**1. Introduction**

Materials science, also known as materials science and engineering, is a multidisciplinary field that deals with the discovery and design of new materials [1]. Always new materials open the door to new technologies, whether they are in chemical, civil, construction, nuclear, agricultural, aeronautical, biomedical, electrical, or mechanical engineering [2, 3]. The materials science engineering includes the study of the relations between the synthesis, forming, properties, structure, and performance of materials that enable an engineering function. The materials properties of interest can be electrical, mechanical, optical or magnetic; the engineering function can affect industries involved in electronics, communications, transportation, manufacturing, medicine, recreation, environment, and energy [4].

Metallurgy is the art and science of making metals and alloys in shapes and with characteristics suitable for practical use [5]. It is an applied science based on a clear understanding of the structures and properties of metals and their alloys. Metallurgy has long occupied the dominant position as the most important engineering materials; steel being by far the most important over the last few centuries. However, increasingly in many areas other materials such as ceramics, plastics and composites are challenging this position. The relatively recent development of materials science and engineering is a merger of metallurgy with others like glass and ceramic technology, mineralogy, physical and inorganic chemistry, solid-state physics, and polymer science in modern techniques to include all structural and functional materials, thus making it one of the widest of study disciplines [1].

### **2. What is metallurgy?**

Metallurgy is the science and technology to extract metals from their ores economically, refining them and preparing them for the end use [6]. It studies the microstructure of a metal, the structural features that are control to observation under a microscope. Microstructure determines mechanical properties of the metal, including their elastic and plastic behavior when applying the force. Chemical composition is the relative content of a particular element within an alloy, usually expressed as a percent weight. Composition, as well as thermal and mechanical processing, will determine microstructure. Metals and their alloys are widely used in our daily lives. They are used for different purposes such as making machines,

bridges, motor vehicles, railways, buildings structure, ships, aircrafts, agricultural tools, etc. Therefore, real economic growth can come from increasing quality and quantity of the metal production in that country [7].

Naturally, most metals occur in the combined state as minerals and they are reactive. Only a few metals like gold, silver, platinum and mercury, etc. are found as Free State in the earth's crust. Metals that have a low reaction show little convergence to air, moisture, carbon dioxide, or non-metals found in nature [8]. Materials that occurring naturally in which a metal or its compound occurs is called a mineral. A mineral from which a metal can be economically extracted is called an ore. The main active components found in nature, especially in the atmosphere are oxygen and carbon dioxide. In the earth's crust, silicon and sulfur are present in large quantities. Seawater also contains large amounts of chloride ions (obtained from dissolved salts). Most active metals are high electrically positive and therefore exist as different ions [9]. For this reason, most of the important ores of these metals occur as different components such as oxides, silicates, carbonates, and halides.


**5. Acknowledgements**

This work was supported in part by the Japan Society for the Promotion of Science Grant 12995 (to K. S.), a fellowship of the Hayashi Memorial Foundation for Female Natural Scientists (to K. S.), a research grant from The Uehara Memorial Foundation (to H.O.), and Grants-in-aid for Scientific Research on Priority Areas 15040209, 17046004, and Research (C) on 17570109 (HO) and 22570111 (HO) from the Ministry of Education, Culture, Sports, Science, and Technology.

We thank K. Ono for editing the English.

#### **6. References**

SDS-PAGE, electrotransferred to a PVDF membrane, and detected with specific antibodies

Key results: The chromatofocusing and immunoblotting of plasma before and after sialidase treatment enabled us to demonstrate when and what alteration of sialylation occurs in each glycoprotein at each different stage during liver regeneration. The changes in pI essentially coincided with that of 2D-PAGE; however, we can determine the pI of a sample under physiological condition by using the chromatofocusing technique. Furthermore, this method is facile, quick, and applicable to recovered samples for activity analyses because they are

This study proposes that alterations of glycosylation, especially decreased sialylation of vitronectin, modulate tissue remodeling processes in multiple steps, especially HSC spreading and survival. Our findings suggest that the removal of sialic acid from vitronectin suppressed activation of stellate cells, indicating the possibility of a new treatment for or method to prevent liver cirrhosis. This will open new windows to the paradigm of glycoregenerative medicine, which is based on the modulatory functions of glycans of ECM

This work was supported in part by the Japan Society for the Promotion of Science Grant 12995 (to K. S.), a fellowship of the Hayashi Memorial Foundation for Female Natural Scientists (to K. S.), a research grant from The Uehara Memorial Foundation (to H.O.), and Grants-in-aid for Scientific Research on Priority Areas 15040209, 17046004, and Research (C) on 17570109 (HO) and 22570111 (HO) from the Ministry of Education, Culture, Sports,

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**5. Acknowledgements**

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We thank K. Ono for editing the English.

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chromatofocusing performance separating protiens on a high-performance DEAE

**6**

*México*

**The Protective Effect of Antioxidants**

Carmen Valadez-Vega1, Eduardo Madrigal-Santillán1, Jaime Esquivel-Soto2,

The term antioxidant was originally utilized to refer specifically to a chemical product that prevented the consumption of oxygen (Burneo, 2009); thus, antioxidants are defined as molecules whose function is to delay or prevent the oxygenation of other molecules. The importance of antioxidants lies in their mission to end oxidation reactions that are found in the process and to impede their generating new oxidation reactions on acting in a type of sacrifice on oxidating themselves. There are endogenous and exogenous antioxidants in nature. Some of the best-known exogenous antioxidant substances are the following: β‐ carotene (pro‐vitamin A); retinol (vitamin (A); ascorbic acid (vitamin C); α‐tocopherol (vitamin E); oligoelements such as selenium; amino acids such as glycine, and flavonoids

Historically, it is known that the first investigations on the role that antioxidants play in Biology were centered on their intervention in preventing the oxidation of unsaturated fats, which is the main cause of rancidity in food (Wolf, 2005). However, it was the identification of vitamins A, C, and E as antioxidant substances that revolutionized the study area of antioxidants and that led to elucidating the importance of these substances in the defense

Due to their solubilizing nature, antioxidant compounds have been divided into hydrophilics (phenolic compounds and vitamin C) and lipophilics (carotenoids and vitamin E). The antioxidant capacity of phenolic compounds is due principally to their redox properties, which allow them to act as reducing agents, hydrogen and electron donors, and individual oxygen inhibitors, while vitamin C's antioxidant action is due to its possessing two free electrons that can be taken up by Free radicals (FR), as well as by other Reactive oxygen species (ROS), which lack an electron in their molecular structure. Carotenoids are deactivators of electronically excited sensitizing molecules, which are involved in the generation of radicals and individual oxygen, and the antioxidant activity of vitamin A is

characterized by hydrogen donation, avoiding chain reactions (Burneo, 2009).

such as silymarin, among other organic compounds (Venereo, 2002).

system of live organisms (Jacob, 1996).

**1. Introduction**

José A. Morales González1 , Liliana Barajas-Esparza1,

Cesar Esquivel-Chirino2, Ana María Téllez-López1, Maricela López-Orozco1 and Clara Zúñiga-Pérez1

**in Alcohol Liver Damage**

*1Instituto de Ciencias de la Salud, UAEH,*

*2Facultad de Odontología, UNAM*

Yoneda, A., Ogawa, H., Matsumoto, I., Ishizuka, I., Hase, S. & Seno, N. (1993) Structures of the *N*-linked oligosaccharides on porcine plasma vitronectin. *Eur. J. Biochem.*, 218, 797-806.


4. Multidimensional risk model of complex systems

Consider the task of ensuring the system sustainable functioning. Here you first need to concretize the "sustainability" concept. Typically the stability of the system functioning is interpreted in terms of its safety. Security issues are resolved with the help of risk analysis [25]. Some authors note that growth rates of damage considerably exceed growth rates of the economy [26, 27]. This can be explained with a constant increase of risk in the conditions of a scientific and technical revolution and the forced development of a technosphere [28]. Therefore, we will assume that the functioning stability of the territorial system is intimately connected with risk; the lower the risk level, the more stable the system state. Thus, the diagnosis of system sustainability can be made on the basis of monitoring its risk. This requires adequate risk models.

Let S be some multidimensional stochastic system. Let us consider an adequate representation of this system as a random vector X = (X1, X2, ..., Xl) with a certain probability density pX(x). The development of a complex system and an increase in the efficiency of its functioning are an inevitable cause of increasing risks. Therefore, it is necessary to assess the risks of such systems. Consider the risk model of multidimensional stochastic systems proposed in [29].

Instead of the generally conventional selection of concrete dangerous situations, we will define the geometric area D of adverse outcomes. Formally this area can look arbitrarily depending on a specific objective.

The concept of dangerous states as larger and improbable deviations of a conception of dangerous conditions as large and improbable deviations of random variables from some best provision Θ is mostly distributed. In this case, D represents an external area of an m-axis ellipsoid.

Setting the function of consequences from dangerous situations (risk function) in the form of g(x), we will receive a model for the quantitative assessment of risk [30]:

$$r(X) = \int\int\_{\mathbb{R}^m} \dots \int\_{\mathbb{R}} g(\boldsymbol{\varkappa}) p\_X(\boldsymbol{\varkappa}) d\boldsymbol{\varkappa}.\tag{9}$$

If in Eq. (9) to accept g xð Þ¼ 1 ∀x∈ D and g xð Þ¼ 0 ∀x ∉ D, that r Xð Þ¼ P Xð Þ ∈ D , that is, the risk is estimated as a probability of an unfavorable outcome.

If at an early stage of system analysis is difficult to describe enough precisely the g(x) function, then Eq. (2) becomes an assessment of P(D) and is a convenient initial approximation of risk model.

To define a function g(x) requires a quantitative assessment of consequences for the studied system depending on values of risk factors. It demands to carry out separate research. Let us note that values of the function g(x) are given in the nominal units. But they are usually quite simply interpreted in the respective subject area. The essence of the function g(x) is as follows. It accepts the least nonnegative (e.g., zero) value in a point of Θ or in its neighborhood of U(Θ). Further in each direction during removal from U(Θ) the g(x) function has to increase monotonously. For scaling on each risk factor, we will set some limit values, at which consequences become dangerous (or irreversible). Let us set values g(x) at each

Seventeen coal mining enterprises were investigated [23]. On the basis of two generalized factors (Y1 is the factor characterizing the organization of safe production; Y2 is the factor reflecting the professionalism of the staff), all enterprises were divided into two groups: (1) enterprises with a low level of injury; (2) enterprises with a high level of injury. For the first and second groups of mines,

¼ ð Þ 2, 42; �0, 31 , H Yð Þ1

¼ ð Þ 3, 74; �0, 70 , H Yð Þ2

In this case, the direction of the entropy change vector will differ from Example 1: at the enterprises with a high level of injury, the randomness entropy needs to be

For example, this can be accomplished as follows: to bring the state of the second group of mines to the state of the first group, it is necessary to reduce the dispersion of the factor characterizing the organization of safe production and reduce interrelation with the factor reflecting the professionalism of the staff. This means a more specific and accurate organization of production. The organization of safe production should not depend on the degree of professionalism and competence of staff

Example 3. We investigate the possibilities of entropy modeling on the example of the population analysis in terms of prevention of chronic noninfectious diseases (CNID) by biological risk factors [23]. For carrying out the analysis of change of population entropy depending on the health status, two equal age groups were formed: 18–26 years and 27–35 years. Four risk factors were identified: "total cholesterol," "systolic blood pressure," "body mass index," and "glucose level." The

As the health of the population deteriorates, the total population entropy and the randomness entropy increase. This can be explained by the fact that the additional damaging influence of CNID, in general, is added to the pathological influence of

> Self-organization entropy H(Y)R

7.131 �0.578 6.553

8.376 �0.542 7.834

Total entropy H(Y)

Conversely, the self-organization entropy as the deterioration of the health status of the population decreases which corresponds to the strengthening of the narrowness of the interrelations between subsystems. This can be explained by the fact that the development of diseases in the organism happens in many respects and it is interdependent. On the other hand, at the development of diseases, some

reduced, and the self-organization entropy needs to be increased.

risk factors on a human body separately and on all population.

Health status Randomness entropy

H(Y)V

18–26 Healthy 5.500 �0.514 4.986

27–35 Healthy 5.731 �0.299 5.432

Patient 7.847 �0.696 7.151

Patient 8.720 �0.781 7.939

¼ 2, 11,

¼ 3, 04:

respectively, we have.

strongly.

Age (years)

Table 1.

28

hð Þ1 V ;hð Þ1 R

Sustainability Assessment at the 21st Century

hð Þ2 V ;hð Þ2 R

results of the analysis are given in Table 1.

Apparently healthy

Apparently healthy

Entropy levels in different groups of people.

such point equal to some value, for example 1. For convenience, it is desirable to impose a number of restrictions on the function g(x): convexity, continuity, etc.

In [30] the variant of the task of the g(x) function in the form of a paraboloid is given. By way of illustration in Figure 4, the example of the risk function for a case m ¼ 2 is shown. The ellipse describing the area D of admissible values of risk factors and lying on the Ox1x2 (r = 0) plane is shown by black color. The paraboloid above the plane represents possible values of risk r(X). White points on the plane are values of risk factors; to them there correspond points on paraboloid surface which set risk values; the image of the border of an ellipse D is shown in the form of the black line. All corresponding couples of points (values of risk factors and risk values) are connected among themselves by vertical dashed lines.

In the problems of risk monitoring, along with risk assessment, r Xð Þ on all risk factors of X1, X2 ..., Xm of the multidimensional system is expedient to estimate the contribution of each factor to total risk. We introduce a random vector X� k ¼ X1, …, Xk�1,Xkþ1 ð Þ , …, Xm . Then the absolute change of risk of the multidimensional system due to the addition of factor Xk is equal:

$$
\Delta r(X\_k) = r(X) - r\left(X\_k^{-}\right). \tag{10}
$$

pXð Þ¼ x

DOI: http://dx.doi.org/10.5772/intechopen.89287

by setting the g(x) function accordingly.

is a coefficient of correlation between X1 and X2.

we will accept A1 ¼ A2 ¼ … ¼ Am ¼ A, Aj ¼ bj=σ j.

sionality factor and narrowness of correlations.

levels are given in Table 2.

Realization of a standard normal random vector.

Figure 5.

31

for monitoring of risk of Sverdlovsk region in 1999–2017.

ance matrix, σii ¼ σ2

1 � j j RX

1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ 2π mj j Σ p exp � 1

where a ¼ ð Þ a1, a2, ::, am T—a vector of expectations, Σ ¼ σij � �

Probabilistic-Entropic Concept of Sustainable Development of the Example of Territories

i —dispersion of factor Xi.

2 ð Þ x � a

The use of a Gaussian random vector is based on the central limit theorem [31]. As approbation on a number of examples has shown, such idealization is not so critical, and if there are any bases to consider that density of probabilities is a component of the vector of X having more extended tails, then this can be corrected

Example 4. Let us consider a two-dimensional Gaussian random vector which components have a zero average and single dispersion. In Figure 5 the example of realization of such accidental vector is shown for: (a) ρ ¼ 0, 4; (b) ρ ¼ 0, 9, where ρ

From Figure 5 we see that the probability of large deviations of the random vector from the origin increases with the increase in the closeness of the correlation. Example 5. Let us estimate the probability of P(D) for a Gaussian random vector

1=m [32], where j j RX is a continuant of a complete correlation matrix (Deð Þ¼ X 0 is independence of components, and Deð Þ¼ X 1 is a rigorous linear relation). Let us consider the following cases: Deð Þ¼ X 0, Deð Þ¼ X 0, 5, Deð Þ¼ X 1. The results of the calculation of P(D) are given in Figure 3. For descriptive reasons

The increase in the probability of an unfavorable outcome is influenced by both the increase in the dimension of m and an increase in the narrowness of correlation communication between the components of a random vector of X. Let us note that even the average narrowness of correlation (Deð Þ¼ X 0, 5) leads to a significant increase in the probability of an unfavorable outcome. The effect increases with the increase in values Aj that correspond to less probable, but more dangerous, adverse outcomes. Therefore, risk modeling should take into account both a multidimen-

Example 6. Approbation of risk model of a multidimensional stochastic system

Let us execute monitoring of Sverdlovsk region on the dynamics of macroeconomic risk factors, taken as an interval of 9 years. Risk factors and their threshold

of X, with the different narrowness of correlation communication Deð Þ¼ X

The analysis of schedules in Figure 6 indicates the following.

TΣ�1

� �,

ð Þ x � a

m�m—a covari-

Dividing Δr Xð Þk of the risk r X� k , we will receive the relative change of risk of the multidimensional system by the addition of factor Xk:

$$
\delta r(X\_k) = \Delta r(X\_k) / r(X\_k^-). \tag{11}
$$

Let us note that along with a contribution to the common risk of one factor, Eqs. (10) and (11) allow us to estimate influence and groups of factors.

Monitoring risk on the basis of the model in Eqs. (9)–(11) consists of serial estimation in time of the actual values of r(X), Δr Xð Þk , δr Xð Þk ,j ¼ 1, 2, …, m, and also dynamics of their change.

Let us consider the most common case when X has a joint normal distribution with a probability density:

Figure 4. An example of a two-dimensional risk functions.

Probabilistic-Entropic Concept of Sustainable Development of the Example of Territories DOI: http://dx.doi.org/10.5772/intechopen.89287

$$p\_X(\mathbf{x}) = \frac{1}{\sqrt{\left(2\pi\right)^m |\Sigma|}} \exp\left\{-\frac{1}{2} (\mathbf{x} - a)^T \Sigma^{-1} (\mathbf{x} - a)\right\},$$

where a ¼ ð Þ a1, a2, ::, am T—a vector of expectations, Σ ¼ σij � � m�m—a covariance matrix, σii ¼ σ2 i —dispersion of factor Xi.

The use of a Gaussian random vector is based on the central limit theorem [31]. As approbation on a number of examples has shown, such idealization is not so critical, and if there are any bases to consider that density of probabilities is a component of the vector of X having more extended tails, then this can be corrected by setting the g(x) function accordingly.

Example 4. Let us consider a two-dimensional Gaussian random vector which components have a zero average and single dispersion. In Figure 5 the example of realization of such accidental vector is shown for: (a) ρ ¼ 0, 4; (b) ρ ¼ 0, 9, where ρ is a coefficient of correlation between X1 and X2.

From Figure 5 we see that the probability of large deviations of the random vector from the origin increases with the increase in the closeness of the correlation.

Example 5. Let us estimate the probability of P(D) for a Gaussian random vector of X, with the different narrowness of correlation communication Deð Þ¼ X 1 � j j RX 1=m [32], where j j RX is a continuant of a complete correlation matrix (Deð Þ¼ X 0 is independence of components, and Deð Þ¼ X 1 is a rigorous linear relation). Let us consider the following cases: Deð Þ¼ X 0, Deð Þ¼ X 0, 5, Deð Þ¼ X 1. The results of the calculation of P(D) are given in Figure 3. For descriptive reasons we will accept A1 ¼ A2 ¼ … ¼ Am ¼ A, Aj ¼ bj=σ j.

The analysis of schedules in Figure 6 indicates the following.

The increase in the probability of an unfavorable outcome is influenced by both the increase in the dimension of m and an increase in the narrowness of correlation communication between the components of a random vector of X. Let us note that even the average narrowness of correlation (Deð Þ¼ X 0, 5) leads to a significant increase in the probability of an unfavorable outcome. The effect increases with the increase in values Aj that correspond to less probable, but more dangerous, adverse outcomes. Therefore, risk modeling should take into account both a multidimensionality factor and narrowness of correlations.

Example 6. Approbation of risk model of a multidimensional stochastic system for monitoring of risk of Sverdlovsk region in 1999–2017.

Let us execute monitoring of Sverdlovsk region on the dynamics of macroeconomic risk factors, taken as an interval of 9 years. Risk factors and their threshold levels are given in Table 2.

Figure 5. Realization of a standard normal random vector.

such point equal to some value, for example 1. For convenience, it is desirable to impose a number of restrictions on the function g(x): convexity, continuity, etc. In [30] the variant of the task of the g(x) function in the form of a paraboloid is given. By way of illustration in Figure 4, the example of the risk function for a case m ¼ 2 is shown. The ellipse describing the area D of admissible values of risk factors and lying on the Ox1x2 (r = 0) plane is shown by black color. The paraboloid above the plane represents possible values of risk r(X). White points on the plane are values of risk factors; to them there correspond points on paraboloid surface which set risk values; the image of the border of an ellipse D is shown in the form of the black line. All corresponding couples of points (values of risk factors and risk

In the problems of risk monitoring, along with risk assessment, r Xð Þ on all risk factors of X1, X2 ..., Xm of the multidimensional system is expedient to estimate the contribution of each factor to total risk. We introduce a random vector

Δr Xð Þ¼ k r Xð Þ� r X�

δr Xð Þ¼ k Δr Xð Þk =r X�

Let us note that along with a contribution to the common risk of one factor,

Monitoring risk on the basis of the model in Eqs. (9)–(11) consists of serial estimation in time of the actual values of r(X), Δr Xð Þk , δr Xð Þk ,j ¼ 1, 2, …, m, and

Let us consider the most common case when X has a joint normal distribution

k

k

, we will receive the relative change of risk of

: (10)

: (11)

values) are connected among themselves by vertical dashed lines.

multidimensional system due to the addition of factor Xk is equal:

k

Eqs. (10) and (11) allow us to estimate influence and groups of factors.

the multidimensional system by the addition of factor Xk:

Dividing Δr Xð Þk of the risk r X�

Sustainability Assessment at the 21st Century

also dynamics of their change.

with a probability density:

Figure 4.

30

An example of a two-dimensional risk functions.

k ¼ X1, …, Xk�1,Xkþ1 ð Þ , …, Xm . Then the absolute change of risk of the

X�

Figure 6.

Dependences of lgP(D) on threshold level A: (a) De(X) = 0; (b) De(X) = 0.5; (c) De(X) = 1. Designations: Row 1 (m = 1), row 2 (m = 2), row 3 (m = 3), row 4 (m = 4), row 5 (m = 5).

#### Table 2.

Macroeconomic risk factors of the region.

We consider that random vector X has a joint normal distribution.

In Figures 7 and 8, results of the calculation of the probability of unfavorable outcome P(D) and risk r(X) for the threshold levels of risk factors K are shown.

Analysis of the results of monitoring of multidimensional risk in the Sverdlovsk region showed the following:

5. Formation of the sustainable development concept

as sustainable development of the complex system.

Figure 7.

Figure 8.

33

Estimates of P(D) in the Sverdlovsk region.

DOI: http://dx.doi.org/10.5772/intechopen.89287

Estimates of r(X) in the Sverdlovsk region.

sustainable development, they need to be taken into account together.

indicators Yj, characterizing the functioning of the system, that is,

Consideration of examples shows that "development" and "sustainability" characterize various aspects of the complex systems operation. And for ensuring

Probabilistic-Entropic Concept of Sustainable Development of the Example of Territories

Hypothesis 2. We will understand dynamics consisting of available trends of the balanced change of a vector entropy while maintaining an acceptable level of risks

For this purpose, we combine the vector entropy model and the risk model of a multidimensional stochastic system. Moreover, it is necessary to consider elements of the system (components of the random vector Z), both as risk factors Xi and as

Z ¼ X∪Y ¼ Z1 ð Þ , Z2, …, Zn , max ð Þ l, m ≤n≤ l þ m:

Probabilistic-Entropic Concept of Sustainable Development of the Example of Territories DOI: http://dx.doi.org/10.5772/intechopen.89287

#### Figure 7. Estimates of P(D) in the Sverdlovsk region.

Figure 8. Estimates of r(X) in the Sverdlovsk region.

## 5. Formation of the sustainable development concept

Consideration of examples shows that "development" and "sustainability" characterize various aspects of the complex systems operation. And for ensuring sustainable development, they need to be taken into account together.

Hypothesis 2. We will understand dynamics consisting of available trends of the balanced change of a vector entropy while maintaining an acceptable level of risks as sustainable development of the complex system.

For this purpose, we combine the vector entropy model and the risk model of a multidimensional stochastic system. Moreover, it is necessary to consider elements of the system (components of the random vector Z), both as risk factors Xi and as indicators Yj, characterizing the functioning of the system, that is,

$$\mathbf{Z} = \mathbf{X} \cup \mathbf{Y} = (Z\_1, Z\_2, \dots, Z\_n), \text{ max } (l, m) \le n \le l + m.$$

We consider that random vector X has a joint normal distribution.

dangerous situations has been growing slightly since 2014.

3. In the Sverdlovsk region until 2010, the main contribution to regional instability was made by factor X6, then X2 became such a factor, and since

2015, the main contribution to instability was made by factor X7.

region showed the following:

Macroeconomic risk factors of the region.

Figure 6.

population

rubles

Table 2.

32

region have increased (decreased risks).

In Figures 7 and 8, results of the calculation of the probability of unfavorable outcome P(D) and risk r(X) for the threshold levels of risk factors K are shown. Analysis of the results of monitoring of multidimensional risk in the Sverdlovsk

Dependences of lgP(D) on threshold level A: (a) De(X) = 0; (b) De(X) = 0.5; (c) De(X) = 1. Designations: Row

Risk factors Threshold levels

X1—real income movement, in % to the previous year 79.93 X2—the ratio of the average size of pension to subsistence minimum of pensioners 0.66 X3—morbidity on 1000 people of the population 960

X5—wear of fixed assets on the end of the year, % 71.33

X7—quantum index of gross regional product, % to the previous year 88.4 X8—unemployment rate, in % 18

X4—mortality from external causes, number of the dead on 100,000 people of the

X6—the volume of budget revenues per capita, in the prices of 2017, thousand

Kj

322.1

21.75

1 (m = 1), row 2 (m = 2), row 3 (m = 3), row 4 (m = 4), row 5 (m = 5).

Sustainability Assessment at the 21st Century

1.During the initial period, the greatest socioeconomic instability (the highest risk values) was observed. Then gradually the dynamics of sustainability in the

2.After the sanctions were imposed, the risk began to increase. The lower rate of growth of r(X) than P(D) indicates that the probability of occurrence of very

The case n <l þ m appears, when X∩Y 6¼ ∅.

Within the framework of the proposed concept, along with the tasks of monitoring complex systems discussed above, it is possible to solve management problems (development of control recommendations).

6. Discussion of results

DOI: http://dx.doi.org/10.5772/intechopen.89287

stability is provided due to an acceptable risk level.

able development of Sverdlovsk region.

multidimensional risk models.

application to territorial systems.

real data.

7. Conclusion

of risk.

Acknowledgements

35

Thus, on the basis of the use of two original models—a vector entropy and multidimensional risk—it was succeeded to formalize the new concept of sustainable development of complex systems. Both models are successfully approved on

Probabilistic-Entropic Concept of Sustainable Development of the Example of Territories

This concept can be implemented by means of monitoring of the studied system. As observed parameters efficiency factors of the functioning of a system and its risk factors are used. The direction of development is given by an entropy vector, and

Management recommendations are formed in the form of the solution to an extreme problem Eq. (12). This problem is solved by methods of penalty functions. Currently, the work is at a stage of practical approbation of monitoring of sustain-

1.The probability-entropy concept of sustainable development of complex stochastic systems is formulated. It is based on vector entropy and

2.According to the formulated concept, the sustainable development of a

3.The proposed concept of sustainable development has been tried out in

The present study was carried out by financial support of the Russian Fund of Fundamental Research grant № 17-01-00315 "Development of models and methods of monitoring, forecasting, and control optimization of multidimensional regional

socioeconomic systems based on entropy and minimax approaches."

complex system will be understood as the dynamics consisting of the tendency of a balanced change in vector entropy while maintaining an acceptable level

The idea of vector entropy control is the transfer of the vector h(Z) from the state h Z0 � � ¼ h0 V; h0 R � � in the state h Zð Þ¼ \* ð Þ hV\*; hR\* , which corresponds to the effective functioning of the stochastic system.

For a Gaussian system, the vector entropy control consists in directing the entropy from some initial point h0 V; h0 R � � ¼ H Z0 � � V; H Z0 � � R � � with the covariance matrix Σ0 to the final point ð Þ hV\*; hR\* with a minimal change of the covariance matrixΣ0 ¼ σ0 ij n o and the expectations vector a0 and acceptable risk (Figure 9).

The problem of the vector entropy control of the Gaussian system to ensure sustainable development will take the form:

$$\begin{cases} \begin{aligned} \mathbf{G}(\boldsymbol{\Sigma}) &= \sum\_{i=1}^{n} \sum\_{j=1}^{n} \left( \sigma\_{\bar{\mathcal{y}}} - \sigma\_{\bar{\mathcal{y}}}^{0} \right)^{2} + \sum\_{i=1}^{j} \left( a\_{i} - a\_{i}^{0} \right)^{2} \to \min\_{a, \boldsymbol{\Sigma}}, \\ \boldsymbol{H}(\boldsymbol{Y})\_{\boldsymbol{V}} &= \boldsymbol{A}, \\ \boldsymbol{H}(\boldsymbol{Y})\_{\boldsymbol{R}} &= \boldsymbol{B}, \\ \boldsymbol{r}(\boldsymbol{X}) &\leq \boldsymbol{r}, \\ \boldsymbol{\Sigma} &\in \mathcal{G}\_{\boldsymbol{\Sigma}}, \quad a \in \mathcal{H}\_{a}, \\ \sigma\_{\bar{\mathcal{y}}}^{2} &< \sigma\_{\bar{\mathcal{u}}} \sigma\_{\bar{\mathcal{y}}}, \ \sigma\_{\bar{\mathcal{y}}} = \sigma\_{\bar{\mathcal{y}}}, \ \sigma\_{\bar{\mathcal{u}}} > 0 \ \ \ \mathbf{1} \leq i, j \leq n, \\ \boldsymbol{\Sigma} &> 0, \end{aligned} \tag{12}$$

where A ¼ hV\*, B ¼ hR\*, a—the expectations vector of the components Xi, i = 1, 2, ..., l.

The last constraint in Eq. (12) means positive definiteness of the matrix Σ. Note that the performance criterion in Eq. (12) may be different, depending upon the characteristics of a particular system S.

Probabilistic-Entropic Concept of Sustainable Development of the Example of Territories DOI: http://dx.doi.org/10.5772/intechopen.89287


**2. Experimental apparatus and data reduction**

#### **2.1. Experimental model**

The schematic diagram of experimental apparatus is depicted in Fig. 1. The facilities were built by Choi et al. [6]. Looking inside, the model mainly comprises the condensing unit, sub-cooler, receiver, pump, mass flow meter, preheater, and test section. The closed refrigeration cycle starts from the receiver. The liquid refrigerant was, at first, pumped by a magnetic gear pump. Then, it was passed through a Coriolis mass flow meter. Before entering the test section, the mass quality of the refrigerant was adjusted by the preheater. At the test section, it was evaporated by applying heat from an AC transformer. The vapor refrigerant at the outlet of the test section was condensed by the condenser and then the liquid refrigerant comes back to the receiver. In this model, the flow rate of the refrigerant can be acquired by controlling the frequency of the gear pump. The photograph of the apparatus is presented in Fig. 2.

The test section was made of stainless steel smooth tube with inner diameters of 1.5mm and 3.0 mm and heated lengths of 1000mm and 2000 mm, respectively. The temperatures of the outside surface of the test section were measured at the top, on both sides, and at the bottom each at 100mm axial intervals from the inlet to the outlet. In this model, Choi et al. [6] used the T-type thermocouples with the outside diameter of 0.1mm. To measure the local pressure in the test section, two pressure gauges were set up at the inlet and outlet. Another differential pressure was used to estimate the exact pressure drop during the evaporation in the test tube. Two sight glasses with the same inner diameter as the test section were installed to visualize Pressure Drop and Heat Transfer during a Two-phase Flow Vaporization of Propane… http://dx.doi.org/10.5772/60813 105

**Figure 1.** The schematic of apparatus

amount of refrigerant charges to as minimal as possible. And one of the best solutions at present is the decrease in the size of the heat exchanger. Compact heat exchangers using minichannels and microchannels have more efficient heat transfer performance than conventional types, due to their higher heat transfer area per refrigerant volume. Given that, the systems could reduce the refrigerant changers but still keep the coefficient of performance. However, since various researches observed the many differences of heat transfer and pressure drop of refrigerants in the mini/microchannels and conventional channels [3, 4]. The studies on these characteristics of propane in mini/microchannels and valuable information will be provided for researchers

In fact, in the past few decades, numerous interesting researches were published in literature such as the studies of [5-9]. Nevertheless, the following chapter does not attempt to review all the available literature but those most focused on the particular heat exchanger design only or the consideration on the heat transfer and pressure drop separately. The intention is to rather present a basis on heat transfer and pressure drop of propane in minichannels, simultaneously under the variation of mass fluxes, heat fluxes, saturated temperature, and tube diameter. The phenomenon will be explained thoroughly by its mechanism that shows some differences observed in the conventional channel. Furthermore, the development of heat transfer coeffi‐ cient and pressure drop correlations are demonstrated. The content is believed to bring up the general understanding, as well as useful information on heat exchanger designs to the readers.

The schematic diagram of experimental apparatus is depicted in Fig. 1. The facilities were built by Choi et al. [6]. Looking inside, the model mainly comprises the condensing unit, sub-cooler, receiver, pump, mass flow meter, preheater, and test section. The closed refrigeration cycle starts from the receiver. The liquid refrigerant was, at first, pumped by a magnetic gear pump. Then, it was passed through a Coriolis mass flow meter. Before entering the test section, the mass quality of the refrigerant was adjusted by the preheater. At the test section, it was evaporated by applying heat from an AC transformer. The vapor refrigerant at the outlet of the test section was condensed by the condenser and then the liquid refrigerant comes back to the receiver. In this model, the flow rate of the refrigerant can be acquired by controlling the

frequency of the gear pump. The photograph of the apparatus is presented in Fig. 2.

The test section was made of stainless steel smooth tube with inner diameters of 1.5mm and 3.0 mm and heated lengths of 1000mm and 2000 mm, respectively. The temperatures of the outside surface of the test section were measured at the top, on both sides, and at the bottom each at 100mm axial intervals from the inlet to the outlet. In this model, Choi et al. [6] used the T-type thermocouples with the outside diameter of 0.1mm. To measure the local pressure in the test section, two pressure gauges were set up at the inlet and outlet. Another differential pressure was used to estimate the exact pressure drop during the evaporation in the test tube. Two sight glasses with the same inner diameter as the test section were installed to visualize

**2. Experimental apparatus and data reduction**

in related fields.

104 Heat Transfer Studies and Applications

**2.1. Experimental model**

**Figure 2.** The Photograph of apparatus

the flow. After all, the test sections and the other components were well isolated by rubber and foams to reduce the effect of the environment. The details of the test section are depicted in Fig. 3.

One of the most important parts in this testing model is to test the heat balance on a test tube. The total electric power was calculated by the root mean square values of electric voltage and current. The heat balance procedure was taken with each tube. The results were then corrected by using logarithmic regression method. The final deviation of electric power was within 5%. The estimated uncertainties of other parameters were also evaluated with the confidence of 95%. The summary is shown in table 1. The testing conditions in study of Choi et al. [6] were also illustrated in table 2. All the properties of refrigerant were queried from REFPROP. 8.

#### **2.2. Data reduction**

#### *2.2.1. Pressure drop*

The total pressure drop of saturated refrigerants comes from the variation of kinetic energy, potential energy, and the friction of the two-phase flow on the wall. In horizontal tubes, the potential energy can be neglected, thus, the total pressure drop is the sum of the momentum pressure drop and the frictional pressure drop. Assuming that the pressure drop linearly increases with the tube length, the saturation pressure at the starting point of saturation regime is interpolated from the measured pressure and the calculated subcooled length. The experi‐ mental two phase frictional pressure drop can be determined as follows:

$$\left(-\frac{dp}{dz}F\right) = \left(-\frac{dp}{dz}\right) - \left(-\frac{dp}{dz}a\right) \tag{1}$$

The refrigerant is evaporated from liquid at the saturation temperature to vapor-liquid mixture at mass quality x with a linear change of test distance over the tube length. Hence, the momentum pressure drop can be calculated with following equation:

$$\left(-\frac{dp}{dz}a\right) = G^2 \frac{d}{dz} \left(\frac{x^2}{a\rho\_g} + \frac{(1-x)^2}{(1-a)\rho\_f}\right) \tag{2}$$

Choi et al. [6] compared various void fraction models including the ones proposed by Steiner [10], CISE [11], and Chisholm [12] with the homogeneous model and their experimental data. The correlation proposed by CISE shows the best agreement with the homogeneous model, while Steiner's correlation shows the best prediction for the experimental data as shown in Fig. 4. Therefore, the void fraction in their study is obtained from the Steiner [10] void fraction as follows:

$$\alpha = \frac{\mathbf{x}}{\rho\_s} \left[ (\mathbf{l} + 0.12(\mathbf{l} - \mathbf{x})) \left( \frac{\mathbf{x}}{\rho\_s} + \frac{\mathbf{l} - \mathbf{x}}{\rho\_f} \right) + \frac{1.18(\mathbf{l} - \mathbf{x}) \left[ \mathbf{g} \sigma (\rho\_f - \rho\_s) \right]^{0.25}}{G \rho\_f^{0.5}} \right]^{-1} \tag{3}$$

The friction factor was determined from the measured pressure drop for a given mass flux by using the Fanning equation,

$$f\_{sp} = \frac{D\_{\overline{\rho}}}{2G^2} \left( -\frac{dp}{dz} F \right) \tag{4}$$

where the average density is calculated with the following equation:

Pressure Drop and Heat Transfer during a Two-phase Flow Vaporization of Propane… http://dx.doi.org/10.5772/60813 107

$$
\overline{\rho} = a \rho\_{\varrho} + (1 - a)\rho\_{f} \tag{5}
$$

The two phase frictional multiplier, *ϕfo* 2 , is determined by dividing the calculated frictional two-phase pressure drop to the pressure drop of the liquid flow assuming the total flow to be liquid. The equation is defined as:

$$
\phi\_{\rho}^{2} = \left(-\frac{dp}{dz}F\right)\_{\rho} / \left(-\frac{dp}{dz}F\right)\_{\rho} = \left(-\frac{dp}{dz}F\right)\_{\rho} / \left(\frac{2f\_{\rho}G^{2}}{D\rho\_{f}}\right) \tag{6}
$$

**Figure 3.** The schematic of test sections

**2.2. Data reduction**

106 Heat Transfer Studies and Applications

*2.2.1. Pressure drop*

as follows:

a

r

using the Fanning equation,

The total pressure drop of saturated refrigerants comes from the variation of kinetic energy, potential energy, and the friction of the two-phase flow on the wall. In horizontal tubes, the potential energy can be neglected, thus, the total pressure drop is the sum of the momentum pressure drop and the frictional pressure drop. Assuming that the pressure drop linearly increases with the tube length, the saturation pressure at the starting point of saturation regime is interpolated from the measured pressure and the calculated subcooled length. The experi‐

mental two phase frictional pressure drop can be determined as follows:

momentum pressure drop can be calculated with following equation:

æ öæ öæ ö - =- -- ç ÷ç ÷ç ÷ è øè øè ø *dp dp dp F a*

æ ö æ ö - -= + ç ÷ ç ÷ è ø - è ø *g f dp d x x a G*

The refrigerant is evaporated from liquid at the saturation temperature to vapor-liquid mixture at mass quality x with a linear change of test distance over the tube length. Hence, the

Choi et al. [6] compared various void fraction models including the ones proposed by Steiner [10], CISE [11], and Chisholm [12] with the homogeneous model and their experimental data. The correlation proposed by CISE shows the best agreement with the homogeneous model, while Steiner's correlation shows the best prediction for the experimental data as shown in Fig. 4. Therefore, the void fraction in their study is obtained from the Steiner [10] void fraction

1 1.18(1 ) ( ) (1 0.12(1 ))

é ù æ ö - - - é ù ë û =+- ++ ê ú ç ÷ è ø ë û

The friction factor was determined from the measured pressure drop for a given mass flux by

 r

2 2 r æ ö = -ç ÷ è ø *tp*

*g g f f*

r

where the average density is calculated with the following equation:

2 2 2 (1 ) ar

 a r(1 )

*dz dz dz* (1)

*dz dz* (2)

0.5

 r

*f g*

*D dp f F G dz* (4)

sr

r

*x x x x g x G* (3)

1 0.25

**Table 1.** Experimental Conditions

**Table 2.** Summary of the estimated uncertainty

**Figure 4.** Void fraction comparison between the homogenous model and the existing correlations

#### *2.2.2. Heat transfer coefficient*

Choi et al. [6] determined the inside tube wall temperature, Twi, by the average inside temper‐ ature of the top, both right and left sides, and bottom wall temperatures. As demonstrated in the apparatus module, the heat was applied on test tubes through uniform current, and the local temperature was evaluated at the cross section where there is an attached the thermo‐ couple, hence, the inside wall temperatures at each point could be determined by using steady state one-dimensional radial conduction heat transfer through the wall with internal heat generation. The quality, x, at the measurement location, z, was determined based on the thermodynamic properties.

Pressure Drop and Heat Transfer during a Two-phase Flow Vaporization of Propane… http://dx.doi.org/10.5772/60813 109

$$\mathbf{x} = \frac{\mathbf{i} - \mathbf{i}\_g}{\mathbf{i}\_{\otimes}} \tag{7}$$

In fact, the refrigerant flow was not completely saturated at the inlet of the test section. The flow becomes totally saturated after a short initial length of tube when the heat, which is called a subcooled length, was applied. Therefore, to ensure the accuracy of data reduction, the subcooled length is calculated using the following equation:

$$z\_{sc} = L \frac{\dot{\mathbf{i}}\_f - \dot{\mathbf{i}}\_g}{\Delta \dot{\mathbf{i}}} = L \frac{\dot{\mathbf{i}}\_f - \dot{\mathbf{i}}\_g}{(Q/W)} \tag{8}$$

The outlet mass quality was then determined using the following equation:

$$\mathbf{x}\_o = \frac{\Delta \mathbf{i} + \mathbf{i}\_{\beta \mathbf{g}} - \mathbf{i}\_f}{\mathbf{i}\_{\beta \mathbf{g}}} \tag{9}$$

#### **3. Results and discussion**

**Parameter Uncertainty at a 95% confidence level**

**Figure 4.** Void fraction comparison between the homogenous model and the existing correlations

Choi et al. [6] determined the inside tube wall temperature, Twi, by the average inside temper‐ ature of the top, both right and left sides, and bottom wall temperatures. As demonstrated in the apparatus module, the heat was applied on test tubes through uniform current, and the local temperature was evaluated at the cross section where there is an attached the thermo‐ couple, hence, the inside wall temperatures at each point could be determined by using steady state one-dimensional radial conduction heat transfer through the wall with internal heat generation. The quality, x, at the measurement location, z, was determined based on the

*Thermocouples* (°C) ±0.62 *P* (kPa) ±2.5 *G* (%) ±5.89 *q* (%) ±2.54 *x* (%) ±6.21 *h(%)* ±6.89

108 Heat Transfer Studies and Applications

**Table 2.** Summary of the estimated uncertainty

*2.2.2. Heat transfer coefficient*

thermodynamic properties.

#### **3.1. Two-phase flow pressure drop**

The study of Choi et al. [6] showed that mass flux, heat flux, inner tube, and saturation temperature all have affected the two-phase flow pressure drop of propane in minichannels. The effect of mass flux on the pressure drop is shown in Fig. 5. It can be seen that the mass flux has a strong influence on the pressure drop. As described above, the two-phase pressure drop is mainly caused by the frictional and acceleration pressure drop. The increase in the mass flux results in a higher flow velocity, which increases both of the two components and, therefore, increases the pressure drop. The similar trends were reported in the studies [13-17]. The strong effect of heat flux on the pressure drop is also described in Fig. 5. An increase in the heat flux causes a higher vaporization. The average vapor quality and flow velocity of higher heat flux conditions increase faster than those of lower ones, and lead to the increase of the pressure drop when the heat flux increases. The results were in well agreement with the study of Zhao et al. [13]. In addition, Fig. 5 illustrates the effect of the tube diameter on the pressure drop. The comparison of the pressure gradient with the inner diameter of 1.5 and 3.0 mm shows that it is higher in smaller diameter. This phenomenon can be explained by saying that the wall shear stress is higher in smaller tubes, which results in both higher frictional and acceleration pressure drop. The effect of saturation temperature on the pressure drop is also observed in Fig. 5. The pressure gradient is higher at a lower saturation temperature. The physical properties are believed to be the reason for this phenomenon. The liquid density and the viscosity of propane increase when the saturation temperature decreases. Hence, at the constant mass flux conditions, the liquid velocity becomes lower, while the vapor velocity is higher. This means the pressure drop increases during evaporation as the saturation temper‐ ature decrease.

**Figure 5.** The effect of mass flux, heat flux, inner tube diameter and saturation temperature on the pressure drop

**Figure 6.** Variation of the two-phase frictional multiplier data with the Lockhart-Martinelli parameter

The comparison of the two-phase frictional multiplier data with the value was predicted from the Lockhart Martinelli correlation with C factors equal to 5 and 20. The value of factor Cs was evaluated follow the study of Chisholm [18]. As shown in figure 6, the presented data is distributed in all regimes: under the baseline C = 5, upper the baseline C = 20, and between the two baselines. That means the flow characteristics include laminar, co-current laminarturbulent, and turbulent flows were observed in the data.

In order to validate the experimental data, the frictional pressure drop of propane was compared with the 13 existing correlations [19 – 31] as shown in Table 3. Among them, the correlations of Mishima and Hibiki [19], Friedel [20] and Chang et al. [21] showed the best prediction with the experimental data. Mishima and Hibiki [19] proposed the frictional pressure drop for air-water in the vertical tube with an inner diameter of 1 to 4 mm, based on the Chisholm's equation. Before that time, using a large database, the model developed by Friedel [20] can be used to predict the frictional pressure drop in both the horizontal and vertical upward flow. The correlation of Chang et al. [21] was developed with the data of R410A and air-water in a 5mm smooth tube. Other correlations, including 4 homogenous models proposed, showed large mean deviations.

**Figure 5.** The effect of mass flux, heat flux, inner tube diameter and saturation temperature on the pressure drop

110 Heat Transfer Studies and Applications

**Figure 6.** Variation of the two-phase frictional multiplier data with the Lockhart-Martinelli parameter

**Table 3.** Deviation of the pressure drop comparison between the present data and the previous correlations.

**Table 4.** Summary of some correlations of two-phase flow pressure drops.

#### **3.2. Heat transfer coefficient**

The mass flux, heat flux, inner tube, and saturation temperature also have affected the twophase flow boiling heat transfer coefficient of propane in the minichannels. Fig. 7 shows the effect of mass flux on heat transfer coefficient. No significant effect of mass flux on the heat transfer coefficient was observed in the low quality region. This phenomenon indicates that the nucleate boiling heat transfer is predominant. The similar tendency of nucleate boiling was performed in numerous studies by Kew and Cornwell [32], Lazarek and Black [33], Wambs‐ ganss et al. [34], Tran et al. [35], and Bao et al. [36], those that used small tubes. The dominant of nucleate boiling heat transfer occurred due to the effect of small channels and the physical properties of the refrigerant. The heat transfer coefficient is higher with higher heat flux at high quality regime since the development of convective boiling heat transfer. The trend also depicts that the drop of heat transfer coefficient appears sooner in higher mass flux. The flow pattern develops faster in the smaller tube and higher mass flux, which lead to the occurrence of dry patches at lower quality.

**Figure 7.** The effect of mass flux on heat transfer coefficient

Lockhart and Martinelli

112 Heat Transfer Studies and Applications

*ϕf* 2 =1− *C Xtt* + 1 *Xtt* 2 ;

*ϕfo* 2 =(1 − *x* )

*Xtt* =( 1− *x x* ) 0.9 ( *ρg ρf* ) 0.5 ( *μf μg* ) 0.1

*A*2 = *x* 0.78( 1 − *x* ) 0.224

2 =1+ (*Γ* 2 −1) *Bx* 0.875(1− *x*)

*dz* )*go* /( *dp dz* ) *fo Γ G*(*kgm*−2

*A*3 =( *ρf ρg* ) 0.91 ( *μf μg* ) 0.19 1−( *μf μg* ) 0.7

*Γ* 2 =( *dp*

≤9.5

*ϕf* 2 =1− *C Xtt* + 1 *Xtt* 2

*ϕfo*

*Γ* 2 = ( *dp dz* )*go* ( *dp dz* ) *fo*

2 =(1− *x*)

**Table 4.** Summary of some correlations of two-phase flow pressure drops.

9.5<*Γ* <28

*X* =( − *dp*

*C* =21(1−*e* −319×10−6

*dz F*) *f* /( − *dp*

; *Nconf* =

2 + 2.87*x* 2 ( *P Pc* )−1

*ϕfo*

;*Cturbulent*−*turbulent* =20

2 + *x* 2 (*ρf f go*)/(*ρg f fo*) + 3.24*A*2*A*3*F r* −0.045*W e* −0.035

0.875 + *x* 0.175

) B

4.8 2400 / *G* 55 / *G* 0.5

520 /(*ΓG* 0.5) 21 / *Γ*

*G* 0.5

*s* −1

≤500 500<*G* <1900 *G* ≥1900

≤600 >600

*dz F* )*g*

)

2 =1+ (4.3*Γ* 2 −1)(*Nconf x* 0.875 + *x* 1.75)

*α g* (*ρf* − *ρg* ) 0.5 *D*

+ 1.68*x* 0.8(1− *x*)

The mass flux, heat flux, inner tube, and saturation temperature also have affected the twophase flow boiling heat transfer coefficient of propane in the minichannels. Fig. 7 shows the effect of mass flux on heat transfer coefficient. No significant effect of mass flux on the heat transfer coefficient was observed in the low quality region. This phenomenon indicates that the nucleate boiling heat transfer is predominant. The similar tendency of nucleate boiling was performed in numerous studies by Kew and Cornwell [32], Lazarek and Black [33], Wambs‐ ganss et al. [34], Tran et al. [35], and Bao et al. [36], those that used small tubes. The dominant of nucleate boiling heat transfer occurred due to the effect of small channels and the physical properties of the refrigerant. The heat transfer coefficient is higher with higher heat flux at high quality regime since the development of convective boiling heat transfer. The trend also depicts that the drop of heat transfer coefficient appears sooner in higher mass flux. The flow pattern develops faster in the smaller tube and higher mass flux, which lead to the occurrence of dry

0.25( *P Pc* )−1.64

≥28 - *Γ* 2

[26]

Friedel [20]

Chisholm [27]

Mishima and Hibiki [19]

Zhang and Webb [28] *ϕfo*

**3.2. Heat transfer coefficient**

patches at lower quality.

Tran et al. [31]

The influence of heat flux on the heat transfer coefficient is illustrated in Fig. 8. The strong effect of heat flux is observed at a low quality regime. It means that the nucleate boiling heat transfer contribution is dominant. When the vapor quality increases, the nucleate boiling heat transfer is suppressed. Therefore, the effect of heat flux becomes limited. The heat transfer coefficient increases continuously in high quality regime, but no difference is found in given conditions. It shows that the convective boiling heat transfer contribution is predominant in this regime.

**Figure 8.** The effect of heat flux on heat transfer coefficient

Fig. 9 shows the effect of a tube diameter on the heat transfer coefficient of propane. The heat transfer coefficient is higher in a smaller tuber diameter at a low quality regime. It can be explained that the nucleate boiling is more active when the tube diameter decreases. In addition, the heat transfer performance is higher in smaller tubes at given conditions due to the increase in contact surface per refrigerant volume.

**Figure 9.** The effect of inner tube diameter on heat transfer coefficient

Fig. 10 illustrates the effect of saturation temperature on the heat transfer coefficient. The increase in saturation temperature leads to the increase in heat transfer coefficient. The reason is that the surface tension is lower and the saturation temperature is higher when the saturation pressure increases. Hence, the heat transfer coefficient is higher when it is considered during the vapor formation in the boiling process.

**Figure 10.** The effect of saturation temperature on heat transfer coefficient

#### **4. Development of new correlations**

#### **4.1. Pressure drop correlation**

Fig. 9 shows the effect of a tube diameter on the heat transfer coefficient of propane. The heat transfer coefficient is higher in a smaller tuber diameter at a low quality regime. It can be explained that the nucleate boiling is more active when the tube diameter decreases. In addition, the heat transfer performance is higher in smaller tubes at given conditions due to

Fig. 10 illustrates the effect of saturation temperature on the heat transfer coefficient. The increase in saturation temperature leads to the increase in heat transfer coefficient. The reason is that the surface tension is lower and the saturation temperature is higher when the saturation pressure increases. Hence, the heat transfer coefficient is higher when it is considered during

the increase in contact surface per refrigerant volume.

114 Heat Transfer Studies and Applications

**Figure 9.** The effect of inner tube diameter on heat transfer coefficient

**Figure 10.** The effect of saturation temperature on heat transfer coefficient

the vapor formation in the boiling process.

The accuracy of predicting two-phase flow pressure drops in heat exchanger devices is quite important in the design and performance optimization of refrigeration systems. For a simple example, if the predicting method shows a higher value compared to the real value, the material design will cost higher. While, on the other hand, if the predicting value is lower than the real value, the performance of the design system would not reach the estimated values. In addition, the comparison of experimental pressure drops of propane and existing correlations described above showed high deviations. Hence, a more precise pressure drop correlation was proposed by Choi et al [6]. The new correlation was developed based on the method of Lockhart and Martinelli [26]. The pressure drop model of Lockhart and Martinelli [26] is defined in the basis pressure drop of three terms including the liquid phase, vapor phase, and the interaction between two phases. The ideal can be expressed in the following equation:

$$\left(-\frac{dp}{dz}F\right)\_p = \left(-\frac{dp}{dz}F\right)\_f + C\left[\left(-\frac{dp}{dz}F\right)\_f\left(-\frac{dp}{dz}F\right)\_8\right]^{1/2} + \left(-\frac{dp}{dz}F\right)\_8\tag{10}$$

The two-phase frictional multiplier based on the pressure gradient for the liquid alone flow, *ϕf* 2 , is calculated by dividing Eq.(10) by the liquid phase pressure drop, as is shown in Eq. (11).

$$\phi\_f^2 = \frac{\left(-\frac{dp}{dz}F\right)\_\wp}{\left(-\frac{dp}{dz}F\right)\_f} = 1 + C \left[\frac{\left(-\frac{dp}{dz}F\right)\_\wp}{\left(-\frac{dp}{dz}F\right)\_f}\right]^{1/2} + \frac{\left(-\frac{dp}{dz}F\right)\_\wp}{\left(-\frac{dp}{dz}F\right)\_f} = 1 + \frac{C}{X} + \frac{1}{X^2} \tag{11}$$

The Martinelli parameter, X, is defined by the following equation:

$$X = \left[ \frac{\left( -\frac{dp}{dz} F \right)\_f}{\left( -\frac{dp}{dz} F \right)\_s} \right]^{1/2} = \left[ \frac{2f\_f G^2 (1-x)^2 \rho\_g / D}{2f\_s G^2 x^2 \rho\_f / D} \right]^{1/2} = \left( \frac{f\_f}{f\_s} \right)^{1/2} \left( \frac{1-x}{x} \right) \left( \frac{\rho\_g}{\rho\_f} \right)^{1/2} \tag{12}$$

The friction factor in Eq. (12) was determined by the flow conditions: laminar (for Re < 1000), *f* =16Re−1 , or turbulent (for Re > 2000), *f* =0.079Re−0.25. For the transition regime, the frictional factor was calculated by interpolating the equations of two regimes.

As discussed in the section above, the experimental result showed that the pressure drop is a function of mass flux, inner tube diameter, surface tension, density, and viscosity. Therefore Choi et al. [6] proposed the factor C in Eq. (11) as a function of the two-phase Weber number, Wetp, and the two-phase Reynolds number, Retp. A new factor C was developed using the regression method, as shown in Eq. (13).

$$C = \left(\phi\_f^2 - 1 - \frac{1}{X^2}\right)X = 1732.953 \times \text{Re}\_p^{-0.323} We\_p^{-0.24} \tag{13}$$

Fig. 11 illustrates the two-phase frictional multiplier comparison between the propane experimental data and the prediction with the new correlation proposed by Choi et al. [6]. The comparison shows a mean deviation of 10.84% and an average deviation of 1.08%.

**Figure 11.** Comparison of the experimental and predicted pressure drop using the new developed correlation

#### **4.2. Heat transfer coefficient correlation**

Choi et al. [6] have compared the experimental heat transfer coefficients of propane with the nine correlations for boiling heat transfer coefficient, as shown in tables 5 and 6. The Shah [37] and Tran et al. [35] correlations provided the better prediction among the other presented correlations. The correlation developed by Shah [37] used a large databank from 19 independ‐ ent studies with various fluids. The correlation can be used to predict the heat transfer coefficient in both horizontal and vertical tubes. However, this correlation was developed for conventional tubes. Tran et al. [35] proposed a nucleation-dominant form of heat transfer coefficient correlation based on the data of R-12 and R-113 in small circular and rectangular channels. Other correlations include the ones proposed by Jung et al. [38], Gungor and Winterton [39], Takamatsu et al. [40], and Kandlikar and Steinke [41], where the mean deviation is around 30%. The Gungor–Winterton's correlation [39] was a modification of the superposition model; it was developed using some fluids in several small and conventional channels with various test conditions. The correlations proposed by Wattelet et al. [42] and Chen [43] showed a large mean deviation, since both of them were developed for large tube diameters. The correlation proposed by Zhang et al. [44] used the data of some working fluids without any hydrocarbon, hence, couldn't predict the present data well. The above compari‐ sons showed the need of developing a more accurate heat transfer coefficient correlations. Based on the one proposed by Chen [43], Choi et al. [6] developed a new correlation for the two-phase flow boiling of propane. He noted that the correlation only used the experimental data prior to the dry-out.

Choi et al. [6] proposed the factor C in Eq. (11) as a function of the two-phase Weber number, Wetp, and the two-phase Reynolds number, Retp. A new factor C was developed using the

2 0.323 0.24

Fig. 11 illustrates the two-phase frictional multiplier comparison between the propane experimental data and the prediction with the new correlation proposed by Choi et al. [6]. The

*X* (13)

 1 1732.953 Re æ ö - - = -- = ´ ç ÷ è ø *C f X tp tp We*

comparison shows a mean deviation of 10.84% and an average deviation of 1.08%.

**Figure 11.** Comparison of the experimental and predicted pressure drop using the new developed correlation

Choi et al. [6] have compared the experimental heat transfer coefficients of propane with the nine correlations for boiling heat transfer coefficient, as shown in tables 5 and 6. The Shah [37] and Tran et al. [35] correlations provided the better prediction among the other presented correlations. The correlation developed by Shah [37] used a large databank from 19 independ‐ ent studies with various fluids. The correlation can be used to predict the heat transfer coefficient in both horizontal and vertical tubes. However, this correlation was developed for conventional tubes. Tran et al. [35] proposed a nucleation-dominant form of heat transfer coefficient correlation based on the data of R-12 and R-113 in small circular and rectangular channels. Other correlations include the ones proposed by Jung et al. [38], Gungor and

2 1

regression method, as shown in Eq. (13).

116 Heat Transfer Studies and Applications

**4.2. Heat transfer coefficient correlation**

f

It is well known that the flow boiling heat transfer is mainly governed by the following two important mechanisms: nucleate boiling and forced convective evaporation. The basic form is described by Chen [43] as follows:

$$h\_{\rm up} = F.h\_{\rm conv} + h\_{\rm nb}.S \tag{14}$$

The convective component was also presented by a Dittus-Boelter type equation.

$$h\_{lo} = 0.023 \left[ \frac{G(1-\mathbf{x})D}{\mu\_f} \right]^{0.8} \left[ \frac{\mu\_f c\_p}{k\_f} \right]^{0.4} \frac{k\_f}{D} \tag{15}$$

**Table 5.** Deviation of the heat transfer coefficient comparison between the present data and the previous correlations.

Chen [43] *htp* =*S*.*h nb* + *F*.*hlo*;*where*:*S* = *fn*(Re*tp*); *F* = *fn*(*Xtt*) *h nb* =0.00122 *kf* 0.79*cpf* 0.45*ρf* 0.49 *σ* 0.5*μf* 0.29*i fg* 0.24*ρg* 0.24 *ΔTe* 0.24*Δ pe* 0.75;*hlo* =0.023 *G*(1− *x*)*D μf* 0.8 *μf cp kf* 0.4 *kf D* Gungor and Winterton [39] *htp* =*Enew* ×*hl Enew* =1 + 3000*Bo* 0.86 + 1.12( *x* 1 − *x* )0.75( *ρl ρv* ) 0.41 If the tube is horizontal and the Froude number (*F rl*) is less than 0.05 then E should be multiplied by the factor *E*2 = *F rl* (0.1−2*F rl* ) Shah [37] *ψ*= *h TP hf* ;*Co* =( 1 *x* −1 ) 0.8( *ρg ρf* ) 0.5 ; *Bo* = *q Gh fg* ; *F rf* = *G* 2 *ρf* 2 *gD hl* =0.023Re *f* 0.8Pr *f* 0.4( *kf Df* ) Jung et al. [38] *htp* =*S*.*h SA* + *F* .*hlo h SA* =207 *k*1 *bd* ( *qbd k*1*Tsat* ) 0.745( *ρg ρf* ) 0.581 Pr *f* 0.533; *hl* =0.023Re *f* 0.8Pr *f* 0.4( *kf Df* ) *bd* =0.0146( 35*π* 180 ) 2*σ g*(*ρf* −*ρg*) 0.5 *S* =4048*Xtt* 1.22*Bo* 1.13 *for* : *Xtt* <1 *F* =2.37(0.29 + 1 *Xtt* ) 0.85 Wattelet et al. [42] *htp* = *h nb n* + *hcb n* 1/2;*n* =2.5 *h nb* =55*pr* 0.12( −0.4343ln*pr*)−0.55*M* −0.5*q* 0.67 *hcb* =0.023Re *f* 0.8Pr *f* 0.4( *kf Df* ) × *F* ×*R F* = *fn*(*Xtt*); *andS* = *fn*(Re*tp*) Tran et al. [35] *h* =(8.4*x*10−5 )(*Bo* 2 *W ef* ) 0.3( *ρf ρg* ) −0.4 Kandlikar and Steinke [41] *h tp h f* =*D C*1 *o D*2 ( 1 − *x* ) 0.8 *fn*(*F rfo*) + *D*3*Bo D*4 ( 1 − *x* ) 0.8*Fn fn*(*F rfo*)=1 Zhang et al. [44] *htp* =*S*.*h nb* + *F*.*hsp h nb* =0.00122 *kf* 0.79*cpf* 0.45*ρf* 0.49 *σ* 0.5*μf* 0.29*i fg* 0.24*ρg* 0.24 *ΔTe* 0.24*Δ pe* 0.75 *hsp* =*MAX* (*h Dittus*−*Boelter*)*if* Re *f* <2300;*hsp* =(*h Dittus*−*Boelter*)*if* Re *f* ≥2300 *S* = *fn*(Re *f* ); *F* = *fn*(*ϕf* )

**Table 6.** Summary of existing heat transfer coefficient correlations

Chen [43]

Gungor and Winterton [39]

Shah [37]

Jung et al. [38]

Wattelet et al.

Kandlikar and Steinke [41]

Zhang et al. [44]

[42]

*htp* =*S*.*h nb* + *F* .*hlo*;*where*:*S* = *fn*(Re*tp*); *F* = *fn*(*Xtt*)

0.24 *ΔTe*

1 − *x* )0.75( *ρl ρv* ) 0.41

0.24*Δ pe*

0.75;*hlo* =0.023

If the tube is horizontal and the Froude number (*F rl*) is less than 0.05 then E should be multiplied by

; *F rf* = *G* 2 *ρf* 2 *gD* *G*(1− *x*)*D μf*

0.8 *μf cp kf*

0.4 *kf D*

*kf* 0.79*cpf* 0.45*ρf* 0.49

*σ* 0.5*μf* 0.29*i fg* 0.24*ρg*

*Enew* =1 + 3000*Bo* 0.86 + 1.12( *x*

(0.1−2*F rl* )

*h nb* =0.00122

118 Heat Transfer Studies and Applications

*htp* =*Enew* ×*hl*

the factor *E*2 = *F rl*

*hl* =0.023Re *f*

*h SA* =207

*hl* =0.023Re *f*

*bd* =0.0146(

*F* =2.37(0.29 +

*htp* = *h nb n* + *hcb*

*h nb* =55*pr*

Tran et al. [35] *h* =(8.4*x*10−5

*h tp h f* =*D C*1 *o*

*fn*(*F rfo*)=1

*h nb* =0.00122

*htp* =*S*.*h nb* + *F* .*hsp*

*S* = *fn*(Re *f* ); *F* = *fn*(*ϕf* )

*hcb* =0.023Re *f*

*htp* =*S*.*h SA* + *F* .*hlo*

*k*1 *bd* ( *qbd k*1*Tsat* ) 0.745 ( *ρg ρf* ) 0.581 Pr *f* 0.533;

;*Co* =( 1 *x* −1 ) 0.8 ( *ρg ρf* ) 0.5 ; *Bo* = *q Gh fg*

> 0.8Pr *f* 0.4( *kf Df* )

> 0.8Pr *f* 0.4( *kf Df* )

35*π* 180 )

*S* =4048*Xtt* 1.22*Bo* 1.13 *for* : *Xtt* <1

1 *Xtt* ) 0.85

*n* 1/2;*n* =2.5

0.8Pr *f* 0.4( *kf Df* ) × *F* ×*R*

)(*Bo* 2 *W ef* ) 0.3( *ρf ρg* ) −0.4

> *kf* 0.79*cpf* 0.45*ρf* 0.49

*σ* 0.5*μf* 0.29*i fg* 0.24*ρg*

*D*2 ( 1 − *x* )

*F* = *fn*(*Xtt*); *andS* = *fn*(Re*tp*)

2*σ g*(*ρf* −*ρg*)

0.12( −0.4343ln*pr*)−0.55*M* −0.5*q* 0.67

0.8 *fn*(*F rfo*) + *D*3*Bo D*4

( 1 − *x* ) 0.8*Fn*

0.24*Δ pe* 0.75

0.24 *ΔTe*

*hsp* =*MAX* (*h Dittus*−*Boelter*)*if* Re *f* <2300;*hsp* =(*h Dittus*−*Boelter*)*if* Re *f* ≥2300

0.5

*ψ*= *h TP hf*

> In small channels, the contribution of force convective heat transfer normally occurs later than it does in conventional channels due to the high contribution of nucleate boiling. Therefore, the enhancement factor F, which describes the increase in the convective heat transfer when the vapor quality increases, should be physically evaluated again. Chen [43] first introduced the enhanced factor F as the function of Lockhart-Martinelli parameter Xtt, *F* = *fn*(*Xtt*). Zhang et al. [44], later, took the flow condition effect in the enhanced factor into account. They introduced the relationship *F* = *fn*(*ϕf* 2 ), where (*ϕf* 2 ) is a general form proposed by Chisholm [18], as shown in Eq. (11). The values of the Chisholm parameter, C, are 20, 12, 10, and 5 corre‐ sponding to the four flow conditions of liquid and vapor: turbulent–turbulent (tt), laminar– turbulent (vt), turbulent–laminar (tv) and laminar–laminar (vv). However, as seen in the evaluation of existing heat transfer correlation above, the correlation of Zhang et al. [44] shows a large deviation with the present data. Hence, the factor F needs to reform to fit in the data.

> Choi et al. [6] modified the enhancement factor F as a function of (*ϕf* 2 ), *F* = *fn*(*ϕf* 2 ), where (*ϕf* 2 ) is obtained from Eq. (11) - (13). The value of C in study of Choi et al. [6] is calculated by an interpolation of the Chisholm parameter with thresholds of Re = 1000 and Re = 2000 for the laminar and turbulent flows, respectively. On the other hand, the enhancement factor F must be reduced to 1 for pure liquid or pure vapor, and be greater than 1 within the two-phase regime. Hence, using a regression method, the formula of F is satisfied as follows:

$$F = MAX(0.5\phi\_f, \mathrm{l})\tag{16}$$

A new factor F is shown in Fig. 12.

Cooper [45] developed a correlation that used a large nucleate boiling data bank. Kew and Cornwell [32] and Jung et al. [46] showed that the Cooper [45] pool boiling correlation best predicted their experimental data. Therefore, the prediction of the nucleate boiling heat transfer for the present experimental data used Cooper [45], which is a pool boiling correlation developed based on an extensive study,

$$h = \\$5 \, p\_r^{0.12} \left( -0.4343 \ln p\_r \right)^{-0.95} M^{-0.5} q^{0.67} \tag{17}$$

To account for the suppression of the nucleate boiling when vapor quality increases, Chen [43] defined the nucleate boiling suppression factor S, which is the ratio of mean superheat temperature, *ΔTe*, to the wall superheat temperature, *ΔTsat*. Jung et al. [38] introduced the suppression factor N as the function of Xtt and boiling number, Bo, to take into account the strong effect of nucleate boiling in flow boiling. On the other hand, to consider the effect of flow conditions, the Lockhart – Martinelli parameter Xtt is replaced by the two-phase frictional multiplier, *ϕf* 2 . Using the regression program and the experimental data, a new nucleate boiling suppression factor was proposed by Choi et al. [6]. It is as follows:

$$LS = 181.458 \left(\phi\_f^2\right)^{0.002} Bo^{0.816} \tag{18}$$

The new heat transfer coefficient correlation is developed using 461 data points. Fig. 13 shows the comparison of the experimental heat transfer coefficient and the predicted one. The new correlation archives a good prediction with a mean deviation of 9.93% and an average deviation of -2.42%. When the C factor of the Chisholm [18] method is used to obtain *ϕf* 2 for equations (11) and (16), the heat transfer coefficient also showed a good comparison mean deviation of 14.40%.

**Figure 12.** Two-phase heat transfer multiplier as a function of *ϕf* 2

**Figure 13.** Comparison of the experimental and predicted heat transfer coefficients using the new developed correla‐ tion


*3.8. White Pigments*

The white pigment employed throughout the manuscript was identified as lead white. In fact, the micro-Raman spectra of the white particles showed the sharp band at 1049 cm−1 that characterises this pigment, whereas XRF data resulted in very high lead signals that emerged in most of the XRF spectra collected from the manuscript. This suggests that lead white was used, by itself or mixed with other pigments, as a primer under the illuminations. The most relevant exceptions are the areas painted with gold or mosaic gold, in which very weak signals of lead were detected.