Temperature Distribution and Heat Transfer in Narrow Channel
during Thermoacoustic Oscillation
TADASHI WATANABE
Research Institute of Nuclear Engineering,
University of Fukui,
Kanawa-cho 1-3-33, Tsuruga-shi, Fukui-ken 914-0055,
JAPAN
Abstract: - Thermoacoustic oscillation is induced when a gas is in a tube with a narrow channel section that has
a large temperature gradient. Although the oscillation in the tube section was studied, that in the narrow
channel has not yet been discussed well. The temperature distribution and the heat transfer in the narrow
channel section, which are the most important parameters for the design of thermoacoustic devices,
are discussed in this study. Numerical calculations of fluid conservation equations are performed, and the
onset of oscillation is obtained by gradually increasing the temperature gradient. The minimum onset
temperature ratio is shown to agree with the existing analytical and experimental results. The average wave
speed is in between the isothermal sound speed for the highest temperature and the adiabatic sound speed for
the lowest temperature. The distribution of oscillating temperature in the narrow channel is found to be
different and reversed from that in the tube. It is shown in the narrow channel that the heat transfer is smaller
for larger diameter and larger for smaller diameter and the heat transfer coefficient or the heat transfer model
should be improved for the oscillating flow.
Key-Words: - Thermoacoustic oscillation, Onset temperature ratio, Temperature distribution, Heat transfer,
Narrow channel
Received: June 9, 2022. Revised: March 19, 2023. Accepted: May 15, 2022. Published: June 29, 2023.
1 Introduction
When a gas is in a tube with a narrow channel
section that has a large temperature gradient,
spontaneous gas oscillation known as
thermoacoustic oscillation is induced, [1]. The
thermoacoustic oscillation is caused by the
irreversible heat exchange between the gas and the
channel wall, [2], and the thermal energy is
converted to the kinetic energy of gas oscillation.
Experimental and analytical studies were performed,
and the early-stage results were reviewed, [3], [4],
[5], [6]. Since the temperature gradient or difference
alone is needed for thermoacoustic devices, the
pressure and temperature conditions, gas type, tube
design, and so on have been studied thereafter
experimentally, [7], [8], [9], [10], [11], [12], [13],
and analytically, [12], [14], [15], [16], [17], [18],
[19], [20], [21]. The applications for thermoacoustic
refrigerators, [22], and electric power generation,
[23], were reviewed as typical examples. The
practical applications for power generation were
proposed such as electric generators for rural
communities, [24], [25], a generator utilizing low-
grade heat sources, [26], and a reliable low-cost
system, [27]. The thermoacoustic temperature
sensor was proposed in the nuclear engineering field
to monitor the neutron flux and fuel condition, [28],
[29]. The applications were of primary interest and
the oscillation in the tube section was mainly
discussed in these studies. The oscillation of
variables in the narrow channel section, however,
has not yet been studied well.
The analytical treatments of thermoacoustic
oscillation are almost based on the linearized
equations representing the oscillation of pressure
and velocity around the mean values with the
assumption of temperature distribution, [30]. Fluid
conservation equations were not solved but the
stability limit was estimated using this acoustic
approximation, [31]. Two-dimensional flow fields
around the narrow channel were discussed using
computational fluid dynamics codes such as CFX,
[32], [33], and Fluent, [34], [35], [36]. Although
fluid equations were solved directly and velocity
and temperature distributions were shown, the
computational region was limited and several
assumptions were needed for boundary and initial
conditions. It was shown by the numerical
simulation that the thermoacoustic oscillation was
induced using one-dimensional fluid conservation
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equations, [37]. The heat transfer coefficient was,
however, assumed to be a constant and suddenly
increased for the onset of oscillation, and the time
variations of oscillating variables alone were shown.
The thermoacoustic oscillation does not need
electricity, and the application is desirable in the
nuclear engineering field during station blackout
accidents, [28], [29]. The mechanism of oscillation
in the narrow channel section is, however, not yet
clearly understood. In this study, the time variations
and spatial distributions of temperature, velocity,
and pressure in the thermoacoustic loop, especially
in the narrow channel section, are made clear.
Numerical simulations are performed by solving the
compressible mass, momentum, and energy
conservation equations for a working fluid to
include nonlinear effects. The heat conduction
equations for wall structures of whole the
thermoacoustic loop are solved simultaneously.
According to the experimental procedure in [31],
realistic initial and boundary conditions are applied.
The dependence of simulated results on the number
of calculation cells is checked, and quantitative
results are obtained. The heat transfer in the narrow
channel section, which is one of the most important
parameters for the design of a thermoacoustic loop,
is discussed.
2 Simulation Method
2.1 Governing Equations
The gas flow is simulated by solving one-
dimensional compressible conservation equations.
The mass conservation equation is given by
, (1)
where A, and u are the flow area, density, and
velocity, respectively. The momentum conservation
equation is given by
, (2)
where p, g, and fw are the pressure, gravitational
acceleration, and wall friction, respectively. The
wall friction is defined by
, (3)
where
and pe are the friction factor and the
perimeter, respectively. The friction factor is given
according to the Reynolds number, Re, by
(4)
(5)
where D is the diameter of a narrow channel or tube
and
is the surface roughness. The Reynolds
number is defined by Re = (
uD/
), where
is the
viscosity. The friction factor between the laminar
and turbulent flows is calculated by interpolation.
The energy conservation equation is given by
, (6)
where U and qw are the internal energy and wall heat
transfer, respectively. The wall heat transfer is
defined by
, (7)
where hw is the heat transfer coefficient, and Tw and
Tg are the wall and gas temperatures, respectively.
The heat transfer coefficient is given as the Nusselt
number,
(8)
(9)
where Nu and Pr are the Nusselt and Prandtle
numbers, respectively defined by Nu = (hwD/k) and
Pr = (
Cp/k), where k and Cp are the thermal
conductivity and specific heat, respectively. The
Nusselt number between the laminar and turbulent
flows is obtained by interpolation.
Tube and channel walls are assumed to be
annular, and a one-dimensional heat conduction
equation in the radial direction is given by
, (10)
where (
Cp)w, T, and kw, respectively, are the
volumetric heat capacity, temperature, and thermal
conductivity of the wall.
The governing equations are calculated using the
RELAP5 code as a numerical solver, which is
designed to simulate the fluid flow and the structural
temperature in nuclear reactors, [38].
2.2 Numerical Model
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The numerical model is based on the closed-loop
device used in the experiment, [31], and in the
simulation, [37], and is briefly described here. The
outline of the numerical model is shown in Figure 1.
The closed loop is 2.8 m in length and consists of
four straight tubes with a length of 0.7 m, an inner
diameter of 40 mm, and a wall thickness of 1.2 mm.
A short section of the left tube in Figure 1 is
replaced by a bundle of narrow channels called a
stack. The stack length is 35 mm, and the diameter
of the narrow channel is from 0.2 mm to 2.0 mm.
The porosity of the stack, which is the ratio of the
flow area between the tube and all of the narrow
channels, is 0.67. One side of the stack is kept at a
high temperature by the hot heat exchanger, HX,
and the other side is kept at a low temperature by
the cold HX. The length of the hot and cold HXs is
13 mm. The flow area of HX is 0.67 of the tube
flow area. The working fluid is air, and the tube and
HXs are made from stainless steel while the stack is
made from ceramic.
The initial temperature is 295 K for the gas,
walls, and outside of the tube. The outer surface
temperature of hot HX, Th, is set at a higher value at
time zero, while that of cold HX, Tc, is unchanged.
The temperature gradient along the outer surface of
the stack is linear between the hot and cold HXs.
The heat transfer coefficient is 5 W/(m2K) for the
outer surface of the tube. After the steady-state flow
field is established, Th is increased by 5 K. The
temperature gradient along the stack is modified
simultaneously. If the oscillating flow is observed,
this Th is regarded as the onset temperature. If the
oscillation is not calculated, the steady-state
calculation is performed again with 5 K higher Th.
These boundary conditions and procedures were
applied for simulating the experimental procedure
[31].
Fig. 1: Outline of numerical model
3 Results and Discussion
3.1 Dependence on Calculation Cells
The dependence of simulated results on the number
of calculation cells, which is not always checked in
numerical approaches, [32], [34], [37], is shown
here for quantitative discussion. A typical example
is shown in Figure 2, where the relative values of
steady-state velocity and pressure are depicted. The
steady-state flow field is established up to 3000 s of
calculation, and the results at 5000 s are shown in
Figure 2. It is seen that the calculated results using a
smaller number of cells are smaller than those using
a larger number of cells and the effect is notable for
the number of cells smaller than 200.
The number of calculation cells in this study is,
thus, determined to be 281, since the calculated
results are not affected when the number of cells is
larger than 250. Each fluid cell has a six-mesh wall
structure, in which the heat conduction is calculated
using the material properties of the walls. The effect
of calculation cells in the wall is found to be
negligibly small, and not shown here. The time step
size used for the steady-state calculation is ranging
from 0.1 ms to 1.0 ms according to the diameter of
the narrow channel to obtain undisturbed flow
fields, and the time step size after the steady state is
0.015 ms in the following simulations. These values
of time step size are determined so that the Courant
number based on the cell size and the sound speed
becomes less than unity.
Fig. 2: Dependence of flow field on calculation
cells
3.2 Onset of Thermoacoustic Oscillation
The variations of gas temperatures at four locations
in the loop are shown in Figure 3 after the steady
state. The diameter of the narrow channel is 0.8 mm,
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and the surface roughness is zero in this case. The
selected locations are the stack mid-elevation,
(1/4)L, (2/4)L, and (3/4)L, where L is the loop
length and (1/4)L, (2/4)L and (3/4)L are the
distances along the loop from the cold HX. The four
locations are almost corresponding to the four
corners in Figure 1. Two cases with different Th,
435 K and 440 K, are indicated in Figure 3 after the
steady state.
The steady-state temperature at 5000 s is
higher in the stack and decreases along the tube
since the outside temperature is the same as the
initial temperature of 295 K and the heat transfer
between the outer surface of the tube and the
environment is calculated. The outer surface is thus
cooled by the environment, and the gas temperature
decreases along the tube.
The maximum Reynolds number, Re, is about
7.9 and 271.1, respectively, in the stack and the tube
due to a small circulating flow. The steady-state
flow is, thus, laminar, and the laminar friction factor,
Eq. (4), and heat transfer coefficient, Eq. (8), are
used up to 5000 s. It is noted that the steady state
was established much earlier when the heat transfer
coefficient was fixed at a small value, [37], while
the heat transfer coefficient is given theoretically
and varied according to the gas temperature in this
study. The heat transfer rate is thus affected by the
circulating flow and the wall temperature, and a
longer time is needed to obtain a steady state.
The thermoacoustic oscillation is shown to occur
at about 5008 s for the case with Th of 440 K in
Figure 3. It is also shown that the oscillation is not
induced when Th is 435 K. After the onset of
oscillation, the oscillation amplitude is larger in the
stack than at the other locations of the tube. This is
because the thermal energy is input to the gas in the
stack, and the oscillation energy is the maximum in
the stack.
The temperatures at (1/4)L, (2/4)L, and (3/4)L
decrease slightly and oscillate. The decrease in
temperature indicates a cooling effect since the
oscillating velocity is much larger than the steady-
state circulating velocity. The maximum Re is about
64.3 and 6307, respectively, in the stack and the
tube during the oscillation. The laminar friction
factor and the heat transfer coefficient are used in
the stack while the laminar and turbulent
correlations are used in the tube after the onset of
oscillation. The heat transfer coefficient is larger for
the turbulent flow, and the cooling effect becomes
larger during the oscillation in the tube.
Fig. 3: Variations of gas temperatures at four
locations after the steady state.
3.3 Temperature Distribution and Wave Speed
The temperature distributions along the loop at 5000
s and 5020 s are shown in Figure 4(a), where the
location is the distance from the cold HX along the
loop normalized by the loop length. The gas
temperature increases steeply in the stack from the
cold HX at the location of 0.0 to the hot HX at about
0.022 and decreases gradually in the loop from the
hot to cold HXs.
The temperature distribution at 5020 s is lower
than that at 5000 s in the tube section. This shows
the cooling effect of oscillation as shown in Figure 3.
The heat transfer rate is larger for the oscillating
flow at 5020 s due to the turbulent heat transfer
coefficient. The temperature distribution thus
becomes lower than the steady-state distribution as
shown in Figure 4(a). The heat transfer rate depends
also on the temperature difference between the gas
and the wall. The gas temperature becomes lower
downstream, where the location becomes larger than
about 0.5. The heat transfer rate is, thus, smaller in
the downstream, and the cooling effect due to the
turbulent heat transfer is not notable in the
downstream.
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Fig. 4: Temperature distributions along the loop (a)
and relation between the sound speed and the
temperature (b).
The relation between the sound speed and the
temperature of air under atmospheric pressure is
shown in Figure 4(b). The oscillation frequency is
125.8 Hz in Figure 3, and the average wave speed
along the loop is estimated to be 352.2 m/s. Under
the adiabatic condition, the temperature
corresponding to this sound speed is 308.3 K, while
432.5 K is under the isothermal condition as shown
in Figure 4(b). These temperatures are between Tc
of 295 K and Th of 440 K. The adiabatic oscillation
is probably established in the tube section of a
relatively large diameter, but the oscillation in the
narrow channel of the stack is close to the
isothermal oscillation.
Fig. 5: Distributions of oscillating pressure during
one oscillation period along the stack and the tube
inlet (a) and along the loop including the stack (b).
The thermal boundary-layer thickness,=
(2/)1/2, where and are, respectively, the
thermal diffusivity and the angular frequency, is
estimated to be 0.284 mm for the average
temperature in the stack, and sufficiently large
compared to the channel radius of 0.4 mm.
Furthermore, the thermal relaxation time, =r2/(2),
where r is the radius of the narrow channel, is about
2.51 ms, and sufficiently large compared to the
oscillation period of 7.95 ms. The heat transfer
similar to the isothermal process is, thus, expected
in the stack. It is noted in Figure 4(b) that the
average wave speed is in between the adiabatic
sound speed of 344.5 m/s for Tc and the isothermal
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sound speed of 355.2 m/s for Th. It is found that
thermoacoustic oscillation could be possible in the
loop when the isothermal sound speed for the
highest temperature exceeds the adiabatic sound
speed for the lowest temperature. This is one of the
necessary conditions for the temperature difference
between the hot and cold HXs.
Fig. 6: Distributions of oscillating velocity during
one oscillation period along the stack and the tube
inlet (a) and along the loop including the stack (b).
Fig. 7: Distributions of oscillating temperature
during one oscillation period along the stack and the
tube inlet (a) and along the loop including the stack
(b).
3.4 Spatial Distribution of Oscillating
Variables
The spatial distributions of oscillating pressure,
velocity, and temperature along the loop are shown
in Figure 5, Figure 6, and Figure 7, respectively.
They are calculated as variations from their time
averages. The oscillation period, T, is divided by
eight, and nine distributions are shown every 0.99
ms after 5020 s in Figure 5, Figure 6, and Figure 7.
The timings are denoted by (n/8)T, where n is from
0 to 8. It is noted that the last timing, (8/8)T, is 7.92
ms, and slightly shifted from the first timing, (0/8)T,
since the oscillation period is 7.95 ms. The
distributions along the stack and tube inlet are
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enlarged in Figs. 5(a), 6(a), and 7(a), while those
along the loop including the stack are in Figs. 5(b),
6(b) and 7(b).
The wavelength of pressure oscillation is shown
to be the same as the loop length in Figure 5(b)
since the loop is closed and the traveling wave is
induced. The amplitude is larger at the location of
the stack and (2/4)L and smaller at (1/4)L and
(3/4)L. The pressure distributions in the stack,
which is from the location of 0.0046 to 0.017, are
almost flat in Figure 5(a). It is thus found that the
pressure is oscillating continuously along the stack
and the tube as a single wave.
The amplitude of velocity oscillation in Figure
6(b) is smaller at the location of the stack and (2/4)L
and larger at (1/4)L and (3/4)L. The velocity
distributions in the stack in Figure 6(a) are almost
flat, as is the case with the pressure distributions
shown in Figure 5(a), though the phase difference
between the pressure and the velocity is 27.17 deg.
The flow area of the stack and HXs are smaller than
that of the tube, and the slight change in velocity is
seen at the tube inlet, which is at about the location
of 0.022 in Figure 6(a). The velocity is thus shown
to be oscillating continuously along the stack and
the tube as a single wave according to the flow area.
The amplitude of temperature oscillation in
Figure 7(b) is larger at the location of the stack and
(2/4)L and smaller at (1/4)L and (3/4)L. The
oscillation timings and the spatial distributions of
the temperature along the tube are similar to those
of the pressure in Figure 5(b).
The temperature distributions in the stack in
Figure 7(a) are, however, much different from the
pressure distributions in Figure 5(a). The
temperature distributions are not flat and the sign of
the temperature variation is opposite to that in the
tube section.
The pressure and the velocity distributions are
similar in the stack as shown in Figs. 5(a) and 6(a).
The gas is moving in the direction from the cold to
hot HXs during the positive velocity period shown
in Figure 6(a), and the temperature becomes lower
due to the cold inflow as shown in Figure 7(a).
When the velocity is negative, on the contrary, the
gas is moving in the direction from the hot to cold
HXs, and the temperature becomes higher due to the
hot reverse flow. The gas is, thus, cooled and heated
during the high and low-pressure periods,
respectively. The sign of the temperature variation is,
thus, opposite to those of the pressure and velocity
variations. Both ends of the stack are open in the
loop and seem to play the role of the node for
temperature oscillation as seen in Figure 7(a). It is
found in the stack that the temperature is oscillating
differently and the distribution of oscillating
temperature is reversed from that in the tube.
3.5 Onset Temperature Ratio
The onset temperature ratio, Th/Tc, is shown in
Figure 8 as a function of the oscillation parameter
. The simulation results with the constant heat
transfer coefficient, [37], and the analytical results
and experimental data, [31], are shown along with
the present results. It is shown by the present
calculations and the existing analytical and
experimental results that the minimum temperature
ratio of about 1.5 is obtained at around of 2.0.
This point corresponds to the calculated results
shown in Figure 3, Figure4, Figure 5, Figure 6, and
Figure 7. It is seen in Figure 8 that the minimum
temperature ratio for the onset of oscillation is
calculated well by the present numerical procedure.
Fig. 8: Onset temperature ratio as a function of
oscillation parameter.
The present results increase as increases
from the minimum point and are larger than the
other results. As decreases, the present results
increase slightly, but smaller than the other results.
The heat transfer area in the stack is
proportional to the product of the perimeter and the
number of channels. The perimeter and the number
of channels are, respectively, proportional to r and
1/r2. The heat transfer area is, thus, proportional to
1/r. The heat transfer coefficient is proportional to
1/r, since it is given by the Nusselt number, Eq. (8),
which is the constant for laminar flows. The heat
transfer between the gas and the wall is proportional
to the product of the heat transfer area and the heat
transfer coefficient, and thus to 1/r2 or 1/(). It is
shown in Figure 8 for larger that the heat transfer
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becomes small and the onset temperature ratio is
larger than the analytical and experimental results.
On the contrary for smaller , the heat transfer
becomes large and the onset temperature ratio is
smaller than the analytical and experimental results.
The simulation results with the constant heat
transfer coefficient, [37], seem to be better for larger
. The Nusselt number used in this study is
obtained analytically for the steady-state non-
oscillating laminar flow and might be different from
that for the oscillating flow. Furthermore, the
distribution of oscillating temperature in the stack is
reversed at a short distance from that in the tube as
shown in Figure 7(a). It should also be noted that
the adiabatic oscillation is established in the tube
section, but the oscillation in the narrow channel of
the stack is close to the isothermal oscillation.
For the practical application and design of
thermoacoustic devices, the heat transfer coefficient
or the heat transfer model should include above
mentioned characteristics of thermoacoustic
oscillation. Although the threshold of oscillation is
simulated well in this study, detailed theoretical and
numerical analyses and experiments would be
necessary to estimate correctly the heat transfer for
the oscillating flow in the narrow channel.
4 Conclusion
The thermoacoustic oscillation in the closed loop
has been numerically simulated by solving the
compressible mass, momentum, and energy
conservation equations for the gas and the heat
conduction equations for the wall. The loop was
composed of the cold heat exchanger, the narrow
channel section, the hot heat exchanger, and the
tube. The onset of oscillation was calculated by
gradually increasing the temperature of a hot heat
exchanger. The minimum onset temperature ratio
was shown to agree with the existing analytical and
experimental results. The average wave speed was
in between the isothermal sound speed for the
highest temperature and the adiabatic sound speed
for the lowest temperature. The distribution of
oscillating temperature in the narrow channel was
shown to be different and reversed from that in the
tube. Although the threshold of oscillation was
simulated, it was shown that the heat transfer in the
narrow channel is smaller for larger channel
diameter and larger for smaller channel diameter,
and the heat transfer coefficient or the heat transfer
model should be improved for the oscillating flow in
the narrow channel.
In this study, several characteristic features of
thermoacoustic oscillation, especially the spatial
distribution of oscillating variables in the narrow
channel, were confirmed using the fluid equations
and heat conduction equations. It was shown that
the safety analysis code for nuclear reactors could
be used for the simulation of thermoacoustic
oscillation. Applications for various configurations
and conditions in nuclear engineering fields would
be evaluated in the same manner, since different
boundaries and initial conditions are set easily. A
similar closed-loop experiment is underway by the
author’s group based on the present simulations for
studying the applicability in the nuclear engineering
fields, and experimental results would be shown in
the near future.
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WSEAS TRANSACTIONS on HEAT and MASS TRANSFER
DOI: 10.37394/232012.2023.18.6
Tadashi Watanabe
E-ISSN: 2224-3461
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Volume 18, 2023
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
Tadashi Watanabe has performed all the research
works in this study; planning, literature survey,
numerical simulation, discussion, writing, and so on.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflict of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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WSEAS TRANSACTIONS on HEAT and MASS TRANSFER
DOI: 10.37394/232012.2023.18.6
Tadashi Watanabe
E-ISSN: 2224-3461
68
Volume 18, 2023
