**4. Waste management processes**

There are different methods of solid waste management which are well known. Among the various methods, bio-process-based are the most suitable for managing the waste in a safe way and they generate the nontoxic degraded foams.

1.**Landfills:** Landfill is the most primitive and commonly used waste disposal method (**Figure 3**). In this method, waste organic materials are simply collected and thrown in depth or pits. There are two main processes involved in an efficient landfilling method: 1. Collection of waste in a very small and confine place. 2. Compaction of collected waste by compactors or bulldozers.

There are five phases involved in this method:

i. Phase I: Adjustment of waste with O2 supply

The waste compacted in this phase with void spacing for oxygen exchange (O2). Initially, O2 level is high due to activity of microbes for aerobic decomposition process but O2 level gradually decreased due to high microbial density.

ii. Phase II: Transition from aerobic to anaerobic

Due to high microbial density, O2 level decreases and CO2 level increases. The anaerobic condition occurs in the layers and primary electron acceptors are sulfates and nitrates instead of O2.

iii. Phase III: Generation of organic acids by acidification

This phase is also known as acidification phase, and in this, volatile fatty acids (VFA) are decomposed to acetic acid (CH3COOH), CO2, and H2. The H2 formation phase occurred in the last and activates the H2 oxidizing bacteria. The acidogenic bacteria increase the decomposition rate of waste.

**Figure 3.** *Landfill mechanism of waste decomposition.*

iv. Phase IV: Methonogenesis

The organic acids products, viz., acetic, propionic, and butyric acids are converted into CH4 and CO2 as intermediate products by methoanogenic microorganisms. In this phase, pH changes from acidic to neutral. This phase is the longest phase of decomposition.

v. Phase V: Humification

All the microbial activity slows down in this phase and CH4 production also completely disappears. All the remaining organic materials are converted into gas phase by the oxidative processes and converted into humic acid compounds.

2.**Anaerobic decomposition:** Anaerobic decomposition is gaining attention in various energy sources to fulfill the demand of increasing population in a safer and economical way [17].  In anaerobic decomposition, microorganisms decompose the solid waste into biodegradable and nontoxic forms in the absence of oxygen (1). There are two types of bacteria which play a significant role in anaerobic decomposition reactions, which include hydrolytic bacteria and the acidogenic bacteria. These two bacteria belong to two kingdoms, the archea and the bacteria [18].

There are four successive stages of anaerobic decomposition which are: hydrolysis, acidogenesis, acetogenesis, and methanogenesis. These four stages are carrying out by the interaction of various microbes [19].

#### a. Hydrolysis

The decomposition pathways start with the hydrolysis of the waste agricultural biomass, which is rich in complex polymeric constituents. The hydrolytical cleavage broke the polymeric interconnecting bonds by the hydrolysis process. Agricultural waste is rich in organic polymers such as cellulose, hemicelluloses, lignin, etc. [20]. Hydrolysis breakdown is an electrochemical process driven by the hydrolytic bacteria by cellulolytic enzymes, lipase enzymes, etc. Hydrolysis cleavage converts these insoluble long chain polymers into soluble simpler derivatives like ammonium and other organic constituents [21].

Kinetically, hydrolysis reactions are the first-order reactions in which complex biodegradable waste material is hydrolyzed at constant temperature and pH [22].

$$(\text{C6H10O5})\text{n} + \text{n H2O} \rightarrow \text{n C6H12O6} + \text{n H2}$$

$$\frac{\text{dXdegr}}{\text{dt}} = -\text{Kb.Xdegr} \tag{1}$$

where Xdegr = Amt. of Decomposed material (kg/m3 )

t = time interval (days)

kh = hydrolysis constant in first order reaction.

Hydrolysis reactions in anaerobic decomposition can be considered as ratedetermining steps, which depend upon the ration of hydrolytic product to microorganisms, pH 5–7, and 30–35°C [23]. The hydrolysis reactions make the agricultural postharvesting biodegradable waste residue into e simpler forms and make it accessible for acidogenic bacteria for further breakup.


**2. Temporal splitting scheme**

and **u** ⊗ **v** ¼ *uiv j*, *i*, *j* ¼ 1, 2*:* Indeed, it holds

*Vortex Dynamics Theories and Applications*

N ð Þ¼ **u**

interior edges of E*h* and Γ*B*

and denote its jump by

and *Mh*ð Þ ⊂ *M* for integer *N* ≥1:

**84**

*Wr*,*s*

*V* ¼ f**v**∈ *L*2

*M* ¼ f*q*∈*L*2

*∂ u*2 ð Þ *∂x* þ *∂*ð Þ *uv ∂y*

1

*h* be the set of all boundary edges. Set Γ*h* ¼ Γ*I*

ð Þ *E* : ∀j*m*j≤ *r*, *∂mv*∈ *Ls* f g ð Þ *E :*

� �

> � �

*E* ∈ *W*2,4*=*3

*E* ∈*W*1,4*=*3

*:*

ð Þ *E* � �2

ð Þg *E :*

*i*,*j TijSij*.

,

g,

A locally conservative DG discretization will be employed for the Navier–Stokes Eq. (9)–(11). We denote by E*h* a shape-regular triangulation of the domain Ω into

any nonnegative integer *r* and *s*≥1, the classical Sobolev space on a domain *E* ⊂ 2 is

ð Þ Ω 2 : ∀*E*∈E*h*, **v**

The jump and average of a function *ϕ* on an edge *e* are defined by:

½ �¼ *ϕ ϕ Ek* Þ �

2 *ϕEk* � �� � *e* þ *ϕ El* Þ � � � � *e*

f g*ϕ* ¼ 1

tensor product of two tensors **T** and **S** is defined as **T** : **S** ¼ P

**V***h* ¼ **v**∈*L*2

*Mh* ¼ *q*∈*L*2

ð Þ Ω : ∀*E* ∈E*h*, *q*

� � � *e* � *ϕ El* Þ � � � � *e* ,

� �

� � � �

Further, let **v** be a piecewise smooth vector-, or matrix-valued function at **x**∈*e*

½ � **v** ≔ **v**þ � **n***E*þ þ **v**� � **n***E*� ,

where *e* is shared by two elements *E*þ and *E*�, and an outward unit normal vector **n***E*þ (or **n***E*� ) is associated with the edge *e* of an element *E*þ (or *E*�). The

Let *N*ð Þ *E* be the set of polynomials on an element *E* with degree no more than *N:* Based on the triangulation, we introduce two approximate subspaces **V***h*ð Þ ⊂*V*

ð Þ Ω 2 : ∀*E*∈ E*h*, **v***h* ∈ ð Þ *N*ð Þ *E* 2 n o

ð Þ Ω : ∀*E*∈ E*h*, *q* ∈*N*�1ð Þ *E* � �*:*

We mainly cite the content of [17], in which was motivated by the work of Girault, Rivière and Wheeler in a series of papers [2, 14]. Some projection methods [6, 18] have been developed to overcome the incompressibility constraints ∇ � **u** ¼ 0. An implementation of the operator-splitting idea for discontinuous Galerkin elements was developed in [2]. We appreciate the advantages of the discontinuous Galerkin methods, such as local mass conservation, high order of approximation, robustness and stability. In this work, we will make use of the underlying physical nature of incompressible flows in the literature and extend the interior penalty discontinuous

CCCA ¼ ∇ � *F*ð Þ **u** *:*

*h* be the set of all

*h* ∪ Γ*B h :* For

0

BBB@

triangles, where *h* is the maximum diameter of elements. Let Γ*I*

ð Þ¼ *E v*∈ *Ls*

We define the spaces of discontinuous functions

*∂*ð Þ *uv ∂x* þ *∂ v*2 ð Þ *∂y*

> We consider here a third-order time-accurate discretization method at each time step by using the previous known velocity vectors. Let *Δt* be the time step, *M* ¼ *T Δt* , and *tn* ¼ *nΔt*. The semi-discrete forms of problem (6)–(8) at time *tn*þ1 is

$$\frac{\gamma\_0 \mathbf{u}^{n+1} - a\_0 \mathbf{u}^n - a\_1 \mathbf{u}^{n-1} - a\_2 \mathbf{u}^{n-2}}{\Delta t} - \nu \Delta \mathbf{u}^{n+1} + \nabla p^{n+1} \tag{13}$$

$$\mathbf{f} = -\beta\_0 \mathcal{N}(\mathbf{u}^n) - \beta\_1 \mathcal{N}\left(\mathbf{u}^{n-1}\right) - \beta\_2 \mathcal{N}\left(\mathbf{u}^{n-2}\right) + \mathbf{f}(t\_{n+1}),$$

$$\nabla \cdot \mathbf{u}^{n+1} = \mathbf{0}, \tag{14}$$

which has a timestep constraint based on the CFL condition (see [19]):

$$
\Delta t \approx O\left(\frac{\mathcal{L}}{l \mathcal{M}^2}\right),
$$

where L is an integral length scale (e.g. the mesh size) and U is a characteristic velocity. Because the semi-discrete system (13)–(14) is linearized, thus, a timesplitting scheme can be applied naturally, i.e., the semi-discretization in time (13)– (14) can be decomposed into three stages as follows.

• The first stage

When **u***n* and **u***n*�1 (*n*≥ 1) are known, the following linearized third-order formula can be used

$$\frac{\gamma\_0 \tilde{\mathbf{u}} - a\_0 \mathbf{u}^n - a\_1 \mathbf{u}^{n-1} - a\_2 \mathbf{u}^{n-2}}{\Delta t} = -\beta\_0 \mathcal{N}(\mathbf{u}^n) - \beta\_1 \mathcal{N}(\mathbf{u}^{n-1}) + \mathbf{f}(t\_{n+1}) \tag{15}$$

with the following coefficients for the subsequent time levels (*n*≥2)

$$y\_0 = \frac{11}{6}, \quad a\_0 = 3,\\ a\_1 = -\frac{3}{2}, \quad a\_2 = \frac{1}{3}, \quad \beta\_0 = 2, \quad \beta\_1 = -1.$$

Especially, by using the Euler forward discretization at the first time step (*n* ¼ 0), we can get a medium velocity field **u**1 by

$$\frac{\mathbf{u}^1 - \mathbf{u}^0}{\Delta t} = -\mathcal{N}(\mathbf{u}^0) + \mathbf{f}(t\_1),$$

and **u**2 by

$$\frac{\gamma\_0 \mathbf{u}^2 - a\_0 \mathbf{u}^1 - a\_1 \mathbf{u}^0}{\Delta t} = -\beta\_0 \mathcal{N} \left( \mathbf{u}^1 \right) - \beta\_1 \mathcal{N} \left( \mathbf{u}^0 \right) + \mathbf{f}(t\_2),$$

which adopts the following coefficients to construct a second-order difference scheme for the time level (*n* ¼ 2) in (15)

$$y\_0 = \frac{3}{2}, \quad a\_0 = 2,\\ a\_1 = -\frac{1}{2}, \quad a\_2 = 0, \ \beta\_0 = 2, \ \beta\_1 = -1.$$

Note that the coefficients in (15) are adjustable, but high-order time discrete schemes need to be verified with stability analysis.

• The second stage

The pressure projection is as follows

$$
\gamma\_0 \frac{\tilde{\tilde{\mathbf{u}}} - \tilde{\mathbf{u}}}{\Delta t} = -\nabla p^{n+1}.\tag{16}
$$

are obviously a type of elliptic and Helmholtz problems at each time step as *n*≥2. We decouple the incompressibility condition and the nonlinearity, then the pressure and velocity semi-discretizations (17)–(18) will be formulated by the interior penalty discontinuous Galerkin methods in spacial discretizations in the

*A Fully Discrete SIPG Method for Solving Two Classes of Vortex Dominated Flows*

For spacial approximations, assume that piecewise polynomials of order *N* are

the approximating polynomial space for the velocity or pressure restricted to each element, a high-order nodal basis can be chosen, consisting of Lagrange interpolating polynomials defined on a reference simplex introduced in [20, 21]. We let **u** be approximated by **u***h* ∈ **V***h* and adopt a suitable approximation for the term *F*, i.e.,

each element of T *h*. Multiplying the nonlinear term by a test function **v***h* ∈ **V***h*, integrating over the computational domain, and applying integration by parts,

ð Þ� *F* � ∇ **v***hdx* þ X

*∂vhi ∂x j*

, respectively, where **u***E*1 and **u***E*2 are the mean values of *uh* on the

correspond to the components of the related vectors. On each edge *e*∈*∂E*1 ∩ *∂E*2 shared by two elements, to ensure the flux Jacobian of purely real eigenvalues, we

*E*2,*e* the largest eigenvalue of the Jacobians *∂*

elements *E*1 and *E*2, respectively. The global Lax-Friedrichs flux is generally more dissipative than the local Lax-Friedrichs flux, therefore, we primarily consider the local flux on each edge. Although the Lax-Friedrichs flux is perhaps the simplest numerical flux and often the most efficient flux, it is not the most accurate scheme. A remedy of the problem is to employ high-order finite elements. By replacing the

**n***e* � ½ �¼ *F* � **v***h* **n***e* � f g*F* � ½ �þ **v***h*

*e*∈Γ*h*

*λe*

� �, one can get a DG discretization for the nonlinear

For the pressure correction step (17) and the viscous correction step (18), we use the SIPG method to approximate the correction steps. Choosing the orthonormal Legendre basis and the Legendre-Gauss-Lobatto quadrature points gives a wellconditioned Vandermonde matrix and the resulting interpolation well behaved, which greatly simplifies the formulas. The *C*0 continuity condition of the basis in the discontinuous Galerkin formulation is not required. Enforcing a weak continu-

� � ¼ *Lp ϕh* ð Þ, *ϕ*∈ *Mh*,

2 ½ �� **u***h* ½ � **v***h* ,

ð *e* *k*¼1*N*ð Þ *Ek*

**n***e* � ½ � *F* � **v***h ds*, (19)

� � **u***E*1 and

, for *i*, *j* ¼ 1, 2 and the indexes *i*, *j*

*∂***u** ð Þ *F* � **n***e*

2 . In

employed, then the approximation space can be rewritten as **V***h* ¼ ⊕*K*

*F*ð Þ **u** ≈ *F*ð Þ **u***h* , where *F*ð Þ **u***h* also can be represented as the *L*2

*Ek*

ð *Ek*

next section.

we have

ð Ω

may define *λ*þ

� � **u***E*2

*∂ ∂***u** ð Þ *F* � **n***e*

**87**

**3. The spatial discretizations**

*DOI: http://dx.doi.org/10.5772/intechopen.94316*

ð Þ� ∇ � *F* **v***hd***x** ¼ �X

*E*1,*e*, *λ*�

integrand in the surface integral as

*E*1,*e*, *λ*� *E*2,*e*

ity on the interior edges by a penalty term, we have for (17)

*a pn*þ1 *h* , *ϕh*

term in (19) by the local Lax-Friedrichs flux.

with *λe* ¼ max *λ*þ

where the term ð Þ� *F* � ∇ **v***h* equals to *Fij*

To seek *pn*þ1 such that ∇ � **u**~~ ¼ 0, we solve the system

$$-\Delta p^{n+1} = -\frac{\chi\_0}{\Delta t} \nabla \cdot \bar{\mathbf{u}},\tag{17}$$

with a Neumann boundary condition being implemented on the boundaries as

$$\begin{split} \frac{\partial p^{n+1}}{\partial \mathbf{n}} &= \mathbf{f}^{n+1} - \beta\_0 \mathbf{n} \cdot \left( \frac{\partial \mathbf{u}^n}{\partial t} + \nabla \cdot F(\mathbf{u}^n) - \nu \Delta \mathbf{u}^n \right) \\ &- \beta\_1 \mathbf{n} \cdot \left( \frac{\partial \mathbf{u}^{n-1}}{\partial t} + \nabla \cdot F(\mathbf{u}^{n-1}) - \nu \Delta \mathbf{u}^{n-1} \right) \\ &- \beta\_2 \mathbf{n} \cdot \left( \frac{\partial \mathbf{u}^{n-2}}{\partial t} + \nabla \cdot F(\mathbf{u}^{n-2}) - \nu \Delta \mathbf{u}^{n-2} \right) \coloneqq G\_n. \end{split}$$

One can compute the vorticity *ωn* ¼ ∇ � **u***n* at time *t n* ¼ *n* � *Δt*. Then we use *pn*þ1 to update the intermediate velocity **u**~~ by (16).

• The third stage is completed by solving

$$
\gamma\_0 \frac{\mathbf{u}^{n+1} - \tilde{\tilde{\mathbf{u}}}}{\Delta t} = \nu \Delta \mathbf{u}^{n+1},
$$

which can be written as a Helmholtz equation for the velocity

$$-\Delta \mathbf{u}^{n+1} + \frac{\mathcal{Y}\_0}{\nu \Delta t} \mathbf{u}^{n+1} = \frac{\mathcal{Y}\_0}{\nu \Delta t} \tilde{\mathbf{u}}.\tag{18}$$

From the three stages given above, we notice that (15) in the semi-discrete systems is presented in a linearized and explicit process, moreover, (17) and (18) are obviously a type of elliptic and Helmholtz problems at each time step as *n*≥2. We decouple the incompressibility condition and the nonlinearity, then the pressure and velocity semi-discretizations (17)–(18) will be formulated by the interior penalty discontinuous Galerkin methods in spacial discretizations in the next section.


**Lignocellulosic Biomass**

Editors

**Alejandro Rodr´ıguez Pascual Eduardo Espinosa V´ıctor**

MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade • Manchester • Tokyo • Cluj • Tianjin

*Editors* Alejandro Rodr´ıguez Pascual Chemical Engineering Department Universidad de Cordoba ´ Cordoba ´ Spain Eduardo Espinosa V´ıctor Inorganic Chemistry and Chemical Engineering Department Universidad de Cordoba ´ Cordoba ´ Spain

*Editorial Office* MDPI St. Alban-Anlage 66 4052 Basel, Switzerland

This is a reprint of articles from the Special Issue published online in the open access journal *Molecules* (ISSN 1420-3049) (available at: www.mdpi.com/journal/molecules/special issues/lignocellulosic).

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