Loading [MathJax]/jax/output/SVG/jax.js
Research article Topical Sections

Effect of different washing conditions on the removal efficiency of selected compounds in biopolymers

  • Received: 21 February 2023 Revised: 29 June 2023 Accepted: 05 July 2023 Published: 18 July 2023
  • 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

    Related Papers:

    [1] Wei Fan, Kangqun Zhang . Local well-posedness results for the nonlinear fractional diffusion equation involving a Erdélyi-Kober operator. AIMS Mathematics, 2024, 9(9): 25494-25512. doi: 10.3934/math.20241245
    [2] Min Jiang, Rengang Huang . Existence of solutions for q-fractional differential equations with nonlocal Erdélyi-Kober q-fractional integral condition. AIMS Mathematics, 2020, 5(6): 6537-6551. doi: 10.3934/math.2020421
    [3] Kangqun Zhang . Existence and uniqueness of positive solution of a nonlinear differential equation with higher order Erdélyi-Kober operators. AIMS Mathematics, 2024, 9(1): 1358-1372. doi: 10.3934/math.2024067
    [4] Dumitru Baleanu, S. Hemalatha, P. Duraisamy, P. Pandiyan, Subramanian Muthaiah . Existence results for coupled differential equations of non-integer order with Riemann-Liouville, Erdélyi-Kober integral conditions. AIMS Mathematics, 2021, 6(12): 13004-13023. doi: 10.3934/math.2021752
    [5] Kishor D. Kucche, Sagar T. Sutar, Kottakkaran Sooppy Nisar . Analysis of nonlinear implicit fractional differential equations with the Atangana-Baleanu derivative via measure of non-compactness. AIMS Mathematics, 2024, 9(10): 27058-27079. doi: 10.3934/math.20241316
    [6] Mohamed Jleli, Bessem Samet . Nonexistence for fractional differential inequalities and systems in the sense of Erdélyi-Kober. AIMS Mathematics, 2024, 9(8): 21686-21702. doi: 10.3934/math.20241055
    [7] Wedad Albalawi, Muhammad Imran Liaqat, Kottakkaran Sooppy Nisar, Abdel-Haleem Abdel-Aty . Qualitative study of Caputo Erdélyi-Kober stochastic fractional delay differential equations. AIMS Mathematics, 2025, 10(4): 8277-8305. doi: 10.3934/math.2025381
    [8] Choukri Derbazi, Hadda Hammouche . Caputo-Hadamard fractional differential equations with nonlocal fractional integro-differential boundary conditions via topological degree theory. AIMS Mathematics, 2020, 5(3): 2694-2709. doi: 10.3934/math.2020174
    [9] Bashir Ahmad, Manal Alnahdi, Sotiris K. Ntouyas, Ahmed Alsaedi . On a mixed nonlinear boundary value problem with the right Caputo fractional derivative and multipoint closed boundary conditions. AIMS Mathematics, 2023, 8(5): 11709-11726. doi: 10.3934/math.2023593
    [10] Manal Elzain Mohamed Abdalla, Hasanen A. Hammad . Solving functional integrodifferential equations with Liouville-Caputo fractional derivatives by fixed point techniques. AIMS Mathematics, 2025, 10(3): 6168-6194. doi: 10.3934/math.2025281
  • 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.



    Generally, one of the purposes of the microwave imaging is to localize or detect unknown objects from measured electromagnetic waves in the high-frequency regime (between 300MHz and 300GHz). Due to this reason, identification of the location and shape of unknown anomaly whose values of dielectric permittivity and electric conductivity differ from the homogeneous background medium from scattering parameter data is an important and interesting research topic in microwave imaging. Many authors have proposed various remarkable algorithms for retrieving information on anomaly for example, Born iterative method for reconstructing permittivity distribution [13] and brain stroke detection [20], level-set method for shape reconstruction of unknown objects [16] and breast cancer detection [21], distorted iterated virtual experiments scheme for imaging unknown scatterers [28], conjugate gradient method for breast imaging [11], Levenberg-Marquardt technique for recovering parameter distribution [17]. We also refer to remarkable mathematical and experimental studies [1,3,12,15,26,45,46,47,48]. However, the success of iteration-based algorithms significantly depends on the priori information and initial guess, which must be close to the unknown anomaly, refer to [25,39].

    For this reason, various non-iterative schemes in inverse scattering problems without a priori information about unknown anomaly have also been developed to retrieve the location and shape, such as a variational algorithm based on the inverse Fourier transform to retrieve small electromagnetic inhomogeneities [5,6], direct sampling method for localizing small electromagnetic inhomogeneities [22,23] and anomaly detection in real-world experiment [43], factorization method for crack detection [10], shape reconstruction of unknown obstacles [18] and numerical study for anomaly imaging [33], linear sampling method for imaging of unknown scatters in limited-aperture problem [8] and crack-like defects [24], MUltiple SIgnal Classification (MUSIC) algorithm for identifying small anomalies [38], fast imaging of small targets in limited-aperture measurement configuration [35] and real-world application of anomaly detection [34], orthogonality sampling method for imaging unknown targets [40], qualitative microwave imaging [2], anomaly detection in microwave imaging [37] and topological derivative strategy for imaging crack-like defects [31] and retrieving unknown scatterers in 3D [27].

    The subspace migration (SM) algorithm is a well-known, non-iterative imaging technique in both inverse scattering problem and microwave imaging. It has been applied successfully to the various problems such as localization of small targets [4], identification of extended objects [9], fast imaging of curve-like cracks [29,30] and anomaly detection in microwave imaging [32,36]. Throughout several studies, it has been confirmed that SM is a fast, robust and effective technique for retrieving unknown anomaly from scattering parameter data. However, for successful application in microwave imaging, accurate values of background permittivity and conductivity must be known because the exact background wavenumber value must be applied. Generally, most researchers have used the statistical values of the background material instead of the true ones and obtained inaccurate locations and shapes of anomaly. This can be examined through the results of numerical simulations but no reliable mathematical theory explaining this phenomenon has yet been developed.

    In this paper, we apply the SM to retrieve unknown anomaly from scattering parameter data without complete information about the background material; that is, inaccurate values of background permittivity and conductivity are applied. To explain the appearance of an inaccurate location and shape of an anomaly, we show that the imaging function of the SM can be written as the infinite series of Bessel function of integer order, antenna arrangement and applied inaccurate values of background permittivity and conductivity. This enables us to theoretically explain the appearance of an inaccurate location and shape of an anomaly through the SM. To confirm the theoretical results, simulation results with inaccurate values of background permittivity and conductivity are also presented.

    The rest of this paper is organized as follows. In Section 2, the two-dimensional direct problem and imaging function of the SM are introduced. The structure of the imaging function with inaccurate values of background permittivity and conductivity is revealed in Section 3. In Section 4, a set of numerical simulation results with synthetic data generated by CST STUDIO SUITE is presented. Finally, a short conclusion is provided in Section 5.

    Let D be a circle-like anomaly with radius α and center r that is surrounded by a circular array of antennas An with location an and |an|=R, n=1,2,,N(>2). Throughout this paper, we use Ω to denote a homogeneous domain filled by matching liquid such that DΩ and assume that all materials D and Ω are nonmagnetic and classified by their value of dielectric permittivity and electric conductivity at a given angular frequency of operation ω, i.e., the value of magnetic permeability is constant for every rΩ say, μ(r)μb=1.257×106H/m, refer to [41]. We use εb and σb to denote the background permittivity and conductivity, respectively. Analogously, let ε and σ be those of D. Then, we introduce the piecewise constant permittivity ε(r) and conductivity σ(r),

    ε(r)={εforrD,εbforrΩD,andσ(r)={σforrD,σbforrΩD,

    respectively. With this, let kb be the background wavenumber that satisfies

    k2b=ω2μb(εbiσbω),

    and further assume that

    ωεbσbandεεb<1+wavelength4α. (2.1)

    With this, the time-dependent, homogeneous, linear Maxwell Equations take the form:

    curl E(r,t)=μbH(r,t)tandcurl H(r,t)=σbE(r,t)+εbE(r,t)t,rΩ, (2.2)

    where E(r,t)R3 and H(r,t)R3 are the electric and magnetic fields, respectively. Here, we consider time-harmonic solutions to the (2.2) such that

    E(r,t)=Re[E(r)eiωt]andH(r,t)=Re[H(r)eiωt],rΩ,t>0.

    Then, E(r)C3 and H(r)C3 satisfy

    curl E(r)=iωμbH(r)andcurl H(r)=(σbiωεb)EinΩ. (2.3)

    For a detailed description, we refer to [7].

    Let Einc(kb,r,am) be the incident electric field due to the point current density J at Am. Then, based on (2.3), it satisfies

    {curl Einc(kb,r,am)=iωμbHinc(kb,am,r),curl Hinc(kb,r,am)=(σbiωεb)Einc(kb,r,am),

    Analogously, let Etot(an,r) be the total field measured at An in the presence of D that satisfies

    {curl Etot(kb,an,r)=iωμbHtot(kb,an,r),curl Htot(kb,an,r)=(σ(r)iωε(r))Etot(kb,an,r),

    with transmission condition on D.

    Let S(n,m) be the S-parameter (or scattering parameter), which is defined as the ratio of the output voltage (or reflected waves) at the An antenna and the input voltage (or incident waves) at the Am (see [41] for instance). We use Sinc(n,m) and Stot(n,m) to denote the incident and total field S-parameters, respectively, due to the absence and presence of D. Correspondingly, we let Sscat(n,m)=Stot(n,m)Sinc(n,m) be the scattered field S-parameter. Then, based on [19], Sscat(n,m) can be represented as follows:

    Sscat(n,m)=ik204ωμbΩ(ε(r)εbεb+iσ(r)σbωεb)Einc(kb,r,am)Etot(kb,an,r)dr,

    where k0 denotes the lossless background wave number that satisfies k20=ω2μbεb.

    In this paper, we adopt the simulation configuration introduced in [32,43,44]. Notice that the height of microwave machine can be said to be long enough, only the z-component of the incident and total fields can be handled based on the mathematical treatment of the scattering of time-harmonic electromagnetic waves from thin infinitely long cylindrical obstacles. Correspondingly, by denoting E(z)inc and E(z)tot as the z-components of the incident and total fields, respectively, Sscat(n,m) can be written as follows:

    Sscat(n,m)=ik204ωμbΩ(ε(r)εbεb+iσ(r)σbωεb)E(z)inc(kb,r,am)E(z)tot(kb,an,r)dr. (2.4)

    Unfortunately, exact expression of the field E(z)tot(kb,an,r) is unknown thus we cannot design the imaging function by using Sscat(n,m) of (2.4) directly. Since the condition (2.1) holds, we can apply the Born approximation such that (see [42] for instance)

    E(z)tot(kb,an,r)=E(z)inc(kb,an,r)+o(α2)=i4H(1)0(kb|ar|)+o(α2),

    where H(1)0 denotes the Hankel function of order zero of the first kind. Correspondingly, Sscat(n,m) of (2.4) can be written as

    Sscat(n,m)=ik204ωμbD(εεbεb+iσσbωεb)E(z)inc(kb,r,am)E(z)tot(kb,an,r)dr+o(α2k20)=ik2064ωμbD(εεbεb+iσσbωεb)H(1)0(kb|amr|)H(1)0(kb|anr|)dr+o(α2k20). (2.5)

    Now, we introduce the imaging function. To this end, let us generate the scattering matrix K such that

    K=[skip0Sscat(1,2)Sscat(1,N1)Sscat(1,N)Sscat(2,1)0Sscat(2,N1)Sscat(2,N)Sscat(N,1)Sscat(N,2)Sscat(N,N1)0]. (2.6)

    See [32] for an explanation of why the diagonal elements of K are set to zero. Since there is a single, small anomaly, the singular value decomposition (SVD) of K can be written as

    K=UDV=Nn=1τnUnVnτ1U1V1, (2.7)

    where denotes the mark of Hermitian, τn are the singular values and Un and Vn are the left and right singular vectors of K, respectively. Then, on the basis of (2.5) and (2.7), we define a unit vector: For each rΩ,

    W(kb,r)=F(kb,r)|F(kb,r)|,F(kb,r)=[H(1)0(kb|a1r|),H(1)0(kb|a2r|),,H(1)0(kb|aNr|)]T. (2.8)

    With this, we introduce the following imaging function of the SM: For each rΩ,

    F(kb,r)=|W(kb,r),U1W(kb,r),¯V1|, (2.9)

    where U,V=UV and ¯V1 denotes the complex conjugate of V1. Then, based on [4]

    W(kb,r),U11andW(kb,r),¯V11whenrD,

    and the orthonormal property of singular vectors, the value of F(kb,r) will be close to 1 when rD and less than 1 at rΩD, so the location and outline shape of D can be identified through the map of F(kb,r).

    Let us emphasize that, to generate the test vector W(kb,r) of (2.8), the exact value of kb must be known, i.e., a priori information of the εb and σb must be available. However, because these values are statistical, the exact value may not be unknown. For this reason, we assume that the exact values of εb and σb are unknown and apply an alternative value ka instead of the true kb and set a unit test vector W(ka,r) from (2.8). Then, by using the imaging function F(ka,r) from (2.9), the exact location and shape of D cannot be retrieved. Fortunately, we can recognize the existence of D and the identified location is shifted in a specific direction.

    In this section, we explore the structure of the imaging function F(ka,r) to explain that retrieved location of D is shifted in a specific direction and size is smaller or larger than the true anomaly. To explain this phenomenon, we explore the structure of the imaging function.

    Theorem 3.1. Let θn=an/|an|=(cosθn,sinθn) and kbrkar=|kbrkar|(cosϕ,sinϕ). If an satisfies |anr|{1/4|ka|,1/4|kb|} for all n=1,2,,N, then F(r,ka) can be represented as follows:

    F(ka,r)=N(N1)area(D)|D(J0(|kbrkar|)+Ψ(kb,ka,r)N)2dr1ND(J0(2|kbrkar|)+Ψ(2kb,2ka,r)N)dr|+o(α2k20), (3.1)

    where Js denotes the Bessel function of order s and

    Ψ(kb,ka,r)=Nn=1s=,s0isJs(|kbrkar|)eis(θnϕ).

    Proof. Since Kτ1U1V1, we can examine that

    F(kb,r)=|W(ka,r),U1W(ka,r),¯V1|=|W(ka,r)U1V1¯W(ka,r)||1τ1W(ka,r)K¯W(ka,r)|.

    Based on the assumption |anr|{1/4|ka|,1/4|kb|} for all n=1,2,,N, the following asymptotic forms of the Hankel function hold (see [14,Theorem 2.5], for instance)

    H(1)0(kb|rr|)=(1i)eikb|r|kbπ|an|eikbθnr+O(1)andH(1)0(ka|rr|)=(1i)eika|r|kaπ|an|eikaθnr+O(1). (3.2)

    Then, W(ka,r) and K can be represented as

    W(ka,r)=1N[eikaθ1r+O(1)eikaθ2r+O(1)eikaθNr+O(1)]

    and

    K=C[0Deikb(θ1+θ2)rdr+o(α2k20)Deikb(θ1+θN)rdr+o(α2k20)Deikb(θ2+θ1)rdr+o(α2k20)0Deikb(θ2+θN)rdr+o(α2k20)Deikb(θN+θ1)rdr+o(α2k20)Deikb(θN+θ2)rdr+o(α2k20)0],

    respectively. Here the constant C is given by

    C=k20e2ikbR32kbωμbπR(εεbεb+iσσbωεb).

    Note that, since the following JacobiAnger expansion formula holds uniformly

    eixcosθ=s=isJs(x)eisθ=J0(x)+s=,s0isJs(x)eisθ, (3.3)

    we have for n=1,2,,N

    Nn=1eiθn(kbrkar)=Nn=1ei|kbrkar|cos(θnϕ)=Nn=1(J0(|kbrkar|)+s=,s0isJs(|kbrkar|)eis(θnϕ))=NJ0(|kbrkar|)+Ψ(kb,ka,r)

    and correspondingly,

    W(ka,r)K=CN[Deikbθ1rNn=1(eiθn(kbrkar)eiθ1(kbrkar))dr+o(α2k20)Deikbθ2rNn=1(eiθn(kbrkar)eiθ2(kbrkar))dr+o(α2k20)Deikbθ1rNn=1(eiθn(kbrkar)eiθN(kbrkar))dr+o(α2k20)]T=CN[Deikbθ1r(NJ0(|kbrkar|)+Ψ(kb,ka,r)eiθ1(kbrkar))dr+o(α2k20)Deikbθ2r(NJ0(|kbrkar|)+Ψ(kb,ka,r)eiθ2(kbrkar))dr+o(α2k20)Deikbθ1r(NJ0(|kbrkar|)+Ψ(kb,ka,r)eiθN(kbrkar))dr+o(α2k20)]T.

    With this, we can evaluate

    W(ka,r)K¯W(ka,r)=CN[Deikbθ1r(NJ0(|kbrkar|)+Ψ(kb,ka,r)eiθ1(kbrkar))dr+o(α2k20)Deikbθ2r(NJ0(|kbrkar|)+Ψ(kb,ka,r)eiθ2(kbrkar))dr+o(α2k20)DeikbθNr(NJ0(|kbrkar|)+Ψ(kb,ka,r)eiθN(kbrkar))dr+o(α2k20)]T[eikaθ1r+O(1)eikaθ2r+O(1)eikaθNr+O(1)]=CNDNn=1eiθn(kbrkar)(NJ0(|kbrkar|)+Ψ(kb,ka,r)eiθn(kbrkar))dr+o(α2k20)=CNDeiθn(kbrkar)(NJ0(|kbrkar|)+Ψ(kb,ka,r))drCNDNn=1e2iθn(kbrkar)dr+o(α2k20)=CND[(NJ0(|kbrkar|)+Ψ(kb,ka,r))2dr(NJ0(2|kbrkar|)+Ψ(2kb,2ka,r))]dr+o(α2k20).

    Hence,

    W(ka,r),U1W(ka,r),¯V1=CNτ1D(J0(|kbrkar|)+Ψ(kb,ka,r)N)2drCτ1D(J0(2|kbrkar|)+Ψ(2kb,2ka,r)N)dr+o(α2k20).

    Since W(r),U1W(r),¯V1=1, J0(|kbrkar|)=J0(2|kbrkar|)=1 and Ψ(kb,ka,r)=Ψ(2kb,2ka,r)=0 when kbr=kar, we have

    CNτ1DdrCτ1Ddr+o(α2k20)=1impliesC=τ1(N1)area(D)+o(α2k20).

    Therefore,

    W(r),U1W(r),¯V1=N(N1)area(D)D(J0(|kbrkar|)+Ψ(kb,ka,r)N)2dr1(N1)area(D)D(J0(2|kbrkar|)+Ψ(2kb,2ka,r)N)dr+o(α2k20).

    With this, we can obtain the structure (3.1).

    From the derived structure (3.1), we can observe that since J0(|kbrkar|)=1 and Ψ(kb,ka,r)=0 when r=(kb/ka)r for rD, an inaccurate location and shape of D must be retrieved through the map of F(ka,r). This is the theoretical reason why an inaccurate location and shape of the anomaly is retrieved when inaccurate values of εb and σb were applied. Further properties will be discussed in the simulation results.

    Here, we present simulation results to support the result in Theorem 3.1. To this end, a circular array of N=16 antennas An is used to transmit and receive signals operated at f=1.24GHz. The location of the antennas was set to

    an=0.09m(cosθn,sinθn),θn=2π(n1)N

    and the search domain Ω was selected as a square region (0.1m,0.1m)×(0.1m,0.1m) with (εb,σb)=(20ε0,0.2S/m). Here, ε0=8.854×1012F/m is the vacuum permittivity. Correspondingly, the exact value of the background wavenumber is kb=116.5273+8.4020i. For anomalies, we selected two small balls D1 and D2 with centers r1=(0.01m,0.03m) and r2=(0.04m,0.02m), same radii α=0.01m and material properties (ε1,σ1)=(55ε0,1.2S/m) and (ε2,σ2)=(45ε0,1.0S/m). With these settings, the measurement data Sscat(n,m) of (2.6) and the incident field data of (2.8) were generated by CST STUDIO SUITE.

    Example 4.1. (Only exact value of εb is unknown) First, we consider the case where only the exact value of εb is unknown. Instead, of the application of εb, we applied alternative values εa and corresponding wavenumber

    ka=ωμb(εaiσbω).

    Note that we already assumed that ωεbσb. Thus, if the condition ωεaσb is satisfied, the identified location becomes

    r=(kbka)r=ωεbiσbωεaiσbrεbεarfor eachrD1. (4.1)

    Hence, identified anomalies will be concentrated at the origin and their retrieved sizes will be smaller than the true one when εa>εb. Otherwise, identified anomalies will be far from the origin and their retrieved sizes will be larger than the true one when εa<εb. See Figure 1 for a related illustration.

    Figure 1.  Description of the simulation result. Yellow-colored circle is the true anomaly and cyan- and violet-colored circles are retrieved anomalies through the map of F(ka,r).

    Figure 2 shows maps of F(ka,r) with various selections of εa in the presence of D1. As we discussed above, the location of the retrieved anomaly gets closer to the origin and it becomes smaller as the value of εa increases (here, εa=3εb,10εb). Otherwise, as the value of εa decreases (here, εa=0.5εb), the identified location becomes far from the origin and the size becomes larger. If εa=0.01εb i.e., the value of εa is very small, it is difficult to distinguish the D1 and artifacts.

    Figure 2.  Maps of F(ka,r) at f=1.24GHz when σa=σb.

    Notice that since

    Js(|kbrkar|)=Js(|ka||(kbka)rr|):=Js(|ka||rr|),

    due to the oscillating property of the Bessel function, several artifacts will be included in the map of F(ka,r) if |ka| is large, i.e., the value of εa is large enough compared to εb. In contrast, the map of F(ka,r) will contain no artifacts but the imaging result will be blurred if the value of εa is small enough compared to εb. This is the reason why several artifacts are included in the map of F(ka,r) when εa=10εb and why the obtained image is blurred when εa=0.1εb and εa=0.01εb. We can observe the same phenomenon in the presence of multiple anomalies D1 and D2, as shown in the Figure 3.

    Figure 3.  Maps of F(ka,r) at f=1.24GHz when εa=εb.

    Example 4.2. (Only exact value of σb is unknown) Next, we consider the case where only the exact value of σb is unknown and apply an alternative one σa such that

    ka=ωμb(εbiσaω).

    Same as the Example 4.1., if σa satisfies ωεaσa, the identified location becomes

    r=(kbka)r=ωεbiσbωεbiσarεbεbr=rfor eachrD1. (4.2)

    Hence, it will be possible to retrieve almost accurate shapes and locations of anomalies when σa is sufficiently small.

    Figure 4 shows maps of F(ka,r) with various selections of σa in the presence of D1. In contrast to the results in Example 4.1., almost the exact location and shape of D1 were retrieved if σa<3σb, i.e., when σa was sufficiently small. The location of the retrieved anomaly gets closer to the origin and its size becomes smaller as the value of εa increases (here, εa=3εb,10εb). Unfortunately, it is very difficult to recognize D1 due to the appearance of a huge artifact with a large magnitude if σa is not small σa=10σb=2S/m.

    Figure 4.  Maps of F(ka,r) at f=1.24GHz.

    We can observe the same phenomenon in the presence of multiple anomalies D1 and D2, as shown in the Figure 5, and conclude that it will be possible to retrieve the accurate shape and location of anomalies by choosing a very small (close to zero) value of σa when the exact value of background permittivity is known.

    Figure 5.  Maps of F(ka,r) at f=1.24GHz.

    Example 4.3. (Identification of circle and rectangular shaped anomalies) Here, we consider the imaging of anomalies with different shapes. To this end, we applied f=1.0GHz, used N=36 antennas An, and selected D1 as a ball of Examples 1 and 2 except (ε1,σ1)=(45ε0,1.0S/m), and D2 as a square with vertices (0.05,0.03), (0.03,0.03), (0.03,0.01) and (0.05,0.01) with (ε2,σ2)=(45ε0,1.0S/m). With this configuration, the scattering parameter data were generated by using the FEKO (Feldberechnung für Körper mit beliebiger Oberfläche).

    Figure 6 shows maps of F(ka,r) when εaεb and σa=σb. Similar to the results in Example 4.1., identified anomalies are concentrated at the origin and their retrieved sizes are smaller than the true one when εa>εb. Moreover, identified anomalies located far from the origin and their retrieved sizes are larger than the true one when εa<εb. However, opposite to the results in Examples 1 and 2, it is hard to recognize the shape of D1 and D2 due to the appearance of several artifacts in the neighborhood of anomalies.

    Figure 6.  Maps of F(ka,r) at f=1GHz when σa=σb.

    Figure 7 shows maps of F(ka,r) when εa=εb and σaσb. Similar to the results in Example 4.2., we can examine that it is possible to recognize the outline shape of anomalies by choosing a very small value of σa. However, exact shape of anomalies cannot be retrieved still.

    Figure 7.  Maps of F(ka,r) at f=1GHz.

    The structure of the imaging function of SM for retrieving small anomalies from scattering matrix is revealed when complete information of the background medium is not available. On the basis of its relationship with the infinite series of Bessel function of the first kind, we have theoretically confirmed why the accurate shape and location of anomalies cannot be retrieved.

    The main subject of this paper is the imaging of small anomaly in two-dimensional microwave imaging. An extension to multiple, small anomalies will be carried out in forthcoming work. Moreover, the development of an effective algorithm for retrieving the exact value of background wavenumber will be an interesting research subject. Finally, we expect that the methodology presented in this paper could be applied to real-world microwave imaging with inhomogeneous background.

    The author declares he has not used Artificial Intelligence (AI) tools in the creation of this article.

    The author would like to acknowledge anonymous reviewers for their comments that help to increase the quality of the paper. The author is also grateful to Sangwoo Kang, Kwang-Jae Lee and Seong-Ho Son for helping in generating scattering parameter data. This work was supported by the research program of the Kookmin University.

    The author declares no conflicts of interest regarding the publication of this paper.



    [1] Plastics Europe, Plastics—the Facts 2022. Plastics Europe, 2022. Available from: https://plasticseurope.org/knowledge-hub/plastics-the-facts-2022/.
    [2] Eze WU, Umunakwe R, Obasi HC, et al. (2021) Plastics waste management: A review of pyrolysis technology. Clean Technol Recy 1: 50–69. https://doi.org/10.3934/ctr.2021003 doi: 10.3934/ctr.2021003
    [3] Soto JM, Blázquez G, Calero M, et al. (2018) A real case study of mechanical recycling as an alternative for managing of polyethylene plastic film presented in mixed municipal solid waste. J Clean Prod 203: 777–787. https://doi.org/10.1016/j.jclepro.2018.08.302 doi: 10.1016/j.jclepro.2018.08.302
    [4] Soto JM, Martín-Lara MA, Blázquez G, et al. (2020) Novel pre-treatment of dirty post-consumer polyethylene film for its mechanical recycling. Process Saf Environ 139: 315–324. https://doi.org/10.1016/j.psep.2020.04.044 doi: 10.1016/j.psep.2020.04.044
    [5] Reclay StewardEdge, Analysis of flexible film plastics packaging diversion systems. Recsource Recycling Systems, and Moore Recycling Associates Inc., 2013. Available from: https://thecif.ca/projects/documents/714-Flexible_Film_Report.pdf.
    [6] Xia D, Zhang FS (2018) A novel dry cleaning system for contaminated waste plastic purification in gas-solid media. J Clean Prod 171: 1472–1480. https://doi.org/10.1016/j.jclepro.2017.10.028 doi: 10.1016/j.jclepro.2017.10.028
    [7] Liu W, Zhang B, Li Y, et al. (2014) An environmentally friendly approach for contaminants removal using supercritical CO2 for remanufacturing industry. Appl Surf Sci 292: 142–148. https://doi.org/10.1016/j.apsusc.2013.11.102 doi: 10.1016/j.apsusc.2013.11.102
    [8] Niaounakis M (2019) Recycling of biopolymers—The patent perspective. Eur Polym J 114: 464–475. https://doi.org/10.1016/j.eurpolymj.2019.02.027 doi: 10.1016/j.eurpolymj.2019.02.027
    [9] Cornell DD (2007) Biopolymers in the existing postconsumer plastics recycling stream. J Polym Environ 15: 295–299. https://doi.org/10.1007/s10924-007-0077-0 doi: 10.1007/s10924-007-0077-0
    [10] Fredi G, Dorigato A (2021) Recycling of bioplastic waste: A review. Adv Ind Eng Polym Res 4: 159–177. https://doi.org/10.1016/j.aiepr.2021.06.006 doi: 10.1016/j.aiepr.2021.06.006
    [11] McAdam B, Fournet MB, McDonald P, et al. (2020) Production of polyhydroxybutyrate (PHB) and factors impacting its chemical and mechanical characteristics. Polymers 12: 2908. https://doi.org/10.3390/polym12122908 doi: 10.3390/polym12122908
    [12] Yu J, Plackett D, Chen LXL (2005) Kinetics and mechanism of the monomeric products from abiotic hydrolysis of poly[(R)-3-hydroxybutyrate] under acidic and alkaline conditions. Polym Degrad Stabil 89: 289–299. https://doi.org/10.1016/j.polymdegradstab.2004.12.026 doi: 10.1016/j.polymdegradstab.2004.12.026
    [13] Pospisilova A, Melcova V, Figalla S, et al. (2021) Techniques for increasing the thermal stability of poly[(R)-3-hydroxybutyrate] recovered by digestion methods. Polym Degrad Stabil 193: 109727. https://doi.org/10.1016/j.polymdegradstab.2021.109727 doi: 10.1016/j.polymdegradstab.2021.109727
    [14] Delva L, Van Kets K, Kuzmanovic M, et al., Mechanical recycling of polymers for dummies. Capture—Plastics To Resource, 2019. Available from: https://www.ugent.be/ea/match/cpmt/en/research/topics/circular-plastics/mechanicalrecyclingfordummiesv2.pdf.
  • This article has been cited by:

    1. Janghoon Jeong, Seong-Ho Son, Localization of small moving objects using only total fields in microwave imaging, 2024, 61, 22113797, 107777, 10.1016/j.rinp.2024.107777
    2. Junyong Eom, Won-Kwang Park, Real-time detection of small objects in transverse electric polarization: Evaluations on synthetic and experimental datasets, 2024, 9, 2473-6988, 22665, 10.3934/math.20241104
    3. Seong-Ho Son, Kwang-Jae Lee, Won-Kwang Park, Real-time tracking of moving objects from scattering matrix in real-world microwave imaging, 2024, 9, 2473-6988, 13570, 10.3934/math.2024662
    4. Won-Kwang Park, On the application of subspace migration from scattering matrix with constant-valued diagonal elements in microwave imaging, 2024, 9, 2473-6988, 21356, 10.3934/math.20241037
    5. Janghoon Jeong, Jang-Moon Jo, Soeng-Ho Son, 2024, Microwave Imaging Method for Object Localization without Background Measurements, 979-8-3503-7581-7, 108, 10.1109/ICITEE62483.2024.10808351
  • Reader Comments
  • © 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2188) PDF downloads(148) Cited by(0)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog