A publication of
CHEMICAL ENGINEERING TRANSACTIONS
VOL. 35, 2013
The Italian Association
of Chemical Engineering
www.aidic.it/cet
Guest Editors: Petar Varbanov, Jiří Klemeš, Panos Seferlis, Athanasios I. Papadopoulos, Spyros Voutetakis
Copyright © 2013, AIDIC Servizi S.r.l.,
ISBN 978-88-95608-26-6; ISSN 1974-9791
The Improvement of the Pneumatic Pulsator Nozzle
according to the Results of the Continuous Adjoint
for Topology Optimization
Krzysztof J. Wolosz*, Jacek Wernik
Department of Process Equipment, Warsaw University of Techology Plock Branch, 09-402 Plock, Jachowicza 2/4,
Poland
krzysztof.wolosz@pw.plock.pl
This paper presents the process of shaping a nozzle which is applied in containers for loose materials
especially as an augmented equipment for pneumatic pulsators. Pneumatic pulsators are applied in
chemical, food, and energy plants to unchoke loose material from container outlets. In pulsators a
pneumatic shock phenomenon is utilized to destroy loose material from structures that product vaults
creations. This process prevents blockages during the transport of materials, decreases the plant
operating costs, limits the energy consumption and improves the work safety.
Directional nozzles are applied to adjust airflow parameters for the loose material specification and
container wall shape. The main task of these nozzles is to enhance efficiency of the pneumatic shock
phenomenon and also appropriate energy transfer of this shock into the loose material. Therefore, it is
necessary to employ methods for the limitation of pressure drop in the nozzle. Pressure drop is a
benchmark of energy loss, which is transferred into the loose material.
A continuous adjoint method is utilized to optimize the topology of the sample pneumatic pulsator nozzle.
While structure mechanics topology optimization with respect to tension or stiffness is a well-established
concept, the application of that concept in the field of computational fluid dynamics in general began since
2003. The method allows to decrease the pressure drop and to increase the flow field uniformity in a
closed channel by applying the Darcy's law of porous media in the computation fluid dynamics equations.
Application of the method is presented in the paper to optimize the shape of the pneumatic pulsator nozzle
by using the two-dimensional numerical model. There are indicators that should be employed to optimize
the nozzle design.
1. Introduction
Finding the best possible solution for a given problem is the meaning of optimization. The issue of
optimization is many times mixed up with a rationalisation or an improvement or, in general,
modernisation. The optimization substance is of finding the best, not only a better, problem solution.
Quoting Urbaniec (1979): “The word optimization is commonly meant as a determining of the best – or
perhaps even ideal – problem solution. The strict meaning consists of a restriction of the best solution at
the given circumstances, and furthermore, the best solution according to certain evaluation criteria.”
Optimization in mathematics is a finding of the function extrema (minimum or maximum) which is
substantially an objective function (Thévenin and Janiga, 2008).
Until recently, the optimization with using numerical methods such as Finite Element or Finite Volume
Method was a calculating a given prime model geometry, and subsequent model parametrization, iterative
calculations till the desire optimal geometry was obtained. That approach can be found, for example, in the
automotive industry at OpenFOAM International Conference – Proceedings and EnginSoft websites.
Whereas the utilization of the optimization methods in the field of solid mechanics according to stiffness
and tension is well-established, the application of direct optimization methods in fluid dynamics has been
started just a few years ago by Borrvall and Petersson (2003) novel work. This work considering Stoke's
flow optimization (Re<<1) is based on, in short, a porosity impact onto fluid flow behaviour and its
mathematical description.
In this work a novel optimization method is used to improve a shape of the pneumatic pulsator directional
nozzle. A continuous adjoint method refers to topology optimization, but not to the shape itself. The
topological space could not always be conformed in a physical meaning. Hence, result of the method
points flow zones which have an impact onto the flow conditions according to the given optimization
criteria. These zones are analysed in detail by using sensitivities that are results of sensitivity analysis.
2. Optimization
2.1 Objective function and restrictions
The optimization problem in the given example of nozzle can be stated as follows:


J α,u opt , popt = mu,ipn J α,u, p 
(1)
where u is velocity, p – pressure, and α is a porosity. For the case of the pulsator nozzle the aim is to reach
the pressure loss as small as possible. Furthermore, the air outflow is desired to be possibly uniform in
order to utilitize a full size of the nozzle. Hence, the objective function of the problem can be written:
J=
 u
n
 u n dSout +
Sout
 p dS
t
Sin
in

 p dS
t
out
(2)
Sout
 
uniformity
pressure loss
Providing as reported by Urbaniec et al. (2009) and Wolosz & Wernik (2011), the first attempt of the
optimization process is an assumption of incompressible air. It is very convenient assumption of a
restriction for the objective function. However, this assumption is only applied while optimizing and for the
sensitivity analysis. The optimization considers the flow area, hence, the restrictions of objective function
are related to equations governing fluid flow behavior:
R 
R   1 
 R2 
(3)
R1 = u  u + p    νu + αu
(4)
R2 =   u
(5)
where R are residuals of Navier-Stokes Eq.(4) and continuity Eq.(5) solutions respectively. The Eq.(3) is
extended by a Darcy's law component with porosity α included that is described for example in Puzyrewski
and Sawicki (1987) and recently numerically tested by Othmer et al. (2006).The residuals set creates the
objective function restrictions, thus Lagrange function can be formed according to Eq.(2) and Eq.(3):
L= J +
 u , p RdV
~~
(6)
V
From within the Langrange function we can extract Lagrange multipliers as follows:
u~, ~p  = u~1 ,u~2 ,u~3 , ~p 
(7)
These multipliers are namely adjoint velocity vector and adjoint pressure respectively.
2.2 Sensitivity Analysis
The continuous adjoint method for topology optimization is a searching for objective function minimum and
determining the conservation laws equations solutions simultaneously. The Eq.(4), as mentioned before, is
extended with the Darcy's law component of porosity. During a fluid flow through a porous medium a
velocity is restricted by porosity of the medium. A similar phenomena is simulated in the continuous adjoint
method where fluid is treated as a porous medium and each cell in considered area is assigned a value of
porosity αi. This value is changed according to the impact of that cell onto the objective function. Cells that
positively impacts on the objective function minimum have the porosity of 0. On the contrary, "bad" cells
are "punished" by assigning the porosity αi > 0. Each cell may be affected by iteratively increasing or
decreasing porosity according to its impact onto the objective function. From an engineering point of view,
the porosity is a design parameter that lets us to specify which zones are "harmful" to flow. The changes of
the objective function are the basis of the sensitivity analysis that can tell which cell are “good” or “bad”.
And consecutively, because the objective function depends on the porosity, velocity and the pressure, we
can state:
ΔL = Δα L + Δv L + Δp L
(8)
The whole idea of the adjoint method is to choose the values of adjoint velocity u~ and adjoint pressure ~
p in
such a way that the objective function changes which depend on velocity and pressure are simultaneously
equal to 0. And therefore, sensitivities can be calculated as follows:
L
J
R
=
+ (u~, ~
p)
dV
αi αi V
αi
(9)
J
R  u 
L
=0,
=  ,
= ui  u~i Vi
αi
αi  0  αi
(10)

3. Numerical simulations of the nozzle
3.1 Reference simulation
The optimized object is a nozzle that is an augmented equipment for pneumatic pulsator. The threedimensional view and a two-dimensional mesh applied is presented in Figure 1 below. Visualization of this
nozzle may also be seen in Wolosz (2013a) short film.
Figure 1. Directional nozzle of pneumatic pulsator with 2-D mesh and flow direction shown.
All numerical simulation is carried out by using open-source CFD toolbox OpenFOAM®, wider presented
at The OpenFOAM® websites. One of the application in this toolbox is sonicFoam which is perfectly suited
for supersonic airflow simulation. There is a reference simulation carried out at first to have a data to which
an optimized ones could be evaluated. This reference simulation was carried out by using the following
parameters and settings:
 air as a perfect gas,
 inlet velocity 500 m/s,
 pressure at inlet boundary 100 kPa,
 standard k-ε turbulence model,
 unsteady airflow,
 time step 10-6 s,
 simulation time 2·10-3 s.
The k-ε model described among others by Ferziger and Peric (2002) was chosen because of the flow
separation. There was also k-ω SST model considered, however instabilities during flow separation that
had to be taken into account constrained the turbulence to standard k-ε. These instabilities often produce
inaccurate and unstable solutions while using the k-ω model (Tu et al., 2008).
Detail results obtained during reference simulation are reporter on the figures below but readers interested
in scientific visualisation may visit Wolosz (2013b) website where animated shockwave expansion within
the nozzle is presented. The shockwave is visualized by using pressure gradient magnitude.
3.2 Optimization and sensitivity analysis
AdjointShapeOptimizationFoam is an application of OpenFOAM® toolbox with the continuous adjoint
optimization method applied. The basic advantage of OpenFOAM® as a whole is that it is open source
package and can be freely adjusted for a particular needs. It is very important for the issue of optimization
and the objective function definitions.
The improvement of pulsator nozzle is based on the result of sensitivity analysis of numerical calculation
with adjointShapeOptimizationFoam. The objective function presented by Eq.(2) was used, and
sensitivities described by Eq.(10) were calculated. Figure 2 shows the sensitivity equal to zero which
means that flow area placed within the isolines is “good” with respect to objective function. Flow areas
outside are “bed” on the contrary, and therefore they are partially removed in improved mesh model.
Comparison between reference and improved meshes is shown in the Figure 3 below.
Figure 2. Zero-sensitivity isoline applied on the Figure 3. Comparison between reference and
background of the reference model geometry.
improved mesh used for simulation.
Figures 4 and 5 shows the comparison between pressure and normal velocity respectively. The plots
visualize values of the pressure and normal velocity along the outlet boundary obtained by using reference
and improved mesh models. A large difference can be noticed at the first sight between the profiles of
each channel. The cause is obviously of the baffle which is unfortunately necessary with respect to
strength of the nozzle.
Uniformity index evaluates the divergence of outlet normal velocity and is calculated according to
Hinterberger and Olesen (2011) and references within with the following equation:
γ =1
1
2S out u n
 u
n
 u n dSout
(11)
Sout
where u n is velocity normal to the outlet boundary and the u n is desired velocity profile and set equal to
inlet velocity. The results of uniformity index calculation according to Eq.(11) for reference and improved
model are γ = 0.8508 and γ = 0.8637 respectively. So the conclusion can be stated that the flow unifomity
has incereased of 1.52%. A bigger improvement with respect to the pressure loss has been noticed. The
differences between mean pressure at inlet and outlet boundary for the reference and improved model are
equal to 77,719 Pa and 73,719 Pa respectively. So the improvement with respect to pressure loss is 6.3%.
Figure 4. Pressure distribution along nozzle outlet
for reference and improved numerical model.
Figure 6. Time dependency of the outlet pressure.
Figure 5. Outlet normal velocity distribution along
nozzle outlet for reference and improved
numerical model.
Figure 7. Time dependency of the outlet normal
velocity.
The optimization results can be observed in Figures 6 and 7, as well, There are the time-dependent
changes of mean outlet pressure and normal velocity shown. Smoother changes of the pressure and
velocity in time means that the loose material has more time to effectively take over the energy of
compressed air.
4. Conclusions
An application of an industrial equipment optimization workflow has been presented in this paper.
Utilization of open-source code and shape impovement has been reported according to the good
engineering praxis. The two-dimensional model has been used to test the methods and veryfiy the results.
To optimize a shape of an equipment there is a lot of work to be done. New methods however such as
continuous adjoint one are the big step to make the optimization really fast process. This evolutions
concerns especially optimization in fluid dynamics. The method of CFD gave to the engineers very
powerful tool to visualize, analize and predict the flow of medium. Continuous adjoint method give even
more: a possibility to obtain the answer of question “Where and what to change to be better then before”.
The presented numerical results of the adjoint method are obtained by using the incompressible model of
flowing gas. It is obvious that this model does not reproduce precisely the particular medium behaviour.
However, the method turns out to be valuable and worth further work.
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