This study examined the potential of electricity generation from biogas and heat energy arising from municipal solid waste (MSW) collected from the year 2021 to 2045 using anaerobic digestion (AD) and incineration (INC) technologies. The goal of this paper is to evaluate the economic and environmental benefits of implementing the aforementioned technologies in Lesotho. The environmental impact was assessed by using the life cycle assessment strategy based on global warming potential for three scenarios, while the economic assessment was carried out by using the net present value (NPV), levelized cost of energy (LCOE) and total life cycle cost. The key findings show that, over 25 years (2021–2045), MSW generation will range from 185.855 to 513.587 kilotons. The methane yield for the duration of the project for AD technology is 44.67–126.56 thousand cubic meters per year. Moreover, the electricity generation will range from 0.336–0.887 GWh for AD technology and 17.15–45.34 GWh for INC technology. Economically, the results demonstrated that the two waste-to-energy technologies are viable, as evidenced by their positive NPV. The NPV for AD was about USD 0.514 million, and that for INC technology was USD 339.65 million. AD and INC have LCOEs of 0.029 and 0.0023 USD/kWh, respectively. The findings demonstrate that AD can minimize the potential for global warming by 95%, signifying a huge environmental advantage. This paper serves to provide the government, as well as the investors, with current and trustworthy information on waste-to-energy technologies in terms of costs, execution and worldwide effect, which could aid optimal decision-making in waste-to-energy projects in Lesotho.
Citation: Tsepo Sechoala, Olawale Popoola, Temitope Ayodele. Economic and environmental assessment of electricity generation using biogas and heat energy from municipal solid waste: A case study of Lesotho[J]. AIMS Energy, 2023, 11(2): 337-357. doi: 10.3934/energy.2023018
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This study examined the potential of electricity generation from biogas and heat energy arising from municipal solid waste (MSW) collected from the year 2021 to 2045 using anaerobic digestion (AD) and incineration (INC) technologies. The goal of this paper is to evaluate the economic and environmental benefits of implementing the aforementioned technologies in Lesotho. The environmental impact was assessed by using the life cycle assessment strategy based on global warming potential for three scenarios, while the economic assessment was carried out by using the net present value (NPV), levelized cost of energy (LCOE) and total life cycle cost. The key findings show that, over 25 years (2021–2045), MSW generation will range from 185.855 to 513.587 kilotons. The methane yield for the duration of the project for AD technology is 44.67–126.56 thousand cubic meters per year. Moreover, the electricity generation will range from 0.336–0.887 GWh for AD technology and 17.15–45.34 GWh for INC technology. Economically, the results demonstrated that the two waste-to-energy technologies are viable, as evidenced by their positive NPV. The NPV for AD was about USD 0.514 million, and that for INC technology was USD 339.65 million. AD and INC have LCOEs of 0.029 and 0.0023 USD/kWh, respectively. The findings demonstrate that AD can minimize the potential for global warming by 95%, signifying a huge environmental advantage. This paper serves to provide the government, as well as the investors, with current and trustworthy information on waste-to-energy technologies in terms of costs, execution and worldwide effect, which could aid optimal decision-making in waste-to-energy projects in Lesotho.
Baló's concentric sclerosis (BCS) was first described by Marburg [1] in 1906, but became more widely known until 1928 when the Hungarian neuropathologist Josef Baló published a report of a 23-year-old student with right hemiparesis, aphasia, and papilledema, who at autopsy had several lesions of the cerebral white matter, with an unusual concentric pattern of demyelination [2]. Traditionally, BCS is often regarded as a rare variant of multiple sclerosis (MS). Clinically, BCS is most often characterized by an acute onset with steady progression to major disability and death with months, thus resembling Marburg's acute MS [3,4]. Its pathological hallmarks are oligodendrocyte loss and large demyelinated lesions characterized by the annual ring-like alternating pattern of demyelinating and myelin-preserved regions. In [5], the authors found that tissue preconditioning might explain why Baló lesions develop a concentric pattern. According to the tissue preconditioning theory and the analogies between Baló's sclerosis and the Liesegang periodic precipitation phenomenon, Khonsari and Calvez [6] established the following chemotaxis model
˜uτ=DΔX˜u⏟diffusion ofactivated macrophages−∇X⋅(˜χ˜u(ˉu−˜u)∇˜v)⏟chemoattractant attractssurrounding activated macrophages+μ˜u(ˉu−˜u)⏟production of activated macrophages,−˜ϵΔX˜v⏟diffusion of chemoattractant=−˜α˜v+˜β˜w⏟degradation∖production of chemoattractant,˜wτ=κ˜uˉu+˜u˜u(ˉw−˜w)⏟destruction of oligodendrocytes, | (1.1) |
where ˜u, ˜v and ˜w are, respectively, the density of activated macrophages, the concentration of chemoattractants and density of destroyed oligodendrocytes. ˉu and ˉw represent the characteristic densities of macrophages and oligodendrocytes respectively.
By numerical simulation, the authors in [6,7] indicated that model (1.1) only produces heterogeneous concentric demyelination and homogeneous demyelinated plaques as χ value gradually increases. In addition to the chemoattractant produced by destroyed oligodendrocytes, "classically activated'' M1 microglia also can release cytotoxicity [8]. Therefore we introduce a linear production term into the second equation of model (1.1), and establish the following BCS chemotaxis model with linear production term
{˜uτ=DΔX˜u−∇X⋅(˜χ˜u(ˉu−˜u)∇˜v)+μ˜u(ˉu−˜u),−˜ϵΔX˜v+˜α˜v=˜β˜w+˜γ˜u,˜wτ=κ˜uˉu+˜u˜u(ˉw−˜w). | (1.2) |
Before going to details, let us simplify model (1.2) with the following scaling
u=˜uˉu,v=μˉu˜ϵD˜v,w=˜wˉw,t=μˉuτ,x=√μˉuDX,χ=˜χ˜ϵμ,α=D˜α˜ϵμˉu,β=˜βˉw,γ=˜γˉu,δ=κμ, |
then model (1.2) takes the form
{ut=Δu−∇⋅(χu(1−u)∇v)+u(1−u),x∈Ω,t>0,−Δv+αv=βw+γu,x∈Ω,t>0,wt=δu1+uu(1−w),x∈Ω,t>0,∂ηu=∂ηv=0,x∈∂Ω,t>0,u(x,0)=u0(x),w(x,0)=w0(x),x∈Ω, | (1.3) |
where Ω⊂Rn(n≥1) is a smooth bounded domain, η is the outward normal vector to ∂Ω, ∂η=∂/∂η, δ balances the speed of the front and the intensity of the macrophages in damaging the myelin. The parameters χ,α and δ are positive constants as well as β,γ are nonnegative constants.
If δ=0, then model (1.3) is a parabolic-elliptic chemotaxis system with volume-filling effect and logistic source. In order to be more line with biologically realistic mechanisms, Hillen and Painter [9,10] considered the finite size of individual cells-"volume-filling'' and derived volume-filling models
{ut=∇⋅(Du(q(u)−q′(u)u)∇u−q(u)uχ(v)∇v)+f(u,v),vt=DvΔv+g(u,v). | (1.4) |
q(u) is the probability of the cell finding space at its neighbouring location. It is also called the squeezing probability, which reflects the elastic properties of cells. For the linear choice of q(u)=1−u, global existence of solutions to model (1.4) in any space dimension are investigated in [9]. Wang and Thomas [11] established the global existence of classical solutions and given necessary and sufficient conditions for spatial pattern formation to a generalized volume-filling chemotaxis model. For a chemotaxis system with generalized volume-filling effect and logistic source, the global boundedness and finite time blow-up of solutions are obtained in [12]. Furthermore, the pattern formation of the volume-filling chemotaxis systems with logistic source and both linear diffusion and nonlinear diffusion are shown in [13,14,15] by the weakly nonlinear analysis. For parabolic-elliptic Keller-Segel volume-filling chemotaxis model with linear squeezing probability, asymptotic behavior of solutions is studied both in the whole space Rn [16] and on bounded domains [17]. Moreover, the boundedness and singularity formation in parabolic-elliptic Keller-Segel volume-filling chemotaxis model with nonlinear squeezing probability are discussed in [18,19].
Very recently, we [20] investigated the uniform boundedness and global asymptotic stability for the following chemotaxis model of multiple sclerosis
{ut=Δu−∇⋅(χ(u)∇v)+u(1−u),χ(u)=χu1+u,x∈Ω,t>0,τvt=Δv−βv+αw+γu,x∈Ω,t>0,wt=δu1+uu(1−w),x∈Ω,t>0, |
subject to the homogeneous Neumann boundary conditions.
In this paper, we are first devoted to studying the local existence and uniform boundedness of the unique classical solution to system (1.3) by using Neumann heat semigroup arguments, Banach fixed point theorem, parabolic Schauder estimate and elliptic regularity theory. Then we discuss that exponential asymptotic stability of the positive equilibrium point to system (1.3) by constructing Lyapunov function.
Although, in the pathological mechanism of BCS, the initial data in model (1.3) satisfy 0<u0(x)≤1,w0(x)=0, we mathematically assume that
{u0(x)∈C0(ˉΩ)with0≤,≢u0(x)≤1inΩ,w0(x)∈C2+ν(ˉΩ)with0<ν<1and0≤w0(x)≤1inΩ. | (1.5) |
It is because the condition (1.5) implies u(x,t0)>0 for any t0>0 by the strong maximum principle.
The following theorems give the main results of this paper.
Theorem 1.1. Assume that the initial data (u0(x),w0(x)) satisfy the condition (1.5). Then model (1.3) possesses a unique global solution (u(x,t),v(x,t),w(x,t)) satisfying
u(x,t)∈C0(ˉΩ×[0,∞))∩C2,1(ˉΩ×(0,∞)),v(x,t)∈C0((0,∞),C2(ˉΩ)),w(x,t)∈C2,1(ˉΩ×[0,∞)), | (1.6) |
and
0<u(x,t)≤1,0≤v(x,t)≤β+γα,w0(x)≤w(x,t)≤1,inˉΩ×(0,∞). |
Moreover, there exist a ν∈(0,1) and M>0 such that
‖u‖C2+ν,1+ν/2(ˉΩ×[1,∞))+‖v‖C0([1,∞),C2+ν(ˉΩ))+‖w‖Cν,1+ν/2(ˉΩ×[1,∞))≤M. | (1.7) |
Theorem 1.2. Assume that β≥0,γ≥0,β+γ>0 and
χ<{min{2√2αβ,2√2αγ},β>0,γ>0,2√2αβ,β>0,γ=0,2√2αγ,β=0,γ>0. | (1.8) |
Let (u,v,w) be a positive classical solution of the problem (1.3), (1.5). Then
‖u(⋅,t)−u∗‖L∞(Ω)+‖v(⋅,t)−v∗‖L∞(Ω)+‖w(⋅,t)−w∗‖L∞(Ω)→0,ast→∞. | (1.9) |
Furthermore, there exist positive constants λ=λ(χ,α,γ,δ,n) and C=C(|Ω|,χ,α,β,γ,δ) such that
‖u−u∗‖L∞(Ω)≤Ce−λt,‖v−v∗‖L∞(Ω)≤Ce−λt,‖w−w∗‖L∞(Ω)≤Ce−λt,t>0, | (1.10) |
where (u∗,v∗,w∗)=(1,β+γα,1) is the unique positive equilibrium point of the model (1.3).
The paper is organized as follows. In section 2, we prove the local existence, the boundedness and global existence of a unique classical solution. In section 3, we firstly establish the uniform convergence of the positive global classical solution, then discuss the exponential asymptotic stability of positive equilibrium point in the case of weak chemotactic sensitivity. The paper ends with a brief concluding remarks.
The aim of this section is to develop the existence and boundedness of a global classical solution by employing Neumann heat semigroup arguments, Banach fixed point theorem, parabolic Schauder estimate and elliptic regularity theory.
Proof of Theorem 1.1 (ⅰ) Existence. For p∈(1,∞), let A denote the sectorial operator defined by
Au:=−Δuforu∈D(A):={φ∈W2,p(Ω)|∂∂ηφ|∂Ω=0}. |
λ1>0 denote the first nonzero eigenvalue of −Δ in Ω with zero-flux boundary condition. Let A1=−Δ+α and Xl be the domains of fractional powers operator Al,l≥0. From the Theorem 1.6.1 in [21], we know that for any p>n and l∈(n2p,12),
‖z‖L∞(Ω)≤C‖Al1z‖Lp(Ω)forallz∈Xl. | (2.1) |
We introduce the closed subset
S:={u∈X|‖u‖L∞((0,T);L∞(Ω))≤R+1} |
in the space X:=C0([0,T];C0(ˉΩ)), where R is a any positive number satisfying
‖u0(x)‖L∞(Ω)≤R |
and T>0 will be specified later. Note F(u)=u1+u, we consider an auxiliary problem with F(u) replaced by its extension ˜F(u) defined by
˜F(u)={F(u)uifu≥0,−F(−u)(−u)ifu<0. |
Notice that ˜F(u) is a smooth globally Lipshitz function. Given ˆu∈S, we define Ψˆu=u by first writing
w(x,t)=(w0(x)−1)e−δ∫t0˜F(ˆu)ˆuds+1,x∈Ω,t>0, | (2.2) |
and
w0≤w(x,t)≤1,x∈Ω,t>0, |
then letting v solve
{−Δv+αv=βw+γˆu,x∈Ω,t∈(0,T),∂ηv=0,x∈∂Ω,t∈(0,T), | (2.3) |
and finally taking u to be the solution of the linear parabolic problem
{ut=Δu−χ∇⋅(ˆu(1−ˆu)∇v)+ˆu(1−ˆu),x∈Ω,t∈(0,T),∂ηu=0,x∈∂Ω,t∈(0,T),u(x,0)=u0(x),x∈Ω. |
Applying Agmon-Douglas-Nirenberg Theorem [22,23] for the problem (2.3), there exists a constant C such that
‖v‖W2p(Ω)≤C(β‖w‖Lp(Ω)+γ‖ˆu‖Lp(Ω))≤C(β|Ω|1p+γ(R+1)) | (2.4) |
for all t∈(0,T). From a variation-of-constants formula, we define
Ψ(ˆu)=etΔu0−χ∫t0e(t−s)Δ∇⋅(ˆu(1−ˆu)∇v(s))ds+∫t0e(t−s)Δˆu(s)(1−ˆu(s))ds. |
First we shall show that for T small enough
‖Ψ(ˆu)‖L∞((0,T);L∞(Ω))≤R+1 |
for any ˆu∈S. From the maximum principle, we can give
‖etΔu0‖L∞(Ω)≤‖u0‖L∞(Ω), | (2.5) |
and
∫t0‖etΔˆu(s)(1−ˆu(s))‖L∞(Ω)ds≤∫t0‖ˆu(s)(1−ˆu(s))‖L∞(Ω)ds≤(R+1)(R+2)T | (2.6) |
for all t∈(0,T). We use inequalities (2.1) and (2.4) to estimate
χ∫t0‖e(t−s)Δ∇⋅(ˆu(1−ˆu)∇v(s))‖L∞(Ω)ds≤C∫t0(t−s)−l‖et−s2Δ∇⋅(ˆu(1−ˆu)∇v(s))‖Lp(Ω)ds≤C∫t0(t−s)−l−12‖(ˆu(1−ˆu)∇v(s)‖Lp(Ω)ds≤CT12−l(R+1)(R+2)(β|Ω|1p+γ(R+1)) | (2.7) |
for all t∈(0,T). This estimate is attributed to T<1 and the inequality in [24], Lemma 1.3 iv]
‖etΔ∇z‖Lp(Ω)≤C1(1+t−12)e−λ1t‖z‖Lp(Ω)forallz∈C∞c(Ω). |
From inequalities (2.5), (2.6) and (2.7) we can deduce that Ψ maps S into itself for T small enough.
Next we prove that the map Ψ is a contractive on S. For ˆu1,ˆu2∈S, we estimate
‖Ψ(ˆu1)−Ψ(ˆu2)‖L∞(Ω)≤χ∫t0(t−s)−l−12‖[ˆu2(s)(1−ˆu2(s))−ˆu1(s)(1−ˆu1(s))]∇v2(s)‖Lp(Ω)ds+χ∫t0‖ˆu1(s)(1−ˆu1(s))(∇v1(s)−∇v2(s))‖Lp(Ω)ds+∫t0‖e(t−s)Δ[ˆu1(s)(1−ˆu1(s))−ˆu2(s)(1−ˆu2(s))]‖L∞(Ω)ds≤χ∫t0(t−s)−l−12(2R+1)‖ˆu1(s)−ˆu2(s)‖X‖∇v2(s)‖Lp(Ω)ds+χ∫t0(R+1)(R+2)(β‖w1(s)−w2(s)‖Lp(Ω)+γ‖ˆu1(s)−ˆu2(s)‖Lp(Ω))ds+∫t0(2R+1)‖ˆu1(s)−ˆu2(s)‖Xds≤χ∫t0(t−s)−l−12(2R+1)‖ˆu1(s)−ˆu2(s)‖X‖∇v2(s)‖Lp(Ω)ds+2βδχ∫t0(R+1)(R+2)t‖ˆu1(s)−ˆu2(s)‖Lp(Ω)+γ‖ˆu1(s)−ˆu2(s)‖Lp(Ω)ds+∫t0(2R+1)‖ˆu1(s)−ˆu2(s)‖Xds≤(CχT12−l(2R+1)(β|Ω|1p+γ(R+1))+2βδχT(R2+3R+γ+2)+T(2R+1))‖ˆu1(s)−ˆu2(s)‖X. |
Fixing T∈(0,1) small enough such that
(CχT12−l(2R+1)(β|Ω|1p+γ(R+1))+2βδχT(R2+3R+γ+2)+T(2R+1))≤12. |
It follows from the Banach fixed point theorem that there exists a unique fixed point of Ψ.
(ⅱ) Regularity. Since the above of T depends on ‖u0‖L∞(Ω) and ‖w0‖L∞(Ω) only, it is clear that (u,v,w) can be extended up to some maximal Tmax∈(0,∞]. Let QT=Ω×(0,T] for all T∈(0,Tmax). From u∈C0(ˉQT), we know that w∈C0,1(ˉQT) by the expression (2.2) and v∈C0([0,T],W2p(Ω)) by Agmon-Douglas-Nirenberg Theorem [22,23]. From parabolic Lp-estimate and the embedding relation W1p(Ω)↪Cν(ˉΩ),p>n, we can get u∈W2,1p(QT). By applying the following embedding relation
W2,1p(QT)↪Cν,ν/2(ˉQT),p>n+22, | (2.8) |
we can derive u(x,t)∈Cν,ν/2(ˉQT) with 0<ν≤2−n+2p. The conclusion w∈Cν,1+ν/2(ˉQT) can be obtained by substituting u∈Cν,ν/2(ˉQT) into the formulation (2.2). The regularity u∈C2+ν,1+ν/2(ˉQT) can be deduced by using further bootstrap argument and the parabolic Schauder estimate. Similarly, we can get v∈C0((0,T),C2+ν(ˉΩ)) by using Agmon-Douglas-Nirenberg Theorem [22,23]. From the regularity of u we have w∈C2+ν,1+ν/2(ˉQT).
Moreover, the maximal principle entails that 0<u(x,t)≤1, 0≤v(x,t)≤β+γα. It follows from the positivity of u that ˜F(u)=F(u) and because of the uniqueness of solution we infer the existence of the solution to the original problem.
(ⅲ) Uniqueness. Suppose (u1,v1,w1) and (u2,v2,w2) are two deferent solutions of model (1.3) in Ω×[0,T]. Let U=u1−u2, V=v1−v2, W=w1−w2 for t∈(0,T). Then
12ddt∫ΩU2dx+∫Ω|∇U|2dx≤χ∫Ω|u1(1−u1)−u2(1−u2)|∇v1||∇U|+u2(1−u2)|∇V||∇U|dx+∫Ω|u1(1−u1)−u2(1−u2)||U|dx≤χ∫Ω|U||∇v1||∇U|+14|∇V||∇U|dx+∫Ω|U|2dx≤∫Ω|∇U|2dx+χ232∫Ω|∇V|2dx+χ2K2+22∫Ω|U|2dx, | (2.9) |
where we have used that |∇v1|≤K results from ∇v1∈C0([0,T],C0(ˉΩ)).
Similarly, by Young inequality and w0≤w1≤1, we can estimate
∫Ω|∇V|2dx+α2∫Ω|V|2dx≤β2α∫Ω|W|2dx+γ2α∫Ω|U|2dx, | (2.10) |
and
ddt∫ΩW2dx≤δ∫Ω|U|2+|W|2dx. | (2.11) |
Finally, adding to the inequalities (2.9)–(2.11) yields
ddt(∫ΩU2dx+∫ΩW2dx)≤C(∫ΩU2dx+∫ΩW2dx)forallt∈(0,T). |
The results U≡0, W≡0 in Ω×(0,T) are obtained by Gronwall's lemma. From the inequality (2.10), we have V≡0. Hence (u1,v1,w1)=(u2,v2,w2) in Ω×(0,T).
(ⅳ) Uniform estimates. We use the Agmon-Douglas-Nirenberg Theorem [22,23] for the second equation of the model (1.3) to get
‖v‖C0([t,t+1],W2p(Ω))≤C(‖u‖Lp(Ω×[t,t+1])+‖w‖Lp(Ω×[t,t+1]))≤C2 | (2.12) |
for all t≥1 and C2 is independent of t. From the embedded relationship W1p(Ω)↪C0(ˉΩ),p>n, the parabolic Lp-estimate and the estimation (2.12), we have
‖u‖W2,1p(Ω×[t,t+1])≤C3 |
for all t≥1. The estimate ‖u‖Cν,ν2(ˉΩ×[t,t+1])≤C4 for all t≥1 obtained by the embedded relationship (2.8). We can immediately compute ‖w‖Cν,1+ν2(ˉΩ×[t,t+1])≤C5 for all t≥1 according to the regularity of u and the specific expression of w. Further, bootstrapping argument leads to ‖v‖C0([t,t+1],C2+ν(ˉΩ))≤C6 and ‖u‖C2+ν,1+ν2(ˉΩ×[t,t+1])≤C7 for all t≥1. Thus the uniform estimation (1.7) is proved.
Remark 2.1. Assume the initial data 0<u0(x)≤1 and w0(x)=0. Then the BCS model (1.3) has a unique classical solution.
In this section we investigate the global asymptotic stability of the unique positive equilibrium point (1,β+γα,1) to model (1.3). To this end, we first introduce following auxiliary problem
{uϵt=Δuϵ−∇⋅(uϵ(1−uϵ)∇vϵ)+uϵ(1−uϵ),x∈Ω,t>0,−Δvϵ+αvϵ=βwϵ+γuϵ,x∈Ω,t>0,wϵt=δu2ϵ+ϵ1+uϵ(1−wϵ),x∈Ω,t>0,∂ηuϵ=∂ηvϵ=0,x∈∂Ω,t>0,uϵ(x,0)=u0(x),wϵ(x,0)=w0(x),x∈Ω. | (3.1) |
By a similar proof of Theorem 1.1, we get that the problem (3.1) has a unique global classical solution (uϵ,vϵ,wϵ), and there exist a ν∈(0,1) and M1>0 which is independent of ϵ such that
‖uϵ‖C2+ν,1+ν/2(ˉΩ×[1,∞))+‖vϵ‖C2+ν,1+ν/2(ˉΩ×[1,∞))+‖wϵ‖Cν,1+ν/2(ˉΩ×[1,∞))≤M1. | (3.2) |
Then, motivated by some ideas from [25,26], we construct a Lyapunov function to study the uniform convergence of homogeneous steady state for the problem (3.1).
Let us give following lemma which is used in the proof of Lemma 3.2.
Lemma 3.1. Suppose that a nonnegative function f on (1,∞) is uniformly continuous and ∫∞1f(t)dt<∞. Then f(t)→0 as t→∞.
Lemma 3.2. Assume that the condition (1.8) is satisfied. Then
‖uϵ(⋅,t)−1‖L2(Ω)+‖vϵ(⋅,t)−v∗‖L2(Ω)+‖wϵ(⋅,t)−1‖L2(Ω)→0,t→∞, | (3.3) |
where v∗=β+γα.
Proof We construct a positive function
E(t):=∫Ω(uε−1−lnuϵ)+12δϵ∫Ω(wϵ−1)2,t>0. |
From the problem (3.1) and Young's inequality, we can compute
ddtE(t)≤χ24∫Ω|∇vϵ|2dx−∫Ω(uϵ−1)2dx−∫Ω(wϵ−1)2dx,t>0. | (3.4) |
We multiply the second equations in system (3.1) by vϵ−v∗, integrate by parts over Ω and use Young's inequality to obtain
∫Ω|∇vϵ|2dx≤γ22α∫Ω(uϵ−1)2dx+β22α∫Ω(wϵ−1)2dx,t>0, | (3.5) |
and
∫Ω(vϵ−v∗)2dx≤2γ2α2∫Ω(uϵ−1)2dx+2β2α2∫Ω(wϵ−1)2dx,t>0. | (3.6) |
Substituting inequality (3.5) into inequality (3.4) to get
ddtE(t)≤−C8(∫Ω(uϵ−1)2dx+∫Ω(wϵ−1)2dx),t>0, |
where C8=min{1−χ2β28α,1−χ2γ28α}>0.
Let f(t):=∫Ω(uϵ−1)2+(wϵ−1)2dx. Then
∫∞1f(t)dt≤E(1)C8<∞,t>1. |
It follows from the uniform estimation (3.2) and the Arzela-Ascoli theorem that f(t) is uniformly continuous in (1,∞). Applying Lemma 3.1, we have
∫Ω(uϵ(⋅,t)−1)2+(wϵ(⋅,t)−1)2dx→0,t→∞. | (3.7) |
Combining inequality (3.6) and the limit (3.7) to obtain
∫Ω(vϵ(⋅,t)−v∗)2dx→0,t→∞. |
Proof of Theorem 1.2 As we all known, each bounded sequence in C2+ν,1+ν2(ˉΩ×[1,∞)) is precompact in C2,1(ˉΩ×[1,∞)). Hence there exists some subsequence {uϵn}∞n=1 satisfying ϵn→0 as n→∞ such that
limn→∞‖uϵn−u∗‖C2,1(ˉΩ×[1,∞))=0. |
Similarly, we can get
limn→∞‖vϵn−v∗‖C2(ˉΩ)=0, |
and
limn→∞‖wϵn−w∗‖C0,1(ˉΩ×[1,∞))=0. |
Combining above limiting relations yields that (u∗,v∗,w∗) satisfies model (1.3). The conclusion (u∗,v∗,w∗)=(u,v,w) is directly attributed to the uniqueness of the classical solution of the model (1.3). Furthermore, according to the conclusion, the strong convergence (3.3) and Diagonal line method, we can deduce
‖u(⋅,t)−1‖L2(Ω)+‖v(⋅,t)−v∗‖L2(Ω)+‖w(⋅,t)−1‖L2(Ω)→0,t→∞. | (3.8) |
By applying Gagliardo-Nirenberg inequality
‖z‖L∞≤C‖z‖2/(n+2)L2(Ω)‖z‖n/(n+2)W1,∞(Ω),z∈W1,∞(Ω), | (3.9) |
comparison principle of ODE and the convergence (3.8), the uniform convergence (1.9) is obtained immediately.
Since limt→∞‖u(⋅,t)−1‖L∞(Ω)=0, so there exists a t1>0 such that
u(x,t)≥12forallx∈Ω,t>t1. | (3.10) |
Using the explicit representation formula of w
w(x,t)=(w0(x)−1)e−δ∫t0F(u)uds+1,x∈Ω,t>0 |
and the inequality (3.10), we have
‖w(⋅,t)−1‖L∞(Ω)≤e−δ6(t−t1),t>t1. | (3.11) |
Multiply the first two equations in model (1.3) by u−1 and v−v∗, respectively, integrate over Ω and apply Cauchy's inequality, Young's inequality and the inequality (3.10), to find
ddt∫Ω(u−1)2dx≤χ232∫Ω|∇v|2dx−∫Ω(u−1)2dx,t>t1. | (3.12) |
∫Ω|∇v|2dx+α2∫Ω(v−v∗)2dx≤β2α∫Ω(w−1)2dx+γ2α∫Ω(u−1)2dx,t>0. | (3.13) |
Combining the estimations (3.11)–(3.13) leads us to the estimate
ddt∫Ω(u−1)2dx≤(χ2γ232α−1)∫Ω(u−1)2dx+χ2β232αe−δ3(t−t1),t>t1. |
Let y(t)=∫Ω(u−1)2dx. Then
y′(t)≤(χ2γ232α−1)y(t)+χ2β232αe−δ3(t−t1),t>t1. |
From comparison principle of ODE, we get
y(t)≤(y(t1)−3χ2β232α(3−δ)−χ2γ2)e−(1−χ2γ232α)(t−t1)+3χ2β232α(3−δ)−χ2γ2e−δ3(t−t1),t>t1. |
This yields
∫Ω(u−1)2dx≤C9e−λ2(t−t1),t>t1, | (3.14) |
where λ2=min{1−χ2γ232α,δ3} and C9=max{|Ω|−3χ2β232α(3−δ)−χ2γ2,3χ2β232α(3−δ)−χ2γ2}.
From the inequalities (3.11), (3.13) and (3.14), we derive
∫Ω(v−β+γα)2dx≤C10e−λ2(t−t1),t>t1, | (3.15) |
where C10=max{2γ2α2C9,2β2α2}. By employing the uniform estimation (1.7), the inequalities (3.9), (3.14) and (3.15), the exponential decay estimation (1.10) can be obtained.
The proof is complete.
In this paper, we mainly study the uniform boundedness of classical solutions and exponential asymptotic stability of the unique positive equilibrium point to the chemotactic cellular model (1.3) for Baló's concentric sclerosis (BCS). For model (1.1), by numerical simulation, Calveza and Khonsarib in [7] shown that demyelination patterns of concentric rings will occur with increasing of chemotactic sensitivity. By the Theorem 1.1 we know that systems (1.1) and (1.2) are {uniformly} bounded and dissipative. By the Theorem 1.2 we also find that the constant equilibrium point of model (1.1) is exponentially asymptotically stable if
˜χ<2ˉw˜β√2Dμ˜α˜ϵˉu, |
and the constant equilibrium point of the model (1.2) is exponentially asymptotically stable if
˜χ<2√2Dμ˜α˜ϵˉumin{1ˉw˜β,1ˉu˜γ}. |
According to a pathological viewpoint of BCS, the above stability results mean that if chemoattractive effect is weak, then the destroyed oligodendrocytes form a homogeneous plaque.
The authors would like to thank the editors and the anonymous referees for their constructive comments. This research was supported by the National Natural Science Foundation of China (Nos. 11761063, 11661051).
We have no conflict of interest in this paper.
[1] |
Carlsson Reich M (2005) Economic assessment of municipal waste management systems—case studies using a combination of life cycle assessment (LCA) and life cycle costing (LCC). J Cleaner Prod 13: 253–263. https://doi.org/10.1016/j.jclepro.2004.02.015 doi: 10.1016/j.jclepro.2004.02.015
![]() |
[2] | Nguyen PH, Nguyen Cao QK, Bui LT (2022) Energy recovery from municipal solid waste landfill for a sustainable circular economy in Danang City, Vietnam. IOP Conference Series: Earth Environ Sci 964: 012015. https://doi.org/10.1088/1755-1315/964/1/012015 |
[3] |
Cudjoe D, Han MS (2021) Economic feasibility and environmental impact analysis of landfill gas to energy technology in African urban areas. J Cleaner Prod 284: 125437. https://doi.org/10.1016/j.jclepro.2020.125437 doi: 10.1016/j.jclepro.2020.125437
![]() |
[4] | Mvuma G (2010) Waste a necessary evil for economically impoverished communities in least developed countries (LCDc): a case study. Available from: https://researchspace.csir.co.za/dspace/handle/10204/4531. |
[5] | Thamae M, Molapo K, Koaleli M, et al. (2006) The baseline assessment for the development of an Integrated Solid Waste Management System (ISWMS) for Maseru City. Lesotho2006. Available from: https://info.undp.org/docs/pdc/Documents/LSO/00058398_PPP-ISWM%20Prodoc.pdf. |
[6] | Hoornweg D, Bhada-Tata P (2012) What a waste: a global review of solid waste management. Available from: https://openknowledge.worldbank.org/handle/10986/17388. |
[7] |
Ayodele TR, Ogunjuyigbe ASO, Alao MA (2017) Life cycle assessment of waste-to-energy (WtE) technologies for electricity generation using municipal solid waste in Nigeria. Appl Energy 201: 200–218. https://doi.org/10.1016/j.apenergy.2017.05.097 doi: 10.1016/j.apenergy.2017.05.097
![]() |
[8] | Independent Group of Scientists (2019) The Future is Now: Science for Achieving Sustainable Development. New York: United Nations. Available from: https://sustainabledevelopment.un.org/content/documents/24797GSDR_report_2019.pdf. |
[9] |
Johari A, Ahmed SI, Hashim H, et al. (2012) Economic and environmental benefits of landfill gas from municipal solid waste in Malaysia. Renewable Sustainable Energy Rev 16: 2907–2912. https://doi.org/10.1016/j.rser.2012.02.005 doi: 10.1016/j.rser.2012.02.005
![]() |
[10] |
Shin HC, Park JW, Kim HS, et al. (2005) Environmental and economic assessment of landfill gas electricity generation in Korea using LEAP model. Energy Policy 33: 1261–1270. https://doi.org/10.1016/j.enpol.2003.12.002 doi: 10.1016/j.enpol.2003.12.002
![]() |
[11] | Sechoala TD, Popoola OM, Ayodele TR (2019) A review of waste-to-energy recovery pathway for feasible electricity generation in lowland cities of Lesotho. IEEE, 1–5. https://doi.org/10.1109/AFRICON46755.2019.9133756 |
[12] |
Scarlat N, Motola V, Dallemand JF, et al. (2015) Evaluation of energy potential of municipal solid waste from African urban areas. Renewable Sustainable Energy Rev 50: 1269–1286. https://doi.org/10.1016/j.rser.2015.05.067 doi: 10.1016/j.rser.2015.05.067
![]() |
[13] |
Cudjoe D, Acquah PM (2021) Environmental impact analysis of municipal solid waste incineration in African countries. Chemosphere 265: 129186. https://doi.org/10.1016/j.chemosphere.2020.129186 doi: 10.1016/j.chemosphere.2020.129186
![]() |
[14] |
Silva LJ de VB da, dos Santos IFS, Mensah JHR, et al. (2020) Incineration of municipal solid waste in Brazil: An analysis of the economically viable energy potential. Renewable Energy 149: 1386–1394. https://doi.org/10.1016/j.renene.2019.10.134 doi: 10.1016/j.renene.2019.10.134
![]() |
[15] | China Statistical Yearbook (2015) China Statistical Yearbook 2014. NBSC Beijing. Available from: http://www.stats.gov.cn/sj/ndsj/2014/indexeh.htm. |
[16] |
Damgaard A, Riber C, Fruergaard T, et al. (2010) Life-cycle-assessment of the historical development of air pollution control and energy recovery in waste incineration. Waste Manage 30: 1244–1250. https://doi.org/10.1016/j.wasman.2010.03.025 doi: 10.1016/j.wasman.2010.03.025
![]() |
[17] |
de Souza Ribeiro N, Barros RM, dos Santos IFS, et al. (2021) Electric energy generation from biogas derived from municipal solid waste using two systems: Landfills and anaerobic digesters in the states of Sao Paulo and Minas Gerais, Brazil. Sustainable Energy Technol Assess 48: 101552. https://doi.org/10.1016/j.seta.2021.101552 doi: 10.1016/j.seta.2021.101552
![]() |
[18] |
El Ibrahimi M, Khay I, El Maakoul A, et al. (2021) Techno-economic and environmental assessment of anaerobic co-digestion plants under different energy scenarios: A case study in Morocco. Energy Convers Manage 245: 114553. https://doi.org/10.1016/j.enconman.2021.114553 doi: 10.1016/j.enconman.2021.114553
![]() |
[19] |
Cudjoe D, Han MS, Nandiwardhana AP (2020) Electricity generation using biogas from organic fraction of municipal solid waste generated in provinces of China: Techno-economic and environmental impact analysis. Fuel Process Technol 203: 106381. https://doi.org/10.1016/j.fuproc.2020.106381 doi: 10.1016/j.fuproc.2020.106381
![]() |
[20] | Tyagi VK, Fdez-Güelfo LA, Zhou Y, et al. (2018) Anaerobic co-digestion of organic fraction of municipal solid waste (OFMSW): Progress and challenges. Renewable Sustainable Energy Rev. https://doi.org/10.1016/j.rser.2018.05.051 |
[21] |
Hapazari I, Ntuli V, Taele B (2015) Waste generation and management in lesotho and waste to clay brick recycling a review. Curr J Appl Sci Technol 8: 148–161. https://doi.org/10.9734/BJAST/2015/11224 doi: 10.9734/BJAST/2015/11224
![]() |
[22] | Sechoala TD, Popoola OM, Ayodele TR (2020) Projection of electricity potential through exploitation of methane gas from landfilled MSW of Lesotho. IEEE, 1–5. https://doi.org/10.1109/PowerAfrica49420.2020.9219870 |
[23] | Sechoala TD, Popoola OM, Ayodele TR (2021) Potential of electricity generation through anaerobic digestion and incineration technology for selected districts in Lesotho. IEEE, 1–7. https://doi.org/10.1109/SAUPEC/RobMech/PRASA52254.2021.9377211 |
[24] |
Adenuga OT, Mpofu K, Modise KR (2020) An approach for enhancing optimal resource recovery from different classes of waste in South Africa: Selection of appropriate waste to energy technology. Sustainable Futures 2: 100033. https://doi.org/10.1016/j.sftr.2020.100033 doi: 10.1016/j.sftr.2020.100033
![]() |
[25] |
Ayodele TR, Alao MA, Ogunjuyigbe ASO (2020) Effect of collection efficiency and oxidation factor on greenhouse gas emission and life cycle cost of landfill distributed energy generation. Sustainable Cities Soc 52: 101821. https://doi.org/10.1016/j.scs.2019.101821 doi: 10.1016/j.scs.2019.101821
![]() |
[26] |
Ayodele TR, Ogunjuyigbe ASO, Amusan TO (2018) Techno-economic analysis of utilizing wind energy for water pumping in some selected communities of Oyo State, Nigeria. Renewable Sustainable Energy Rev 91: 335–343. https://doi.org/10.1016/j.rser.2018.03.026 doi: 10.1016/j.rser.2018.03.026
![]() |
[27] |
Ayodele TR, Alao MA, Ogunjuyigbe ASO (2018) Recyclable resources from municipal solid waste: Assessment of its energy, economic and environmental benefits in Nigeria. Resour, Conserv Recycl 134: 165–173. https://doi.org/10.1016/j.resconrec.2018.03.017 doi: 10.1016/j.resconrec.2018.03.017
![]() |
[28] | Bureau of Statistics (2016) 2016 Lesotho Population Census. Maseru Lesotho: Ministry of Development Planning. Available from: https://searchworks.stanford.edu/view/13170410. |
[29] |
Alao MA, Popoola OM, Ayodele TR (2021) Selection of waste-to-energy technology for distributed generation using IDOCRIW-Weighted TOPSIS method: A case study of the City of Johannesburg, South Africa. Renewable Energy 178: 162–183. https://doi.org/10.1016/j.renene.2021.06.031 doi: 10.1016/j.renene.2021.06.031
![]() |
[30] |
Ryu C (2010) Potential of municipal solid waste for renewable energy production and reduction of greenhouse gas emissions in South Korea. J Air Waste Manage Assoc 60: 176–183. https://doi.org/10.3155/1047-3289.60.2.176 doi: 10.3155/1047-3289.60.2.176
![]() |
[31] |
Ogunjuyigbe A, Ayodele T, Alao M (2017) Electricity generation from municipal solid waste in some selected cities of Nigeria: An assessment of feasibility, potential and technologies. Renewable Sustainable Energy Rev 80: 149–162. https://doi.org/10.1016/j.rser.2017.05.177 doi: 10.1016/j.rser.2017.05.177
![]() |
[32] | Moeketsi M (2019) 2018 Annual Report by Central Bank of Lesotho. Lesotho: Mininstry of Finance. Available from: https://www.centralbank.org.ls/images/Publications/ANNUAL_REPORTS/2019_CBL_Annual_Report_-_07.09.2020.pdf. |
[33] | Seleteng M (2010) Inflation and Economic Growth: An estimate of an optimal level of inflation in Lesotho. Maseru Lesotho: Central Bank of Lesotho. 16. Available from: https://www.centralbank.org.ls/images/Publications/Research/Papers/Working/Inflation__Econo_Growth.pdf. |
[34] |
Gonzalez R, Daystar J, Jett M, et al. (2012) Economics of cellulosic ethanol production in a thermochemical pathway for softwood, hardwood, corn stover and switchgrass. Fuel Process Technol 94: 113–122. https://doi.org/10.1016/j.fuproc.2011.10.003 doi: 10.1016/j.fuproc.2011.10.003
![]() |
[35] |
Fernández-González JM, Grindlay AL, Serrano-Bernardo F, et al. (2017) Economic and environmental review of Waste-to-Energy systems for municipal solid waste management in medium and small municipalities. Waste Manage 67: 360–374. https://doi.org/10.1016/j.wasman.2017.05.003 doi: 10.1016/j.wasman.2017.05.003
![]() |
[36] |
Ayodele TR, Ogunjuyigbe ASO, Alao MA (2018) Economic and environmental assessment of electricity generation using biogas from organic fraction of municipal solid waste for the city of Ibadan, Nigeria. J Cleaner Prod 203: 718–735. https://doi.org/10.1016/j.jclepro.2018.08.282 doi: 10.1016/j.jclepro.2018.08.282
![]() |
[37] | Short W, Packey DJ, Holt T (1995) A manual for the economic evaluation of energy efficiency and renewable energy technologies. National Renewable Energy Lab. (NREL), Golden, CO (United States). https://doi.org/10.2172/35391 |
[38] |
Cudjoe D, Han MS (2020) Economic and environmental assessment of landfill gas electricity generation in urban districts of Beijing municipality. Sustainable Prod Consumption 23: 128–137. https://doi.org/10.1016/j.spc.2020.04.010 doi: 10.1016/j.spc.2020.04.010
![]() |
[39] |
Leme MMV, Rocha MH, Lora EES, et al. (2014) Techno-economic analysis and environmental impact assessment of energy recovery from Municipal Solid Waste (MSW) in Brazil. Resour, Conserv Recycl 87: 8–20. https://doi.org/10.1016/j.resconrec.2014.03.003 doi: 10.1016/j.resconrec.2014.03.003
![]() |
[40] |
Dong J, Tang Y, Nzihou A, et al. (2018) Life cycle assessment of pyrolysis, gasification and incineration waste-to-energy technologies: Theoretical analysis and case study of commercial plants. Sci Total Environ 626: 744–753. https://doi.org/10.1016/j.scitotenv.2018.01.151 doi: 10.1016/j.scitotenv.2018.01.151
![]() |
[41] |
Shabib A, Abdallah M (2020) Life cycle analysis of waste power plants: systematic framework. Int J Environ Stud 77: 786–806. https://doi.org/10.1080/00207233.2019.1708146 doi: 10.1080/00207233.2019.1708146
![]() |
[42] | Selibe Mochoboroane (2015) Lesotho Energy Policy. Maseru Lesotho: Ministry of Energy and Meteorology. Available from: https://worldcat.org/title/1033543233. |
[43] | International Renewable Energy Agency (2012) Biomass for power generation. IRENA Abu Dhabi, UAE. Available from: https://www.irena.org/publications/2012/Jun/Renewable-Energy-Cost-Analysis---Biomass-for-Power-Generation. |
[44] |
Cudjoe D, Nketiah E, Obuobi B, et al. (2021) Forecasting the potential and economic feasibility of power generation using biogas from food waste in Ghana: Evidence from Accra and Kumasi. Energy 226: 120342. https://doi.org/10.1016/j.energy.2021.120342 doi: 10.1016/j.energy.2021.120342
![]() |
[45] |
Suryati I, Farindah A, Indrawan I (2021) Study to reduce greenhouse gas emissions at waste landfill in Medan City. IOP Conference Series: Earth Environ Sci 894: 012005. https://doi.org/10.1088/1755-1315/894/1/012005 doi: 10.1088/1755-1315/894/1/012005
![]() |
[46] |
Mohareb AK, Warith MA, Diaz R (2008) Modelling greenhouse gas emissions for municipal solid waste management strategies in Ottawa, Ontario, Canada. Resour, Conserv Recycl 52: 1241–1251. https://doi.org/10.1016/j.resconrec.2008.06.006 doi: 10.1016/j.resconrec.2008.06.006
![]() |
[47] | Change I (2006) 2006 IPCC guidelines for national greenhouse gas inventories. Institute for Global Environmental Strategies, Hayama, Kanagawa, Japan. |
[48] | Assamoi B, Lawryshyn Y (2012) The environmental comparison of landfilling vs. incineration of MSW accounting for waste diversion. Waste Manage 32: 1019–1030. https://doi.org/10.1016/j.wasman.2011.10.023 |
[49] |
Moberg Å, Finnveden G, Johansson J, et al. (2005) Life cycle assessment of energy from solid waste—part 2: landfilling compared to other treatment methods. J Cleaner Prod 13: 231–240. https://doi.org/10.1016/j.jclepro.2004.02.025 doi: 10.1016/j.jclepro.2004.02.025
![]() |
[50] |
Finnveden G, Johansson J, Lind P, et al. (2005) Life cycle assessment of energy from solid waste—part 1: general methodology and results. J Cleaner Prod 13: 213–229. https://doi.org/10.1016/j.jclepro.2004.02.023 doi: 10.1016/j.jclepro.2004.02.023
![]() |
[51] |
Finnveden G, Moberg Å (2005) Environmental systems analysis tools–an overview. J Cleaner Prod 13: 1165–1173. https://doi.org/10.1016/j.jclepro.2004.06.004 doi: 10.1016/j.jclepro.2004.06.004
![]() |
[52] | Guendehou S, Koch M, Hockstad L, et al. (2006) Incineration and Open Burning of Waste. 2006 IPCC Guidelines for National Greenhouse Gas Inventories. Japan: Institute for Global Environmental Strategies (IGES). Available from: https://nswmc.emb.gov.ph/wp-content/uploads/2022/08/2006-IPCC-Guidelines-for-National-Greenhouse-Gas-Inventories.pdf. |
[53] | EPA U (2018) Emission factors for greenhouse gas inventories Stationary combustion emission factors. Go to reference in article. Available from: https://www.epa.gov/sites/default/files/2018-03/documents/emission-factors_mar_2018_0.pdf. |
[54] |
Kweku DW, Bismark O, Maxwell A, et al. (2018) Greenhouse effect: greenhouse gases and their impact on global warming. J Sci Res Rep 17: 1–9. https://doi.org/10.9734/JSRR/2017/39630 doi: 10.9734/JSRR/2017/39630
![]() |
[55] | Guo Y, Glad T, Zhong Z, et al. (2018) Environmental life-cycle assessment of municipal solid waste incineration stocks in Chinese industrial parks. Resources, Conservation and Recycling 139: 387–395. https://doi.org/10.1016/j.resconrec.2018.05.018 |
[56] |
Yao X, Guo Z, Liu Y, et al. (2019) Reduction potential of GHG emissions from municipal solid waste incineration for power generation in Beijing. J Cleaner Prod 241: 118283. https://doi.org/10.1016/j.jclepro.2019.118283 doi: 10.1016/j.jclepro.2019.118283
![]() |
[57] | EPA U (1996) Solid Waste Disposal. Refuse Combustion 42. Available from: https://www3.epa.gov/ttnchie1/ap42/ch02/. |
[58] |
Kumar A, Sharma M (2014) Estimation of GHG emission and energy recovery potential from MSW landfill sites. Sustainable Energy Technol Assess 5: 50–61. https://doi.org/10.1016/j.seta.2013.11.004 doi: 10.1016/j.seta.2013.11.004
![]() |
[59] |
Kale C, Gökçek M (2020) A techno-economic assessment of landfill gas emissions and energy recovery potential of different landfill areas in Turkey. J Cleaner Prod 275: 122946. https://doi.org/10.1016/j.jclepro.2020.122946 doi: 10.1016/j.jclepro.2020.122946
![]() |
[60] |
Shunda l, Jiang X, Zhao Y, et al. (2022) Disposal technology and new progress for dioxins and heavy metals in fly ash from municipal solid waste incineration: A critical review. Environ Pollut 311: 119878. https://doi.org/10.1016/j.envpol.2022.119878 doi: 10.1016/j.envpol.2022.119878
![]() |
[61] |
Yang L, Liu G, Zhu Q, et al. (2019) Small-scale waste incinerators in rural China: Potential risks of dioxin and polychlorinated naphthalene emissions. Emerging Contam 5: 31–34. https://doi.org/10.1016/j.emcon.2019.01.001 doi: 10.1016/j.emcon.2019.01.001
![]() |
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