HUNGARIAN JOURNAL OF
INDUSTRY AND CHEMISTRY
Vol. 47(2) pp. 63–70 (2019)
hjic.mk.uni-pannon.hu
DOI: 10.33927/hjic-2019-21

INVESTIGATION OF MIXING IN TANKS OF A SPECIAL GEOMETRY

BÁLINT LEVENTE TARCSAY*1 , ATTILA EGEDY1 , JANKA BOBEK2 , AND DÓRA RIPPEL-PETHŐ2

1Department of Process Engineering, University of Pannonia, Egyetem u. 10, Veszprém, 8200, HUNGARY
2Department of Chem. Eng. Sci.s, University of Pannonia, Egyetem u. 10, Veszprém, 8200, HUNGARY

Mixing is one of the most crucial processes in the chemical industry. Homogeneity is a requirement for all feedstocks and
industrial products. The degree of mixing depends on the hydrodynamic properties of the fluid in the units. The Residence
Time Distribution (RTD) was investigated in a tank of a special geometry. Mixing was investigated using various geome-
tries of the tank by applying the Heaviside function in step-response experiments. After obtaining experimental results,
the RTD function was calculated. The flow structure in the tank was approximated by fitting black-box transfer function
models onto the RTD function of the system. Two general model structures were defined and their fitness compared. By
evaluating the fitted models, a relationship was established between the flow structure in the tank and its geometry.

Keywords: tanks with of special geometry, mixing, flow pattern, transfer function model

1. Introduction

In industrial processes, the thorough homogeneity of ma-
terials is a necessity to ensure the quality of the fin-
ished products and the safety of the production process.
Therefore, various industrial-scale techniques for the ho-
mogenisation of materials of various densities and phases
have been developed over the years. In the case of fluids,
homogenization can be achieved through the application
of mixers in a device. The optimal type and geometry of
these stirred tanks can be approximated by specific mod-
els which take the properties of the materials to be mixed
and the desired degree of homogeneity into considera-
tion [1]. In the case of the homogenisation of large vol-
umes of liquids (petrochemical and radiochemical indus-
tries), external recirculation can be more advantageous
since it is considerably more inexpensive in terms of both
installation and operation. The operation of stirred tanks
has been thoroughly investigated by using mathematical
and experimental methods, however, research on mixing
achieved by external recirculation is lacking [2]. Opti-
mization of the mixing process to achieve the most ho-
mogenous product possible in an external recirculation
tank can only be achieved after the flow structure of the
unit is known. The aim of this research was to analyze the
flow structure in a tank of special geometry, moreover, to
determine the connection between the flow structure and
geometry of the tank.

This research forms the basis for future investigations
where the flow structure and homogenization in a tank
will be tested through an external recirculation mixing

*Correspondence: tarcsaybalint95@gmail.com

process. Mixing in stirred tanks can traditionally be ana-
lyzed by step-response experiments.

Step-response experiments are a widely accepted and
simple way of determining the flow structure of process
units [3]. During these experiments, the input of the sys-
tem is a tracer fed according to the Heaviside function.
The time evolution of the tracer concentration in the out-
put of the tank holds information regarding the flow and
mixing properties of the system. The Heaviside function
is the integral of the Dirac delta function δ(t):

S(t) =

∫ ∞
−∞

δ(t)dt =

{
0 if t < t∗

1 if t ≥ t∗ (1)

used as an input in pulse-response experiments [4]. The
Heaviside function, S(t), as a function of time takes on a
value of 0 if the investigated time t is less than the time of
the tracer feed t∗ and a value of 1 during and after being
fed.

After experimental data was collected on the response
function of the system, the residence time distribution
function (RTD) could be calculated. The RTD function
approximates the probability that the residence time of
the fed tracer agent is less than a specified residence time
[4]. The RTD function, F , was calculated from

F(t) =
Rt − Rt−1
Rsteady − R0

, (2)

where Rsteady and R0 are the values of the response func-
tion in the steady state and at the beginning of the exper-
iment, respectively.

By analyzing the RTD curve in the studied unit, mix-
ing and flow can be characterised and compared to the

https://doi.org/10.33927/hjic-2019-21
mailto:tarcsaybalint95@gmail.com


64 TARCSAY, BOBEK, EGEDY, AND RIPPEL-PETHŐ

Figure 1: Response functions of various ideal flow models
[3].

flow structure of ideal hydrodynamic models [5, 6]. In
this study, the ideal hydrodynamic models, which were
used to describe the response, include the plug flow reac-
tor (PFR) and tanks-in-series (TIS) model.

The PFR model is useful for modelling systems with
a continuous flow of cylindrical geometry. It works under
the assumption that the flow through a PFR is perfectly
mixed in the radial direction but not in the axial direc-
tion. It is also assumed that the residence time of each
fluid unit within the system is equal to the mean residence
time, τ. Because of these conditions, the step response
function of a system with PFR tendencies is identical to
the step function which was used as an input but with a
time delay equal to the mean residence time in the unit.
An altered version of the PFR model is the laminar flow
reactor (LFR) model. The LFR model assumes a laminar
flow structure within the system. Therefore, the residence
time of the fluid phase within the system has a distribu-
tion. This causes the shape of the response function to be-
come slightly altered compared to the step function. The
TIS model is an extension of the continuous stirred tank
reactor model (CSTR) which defines a system that is per-
fectly mixed and, therefore, homogeneous. Upon entering
the unit, the tracer is dispersed throughout the whole tank,
therefore, it immediately appears in the output without a
time delay. The TIS model examines a series of CSTR
units. The RTD function in this model is dependent on the
number of tanks that have been linked. The RTD function
of the TIS model converges towards the PFR model as the
number of units increases and towards the CSTR distri-
bution as the number of units decreases. One tank returns
the RTD function of a CSTR and as the number of tanks
in series approaches infinity, the RTD function converges
towards the PFR model. The characteristic RTD curves of
the aforementioned ideal flow models are shown in Fig.
1 [3]. In this figure, t∗ denotes the duration of tracer in-
jection and dt represents the time delay when the tracer
appears in the outlet.

In this investigation, black-box models were used to
approximate the behavior of the tank. Black- and white-
box modelling are two distinct modelling techniques to

describe the traits of various systems. White-box models
utilize well-known physical and chemical laws to charac-
terize a system. These laws come in the form of differen-
tial equations, algebraic equations, etc. The model defines
the internal functioning of the system and calculates the
specific output from the input by taking these into con-
sideration. Generally in the case of complex mixing prob-
lems, computational fluid dynamics (CFD) simulators are
utilized to solve the white-box model of the system [7, 8].
CFD simulations can determine the flow field in various
units [9].

These simulators can be used to solve scientific prob-
lems from a diverse spectrum of disciplines, e.g. aero-
dynamics, hydrodynamics, ocean engineering, chemical
engineering, etc. [10]. To solve these problems, various
numerical methods have been invented over the years
among which the finite element method (FEM) is one of
the more frequently applied approaches [11]. However, in
cases where the FEM is insufficiently advanced, alterna-
tive numerical methods can also be implemented, e.g. the
spectral element method (SEM) [12]. Their disadvantage
is, however, that since they model the systems so thor-
oughly, they require considerable computational power
and computation time. Through these methods, it is pos-
sible to analyze processes which are more complex or re-
quire long experimentation times to be dissected. Despite
this, these methods also require experimental data for val-
idation.

Black-box models, on the other hand, provide gener-
alized means for the modelling of various problems [13].
They often employ empirical methods to calculate the
output of a system from the input without thoroughly de-
scribing the internal properties of the system [14]. The
relationship between the input and output can be mod-
elled using various techniques with the most popular be-
ing neural networks as well as state-space and transfer
function models [15]. In this study, the transfer function
model was utilized to approximate the behavior of a tank
of special geometry.

2. Experimental and the modelling pro-
cess

The geometry of the investigated tank is shown in Fig. 2.
The tank consists of 5 cylindrical units integrated with
each other. A removable wall was implemented which
makes it possible to create the desired number of cylin-
ders from the main tank. Therefore, the number of cylin-
drical units can be varied between 1 and 5. During the
investigation, geometries consisting of 1, 3 and 5 cylin-
drical units were investigated. The residence time, cal-
culated from the inlet volume flow rate and volume of
the liquid, was 3 h for all geometries. Therefore, the in-
let volume flow rate changed as the volume of liquid and
number of cylindrical units varied. To obtain the response
function of the system, a Heaviside function was used.
The utilized tracer was a borax solution of low concentra-
tion (6.24 gl−1). The tracer was fed into the tank which

Hungarian Journal of Industry and Chemistry



MIXING IN TANKS OF A SPECIAL GEOMETRY 65

Figure 2: The investigated tank.

was filled with a high concentration of borax solution
(29.16 gl−1).

Conductivity measurements were used to monitor the
change in concentration and obtain the response function.
The experimental equipment is shown in Fig. 3. The ex-
perimental conditions for the feed and residence time are
shown in Table 1.

The positions of the inlet and outlet during the exper-
iments are shown in Table 2. The (0,0,0) coordinates of
the system were defined as the bottom of the tank at the
position of the outlet. The coordinates x, y and z cor-
respond to the longitudinal, vertical and horizontal dis-
tances, respectively.

The RTD curves of the three different geometries of
the tank were investigated. After experimentally obtain-
ing the data from the RTD curves, transfer function mod-
els were fitted to the experimental results. Two types of
fitted functions were investigated. Both were black-box
convolution models of the PFR and TIS. However, one
contained only serial links between the perfectly mixed
units, the other assumed parallel connections as well. Dif-
ferential equations describe the change in concentration
of the output of the tank as a function of the input. In var-
ious ideal flow models, these equations were converted
into algebraic equations by using the Laplace transform.
After this the transfer function of the ideal flow mod-

Figure 3: Setup of the experimental equipment (1–
experimental tank, 2–tracer containment tank, 3–
peristaltic pump, 4–buffer unit, 5–rotameter, 6–output
containment tank).

Table 1: Experimental parameters.

Number of Cylinders τ (h) B (lh−1)
1 0.4

3 3 1.1

5 1.7

Table 2: Inlet and outlet positions.

Number of
Cylinders

Inlet position Outlet position

x
(mm)

y
(mm)

z
(mm)

x
(mm)

y
(mm)

z
(mm)

1 75 100 0 0 2 0
3 225 100 0 0 2 0
5 375 100 0 0 2 0

els was defined as the ratio of the output (the Laplace
transformed response function) to the input of the system
(the step function of the tracer injection). If the product
of multiple elementary transfer functions is computed, a
TIS model can be generated. A general model structure
for this network can be seen in Fig. 4.

On the other hand, by combining the elementary
transfer functions of ideal flow models, a network of
flow models with parallel links can be created. A general
model for this structure is shown in Fig. 5. In both cases,
a PFR unit has been included in the structure, in general
this is useful for modelling systems with a time delay. The
time delay in the RTD function could point to either PFR
tendencies in the flow structure or the presence of dead
volumes within the tank. To distinguish which is present,
further investigations with a CFD simulator are required.

The general form of the transfer function that only
uses serial connections is

Gfitted =
a

bnsn + bn−1sn−1 + · · · + b0s0
e−tds, (3)

where n denotes the order of the denominator and bn, a,
and td are parameters of the transfer function.

In the case of the model containing both parallel and
serial links, the general transfer function is

Gfitted=
an−1s

n−1 + · · · + a0s0

bnsn + bn−1sn−1 + · · · + b0s0
e−tds, (4)

where n is the order of the denominator and bn, an, and
td are parameters of the transfer function. However, due
to the parallel connections, the elementary transfer func-
tions are not just multiplied but totalled. Therefore, the

Figure 4: General model structure involving only serial
links [5, 6].

47(2) pp. 63–70 (2019)



66 TARCSAY, BOBEK, EGEDY, AND RIPPEL-PETHŐ

Figure 5: General model structure involving both serial
and parallel links [5, 6].

nominator also contains a higher order polynomial. The
order of this polynomial can be one at most below the
order of the denominator for the system to be suitable.

In both cases, the fitted RTD curve Ffitted was cal-
culated from the step response of the system which was
characterized by the transfer function, Gfitted.

The overall parameters of the transfer function were
determined by numerically solving a minimization prob-
lem. The minimization problem is expressed in

min [E (n,a,b,d)] =

∑
[Fexp − Ffitted (n,a,b,d)]

2∑
(Fexp)

2

(5)
where E is a function denoting the sum of the squared
difference of the experimental and fitted RTD functions
divided by the sum of the square of the experimental RTD
curve. As such, E is a function similar to the relative error
of the fitting which is dependent on the parameters of the
transfer function, Gfitted.

Inversely, the goodness of the fit, I, can be defined as
the complementer of the fitting error:

I (n,a,b,d) = 1 − E (n,a,b,d) . (6)

For the minimization of the function, the interior-point
method was applied in MATLAB R2011. Constraints
were defined for the lower and upper boundaries of the
parameters, namely 0 and 100 for all parameters, respec-
tively. The fitting algorithm was modified to favor trans-
fer functions with the smallest possible order and smallest
relative error. The algorithm can be seen in Fig. 6.

The algorithm follows a general route for fitting both
parallel and serial models. During the first step, a low-
order (first-order) transfer function is created. After that
the parameters of this transfer function are identified us-
ing the interior point method. After obtaining the param-
eters, the E function is calculated to evaluate the error of
the fitting. If the error is less than a predefined threshold

(Erropt), then the fitting will stop and the current trans-
fer function be defined as the best possible fit. If the error
of the fit exceeds the defined threshold, then the fitting
continues. In this case, the order of the denominator is
increased and the error of the higher order transfer func-
tion evaluated as well. If the error of fitting is less than
the lower order transfer function, then the fitting contin-
ues until either a transfer function which satisfies the er-
ror boundary is identified or the error of the fitting does
not decrease significantly even though the order of the
denominator increases.

In the case of models which include parallel links,
this algorithm contains an additional iteration step. Dur-
ing this step, the algorithm investigates the fit of an nth
order transfer function (if n ≥ 2) with varying orders
of the nominator. To achieve this, the fitting is evaluated
by applying increasing orders of the nominator. During
this process, either the maximum order of the nominator
(n − 1) is reached or a point where the increase in the
order of the nominator is identified that does not signif-
icantly impact the fitting error. In either case, fitting is
stopped and the solution accepted as the best possible fit
for a transfer function with order of the denominator, n.

To evaluate the proposed fitting algorithm for both
models, a custom-defined transfer function was analyzed.
This reference transfer function (Gref is defined by

Gref =
1

1.6s2 + 2.6s + 1
e−5s (7)

in the case of the model contains only serial links.
The step response function for this transfer function

was used as reference data. The extrema of function E
were numerically determined by using the devised algo-
rithm. The fitted transfer function is

Gref =
1

1.58s2 + 2.59s + 1
e−5s. (8)

Fig. 7 shows the fitting of the RTD of the identified trans-
fer function to the reference RTD function. The goodness
of the fit was 98.2 %.

The testing was also conducted in cases where paral-
lel links are present besides serial links. In this case, the
reference system is characterized by the transfer function:

Gref =
8.6s + 1

s4 + 2.5s3 + 11s2 + 10s + 1
e−2s. (9)

The fitting algorithm was augmented to identify both the
optimal orders of the nominator and denominator. The
fitted transfer function is

Gfitted =
8.95s + 1

s4 + 2.37s3 + 11.14s2 + 10.33s + 1
e−2s.

(10)
The results of the fitting are displayed in Fig. 8. In this
case, the goodness of the fit exceeded 99 %.

Both Fig. 8 and the measurements of the parameters
show that the algorithm is capable of precisely fitting both
models, therefore, it is suitable for identifying the trans-
fer functions of the system. After validating the fitting

Hungarian Journal of Industry and Chemistry



MIXING IN TANKS OF A SPECIAL GEOMETRY 67

Figure 6: The general applied fitted algorithm for a model containing both serial and parallel links.

Figure 7: The results of the algorithm validation for fitting
the model containing only serial connections.

Figure 8: The results of the algorithm validation for fitting
the model containing parallel and serial links.

method, transfer functions were identified for the tank by
using the experimental RTD curves.

3. Results and Analysis

The results of the fittings in the case of the model with
only serial connections can be seen in Figs. 9–11 for all
three geometries of the tank that were investigated.

The geometry consisting of 5 cylinders was origi-
nally modeled using a second order transfer function that
yielded the best fit for the experimental RTD curve (92
%). However, to compare the parameters of the fitting

Figure 9: Results of the fitting of the serial model for 1
cylindrical unit.

Figure 10: Results of the fitting of the serial model for 3
cylindrical units.

and characterise the system, it was more favorable to fit a
third order transfer function here as well since it describes
the other two RTD curves best. In this case, the average
goodness of the fit was 91 % with the goodness of the
fit in the case of the geometry consisting of 5 cylindrical
units (86 %) being the lowest. The numerical parameters
of the fit are shown in Table 3.

The results are as follows: to describe the system in
the cases of 1 and 3 cylindrical units, a third order transfer
function was used. In the case of 5 cylindrical units, a
second order transfer function was optimal.

In the case of the model containing both parallel and

47(2) pp. 63–70 (2019)



68 TARCSAY, BOBEK, EGEDY, AND RIPPEL-PETHŐ

Figure 11: Results of the fitting of the serial model for 5
cylindrical units.

Table 3: Fitted parameters for the model containing only
serial links.

Parameters
N n a0 b3 b2 b1 b0 td (h) I (%)
1 3 1 0.1 0.3 1.0 1 0.9 93.43
3 3 1 0.2 0.6 1.5 1 0.6 94.00
5 3 1 0.3 0.7 2.0 1 0.7 85.76
5 2 1 - 0.9 2.3 1 0.7 92.00

serial links, the fitted curves are displayed in Figs. 12–14.
In the case of the model containing both parallel and

serial connections, the tendencies are similar to the re-
sults of the fitting conducted using the model consisting
of only serial links. However, the goodness of the fit was
enhanced compared to the simpler models (over 99 % for
all investigated geometries). In this case, the behavior of
the tank consisting of 1 cylindrical unit was described by
a transfer function where the orders of the nominator and
denominator were 4 and 2, respectively. In the case of 3
and 5 cylindrical units, the orders of the denominator and
nominator were 3 and 1, respectively. The order of the
transfer function (the number of CSTR units in the TIS
model) increases as the number of cylinders decreases.
This is evident of an increase in the PFR/LFR tenden-
cies of the functions which can also be seen in the fig-

Figure 12: Results of the fitting of the parallel model for
1 cylindrical unit.

Figure 13: Results of the fitting of the parallel model for
3 cylindrical units.

Figure 14: Results of the fitting of the parallel model for
5 cylindrical units.

ures. It should also be noted, however, that the fitting in
all models was exceedingly high (over 95 % on average).
This might suggest that the model algorithm is too pre-
cise and the model was also fitted according to the exper-
imental noise remaining in the data. In the absence of the
noise, such tendencies might be clearer and the decrease
in the order of the nominator between the different ge-
ometries more prominent. Further investigations carried
out by CFD simulations may clarify these concerns.

The parameters of the fit are displayed in Table 4.

4. Conclusion

In this paper, the velocity field of a tank of special ge-
ometry was investigated. In order to determine the re-
lationship between the flow profile and geometry of the
tank, step-response experiments were conducted. The
RTD curves of the system were measured by applying
various geometries of the tank. By analyzing the exper-
imental RTD curves, a model was proposed to be fitted
onto the results. The transfer function of the system was
optimized by minimalizing the squared difference of the
experimental and fitted RTD curves. The interior-point
method was applied to minimalize the difference. The
following conclusions were drawn from the fitted RTD
curves:

Hungarian Journal of Industry and Chemistry



MIXING IN TANKS OF A SPECIAL GEOMETRY 69

Table 4: Fitted parameters of the model containing both parallel and serial links.

Parameters
N n m a2 a1 a0 b4 b3 b2 b1 b0 td (h) I (%)
1 4 2 6 2.4 1 1 4.3 8 3.12 1 1.2 98.28

3 4 1 − 5.5 1 1 3.4 8.5 7.1 1 0.6 99.77
5 3 1 − 12 1 1 3.5 18 14.4 1 0.5 99.43

In the case of models containing only serial links, 1
and 3 cylindrical units could be optimally described us-
ing a third order transfer function. The geometry consist-
ing of 5 cylindrical unitscan be described by a second
order transfer function. The average goodness of the fit in
this case was 91 %. In the case of the model utilizing par-
allel links, the RTD function of the tank consisting of 1
cylindrical unit was approximated by a transfer function
where the orders of the denominator and nominator were
4 and 2, respectively. In the cases of 3 and 5 cylindrical
units, the orders of the denominator and nominator were
3 and 1, respectively. The goodness of fit on average in
the case of this model was over 99 %.

It can be concluded that all geometries of the tank can
be described using fourth order or lower transfer func-
tions with increasing model parameters according to the
number of cylinders.

Further investigations to characterise the reasons for
such behavior will be carried out by using CFD meth-
ods. Simultaneously, different step-response experiments
will be conducted to observe the behavior of the system.
During these experiments, instead of feeding the solution
of lower concentration into that of higher concentration
within the tank, a positive step function will be used. As
a result, the orders will be reversed with the solution of
low concentration being in the tank and that of high con-
centration in the feed.

Notations

Latin letters
a nominator parameter of transfer function
b denominator parameter of transfer function
dt time delay [h]
h order of the nominator of transfer function (1,n-1)
n order of the denominator of transfer function [-]
t time [h]
td time delay of transfer function [h]
x longitudinal coordinate [mm]
y vertical coordinate [mm]
z horizontal coordinate [mm]

Capital letters
B volume flow rate [lh−1]
E error of fitting [%]
F residence time distribution function
G transfer function
I goodness of fit [%]
N number of cylinders
R response function [gl−1]
S step function [gl−1]

Greek letters
δ Dirac delta function
τ mean residence time [h]

Indices
∗ time of tracer injection
d delay
exp experimental
fitted fitted
ref reference
steady steady
0 initial

Acknowledgement

This research was supported by the project EFOP-3.6.1-
16-2016-00015 “Smart Specialisation Strategy (S3) -
comprehensive institutional development program at the
University of Pannonia to promote sensible individual ed-
ucation and career choices”.

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Hungarian Journal of Industry and Chemistry

https://doi.org/10.1016/j.cej.2018.06.017
https://doi.org/10.1016/j.ces.2013.11.033
https://doi.org/10.1016/j.ces.2013.11.033
https://doi.org/10.1016/j.is.2019.01.006
https://doi.org/10.1016/j.is.2019.01.006
https://doi.org/10.1086/287954

	Introduction
	 Experimental and the modelling process
	 Results and Analysis
	 Conclusion