**3.1 Formation of the structure of the combined methods**

The generalized structure of the standard basic and calibration determination of the strength parameters of main oil pipeline systems discussed above (Section 2) reflects the theory and practice of computational and experimental substantiation of strength developed in our country and abroad for six to seven decades. The focus is on the trunk pipelines for the transportation of oil and oil products. The calculation of strength analysis is based on two methods—the method of calculation for permissible stresses (adopted in foreign practice) and the method of calculation for limiting states and limit resistances (adopted in Russian practice).

The most developed and applied is the deterministic strength calculation at the design stage. This solves the direct main problems of determining the wall thickness of the pipeline for given pressures, throughput of pipes and selected pipe steels. The same method is used at the stage of calibration calculations of the strength of the pipelines under construction and the majority of the pipelines being operated.

In those cases when it is necessary to calculate the substantiation of the strength of functioning pipelines with deviations from the design decisions and when defects in pipes occur outside the established norms, it is necessary to carry out calibration calculations using actual statistical information on all the calculated parameters. One of the tasks solved at the same time is the appointment of all the main design parameters according to the obtained statistical information. In these cases the preservation of regulatory reserves is typical.

For the most critical sections of pipelines, statistical strength analysis may be insufficient and unacceptable. Then probabilistic estimates of strength are required using the functions of the distribution of operational loading and the mechanical properties of pipe steels by the parameter of operation time. For these situations, it becomes possible to change the safety margins for the required probabilities of the occurrence of dangerous states.

*Combined Calculated, Experimental and Determinated and Probable Justifications… DOI: http://dx.doi.org/10.5772/intechopen.89036*

#### **Figure 7.**

**3. Implementation of combined methods to substantiate strength**

*Relative decrease in strength margins n with changing probabilistic characteristics of loading and mechanical*

The generalized structure of the standard basic and calibration determination of the strength parameters of main oil pipeline systems discussed above (Section 2) reflects the theory and practice of computational and experimental substantiation of strength developed in our country and abroad for six to seven decades. The focus is on the trunk pipelines for the transportation of oil and oil products. The calculation of strength analysis is based on two methods—the method of calculation for permissible stresses (adopted in foreign practice) and the method of calculation for

The most developed and applied is the deterministic strength calculation at the design stage. This solves the direct main problems of determining the wall thickness of the pipeline for given pressures, throughput of pipes and selected pipe steels.

strength of the pipelines under construction and the majority of the pipelines being

For the most critical sections of pipelines, statistical strength analysis may be insufficient and unacceptable. Then probabilistic estimates of strength are required using the functions of the distribution of operational loading and the mechanical properties of pipe steels by the parameter of operation time. For these situations, it becomes possible to change the safety margins for the required probabilities of the

In those cases when it is necessary to calculate the substantiation of the strength of functioning pipelines with deviations from the design decisions and when defects in pipes occur outside the established norms, it is necessary to carry out calibration calculations using actual statistical information on all the calculated parameters. One of the tasks solved at the same time is the appointment of all the main design parameters according to the obtained statistical information. In these cases the

**3.1 Formation of the structure of the combined methods**

limiting states and limit resistances (adopted in Russian practice).

The same method is used at the stage of calibration calculations of the

preservation of regulatory reserves is typical.

occurrence of dangerous states.

operated.

**156**

**Figure 6.**

*Probability, Combinatorics and Control*

*properties.*

*Block diagram of regulatory foreign and domestic calculations.*

The scientific basis of these calculations is the entire system of calculation expressions (1)–(42) (**Figure 7**).

This system has been and remains basic in all international practice [1–4] to the present time with the development of methods for the design, construction, and operation of trunk pipelines to ensure their strength and deformability expressed in a gradual decrease in margins *n* (1.8 ≥ *nT* ≥ 1.2; 2.5 ≥ *nв* ≥ 1.7) и and an increase in the strength characteristics *σ*т, *σ*в (200 ≤ *σТ* ≤ 800; 420 ≤ *σв* ≤ 920 MPa).

All uncertainty factors included in the calculations and reflecting the operating conditions, design, and construction technologies were taken into account by the coefficients (*nσ*, *nT*, *nв*) and the standard purpose of guaranteed mechanical properties (*σ*в, *σ*Т).

A generalized analysis of trends and parameters of the development of pipeline transport of oil and oil pipelines and methods for calculating the strength is made in [6, 7].

Expressions (1) and (2) are initial in assessing the strength of pipelines at all the main stages of the life cycle—design, construction, operation, and decommissioning. Currently two tasks are being solved:

• The direct task of a deterministic basic calculation of the wall thickness *δ* of the pipeline at the design stage with a preliminary feasibility study of the diameter *Dв* and pressure *р* as well as with the selected structural material *σоп*,ð*σв*, *σТ* ) and assigned margin *nσ*,ð Þ *nT*, *nв* :

$$\delta \ge \frac{pD\_o}{2[\sigma]} = \frac{pD\_o \cdot n\_\sigma}{2\sigma\_{on}}.\tag{43}$$

Under these conditions, the wall thickness *δ* cannot be less than the value calculated by expression (6) (**Figure 8**).

At the stages of construction, operation, and decommissioning on the basis of (43), deterministic calibration calculations are performed with the following objectives:

• Check of permissible operating pressure [*р*] at specified

$$\mathbb{E}[p] \le 2\frac{\sigma\_{on}}{n\_{\sigma}}.\tag{44}$$

**Figure 8.**

*Regulatory basic and verification calculations for different stages of the pipeline life cycle.*

• Validation of selected and assigned mechanical properties *σоп*,ð Þ *σТ* , *σв* with known *р*, *Dв*, *δ*, *nσ*

$$
\sigma\_{on} \ge \frac{pD\_a}{2\delta} \cdot n\_\sigma. \tag{45}
$$

where *n*п is the number of measurements of calculated parameters *R*п*i*. According to the obtained statistical information on the parameters *Rпi*, the corresponding histograms are constructed by the intervals of their values. For example, **Figure 9** shows the change in the main design parameters—pressure *р* and

*Combined Calculated, Experimental and Determinated and Probable Justifications…*

Inequality (47) with the parameters included in it, as well as the data from **Figure 9**, are the basis for calculating the determination of *Rp* taking into account the statistical and probabilistic dispersion characteristics. At the same time, the

For the design stage, statistical analysis of the design parameters using expressions (8)–(12) is done using factory test data for sheet blanks for pipes (*δ*), pipes (*δ*, *D*), and laboratory samples for static tension (*σ*Т, *σ*в).The values obtained *δр*, *Dp*, (*σ*Т, *σ*в)*p* are entered in the technical conditions or standards. They are the basis of

If these measurements are carried out at the stage of construction or operation, then the data obtained from (49)–(52) are included in deterministic calibration

Technical diagnostics of trunk pipelines (mainly using in-line diagnostics [11]) shows that the most significant from the point of view of strength is the decrease in time *τ* wall thickness parameters *δ* due to such processes as uniform and uneven corrosion, formation and development of cracks of corrosion, and cyclical nature.

The statistical variation of diameters in (11) at the stage of manufacturing, construction, and operation of the linear part of trunk pipelines is small (0.99 ≤ *Dв/ Dm* ≤ 1.01) and can be neglected in deterministic and statistical strength calculations according to (1). However, if during operation there are significant reductions

*P*min ≤*Pm* ≤*Pp* ≤*P*max (49)

*D*min ≤ *Dm* ≤ *Dp* ≤ *D*max (51)

*δ*min ≤*δp* ≤*δm* ≤*δ*max (52)

f g *σT*, *σв* min ≤f g *σT*, *σв p* ≤f g *σT*, *σв m* ≤f g *σT*, *σв* (50)

assigned parameters *Rр* should correspond to the inequality systems

ultimate strength *σв* [10].

*Histograms of pressures and strengths.*

*DOI: http://dx.doi.org/10.5772/intechopen.89036*

**Figure 9.**

deterministic calculations.

**159**

calculations and expressions (44)–(47) (**Figure 10**).

These processes, as a rule, increase the variation of values *δ* in (52).

• Check by (6) the permissible wall thickness ½ � *δ* at

$$\left[\delta\right] \geq \frac{pD\_o n\_\sigma}{2\sigma\_{on}}.\tag{46}$$

• Checking the allowable strength margin[*nσ*]with known

$$m\_{\sigma} \geq \frac{\sigma\_{\text{on}} \cdot 2\delta}{pD\_{\sigma}}.\tag{47}$$

In deterministic calculations according to (1) and (2), a systematic analysis of uncertainty factors affecting the quantities *nσ*, *n*, *m*, *KI*, and *Kн* is carried out. These factors [6–10] included such factors as:

#### **3.2 Databases for calculations**

On the basis of statistical measurements and estimates of all specified design parameters (pressures *р*, mechanical properties *σТ* , *σв*, geometrical dimensions *δ* and *Dв* with variations within {min, max}), first of all, the determination of their average (median) values becomes important

$$R\_{\rm int} = \frac{1}{n\_{\rm in}} \sum R\_{\rm ni},\tag{48}$$

*Combined Calculated, Experimental and Determinated and Probable Justifications… DOI: http://dx.doi.org/10.5772/intechopen.89036*

#### **Figure 9.**

• Validation of selected and assigned mechanical properties *σоп*,ð Þ *σТ* , *σв* with

½ � *δ* ≥ *рDвnσ* 2*σоп*

*nσ* ≥ *σоп* � 2*δ рDв*

In deterministic calculations according to (1) and (2), a systematic analysis of uncertainty factors affecting the quantities *nσ*, *n*, *m*, *KI*, and *Kн* is carried out. These

• Temperature–time change in mechanical properties *σТ* , *σв*, which determines

On the basis of statistical measurements and estimates of all specified design parameters (pressures *р*, mechanical properties *σТ* , *σв*, geometrical dimensions *δ* and *Dв* with variations within {min, max}), first of all, the determination of their

> *R*п*m* ¼ 1 *n*п

2*δ* � *nσ:* (45)

*:* (46)

*:* (47)

X*R*п*i*, (48)

*σоп* ≥ *pDв*

*Regulatory basic and verification calculations for different stages of the pipeline life cycle.*

• Check by (6) the permissible wall thickness ½ � *δ* at

factors [6–10] included such factors as:

**3.2 Databases for calculations**

**158**

the processes of aging and degradation

average (median) values becomes important

• Availability of welded joints with altered properties

• Checking the allowable strength margin[*nσ*]with known

• The effect of the absolute dimensions of the sections ð Þ *Dв*, *δ*

• Type of stress–strain state (components of the main stress *σ*1, *σ*2, *σ*3)

known *р*, *Dв*, *δ*, *nσ*

*Probability, Combinatorics and Control*

**Figure 8.**

*Histograms of pressures and strengths.*

where *n*п is the number of measurements of calculated parameters *R*п*i*.

According to the obtained statistical information on the parameters *Rпi*, the corresponding histograms are constructed by the intervals of their values. For example, **Figure 9** shows the change in the main design parameters—pressure *р* and ultimate strength *σв* [10].

Inequality (47) with the parameters included in it, as well as the data from **Figure 9**, are the basis for calculating the determination of *Rp* taking into account the statistical and probabilistic dispersion characteristics. At the same time, the assigned parameters *Rр* should correspond to the inequality systems

$$P\_{\min} \le P\_m \le P\_p \le P\_{\max} \tag{49}$$

$$\{\sigma\tau,\sigma\_{\theta}\}\_{\min} \leq \{\sigma\tau,\sigma\_{\theta}\}\_{p} \leq \{\sigma\tau,\sigma\_{\theta}\}\_{m} \leq \{\sigma\tau,\sigma\_{\theta}\} \tag{50}$$

$$D\_{\min} \le D\_m \le D\_p \le D\_{\max} \tag{51}$$

$$
\delta\_{\min} \le \delta\_p \le \delta\_m \le \delta\_{\max} \tag{52}
$$

For the design stage, statistical analysis of the design parameters using expressions (8)–(12) is done using factory test data for sheet blanks for pipes (*δ*), pipes (*δ*, *D*), and laboratory samples for static tension (*σ*Т, *σ*в).The values obtained *δр*, *Dp*, (*σ*Т, *σ*в)*p* are entered in the technical conditions or standards. They are the basis of deterministic calculations.

If these measurements are carried out at the stage of construction or operation, then the data obtained from (49)–(52) are included in deterministic calibration calculations and expressions (44)–(47) (**Figure 10**).

Technical diagnostics of trunk pipelines (mainly using in-line diagnostics [11]) shows that the most significant from the point of view of strength is the decrease in time *τ* wall thickness parameters *δ* due to such processes as uniform and uneven corrosion, formation and development of cracks of corrosion, and cyclical nature. These processes, as a rule, increase the variation of values *δ* in (52).

The statistical variation of diameters in (11) at the stage of manufacturing, construction, and operation of the linear part of trunk pipelines is small (0.99 ≤ *Dв/ Dm* ≤ 1.01) and can be neglected in deterministic and statistical strength calculations according to (1). However, if during operation there are significant reductions in wall thickness *δ*, then a significant local increase in diameter *Dв* (by 5–10 due to plastic deformations with the formation of shape defects) is possible. Similar processes of loss of shape and increase in *Dв* are possible with nonstandard bending of pipelines with loss of stability and formation of corrugations.

In the process of development (in time ) of pipeline transportation of hydrocarbons in Russia and abroad, three trends remain dominant using deterministic (D), statistical (C), and probabilistic (P) methods (**Table 1** and **Figure 13**).

*Combined Calculated, Experimental and Determinated and Probable Justifications…*

*DOI: http://dx.doi.org/10.5772/intechopen.89036*

**No. Description Symbols Value** 1. Coefficient of working conditions *m* 0.6–0,9 2. Load reliability factor *K*1 1.1–1,5 3. Material reliability factor *K*2 1.34–1,55 4. Reliability factor to destination *K*н 1.0–1,05

**Figure 12.**

**Table 1.**

**Figure 13.**

**161**

*Calculated standard values of coefficients.*

*Distribution function of the ultimate strength of pipe steels 17G1S.*

*The main regularities of changes in the estimated parameters of pipelines.*

The change in the average values and variation of the design characteristics of strength ð*σ*в, *σ*Т) according to (50) is associated with the instability of technological processes for the production of pipe steels, rolling and heat treatment of sheets, pipe manufacturing, construction of pipeline systems, as well as temporary processes of aging and degradation.

**Figures 11** and **12** show histograms and distribution functions of the mechanical properties of a long-term (up to 50 years) operated 17G1S tubular steel. This information is used in the implementation of calculations for paragraphs 2.1–2.3.

**Figure 10.**

*Statistics on the relative decrease or increase in operating time.*

**Figure 11.** *Strength tensile histograms (total number of tests n = 160).*

*Combined Calculated, Experimental and Determinated and Probable Justifications… DOI: http://dx.doi.org/10.5772/intechopen.89036*

In the process of development (in time ) of pipeline transportation of hydrocarbons in Russia and abroad, three trends remain dominant using deterministic (D), statistical (C), and probabilistic (P) methods (**Table 1** and **Figure 13**).

**Figure 12.** *Distribution function of the ultimate strength of pipe steels 17G1S.*

#### **Table 1.**

in wall thickness *δ*, then a significant local increase in diameter *Dв* (by 5–10 due to plastic deformations with the formation of shape defects) is possible. Similar processes of loss of shape and increase in *Dв* are possible with nonstandard bending of

The change in the average values and variation of the design characteristics of strength ð*σ*в, *σ*Т) according to (50) is associated with the instability of technological processes for the production of pipe steels, rolling and heat treatment of sheets, pipe manufacturing, construction of pipeline systems, as well as temporary processes of

**Figures 11** and **12** show histograms and distribution functions of the mechanical properties of a long-term (up to 50 years) operated 17G1S tubular steel. This information is used in the implementation of calculations for paragraphs 2.1–2.3.

pipelines with loss of stability and formation of corrugations.

aging and degradation.

*Probability, Combinatorics and Control*

**Figure 10.**

**Figure 11.**

**160**

*Statistics on the relative decrease or increase in operating time.*

*Strength tensile histograms (total number of tests n = 160).*

*Calculated standard values of coefficients.*

**Figure 13.** *The main regularities of changes in the estimated parameters of pipelines.*

**References**

USSR; 1986

2004. 1104p

47-59

**163**

Service. USA; 2007

17457. Germany; 1985

[1] SNIP 2.05.06-85. Trunk Pipelines.

*DOI: http://dx.doi.org/10.5772/intechopen.89036*

*Combined Calculated, Experimental and Determinated and Probable Justifications…*

[2] API 579/ASMEFFS-1 Fitness for

[3] DIN. Stainless Welded Pipe DIN

[4] Mazur II, Ivantsov OM. Safety of Pipeline Systems. Moscow: ITS-ELIMA;

Маkhutov NА, Gadenin ММ, Lisin YV, Neganov DА, et al. Prospects for research in the field of risk analysis to improve government regulation and improve the safety of petrochemical complex. Safety in Industry. 2017;**9**:5-13

[6] Маkhutov NА. Strength and Safety:

[7] Lisin YV, Makhutov NA, Nadein VA, Neganov DA. In: Kostogryzov A, editor. Probabilistic Modeling in System Engineering. Рrobabilistic Analysis of Transportation Systems for Oil and Natural Gas. Rijeka, Croatia: InTechOpen; 2018. pp. 81-103

[8] Lisin YV, Маkhutov NА, Neganov DА, Skorodumov SV, Studenov ЕP. Comprehensive mechanical tests to calculate the strength of the main pipeline transport and petroleum products. Factory Laboratory. Diagnostics of Materials. 2018;**84**(4):

[9] Security of Russia. Security of Energy Storage and Transportation Facilities.

[10] Маkhutov NА, Permyakov VV. Resource Safe Operation of Pipelines. Novosibirsk: Nauka; 2005. 516p

МGОF/Znanie; 2019. 928p

Basic and Applied Research. Novosibirsk: Nauka; 2008. 528p

[5] Radionova SG, Zhulina SА,

• Reduction of strength margins *nТ* (от 1.8–3,2 до 1.2–1,5) and (от 2.4–3,5 до 1.6–1,8) in expression (3) and the estimated coefficients in expression (5)


**Abstract**

In many of the active shale plays, the extremely low permeability of the shale means simple, bi-planar hydraulic fractures do not provide enough surface area to make an economic well. In these cases, the optimal, economic completion requires stimulation of the natural fracture system - often called increasing the 'complexity' of the stimulation. A number of different multi-well completion techniques have been proposed to enhance shale complexity. The 'simul-frac' technique is where companion wells are stimulated at the same location at the same time, whereas the 'zipper-frac' technique employs companion wells that are stimulated in staggered locations at the same time. The intention with these techniques is to alter either or both the stress field and the pore pressure field to enhance the shearing of natural fractures.

In this paper, we present the results of a numerical study to quantitatively evaluate the effectiveness of multi-well completion techniques, particularly the 'modified zipper-frac' technique, to optimize shale completions. The study includes a parametric study of the effects of in-situ stress conditions, natural fracture orientation and fracture friction, and hydraulic fracture layout on changing near and far-field natural fracture shear (complexity). Changes in the stress field, particularly shear stress, are considered the primary means of increasing fracture complexity. The quantitative results of the study provide a means to optimize the application and design of different multi-well completion techniques as a function of the presented parameters. Optimized completion designs mean lower well costs, greater produc‐ tion and, ultimately, improved well economics.

**Keywords:** Hydraulic fracturing, stimulation, unconventional, complexity, well comple‐ tion, shale, numerical simulation, simul-frac, zipper-frac, discrete element model, DEM, microseismicity

© 2013 Nagel et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


**Author details**

Muhammad Asraf Hairuddin1 , Nur Dalila Khirul Ashar1 \*, Amar Faiz Zainal Abidin2 and Nooritawati Md Tahir3

1 College of Engineering, Universiti Teknologi MARA, Cawangan Johor, Kampus Pasir Gudang, Malaysia

2 Faculty of Electrical and Electronics Engineering Technology, Universiti Teknikal Malaysia Melaka, Melaka, Malaysia

3 College of Engineering, Universiti Teknologi MARA, Shah Alam Selangor, Malaysia

\*Address all correspondence to<EMAIL_ADDRESS>
© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


**3. Climate Change Adaptation**

It is recognised that adaptation activities can be specific to locations and context because of variability in the social and natural environment, climatic impacts, and socioeconomic and cultural conditions (Füssel and Klein 2006). For example, the spatial heterogeneity of climate change vulnerability and effects (IPCC 2022) is expected to lead to variation in adaptation activities. Different regions will have to adapt to the specific climate vulnerabilities and challenges that exist in that area. Climate change is expected to lead to varying degrees of precipitation across the African continent, with some areas projected to have more intense rainfall and others, less rainfall (Dunning et al. 2018). Additionally, within a specific geographical location, there may be variations in adaptation activity owing to differences in perception (Berkhout 2012). Farmers are sometimes challenged to develop novel means of adaptation. Moreover, non-traditional cultural measures may have been developed to adapt to climate change. The type and extent to which various adaptation measures are implemented, the underlying factors driving adaptation decision making, the lessons learnt, and the success achieved can provide policymakers with information on the practical experiences of the farming community.

The Ministry of Planning and Development (2019) recommends farmers in Trinidad and Tobago implement adaptation measures such as water management, land distribution and management, research and development, climate-sensitive farming systems, increased awareness, and communication. Based on the cost–benefit ratio, suggested high-priority adaptation measures include on-farm water storage, mainstreaming climate change issues into agricultural management and drip irrigation. In addition, crop diversification was recognised as a suitable adaptation measure (Hutchinson 2011). Eitzinger et al. (2015) also identified cassava, sweet potato, and yam as alternative crops that might be more resilient to the changing environmental conditions in Trinidad and Tobago. Climate monitoring was also suggested as a suitable adaptation measure. Climate monitoring can inform farmers of optimal planting times to maximise the possibility of having the best possible crop yield. Climate monitoring can also ensure farmers are not overexposed to harmful weather conditions that threaten their health and safety.


*6.4.2 IPEN*

The cut off value of 318 mOsm/L showed a sensitivity of 90.9% and specificity of 90.6% for diagnosing DED. The IPen osmometer can be considered suitable for use in the clinical setting, with good performance in DED diagnosis [38].


Sorption of Heavy Metals onto Biochar

*Ramalingham Senthilkumar and Donipathi Mogili Reddy Prasad*

### **Abstract**

Biochar is a stable carbon-rich product synthesized from biological materials through different heating methods above the decomposition temperature. The potential uses of biochar in various fields include soil fertility improvement, C sequestration, pollutant removal and waste minimization/reuse. In recent years, large number of research has confirmed that biochar can be used successfully for the removal of heavy metal ions from aqueous solutions. The main aim of this chapter is to summarize and assess the sorption capacity of biochar toward various heavy metal ions. Considering that sorption is a surface phenomenon, the key parameters controlling the formation of biochar including pyrolysis temperature, residence time, and feedstock type will be discussed in detail. In addition, the mechanism associated with remediation of heavy metal ions and the physicochemical factors affecting the sorption potential will be discussed. Mathematical models employed in the sorption studies will be given special importance. The modification procedures used to enhance the sorption capacity of biochar will also be highlighted.

**Keywords:** biochar, adsorption, heavy metals, water quality, pyrolysis

### **1. Introduction**

Water usage has been rising immensely with growing population and industrial activities in both developed and developing countries [1]. This resulted in deterioration of water sources as various contaminants such as dyes [2], toxic heavy metals [3], organic compounds like detergents, phenols, dyes, pesticides in addition to the other persistent organic pollutants [4] are increasingly being dumped into the water bodies [5]. Among these contaminants, heavy metals are of high priority because they persist in soils and do not undergo biodegradation [6]. This might affect significantly the suitability and sustainability of the water resources [7]. These contaminants reach water bodies through various industrial activities, including mining, electrolysis, metallurgy, battery manufacture, metal finishing, electroplating, electro-osmosis, pigment manufacture, tanneries, etc. [8, 9]. Heavy metals are then taken up by the biological systems through food intake and thereby cause major hazardous health impacts [10, 11]. Owing to this, different biological and physico-chemical treatment techniques have been proposed to remediate heavy metal-bearing contaminated waters. These remediation technologies include adsorption, biosorption, ion-exchange, electrocoagulation, membrane technologies and precipitation [12]. Adsorption is one of the widely used remediation

approaches, which has its unique advantages including cost-effectiveness, high performance toward metal ion of interest, flexibility and ease in operation. [13]. Some of the widely used adsorbents for heavy metal remediation include fly ash, activated carbon, sorbents prepared from agricultural, industrial and biological waste materials [14]. Recently, char derived from biological materials under oxygen free condition, popularly known as "biochar" has been recently been introduced as an effective sorbent for various toxins [4, 15].

Biochar is a stable carbon-rich product synthesized from biological materials through different heating methods above the decomposition temperature. Biochar is produced through thermal degradation of organic components in absence of O2 or under limited oxygen conditions (pyrolysis) [16]. In recent years, owing to the inherent biochar properties such as surplus surface binding sites (hydroxyl, carboxyl, phenolic hydroxyl and carbony groups), porous surface, high cation exchange capacity and its surface area, this organic amendment be utilized as an efficient and practical sorbent for remediation [17]. Biochars from different feed stock materials were prepared and successfully examined for their adsorption potential toward various metal ions [3] and nutrients [18]. The pyrolysis conditions such as temperature, rate of heating and residence time are all critical factors influenceing the potentiality of metal sorption on biochar. In addition, changes in sorption potential can also be obtained through physico-chemical techniques. Thus through this chapter, we critically explore the state of knowledge on different methods to modify biochar with preferred properties. We focused on various aspects including biochar preparation methods, factors affecting the biochar sorption potential and metal removal mechanisms.


**Advances in Legumes for Human Nutrition**

Editors

**Stuart Johnson Rewati Bhattarai**

Basel • Beijing • Wuhan • Barcelona • Belgrade • Novi Sad • Cluj • Manchester

*Editors* Stuart Johnson Curtin University Perth, Australia

Rewati Bhattarai Curtin University Perth, Australia

*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 *Nutrients* (ISSN 2072-6643) (available at: https://www.mdpi.com/journal/nutrients/special\_ issues/Advances\_in\_Legumes\_for\_Human\_Nutrition).

For citation purposes, cite each article independently as indicated on the article page online and as indicated below:

Lastname, A.A.; Lastname, B.B. Article Title. *Journal Name* **Year**, *Volume Number*, Page Range.

**ISBN 978-3-0365-9620-4 (Hbk) ISBN 978-3-0365-9621-1 (PDF) doi.org/10.3390/books978-3-0365-9621-1**

© 2023 by the authors. Articles in this book are Open Access and distributed under the Creative Commons Attribution (CC BY) license. The book as a whole is distributed by MDPI under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs (CC BY-NC-ND) license.

## **Contents**

Reprinted from: *Nutrients* **2023**, *15*, 89, doi:10.3390/nu15010089 ................... **163**

of Enzymatic Hydrolysis and Thermal Processing on Proteins and Phenolics

### **Duc Toan Do, Jaspreet Singh, Stuart Johnson and Harjinder Singh**


**1. Introduction**

In the framework of the "20–20–20" targets [1], measures adopted by the European Union to mitigate climate change include the improvement of conventional prime movers' conversion efficiency. As indicated in the European directive 2012/27/EU [2], the promotion of cogeneration, as well as waste heat recovery (WHR) strategies, could significantly contribute to achieving this objective.

Internal combustion engines typically convert just the 30–45% of the fuel primary energy into mechanical or electric power, while the remaining part is rejected as thermal energy with exhaust gases, cooling water, oil, etc. This heat, which would be wasted otherwise, can be used for combined heat and power (CHP) applications, thus improving the overall first law conversion efficiency, but also

for additional electric production, via a bottomer thermodynamic cycle. Heat released by internal combustion engine (ICE) exhaust gases is usually characterized by (i) limited thermal power size (varying typically in the range from kW to tens of MW as order of magnitude), (ii) by temperature often quite lower than 500 ◦C, and (iii) both strongly variable with the engine load. These conditions are not always compatible with the adoption of conventional steam Rankine cycle [3] since problems related to the use of water as working fluid can arise as (i) difficulty in superheating the fluid in order to avoid condensate formation during the expansion process and the risk of erosion of the turbine blades [3]; (ii) working with excessive vapor operating pressure that impose to install complex multi-stage and expensive turbines [3]. A solution is identified in changing the operating fluid with an organic compound in an organic Rankine cycle (ORC). Organic fluids are characterized by higher molecular mass and lower critical temperature compared to water that allows exploiting efficiently, also, low-grade heat sources. Even though the conversion efficiency of ORC is typically limited, compared to steam cycle, this technology features a number of advantages: lower O&M and personnel costs; high molecular weight; low enthalpy drop in the expander, and, as a consequence, higher mass flow rates, if compared with water. Moreover, ORCs are characterized by a wider regulation range, and performances are not particularly penalized at part-load conditions [4].

WHR from stationary ICE prime movers is quite a consolidated option, and several investigations have been proposed on this topic (see for example [5–14]). The studies point out that ORC performance is significantly affected by the choice of the working fluid, the cycle architecture, and the setting of the operating parameters. Investigated fluids are hydrocarbons, refrigerants, and zeotropic mixtures, while the other most important decision variables are identified in the evaporating pressure and the superheating degree [5–7]. In particular, Yang et al. analyzed an ORC system in [5] to recover waste heat from the diesel engine exhaust, where the zeotropic mixture R416A was used as the working fluid for the ORC. Considering various operating conditions of the diesel engine, this study investigated the effects of the degree of superheating on the running performance of the ORC waste heat recovery. In the thermodynamic study by Han et al. [6], a regenerative ORC system was established to recover the waste flue gas of 160 ◦C, focusing on thermodynamic and economic performance while simultaneously considering the effect of superheating and working fluid selection. The optimization of the evaporation temperature was carried out by analyzing the variation of net power output and specific investment cost. Energy and exergy analysis of three ORC–WHR configurations that use a coupling thermal oil circuit was performed by Valencia et al. in [7], where a simple ORC, an ORC with a recuperator, and an ORC with double-pressure configuration were considered; cyclohexane, toluene, and acetone were simulated as ORC working fluids. In addition, the effect of evaporating pressure on the net power output, thermal efficiency increase, specific fuel consumption, overall energy conversion efficiency, and exergy destruction was also investigated.

These studies demonstrated also the importance of taking into account the part-load operation of the ICEs when evaluating the performance of the ORC bottoming solution. For example, in [8] and [9], the performance of combined systems with ORC and ICE was analyzed at steady-state under different typical working conditions, by means of a numerical simulation model of the system. These papers analyzed, in addition, the effect of adjustable parameters on the system performance, giving effective control directions under various conditions. They proved that to get a better system performance under different working conditions, the system should be operated with a slight degree of superheating.

Valorization of waste heat through ORC creates interesting opportunities, especially to combined heat and power generation, in the residential sector and in energy-intensive industries. In this context, a series of hybrid power plants comprising a CHP prime mover integrated with ORC have been investigated [10–13], with the aim of maximizing the share of electricity production of CHP plants rather than a higher heat production rate in a cost-effective manner. The analyzed heat sources are usually cooling water from steam cycles or exhaust gas from the industrial process. An example of a hybrid CHP plant was proposed by Yi et al. in [10], where an ORC and a hydraulic turbine were introduced to reduce exergy losses, which occurred in a thermal storage process for district heating (DH) purpose. Results showed a payback period equal to just 3.5 years. The study proposed by Arabkoohsara and Namib [11] analyzed a waste-driven CHP plant. In addition to the electricity generated by the main steam cycle, the heat withdrawn from the condenser was employed for additional heat and electricity production by employing a small-scale ORC unit and a heat exchanger connected to the local district heating system. This study evaluated an electric efficiency improvement of 20% and a favorable period of return on investment. Marty et al. [12] analyzed the optimization of parallel distribution between electricity and heat production for a geothermal plant. The geothermal fluid was split into two streams, one used for an ORC system, and the other for the DH. An optimization tool was implemented to obtain the sizing of the ORC and the best distribution between electricity and heat production. In the project by Ramireza et al. [13], a large-scale ORC pilot plant along with a waste heat recovery unit in a steel mill was designed. Waste heat was recovered from the fumes of the furnace to produce saturated steam, which was then delivered to a DH network during the cold season and to the ORC for electricity generation during the rest of the year. Results on the performance of the plant during the first months of operation showed promising values. The study by Grljuši´c et al. [14] showed instead an application involving a diesel engine for maritime application. A standard operating propulsion engine at maximum efficiency was assumed to supply the auxiliary power. The aim of this research was to investigate the possibility of using the CHP plant over a ship to meet all heating and electricity requirements during navigation.

All these studies make evident that the ORC design for CHP hybrid systems has to be determined in accordance with the specific application. In particular, in the case of a CHP ICE-ORC hybrid system, the optimal ORC sizing, the optimal load allocation, and the resulting off-design performance should be explored more in depth, especially if it is called to operate under variable thermal demand conditions.

#### *The Objective of the Study*

The main objective of this work was the energy and economic assessment of a CHP power plant based on ICEs, feeding a DH network, and coupled with an ORC to optimally recover the ICEs' residual heat. In this work, a specific case study of an existing CHP system was analyzed; however, the same presented approach and similar considerations could be applied to any CHP feeding a district heating network (DHN) as a thermal user.

In detail, as the first step, the optimal operation of the CHP power plant was assessed without ORC: the ICEs load profile was determined, in order to fulfill the DH thermal demand, while minimizing the heat dissipation to the ambient. Then, in order to increase the amount of produced electrical energy and to better exploit the available ICEs residual heat, the opportunity of integrating an ORC as a bottomer was investigated.

The original power plant optimal operation was compared with the new arrangement, where the ICEs were operated at full-load, and the ORC was integrated also with the DH demand, to recover the ICEs' residual heat. The ORC as bottomer of the CHP was an interesting solution also because it could represent a flexibilization tool in the regulation of the whole plant. The ORC allowed operating the ICEs at full-load (i.e., at maximum efficiency point) while not dissipating the surplus of heat production when it is not required by the thermal user. Indeed, the surplus of high enthalpy-heat production was recovered by the ORC and converted into additional electric power, sold to the electric grid.

In detail, three different system architectures were proposed and analyzed. A sensitivity analysis of organic fluid and key cycle parameters was performed in order to identify, for each proposed arrangement, the optimum design of the ORC. A thermodynamic design and off-design model of the ORC system has been developed in the *Thermoflex*™ environment [15]. The additional electrical energy produced during the year, the ORC operating hours, and total investment costs were quantified and compared among the analyzed layouts. Finally, the original optimal operation of the CHP plant was compared with the modified one comprising the ORC, in order to estimate, for each proposed arrangement, the amount of additional generated electric power and the corresponding increase in primary energy consumption.

An energetic index accounting for the global primary energy saving was introduced, in order to evaluate the convenience in producing the additional electric power with the modified CHP plant rather than with the original plant with the help of the electric generating mix. From the economic point of view, considerations on the electric energy sell price to return on investment were also presented.

#### **2. Cogeneration Power Plant Description**

This section presents the description of the CHP plant feeding the DH network and the control logic adapted to regulate the system when working at part-load conditions.

#### *2.1. CHP Plant Arrangement*

The case study system was an existing cogeneration plant intended to satisfy the heat demand of an international hub airport; it comprised three ICEs (model Rolls Royce Bergen engine B35: 40 V20 AG), fed with natural gas, named GR1, GR2, and GR3, as depicted in Figure 1. The CHP single unit prime movers were conceived to provide a nominal electric power equal to 8.5 MW and thermal power of about 4.6 MW (only heat recovery from the exhaust gas is considered). Thus, the nominal electric and thermal efficiency were equal to 47% and 25%, respectively. The DH demand was fulfilled by the ICEs discharged heat and by three additional boilers, producing 8 MW of nominal thermal power output each, also fed with natural gas. The generated electric power was fed into the grid at a connection voltage of 150 kV.

The DH network (DHN) was designed to provide hot water for various types of tertiary building users. It included heat exchangers for heating purposes in winter and heat exchangers working all along the year, providing high-temperature heat to produce food use steam and to feed absorption refrigeration units, but also sanitary water.

For what regards the management strategy, a DHN could be regulated (i) varying the mass flow rate but keeping constant the temperature difference (Δ*T*) on the utilities (this approach is particularly suitable for small size DHNs) or (ii) varying the Δ*T* on the utilities with a constant mass flow rate (usually for large DHNs) [16]. The regulation adopted in this study was the first one: in particular, it was assumed that the delivery temperature was kept equal to 130 ◦C and the return equal to 70 ◦C. This supposition could be reasonable if considering that the DHN inertia was typically such high to presume to be neglectable the return temperature oscillation when relying on an hourly basis (which was the timestep considered in this analysis).

The yearly required thermal production profile is shown in Figure 2. Higher thermal power requests were observed during winter, with a demand peak close to 20 MW; lower heat demand values, around 3 MW on average, were registered during summer months.

**Figure 1.** Schematic of the cogeneration power plant.

**Figure 2.** Heat demand yearly profile.


Mode

The granodiorite contains 89.3% of quartz-feldspar mass and 7.7% of mica according to the CT results. Mica is fairly evenly distributed throughout the rock volume, forming slightly oriented flattened grains of 0.594 mm in size (Figure 11a). There are also individual aggregates of 1.8–3.0 mm (Figure 11b).

**Figure 11.** (**a**) Mica in the volume of granodiorite and (**b**) their distribution according to the maximum diameter.


**2.3 Entropy of the mesoscopic process resulting from deterministic, microscopic system**

Kolmogorov and other authors [19] studied the entropy and ergodic properties of the stationary mesoscopic process defined previously, following methods introduced by Shannon in the framework of signal theory [27–30]. These methods, and part of Kolmogorov's results, can be extended to the nonstationary process (11).

#### *2.3.1 The* n*-times entropy and the instantaneous entropy of the mesoscopic system*

Following Kolmogorov, we consider y the Shannon entropy [27–30] of the trajectory (*i*)*n* = (*i*0, … , *in-*1) in the phase space

$$S(p\_n) = -\sum\_{i\_0,\dots,i\_{n-1}} p\_n(i\_0, 0; \dots; i\_{n-1}, n-1) \ln p\_n(i\_0, 0; \dots; i\_{n-1}, n-1). \tag{15}$$

*Stochastic Theory of Coarse-Grained Deterministic Systems: Martingales and Markov… DOI: http://dx.doi.org/10.5772/intechopen.95903*

On the other hand, the new information obtained by observing the system in the mesoscopic state *in* at time *tn*, knowing that it was in the respective states *i*0, … *in*-1 at the prior times 0, … *n*-1, will be called the instantaneous entropy

$$\begin{split} s\_n(p) = \mathbb{S}\_{n+1}(p) - \mathbb{S}\_n(p) &= -\sum\_{i\_0, \ldots; i\_n} p(i\_0, 0; \ldots i\_n, n) \ln p\left(i\_n, n|i\_{n-1}, n-1; \ldots; i\_0, 0\right) \geq 0, \\ &= \sum\_{i\_0, \ldots; i\_{n-1}} p(i\_0, 0; \ldots i\_{n-1}, n-1) \operatorname{\mathbf{S}}\left(p\left(\bullet, n|i\_{n-1}, n-1; \ldots; i\_0, 0\right)\right). \end{split} \tag{16}$$

where *p* denotes the infinite process. The properties of *S*(*pn*) and *sn*(*p*) have been extensively studied by Kolmogorov and other authors in the case of the stationary process (6) [19]: they are summarily mentioned in 2.5. They are not necessarily valid for the nonstationary process.

### *2.3.2 Maximizing the* n*-times entropy of the mesoscopic system: The "Markov scheme"*

If one knows the first two distributions *p*1 and *p*2, one can mimics the exact mesoscopic distributions *pn* by using the Jaynes' principle, maximizing the entropy *S*(*qn*) of a ditribution *qn* under the constraints *q*1 = *p*1 and *q*2 = *p*2. Then it is found that optimal distribution *qn* is the Markov distribution *q*n satifying these constraints [18].

It is shown in Ref. [18] that for n > 2, both the *n*-times entropy *Sn*ð Þ*q* and the instantaneous entropy *sn*ð Þ*q* are larger than the correponding entropies *Sn*ð Þ *p* and *sn*ð Þ *p* of the exact process *p*, except if *p* is Markov: *p = q*.

The Markov process *qn* is not really an approximation of the mesoscopic process *p*, because *q*n does not tend to *pn* when *n* ! ∞. Approximating the exact mesoscopic process by a Markov process will be the main purpose of the next section.

### **2.4 Entropy and memory in the stationary situation**

### *2.4.1 Kolmogorov entropy of the stationary process*

Here we consider the stationary process arising from the initial uniform microscopic distribution *μ*(*x*), when the *n*-times stationary probability is *p*0 *n* given by (7). For the sake of simplicity we omit the index 0 in the present Section, unless otherwise specified. It can be shown [19] that the entropy *Sn*(*p*) is an increasing, concave function of *n*

$$\mathfrak{s}\_{n} \equiv \mathfrak{S}\_{n+1}(p) - \mathfrak{S}\_{n}(p) \ge 0. \tag{17}$$

$$
\mathfrak{L}\_{n+1} - \mathfrak{s}\_n = \mathfrak{S}\_{n+1}(p) - 2\mathfrak{S}\_n(p) + \mathfrak{S}\_{n-1}(p) \le \ 0. \tag{18}
$$

It results from (17) and (18), and also from 2.5.2, that the limits

$$\lim\_{n \to \infty} \frac{1}{n} \operatorname{S}\_{\mathfrak{n}}(p) \; = \lim\_{n \to \infty} \mathfrak{s}\_{\mathfrak{n}}(p) = \mathfrak{s}(p). \tag{19}$$

exist: *s*(*p*) is the Kolmogorov entropy of the evolution function *f* with respect to the partition (*i*) of the mesoscopic states [19]. More simply, we can call it entropy of the mesoscopic process.

#### *2.4.2 Memory decrease in the stationary mesoscopic process*

It has been proved recently [18] that, although it is infinite, the memory of the mesoscopic process fades out with time: for *n* large enough, if *N* > *n* the probability

*N* being the total number of microstates in *X*.

**The** *n***-times nonstationary mesoscopic probabilities** are obtained from (9)

where *n*(*A*) is the number of microstates belonging to some subset *A* of *X*. So

*n i*0∩*φ*�1*i*1∩ … ∩*φ*�*n*þ1*in*�1 � �

� � *p i*ð Þ0

*μ i*0∩*φ*�1*i*1∩ … ∩*φ*�*nin* � � *p i*ð Þ0

*N :* (10)

*μ*ð Þ *i*0

*μ*ð Þ *i*0

� �*:* (13)

*:* (11)

*:* (12)

*pn*ð*i*0, 0; *i*1, 1, ; *in*�1, *n* � 1Þ ¼ *p i*0∩*φ*�1*i*1∩ … ∩*φ*�*n*þ1*in*�1, 0 � � ¼ *p i*ð Þ0 *μ*ð Þ *i*0

*Advances in Dynamical Systems Theory, Models, Algorithms and Applications*

*pn*ð*i*0, 0; *i*1, 1, ; *in*�1, *n* � 1Þ ¼ *μ i*0∩*φ*�1*i*1∩ … ∩*φ*�*n*þ1*in*�1

*i*0

From the definition of the relative probabilities, one can formally write

*i*1

*p i*ð *n*, *n i*j *n*�1, *n* � 1; … ; *i*0, 0Þ ¼ *p*0ð Þ *in*, *n i*j *n*�1, *n* � 1; … ; *i*0, 0

We will make use of this simple but important property later.

**2.3 Entropy of the mesoscopic process resulting from deterministic,**

*2.3.1 The* n*-times entropy and the instantaneous entropy of the mesoscopic system*

Following Kolmogorov, we consider y the Shannon entropy [27–30] of the

The corresponding process is generally not Markovian. For instance, if *i*0 ∩ *ϕ*�1

*p i*2, *t*2 j*i*1, *t*1

but in general this equation is useless, since the conditional probability *p*(*i*2, *t*2|

It results from (11) that the nonstationary conditional probabilities, *conditioned by the whole past up to time* 0, are identical to the corresponding stationary proba-

Kolmogorov and other authors [19] studied the entropy and ergodic properties of the stationary mesoscopic process defined previously, following methods introduced by Shannon in the framework of signal theory [27–30]. These methods, and part of Kolmogorov's results, can be extended to the nonstationary process (11).

� �*p i*1, *t*1

¼ *μ i*0∩*φ*�1*i*1∩ … ∩*φ*�*n* ð Þ *in μ i*0∩*φ*�1*i*1∩ … ∩*φ*�*n*þ1*in*�1

*pn*ð Þ *i*0, 0; … ; *in*�1, *n* � 1 ln *pn*ð Þ *i*0, 0; … ; *in*�1, *n* � 1 *:* (15)

� � *:* (14)

and all multiple probabilities follow, for instance

*pn*�1ð Þ¼ *i*1, 1; … ; *in*, *n* X

*p i*2, *t*2 � � ¼ X

*i*1, *t*1) cannot be computed independently of *p*(*i*1, *t*1).

� ¼ 0 but *p i*2, 2 *i*1, 1Þ � � 6¼ 0 � � .

*p :i*2, 2 *i*1, 1; *i*0, 0Þ �

bilities: as an example

**microscopic system**

*S pn*

**88**

� � ¼ � X

trajectory (*i*)*n* = (*i*0, … , *in-*1) in the phase space

*i*0, … *in*�1

*i*1 6¼ ∅, *i*1 ∩ *ϕ*�1 *i*2 6¼ ∅ and *i*0 ∩ *ϕ*�2 *i2* = ∅, it is easily seen that

of *iN* at time *N* conditioned by the *n* last events, is practically equal to the probability at time *N*, conditioned by the *whole past* down to time 0.

$$p(i\_N, N \mid i\_{N-1}, N-1; \dots i\_n, N-n) \approx p(i\_N, N \mid i\_{N-1}, N-1; \dots i\_0, 0) \text{ when } n \to \infty. \tag{20}$$

More precisely, for any *ε* > 0, there exists a positive integer *n* such that for any *N* > *n*

$$0 < \mathfrak{s}\_N < \mathfrak{s}\_n < \varepsilon. \tag{21}$$

*d*0, … *m*þ*n*�1 *Πm*þ*n*, *Π*ð Þ *n*

*Stochastic Theory of Coarse-Grained Deterministic Systems: Martingales and Markov…*

stationary conditional probability *Πm* + *n* and its approximation *Π*ð Þ *n*

*d*0, … *m*þ*n*�1 *Πm*þ*n*, *Π*ð Þ *n*

**2.5 Martingale theory and almost sure convergence**

systems. We refer to [20] for adressing more general cases.

as small as desired by choosing *n* large enough.

*DOI: http://dx.doi.org/10.5772/intechopen.95903*

*almost surely* [18], as well as and in probability

So, the approximation *Π*ð Þ *n*

a martingale if for all *n*:

*2.5.1 Definitions*

variable *X*.

⊂ … ⊂ F), and.

theorem holds [20, 21]:

stochastic variable *X*.

**91**

*m*þ*n* !

So, the probability that this distance exceeds a given accuracy *a* > 0 can be made

Further results can be obtained by newly using the sationnarity of process *p*. In fact, it can be shown [18] that *p* is a martingale [20–22]. Then, general results from martingales theory (see below) show that when *n* ! ∝ the distance between the

> *m*þ*n* !

For convenience, we first summarize some definitions and results of martingale

i. *simplified definition*: a (discrete time) sequence of stochastic variables *Xn* is

We now sketch the derivation of this conclusion from martingale theory.

theory [20–22], before applying them to the mesoscopic laws of deterministic

h i j j *Xn* < ∞ and *Xn*þ1j*Xn*, … *X*1

ii. *more generally* (see the general definition, for instance, in [20])

where h i *X* denotes the average (mathematical expectation) of the stochastic

If • (Ω, F, *P*) is a probability space (where Ω is the state space, *P* is the probability law, and F is the set of all subspaces (*σ*-algebra) for which *P* is defined),

• F n is an increasing sequence of *σ*-algebras extracted from F (F n ⊂ F *n* + 1

Among the remarkable properties of martingales, the following convergence

If (*Xn*) is a positive martingale, the sequence *Xn* converges almost surely to a

Stronger and more general results can be found in the references.

So, for almost all trajectories *ω*, *Xn*(*ω*) ! *X*(*ω*) with probability 1 when *n* ! ∞.

• for all *n* ≥ 0, *Xn* is a stochastic variable defined on (Ω,F n, *P*),

the sequence *X*n is a martingale if h i j j *Xn* < ∞ and *Xn*þ1jF*n*

*2.5.2 Convergence theorem for martingales*

*a:s:*

*m*þ*n* converges to *Πm* + *n* for almost all trajectories [18].

¼ *Xn:* (30)

¼ *Xn*.

*p*

0 if *n* ! ∞*:* (28)

0 if *n* ! ∞*:* (29)

*m*þ*n* tends to 0

where *sn* is the instantaneous entropy given by (14). In fact, let us write

$$\Pi\_N(i\_N) = p\left(i\_N, N|i\_{N-1}, N-1; \dots \newline \ldots \newline 0\right) = \mu\left(\left.f\_{-N}i\_N\right|f\_{-N+1}i\_{N-1}\cap \dots \cap i\_0\right); \tag{22}$$

$$\mu^{(n)}(i\_1) = \mu(i\_1, N|i\_1, N, \dots \newline \ldots \newline \ldots \newline \ldots \newline \ldots \newline \ldots) = \mu(f\_{-N}|f\_1, \dots |f\_0) = \mu(f\_{-N}|f\_0) = \mu(f\_{-N}|f\_0) = \mu(f\_0) = \mu(f\_{-N}) = \mu(f\_0) = \mu(f\_{-N}) = \mu(f\_0) = \mu(f\_{-N}) = \mu(f\_0) = \mu(f\_0) = \mu(f\_0) = \mu(f\_0) = \mu(f\_0) = \mu(f\_0) = \mu(f\_0) = \mu(f\_0) = \mu(f\_0) = \mu(f\_0) = \mu(f\_0) = \mu(f\_0) = \mu(f\_0) = \mu(f\_0) = \mu(f\_0) = \mu(f\_0) = \mu(f\_0) = \mu(f\_0) = \mu(f\_0) = \mu(f\_0) = \mu(f\_0) = \mu(f\_0) = \mu(f\_0) = \mu(f\_0) = \mu(f\_0) = \mu(f\_0) = \mu(f\_0) = \mu(f\_0) = \mu(f\_0) = \mu(f\_0) = \mu(f\_0) = \mu(f\_0) = \mu(f\_0) = \mu(f\_0) = \mu(f\_0) = \mu(f\_0) = \mu(f\_0) = \mu(f\_0)$$

$$I\_N^{(n)}(i\_N) = p\left(i\_N, N \middle| i\_{N-1}, N-1; \dots; i\_{N-n}, N-n\right) = \mu\left(f\_{-N} i\_N \middle| f\_{-N+1} i\_{N-1} \cap \dots \cap f\_{-N+n} i\_{N-n}\right). \tag{23}$$

For a given *n*, formula (23) allows one to define a new process *p*(*n*) from the original process *p*, which can be called "the approximate process of order *n*" of *p* (see Section 2.6). It results from (21) and from the stationarity of *p* that for any *ε* > 0, there is an integer *n*(*ε*) depending only on *ε*, such that for any integers *N*, *n* > *n*(*ε*)

$$0 < s\_n(p) - s\_N(p) = \sum\_{i\_0, \dots, i\_{N-1}} \mu\left(i\_0 \cap f\_{-1}(i\_1) \cap \dots \cap f\_{-N+1}(i\_{N-1}) \cdot \mathbb{S}\_{0,\dots,N-1}\left(\Pi\_N \middle| \begin{matrix} n \\ N \end{matrix}\right)\right)$$

where *S*0, … *N*�1 *ΠN*|*Π*ð Þ *n N* � � is the relative entropy of *ΠN* with respect to *Π*ð Þ *n N* : the last right hand member of Eq. (22) is the average of this relative entropy on the past of *N.* Because *sN*(*p*) decreases to a limit *s p*ð Þ when *N* ! ∞, it results that

$$0 < \delta \mathfrak{s}\_n(p) \equiv \mathfrak{s}\_n(p) - \mathfrak{s}(p) \le \varepsilon \text{ if } n > n(\varepsilon). \tag{25}$$

The total variation distance *d*(*P*,*Q*) between two distributions *Pj* and *Qj* over the states *j* of a finite set (*j*) is

$$d(P, Q) = \frac{1}{2} \sum\_{j} \left| P\_j - Q\_j \right|. \tag{26}$$

Then, the total variation distance *d*0, … *N*�1 *ΠN*, *Π*ð Þ *n N* � � between *ΠN* and *Π*ð Þ *n N* (for a given past trajectory between times 0 and *N*-1) is related to the relative entropy [18, 31] and it can be concluded that

$$\left\langle d\_{0,\ldots,N-1}\left(\Pi\_N, \Pi\_N^{(n)}\right)\right\rangle^2 \le \left\langle \left[d\_{0,\ldots,N-1}\left(\Pi\_N, \Pi\_N^{(n)}\right)\right]^2 \right\rangle < \varepsilon/2 \text{ if } n(\varepsilon) < n < N. \tag{27}$$

#### *2.4.3 Convergence properties of the approximate process*

Let us write *m* = *N-n* > 0. It follows [18] from (25) that for any fixed *m*, the total variation distance between the exact and the approximate probabilities *d*0, … *m*þ*n*�1 *Πm*þ*n*, *Π*ð Þ *n m*þ*n* � � *tends to* 0 *in probability* when *n* ! ∞

*Stochastic Theory of Coarse-Grained Deterministic Systems: Martingales and Markov… DOI: http://dx.doi.org/10.5772/intechopen.95903*

$$d\_{0,\ldots m+n-1} \left(\varPi\_{m+n}, \varPi\_{m+n}^{(n)}\right) \xrightarrow{p} \mathbf{0} \text{ if } n \to \infty. \tag{28}$$

So, the probability that this distance exceeds a given accuracy *a* > 0 can be made as small as desired by choosing *n* large enough.

Further results can be obtained by newly using the sationnarity of process *p*. In fact, it can be shown [18] that *p* is a martingale [20–22]. Then, general results from martingales theory (see below) show that when *n* ! ∝ the distance between the stationary conditional probability *Πm* + *n* and its approximation *Π*ð Þ *n m*þ*n* tends to 0 *almost surely* [18], as well as and in probability

$$d\_{0,\ldots m+n-1} \left(\varPi\_{m+n}, \varPi\_{m+n}^{(n)}\right) \stackrel{a.s.}{\rightarrow} \mathbf{0} \text{ if } n \rightarrow \infty. \tag{29}$$

So, the approximation *Π*ð Þ *n m*þ*n* converges to *Πm* + *n* for almost all trajectories [18]. We now sketch the derivation of this conclusion from martingale theory.

#### **2.5 Martingale theory and almost sure convergence**

For convenience, we first summarize some definitions and results of martingale theory [20–22], before applying them to the mesoscopic laws of deterministic systems. We refer to [20] for adressing more general cases.


**Jordi Pons-Prats**

Dr. Jordi Pons-Prats is a Mechanical Engineer, PhD in Aerospace Science and Technology by Polytechnical University of Catalonia (UPC, BarcelonaTech). He is lecturer at UPC since 2020. Before he spent 12 years at CIMNE (International Center for Numerical Methods in Engineering), a research center of UPC. From 2008 he is the responsible of the aerospace engineering group of CIMNE. He also had previous experience as project manager at industrial companies. His research career focused on the field of optimization, robust design and stochastic analysis, while with the integration into UPC it moved to Air Transport, Air Operations and Urban Air Mobility. He has published 21 scientific papers in JCR journals, among other publications like 10 book chapters and 1 conference proceedings. He has more than 250 citations and a h-index of 7 (Source: Web of Knowledge). He is the coordinator of the Iberia East Pilot Center of ERCOFTAC (www.ercoftac.org), and the Spanish National Contact Point and member of the Board of Directors of EASN (www.easn.net).


*Edited by Josette Bettany-Saltikov and Sanja Schreiber*

Innovations in Spinal Deformities and Postural Disorders presents a compendium of innovative work in the management of spinal deformities and postural disorders. The chapters were carefully selected with clinicians, researchers, patients and parents in mind. All of these stakeholders are important links in the management of spinal deformities and disorders. It is our hope that all will remain open to new ideas in the field and will be able to evaluate the material carefully and in ways that are objective and evidence based. We hope that the different chapters in the book will stimulate readers to be original and innovative in their own centers in order to help our patients in the best way possible. This book contains new information on the 3D measurement of, as well as new approaches to, the 3D conservative, including exercises and braces, and surgical treatments for patients with spinal deformities and postural disorders.

ISBN 978-953-51-3541-8

Innovations in Spinal Deformities and Postural Disorders

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**7. Conclusions**

The physically-based EROSION-3D model was applied to simulate the runoff-erosion processes in the Slovak catchment. The model helps to identify the most sensitive erosion and deposition zones within a catchment. The long-term simulations were done for two periods (2015–2016 and 2016–2017), which were evaluated according to the time of the bathymetric measurements conducted. The results of the model pointed to an underestimation of the actual processes. The initial soil moisture plays an important role through all the input parameters of the EROSION-3D model. Finally, the rainfallerosion events were able to erode and produce a significant part of the sediments, especially during a drier year or season.

**Author Contributions:** Conceptualization, Z.N. and D.H.; methodology, Z.N., D.H., S.K. and K.H.; software, Z.N., V.S. and Y.V.; validation, Z.N. and D.H.; formal analysis, Z.N. and D.H.; investigation, Z.N., D.H., V.S. and Y.V.; writing—original draft preparation, Z.N., D.H and M.Š.M.; writing—review and editing, S.K., K.H. and M.Š.M.; visualization, D.H. All authors have read and agreed to the published version of the manuscript.

**Funding:** This project was funded by Contracts No. APVV-15-0497 and VEGA Grant Agency No. 1/0632/19. This work was supported by the Specific Research project at Masaryk University (MUNI/A/1356/2019), the Slovak Research and Development Agency under Contracts No. APVV-15-0497, APVV-18-0347 and APVV-18-0205, the VEGA Grant Agency No. 1/0632/19 and No. 2/0025/19. The authors thank the agency for its research support

**Conflicts of Interest:** The authors declare no conflict of interest.


**Abstract**

Due to the tension of urban uniformity and environmental setting, what emerges in most patterns of use in the city is lost space that usually ignored as insignificant. They emerge as if by accident during the design process, sometimes, the result of negligence or simply a feature of time flow in the city. To understand the phenomenon of lost space, this research seeks to identify the prescriptive and descriptive character of lost space from the local perception in Kuala Lumpur. This research also investigates the varied forces, circumstances, and the mechanisms that are responsible for their production. A qualitative analysis is made on selected parameters in the theory of lost space, activities, accessibility, connectivity, maintenance, and design. The result indicates that there were various new perceptions of descriptive lost space includes economic and social activity as these are an important strategy for maintaining the vitality and robustness of urban space. It is proposed in this paper the need of tools and effective ways of (re)designing a space to be applied in a new context and can act as a resource for urban life by hosting the intervention or alternative uses.

**Keywords:** lost space, urban fabrics, vitality, local perceptions