Recycling of plastic materials is a key sustainability topic. Hence, the scope of this study is to evaluate the potential of this purification step for achieving high-purity recyclates via mechanical recycling. In this study, the focus is set on the revalorization of poly(3-hydroxy butyrate) and poly(3-hydroxy butyrate-co-3-hydroxy valerate)—two biobased and biodegradable polymers that have properties similar to those of polyolefins and are therefore possible eco-friendly alternatives. Specifically, the washing process as an important part of polymer recycling processes is evaluated regarding different washing conditions on a laboratory scale. For this purpose, several virgin polymers were contaminated with volatile organic compounds that differed in functionality and molecular weight. Regarding contamination, concentration correlates with contamination time. Moreover, the contamination degree was found to be higher for polar contaminants since polar compounds show higher compatibility with the polymer. General beneficial effects of higher temperatures and longer washing times were observed. The choice of washing medium was relevant for different polarities of the contaminants. At higher process temperatures, material degradation occurred. Hence, recyclers have to pay attention to the difference in the interaction between impurities and the polymer and to the degradation of the polymer during recycling and the subsequent formation of degradation products. Since these biopolymers display comparable properties to polyolefins, great potential in packaging applications is apparent. Moreover, the method of analyzing the removal efficiency of volatile organic compounds via washing can be applied to all recyclable polymers.
Citation: Konstanze Kruta, Jörg Fischer, Peter Denifl, Christian Paulik. Effect of different washing conditions on the removal efficiency of selected compounds in biopolymers[J]. Clean Technologies and Recycling, 2023, 3(3): 134-147. doi: 10.3934/ctr.2023009
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Abstract
Recycling of plastic materials is a key sustainability topic. Hence, the scope of this study is to evaluate the potential of this purification step for achieving high-purity recyclates via mechanical recycling. In this study, the focus is set on the revalorization of poly(3-hydroxy butyrate) and poly(3-hydroxy butyrate-co-3-hydroxy valerate)—two biobased and biodegradable polymers that have properties similar to those of polyolefins and are therefore possible eco-friendly alternatives. Specifically, the washing process as an important part of polymer recycling processes is evaluated regarding different washing conditions on a laboratory scale. For this purpose, several virgin polymers were contaminated with volatile organic compounds that differed in functionality and molecular weight. Regarding contamination, concentration correlates with contamination time. Moreover, the contamination degree was found to be higher for polar contaminants since polar compounds show higher compatibility with the polymer. General beneficial effects of higher temperatures and longer washing times were observed. The choice of washing medium was relevant for different polarities of the contaminants. At higher process temperatures, material degradation occurred. Hence, recyclers have to pay attention to the difference in the interaction between impurities and the polymer and to the degradation of the polymer during recycling and the subsequent formation of degradation products. Since these biopolymers display comparable properties to polyolefins, great potential in packaging applications is apparent. Moreover, the method of analyzing the removal efficiency of volatile organic compounds via washing can be applied to all recyclable polymers.
1.
Introduction
Fractional calculus is a subdivision of classical calculus that concerns itself with derivatives as well as integrals of nearly any fractional order. This mathematical field has gained significant attention due to its ability to model complex phenomena more accurately than integer-order calculus. Fractional calculus offers robust methods for characterizing memory and hereditary features of different materials and processes, rendering it essential in disciplines such as control theory [14], signal processing [4], and differential equations [11,12,13]. The foundations of fractional calculus were laid by early mathematicians such as Leibniz, Liouville, and Riemann, who explored the concept of generalizing the order of differentiation and integration beyond integers [3,7]. Modern advancements have further developed these ideas, leading to a robust theoretical framework and numerous practical applications.
The nabla difference operator is a discrete counterpart of the continuous derivative, employed in discrete calculus, specifically in the context of time-scale calculus. This operator, often denoted by ∇, operates on functions defined on discrete domains, such as sequences or time scales. The nabla difference operator plays a crucial role in various fields, including exact analysis, discrete dynamic systems, and operational calculus in [9,21,22]. It facilitates the formulation and solution of difference equations, which are the discrete counterparts of differential equations. The papers "On the definitions of nabla fractional operators" and "Discrete fractional calculus includes the nabla operator" go into great detail about what nabla fractional operators are and how they can be used in real life. They stress how important they are in the field of discrete mathematics [1,2]. Recently, the nabla difference operator has been seen in sequential differences in the nabla fractional calculus, the combined delta-nabla sum operator in discrete fractional calculus, and discrete fractional calculus consisting of the nabla operator in [2,20]. These studies contribute to a deeper understanding of the interplay between discrete and continuous analysis. In research [15], such as on K¨othe-Toeplitz duals of generalized difference sequence spaces and Laplace transforms for the nabla-difference operator, investigations into the theoretical underpinnings and applications of these operators in discrete calculus [6].
The Fibonacci sequence is a widely recognized integer sequence defined by the recurrence relation Fr=Fr−1+Fr−2, where F0 is 0 and F1 is 1. This sequence appears in various natural phenomena and has numerous applications in computer science, mathematics, and financial modeling. The study of sequence spaces derived from difference operators has gained significant attention, particularly with generalized Fibonacci difference operators. The derivation of sequence spaces arising from triple-band generalized Fibonacci difference operator [18,24], investigates the structural properties of these spaces, while generalized Fibonacci difference sequence spaces and compact operators [17,23] explores their impact on the boundedness and compactness of sequences. These works extend classical sequence space theory by integrating the recursive properties of Fibonacci sequences, highlighting new connections in functional analysis.
The motivation for this study arises from the need to further explore and expand the theoretical systems and applications of the nabla difference operator in generating and analyzing Fibonacci sequences. We present the trigonometric nabla difference operator of order 2 and its discrete integral, analyzing the ¯θ(t)-sequences, their summation, and the proportional derivative of the ¯θ(t)-polynomials.
The contributions of the research are delineated as follows:
1. By utilizing various trigonometric coefficients in the ¯θ(t)-Fibonacci equation and its inverses, we formulate new sequences and analyze their characteristics.
2. The nabla difference operator of order 2 facilitates the generation of ¯θ(t)-Fibonacci sequences and their exact and numerical solutions.
3. We provide chaotic behavior of the generating function of the second-order Fibonacci sequence with the coefficients of trigonometric functions.
4. The study includes MATLAB examples to demonstrate the practical applications of our theoretical findings.
Throughout this article, we make use of the notations and elucidations from the following Table 1.
Table 1.
The notations and elucidations.
ξ
Shift (translation) value (i.e., ξ∈[0,∞))
t(x)
t(t−ξ)(t−2ξ)(t−3ξ)...(t−(x−1)ξ)
t−(n+r)ξ
tn,r, where n, r are integers and t∈(−∞,∞).
E∗-solution
Exact solution
N∗-solution
Numerical solution
¯θ(t)-sequence
Fibonacci sequence derived from general difference equation (Nabla) with trigonometric coefficients of order 2.
¯θ(t)-equation
General difference equation (Nabla) with trigonometric coefficients of order 2.
2.
Second-order ¯θ(t)-Fibonacci sequence and its sum
Here, we formulate a generic nabla-difference operator that includes a trigonometric co-efficient ∇¯θ(t)v(t)=v(t)−α1sin(b1t)v(t0,1)−α2sin(b2t)v(t0,2) which generates second-order ¯θ(t)-Fibonacci sequence and its sum.
Definition 2.1.Let t be any positive real number and n≥2; a second-order generic ¯θ(t)-sequence is defined recurrently as Ft,0=1,Ft,1=α1sin(b1t), and
Ft,n=α1sin(b1tn,1)Ft,n−1+α2sin(b2tn,2)Ft,n−2.
(2.1)
Definition 2.2.For any positive real number t, a generic nabla difference operator of order two using sine (any trigonometric) function coefficients on v(t), denoted as ∇¯θ(t)v(t), is defined as
Proof: For the function v(t), the Definition 2.2 yields
v(t)=u(t)+α1sin(b1t)v(t0,1)+α2sin(b2t)v(t0,2).
(2.11)
By changing k by k0,1 and then put the value of v(t0,1) into (2.11), we get
v(t)=u(t)+Ft,1u(t0,1)+[Ft,1α1sin(b1t0,1)+
α2sin(b2t)]v(t0,2)+Ft,1α2sin(b2t0,1)v(t0,3),
(2.12)
which leads to v(t)=Ft,0u(t)+Ft,1u(t0,1)+
Ft,2v(t0,2)+α2sin(b2t0,1)Ft,1v(t0,3),
(2.13)
where Ft,0=0, Ft,1=α1sin(b1t) and Ft,2=Ft,n+1=α1sin(b1t1,0)Ft,1+α2sin(b2t1,−1)Ft,0. By changing k by k0,2 in (2.11) and then putting the value of v(t0,2) into (2.13), we observe
Proof: By doing α1=α2=1 in (2.23), we obtain (2.24).
3.
Illustrative numerical examples
The objective of this section is to demonstrate the efficacy of the primary findings presented in this paper by employing precise examples from the literature. Also, we have investigated graphical representations of Fibonacci sequences with the co-efficients of the Sine, Cosine and Co-secant functions for different values of t in the following Figures 1–3.
Figure 1.
Sine and Cosine-Fibonacci Sequences for t=10,ξ=0.4, and t=15, ξ=0.5, respectively.
The validity of the definition 2.1 is confirmed by the subsequent illustrative example 3.1.
Example 3.1.(i) By doing t=10, b1=2, b2=1, ξ=0.4, α1=2, and α2=1 in (2.1), we obtain a Sine-Fibonacci sequence {1,1.8259,0.7097,−0.9351,1.9327,−3.9778,⋯}.
(ii) When t=15, ξ=0.5, α1=0.2, α2=0.1, r1=3, and r2=1 in (2.1), we have a Cosine-Fibonacci sequence {1.0000,−1.5194,1.9182,−3.9008,3.4203,0.8856,5.1684,⋯}.
Also, the sine Fibonacci polynomials are observed by
F0(t)=1,
F1(t)=2sin(2t),
F2(t)=4sin(2t)sin(2t−0.8)+sin(t),
F3(t)=8sin(2t)sin(2t−1.6)sin(2t−0.8)+2sin(t)sin(2t−1.6)+2sin(t−0.4)sin(2t), etc.
Furthermore, using (2.4), we have the following proportional α-derivative of the sine Fibonacci polynomials;
One can derive second-order Fibonacci polynomials and sequences for each pair ¯θ(t)∈R2. The validity of the result 2.5 is confirmed by the subsequent illustrative example 3.2.
Example 3.2.Setting t=11, b1=2, α1=3, s=2, α2=2, n=2, ξ=0.3, b2=1, and a=3 in (2.15), we obtain the following:
F3(t)=27cos(2t−1.2)cos(2t−0.6)cos(2t)+6cos(2t−1.2)cos(t)+2cos(t−0.4)cos(2t), etc. Furthermore, using (2.4), we have the following proportional α-derivative of the co-secant Fibonacci polynomials;
Fα0(t)=0,Fα1(t)=α[−6sin(2t)]+3(1−α)cos(2t),
Fα2t)=α[−18sin(2t−0.6)cos(2t)−18cos(2t−0.6)
sin(2t)−2sint]+(1−α)9cos(2t−0.6)cos(2t)+2cos(t)], and etc.
The validity of the definition 2.7 is confirmed by the subsequent illustrative example 3.3.
Example 3.3.By doing α1=2, t=10, α2=1, ξ=0.4, b1=2, n=2, and b2=1 in (2.16), we have the following:
F3(t)=cosec(t−1)cosec(t−0.5)cosec(2t)+cosec(t−1)cosec(3t)+cosec(t−0.5)cosec(2t), etc. Further-more, using (2.4), we have the following proportional α-derivative of the co-secant Fibonacci polynomials;
Fα2(t)=α[−2cosec(t−0.5)cot(t−0.5)cosec(2t)−2cosec(t−0.5)cosec(2t)cot(2t)−3cosec(3t)cot(3t)]+(1−α)[cosec(t−0.5)cosec(2t)+cosec(3t)], and etc.
4.
Bifurcation behavior of ¯θ(t)-Fibonacci generating function
The objective of this section is to demonstrate the efficacy of the bifurcation analysis of the ¯θ(t)-Fibonacci generating function and primary findings presented in this paper by employing analysis from the following precise examples.
A discrete one-dimensional dynamical system is a system subjected to a single equation of this type
x(t+1)=f(t)
(4.1)
where x∈Z and f is a function of x. The variable t is in general considered as the time, but in discrete systems the time takes only discrete values, so that it is possible to take t∈Z.
A generalized discrete two-dimensional dynamical system is a system subjected to a single equation of this type
x(t+2ξ)=x(t+ξ)+x(t)
(4.2)
where t∈R. When we reformulate Eq (4.2), we obtain a two-dimensional discrete system
x(t+ξ)=x1(t)+x2(t)x2(t+ξ)=x1(t).
A trajectory is a set {x(t)}∞t=0 of points satisfying the above equation (4.1). It is evident that the initial point x0=x(0) determines the entire trajectory. The behaviour of the dynamical system is therefore given by all the trajectories {x(t):x(0)=x0} for all initial values x0∈I. When the value of the parameter changes continuously, the behaviour of the system may change in a discontinuous way. One says that a bifurcation occurs for an isolate value of the parameter at which the type of dynamic changes. In bifurcation analysis, the region of stable operation is determined through the search of Hopf bifurcation points. This gives an insight into how the variations in the system parameters influence region of stable operation. This knowledge can be effectively used by the system designers to ensure the stability of the actual system.
A bifurcation diagram is a traditional and visual way to look into how dynamical systems, difference equations, and differential equations change over time [10]. This tool is excellent for looking at how the system reacts to changes in parameters [19]. This diagram illustrates the system's different dynamic patterns and phase transitions by plotting the link between the system reaction and parameters [5,8]. This section employs bifurcation theory to determine the existence of the period-doubling (flip) bifurcation. We discuss the ¯θ(t)-Fibonacci generating function and investigate the bifurcation analysis of the ¯θ(t)-Fibonacci sequences.
¯θ(t)-Fibonacci sequences are generated by
f(t)=11−θ1(t)t−θ2(t)t2,
(4.3)
where θ1(t) and θ2(t) are any trigonometric and hyperbolic functions. If θ1(t)=sin(b1t) and θ2(t)=sin(b2t) in (4.3), then we have the Fibonacci generating function
f(t)=11−sin(b1t)t−sin(b2t)t2.
(4.4)
(ii). If θ1(t)=cos(b1t) and θ2(t)=cos(b2t) in (4.3), then we have the Fibonacci generating function
f(t)=11−cos(b1t)t−cos(b2t)t2.
(4.5)
4.1. Chaotic behavior of the fibonacci generating function
In dynamical systems theory, a period-doubling bifurcation occurs when a slight change in a system's parameters causes a new periodic trajectory to emerge from an existing periodic trajectory. With the doubled period, it takes twice as long (or, in a discrete dynamical system, twice as many iterations) for the numerical values visited by the system to repeat themselves. A period-halving bifurcation occurs when a system switches to a new behavior with half the period of the original system. A period-doubling cascade is an infinite sequence of period-doubling bifurcations. Such cascades are a common route by which dynamical systems develop chaos.
Now, we consider the recurrent form of (4.3)
tr=11−θ1(tr−1)tr−1−θ2(tr−1)t2r−1,
(4.6)
and we consider the recurrent form of (4.4)
tr=11−sin(b1tr−1)tr−1−sin(b2tr−1)t2r−1.
(4.7)
We consider the recurrent form of (4.5)
tn=11−cos(b1tn−1)tn−1−cos(b2tn−1)t2n−1.
(4.8)
We examine the orbit {tr}∞r=0 for any point t0 within the domain of the map.
Figure 4(a) displays a periodic doubling bifurcation diagram of the Fibonacci generating function of the co-efficient of the sine function with the initial condition t0=0.8 at the intrinsic growth bifurcation parameter b2=2 and b1∈[−8,11].Figure 4(b) displays periodic doubling bifurcation diagram of the Fibonacci generating function of the co-efficient of the cosine function with the initial condition t0=0.8 at the intrinsic growth bifurcation parameter b2=2 and b1∈[−5,5].Figure 4(c) displays periodic doubling bifurcation diagram of the Fibonacci generating function of the co-efficient of the tangent function with the initial condition t0=0.8 at the intrinsic growth bifurcation parameter b2=1 and b1∈[−5,1].Figure 4(d) displays periodic doubling bifurcation diagram of the Fibonacci generating function of the co-efficient of the cosine function with the initial condition t0=0.8 at the intrinsic growth bifurcation parameter b2=1 and b1∈[−10,5].
Figure 4.
Period doubling bifurcation diagram for ¯θ(t) -Fibonacci generating function.
Figure 5(a) displays sub periodic doubling bifurcation diagram of the Fibonacci generating function of the co-efficient of the sine function with the initial condition t0=0.8 at the intrinsic growth bifurcation parameter b=2 and b1∈[0,0.3].Figure 5(b) displays sub periodic doubling bifurcation diagram of the Fibonacci generating function of the co-efficient of the cosine function with the initial condition t0=0.8 at the intrinsic growth bifurcation parameter b2=2 and b1∈[0.58,2].Figure 5(c) displays sub periodic doubling bifurcation diagram of the Fibonacci generating function of the co-efficient of the tangent function with the initial condition t0=0.8 at the intrinsic growth bifurcation parameter b2=1 and b1∈[−2.2,−1.8].Figure 5(d) displays sub periodic doubling bifurcation diagram of the Fibonacci generating function of the co-efficient of the cosine function with the initial condition t0=0.8 at the intrinsic growth bifurcation parameter b2=1 and b1∈[−3,−2.1].
Figure 5.
Sub period doubling cascade for ¯θ(t) -Fibonacci generating function.
This paper deduced an inverse formula for the ¯θ(t)-Fibonacci sequence. The inverse of a generic difference (nabla) operator with trigonometric coefficients of order 2 was used to derive this formula. The results we have obtained regarding the E∗ and N∗ solutions, Fibonacci polynomials, and the proportional derivative of the generic difference equation with trigonometric coefficients of order 2 will be applied to our future research. Additionally, we have conducted a bifurcation analysis of the ¯θ(t)-Fibonacci generating function.
Author's contributions
R.P: Conceptualization; R.P, S.T.M.T, and I.K.: Data curation; R.P, S.T.M.T, I.K, and M.V.C.: Formal analysis; P.R, I.K, and M.V.C.: Investigation; R.P, S.T.M.T, and I.K.: Methodology; R.P and S.T.M.T.: Writing-original draft; R.P, I.K and M.V.C.: Writing-review and editing. All authors have read and agreed to the published version of the article.
Use of AI tools declaration
The authors declare they have not used artificial intelligence (AI) tools in the creation of this article.
Acknowledgments
"La derivada fraccional generalizada, nuevos resultados y aplicaciones a desigualdades integrales" Cod UIO-077-2024. This study is supported via funding from Prince Sattam bin Abdulaziz University project number (PSAU/2025/R/1446).
Conflict of interest
The authors declare that they have no conflicts of interest.
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Konstanze Kruta, Jörg Fischer, Peter Denifl, Christian Paulik. Effect of different washing conditions on the removal efficiency of selected compounds in biopolymers[J]. Clean Technologies and Recycling, 2023, 3(3): 134-147. doi: 10.3934/ctr.2023009
Konstanze Kruta, Jörg Fischer, Peter Denifl, Christian Paulik. Effect of different washing conditions on the removal efficiency of selected compounds in biopolymers[J]. Clean Technologies and Recycling, 2023, 3(3): 134-147. doi: 10.3934/ctr.2023009