The issue of climate change and management of municipal solid waste (MSW) necessitates transition to renewable energy, including bioenergy. This work assessed energy production from organic fraction of municipal solid waste (OFMSW) in the thirty-six state capitals and Federal Capital Territory (FCT), Abuja, Nigeria. Secondary research method (qualitative and quantitative analysis) was adopted. The four valorization methods considered were incineration, anaerobic digestion (AD), landfill gas-to-energy (LFGTE) and densification. MSW and OFMSW generation rate (kg/cap/day) for the thirty-six state capitals and the FCT, Abuja were obtained. The paper estimated that about 4.7 million tons per year (TPY) of OFMSW is generated in the 37 cities. Daily OFMSW generation ranges from 10416 tons per year (TPY) in Damaturu, to 1.6 million TPY in Lagos. The estimates show that about 1.82 billion Nm3 of biogas could be obtained from anaerobic digestion (AD) of OFMSW generated in the cities each year; about 984 Gg (1085688 tons) of methane can be recovered from the landfill gas technology, while drying and densification will produce about 1.82 million tons of solid fuel. Based on secondary sources, the cost per ton waste and emissions (kg/ton) processed were also presented.
Citation: Charles C. Ajaero, Chukwuebuka C. Okafor, Festus A. Otunomo, Nixon N. Nduji, John A. Adedapo. Energy production potential of organic fraction of municipal solid waste (OFMSW) and its implications for Nigeria[J]. Clean Technologies and Recycling, 2023, 3(1): 44-65. doi: 10.3934/ctr.2023003
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The issue of climate change and management of municipal solid waste (MSW) necessitates transition to renewable energy, including bioenergy. This work assessed energy production from organic fraction of municipal solid waste (OFMSW) in the thirty-six state capitals and Federal Capital Territory (FCT), Abuja, Nigeria. Secondary research method (qualitative and quantitative analysis) was adopted. The four valorization methods considered were incineration, anaerobic digestion (AD), landfill gas-to-energy (LFGTE) and densification. MSW and OFMSW generation rate (kg/cap/day) for the thirty-six state capitals and the FCT, Abuja were obtained. The paper estimated that about 4.7 million tons per year (TPY) of OFMSW is generated in the 37 cities. Daily OFMSW generation ranges from 10416 tons per year (TPY) in Damaturu, to 1.6 million TPY in Lagos. The estimates show that about 1.82 billion Nm3 of biogas could be obtained from anaerobic digestion (AD) of OFMSW generated in the cities each year; about 984 Gg (1085688 tons) of methane can be recovered from the landfill gas technology, while drying and densification will produce about 1.82 million tons of solid fuel. Based on secondary sources, the cost per ton waste and emissions (kg/ton) processed were also presented.
The theory of sub-manifolds of finite type has led to significant insights and results in differential geometry, helping to identify and characterize sub-manifolds with special geometric properties.
In [1], Chen mentioned the concept of Euclidean immersions of finite type concerning the first fundamental form Ⅰ of a surface ℵ. According to Chen's theory, a surface ℵ is said to be of finite type if its coordinate functions can be expressed as a finite sum of eigenfunctions of the Beltrami operator ΔI.
For instance, Chen [2] posed the problem of classifying finite type sub-manifolds in 3-dimensional Euclidean space E3. This initiated a comprehensive study of the spectral properties of the Laplacian on these sub-manifolds, leading to the classification of minimal surfaces, spheres, and circular cylinders as specific examples of finite-type sub-manifolds.
If we consider the surface ℵ in E3, its position vector
X=X(v1, v2) |
can be written as:
X=n∑i=1Fi(v1, v2)ei, |
where Fi(v1, v2) are eigenfunctions of the Beltrami operator ΔI, and ei are constant vectors in E3.
For a surface to be of finite Ⅱ-type, its shape operator (related to the second fundamental form) must also have a similar decomposition into a finite sum of eigenfunctions of the Beltrami operator.
To understand the implications of a surface ℵ being of finite type l, we start by considering the relevant equation involving the second fundamental form, ΔII, which is the Laplacian operator applied to the components of the second fundamental form.
When ℵ is of finite type l, there exists a monic polynomial
F(x)≠0, |
such that
F(ΔII)(X−c)=0. |
Here, X represents the position vector of a point on the surface, and c is a constant vector.
Suppose the polynomial
F(x)=xl+γ1xl−1+...+γl−1x+γl. |
Then the coefficients γi are determined by the specific relationship between the eigenfunctions of ΔII, and the position vector components. These coefficients γi are related to the eigenvalues μi of ΔII acting on the coordinate functions of X. In detail, γi are typically given in terms of symmetric polynomials of these eigenvalues. Specifically, they can be expressed as follows:
γ1=−(μ1+μ2+...+μl),γ2=(μ1μ2+μ1μ3+...+μ1μl+μ2μ3+...+μ2μl+...+μl−1μl),γ3=−(μ1μ2μ3+...+μl−2μl−1μl), …γl=(−1)lμ1μ2...μl. |
Therefore the position vector X satisfies the following equation (see [3]):
(ΔII)lX+σ1(ΔJ)l−1X+...+σl(X−c)=0. | (1.1) |
Finite-type immersions involve studying sub-manifolds whose coordinate functions are finite sums of eigenfunctions of the Laplace-Beltrami operator. This notion provides a way to classify sub-manifolds based on the spectral properties of the Laplacian acting on the coordinate functions.
These classifications help in understanding the geometric and topological properties of sub-manifolds. For instance, the result that spheres are the only quadric surfaces of finite Ⅱ-type in E3 provides a clear distinction between spheres and other quadric surfaces like ellipsoids, hyperboloids, and paraboloids, based on their curvature properties.
A recent study in [4] authors investigated the Hasimoto surfaces according to their finite Chen type, while in [5,6] interesting researches were done by studying the class of translation surfaces according to it's finite Chen Ⅲ-type once in E3, and on the other hand in Sol3.
Takahashi in [7] mentioned that a surface M2 whose position vector X satisfies
ΔIX=μX |
is either a minimal with μ=0 or M2 lies in an ordinary sphere S2 with a fixed nonzero eigenvalue.
Garay in his article [8] made a generalization of Takahashi's condition. In his study, he considered surfaces in E3 satisfying
ΔIXi=μiXi, i=1,2,3, |
where (X1,X2,X3) are the coordinate functions of the position vector X and μi, are different eigenvalues. Garay's work expands on the problem of identifying surfaces in E3 that satisfy this eigenvalue condition. The coordinate functions of these surfaces are expressed as eigenfunctions of the Laplace-Beltrami operator associated with distinct eigenvalues, contributing to the understanding of surfaces of finite type in a more general context.
Another related general problem was presented in [9], which investigated surfaces in E3 satisfying
ΔIX=KX+L, | (1.2) |
where K is a 3×3 matrix; L is a 3×1 matrix. It was proven that minimal surfaces, spheres, and circular cylinders are the only surfaces in E3 satisfying Eq (1.2). Surfaces meeting this criterion are said to be of coordinate finite type.
As an application, the alignment of molecules in relation to quadric surfaces has meaningful applications in understanding molecular orientations, interactions, and behaviors under external influences. Quadric surfaces, such as ellipsoids, hyperboloids, and paraboloids, serve as mathematical representations of properties like potential energy distributions, molecular shapes, and field effects. The shapes of molecules can often be described using quadric surfaces such as ellipsoids which is common for anisotropic molecules like liquid crystal rods or elongated organic molecules, or spheres which represent isotropic molecules such as noble gases or symmetric compounds like CH4. These quadric shapes help model how molecules orient or align in space (see [10,11]).
We consider a (connected) surface ℵ in a Euclidean 3-space E3 referred to any system of coordinates v1,v2, whose Gaussian curvature never vanishes. Let Bst be the components of the second fundamental form
II=Bstdvsdvt |
of ℵ. For any two sufficiently differentiable functions φ(v1,v2) and ψ(v1,v2) on ℵ, the first Beltrami operator with respect to the second fundamental form of ℵ is given by
∇II(ψ,φ):=Bstψ/sφ/t, |
where
ψ/s:=∂ψ∂vs, |
and (Bst) denotes the inverse tensor of (Bst).
The second Beltrami operator regarding the second fundamental form of ℵ is defined by [12]
ΔIIφ:=−1√B(√BBstφ/s)/t, |
where
B:=Det(Bst). |
In [13], authors proved that, for the position vector
X=X(v1,v2) |
of ℵ, the relation is
ΔIIX=12K∇IIIK−2G, |
where G is the Gauss map, K the Gauss curvature, and H the mean curvature of ℵ.
The main result in this study presents the following as detailed below.
Theorem 1. Among all quadric surfaces in E3, the only one that satisfies the finite Ⅱ-type condition is the sphere.
This result highlights a unique geometric property of spheres compared to other quadric surfaces such as ellipsoids, hyperboloids, and paraboloids.
Let ℵ be a Cr quadric surface in E3 defined on a region U⊂R2. Then, ℵ is one of the following three kinds [14,15]:
1stKind:ℵ is a ruled surface.
2ndKind:ℵ is of the form
z2=γ+αX2+βY2,α,β,γ∈R,αβ≠0,γ>0. |
3rdKind:ℵ is of the form
z=α2X2+β2Y2,α,β∈R,α,β>0. |
The class of ruled surfaces has been studied in [16], so we will complete our study by investigating the second and third kinds of surfaces mentioned above in terms of their finite Chen type.
A parametrization of a part of a quadric of this kind is [15]
x(υ,ν)=(υ,ν,√αυ2+βν2+γ),αυ2+βν2+γ>0. | (2.1) |
For simplicity, we use
αυ2+βν2+γ:=ω. |
The metrics I, II of ℵ, are respectively,
I=(α2υ2ω+1)dυ2+2αβυνωdυdν+(β2v2ω+1)dν2,II=1ω√T(α(γ+βν2)dυ2−2αβυνdυdν+β(γ+αυ2)dν2), |
where
T=γ+β(β+1)ν2+α(α+1)υ2. |
The Laplacian △II of ℵ can be expressed as follows:
△II=−√Tγ[γ+αυ2α∂2∂υ2+2υν∂2∂υ∂ν+γ+βν2β∂2∂ν2+2υ∂∂υ+2ν∂∂ν]. | (2.2) |
For a function φ(υ)∈C∞(U), on account of Eq (2.2), we get
△IIφ=−√Tγ[γ+αυ2α∂2φ∂υ2+2υ∂φ∂υ]. | (2.3) |
On use of Eq (2.2), it can be easily proved:
Lemma 1. The relation
△II(αr(α+1)rυmTn)=−αr+2(α+1)r+2[m(m+1)+4n(n−m)−2n]υm+4γTn+32+1γTn+32F(υ,ν) |
holds true, where F(υ,ν) is a polynomial of degree at most m+4, and when ν=0, deg(F(υ,0)) is at most m+2.
We denote by (x1,x2,x3) the components of x(υ,ν). On account of Eq (2.3), we have
△IIx1=△IIυ=−2υ√Tγ. | (2.4) |
Applying Eq (2.2) for the relation (2.4), we find
(△II)2x1=(△II)2υ=2γ2T[6α2(α+1)2υ5+f2(υ,ν)], | (2.5) |
where
f2(υ,ν)=αγ(α+1)(γ(β+1)+2α+11)υ3+αβ(α+1)(β+1)(γ+12)υ3ν2+β2(β+1)2(γ+6)υν4+βγ(β+1)(βγ+2γ+3α+11)υν2+γ2(γ(β+1)+3α+5)υ. |
Note that f2(υ,ν) is a polynomial of degree at most 5, and if we put ν=0, then f2(υ,0) is a polynomial in υ of degree at most 3.
From Lemma 1, we get
(△II)3υ=−2γ3T52[72α4(α+1)4υ9+f3(υ,ν)], | (2.6) |
where f3(υ,ν) is a polynomial of degree at most 9, with
deg(f3(υ,0))≤7. |
We will also prove:
Lemma 2. The relation
(△II)lυ=(−1)l(∏i=1li(i+1))(α2l−2(α+1)2l−2υ4l−3+Pl(υ,ν)γlT32l−2), |
holds true, where
deg(Pl(υ,0))≤4l−5. |
Proof. The proof goes by induction on l.
Base case: For l=1, the formula comes true from Eq (2.4) applied to φ=υ.
Inductive step: Assume that the lemma is true for l−1. So,
(△II)l−1υ=(−1)l−1(∏i=1l−1i(i+1))(α2l−4(α+1)2l−4υ4l−7+Pl−1(υ,ν)γl−1T32l−72). |
Proof for l. Taking into account ν=0, relation (2.3), and Lemma 1, we obtain
(△II)lυ=△II((△II)l−1υ)=(−1)l−1(l−1∏i=1i(i+1))(1γl−1)(△II(α2l−4(α+1)2l−4υ4l−7T32l−72))+△II(Pl−1(υ,ν)γl−1T32l−72)=(−1)l−1γl−1(l−1∏i=1i(i+1))(−l(l+1)α2l−4+2(α+1)2l−4+2υ4l−7+4+Pl(υ,ν)γT32l−72+32)=(−1)lγl(l(l+1)l−1∏i=1i(i+1))(α2l−2(α+1)2l−2υ4l−3+Pl(υ,ν)T32l−2)=(−1)l(l∏i=1i(i+1))(α2l−2(α+1)2l−2υ4l−3+Pl(υ,ν)γlT32l−2). |
This completes the proof.
For the second component x2, we have
△IIx2=△IIν=−2ν√Tγ. | (2.7) |
Also
(△II)2x2=(△II)2ν=2γ2T[6β2(β+1)2ν5+g2(υ,ν)], | (2.8) |
and
g2(υ,ν)=βγ(β+1)(γ(α+1)+2β+11)ν3+αβ(α+1)(β+1)(γ+12)υ2ν3+α2(α+1)2(γ+6)νυ4+αγ(α+1)(αγ+2γ+3β+11)υ2ν+γ2(γ(α+1)+3β+5)ν. |
Similarly, g2(υ,ν) is a polynomial of degree at most 5, and if we put υ=0, then g2(0,ν) is a polynomial in ν of degree at most 3.
Lemma 3. The relation
△II(βr(β+1)rνmTn)=−βr+2(β+1)r+2[m(m+1)+4n(n−m)−2n]νm+4γTn+32+1γTn+32G(υ,ν), |
holds true, where G(υ,ν) is a polynomial of degree at most m+4, and when υ=0, then deg(G(0,ν)) is at most m+2.
So, using the above lemma, one can find that
(△II)3ν=−2γ3T52[72β4(β+1)4ν9+g3(υ,ν)], | (2.9) |
where g3(υ,ν) is a polynomial of degree at most 9, with
deg(g3(0,ν))≤7. |
By induction, one can also obtain:
Lemma 4. The relation
(△II)lν=(−1)l(∏i=1li(i+1))(β2l−2(β+1)2l−2ν4l−3+Ql(υ,ν)γlT32l−2), |
is valid, and
deg(Ql(0,ν))≤4l−5. |
Let now ℵ be of finite Ⅱ-type l. Then, there exist real numbers ci,i=1,…,l such that
(△II)l+1x+c1(△II)lx+…+cl△IIx=0. | (2.10) |
Applying Eq (2.10) to the coordinate functions x1=υ and x2=ν of the position vector (2.1) of ℵ, we obtain
(△II)l+1υ+c1(△II)lυ+⋯+cl△IIυ=0, | (2.11) |
(△II)l+1ν+c1(△II)lν+⋯+cl△IIν=0. | (2.12) |
From Lemma 2, relations (2.4)–(2.6), and (2.11), it follows that
(−1)l+1(l+1∏i=1i(i+1))(α2l(α+1)2lυ4l+1+Pl+1(υ,ν)γl+1T32l−12)+c1(−1)l(l∏i=1i(i+1))(α2l−2(α+1)2l−2υ4l−3+Pl(υ,ν)γlT32l−2)+⋯+cl−12γ2T(6α2(α+1)2υ5+P2(υ,ν))+cl2υ√Tγ=0, |
which can be written as
(−1)l+1(l+1∏i=1i(i+1))(α2l(α+1)2lυ4l+1γl+1)+Pl+1(υ,ν)+c1(−1)l(l∏i=1i(i+1))(α2l−2(α+1)2l−2υ4l−3T32Pl(υ,ν)γl)+⋯+cl−112α2(α+1)2υ5T32(l−1)γ2+cl−1T32(l−1)P2(υ,ν)+cl2υT32lγ=0. | (2.13) |
Inserting ν=0 in (2.13), we obtain a nontrivial polynomial in υ with constant coefficients. Therefore, the above equation can be rewritten as
(−1)l+1(∏i=1l+1i(i+1))(α2l(α+1)2lυ4l+1γl+1)+P(υ,ν)=0, | (2.14) |
with
deg(P(υ,ν))≤4l. |
Since α≠0, the relation (2.14) implies that α must be equal −1.
Following the same procedure for the second component x2, by using relations (2.7)–(2.9), (2.12), and Lemma 4, we get
(−1)l+1(∏i=1l+1i(i+1))(β2l(β+1)2lν4l+1γl+1)+Q(υ,ν)=0, | (2.15) |
where Q(υ,ν) is a polynomial of degree at most 4l. Putting υ=0, then Eq (2.15) is a nontrivial polynomial in ν with constant coefficients. However β≠0, so from (2.15) β equals −1. Therefore, ℵ is a sphere.
Let
α=β=−1. |
Then,
T=γ. |
Thus, relation (2.2) reduces to
△II=−1√γ[(υ2−γ)∂2∂υ2+2υν∂2∂υ∂ν+(ν2−γ)∂2∂ν2+2υ∂∂υ+2ν∂∂ν]. |
So, relations (2.4) and (2.7) become
△IIυ=−2√γυ. |
△IIν=−2√γν. |
For the third coordinate
x3=√ω=√γ−υ2−ν2, |
after simple calculation, we conclude
△II√ω=−2√γ√ω. |
Thus, we find that
△IIx=−2√γx. |
That is, spheres are the only quadric surfaces of the kind (2) of finite Ⅱ-Chen type.
A parametrization of a part of a quadric of this kind is
x(υ,ν)=(υ,ν,α2υ2+β2ν2). | (2.16) |
The matrix of the components of the first fundamental form of ℵ is the following:
(gij)=[1+α2υ2αβυναβυν1+β2ν2]. |
Denote
g:=Det(gij)=1+(αυ)2+(βν)2 |
The matrix of the components of the second fundamental form Ⅱ is given as follows:
(bij)=[α√g00β√g]. |
Thus, △II of ℵ becomes
△II=−√g(1α∂2∂υ2+1β∂2∂ν2). | (2.17) |
By applying the operator △II to the components
x1=υandx2=ν, |
we get
△IIx1=△IIx2=0. |
For the third coordinate
x3=α2υ2+β2ν2, |
we find
△II(α2υ2+β2ν2)=−2√g. | (2.18) |
Applying Eq (2.17) for the relation (2.18), we find
(△II)2x3=2(α+β)g+2αβgf1(υ,ν), | (2.19) |
where
f1(υ,ν)=αυ2+βν2, |
with
deg(f1)=2. |
On one hand, using (2.17), one can find:
Lemma 5. For n>0, we find
△II(g−n)=2n(α+β)g−n−12−4n(n+1)g−n−32(α3υ2+β3ν2), | (2.20) |
and, on the other hand, we have:
Lemma 6.
△II(αrυt+βrνtgn)=−(t(t−1)(αr−1υt−2+βr−1νt−2)gn−12)+(2n(2t+1)(αr+1υt+βr+1νt)+2nαβ(αr−1υt+βr−1νt)gn+12)−(4n(n+1)(αr+3υt+2+βr+3νt+2)gn+32)−(4n(n+1)α3β3υ2ν2(αr−3υt−2+βr−3νt−2)gn+32). | (2.21) |
From Lemma 6, we obtain that
△II(αrυt+βrνtgn)=−(1gn+32)G(υ,ν), | (2.22) |
where deg(G(υ,ν)) is at most t+2.
Taking into account (2.20), and (2.22), we get
(△II)3x3=4(α+β)2g32+1g52G2(υ,ν), | (2.23) |
where G2(υ,ν) is a polynomial in υ,ν of degree at most 4. Similarly, we get
(△II)4x3=12(α+β)3g2+1g3G3(υ,ν), | (2.24) |
where G3(υ,ν) is a polynomial in υ,ν of degree at most 6. In general, we have
Lemma 7. The relation
(△II)lx3=2(l−1)!(α+β)l−1g12l+1g1+12lGl−1(u,v), |
holds true for l>2, with
deg(Gl−1(u,v))≤2l−2. |
On account of relation (1.1) applied to the component x3, we have
(△II)l+1x3+c1(△II)lx3+⋯+cl△IIx3=0. |
From (2.18), (2.19), (2.23), (2.24), and Lemma 7, we get
2(l)!(α+β)lg12(l+1)+1g32+12lGl(u,v)+2c1(l−1)!(α+β)l−1g12l+c1g1+12lGl−1(u,v)+⋯+2cl√g=0 |
or
2(l)!(α+β)l+2c1(l−1)!(α+β)l−1g12+2c2(l−2)!(α+β)l−2g32+⋯+2clg1+12l+1gG(u,v)=0, | (2.25) |
where
deg(G(u,v))≤2l. |
Relation (2.25) must hold true for all (υ,ν). This is clearly impossible since the first term of (2.25), which is the constant term of (2.25), must equal 0, something that cannot be satisfied since α,β>0.
This research article was divided into three sections, where after the introduction, the needed definitions and relations regarding this interesting field of study were given. Then, a formula for the Laplace operator corresponding to the second fundamental form Ⅱ was proved once for the position vector and another for the Gauss map of a surface. Finally, we classify the quadric surfaces of finite Chen type regarding the second fundamental form. An interesting study can be drawn if this type of study can be applied to other classes of surfaces that have not been investigated yet, such as spiral surfaces, or tubular surfaces.
The author would like to thank the referees for their useful remarks.
The author declares that he has no conflict of interest.
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