combustion thermo-acoustic

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ScienceDirect
Energy Procedia 00 (2017) 000–000
www.elsevier.com/locate/procedia
1876-6102 © 2017 The Authors. Published by Elsevier Ltd.
Peer-review under responsibility of the Scientific Committee of The 15th International Symposium on District Heating and Cooling.
The 15th International Symposium on District Heating and Cooling
Assessing the feasibility of using the heat demand-outdoor
temperature function for a long-term district heat demand forecast
I. Andrića,b,c
*, A. Pinaa
, P. Ferrãoa
, J. Fournierb
., B. Lacarrièrec
, O. Le Correc
a
IN+ Center for Innovation, Technology and Policy Research - Instituto Superior Técnico, Av. Rovisco Pais 1, 1049-001 Lisbon, Portugal
b
Veolia Recherche & Innovation, 291 Avenue Dreyfous Daniel, 78520 Limay, France
c
Département Systèmes Énergétiques et Environnement - IMT Atlantique, 4 rue Alfred Kastler, 44300 Nantes, France
Abstract
District heating networks are commonly addressed in the literature as one of the most effective solutions for decreasing the
greenhouse gas emissions from the building sector. These systems require high investments which are returned through the heat
sales. Due to the changed climate conditions and building renovation policies, heat demand in the future could decrease,
prolonging the investment return period.
The main scope of this paper is to assess the feasibility of using the heat demand – outdoor temperature function for heat demand
forecast. The district of Alvalade, located in Lisbon (Portugal), was used as a case study. The district is consisted of 665
buildings that vary in both construction period and typology. Three weather scenarios (low, medium, high) and three district
renovation scenarios were developed (shallow, intermediate, deep). To estimate the error, obtained heat demand values were
compared with results from a dynamic heat demand model, previously developed and validated by the authors.
The results showed that when only weather change is considered, the margin of error could be acceptable for some applications
(the error in annual demand was lower than 20% for all weather scenarios considered). However, after introducing renovation
scenarios, the error value increased up to 59.5% (depending on the weather and renovation scenarios combination considered).
The value of slope coefficient increased on average within the range of 3.8% up to 8% per decade, that corresponds to the
decrease in the number of heating hours of 22-139h during the heating season (depending on the combination of weather and
renovation scenarios considered). On the other hand, function intercept increased for 7.8-12.7% per decade (depending on the
coupled scenarios). The values suggested could be used to modify the function parameters for the scenarios considered, and
improve the accuracy of heat demand estimations.
© 2017 The Authors. Published by Elsevier Ltd.
Peer-review under responsibility of the Scientific Committee of The 15th International Symposium on District Heating and
Cooling.
Keywords: Heat demand; Forecast; Climate change
Energy Procedia 142 (2017) 1488–1495
Peer-review under responsibility of the scientific committee of the 9th International Conference on Applied Energy.
10.1016/j.egypro.2017.12.597
ScienceDirect
Energy Procedia 00 (2017) 000–000
www.elsevier.com/locate/procedia
9th International Conference on Applied Energy, ICAE2017, 21-24 August 2017, Cardiff, UK
Analysis of Pressure Wave Development in a Thermo-acoustic
Engine and Sensitivity Study
Mohammed Noorul Hussaina
, Isam Janajreha
*
a
Mechanical Engineering Dept, Khalifa University of Science and Technology, Masdar Institute, UAE 54224
Abstract
Thermoacoustic engines (TAE) are special devices that convert thermal energy into mechanical energy in the form of a high decibel
acoustic wave. These devices have a huge potential to revolutionize fields like refrigeration, if efficiently designed. They are simple
in construction, having two main parts, i.e. the stack and the resonator tube, and work on a process similar to the Stirling cycle.
Although, this technology exists for several years, there is still no strong design available that can be used for actual practical
purposes. They have been tested through experiments and analytical techniques and recently through CFD based numerical
simulations. In this work, a TAE has been analyzed through CFD simulations using a commercial package. Taking a step further
from the existing research related to TAEs, a sensitivity study has been carried out. The study aims to analyze the pressure wave
development in a TAE, and its dependency on the temperature gradient at the stack and the length of the resonator. It is seen that
higher temperature gradient causes a higher pressure amplitude and early pressure wave development. The critical temperature
gradient is found in the range 200 – 300 K. With resonator length, it is noticed that, very small resonator length results in failure,
while very long resonator length causes delay in pressure wave initiation although it provides high amplitude. Medium lengths are
better since they provide high pressure amplitude and faster wave initiation. Analysis of frequency showed that temperature
gradient has minor effect whereas length has major effects.
© 2017 The Authors. Published by Elsevier Ltd.
Keywords: Thermoacoustic Engine, Stirling cycle, Refrigeration,
* Corresponding author. Tel.: +971-2-8109130; fax: +971-28109901.
E-mail address: ijanajreh@masdar.ac.ae

Mohammed Noorul Hussain et al. / Energy Procedia 142 (2017) 1488–1495 1489
2 Mohammed Noorul Hussain, Isam Janajreh / Energy Procedia 00 (2017) 000–000
1. Introduction
Thermoacoustic heat engines (TAE) are devices that convert thermal energy to acoustic energy, and they are mostly
based on stationary parts [1], using the Stirling cycle principle. In the Stirling cycle, a working gas is compressed in a
piston cylinder arrangement, while a heat sink is actively absorbing the excess heat to keep the temperature of the gas
constant. The gas is then passed through a regenerator where it is heated at constant volume. It is further heated at the
heat source where it expands to deliver power to a piston [2]. Researchers like Ceperley [3] worked on replacing the
pistons in a Stirling engine with sound waves and created the very first thermoacoustic heat engine technology. This
concept is complicated in terms of the fluid dynamics involved.
The thermoacoustic engines are of 2 basic types, i.e. the standing wave and the moving/traveling wave types. In
both systems the main component is the regenerator stack which is a porous stack of heating plates. In the standing
wave TAE, the stack is placed in a cylindrical resonator with one end is sealed and the other end completely is open.
The resonator tube is filled with a working gas such as air, argon or nitrogen and the stack is placed near the closed
end. The stack is heated in such a way that there is a linearly decreasing temperature profile along its length. This
temperature gradient across this porous stack amplifies the pressure disturbance in the fluid causing the pressure in the
region after the stack to oscillate and produce a sound wave. However, this happens after sometimes in reaching out
to this stationary oscillation operation. In experiments, the stack is usually made of ceramic monolith structures. For
applications like refrigeration and in replacement of the compressor, high frequency of this sound wave is required.
The length and shape of the resonant tube affect factors like operating frequency, oscillatory pressure, and velocity
profiles. Simulation of standing wave TAE’s has been carried out initially by linear models [4] which were then
developed into non-linear 2-dimensional models [5]. The linear theory is limited to small acoustic perturbations and
under laminar flow conditions [1]. The best non-linear transient models for TAE’s were initially developed by Kaprov
et al. [6] that closely captured the oscillatory non-linear pressure variations. Further non-linear models were developed
by Hamilton et al [7]. Although such non-linear models are an advancement compared to linear modelling, they still
fall behind in capturing phenomena like streaming. These can only be used for deeper understanding of the subject but
for accurate simulation CFD based numerical techniques are required.
In the field of numerical analysis, early modeling work was done by Cao et al [8] through a simple model that
considered a reduced domain comprised of a single stack plate and a section of the resonator tube. Their investigation
was focused on the temperature behavior of the stack plate. A much more advanced model was made by Besnoin et al
[9] which included heat exchangers on the stack, which was assumed to be only a single plate. Their work provided a
good understanding about the placement of the stack and the heat exchangers. The use of compressible Navier-Stokes
equations was first carried out by Ishikawa et al [10]. The most sophisticated initial work that used CFD was carried
out by Zoontjens et al [11]. They used the Ansys Fluent commercial code and still maintained a single plate stack.
Using oscillatory pressure boundary conditions, they simulated the pressure variations in the fluid in a Thermoacoustic
cycle. Similar works were also carried out by [12] [13]. In recent works much advancement has been carried out in
CFD based numerical simulation of standing wave TAEs. Zink et al [14] carried out transient, turbulent, 2D simulation
of a standing wave TAE in Ansys fluent. Their stack was made up of multiple horizontal plates. The geometrical
extremities were not greater than 15 cm x 5cm. They succeeded in simulating a perfect standing wave in the resonator
section after the stack. The obtained amplitude of the standing wave was close to 5,000 Pa which corresponds to nearly
120 dB, indicating the high effectiveness of the TAE. In a recent work, Zink et al [15] simulated the cooling effect of
a TAE and extensively analyzed the temperature behavior at the stack and in the resonator and the velocity in the
resonator tube. Yu et al [1] also conducted simulation of high frequency standing wave TAE that suitable for space
applications. The TAE had coaxially stacked plated and had an operating frequency of 300 Hz. They also made
comparisons between simulation and experimental works. The largest pressure amplitude in their model simulation
model rose as high as 0.279 MPa.
In this work, CFD based numerical simulation of a standing wave TAE has been carried out. The TAE simulated
has an axisymmetric cylindrical domain with a horizontal plate stack as the regenerator. The boundary conditions are
similar to previous works like [14]. However in this work expand on the cited work by undertaking sensitivity study
on two most important factors which are the length of the resonator and the temperature gradient at the stack has been.
This works is aimed at analyzing the pressure fluctuation, its development and its amplitude.

1490 Mohammed Noorul Hussain et al. / Energy Procedia 142 (2017) 1488–1495
Mohammed Noorul Hussain, Isam Janajreh / Energy Procedia 00 (2017) 000–000 3
Fig. 1. Modelled TAE baseline geometry in Fluent (Axisymmetric tube section)
2. Methodology
A cylindrical TAE with working fluid as air is modelled as an axisymmetric geometry in Ansys Fluent. A stack of
horizontal plates is modelled with the thickness of plates as 5mm and a gap of 5 mm between each plate. The mesh is
created in Gambit geometry before is imported into Fluent solver. Fig. 1 shows the asymmetrical geometry of the
baseline model mesh. The numerical model is based on the transient, none-isothermal and 2D cylindrical Navies-
Stokes flow. The ideal gas model is assumed that governs the fluid state as the developed pressure wave is relatively
small. Turbulence is accounted for following the averaging of these equations where the resulted Reynolds stresses
are modeled via the common eddy viscosity ( )
/
/
(
'
'
i
j
j
i
t
j
i V
V
V
V
V
V 





 
 ) and k- transport model.
Eq. 1-6 describe the overall governing equations (Eq. 1 the continuity, Eq. 2, 3, and 4 the momentums, Eq. 3 and Eq.
6 is the transport equations for scalar quantities such as those that governs the turbulence (K and  following the
common eddy viscosity model. Eq. 7 governs the energy equation.
0
)
.( 




V
t


 and for axisymmetric geometries is written as 0
)
(
)
(










r
V
r
V
x
V
t
r
r
x 


 (1)
Where x is the axial coordinate, r is the radial coordinate, Vx is the axial velocity, and Vr is the radial velocity. The
momentum equation is written as:
g
I
V
V
V
p
V
V
t
V T 
























]
.
3
2
)
[(
.
)
.(
)
( (2)
Where p is the static pressure, is the molecular viscosity, and I is the unit tensor and the term is the effect of
volume dilation and g is the gravitational body forces. In 2D axisymmetric geometries the axial and radial
conservation of momentum are written as:
x
r
x
x
x
r
x
x
x
g
x
V
r
V
r
r
r
V
x
V
r
x
r
x
p
V
V
r
r
r
V
V
r
x
r
t
V


































)]
(
[
1
)]
.
(
3
2
2
(
[
1
)
(
1
)
(
1  (3)
r
r
r
x
r
r
r
r
x
r
g
V
r
r
V
V
r
V
r
r
r
r
V
x
V
r
x
r
r
p
V
V
r
r
r
V
V
r
x
r
t
V







































)
.
(
3
2
2
))]
.
(
3
2
2
(
[
1
)]
(
[
1
)
(
1
)
(
1
2

 (4)
Where in axisymmetric the divergence is written as:
r
V
r
V
x
V
V r
r
x









. (5)
The transport equations in terms of dependant variable and in particular for the two turbulence scalars are written
following the common four tem formulation, i.e. temporal, adjective, diffusive and any the additional sources as:
k
S
V
t













))
.(
)
(  (6)
Where takes quantities such as turbulent kinetic energy (k) and its dissipation rate (;  is the diffusion
coefficient and S is the source term corresponding to each of the scalar equations. The k and  equations are related
by the eddy viscosity term such that 𝜇𝜇𝑇𝑇 = 𝐶𝐶𝑒𝑒2ρ
𝜀𝜀2
𝑘𝑘
where 𝜇𝜇𝑇𝑇 is the turbulent viscosity. The internal energy (E)
equation is written as:
]
).
.
3
2
)
(
(
.[
))
(
.(
)
(
V
I
V
V
V
T
K
p
E
V
t
E T






















 (7)

Where ρ is the density, V represents the velocity field, p is the flow pressure, 𝜇𝜇 is the dynamic viscosity, 𝜇𝜇𝑇𝑇 is the
turbulent viscosity. The internal system energy (E) can be described as:
V
V
p
H
E


.
/ 2
1


  (8)
Where H is the system enthalpy which related to internal energy by the static pressure and density term per eq. 8. The
boundary condition at the closed end was wall with a no-slip condition, which similar for all other walls of the
geometry except the outlet where the pressure is set to zero indicating a free pressure outflow. The most important
boundary condition is at the walls of the stack. For each horizontal stack there are four edges the top and the bottom
edges are given a temperature gradient boundary condition. This temperature gradient is created as a profile in Fluent
and assigned to the respective edge. For the left and the right edges of each stack plate the heat transfer coefficient of
50 W/m2
K is assigned [14]. These are the most crucial boundary conditions that determine the occurrence of an
oscillatory pressure standing wave in the resonator. Ideally, in a transient simulation the closed end is a wall with a
general no slip condition. In practical application there is a very minute velocity even though there is no inlet and
hence when the TAE is operational, sound wave is achieved. However in a simulation model, a small disturbance in
the flow profile of the field has to be created. This disturbance is achieved by running the model in a steady state
solver with pressure boundary condition, instead of a wall at the closed end. The specified pressure is 10 Pa [14],
alternatively one can proceed with the transient solution under specific initial conditions to trigger the needed
perturbation. Therefore, the simulation starts with first 500 iterations running at steady state solver with a pressure
inlet boundary condition. Once the initial disturbance is created, the transient simulation with a wall boundary
condition at the closed end is actuated. For this part the time step is set to 1 x 10-5
s govern by the stability of Courant
Friedrichs Lewy number (CFL) and ran for 60,000 time steps, therefore the simulation represents a physical time of
0.6 s. To monitor the pressure fluctuation, points were created at varying distances after the stack and the average
pressure of each iteration was plotted for the simulation.
The main analysis in this work is the sensitivity study that aims to analyze the pressure and frequency variation with
varying temperature gradient of the stack and the length of the resonator. For the temperature analysis the temperature
gradient is varied with 4 cases being considered. In all cases the lower temperature of 300 K is constant while the
higher temperature is varied as 500 K, 600 K, 700 K and 1000 K. This study helps in identifying the critical
temperature gradient which is characteristic of TAEs [15]. In the second study, length sensitivity to analyze the
pressure amplitude and frequency has been carried out. This study is aimed more at the practical side of a TAE. Since
the pressure amplitude and the frequency are the most sought after effects of any TAE irrespective of its end use
application, therefore it is necessary to know how the length of the resonator tube affects these variables. It is known
the characteristic frequency of the TAE is related to the tube length according to eq.9.
𝑓𝑓 = 𝐶𝐶/ 𝜆𝜆 𝑎𝑎𝑎𝑎𝑎𝑎 𝜆𝜆 = 4 𝑥𝑥 𝐿𝐿 (9)
Where c is the sound speed (≈360m/s) and  and L are the characteristic and tube length, respectively. In this study
two other cases are considered in addition to the baseline, one shorter length which is half of the baseline length and
second one which is double the baseline length. Therefore the 3 cases considered were at lengths of 7.5 cm, 15 cm
and 30 cm without changing the stack position with anticipated frequency of 300 Hz, 600 Hz, and 1,200 Hz,
respectively. Also with pressure monitors placed at different distances in the resonator one also gets an analysis of
the pressure development of the length of the TAE.
3. Results and Discussion
3.1 Mesh sensitivity
Four level of meshes has been implemented to assess the discretized mesh independency on the solution. These meshes
are denoted as fine, baseline, and coarse I and coarse II. Results of the weighted area average temperature of the stack
upstream and downstream and the absolute relative errors on the temperature difference are presented in table 3. The
fine mesh seems to be the best choice as far as accuracy, but at longer computation time especially when one considers
the unsteady solution. The deployed baseline mesh is appropriately accurate with an absolute error in the evaluated
temperature of 0.7% while the coarse I and course marked 1.5% and 10.7% respectively. It should be noted that a very
strict residual of 10-11
is achieved for the four considered mesh levels.

Table 3. Mesh sensitivity results on the evaluated Temperature difference between the up- and downstream of the stack
The initial run was carried out for the baseline case where the length of the resonator was 15 cm and the temperature
gradient at the horizontal edges of the stack was 700 K to 300 K. This was done to test whether the applied boundary
conditions, the geometry and the temperature gradient was appropriate in order to produce a sound wave in the
resonator. The model after the initial creation of disturbances was set to run for 60,000 time steps and each time step
of 1 x 10-5
s. The model setup was successfully able to produce pressure oscillations in the resonator thus implying
the applicability of the chosen boundary conditions and parameters.
3.2 Sensitivity Study
Temperature Sensitivity: The temperature sensitivity was carried out initially for three cases. The first case (T1) was
with a temperature gradient from 500 K to 300 K, the second (T2) was the baseline case with gradient of 700 K to 300
K and the third (T3) was a high temperature case with gradient from 1,000 K to 300 K. in all the cases there were 4
pressure monitors placed up- and downstream at 5 cm, 10 cm, 14 cm and 15 cm respectively. In cases T2 and T3, it
was noticed that there is clear formation of a standing wave in the resonator, however in T1 with the lowest temperature
gradient, there is a bleak pressure fluctuation in first few time steps which quickly dies out and there is temporally
constant pressure in the resonator. This was noticed in all the 4 pressure monitoring points. This behavior corresponds
to the presence of a characteristic critical temperature gradient across the stack plates which is a major factor that
determines the development of the standing wave. To further investigate this critical temperature gradient for the
geometry modelled in this work, an additional case of temperature gradient was added that was intermediate between
T1 and T2. In this intermediate gradient case, the highest temperature was set to 600 K while keeping the lowest
temperature constant. Upon completion of simulation it was noticed that there is a formation of standing wave for this
temperature gradient. This indicates that the critical temperature gradient for this geometry or this TAE lies around
200 to 300 K. Fig. 2 shows the pressure fluctuation for different cases with respect to the time step.
Fig. 2. Static pressure wave development in the resonator for different cases of temperature gradient.
Mesh level Cells number Upstream Temp
(o
C)
Downstream Temp
(o
C)
Del Temp (o
C) Del Temp Rel. Err (%)
Fine 76,262 405.35684 322.7341 82.62274 -----
Baseline 30,003 405.93158 322.7221 83.20948 0.7101
Coarse I 21,293 406.58249 322.7018 83.88069 1.5225
Coarse II 17,617 414.23816 322.7408 91.49736 10.7411

Fig. 3. The pressure observed for different lengths of the TAE recorded at 5 cm of the resonator.
Along with identifying the critical temperature gradient of the TAE, an analysis of the peak amplitudes of the
oscillatory pressure at the resonator tube was also carried out. Interestingly it was noticed that with increase in the
temperature gradient at the stack, the peak pressure amplitude of the TAE also increased. The least pressure amplitude
at T1 was 53 Pa, growing to 2,813 Pa at intermediate temperature gradient, and then to 5,052 Pa at T2 and the highest
amplitude of 8,100 at T3. The cases of intermediate temperature gradient, T2 and T3 correspond to a sound pressure
level above of 120 dB. This clearly indicates the good working conditions of the TAE thereby making it suitable for
further applications.
Length Sensitivity: In many applications for TAEs there are very often space restrictions which must be adhered to
while keeping the performance of the TAE. In order to tackle such a practical issue, there must be studies that analyze
the effect of difference in geometry on the performance of a TAE.
Fig. 4. Temperature distribution plots for the three considered lengths Temperature in K
In this study, the sensitivity to length of the TAE was carried out and also the pressure profile in different pressure
monitor points at the different distance in the resonator tube was analyzed. Three cases of length 7.5 cm (L1), 15 cm
(L2) and 30 cm (L3) were considered. Fig. 3 shows pressure amplitude for different TAE lengths. The temperature
gradient in this case was 100 K to 300 K. In case L1, there were two pressure monitoring points, the first one at 5 cm
length and the second one at 7 cm. In case L2, which is the baseline case the pressure monitors were in the same
position as in temperature gradient sensitivity study. In case L3, which is the longest length, there were 4 pressure
monitors at 5 cm, 10 cm, 15 cm and 29 cm. In the case L1 with the smallest length, it was noticed that there is very
bleak standing wave formation with a very small pressure amplitude. At the second monitoring point the pressure was
near zero with no fluctuation. This behavior is understood from the fact that with the exit of the resonator being close
to the stack, the boundary condition of free pressure flow affected the development of the standing wave in the
resonator. This implies that shorter TAE’s are not suitable due to the non-formation of a clear high amplitude standing

wave. The maximum pressure amplitude in this case was 50 Pa. In the cases L2 and L3, there was a clear standing
wave formation. Two important points are noticed in this study. Firstly, with the data obtained from the four monitors
a common trend of decreasing pressure amplitude is observed in both cases of L2 and L3. This indicates that the
pressure wave formation is best in the area close to the stack. The second important point observed here was that, in
the case of length L2 the standing wave formation started at nearly the 1,000th
time step where in the case of L3 there
was a delay in formation of the standing wave since clear fluctuations were observed only after the 5,000th
time step.
This implied that longer length of the TAE resonator tubes leads to a delay in the development of the standing wave,
which in many practical cases might not be desirable. However, analyzing the peak amplitudes in both cases showed
that the peak pressure amplitude for case L3 was slightly higher at 8,776 Pa as compared to 8,100 for case L2. This
study shows a possible trade-off situation between space constraint, peak pressure and standing wave initialization
time.
Fig. 4 shows the temperature distribution plots for the considered three lengths of the TAE. While in the short and the
medium length one can see higher temperature in the resonator after the stack, the temperature in the long TAE
resonator is largely at the ambient of 300 K.
Fig. 5. Analyzed frequencies for different length cases and temperature cases.
Frequency Analysis: Frequency of a thermoacoustic heat engine is a parameter that determines its functional ability
as a prime mover in post resonator applications such as refrigeration. It is quite difficult to produce high frequencies
in conventional Stirling Engines, and this is specifically where the advantage of a TAE lies. Frequency of a TAE is a
major dependent of its length. Theoretically the frequency can be predicted through the resonator length and the wave
length. The wave length is estimated as a quarter of the resonator length and then the frequency is calculated as a ratio
of speed of sound and wavelength [14]. Theoretically, the calculated frequency for a 15 cm resonator length must be
in the range of 600 Hz. Achieving this is also a validation of the simulation model.
The pressure wave profiles obtained in all the cases were processed externally to obtain a fast Fourier transform
spectrum and analysing the frequency. Fig. 5 shows the obtained frequencies for all the cases considered. Analysis of
frequency for the baseline case with 15 cm length and temperature gradient from 700 K to 300 K shows a result of
600 Hz, indicating the validity of the TAE and the computational model. Further analysing frequency for different
cases, beginning with temperature we see that excluding the case for the lowest temperature gradient for all the other
three cases it is seen that the frequency is in the range of 600 Hz with slight variations. For the case of least temperature
gradient the analysed frequency is much higher, however this is not a representative of the pressure wave, since there
was no pressure wave development, but it is a reflection of haphazard data. This frequency sensitivity study in respect
to temperature indicates that temperature has a negligible effect on the frequency of the pressure wave. With respect
to increase in length of resonator, the obtained frequencies are adherent to theoretical values. The frequency was
highest for the case of shortest length at 1200 Hz and decreased to 300 Hz for the longest length.

4. Conclusion
In this work, numerical simulation through commercial CFD package was carried out. Advancing from the existing
literature works related to CFD simulations of TAE, a sensitivity study was carried out to analyze the effect of
temperature gradient at stack and length of resonator on the pressure wave development. Four temperature gradient
cases and 3 length cases were considered. It is seen that higher temperature gradient causes a higher pressure amplitude
and early pressure wave development. The critical temperature gradient is found in the range 200 – 300 K. With length
it is noticed that, very small resonator length is a failure, while very long resonator length causes delay in pressure
wave initiation although it provide high amplitude. Medium lengths are better since they provide high pressure
amplitude and quicker wave initiation. In addition to these studies, the frequency was also analyzed for all the cases.
With variation in temperature gradient it is seen that there is very minor change in frequency. However with variation
in length the frequency changes are large. The largest frequency (1200 Hz) obtained was for shortest length and the
smallest frequency (300 Hz) was for the longest length.
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