Research article Special Issues

Effect of electrode modification on the production of electrical energy and degradation of Cr (Ⅵ) waste using tubular microbial fuel cell

  • Carcinogenic hexavalent chromium is increasing worldwide due to the increased electroplating, welding and textile industry. On the other hand, molasses, the sugar factory's byproduct with high organic compounds (sugars), may pollute the environment if it is not processed. However, microbial fuel cell (MFC) seems to be a promising technology due to its ability to produce electrical energy from pollutant degradation using microbes while reducing hexavalent chromium to trivalent chromium with less toxicity. Carbon felt was used at both electrodes. This research aimed to determine the effect of modifying the anode with rice bran and cathode with Cu catalyst towards electricity generation and pollutant removal in molasses and reducing Cr (Ⅵ) into Cr (Ⅲ) using tubular microbial fuel cells. Moreover, the effect of mixing Sidoarjo mud and Shewanella oneidensis MR-1 as electricigen bacteria toward electrical energy production and pollutant removal was determined. Experiments revealed that the S/CM/AM variable, which only used Shewanella oneidensis MR-1 as an electricigen bacteria with both modified electrodes, produced the highest total power density of 530.42 mW/m2 and the highest percentage of Cr (Ⅵ) reduction of 98.87%. In contrast, the highest microbial population of 66.5 × 1010 cells/mL, 61.28% of Biological Oxygen Demand (BOD5) removal and 59.49% of Chemical Oxygen Demand (COD) were achieved by SSi/CM/AM variable, mixing Shewanella oneidensis MR-1 and Sidoarjo mud as an electricigen bacteria with both modified electrodes. Therefore, this study indicates that double chamber tubular microbial fuel cells may be a sustainable solution for managing molasses and carcinogen hexavalent chromium.

    Citation: Raden Darmawan, Sri Rachmania Juliastuti, Nuniek Hendrianie, Orchidea Rachmaniah, Nadila Shafira Kusnadi, Ghassani Salsabila Ramadhani, Yawo Serge Marcel, Simpliste Dusabe, Masato Tominaga. Effect of electrode modification on the production of electrical energy and degradation of Cr (Ⅵ) waste using tubular microbial fuel cell[J]. AIMS Environmental Science, 2022, 9(4): 505-525. doi: 10.3934/environsci.2022030

    Related Papers:

    [1] Rehab Alharbi, S. E. Abbas, E. El-Sanowsy, H. M. Khiamy, Ismail Ibedou . Soft closure spaces via soft ideals. AIMS Mathematics, 2024, 9(3): 6379-6410. doi: 10.3934/math.2024311
    [2] Orhan Göçür . Amply soft set and its topologies: AS and PAS topologies. AIMS Mathematics, 2021, 6(4): 3121-3141. doi: 10.3934/math.2021189
    [3] Tareq M. Al-shami, El-Sayed A. Abo-Tabl . Soft α-separation axioms and α-fixed soft points. AIMS Mathematics, 2021, 6(6): 5675-5694. doi: 10.3934/math.2021335
    [4] Dina Abuzaid, Samer Al-Ghour . Supra soft Omega-open sets and supra soft Omega-regularity. AIMS Mathematics, 2025, 10(3): 6636-6651. doi: 10.3934/math.2025303
    [5] Arife Atay . Disjoint union of fuzzy soft topological spaces. AIMS Mathematics, 2023, 8(5): 10547-10557. doi: 10.3934/math.2023535
    [6] Mesfer H. Alqahtani, Zanyar A. Ameen . Soft nodec spaces. AIMS Mathematics, 2024, 9(2): 3289-3302. doi: 10.3934/math.2024160
    [7] Dina Abuzaid, Samer Al Ghour . Three new soft separation axioms in soft topological spaces. AIMS Mathematics, 2024, 9(2): 4632-4648. doi: 10.3934/math.2024223
    [8] Tareq M. Al-shami, Salem Saleh, Alaa M. Abd El-latif, Abdelwaheb Mhemdi . Novel categories of spaces in the frame of fuzzy soft topologies. AIMS Mathematics, 2024, 9(3): 6305-6320. doi: 10.3934/math.2024307
    [9] Ahmad Al-Omari, Mesfer H. Alqahtani . Some operators in soft primal spaces. AIMS Mathematics, 2024, 9(5): 10756-10774. doi: 10.3934/math.2024525
    [10] Tareq M. Al-shami, Abdelwaheb Mhemdi, Radwan Abu-Gdairi, Mohammed E. El-Shafei . Compactness and connectedness via the class of soft somewhat open sets. AIMS Mathematics, 2023, 8(1): 815-840. doi: 10.3934/math.2023040
  • Carcinogenic hexavalent chromium is increasing worldwide due to the increased electroplating, welding and textile industry. On the other hand, molasses, the sugar factory's byproduct with high organic compounds (sugars), may pollute the environment if it is not processed. However, microbial fuel cell (MFC) seems to be a promising technology due to its ability to produce electrical energy from pollutant degradation using microbes while reducing hexavalent chromium to trivalent chromium with less toxicity. Carbon felt was used at both electrodes. This research aimed to determine the effect of modifying the anode with rice bran and cathode with Cu catalyst towards electricity generation and pollutant removal in molasses and reducing Cr (Ⅵ) into Cr (Ⅲ) using tubular microbial fuel cells. Moreover, the effect of mixing Sidoarjo mud and Shewanella oneidensis MR-1 as electricigen bacteria toward electrical energy production and pollutant removal was determined. Experiments revealed that the S/CM/AM variable, which only used Shewanella oneidensis MR-1 as an electricigen bacteria with both modified electrodes, produced the highest total power density of 530.42 mW/m2 and the highest percentage of Cr (Ⅵ) reduction of 98.87%. In contrast, the highest microbial population of 66.5 × 1010 cells/mL, 61.28% of Biological Oxygen Demand (BOD5) removal and 59.49% of Chemical Oxygen Demand (COD) were achieved by SSi/CM/AM variable, mixing Shewanella oneidensis MR-1 and Sidoarjo mud as an electricigen bacteria with both modified electrodes. Therefore, this study indicates that double chamber tubular microbial fuel cells may be a sustainable solution for managing molasses and carcinogen hexavalent chromium.



    Most real-world problems in engineering, medical science, economics, the environment, and other fields are full of uncertainty. The soft set theory was proposed by Molodtsov [25], in 1999, as a mathematical model for dealing with uncertainty. This is free of the obstacles associated with previous theories including fuzzy set theory, rough set theory, and so on. The nature of parameter sets related to soft sets, in particular, provides a uniform framework for modeling uncertain data. This results in the rapid development of soft set theory in a short period of time, as well as diverse applications of soft sets in real life.

    Influenced by the standard postulates of traditional topological space, Shabir and Naz [29], and Çağman et al. [18], separately, established another branch of topology known as "soft topology", which is a mixture of soft set theory and topology. This work was essential in building the subject of soft topology. Despite the fact that many studies followed their directions and many ideas appeared in soft contexts such as those discussed in [2,3,12,13,14]. However, significant contributions can indeed be made.

    The separation axioms are just axioms in the sense that you could add these conditions as extra axioms to the definition of topological space to achieve a more restricted definition of what a topological space is. These axioms have a great role in developing (classical) topology. Correspondingly, soft separation axioms are a significant aspect in the later development of soft topology; see for example [4,6,7,8,19,24,29]. A specific type of separation axioms was defined by Aull and Thron [15]. This axiom performs as an important part in the development other disciplines like Locale Theory [28], Logic and Information Theory [16] and Philosophy [27]. First, motivating the role of "TD-spaces", we generalize this separation axiom in the language of soft set theory under the name of "soft TD-spaces", and study their primary properties. Second, most of the given soft separation axioms were characterized by soft open, soft closed, or soft closure, we want to describe them differently. As a result, this work is demonstrated. Finally, the desire of describing some soft Ti-spaces using new soft operators motivates us to present the operators of "soft kernel" and "soft shell".

    The body of the paper is structured as follows: In Section 2, we present an overview of the literature on soft set theory and soft topology. Section 3 focuses on the concepts of soft topological operators and their main properties for characterization of soft separation axioms. Section 4 introduces a new soft separation axiom called a soft "TD-space". The relationships of soft TD-spaces with known soft separation axioms are determined. Furthermore, we characterize soft TD-spaces via soft operators proposed in Section 3. In Section 5, we offer characterizations of soft T0-spaces and soft T1-spaces through the given operators. We end our paper with a brief summary and conclusions (Section 6).

    Let X be a domain set and E be a set of parameters. A pair (F,E)={(e,F(e)):eE} is said to be a soft set [25] over X, where F:E2X is a set-valued mapping. The set of all soft sets on X parameterized by E is identified by SE(X). We call a soft set (F,E) over X a soft element [29], denoted by ({x},E), if F(e)={x} for each eE, where xX. It is said that a soft element ({x},E) is in (F,E) (briefly, x(F,E)) if xF(e) for each eE. On the other hand, x(F,E) if xF(e) for some eE. This implies that if ({x},E)˜(F,E)=Φ, then x(F,E). We call a soft set (F,E) over X a soft point [10,26], denoted by xe, if F(e)={x} and x(e)= for each eE with ee, where eE and xX. An argument xe(F,E) means that xF(e). The set of all soft points over X is identified by PE(X). A soft set (X,E)(F,E) (or simply (F,E)c) is the complement of (F,E), where Fc:E2X is given by Fc(e)=XF(e) for each eE. If (F,E)SE(X), it is denoted by Φ if F(e)= for each eE and is denoted by ˜X if F(e)=X for each eE. Evidently, ˜Xc=Φ and Φc=˜X. A soft set (F,E) is called degenerate if (F,E)={xe} or (F,E)=Φ. It is said that (A,E1) is a soft subset of (B,E2) (written by (A,E1)˜(B,E2), [22]) if E1E2 and A(e)B(e) for each eE1, and (A,E1)=(B,E2) if (A,E1)˜(B,E2) and (B,E2)˜(A,E1). The union of soft sets (A,E),(B,E) is represented by (F,E)=(A,E)˜(B,E), where F(e)=A(e)B(e) for each eE, and intersection of soft sets (A,E),(B,E) is given by (F,E)=(A,E)˜(B,E), where F(e)=A(e)B(e) for each eE, (see [9]).

    Definition 2.1. [29] A collection T of SE(X) is said to be a soft topology on X if it satisfies the following axioms:

    (T.1) Φ,˜XT.

    (T.2) If (F1,E),(F2,E)T, then (F1,E)˜(F2,E)T.

    (T.3) If {(Fi,E):iI}˜T, then ˜iI(Fi,E)T.

    Terminologically, we call (X,T,E) a soft topological space on X. The elements of T are called soft open sets. The complement of every soft open or elements of Tc are called soft closed sets. The lattice of all soft topologies on X is referred to TE(X), (see [1]).

    Definition 2.2. [11] Let F˜SE(X). The intersection of all soft topologies on X containing F is called a soft topology generated by F and is referred to T(F).

    Definition 2.3. [29] Let (B,E)SE(X) and TTE(X).

    (1) The soft closure of (B,E) is cl(B,E):=˜{(F,E):(B,E)˜(F,E),(F,E)Tc}.

    (2) The soft interior of (B,E) is int(B,E):=˜{(F,E):(F,E)˜(B,E),(F,E)T}.

    Definition 2.4. [18] Let (B,E)SE(X) and TTE(X). A point xePE(X) is called a soft limit point of (B,E) if (G,E)˜(B,E){xe}Φ for all (G,E)T with xe(G,E). The set of all soft limit points is symbolized by der(B,E). Then cl(F,E)=(F,E)˜der(F,E) (see Theorem 5 in [18]).

    Definition 2.5. [21] Let TTE(X). A set (A,E)SE(X) is called soft locally closed if there exist (G,E)T and (F,E)Tc such that (A,E)=(G,E)˜(F,E). The family of all soft locally closed sets in X is referred to LC(X).

    Definition 2.6. [20] Let TTE(X) and let (A,E)SE(X). A point xe(A,E) is called soft isolated if there exists (G,E)T such that (G,E)˜(A,E)={xe}. It is called soft weakly isolated if there exists (G,E)T with xe(G,E) such that (G,E)˜(A,E)˜cl(xe). Let I(A,E), WI(A,E) respectively denote the set of all soft isolated and soft weakly isolated points of (A,E).

    Definition 2.7. [17] A soft space (X,E,T) (or simply soft topology TTE(X)) is called

    (1) Soft T0 if for every xe,yePE(X) with xeye, there exist (U,E),(V,E)T such that xe(U,E), ye(U,E) or ye(V,E), xe(V,E).

    (2) Soft T1 if for every xe,yePE(X) with xeye, there exist (U,E),(V,E)T such that xe(U,E), ye(U,E) and ye(V,E), xe(V,E).

    The above soft separation axioms have been defined by Sabir and Naz [29] with respect to soft elements.

    Lemma 2.8. [17,Theorem 4.1] Let TTE(X). Then T is soft T1 iff cl(xe)={xe} for every xePE(X).

    In this section, we define "soft kernel" and "soft shell" as two topological operators. Then the connections between these operators and soft closure and soft derived set operators are obtained. The presented results will be used to characterize several soft separation axioms.

    Definition 3.1. Let (F,E)SE(X) and let TTE(X). The soft kernel of (F,E) is defined by:

    ker(F,E):=˜{(G,E):(G,E)T,(F,E)˜(G,E)}.

    Lemma 3.2. Let (F,E),(G,E)SE(X) and TTE(X). The following properties are valid:

    (1) (F,E)˜ker(F,E).

    (2) ker(F,E)˜ker(ker(F,E)).

    (3) (F,E)˜(G,E)ker(F,E)˜ker(G,E).

    (4) ker[(F,E)˜(G,E)]˜ker(F,E)˜ker(G,E).

    (5) ker[(F,E)˜(G,E)]=ker(F,E)˜ker(G,E).

    Proof. Standard.

    From Definitions 2.3 and 3.1, it is obtained that

    Definition 3.3. Let xePE(X) and TTE(X). Then

    (1) ker({xe}):=˜{(G,E):(G,E)T,xe(G,E)}.

    (2) cl({xe}):=˜{(F,E):(F,E)Tc,xe(F,E)}.

    In the sequel, we interchangeably use xe or {xe} for the one point soft set containing xe.

    Lemma 3.4. For (F,E)SE(X) and TTE(X), we have

    ker(F,E)={xePE(X):cl(xe)˜(F,E)Φ}.

    Proof. Let xeker(F,E). If cl(xe)˜(F,E)=Φ, then (F,E)˜˜Xcl(xe). Therefore, ˜Xcl(xe)T such that it contains (F,E) but not xe, a contradiction.

    Conversely, if xeker(F,E) and cl(xe)˜(F,E)Φ, then there is (G,E)T such that (F,E)˜(G,E) but xe(G,E) and yecl(xe)˜(F,E). Therefore, ˜X(G,E)Tc including xe but not ye. This contradicts to yecl(xe)˜(F,E). Thus, xeker(F,E).

    Definition 3.5. Let (F,E),(G,E)SE(X) and TTE(X). It is said that (F,E) is separated in a weak sense from (G,E) (symbolized by (F,E)(G,E)) if there exists (H,E)T with (F,E)˜(H,E) such that (H,E)˜(G,E)=Φ.

    We have the following observation in light of Lemma 3.4 and Definition 3.5.

    Remark 3.6. For xe,yePE(X) and TTE(X), we have

    (1) cl(xe)={ye:ye  xe}.

    (2) ker(xe)={ye:xe  ye}.

    Definition 3.7. For xePE(X) and TTE(X), we define:

    (1) The soft derived set of xe as der(xe)=cl(xe){xe}.

    (2) The soft shell of xe as shel(xe)=ker(xe){xe}.

    (3) The soft set xe=cl(xe)˜ker(xe).

    We have the following remark in view of Definition 3.7 and Remark 3.6.

    Remark 3.8. For xe,yePE(X) and TTE(X), we have

    (1) der(xe)={ye:yexe,ye  xe}.

    (2) shel(xe)={ye:yexe,xe  ye}.

    Example 3.9. Let X={0,1,2} and let E={e1,e2} be a set of parameters. Consider the following soft topology on X:

    T={Φ,(F,E),G,E),(H,E),˜X},

    where, (F,E)={(e1,{0}),(e2,)}, (G,E)={(e1,{0,1}),(e2,)}, and (H,E)={(e1,{0,2}),(e2,X)}. By an easy computation, one can conclude the following:

    ker({1e1})=(G,E)ker({1e2})=(H,E)shel({1e1})=(F,E)shel({1e2})={(e1,{0,2}),(e2,{0,2})}cl({1e1})={1e1}cl({1e2})={(e1,{2}),(e2,X)}der({1e1})=Φder({1e2})={(e1,{2}),(e2,{0,2})}.

    Lemma 3.10. The following properties are valid for every xe,yePE(X) and TTE(X):

    (1) yeker(xe)xecl(ye).

    (2) yeshel(xe)xeder(ye).

    (3) yecl(xe)cl(ye)˜cl(xe).

    (4) yeker(xe)ker(ye)˜ker(xe).

    Proof. (1) and (2) follow, respectively, from Remarks 3.6 and 3.8.

    (3) Straightforward.

    (4) Let zeker(ye). By (1), yecl(ze) and so cl(ye)˜cl(ze) (by (3)). By hypothesis, yeker(xe) and so xecl(ye). Therefore, cl(xe)˜cl(ye). Finally, we get cl(xe)˜cl(ze) and then xecl(ze). By (1), zeker(xe). Thus, ker(ye)˜ker(xe).

    Lemma 3.11. Let TTE(X) and let xePE(X). Then

    (1) shel(xe) is degenerate iff for every yePE(X) with yexe, der(xe)˜der(ye)=Φ.

    (2) der(xe) is degenerate iff for every yePE(X) with yexe, shel(xe)˜shel(ye)=Φ.

    Proof. (1) If der(xe)˜der(ye)Φ, then there exists zePE(X) such that zeder(xe), zeder(ye). Therefore, zeyexe for which zecl(xe) and zecl(ye). By Lemma 3.10 (1), xe,yeker(ze). Thus, xe,yeker(ze)ze=shel(ze). This proves that shel(xe) is not degenerate.

    Conversely, if xe,yeshel(ze), then xeye, xeze and so xeker(ze), yeker(ze). Therefore, zecl(xe)˜cl(ye) and thus zeder(xe)˜der(ye). But this is impossible, hence der(xe)˜der(ye)=Φ.

    Lemma 3.12. Let TTE(X) and let xe,yePE(X). Then

    (1) If yexe, then ye=xe.

    (2) Either ye=xe or ye˜xe=Φ.

    Proof. (1) If yexe, then yecl(xe) and yeker(xe). When yecl(xe), by Lemma 3.10 (1), xeker(ye). By Lemma 3.10 (3) and (4), cl(ye)˜cl(xe) and ker(xe)˜ker(ye). When yeker(xe), by Lemma 3.10 (2), xecl(ye). By Lemma 3.10 (3) and (4), cl(xe)˜cl(ye) and ker(ye)˜ker(xe). Summing up all these together, we get cl(xe)=cl(ye) and ker(xe)=ker(ye). Thus, ye=xe.

    (2) It can be deduced from (1).

    Lemma 3.13. Let TTE(X) and let xe,yePE(X). Then ker(xe)ker(ye) iff cl(xe)cl(ye).

    Proof. If ker(xe)ker(ye), then one can find zeker(xe) but zeker(ye). From zeker(xe), we get xecl(ze) and then cl(xe)˜cl(ze). Since zeker(ye), by Lemma 3.10 (1), cl(ze)˜ye=Φ. Therefore, cl(ze)˜ye=Φ implies yecl(xe). Hence, cl(ye)cl(xe).

    The converse can be proved in a similar manner to the first part.

    Definition 4.1. Let TTE(X). We call T a soft TD-space if der(xe) is a soft closed set for every xePE(X).

    Theorem 4.2. Let TTE(X). Then

    (1) If T is soft T1, then it is soft TD.

    (2) If T is soft TD, then it is soft T0.

    Proof. (1) If T is soft T1, by Lemma 2.8, for every xePE(X), cl(xe)={xe}, so der(xe)=ΦTc. Thus, T is soft TD.

    (2) Let xe,yePE(X) with xeye. If yeder(xe), then [der(xe)]c is a soft open set that includes xe but not ye. If yeder(xe) and since xeye, then ye[cl(xe)]c and [cl(xe)]cT with xe[cl(xe)]c. Consequently, T is soft T0.

    The reverse of the above implications may not be true, as illustrated by the examples below.

    Example 4.3. Let X be an infinite and let E be a set of parameters. For a fixed pePE(X), the soft topology T on X is given by T={(F,E)SE(X):pe(F,E)or(F,E)=˜X}. We first need to check T is soft TD. Indeed, take xePE(X), if xe=pe, then der(xe)=Φ. If xepe, then der(xe)={pe}. Therefore, in either cases, der(xe) is soft closed. On the other hand, for any xepe, cl(xe)={xe,pe}{xe}, which means {xe} is not a closed set. Hence T is not soft T1.

    Example 4.4. Let E={e1,e2} be a set of parameters and let T be a soft topology on the set of real numbers R generated by

    {{(e1,B(e1)),(e2,B(e2))}:B(e1)=(a,b),B(e2)=(c,);a,b,cR;a<b}.

    Let xe1,ye2PE(X) with xe1ye2. W.l.o.g, we assume x<y. Take (G,E)={(e1,),(e2,(x,))}. Then (G,E) is a soft open set containing ye2 but not xe1 and hence T is soft T0. But then der(ye2)={(e1,),(e2,(,y))} is not soft closed, and consequently T is not soft TD.

    Proposition 4.5. Let TTE(X). Then T is soft TD iff {xe}LC(X) for every xePE(X).

    Proof. Let xePE(X). We need to prove that {xe} can be written as an intersection of a soft open set with a soft closed set. Set (G,E)=[der(xe)]c and (F,E)=cl(xe). Then (G,E)T and (F,E)Tc such that {xe}=(G,E)˜(F,E).

    Conversely, (w.l.o.g) we set {xe}=(G,E)˜cl(xe). Now, der(xe)=cl(xe){xe}=cl(xe)[(G,E)˜cl(xe)]= cl(xe)˜(G,E)c. Since finite intersections of soft closed sets are soft closed, so der(xe) is soft closed.

    Proposition 4.6. Let TTE(X). Then T is soft TD iff for every xePE(X), there exists (G,E)T including xe such that (G,E){xe}T.

    Proof. Take a point xePE(X). If we set (G,E)=[der(xe)]c, then (G,E)T containing xe. Now,

    (G,E){xe}=˜Xder(xe)˜{xe}c=˜X˜(der(xe))c˜{xe}c=˜X˜[der(xe)˜{xe}]c=˜Xcl(xe).

    Thus, (G,E){xe}T.

    Conversely, suppose for every xePE(X), there exists xe(G,E)T such that (G,E){xe}T. Therefore, {xe}=(G,E)˜[(G,E){xe}]c. By Proposition 4.5, T is soft TD.

    Proposition 4.7. For a soft topology TTE(X), the following properties are equivalent:

    (1) T is soft TD.

    (2) der(der(A,E))˜der(A,E) for every (A,E)SE(X).

    (3) der(A,E)Tc for every (A,E)SE(X).

    Proof. (1)(2) Let xeder(der(A,E)). Then every (G,E)T with xe(G,E) includes some points of der(A,E). Since T is soft TD,

    (H,E){xe}˜der(A,E)Φ,

    where (H,E)=[der(xe)]c˜(G,E). Suppose yeder(A,E) with yexe. Then ye(H,E)˜(G,E). Since yeder(A,E), then (H,E)T contains a point ze of (A,E) except ye. Indeed, zexe and then every (G,E) with xe(G,E) contains some points of (A,E) except xe. Hence, xeder(A,E).

    (2)(3) Since cl(der(A,E))=der(der(A,E))˜der(A,E)˜der(A,E), so der(A,E)Tc.

    (3)(1) It is evident.

    Proposition 4.8. Let (A,E)SE(X), TTE(X) and (F,E)Tc. The following properties are equivalent:

    (1) T is soft TD.

    (2) For every xePE(X), [cl(xe)]c˜{xe}T.

    (3) Every xeWI(A,E)xeI(A,E).

    (4) Every xeWI(F,E)xeI(F,E).

    Proof. (1)(2) Given xePE(X), by Proposition 4.6, there is (G,E)T such that xe(G,E) and (G,E){xe}T. Therefore, (G,E){xe}=(G,E)cl(xe). Since T is soft TD, so (G,E)der(xe)=(G,E)cl(xe)˜{xe}T. But, for every xe[cl(xe)]c˜{xe}, we have

    xe(G,E)=(G,E)der(xe)˜{xe}˜[cl(xe)]c˜{xe}.

    Thus, [cl(xe)]c˜{xe}T.

    (2)(3) Suppose xeWI(A,E). Then there is (G,E)T such that

    xe(G,E)˜(A,E)˜cl(xe).

    By (2), [cl(xe)]c˜{xe}T. But,

    (G,E)˜(A,E)˜[[cl(xe)]c˜{xe}]={xe}.

    Hence, xeI(A,E).

    (3)(4) Clear.

    (4)(1) Given xePE(X), we can easily conclude from the definition that xeWI(cl(xe)). By (4), xeI(cl(xe)), and so there exists (G,E)T such that (G,E)˜cl(xe)={xe}. Therefore, (G,E){xe}=(G,E)cl(xe)T. By Proposition 4.6, T is soft TD.

    Summing up all the above findings yields the following characterization:

    Theorem 4.9. For a soft topology TTE(X), the following properties are equivalent:

    (1) T is soft TD.

    (2) {xe}LC(X) for every xePE(X).

    (3) der(A,E)Tc for every (A,E)SE(X).

    (4) der(der(A,E))˜der(A,E) for every (A,E)SE(X).

    (5) xePE(X), there exists (G,E)T with xe(G,E) such that (G,E){xe}T.

    (6) xePE(X), [cl(xe)]c˜{xe}T.

    (7) xeWI(A,E)xeI(A,E), where (A,E)SE(X).

    (8) xeWI(F,E)xeI(F,E), where (F,E)Tc.

    The properties of soft topological operators derived in Section 2 are used to develop new characterizations of soft Ti-spaces for i=0,1.

    Proposition 5.1. For a soft topology TTE(X), the following properties are equivalent:

    (1) T is soft T0.

    (2) For every xe,yePE(X) with xeye, either xeye or yexe.

    (3) yecl(xe)xecl(ye).

    (4) For every xe,yePE(X) with xeye, cl(xe)cl(ye).

    Proof. (1)(2) It is just a reword of the definition.

    (2)(3) Let yecl(xe). For every (G,E)T that contains ye, (G,E)˜{xe}Φ and so yexe. If xe=ye, then there is nothing to prove. Otherwise, by (2), xeye. Therefore, there exists (H,E)T such that xe(H,E) and (H,E)˜{ye}=Φ. Hence, xecl(ye).

    (3)(4) Suppose the negative of (4) holds. Then cl(xe)˜cl(ye) and cl(ye)˜cl(xe). Since yecl(ye), then it implies that cl(ye)cl(xe) and so yecl(xe). By (3), xecl(ye) implies xecl(xe) which is impossible.

    (4)(1) Suppose xe,yePE(X) with xeye, cl(xe)cl(ye). This means that there is zePE(X) for which zecl(xe) but zecl(ye). We claim that xecl(ye). Otherwise, we will have {xe}cl(ye) and so cl(xe)cl(ye). This implies that zecl(ye), a contradiction to the selection of ze. Set (G,E)=[cl(ye)]c. Therefore, (G,E)T such that xe(G,E) and ye(G,E). Hence, T is soft T0.

    Proposition 5.2. For a soft topology TTE(X), the following properties are equivalent:

    (1) T is soft T0.

    (2) For every xe,yePE(X), yeker(xe)xeker(ye).

    (3) For every xe,yePE(X) with xeye, ker(xe)ker(ye).

    Proof. Follows from Lemma 3.10 (1) and Proposition 5.1.

    Proposition 5.3. A soft topology TTE(X) is soft T0 iff yeder(xe) implies cl(ye)˜der(xe) for every xe,yePE(X).

    Proof. Given xe,yePE(X). If yeder(xe), then yexe and xecl(ye) (as T is soft T0), then cl(ye)˜der(xe).

    Conversely, let xe,yePE(X) be such that xeye. If yeder(xe), then cl(ye)˜der(xe). This means that yecl(xe) and xecl(ye). From Proposition 5.1, T is soft T0.

    Proposition 5.4. A soft topology TTE(X) is soft T0 iff yeshel(xe) implies ker(ye)˜shel(xe) for every xe,yePE(X).

    Proof. By Proposition 5.3 and Lemma 3.10, we can obtain the proof.

    Proposition 5.5. A soft topology TTE(X) is soft T0 iff [cl(xe)˜{ye}]˜[{xe}˜cl(ye)] is degenerate for every xe,yePE(X).

    Proof. Assume xe,yePE(X) and T is soft T0. By Proposition 5.1, for every xe,yePE(X), if yecl(xe), then xecl(ye). Therefore, [cl(xe)˜{ye}]˜[{xe}˜cl(ye)]={ye} is a degenerated soft set. Otherwise, [cl(xe)˜{ye}]˜[{xe}˜cl(ye)]={xe} which is also degenerate.

    Conversely, if the given condition is satisfied, then the result is either Φ,{xe}, or {ye}. For the case of Φ, the conclusion is obvious. If [cl(xe)˜{ye}]˜[{xe}˜cl(ye)]={xe} implies xecl(ye) and cl(xe)˜{ye}=Φ. Therefore, yecl(xe). The case of {ye} is similar to the latter one. Hence, T is soft T0.

    Proposition 5.6. A soft topology TTE(X) is soft T0 iff der(xe)˜shel(xe)=Φ for every xePE(X).

    Proof. If der(xe)˜shel(xe)Φ, then there is xePE(X) such that zeder(xe) and zeshel(xe). Indeed, zexe and so zecl(xe) and zeker(xe). By Remark 3.6, zexe and xeze implies that T cannot be soft T0, a contradiction.

    Conversely, if der(xe)˜shel(xe)=Φ, then for each zexe, either zecl(xe) or zeker(xe). Therefore, either zecl(xe) or xecl(ze). By Proposition 5.1 (3), T is soft T0.

    Proposition 5.7. A soft topology TTE(X) is soft T0 iff xe={xe} for every xePE(X).

    Proof. It is a consequence of Definition 3.7 and Proposition 5.6.

    Proposition 5.8. A soft topology TTE(X) is soft T0 iff der(xe) is a union of soft closed sets for every xePE(X).

    Proof. Since, for every xePE(X), der(xe)Tc, then for every zeder(xe) we must have (G,E)T such that xe(G,E) and ze(G,E). Therefore, (F,E)=(G,E)cTc with with ze(F,E) but xe(F,E). This means that zeder(xe), we have

    ze(F,E)˜cl(xe)˜der(xe).

    Since (F,E)˜cl(xe)Tc, so der(xe) is a union of soft closed sets.

    Conversely, let der(xe)=˜iI(Fi,E), where (Fi,E)Tc. If zeder(xe), then ze(Fi,E) for some i but xe(Fi,E). Therefore, (Fi,E)cT such that xe(Fi,E)c but ze(Fi,E)c. If zeder(xe) and zexe, then ze[cl(xe)]c and [cl(xe)]cT for which xe[cl(xe)]c. This proves that T is soft T0.

    Summing up all the above propositions yields the following characterization:

    Theorem 5.9. For a soft topology TTE(X), the following properties are equivalent:

    (1) T is soft T0.

    (2) For every xe,yePE(X) with xeye, either xeye or yexe.

    (3) For every xe,yePE(X), yecl(xe)xecl(ye).

    (4) For every xe,yePE(X), yeder(xe)cl(ye)˜der(xe)

    (5) For every xe,yePE(X) with xeye, cl(xe)cl(ye).

    (6) For every xe,yePE(X), yeker(xe)xeker(ye).

    (7) For every xe,yePE(X), yeshel(xe)ker(ye)˜shel(xe)

    (8) For every xe,yePE(X) with xeye, ker(xe)ker(ye).

    (9) For every xe,yePE(X) [cl(xe)˜{ye}]˜[{xe}˜cl(ye)] is degenerate.

    (10) For every xePE(X), der(xe)˜shel(xe)=Φ.

    (11) For every xePE(X), der(xe) is a union of soft closed sets.

    (12) For every xePE(X), xe={xe}.

    Theorem 5.10. For a soft topology TTE(X), the following properties are equivalent:

    (1) T is soft T1.

    (2) For every xe,yePE(X) with xeye, xeye.

    (3) For every xePE(X), cl(xe)={xe}.

    (4) For every xePE(X), der(xe)=Φ.

    (5) For every xePE(X), ker(xe)={xe}.

    (6) For every xePE(X), shel(xe)=Φ.

    (7) For every xe,yePE(X) with xeye, cl(xe)˜cl(ye)=Φ.

    (8) For every xe,yePE(X) with xeye, ker(xe)˜ker(ye)=Φ.

    Proof. One can easily notice that all the statements are rephrases of (1) with the help of Lemmas in Section 3. Last statement means xeker(ye) and yeker(xe). Equivalently, yecl(xe) and xecl(ye). This guarantees the existence of two sets (G,E),(H,E)T such that xe(G,E),ye(G,E) and ye(H,E),xe(H,E). Thus, T is soft T1.

    We close this investigation with the following remark:

    Remark 5.11. Section 2 recalls soft points and soft elements, which are two distinct types of soft point theory. We have employed the concept of soft points throughout this paper, although most of the (obtained) results are invalid for soft elements. The reasons can be found in [30], Examples 3.14–3.21. The divergences between axioms via classical and soft settings were studied in detail in [5].

    Soft separation axioms are a collection of conditions for classifying a system of soft topological spaces according to particular soft topological properties. These axioms are usually described in terms of soft open or soft closed sets in a topological space.

    In this work, we have proposed soft topological operators that will be used to characterize certain soft separation axioms and named them "soft kernel" and "soft shell". The interrelations between the latter soft operators and soft closure or soft derived set operators have been discussed. Moreover, we have introduced soft TD-spaces as a new soft separation axiom that is weaker than soft T1 but stronger than soft T0-spaces. It should be noted that TD-spaces have applications in other (applied) disciplines. Some examples have been provided, illustrating that soft TD-spaces are at least different from soft T1 and soft T0-spaces. The soft topological operators mentioned above are used to obtain new characterizations of soft Ti-spaces for i=0,1, and D. Ultimately, we have analyzed the validity of our findings in relation to two different theories of soft points.

    In the upcoming work, we shall define the axioms given herein and examine their properties via other soft structures like infra soft topologies and supra soft topologies. We will also conduct a comparative study between these axioms and their counterparts introduced with respect to different types of belonging and non-belonging relations. Moreover, we will generalize the concept of functionally separation axioms [23] to soft settings and investigate its relationships with the other types of soft separation axioms.

    The authors declare that they have no competing interests.



    [1] Ucal M, Xydis G (2020) Multidirectional relationship between energy resources, climate changes and sustainable development: Technoeconomic analysis. Sustain Cities Soc 60: 102210. https://doi.org/10.1016/j.scs.2020.102210 doi: 10.1016/j.scs.2020.102210
    [2] Yan JC (2021) The impact of climate policy on fossil fuel consumption: Evidence from the Regional Greenhouse Gas Initiative (RGGI). Energ Econ 100: 105333. https://doi.org/10.1016/j.eneco.2021.105333 doi: 10.1016/j.eneco.2021.105333
    [3] Lawati MJA, Jafary T, Baawain TMS, et al. (2019) A mini review on biofouling on air cathode of single chamber microbial fuel cell; prevention and mitigation strategies. Biocatal Agric Biotechnol 22: 101370. https://doi.org/10.1016/j.bcab.2019.101370 doi: 10.1016/j.bcab.2019.101370
    [4] Bist N, Sircar A, Yadav K (2020) Holistic review of hybrid renewable energy in circular economy for valorisation and management. Environ Technol Inno 20: 101054. https://doi.org/10.1016/j.eti.2020.101054 doi: 10.1016/j.eti.2020.101054
    [5] Gul H, Raza W, Lee J, et al. (2021) Progress in microbial fuel cell technology for wastewater treatment and energy harvesting. Chemosphere 281: 130828. https://doi.org/10.1016/j.chemosphere.2021.130828 doi: 10.1016/j.chemosphere.2021.130828
    [6] Yaqoob AA, Ibrahim MNM, Rodríguez-Couto S (2020) Development and modification of materials to build cost-effective anodes for microbial fuel cells (MFCs): An overview. Biochem Eng J 164: 107779. https://doi.org/10.1016/j.bej.2020.107779 doi: 10.1016/j.bej.2020.107779
    [7] Wang HM, Park JD, Ren ZJ (2015) Practical energy harvesting for microbial fuel cells: A review. Environ Sci Technol 49: 3267–3277. https://doi.org/10.1021/es5047765 doi: 10.1021/es5047765
    [8] Jatoi AS, Akhter F, Mazari SA, et al. (2021) Advanced microbial fuel cell for waste water treatment—a review. Environ Sci Pollut Res 28: 5005–5019. https://doi.org/10.1007/s11356-020-11691-2 doi: 10.1007/s11356-020-11691-2
    [9] James C, Meenal SH, Elakkiya S, et al. (2020) Sustainable environment through treatment of domestic sewage using MFC. Mater Today: Proc 37: 1495–1502. https://doi.org/10.1016/j.matpr.2020.07.110 doi: 10.1016/j.matpr.2020.07.110
    [10] Kumar SS, Kumar V, Maylan SK, et al. (2019) Microbial fuel cells (MFCs) for bioelectrochemical treatment of different wastewater streams. Fuel 254: 115526. https://doi.org/10.1016/j.fuel.2019.05.109 doi: 10.1016/j.fuel.2019.05.109
    [11] Zhou J, Li M, Zhou W, et al. (2020) Efficacy of electrode position in microbial fuel cell for simultaneous Cr(Ⅵ) reduction and bioelectricity production. Sci Total Environ 748: 141425. https://doi.org/10.1016/j.scitotenv.2020.141425 doi: 10.1016/j.scitotenv.2020.141425
    [12] Sciarria TP, Arioli S, Gargari G, et al. (2019) Monitoring microbial communities' dynamics during the start-up of microbial fuel cells by high-throughput screening techniques. Biotechnol Rep 21: e00310. https://doi.org/10.1016/j.btre.2019.e00310 doi: 10.1016/j.btre.2019.e00310
    [13] Jung SP, Pandit S (2018) Important factors influencing microbial fuel cell performance, In: Mohan SV, Varjani S, Pandey A (Eds.), Biomass, biofuels, biochemicals: Microbial electrochemical technology, chemicals and remediation, Amsterdam: Elsevier, 377–406. https://doi.org/10.1016/B978-0-444-64052-9.00015-7
    [14] Thygesen A, Poulsen FW, Min B et al. (2009) The effect of different substrates and humic acid on power generation in microbial fuel cell operation. Bioresource Technol 100: 1186–1191. https://doi.org/10.1016/j.biortech.2008.07.067 doi: 10.1016/j.biortech.2008.07.067
    [15] Hwang JH, Kim KY, Lee WH, et al. (2019) Surfactant addition to enhance bioavailability of bilge water in single chamber microbial fuel cells (MFCs). J Hazard Mater 368: 732–738. https://doi.org/10.1016/j.jhazmat.2019.02.007 doi: 10.1016/j.jhazmat.2019.02.007
    [16] Bai X, Lin T, Liang N, et al. (2021) Engineering synthetic microbial consortium for efficient conversion of lactate from glucose and xylose to generate electricity. Biochem Eng J 172: 108052. https://doi.org/10.1016/j.bej.2021.108052 doi: 10.1016/j.bej.2021.108052
    [17] Darmawan R, Widjadja A, Juliastuti SR, et al. (2017) The use of mud as an alternative source for bioelectricity using microbial fuel cells. AIP Conf Proc 1840: 040006. https://doi.org/10.1063/1.4982273 doi: 10.1063/1.4982273
    [18] Purnomo T, Rachmadiarti F (2018) The changes of environment and aquatic organism biodiversity in east coast of Sidoarjo due to Lapindo hot mud. Int J GEOMATE 15: 181–186. https://doi.org/10.21660/2018.48.IJCST60 doi: 10.21660/2018.48.IJCST60
    [19] Wang HM, Song XY, Zhang HH, et al. (2020) Removal of hexavalent chromium in dual-chamber microbial fuel cells separated by different ion exchange membranes. J Hazard Mater 384: 121459. https://doi.org/10.1016/j.jhazmat.2019.121459 doi: 10.1016/j.jhazmat.2019.121459
    [20] GracePavithra K, Jaikumar V, Kumar PS, et al. (2019) A review on cleaner strategies for chromium industrial wastewater: Present research and future perspective. J Clean Prod 228: 580–593. https://doi.org/10.1016/j.jclepro.2019.04.117 doi: 10.1016/j.jclepro.2019.04.117
    [21] Costello RB, Dwyer JT, Merkel JM (2019) Chromium supplements in health and disease, In: Vincent JB (Eds.), The nutritional biochemistry of chromium (Ⅲ), 2 Eds., Amsterdam: Elsevier, 219–249. https://doi.org/10.1016/B978-0-444-64121-2.00007-6
    [22] Rajaeifar MA, Hemayati SS, Tabatabaei M, et al. (2019) A review on beet sugar industry with a focus on implementation of waste-to-energy strategy for power supply. Renew Sust Energ Rev 103: 423–442. https://doi.org/10.1016/j.rser.2018.12.056 doi: 10.1016/j.rser.2018.12.056
    [23] Aghbashlo M, Tabatabaei M, Karimi K, et al. (2017) Effect of phosphate concentration on exergetic-based sustainability parameters of glucose fermentation by Ethanolic Mucor indicus. Sustain Prod Consum 9: 28–36. https://doi.org/10.1016/j.spc.2016.06.004 doi: 10.1016/j.spc.2016.06.004
    [24] Aghbashlo M, Tabatabaei M, Karimi K (2016) Exergy-based sustainability assessment of ethanol production via Mucor indicus from fructose, glucose, sucrose, and molasses. Energy 98: 240–252. https://doi.org/10.1016/j.energy.2016.01.029 doi: 10.1016/j.energy.2016.01.029
    [25] Xiao N, Wu R, Huang JJ, et al. (2020) Anode surface modification regulates biofilm community population and the performance of micro-MFC based biochemical oxygen demand sensor. Chem Eng Sci 221: 115691. https://doi.org/10.1016/j.ces.2020.115691 doi: 10.1016/j.ces.2020.115691
    [26] Srivastava P, Yadav AK, Mishra BK (2015) The effects of microbial fuel cell integration into constructed wetland on the performance of constructed wetland. Bioresource Technol 195: 223–230. https://doi.org/10.1016/j.biortech.2015.05.072 doi: 10.1016/j.biortech.2015.05.072
    [27] Yaqoob AA, Ibrahim MNM, Rafatullah M, et al. (2020) Recent advances in anodes for microbial fuel cells: An overview. Materials 13: 2078. https://doi.org/10.3390/ma13092078 doi: 10.3390/ma13092078
    [28] Yaqoob AA, Ibrahim MNM, Rodríguez-Couto S (2020) Development and modification of materials to build cost-effective anodes for microbial fuel cells (MFCs): An overview. Biochem Eng J 164: 107779. https://doi.org/10.1016/j.bej.2020.107779 doi: 10.1016/j.bej.2020.107779
    [29] Liu JH, Ma ZL, Zhu HJ, et al. (2017) Improving xylose utilisation of defatted rice bran for nisin production by overexpression of a xylose transcriptional regulator in Lactococcus lactis. Bioresource Technol 238: 690–697. https://doi.org/10.1016/j.biortech.2017.04.076 doi: 10.1016/j.biortech.2017.04.076
    [30] Hou TT, Chen N, Tong S, et al. (2019) Enhancement of rice bran as carbon and microbial sources on the nitrate removal from groundwater. Biochem Eng J 148: 185–194. https://doi.org/10.1016/j.bej.2018.07.010 doi: 10.1016/j.bej.2018.07.010
    [31] Fernandes IJ, Calheiro D, Sánchez FAL, et al. (2017) Characterization of silica produced from rice husk ash: Comparison of purification and processing methods. Mat Res 20: 519–525. http://doi.org/10.1590/1980-5373-MR-2016-1043 doi: 10.1590/1980-5373-MR-2016-1043
    [32] Si FZ, Zang YW, Yan L, et al. (2014) Electrochemical Oxygen Reduction Reaction, In: Xing W, Yin GP, Zhang JJ (Eds.), Rotating electrode methods and oxygen reduction electrocatalysts, Amsterdam: Elsevier, 133–170. https://doi.org/10.1016/B978-0-444-63278-4.00004-5
    [33] Fan LP, Xu DD, Li C, et al. (2016) Molasses wastewater treatment by microbial fuel cell with MnO2-modified cathode. Pol J Environ Stud 25: 2349–2356. https://doi.org/10.15244/pjoes/64197 doi: 10.15244/pjoes/64197
    [34] Scott K, Murano C, Rimbu G (2007) A tubular microbial fuel cell. J Appl Electrochem 37: 1063–1068. https://doi.org/10.1007/s10800-007-9355-8 doi: 10.1007/s10800-007-9355-8
    [35] Flimban SE, Sami GA, Ismail, et al. (2019) Review overview of recent advancements in the microbial fuel cell from fundamentals to applications. Energies 12: 3390. https://doi.org/10.3390/en12173390 doi: 10.3390/en12173390
    [36] Ye YY, Ngo HH, Guo WS, et al. (2019) Effect of organic loading rate on the recovery of nutrients and energy in a dual-chamber microbial fuel cell. Bioresource Technol 281: 367–373. https://doi.org/10.1016/j.biortech.2019.02.108 doi: 10.1016/j.biortech.2019.02.108
    [37] Li WW, Sheng GP, Liu XW, et al. (2011) Recent advances in the separators for microbial fuel cells. Bioresource Technol 102: 244–252. https://doi.org/10.1016/j.biortech.2010.03.090 doi: 10.1016/j.biortech.2010.03.090
    [38] Luo Y, Zhang F, Wei B, et al. (2013) The use of cloth fabric diffusion layers for scalable microbial fuel cells. Biochem Eng J 73: 49–52. https://doi.org/10.1016/j.bej.2013.01.011 doi: 10.1016/j.bej.2013.01.011
    [39] Zhuang L, Feng CH, Zhou SG (2010) Comparison of membrane- and cloth-cathode assembly for scalable microbial fuel cells: Construction, performance and cost. Process Biochem 45: 929–934. https://doi.org/10.1016/j.procbio.2010.02.014 doi: 10.1016/j.procbio.2010.02.014
    [40] Absher M (1973) Hemocytometer Counting, In: Kruse PK, Pattersom MK (Eds.), Tissue Culture, London: Academic Press, 395–397.
    [41] Muhaimin, Hawa RII, Hidayati ER, et al. (2021) Test method verification of chrome heksavalen (Cr-Ⅵ) test in waste water using UV-visible spectrophotometer. AIP Conf Proc 2370: 030001. https://doi.org/10.1063/5.0062198 doi: 10.1063/5.0062198
    [42] Wang YJ, Zhao NN, Fang BZ, et al. (2015) Effect of different solvent ratio (ethylene glycol/water) on the preparation of Pt/C catalyst and its activity toward oxygen reduction reaction. RSC Adv 5: 56570–56577. https://doi.org/10.1039/C5RA08068A doi: 10.1039/C5RA08068A
    [43] Raad NK, Farrokhi F, Mousavi SA, et al. (2020) Simultaneous power generation and sewage sludge stabilisation using an air cathode-MFCs. Biomass Bioenerg 140: 105642. https://doi.org/10.1016/j.biombioe.2020.105642 doi: 10.1016/j.biombioe.2020.105642
    [44] Tashiro T, Yoshimura F (2019) A neo-logistic model for the growth of bacteria. Physica A 525: 199–215. https://doi.org/10.1016/j.physa.2019.03.049 doi: 10.1016/j.physa.2019.03.049
    [45] Microbial Growth. Bruslind L, 2021. Available from: https://bio.libretexts.org/Bookshelves/Microbiology/Book%3A_Microbiology_(Bruslind)/09%3A_Microbial_Growth.
    [46] Nicola L, Bååth E (2019) The effect of temperature and moisture on lag phase length of bacterial growth in soil after substrate addition. Soil Biol Biochem 137: 107563. https://doi.org/10.1016/j.soilbio.2019.107563 doi: 10.1016/j.soilbio.2019.107563
    [47] Maier RM, Pepper IL (2015) Bacterial growth, In: Pepper IL, Gerba CP, Gentry TJ (Eds.), Environ microbiol, 3 Eds., London: Academic Press: 37–56. https://doi.org/10.1016/B978-0-12-394626-3.00003-X
    [48] Silveira G, Aquino NS, Schneedorf JM (2020) Development, characterisation and application of a low-cost single chamber microbial fuel cell based on hydraulic couplers. Energy 208: 118395. https://doi.org/10.1016/j.energy.2020.118395 doi: 10.1016/j.energy.2020.118395
    [49] Yarmush M, Pedersen H (1995) Biochemical engineering. Curr Opin Biotech, 6: 189–191. https://doi.org/10.1016/0958-1669(95)80030-1 doi: 10.1016/0958-1669(95)80030-1
    [50] Nuryana IF, Puspitasari R, Juliastuti SR (2020) Study of electrode modification and microbial concentration for microbial fuel cell effectivity from molasses waste and reduction of heavy metal Cr (Ⅵ) by continue dual chamber reactor. IOP Conf Ser: Mater Sci Eng 823: 012016. https://doi.org/10.1088/1757-899X/823/1/012016 doi: 10.1088/1757-899X/823/1/012016
    [51] Kaur R, Marwaha A, Chhabra VA, et al. (2020) Recent developments on functional nanomaterial-based electrodes for microbial fuel cells. Renew Sust Energ Rev 119: 109551. https://doi.org/10.1016/j.rser.2019.109551 doi: 10.1016/j.rser.2019.109551
    [52] Slate AJ, Whitehead KA, Brownson DAC, et al. (2019) Microbial fuel cells: An overview of current technology. Renew Sust Energ Rev 101: 60–81. https://doi.org/10.1016/j.rser.2018.09.044 doi: 10.1016/j.rser.2018.09.044
    [53] Dutta K, Kundu PP (2018) Introduction to microbial fuel cells, In: Progress and recent trends in microbial fuel cells, Amsterdam: Elsevier, 1–6. https://doi.org/10.1016/B978-0-444-64017-8.00001-4
    [54] Wang G, Huang LP, Zhang YF (2008) Cathodic reduction of hexavalent chromium [Cr(Ⅵ)] coupled with electricity generation in microbial fuel cells. Biotechnol Lett 30: 1959. https://doi.org/10.1007/s10529-008-9792-4 doi: 10.1007/s10529-008-9792-4
    [55] Lokman NA, Ithnin AM, Yahya WJ, et al. (2021) A brief review on biochemical oxygen demand (BOD) treatment methods for palm oil mill effluents (POME). Environ Technol Inno 21: 101258. https://doi.org/10.1016/j.eti.2020.101258 doi: 10.1016/j.eti.2020.101258
    [56] Yan FF, Wu C, Cheng YY, et al. (2013) Carbon nanotubes promote Cr(Ⅵ) reduction by alginate-immobilised Shewanella oneidensis MR-1. Biochem Eng J 77: 183–189. https://doi.org/10.1016/j.bej.2013.06.009 doi: 10.1016/j.bej.2013.06.009
    [57] Aoki Y, Sidiq TP (2014) Ground deformation associated with the eruption of Lumpur Sidoarjo mud volcano, east Java, Indonesia. J Volcanol. Geoth Res 278–279: 96–102. https://doi.org/10.1016/j.jvolgeores.2014.04.012 doi: 10.1016/j.jvolgeores.2014.04.012
  • This article has been cited by:

    1. Arife Atay, Disjoint union of fuzzy soft topological spaces, 2023, 8, 2473-6988, 10547, 10.3934/math.2023535
    2. Tareq M. Al-shami, Abdelwaheb Mhemdi, Radwan Abu-Gdairi, Mohammed E. El-Shafei, Compactness and connectedness via the class of soft somewhat open sets, 2023, 8, 2473-6988, 815, 10.3934/math.2023040
    3. Samer Al Ghour, Soft Complete Continuity and Soft Strong Continuity in Soft Topological Spaces, 2023, 12, 2075-1680, 78, 10.3390/axioms12010078
    4. Samer Al Ghour, Soft Regular Generalized ω-Closed Sets and Soft ω-T1/2 Spaces, 2022, 11, 2075-1680, 529, 10.3390/axioms11100529
    5. Samer Al Ghour, Between Soft θ-Openness and Soft ω0-Openness, 2023, 12, 2075-1680, 311, 10.3390/axioms12030311
    6. Rehab Alharbi, S. E. Abbas, E. El-Sanowsy, H. M. Khiamy, Ismail Ibedou, Soft closure spaces via soft ideals, 2024, 9, 2473-6988, 6379, 10.3934/math.2024311
    7. Samer Al Ghour, Soft ωs-irresoluteness and soft pre-ωs-openness insoft topological spaces, 2023, 45, 10641246, 1141, 10.3233/JIFS-223332
    8. Samer Al-Ghour, Soft ωβ
    -open sets and their generated soft topology, 2024, 43, 2238-3603, 10.1007/s40314-024-02731-5
    9. Mohammed Abu Saleem, On soft covering spaces in soft topological spaces, 2024, 9, 2473-6988, 18134, 10.3934/math.2024885
    10. Tareq M. Al-shami, Abdelwaheb Mhemdi, On soft parametric somewhat-open sets and applications via soft topologies, 2023, 9, 24058440, e21472, 10.1016/j.heliyon.2023.e21472
    11. Harzheen D. Abdulkareem, Ramadhan A. Mohammed, Zanyar A. Ameen, Connectedness of Soft r-Topological Spaces, 2024, 20, 1793-0057, 27, 10.1142/S1793005724500030
    12. Tareq M. Al-shami, Zanyar A. Ameen, Abdelwaheb Mhemdi, The connection between ordinary and soft σ-algebras with applications to information structures, 2023, 8, 2473-6988, 14850, 10.3934/math.2023759
    13. Fatma Onat Bulak, Hacer Bozkurt, Soft quasilinear operators in soft normed quasilinear spaces, 2023, 45, 10641246, 4847, 10.3233/JIFS-230035
    14. Samer Al Ghour, Soft C-continuity and soft almost C-continuity between soft topological spaces, 2023, 9, 24058440, e16363, 10.1016/j.heliyon.2023.e16363
    15. Dina Abuzaid, Samer Al Ghour, Monia Naghi, Praveen Kumar Donta, Soft super-continuity and soft delta-closed graphs, 2024, 19, 1932-6203, e0301705, 10.1371/journal.pone.0301705
    16. Zanyar A. Ameen, Tareq M. Al-shami, Abdelwaheb Mhemdi, Mohammed E. El-Shafei, Naeem Jan, The Role of Soft θ‐Topological Operators in Characterizing Various Soft Separation Axioms, 2022, 2022, 2314-4629, 10.1155/2022/9073944
    17. Dina Abuzaid, Samer Al-Ghour, Supra soft Omega-open sets and supra soft Omega-regularity, 2025, 10, 2473-6988, 6636, 10.3934/math.2025303
    18. Merve Aliye Akyol, Özlem Küçükgüçlü, Ahmet Turan Işık, Görsev Yener, The Efficacy of the Computer-Based Multi-Domain Cognitive Training Program on the Cognitive Performance of Healthy Older Adults: A Pilot Randomized Controlled Study, 2025, 26872625, 10.4274/ejgg.galenos.2024.2023-10-5
    19. Jean-Pierre Tardieu, Los esclavizados en Hispanoamérica a mediados del siglo XVII. Estimaciones de un memorial de 1644, 2024, 84, 1988-3188, 1755, 10.3989/revindias.2024.1755
  • Reader Comments
  • © 2022 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(2595) PDF downloads(151) Cited by(5)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog