CET Volume 86


 
 

 

                                                                    DOI: 10.3303/CET2186220 
 

 
 
 
 
 
 
 
 
 
 

 
 
 
 
 
 
 
 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Paper Received: 24 August 2020; Revised: 2 March 2021; Accepted: 8 May 2021 
Please cite this article as: Orellano Lasprilla J.L., Bayona Roa C.A., Lopez Gomez M., Gonzalez Caranton A.R., 2021, Analysis of a Static Mixer 
for an Aeronautical Test Bench of the Colombian Air Force: a Numerical Study of the Process Variables, Chemical Engineering Transactions, 
86, 1315-1320  DOI:10.3303/CET2186220 

CHEMICAL ENGINEERING TRANSACTIONS 

VOL. 86, 2021 

A publication of 

The Italian Association 
of Chemical Engineering 
Online at www.cetjournal.it 

Guest Editors: Sauro Pierucci, Jiří Jaromír Klemeš
Copyright © 2021, AIDIC Servizi S.r.l. 
ISBN 978-88-95608-84-6; ISSN 2283-9216 

Analysis of a Static Mixer for an Aeronautical Test Bench of 
the Colombian Air Force: a Numerical Study of the Process 

Variables 
Jose Luis Orellano Lasprillaa, Camilo Bayona-Roaa, Mauricio López Gomezb, 
Alberth Renne Gonzalez Carantóna* 
aGrupo de Investigación en Aprovechamiento Tecnológico de Materiales y Energía, Universidad ECCI, Bogotá, Colombia 
bGrupo de Investigación en Electrónica y Tecnologías Para la Defensa, TESDA. Escuela de Suboficiales de la Fuerza 
Aérea Colombiana, ESUFA Fuerza Aérea Colombiana, FAC, Bogotá, Colombia  
argonzalezc@ecci.edu.co 

Static mixing in conventional fuels is an industrial practice to obtain homogeneous liquid mixtures; due to an 
obstruction in the flow, the liquid-liquid phase mixtures become homogeneous, since a higher intimate contact is 
allowed between the mixing substances. This work displays a hydrodynamic simulation of a static mixer using 
FAME biodiesel and JETA-1 blends with a percentage in volume up to 25% of biodiesel. The main effects on 
the concentration distribution are obtained as a function of the % of substitution of biofuel and the internal 
mixer geometry, thus exhibiting different hydrodynamic responses and output properties for the blends. These 
properties serve as a preliminary reference before achieving experimental measurements in the laboratory. 
Keywords: static mixer; JetA-1; Biodiesel, hydrodynamic distribution. 

1. Introduction
Mixing of fluids is a relevant industrial operation, especially when it is applied to blending systems in refineries 
or food and plastic processing. In industrial applications, where a mixed homogeneous system is required to 
obtain the final product, mixing plays an important role in terms of the quality of the final mixture, energy costs, 
and turnover of the process. Mixing is carried out in two different ways: one includes a mechanical agitator —
known, as dynamic mixers—, and another without the agitator, known, as static mixers (Singh et al., 2009). 
Static mixers are, thus, efficient devices used for promoting the homogeneous mixing of single and multiphase 
fluids (Hobbs & Muzzio, 1997). Dynamic mixing  (Rauline et al., 2000) is frequently applied when the inlet flow 
enters in a turbulent regime (Theron & Sauze, 2011). On the other hand, for laminar flow regimes and highly 
viscous fluids, dynamic mixers require a huge amount of power to mix the fluid which makes them an expensive 
choice compared to motionless mixers. For viscous fluids in the laminar region, static mixers need less power 
and perform better than the conventional agitator mixers in terms of mixing operation (Lindenberg et al., 2008). 
The driven force of static mixers is the kinetic and potential energy of the fluid, based on the momentum of the 
flowing fluid (Zalc et al., 2002). Additionally, several reductions in costs can be achieved for multiple applications 
of interest, where cost reduction can be significant. Understanding the physical phenomena of the flow inside 
the static mixer is a key ingredient in the development of energy reduced technologies involving the mixing of 
two fluids (Saatdjian et al., 2012). This effort is related to completely describing the separated flow fields 
generated by the different transport properties of both fluids and the flow patterns between them under the 
complex conditions of viscous flows (Kumar et al., 2008). Several configurations of static mixers have been 
proposed, however, the fluid dynamics effects inside this equipment still need to be studied (Berkman & 
Calabrese, 1988). T his is required in order to assure a perfect blend, where the most diluted phase must be 
distributed homogeneously within the solvent (Rabha et al., 2015). The incompressible Navier-Stokes 
equations portray a mathematical model that completely describes the two fluid flow. The description made 
by these equations have the advantage of representing the turbulent Eddies in the continuum limit. However, 
the main disadvantage is that the analytical solution of those equations is almost impossible for real applications 

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(Coroneo et al., 2012). Thus, two different approaches can be used instead for representing the turbulent flow inside 
the static mixer: the first is by means of experimental methods (Theron & Sauze, 2011) whilst the second uses 
numerical simulations (Hanada et al., 2016). In the present work, the flow description can also be achieved with 
the second approach: computational simulations, which are inexpensive and fast when used for new designs 
to predict the system integrity, and calculate real time responses. A finite element variational multiscale 
formulation is developed to stabilize the numerical solution of the fluid flow problem (Hughes, 1995), particularly, 
for the spatial approximation, which may suffer from instabilities arising from equal interpolation spaces for 
unknown velocity and pressure, and convection dominant type of flows. This approach shows accurate and 
stable numerical descriptions of the two fluid flows modelled by the concentration transport equation, and by 
including an implicit turbulence modelling in the line of the Implicit Large Eddy Simulation methods in (Bayona 
et al., 2018). Numerical developments are applied to simulate a static fuel mixer which reduces the risk of 
saturation present in conventional agitators. This article is organized as follows. In Section 2 we present the 
static mixer flow setting. Next, in Section 3, we describe the computational methodology used, to numerically 
simulate the annular flow. Then, in Section 4 we present the numerical results of a complete study of the 
resulting variables inside the static mixer for different blends. Finally, some concluding aspects about the 
computational methodology of studying the flow setting are stated in Section 5. 

2. Static mixer and operation procedure
2.1 Simplified geometry and operative conditions 

The static mixer geometry is presented in Figure 1, as well as the main components of the mixer, which are the 
pipe, the injection nozzles, and the obstructions. The pipe has a diameter d of 1m and a total length L of 10m. It 
is considered as a two dimensional flow, with an infinite extension in the depth dimension and the baffles 
perpendicularly located with respect to the axial direction of the flow. The inlet for the JETA-1 fuel is located at 
the left wall, where the velocity data is known and prescribed. Two inlets for the biodiesel fuel are located at the 
upper and lower channel walls and defined from the left-most wall up to a certain horizontal distance b: the 
Biodiesel is injected in a perpendicular fashion through these inlets in order to represent a jet injection of the 
solute into the solvent. The first baffle is horizontally separated l 1 = 2m from the inlet. Each baffle has a width 
of 0.05 m and the axial distance between consecutive baffles is set to be l 2 = 0.9m, or exactly l 3 = 1.8 m 
between baffles at the same side. The turbulent regime load is of main interest, especially the Reynolds 10000 
case, for which turbulent effects are relevant in the mixing process. Although turbulence is a three-dimensional 
process, as commented before, the numerical method has an implicit model of turbulence despite the two 
dimensional solution. Thus, the JETA-1 velocity at the inlet is fixed at u = (0.01399) m/s. This same speed is set 
for the biodiesel injection. Nevertheless, the horizontal length b (meaning the transversal area of the injected 
jet) is modified according to the mass concentration ratio. This study evaluates fuel blends with a percentage 
in volume up to 25% of biodiesel. In this sense, four different blends are simulated: 5%, 10%, 15% and 25% 
percentage in volume of biodiesel in JETA-1.The horizontal length b gives 0.026m, 0.055m, 0.088m, and 
0.166m, respectively, for these blends and the fixed inlet velocities. 

2.2 Mixing Rules 

Some mixing rules are used for the description of some mechanical fluid properties of interest, such as: 
density ρ, kinematic viscosity ν, viscosity µ and Lower Heating Value (LHV). The most important quantity is the 
mass fraction of each fluid content. In this sense, the mass fraction of JETA-1 is given by the X J variable, while 
the mass fraction of FAME biodiesel is X B for the mixed compound, the fluid properties are given in terms of 
the mass fractions as, 
ln(ρM) = XJ ln�ρJ� + XB ln(ρB) (1) 
ln(µM) = XJ ln�µJ� + XB ln(µB) (2) 
ln(LHVM ) = XJ ln�LHVJ� + XB ln(LHVB) (3) 
Where the subscripts J and B refer to the JETA-1 fuel and FAME biodiesel, respectively.  
Even though the mixer geometry is not axisymmetric due to the flow obstructions, it is considered as a two 
dimensional flow only for engineering purposes. This simplification aims to produce a fast output of the mixed 
bulk properties at the outlet, without resorting to a costly three-dimensional simulation. Also, by assuring the 
turbulent description of the flow in the two-dimensional framework given by the Implicit Large Eddy Simulation 
from the Variational Multiscale stabilized finite element formulation (Bayona et al., 2018). Hence, the two-
dimensional geometry is considered with an infinite extension in the depth dimension and the baffles 
perpendicularly located with respect to the axial direction of the flow. The fluid properties for each pure 
compound are given in Table 1. 

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Table 1: Pure fluid properties 

. 

Figure 1: Geometry of static mixer. 

2.3 Mass diffusivity relation 

The Wilke-Chang equation is used for the diffusivity calculation of liquid biodiesel into liquid JETA-1. This 
relation is given by ϵB = CD

ψJMJT
µMVB

, where CD = 7.4 x 10
-16 m4/s2K is a chemical constant, ψ J = 1 is the molecular 

association parameter relating the molecular interaction between solvent and solute, M J  = 0,186 kg/mol is the 
molecular weight of JETA-1, T = 298.15 K is the flow temperature, and V B  = 0.05181068 m3/mol is the molar 
volume of biodiesel (solute) at its normal boiling point. 

3. Computational methodology
The Python library dolfin, from the FEniCS Project (Alnæs, M et al., 2015), is used to formulate and 
numerically solve the problem in variational form. The FEniCS Project provides a novel tool for the automated 
solution of partial differential equations by the finite element method that we exploit. Next, we present the finite 
element formulation which is capable of representing the transient two fluid flow. 

3.1 Navier-Stokes equations for the fluid flow 

The Wilke-Chang equation is used for the diffusivity calculation of liquid biodiesel into liquid JETA-1. In terms 
of the hydraulic description, the flow velocity and pressure fields are completely described by the 
incompressible Navier-Stokes equations. These relations are given by 

ρM  ∂t𝐮𝐮 +  ρM (𝐮𝐮 ∙ ∇)𝐮𝐮 + ∇p − 2µM∇ ∙ (∇u) = ρM𝐛𝐛, ∀𝐱𝐱 ∈ Ω, t ∈ (0, tf ), (4) 

∇ ∙ 𝐮𝐮, ∀𝐱𝐱 ∈ Ω, t ∈ (0, tf ), (5) 

where Ω ⊂ R2 is the computational domain, and (0, tf) is the time interval in which the problem is solved. The 
bulk properties of the mixed flow are calculated using relations (1) - (3). The fluid velocity is ui  : Ωi  × (0, tf ) → 
R2, the pressure is p : Ωi  × (0, tf ) → R, and the force vector is ρ M b : Ω → R2, which is currently neglected. 

3.2 Transport equation for the fluid concentrations 

The mixing process is modelled by using a Convection-Diffusion-Reaction transport equation for the fluid 
concentrations of Biodiesel as a miscible fluid in the JETA-1 flow. Thus, the equation for the biodiesel 
concentration in the JETA-1 flow is the following: 

∂txB +  𝐮𝐮 ∙ ∇xB − ∇ ∙ (ϵB∇xB) = 0, ∀𝐱𝐱 ∈ Ω, t ∈ (0, tf ). (6) 

Complementary, the Jet A-1 mass fraction X J can be calculated from the total mass relationship X J + X B  = 1. 
The initial condition of the transport equation is chosen to define the initial concentration ratio between the two 
fluids. In this sense, a strong Dirichlet boundary condition ζG is imposed on the inlet boundaries ΓG: a 
separate injection for the FAME biodiesel is set along the pipe radial contour, where χB = 1. In the main 
JETA­1 injection nozzle, this variable is strongly set to χB = 0. These strong conditions define the inlet blend 
ratios for the static mixer setting. The mixing rules, together with the mass and momentum balances that 
include the mixed properties in the bulk flow, guarantee a complete description of both miscible fluids in the 
mechanical problem. 

Property Units JETA-1 FAME Biodiesel 
Density kg/m3 809,49 841,26 

Kinematic viscosity mm2/s 1.4 4.5 
LHV Pa∙s 1.133x10-3 3.786x10-3 

Density MJ/kg 42.9 37.08 

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3.3 Finite Element approximation of the coupled problem 

Let us first recall the space of functions H(div, Ω) ≑ (u ∈ (L2 (Ω))d ∣ ∇∙u∈L2 (Ω)), which is the space where the 
velocity u lives V ≑ H(div,Ω). Now, let us denote by h = Ωe the finite element partition of the domain Ω, with 
index e ranging from 1 to the number of elements n el  in the finite mesh. The diameter of the element 
partition is denoted by h. We define the finite test function spaces V h  ⊂ V and Q h  ⊂ Q ≑ L2 (Ω) as made of 
continuous piecewise polynomial functions in space. The Galerkin approximation of the coupled version of (4), 
(5) and (6) is to find [µ h , p h , χ B,h ] ∈ V h × Q h  × Q h  such that 

(ρM ∂t𝐮𝐮h, 𝐯𝐯h) + (∂txB,h, ηh) + A([𝐮𝐮h, ph, χB,h]; [𝐯𝐯h, qh, ηh]) = L([𝐯𝐯h, qh, ηh]), ∀[𝐯𝐯h, qh, ηh] ∈ Vh × Qh × Qh,  (7) 

where (∙,∙) is the integral of the product of two functions (scalar or vector valued) in the Ω domain. Then the 
variational forms are defined as: 

A([𝐮𝐮, p, χB]; [v, q, η]) = a(𝐮𝐮, 𝐯𝐯) − b (p, 𝐯𝐯)  +  c (𝐮𝐮; 𝐮𝐮, 𝐯𝐯) + b (q, 𝐮𝐮) + ac(χB, η) + cc(u; χB, η), (8) 

L([𝐯𝐯, q, η]) = l(𝐯𝐯), (9) 

and: a(u, v) = µM ∫ ∂jui ∂jvidΩΩ , ac(χB, η) = ϵB ∫ ∂iη ∂iχBdΩΩ , b(q, 𝐯𝐯) = ∫ q ∂ividΩΩ , 

c(𝐮𝐮,� 𝐮𝐮, 𝐯𝐯) = ρM ∫ vjv�i ∂iujdΩΩ , cc(𝐮𝐮; χB, η) = ∫ ηui ∂ixBdΩΩ , l(𝐯𝐯) = ρM ∫ fvidΩΩ  with u, v ∈ V and p, q, χB, η ∈ Q. 

In the discrete sense, the problem is to seek the discrete solution [u h , p h , χ B,h ] ∈ Vh × Qh × Qh, such that: 

(ρM ∂tuh, vh) + (∂tχB,h, ηh) + A([uh, ph, χB,h] ; [vh, qh, ηh])
+ AS(𝛕𝛕([𝐮𝐮h, χB,h])𝐑𝐑([𝐮𝐮h, ph, χB,h]) ;  [𝐯𝐯h, qh, ηh]) = L([𝐯𝐯h, qh, ηh]) , (10) 

for all [v h , q h , η h ] ∈ V h  × Q h  × Q h . The term A S (τ([u h , x B,h ])R([u h , p h , x B,h ]); [v h , q h , η h]) is a consistent 
Variational Multi-Scale stabilization term, which is added to overcome the instability problems when the 
standard Galerkin formulation is applied to the Navier-Stokes problem (Codina & Blasco, 1997). The 
stabilization term incorporates a matrix of stabilization parameters τ([u h , x B,h ]) that depends on the unknown 
values, and the residual of the finite element approximation (in vector form) R(·). We adopt the diagonal 
definition τ([u h , x B,h ]) = diag (τ 1 (u h , χ B,h )I,τ 2 (u h , χ B,h ),τ 3 (u h , χ B,h )) that was proposed in (Codina & Blasco, 
1997), with the 2×2−identity tensor as I, and the components of the diagonal matrix of stabilization parameters 
as τ1−1�𝐮𝐮h,xB,h� = C1µM h2⁄ + C2ρM |𝐮𝐮h| h, τ2−1�𝐮𝐮h,xB,h�⁄ =  C3µM +  C4ρM|𝐮𝐮h|h, and τ3−1�𝐮𝐮h,xB,h� =
C1B h2⁄ + C2�𝐮𝐮h,� h,⁄  respectively. The term h is the diameter of the element partition, and C1  = 12, C2  = 2, C 3
= 1, and C 4  = 1 are numerical parameters. Since the present work is restricted to linear order finite elements, 
the stabilization variational form is defined as follows: 

AS �𝛕𝛕��𝐮𝐮h, χB,h�� 𝐑𝐑��𝐮𝐮h, ph, χB,h��; [𝐯𝐯h, qh, ηh]�  
≑ τ1(−ρM𝐮𝐮h ∙ ∇𝐯𝐯h − ∇qh, ρM ∂t𝐮𝐮h − ρM𝐮𝐮h ∙ ∇𝐮𝐮h − ∇ph) + τ2(−∇ ∙ 𝐯𝐯h, −∇ ∙ 𝐮𝐮h)  
+ τ3�−𝐮𝐮h ∙ ∇ηh, ∂tχB,h − 𝐮𝐮h − ∇ ∙ χB,h�. 

(11) 

In the case of the first-order temporal derivative of the fluid flow problem (4) and (6), the second order 
Backward Differentiation Formula (BDF) scheme is used. The Newton method as the linearization 
scheme for solving the non-linear coupling at each time step. 

4. Results
A finite element mesh composed of 150463 linear triangular elements and 76394 total nodes is used to solve 
the discrete fluid flow problem. According to the time-averaged numerical results, which are presented in 
figures 2-5, the following observations are stated. Regarding the flow field, some variations on the velocity and 
pressure profiles for values up to 25% of biodiesel in the blend are displayed in Figure 2(a). These variations 
are visible when contrasting the lower (top) and higher (bottom) concentrations of biodiesel in the blends. 
According to the time-averaged results, the edge effects of flow patterns can be considered as important 
before the obstructions. A maximum magnitude of the fluid velocity appeared in the regions where the fluid 
gets in contact with the obstruction, thus validating the efficiency of the static mixer operation, as shown in 
Figure 2(a). 

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(a) (b) 

Figure 2: Time-averaged flow field results for the 5%, 10%, 15%, and 25% blends: (a) flow field (b) mass 
diffusion. 

The increased fluid velocity can be considered as a significant consequence of the convective processes in 
the flow. The present approach, in which different concentrations and several obstructions in the flow are 
implemented, guarantees a perfect mixing produced by the convective effects. Indeed, these effects are tightly 
associated with mass, heat, and momentum transfer. The fluid flow results show that for the inlet region of the 
tube, large gradients occur in a finite and small region near to the perpendicular injection zone where the fluids 
are in contact. It can be observed that diffusion occurs right from the start of this mixing process. However, 
Figure 2(b) evidences that the expected theoretical mixing between FAME biodiesel and JETA-1 is not 
achieved with the geometric configuration used for the computational simulation. This deficiency in the mixing 
becomes more evident as the percentage of FAME biodiesel increases in the mixture. In consequence, the 
geometric configuration needs to be improved in terms of speed profile changes and moment transfer. Density 
profiles along the tube demonstrate that the predominant value is given by the JETA-1 major compound in the 
mixture. In the case of the 25% of biodiesel injection, significant changes were demonstrated by the results. A 
maximum value of mass density of 815 Kg/m3 is evidenced in the mixture. These results are obtained, despite 
the fact that Figure 3 (a) and (b) validate the mathematical model used to carry out the computational 
simulations obtaining physically-accurate results. However, for the estimation of the mass density, a 
correlation of mixture with shrinkage loss parameters could be used along with the fluid flow. This may enable 
a complete fluid flow visualization of mass density during the convective phenomena and the main effect of 
obstruction in the mixing process. 

(a) (b) 

Figure 3: Time-averaged density (a) and viscosity (b) results for the 5%, 10%, 15%, and 25% blends. 

The viscosity of the blend demonstrates more significant gradients: for values up to 25% of biodiesel, it 
exhibits viscosity gradients near to the region of injection of FAME biodiesel. Since the proposed model 
assumes that the tube is initially filled with the majority compound JETA-1, it is not possible to notice a 
shrinkage or a deformation effect of one fluid into the other. In the corners of the tube, large viscosity gradients 
can be appreciated, however, the viscosity profile is homogeneous without significant effects of baffles and with 
maximum values of 0,0018 Pa*s for dynamic viscosity in the axial length. For each of the FAME biodiesel and 
JETA-1 blends evaluated, the decrease in the Lower Heating Calorific value is minimal. This variable is 

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considered of great relevance in this type of analysis, because it allows determining the energetic parameters 
of the aviation test bench. The results are concordant with experimental measurements made over 
JETA-1/Biodiesel blends with the same blending proportions, exhibiting values between 41 and 43 MJ/Kg. 

5. Conclusions
Some highlights in the mixing process of a static mixer have been observed when numerically solving two-fluid 
dynamics models given by the Navier-Stokes equations for incompressible fluids. The developed numerical 
strategy enables computational fluid dynamic results, which showed spatial gradients in the convective 
phenomena of biodiesel mixing in JETA-1 fuel. Velocity gradients can be associated with the effect of baffles 
over the natural path of both mixed phases, however, the computational model exhibited no significant 
distortions and contractions of both mixing fluids, nor the consequences of this phenomena over transport 
properties, like mass density and dynamic viscosity. This can be explained by the mass correlation that have 
been used for these properties in the physical and mixing model.  

Acknowledgments 

This research was financed by COLCIENCIAS with resources from the autonomous patrimony National Fund 
of Financing for Science, Technology and Innovation Francisco José de Caldas, within the framework of the 
project entitled "Evaluation of the behaviour of Colombian biofuel mixtures in aeronautical turbines”. The 
authors also acknowledge the Colombian Air Force and ECCI University support.  

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