Ministry of Research, Technology, and Higher Education, Indonesia, via
the Engineering Faculty of Brawijaya University, for the financial support to
the present research under the grant number 134/UN10.F07/PN/2018.

The
authors would also like to thank all the members of the porous media team, with
particular mention to Rizki Ferdiantara, for their fundamental role in
providing data.

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