Specifications for the counter-flow regenerative IEHX.
\r\n\t
",isbn:"978-1-83962-547-3",printIsbn:"978-1-83962-546-6",pdfIsbn:"978-1-83962-548-0",doi:null,price:0,priceEur:0,priceUsd:0,slug:null,numberOfPages:0,isOpenForSubmission:!1,hash:"e5ba02fedd7c87f0ab66414f3b07de0c",bookSignature:"Dr. John P. Tiefenbacher",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/10765.jpg",keywords:"Managing Urbanization, Managing Development, Managing Resource Use, Drought Management, Flood Management, Water Quality Monitoring, Air Quality Monitoring, Ecological Monitoring, Modeling Extreme Natural Events, Ecological Restoration, Restoring Environmental Flows, Environmental Management Perspectives",numberOfDownloads:24,numberOfWosCitations:0,numberOfCrossrefCitations:0,numberOfDimensionsCitations:0,numberOfTotalCitations:0,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"January 12th 2021",dateEndSecondStepPublish:"February 9th 2021",dateEndThirdStepPublish:"April 10th 2021",dateEndFourthStepPublish:"June 29th 2021",dateEndFifthStepPublish:"August 28th 2021",remainingDaysToSecondStep:"2 months",secondStepPassed:!0,currentStepOfPublishingProcess:4,editedByType:null,kuFlag:!1,biosketch:"A geospatial scholar working at the interface of natural and human systems, collaborating internationally on innovative studies about hazards and environmental challenges. Dr. Tiefenbacher has published more than 200 papers on a diverse array of topics that examine perception and behaviors with regards to the application of pesticides, releases of toxic chemicals, environments of the U.S.-Mexico borderlands, wildlife hazards, and the geography of wine.",coeditorOneBiosketch:null,coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"73876",title:"Dr.",name:"John P.",middleName:null,surname:"Tiefenbacher",slug:"john-p.-tiefenbacher",fullName:"John P. Tiefenbacher",profilePictureURL:"https://mts.intechopen.com/storage/users/73876/images/system/73876.jfif",biography:"Dr. John P. Tiefenbacher (Ph.D., Rutgers, 1992) is a professor of Geography at Texas State University. His research has focused on various aspects of hazards and environmental management. Dr. Tiefenbacher has published on a diverse array of topics that examine perception and behaviors with regards to the application of pesticides, releases of toxic chemicals, environments of the U.S.-Mexico borderlands, wildlife hazards, and the geography of wine. More recently his work pertains to spatial adaptation to climate change, spatial responses in wine growing regions to climate change, the geographies of viticulture and wine, artificial intelligence and machine learning to predict patterns of natural processes and hazards, historical ethnic enclaves in American cities and regions, and environmental adaptations of 19th century European immigrants to North America's landscapes.",institutionString:"Texas State University",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"1",totalChapterViews:"0",totalEditedBooks:"6",institution:{name:"Texas State University",institutionURL:null,country:{name:"United States of America"}}}],coeditorOne:null,coeditorTwo:null,coeditorThree:null,coeditorFour:null,coeditorFive:null,topics:[{id:"12",title:"Environmental Sciences",slug:"environmental-sciences"}],chapters:[{id:"76073",title:"Integrating Ecological Site Descriptions with Soil Morphology to Optimize Forest Management: Three Missouri Case Studies",slug:"integrating-ecological-site-descriptions-with-soil-morphology-to-optimize-forest-management-three-mi",totalDownloads:25,totalCrossrefCites:0,authors:[{id:"185895",title:"Dr.",name:"Michael",surname:"Aide",slug:"michael-aide",fullName:"Michael Aide"},{id:"269286",title:"Dr.",name:"Christine",surname:"Aide",slug:"christine-aide",fullName:"Christine Aide"},{id:"269287",title:"Dr.",name:"Indi",surname:"Braden",slug:"indi-braden",fullName:"Indi Braden"}]},{id:"76399",title:"Hunting and Deforestation: A Threat to the Existence of the Niger Delta Red Colobus Monkey (Procolobus epieni)",slug:"hunting-and-deforestation-a-threat-to-the-existence-of-the-niger-delta-red-colobus-monkey-procolobus",totalDownloads:0,totalCrossrefCites:0,authors:[{id:"142349",title:"Dr.",name:"Gbolagade Akeem",surname:"Lameed",slug:"gbolagade-akeem-lameed",fullName:"Gbolagade Akeem Lameed"}]}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"},personalPublishingAssistant:{id:"194667",firstName:"Marijana",lastName:"Francetic",middleName:null,title:"Ms.",imageUrl:"https://mts.intechopen.com/storage/users/194667/images/4752_n.jpg",email:"marijana@intechopen.com",biography:"As an Author Service Manager my responsibilities include monitoring and facilitating all publishing activities for authors and editors. From chapter submission and review, to approval and revision, copyediting and design, until final publication, I work closely with authors and editors to ensure a simple and easy publishing process. I maintain constant and effective communication with authors, editors and reviewers, which allows for a level of personal support that enables contributors to fully commit and concentrate on the chapters they are writing, editing, or reviewing. I assist authors in the preparation of their full chapter submissions and track important deadlines and ensure they are met. I help to coordinate internal processes such as linguistic review, and monitor the technical aspects of the process. As an ASM I am also involved in the acquisition of editors. 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Venkateswarlu",coverURL:"https://cdn.intechopen.com/books/images_new/371.jpg",editedByType:"Edited by",editors:[{id:"58592",title:"Dr.",name:"Arun",surname:"Shanker",slug:"arun-shanker",fullName:"Arun Shanker"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}]},chapter:{item:{type:"chapter",id:"62339",title:"Energy Efficient Indirect Evaporative Air Cooling",doi:"10.5772/intechopen.79223",slug:"energy-efficient-indirect-evaporative-air-cooling",body:'\nThe evaporative cooling technique takes the advantage of water evaporation to achieve cooling effect. As a potential alternative to the conventional mechanical vapor compression system, it has drawn great attention for building cooling applications.
\nAn indirect evaporative cooling system is able to produce the cool air without moist change. For a typical indirect evaporative heat exchanger (IEHX), the primary air (or product air) and the secondary air (or working air) flow in separate passages. The secondary air in wet channel acts as a heat sink by absorbing heat due to water evaporation.
\nThe evaporative cooling system has the following advantages over the current mechanical vapor compression system [1, 2, 3, 4]: (1) energy and cost savings; (2) reducing peak power requirements; (3) no CFCs; (4) reducing pollutant releases; and (5) easily incorporation with existing systems. On the other hand, the traditional IEHX shows the following limitations: (1) the air humidity increases in direct evaporative cooling systems resulting in uncomfortable indoor thermal environment for humans; (2) the indirect evaporative cooling system generally has a low cooling efficiency [5]; and (3) the theoretical ultimate cooled air temperature is the wet-bulb temperature.
\nTo further enhance the cooling performance of conventional IEHX, research works have proposed a novel regenerative IEHX, which can cool the air below its wet-bulb temperature [1]. Since this design was proposed by Maisotsenko, the airflow arrangement is also named as M-cycle [2, 6]. Several studies have theoretically and experimentally investigated the dew-point evaporative cooling system [7, 8, 9]. This type of IEHX is able to branch part of the pre-cooled primary air into the wet channel [10, 11].
\nIn order to investigate the cooling performance of a regenerative IEHX, analytical models have been developed based on modified effectiveness-NTU method [10, 12]. In addition, numerical simulations have been carried out to study the impact of key parameters on a counter-flow IEHX [13, 14, 15]. Anisimov and Pandelidis [16] presented a numerical study analyzing the indirect evaporative cooler with four different configurations. Zhao et al. [17] conducted a numerical study on a novel dew-point IEHX. According to the simulation results, the wet-bulb effectiveness of the IEHX was greatly influenced by the dimension of the airflow passages, the inlet air velocity, and working-to-intake-air ratio. The cooler achieved the highest wet-bulb effectiveness of 1.3 for a typical summer condition in the UK. Moshari and Heidarinejad [18] developed a numerical model and solved the governing equations by using finite difference method in MATLAB. The regenerative evaporative heat exchanger has been demonstrated to provide sub-wet-bulb cooling. Jradi and Riffat [19] conducted an experimental and numerical investigation on a dew-point evaporative cooling system. An optimized design of the indirect evaporative heat exchanger has been presented to obtain a dew-point effectiveness of 78%. Ham and Jeong [20] proposed a novel design on dew-point evaporative heat exchanger to address the issues of complex ventilation control and energy waste. Moshari et al. [21] investigated indirect evaporative cooling systems with one- and two-stage to analyze the optimum configuration by considering the cooling effectiveness, water consumption and thermal comfort. Chen et al. [22] developed an analytical model to study the heat and mass transfer process in an indirect evaporative cooler as a pre-cooling device in tropical areas. The model was established taking account of the condensation in the dry channels. Woods and Kozubal [23] conducted an experimental and numerical study on a desiccant-enhanced air conditioner by combining a dew-point evaporative heat exchanger and a desiccant dehumidifier. The second stage of this air conditioner was a counter-flow regenerative IEHX, which had a similar configuration compared with the IEHX designed by Zhao et al. [17] and Riangvilaikul and Kumar [24].
\nIn this chapter, we first introduce the design of novel indirect evaporative heat exchangers, followed by the mathematical description, which was employed to study the air treatment process in the IEHX. An experimental study was then conducted to validate the computational model. Finally, the validated model was used to study the cooling performance of the novel dew-point IEHX and to investigate the impact of several influential parameters.
\nIn this section, we introduce two types of IEHX. The first type is a counter-flow regenerative IEHX. The other one is a novel dew-point IEHX based on the modification of the M-cycle arrangement.
\nThe schematic of a one-unit channel pair of a typical counter-flow regenerative IEHX is shown in Figure 1. A number of this type of channel pairs is stacked to form the structure of the IEHX. The intake air flows in alternative primary channels (dry channels). Before the outlet of the dry channel, a part of the primary air is diverted into the secondary channel (wet channel). The primary air in the IEHX can be cooled along the flow passages without changing the humidity ratio.
\nSchematic of a one-unit channel pair of a typical counter-flow regenerative indirect evaporative heat exchanger.
Figure 2 illustrates the airflow in novel dew-point IEHX in terms of a one-unit channel pair. It comprises a product channel and adjacent working channels. By spraying water in working wet channels, inner surfaces are maintained in wet conditions. The working channel employs a closed-loop arrangement. The working air firstly enters the working dry channels, and it is then recirculated into the wet channels. The configuration is able to pre-cool the working air before entering the wet side. Therefore, at the recirculation point of the working channel, the air stream achieves a high cooling capacity as a result of a lower wet-bulb temperature compared with the inlet air. Theoretically, the outlet temperature of product air can be reduced toward its dew-point temperature.
\nSchematic of the novel dew-point IEHX (one-unit channel pair).
To develop a mathematical model predicting the air treatment performance in the IEHX, the following assumptions are considered: (1) the airflow is steady and incompressible; (2) the channel height is comparatively small; (3) the airflow is fully developed and laminar; (4) the surface of wet channel is covered with a thin water film; and (5) the outer surface is insulated.
\nThe moist air flowing in both the working channel and the product channel is governed by the following equations [9].
\nThe continuity equation:
\nMomentum conservation equation:
\nEnergy conservation equation:
\nEquation of species diffusion for water vapor is expressed as
\nThe inlet boundary conditions of the air are indicated as
\nAt the air-water interface, the moist air is assumed to be saturated with the water film temperature. The vapor pressure of the saturated moist air can be expressed as a function of temperature using the following equation [25]:
\nwhere
The water vapor concentration is specified as
\nwhere \n
As a result, the interfacial condition at the water film surface in the working wet channel is given as
\nTo evaluate the cooling efficiency of the indirect evaporative heat exchangers, following expressions (wet-bulb effectiveness and dew-point effectiveness) are defined.
\nA prototype of the plate-type counter-flow regenerative IEHX was fabricated as schematically illustrated in Figure 3. The one-unit channel pair comprises a product channel and two adjacent working channels. The intake air, which consists of both the working air and the product air, flows into the product channel (dry channel). At the end of the product channel, part of the product air is branched into the adjacent working wet channels by passing through the perforated plates. The dimensions for the counter-flow regenerative IEHX are listed in Table 1.
\nSchematic of the counter-flow regenerative IEHX.
Dimension | \nSymbol | \nValue | \nUnits | \n
---|---|---|---|
Product channel gap | \n10 | \nmm | \n|
Working channel gap | \n6 | \nmm | \n|
Channel length | \n750 | \nmm | \n|
Channel width | \n300 | \nmm | \n|
Wall thickness | \n0.3 | \nmm | \n|
Wick thickness | \n0.2 | \nmm | \n
Specifications for the counter-flow regenerative IEHX.
The schematic of the experimental setup for the IEHX is shown in Figure 4. A variable speed blower, which was used to control the intake airflow rate, was equipped at the inlet of the IEHX. The intake air temperature was adjusted by a heater before the blower. The measured parameters included the air temperature, humidity, and velocity.
\nSchematic diagram of the experimental setup.
Thermistors with accuracies of ±0.1°C were employed to measure the air dry-bulb temperature. The probes were inserted into the center of the airflow passages to measure the relative humidity and velocity of the air stream. The measured data were recorded using a data acquisition unit.
\nThe experimental study was carried out to study the performance of the IEHX under varying inlet conditions. The inlet air velocity and temperature were adjusted to a stable condition by controlling the air blower and the heater. The operating conditions in this study were presented in Table 2.
\n\n | Flow rate I | \nFlow rate II | \n
---|---|---|
Flow rate of intake air (L/s) | \n4.5 | \n6 | \n
Velocity of product air (m/s) | \n1.5 | \n2.0 | \n
Velocity of working air (m/s) | \n1.0 | \n1.3 | \n
Inlet air humidity ratio (g/kg) | \n10 | \n10 | \n
Inlet air temperature (°C) | \n22–29 | \n22–29 | \n
Operating condition of the counter-flow regenerative IEHX.
The experimental data were first employed to validate the numerical model for the regenerative IEHX. The experimental condition was replicated in the simulation. Figure 5 illustrates the comparison between the calculated outlet air temperature and the experimental results. The numerical model shows a good agreement within a maximum discrepancy about 5%.
\nValidation 1: comparison between modeled results and experimental data on the counter-flow regenerative IEHX (a) flow rate of intake air is 4.5 L/s; (b) flow rate of intake air is 6 L/s.
The numerical model was further validated against experimental data presented by Woods and Kozubal [23, 26]. The experimental study was conducted for an M-cycle indirect evaporative heat exchanger. The supply air temperatures were measured under different operating conditions. The validation was performed by comparing the simulated supply air temperature reduction with the experimental data. The discrepancy was within ±10% as shown in Figure 6.
\nValidation 2: compare the supply air temperature change.
By comparing the simulated results with experimental data, we can draw the conclusion that the validation has demonstrated the capability of the numerical model to theoretically investigate the cooling process for the IEHX.
\nBy using the validated model, we investigated the air treatment process of the novel dew-point IEHX. Simulations were carried out to predict the states of air stream in the flow passages of the cooler [5]. The assumed inlet conditions were as follows: the dry-bulb temperature was 35°C and the humidity ratio was 10 g/kg for both working air and product air. The geometry parameter and the air velocity were maintained as the pre-set conditions as shown in Table 3.
\nParameters | \nValue | \nUnits | \n
---|---|---|
1 | \nm | \n|
6 | \nmm | \n|
3 | \nmm | \n|
35 | \n°C | \n|
10 | \ng/kg | \n|
1 | \nm/s | \n|
1 | \nm/s | \n
Pre-set conditions for the novel dew-point IEHX.
The psychrometric analysis of the air conditions in the novel dew-point IEHX is shown in Figure 7. The inlet conditions for both the working air (point W1 in Figure 7) and the product air (point P1 in Figure 7) are the same. In the product channel, the air dry-bulb temperature decreases with constant humidity ratio, resulting in a change of the condition from point P1 to point P2. In the working dry channel, sensible heat is transferred to the wet channel so that the working air temperature is reduced from state W1 to state W2. Since the air (at point W2 in Figure 7) is directed into the adjacent wet channel, it is humidified in the working wet channel where the heat is absorbed as a result of vaporizing water. The air in the working wet channel is finally exhausted at state W3.
\nIllustration of air conditions on the psychrometric chart.
Figure 8 illustrates the dry-bulb temperature profiles in airflow passages including one product channel and two working channels. In both the working dry channel and the product channel, the air temperature decreases in the direction of the airflow. In the working wet channel, the lowest air temperature is achieved. It can be inferred from Figure 8 that the heat is transferred from dry channels to working wet channels. The working air absorbs heat in the working wet channel resulting in a temperature increase along the flow direction in the wet channel [11].
\nTemperature profiles of air in dry working channel, wet working channel, and product channel.
Figure 9 shows the wet-bulb effectiveness and dew-point effectiveness of the novel dew-point IEHX under different inlet air temperature and humidity conditions. In this study, the inlet air temperature varied in a range of values from 25 to 40°C and the humidity ratio was changed from 8 to 20 g/kg. The inlet air velocity and other geometry parameters were maintained as specified in Table 3.
\nCooling effectiveness for different inlet air conditions.
As shown in Figure 9, the dew-point effectiveness ranges 81–93%, and the wet-bulb effectiveness spans from 122 to 132%. Simulation results illustrate that the wet-bulb effectiveness of the IEHX is above 100%, which demonstrates the capability of the IEHX to produce air with a temperature lower than the wet-bulb temperature of inlet air.
\nWhen the inlet air humidity ratio is low, the vapor pressure gradient at the air-water interface is large resulting in a greater driving force for mass transfer. As a consequence, the working air is able to absorb more moisture during the process of vaporizing water, and the working air has a greater capacity to cool the air in adjacent dry channels. Therefore, a higher wet-bulb effectiveness can be reached by decreasing the inlet air humidity ratio.
\nAnother finding from Figure 9 is that the dew-point effectiveness may be decreased for a lower inlet air humidity ratio. It can be attributed to the following reason. As shown in the psychrometric chart, the dew-point temperature decreases significantly for a lower humidity ratio. Therefore, the temperature difference between the dew-point and the inlet dry bulb temperature shows a larger value at low humidity ratio [11].
\nThe cooling efficiency of the IEHX is also affected by the channel height and channel length. Simulations have been conducted by using two types of the channel dimensions. For the first type, the height of product channel (
Figure 10 illustrates the correlation between the dimensionless channel length (
Impact of the dimensionless channel length on the cooling effectiveness.
Table 4 indicates the thermal resistances with examples values for the counter-flow regenerative IEHX. The channel dimension significantly influences the thermal resistances due to convective heat transfer (
Thermal resistance | \nExpression | \nExample values | \n
---|---|---|
\n\n | \n||
\n\n | \n||
\n\n | \n||
\n\n | \n
Thermal resistances in the IEHX.
The impact of channel height on the thermal resistances for convective heat transfer (
Thermal resistance for convective heat transfer under varying channel height.
Thermal resistance for conduction in plate and water film.
According to the analysis on thermal resistances, following suggestions are obtained in order to enhance the cooling performance of the IEHX. The airflow channels may be redesigned to generate larger flow turbulence resulting in a lower thermal resistance so that a larger heat transfer coefficient can be achieved [30]. The Reynolds number of the airflow in this IEHX is close to 1100. Therefore, the channel can be modified to create the flow turbulence by installing physical ribs in the airflow channels, which could potentially reduce the thermal resistance and increase the Reynolds number above 3000. In addition, the IEHX with a smaller hydraulic diameter will positively influence its cooling effectiveness due to a higher Nusselt number. To enhance the mass transfer process due to water evaporation in wet channels, some techniques may be employed involving increasing the water vapor concentration gradient between the water surface and the air stream.
\nThe performance of novel indirect evaporative heat exchangers has been investigated. We developed a numerical model to study the influences of several parameters on the cooling performance. Wet-bulb and dew-point effectiveness were employed to evaluate the cooling efficiency of the IEHX. The cooling performance was impacted by the operating conditions and the structure of airflow passages. To promote cooling effectiveness, suggestions are provided to improve the thermal performance of the cooler. To achieve a lower thermal resistance, it is possible to modify the IEHX’s channel and to create larger airflow turbulence. In addition, the performance of the IEHX can be positively improved by employing a geometry with a smaller hydraulic diameter.
\nIn statistical physics only a few problems can be solved exactly. For complex problems, numerical methods can give exact results for problems that could only be solved in an approximate way. Numerical simulation can be a way to test the theory. The numerical results can be compared to the experimental results. The numerical simulation is placed between the fundamental and the experimental treatment; it has a quasi-experimental character (numerical experience). For problems of statistical physics, the most widely used simulation methods are the Monte Carlo method and the molecular dynamics method.
The first Monte Carlo simulation (MCS) was proposed by Metropolis et al. in 1953 [1]. The second Monte Carlo simulation was proposed by Wood and Parker in 1957 [2]. The obtained results were in good agreement with the experimental results of Bridgman [3] and those of Michels et al. [4]. In this method we attribute a series of initial positions chosen randomly to a system of N particles interacting through a defined potential. A sequence of particle configurations is generated by giving successive displacements to particles; we only retain configurations to ensure that the probability density is that of the chosen.
Molecular dynamics simulation (MDS) has been first introduced to simulate the behavior of fluids and solids at the molecular or atomic level. MDS was used for the first time by Alder and Wainwright in the late 1950s [5, 6] to study the interactions of hard spheres. The principle is the resolution of equations of motion for a hard sphere system in a simulation cell. The basic algorithm is Verlet’s algorithm [7].
In this chapter, we will present techniques of numerical simulations using the Monte Carlo method. We will present an application on the gas phase during plasma-enhanced chemical vapor deposition (PECVD) of thin films. The application concerns collisions between particles. Particles are in Brownian motion. Collisions, elastic or inelastic, are considered to be binary. Non-elastic collisions result in effective chemical reactions.
In Section 2, we cite some MCS and MDS works on PECVD processes. Section 3 presents general rules on numerical simulation methods. Section 4 presents how to simulate a physical problem using MCS? We present the Metropolis algorithm as a scheme to trait random configurations and different modules related to elaborate an MCS code. In Section 5, we apply the MCS on SiH4/H2 gas mixture during a PECVD process. Finally the conclusion summarizes the contents of the chapter.
The PECVD is the most widely used technique to produce hydrogenated amorphous silicon thin films (a-Si:H) for solar cells and for film transistors and electronic devices [8, 9]. Reactions during plasma deposition are complex and are not understood completely.
Gorbachev et al. [10, 11, 12] have developed a model that is based on chemical reactions and different processes in a PECVD reactor. The model takes into account the formation of SinHm oligomers (n ≤ 5). It presents a simulation of the growth of the films. Gorbachev et al. found that Si2H5 and Si3H7 strongly influence the growth of the film [11].
Valipa et al. [13] calculated the β reactivity of the SiH3 radical on a surface of a silicon lattice plane during the growth of a-Si:H using MDS. The mechanisms of physical and chemical interactions of low temperature plasmas with surfaces can be explored using MDS [14].
For a CH4/H2 mixture, Farouk et al. used the Monte Carlo method (PIC/MC); they calculated the ionization rate of the plasma and the deposition rate of the thin layer [15]. Rodgers et al. [16] have developed three-dimensional Monte Carlo simulations of diamond (100) surface CVD. Other works on MCS are in [17, 18, 19].
In our previous works [20, 21, 22, 23, 24], we were interested in the study of the gas phase and the interaction of plasmas with the surface, for SiH4/H2 and CH4/H2 gas mixtures during PECVD processes. The used numerical simulation techniques were MCS and MDS. To complete the studies, we used the fluid model [25].
The starting point of numerical simulation is a physical phenomenon; its purpose is to obtain useful physical results. Between these two points, several steps can be identified. These steps are general and they are applicable for MCS. The steps can be summarized as follows:
The physical phenomenon must be defined by the description of the dominant domain of physics. The main assumptions and simplifying approximations are necessary to understand the physical phenomenon and the design of the first model.
Mathematical model requires a mathematical formulation of the problem. It may be a problem of elements or discrete object or a problem of a continuous medium; it may be a spatiotemporal problem or frequency problem and may be a deterministic or probabilistic problem.
It would be interesting to know the mathematical equations that govern the phenomenon:
The forces between particles and elements
The potential interaction
The determination of a time scale
The determination of a length scale
Definition of constant magnitudes of motion and equilibrium magnitudes
Continuity equations, balance equations, transfer equations, etc.
The MCS technique has been chosen for this work; knowing its basic algorithm is necessary for elaborating the simulation. This step requires some actions:
Validation of the model on simple cases
Simulation calculation on complex phenomena
The MCS is based on a probabilistic process with a random choice of configurations and samples of the situation of the physical system. The two pedagogical examples most cited in the literature are the integration of a single variable function and Ising’s model of spin. In the following subsection, we define the integration of a single variable function. We introduce the Ising model at the end of Section 4.2.2.
Calculation of the definite integral for a function f(x) of a single variable x on domain {a, b} has been proposed (Figure 1):
The integral of a function f(x).
Let:
Let xi and yi be real random numbers (i = 1, 2,…, N), and let H be a real number greater than the f(x) for x belonging to the domain {a, b} (or x ∈ {a, b}).
Let r1 and r2 be two random numbers belonging to the domain {0, 1} according to a uniform distribution law. Generators (e.g., Ran, RANDOM, RANDUM, or other IMSL mathematical libraries) of random numbers can be used:
where xi and yi are random numbers (xi ∈ {a, b} and yi ∈ {0, H}).
The Monte Carlo (MC) method is based on a probabilistic process. Let N be the total number of cases chosen (possible cases). It is necessary to count the number of favorable cases (or the number of points below the curve y = f(x)); let yi ≤ f(xi)). The number of favorable cases is Nfav. When N➔∞, the value
An example [26] is the calculation of the value π by calculating the integral
We take a = 0.0, b = 1.0, and H = 1.0.
For different values of N, we show that the numerical solution tends to π = 4
Although this integral is simple, it shows the strength and simplicity of the method. The technique can be generalized for the integration of multivariate functions.
We note that integration by the MC method is based on:
The choice of
Each configuration chosen is either
For statistical physics problems, the probabilistic choice of configurations is not always deterministic; the favorable and unfavorable cases are not exclusive. According to the Metropolis algorithm [26, 27], the steps of the simulation are:
Choice of a simulation cell of adequate shape to the studied phenomena. The size of the simulation cell is related to a scale of length characteristic of the forces and interaction potential of the studied phenomenon. This cell may contain Npc particles (and/or elements).
Choice of an initial configuration that responds to some physical and thermodynamic properties. The total or internal energy of the system is Ei.
Infinitesimal random displacement of a particle (or element of the system) and calculation of the new internal energy of the system Ef. This displacement is related to the physical magnitudes: time scale and length scale.
If
If
If
Figure 2 shows how to choose between the selected configurations. Let
Configuration choice according to Metropolis scheme.
Numerical simulation using the MC method is a very important tool for the study of static properties. The basic algorithm is based on probability notions. Understanding of the distribution function and/or interaction potentials is the heart of the calculation.
In equilibrium statistical physics, the system has a certain probability that can be in any states. The probability of being in a state
where T is the absolute temperature and kB is called Boltzmann’s constant. It is conventional to denote the quantity (kBT)−1 by the symbol β. The normalizing factor Z, or partition function, is given by:
The average of a quantity Q fora system in equilibrium is:
The internal energy U, is given by:
which can be written in terms of a derivative of the partition function:
From thermodynamics we have expressions for the specific heat C, the entropy S, and the Helmholtz free energy F:
or
and
and
We can calculate other parameters affecting the system.
The Monte Carlo method is an excellent technique for estimating probabilities, and we can take advantage of this property in evaluating the results. The simplest and most popular model of a system of interacting variables in statistical physics is the Ising model. It consists of spins
The Ising model has been studied in one and two dimensions to obtain results of thermal properties, phase transition, and magnetic properties [26, 27, 28]. For chosen values of J and/or B, different steps may be taken for the calculations (simulation cell, initialization, configurations, boundary conditions, calculation algorithms). For any configuration, each spin takes the two possible directions. The detail of the calculation procedure is not the purpose of this chapter.
We give a system of N particles (atoms, molecules, ions or particles) placed in a cell of fixed volume, generally of cubic form. The initial positions may, depending on the case, be distributed randomly according to a certain law (uniform or otherwise) or have a given symmetry. In a fluid, a gas, or a plasma, the particles may have random positions in general; in a solid or surface, with a crystal structure, the particles take ordered positions. The choice of random initial positions allows great freedom on the choice of the number of particles in the cell.
At the first step, the particles are given velocities that are generally selected to have a zero total momentum. If the system is in thermodynamic equilibrium, the initial velocities will be randomly chosen according to a Maxwell-Boltzmann law. In the general case, the velocity distribution is according to the problem dealt with. All other phase properties can be initialized to the particles; the main thing is the conservation of the total quantities of the system.
The particles interact with each other according to chosen interaction potentials. Since the interaction potentials are specific for each “numerical experiment,” the main part of the work consists in calculating the interaction energies for each proposed configuration.
The choice of interaction potentials is directly related to the mathematical formulation of the problem according to the state of the medium: fluid, gas, plasma, or solid. It can be Lennard-Jones potential, Coulomb potential, Debye potential, Morse potential, Stillinger-Weber potential, Born-Mayer potential, Moliere potential, or others.
In general, two main boundary conditions are used: periodic boundary conditions (PBC) and minimum image convention (MIC) [29].
To minimize the surface effect, periodic boundary conditions (PBC) [30] are invariably imposed. The simulation cell is reproduced throughout the space to form an infinite mesh. We can simulate the properties of an infinite system. The particles that we follow are in the central cell; if a particle crosses a wall with a certain velocity, its image returns with the same velocity by the opposite wall. Under these conditions, the number of particles in the central cell, and consequently the density, is constant. These conditions also allow the conservation of the energy and the momentum of the system and do not introduce periodic effects (because of the interaction between particles).
According to the hypotheses and according to the geometry of the problem, other boundary conditions are proposed [26]. For example, in order to model thin films, the simulation cells are longitudinal and parallel to the film; one uses PBC in the directions parallel to the film. In the direction normal to the film, free edge boundary conditions can be used. In such cases, it may be appropriate to also include surface fields and surface interactions. In this way, one can study phenomena such as wetting, interface localization-delocalization transitions, surface-induced ordering and disordering, etc.
The core of the program includes calculating the potential energies of particle configuration and particle collisions. The interactions and collisions between particles can be elastic or inelastic; they can be binary or collective. For computation, the interaction energy of a particle with its neighbors is carried out by refocusing a base cell on the particle. This particle only interacts with particles in this region. This is called the “minimal image convention” (MIC) [1].
Generally, a RANDOM generator of real random numbers ri belonging to the domain {0, 1} (or ri ∈ {0, 1} is available. This distribution law is uniform.
To have a real random number xi belonging to the domain {a, b} (or
To have a real random number xi belonging to the domain {a, b} (or xi ∈ {a,b}) according to a formula (or law) of nonuniform distribution f(x), a histogram technique is used. Let Nm be the number of intervals. If the mesh is regular (Figure 3):
Random number selection according to f (x) distribution.
We define:
We define the sequence:
and the sequence:
Hence each real random number ri belongs to the domain {0, 1} (where ri ∈ {0, 1}) (according to the uniform law); this number belongs to the domain {rxj-1, rxj}. It corresponds to a random value xran of the domain {xj-1, xj}; this number satisfies the formula (or the law) of nonuniform distribution f(x).
This technique can be generalized for a nonuniform distribution law f(x) with an irregular mesh Δxi, or with tabular data f(xi) with
The technique can be generalized, too, for a discrete distribution law f(i) with
In the literature, the reader can find simple algorithms for the choice of random numbers of some simple functions (Gaussian, etc.).
It is necessary to find some parameters allowing the control of the smooth course of the evolution of the system. We must look for the constants of movement. For example for an isolated system, we have the conservation of the total energy and the quantity of matter.
By using the numerical simulation, it is possible to calculate many spatiotemporal quantities F(r,t). These quantities can be positions, speeds, kinetic moments, particle energies, concentrations, transport coefficients, etc. It would then be possible to calculate all other quantities related to F(r,t).
For the calculation of the averages, one can note the quantities on the space, on the time or on both. The histogram methods can be used. Static or dynamic distribution functions and spatial or temporal correlation functions can be calculated. It should be noted that the SMC is much more adequate for static properties because of the probabilistic choice of configurations.
Any calculated function or parameter F(r,t) can be used for another application in another calculation program.
In the MCS model discussed extensively in this chapter, it’s more about collisions between particles. It’s
Other MCS models, named
For statistical physics problem solving (such as thin film deposition problems), MCS models use experimental, numerical, or theoretical data from other methods and models. Models can be improved to
Schematic of a hybrid model of three modules used to study gas mixtures in the PECVD [
To solve statistical physics problems with evolutions as a function of time,
Other CVD and PECVD works on MCS are presented in Ref.s [15, 34, 35, 36, 37, 38]. They show how MCS methods can study properties of gas mixtures and properties of the growth of thin films.
In this section, we present an example of PP-MCS of collisions and reactions in gas phase of SiH4/H2 mixture used in PECVD process. Some paragraphs have been treated in previous works [21, 24].
We use a MCS to study collisions and chemical reactions in gas phase of SiH4/H2 mixture used in the PECVD process. In this phase, important reactions have been identified that contribute to the production and the consumption of hydrogen (H), silylene (SiH2), and silyl (SiH3). The hydrogen consumption reactions SiH4 + H → SiH3 + H2 and SiH3 + H → SiH2 + H2 are found to play a central role in deciding the distribution of hydrogen [39].The plasma chemistry indicates that H atoms and SiH3 radicals play an important role in the a-Si:H deposition process [40]. Experimentally, it is generally accepted that SiH3 radicals dominate a-Si:H and μc-Si film growth from SiH4 plasmas in the PECVD; it is the key precursor of a-Si:H deposition [41]. The proposed MCS allowed to get the ratio SiH2/SiH3 and mean value of densities of species. It provides information on SiH4 dissociation and on the production of SiH3, H, SiH2, and Si2H6 and other important parameters.
The plasma in the PECVD reactor is weakly ionized. For our study, the mixture gas contains 22% of SiH4 and 78% of H2; the pressure is 100 mtorr, the temperature of the gas ranges from 373 to 723 K, the electron temperature is about 2.5 eV, and the electron density is 3. 108 cm−3. The process is considered to be stationary. We take into account electrons and eight neutral species (SiH4, SiH3, SiH2, H, H2, Si2H6, Si2H5, SiH).
Symbol | Reactions | Kreac (cm3/s) |
---|---|---|
R1 | SiH4 + e→SiH3 + H+e | k1 = 3 × 10−11 [42] |
R2 | SiH4 + e→SiH2 + 2H + e | K2 = 1.5 × 10−10 [42] |
R3 | SiH4 + e→SiH + H + H2 + e | K3 = 9.34 × 10−12 [42] |
R4 | SiH4 + e→SiH2 + H2 + e | K4 = 7.19 × 10−12 [42] |
R5 | H2 + e→2H + e | K5 = 4.49 × 10−12 [42] |
R6 | Si2H6 + e→SiH3 + SiH2 + H + e | K6 = 3.72 × 10−10 [42] |
R7 | Si2H6 + e→SiH4 + SiH2 +e | K7 = 1.1 × 1010× (1.(1./(1. + (0.63 × P)))) [43] |
R8 | SiH4 + H→SiH3 + H2 | K8 = 2.8 × 10−11 × exp.(−1250/T) [44] |
R9 | SiH4 + SiH2→Si2H6 | K9 = 1.1 × 1010 × (1.−(1./(1. + (0.63 × P)))) [43] |
R10 | SiH3 + SiH3→SiH4 + SiH2 | K10 = 0.45 × 1.5 × 10−10 [44] |
R11 | SiH4 + Si2H5→SiH3 + Si2H6 | K11 = 5 × 10−13 [42] |
R12 | SiH3 + H→SiH2 + H2 | K12 = 2 × 10−11 [44] |
R13 | SiH3 + Si2H6→SiH4 + Si2H5 | K13 = 4 × 10−10 × exp. (−2500/T) [44] |
R14 | SiH2 + H→SiH + H2 | k14 = 2 × 10−11 [44] |
R15 | Si2H6 + H→Si2H5 + H2 | K15 = 0.66 × 2.4 × 10−10 × exp. (−1250/T) [43] |
R16 | Si2H6 + H→SiH4 + SiH3 | K16 = 0.34 × 2.4 × 10−10 × exp. (−1250/T) [44] |
R17 | SiH + H2→SiH3 | K17 = 2 × 10−12 [43] |
R18 | SiH2 + SiH3→Si2H5 | K18 = 3.77 × 10−13 [43] |
R19 | SiH2 + H2→SiH4 | K19 = 3 × 10−12 × (1. + (1./1. + (0.03 × P))) [43] |
R20 | 2SiH3→Si2H6 | K20 = 0.1 × 1.5 × 10−10 [43] |
R21 | SiH4 + SiH→Si2H5 | K21 = (1.−(1./(1. + (0.33 × P)))) × (6.9 × 10−10) [43] |
List of gas phase reactions and corresponding rate constants [24].
Let
And chemical reaction for the production of A is as:
Rate production and consumption for any species A are taken as:
The MCS is based on binary collisions at the microscopic level. Elastic collisions are between all particles, and inelastic collisions (or effective collisions) are those that result in a chemical reaction. A chemical reaction needs a collision involving at least two particles (atoms, ions, electrons, or molecules). According to kinetic theory, gases consist of particles in random motion. These particles are
Form of the simulation cell.
Let ni be the density of neutral spice
The chosen particle takes randomly three components of space in cell
Let ni and nj be the densities of species
According to the kinetic theory of gases, we have for an incident particle
where <sij> is the cross section of the particle
The mean free path
The time between two collisions τij is then:
For chemical effective reactions (inelastic collisions) between two reactive species
General rules of collision theory are applied:
The new velocities of the colliding particles are calculated using conservation of energy and momentum for elastic collisions.
Conservation of total energy as isolated system.
Movement of the center of mass and relative motion around the center of mass.
The reader can refer to some fundamental physics books that deal with general notions of collisions and corresponding parameters [45, 46, 47, 48].
The plasma in the PECVD reactor is weakly ionized. At low temperature, particles interact occasionally with each other and move under the effect of thermal agitation. In reality, only a small fraction of collisions are effective (result in a chemical reaction) [21].
In our MCS, after traveling a random walk given by a Gaussian distribution, the first chosen particle collides with a second particle (molecule, atom, radical, or electron). The last particle
where
The activation energy is given by:
where the pre-exponential factor is assumed to be the collision frequency factor and Kreac is the rate constant of the gas phase reaction.
The two colliding particles (e.g., the electron and SiH4 molecule) can interact by several reactions (R1, R2, R3, and R4 in Table 1); we choose randomly one of gas phase reactions occurring according to a,
where
All chemical systems go naturally toward states of minimum Gibbs free energy [21, 24]. A chemical reaction tends to occur in the direction of lower Gibbs free energy. To determine the direction of the reaction that is taking place, we use the old and new values of Kreac and the equilibrium constant with reactants and product concentrations. Each set of binary collisions can be related or converted into time. As cited in section (a), Table 1 gives gas phase reactions and corresponding rate constants used in this MCS.
To continue the simulation, after the elastic collision, particle
From Metropolis algorithm, the scheme of this MCS is as follows:
Choices of particle of spice
Choices of random collision with a spice j.
Study of collision type (elastic, inelastic). If the collision is elastic the particle i move with a new velocity and mean free path, and we return to step (b). If the collision is inelastic particles i and j give new particles i’ and j’, according to Metropolis scheme, and we return to step (a) or (b). Periodic boundary conditions are used to keep particles in the elementary cell.
At each step, we can note the different statistics.
Once the species are selected for the simulation model, an estimate of species densities should be made. Following the model of interaction and collisions between particles (binary, collective, etc.), a first choice of the minimum number Ni of particles of each species is made. A first estimate of the sizes (Lx, Ly, Lz) of the elementary cell is made.
The study of the types of interaction potentials and the calculation of the approximate values of the force ranges, the kinetic energies, the internal energies, and the energies of activation make it possible to correct the minimal numbers Ni of particles and the sizes (Lx, Ly, Lz) of the elementary cell.
Let kp be the number of a species, kp = 1,…, 9. The minimal numbers Qnp(kp) and the sizes (Lx, Ly, Lz) have to be discussed for statistical calculations.
For numerical programming, according to the programming language used and according to the size (or the computational capacity) of the computer, it is necessary to find a judicious choice of the tables of integer or real values and which values would be useful to save all during simulation. Let Ncol,m be the maximum number of elastic collisions per particle, and let Ncycle be the number of cycles to average the simulation calculations.
For this MCS, the numerical chosen values are in Table 2.
Cell dimensions and steps for collisions | Number of species Kp | Initial number of particles in cell | ||
---|---|---|---|---|
Lx (m) | 4.68 10−6 | 1 | Qnp(SiH4) | Qnp1 |
Ly (m) | 4.68 10−6 | 2 | Qnp(SiH3) | 10 |
Lz (m) | 20.0 10−3 | 3 | Qnp(SiH2) | 10 |
4 | Qnp(H) | 10 | ||
Ncol,m | 500 | 5 | Qnp(H2) | Qnp5 |
Ncycle internal cycle | 2000 | 6 | Qnp(Si2H6) | 10 |
Ncycle external cycle | 200,000 | 7 | Qnp(SiH) | 10 |
8 | Qnp(Si2H5) | 10 | ||
9 | Qnp(e) | Qnp9 |
Used quantities and parameters in calculations for the gas temperature Tg = 520 K.
For radicals (e.g., SiH3), particle numbers Qnp(k) are very small; we take Qnp(k) = 10. These numbers cannot take value 1 or 0, even if a species k is in trace form in the gas. The value 0 for a species k means that any other species k’ does not make a collision with the species k; and the value 1 means that we have no collisions between particles of the same species in the cell.
Qnp1, Qnp5, and Qnp9 are calculated from the volume of cell, the pressure, the temperature, and the total number of particles in the cell (Qnp1 = 0.81187824 * 109; Qnp5 = 0.20296956 * 109; Qnp9 = 131).
As we have chosen a stationary regime, we must reach the values and properties at equilibrium. The results of the simulation show this trend. In MCS, averaged values, distribution functions, autocorrelation functions, and correlation functions can be calculated. To ensure rapid convergence of calculations, it would be useful to look for statistically symmetric (or stationary or unsteady) parameters [26, 50].
As an example for our MCS calculation, we have:
The number of Si2H6, SiH, and Si2H5 particles reaching the surface is negligible.
Let Ns,i and Ns, H2 be the densities of a species
Let Ns,i be the density of a species
The reactions begin with the dissociation (consumption) of H2 and SiH4 by R5, R1, and R2 reactions.
The production of SiH3 is done by R8, and then there is production of SiH2 by R12.
The reaction R2: SiH4 + e → SiH2 + 2H + e plays the central role in SiH4 dissociation by electron impact [24]. This result is compatible with [39].
The second important chemical reaction in the SiH4 dissociation is R1: SiH4 + e → SiH3 + H + e [24]. This result is compatible with that of Perkins et al. [51] and that of Doyle et al. [52].
Type | H2 | SiH4 | H | SiH3 | SiH2 |
---|---|---|---|---|---|
Ns,i/Ns, H2 | 1 | 0.23 | 1.67 10−4 | 8.60 10−5 | 9.86 10−6 |
Ratios Ns,i/Ns, H2 of particles reaching the surface compared to H2.
Type | SiH4 | SiH3 | SiH2 |
---|---|---|---|
6.695 10−6 | 7.965 10−6 | 775 10−6 |
Ratios Ns,i/Nv,i of particles reaching the surface compared to volume.
MCS is a widely used method in statistical physics to study thermodynamic, structural, or phase properties. It is based on random and probabilistic processes. The purpose of this chapter is to present the technique for general use in physics for the study of thin film deposition problems. The technique can be generalized to other fields of science: biology, economics, transportation, and social sciences.
We started by presenting general rules for numerical simulation methods. Metropolis algorithm has been considered as the basic algorithm. After, we presented the different steps for the realization of a MCS code. We chose the particle-particle model MCS (PP-MCS) to explain the different steps and procedures to be applied in the deposition of thin layers by PECVD processes. We have shown that this technique can be generalized to the particle-in-cell MCS (PIC-MCS) case or kinetic MCS (kMCS), as it can be joined with other modules to give hybrid models. It is important to know how to choose random configurations from the laws or probability distributions in the system.
A numerical application is presented for collisions in a SiH4/H2 gas mixture in the PECVD process. A preliminary work of determination of the chemical reactions between molecules and radicals is made. A choice of the simulation cell is made, and the definition of the probabilities of the collisions between peers is made. The Metropolis algorithm makes it possible to follow the various elastic and inelastic collisions; it also makes it possible to make the statistics of the interactions with the surface. The results are compatible with [39, 51, 52].
Other questions may be asked to account for molecular ions, surface and volume kinetics, or thin film formation. The techniques and different models of the MCS (PP-MCS, MCS-PIC, kMCS) allow taking care of these questions.
The interconnection of the MCS with other models (MDS, hybrid model, fluid model, electromagnetic model, etc.) would allow answering more questions. The methods can be applied to other specialties than the physical sciences.
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