Dilution mixing and incubation time.
\\n\\n
More than half of the publishers listed alongside IntechOpen (18 out of 30) are Social Science and Humanities publishers. IntechOpen is an exception to this as a leader in not only Open Access content but Open Access content across all scientific disciplines, including Physical Sciences, Engineering and Technology, Health Sciences, Life Science, and Social Sciences and Humanities.
\\n\\nOur breakdown of titles published demonstrates this with 47% PET, 31% HS, 18% LS, and 4% SSH books published.
\\n\\n“Even though ItechOpen has shown the potential of sci-tech books using an OA approach,” other publishers “have shown little interest in OA books.”
\\n\\nAdditionally, each book published by IntechOpen contains original content and research findings.
\\n\\nWe are honored to be among such prestigious publishers and we hope to continue to spearhead that growth in our quest to promote Open Access as a true pioneer in OA book publishing.
\\n\\n\\n\\n
\\n"}]',published:!0,mainMedia:null},components:[{type:"htmlEditorComponent",content:'
Simba Information has released its Open Access Book Publishing 2020 - 2024 report and has again identified IntechOpen as the world’s largest Open Access book publisher by title count.
\n\nSimba Information is a leading provider for market intelligence and forecasts in the media and publishing industry. The report, published every year, provides an overview and financial outlook for the global professional e-book publishing market.
\n\nIntechOpen, De Gruyter, and Frontiers are the largest OA book publishers by title count, with IntechOpen coming in at first place with 5,101 OA books published, a good 1,782 titles ahead of the nearest competitor.
\n\nSince the first Open Access Book Publishing report published in 2016, IntechOpen has held the top stop each year.
\n\n\n\nMore than half of the publishers listed alongside IntechOpen (18 out of 30) are Social Science and Humanities publishers. IntechOpen is an exception to this as a leader in not only Open Access content but Open Access content across all scientific disciplines, including Physical Sciences, Engineering and Technology, Health Sciences, Life Science, and Social Sciences and Humanities.
\n\nOur breakdown of titles published demonstrates this with 47% PET, 31% HS, 18% LS, and 4% SSH books published.
\n\n“Even though ItechOpen has shown the potential of sci-tech books using an OA approach,” other publishers “have shown little interest in OA books.”
\n\nAdditionally, each book published by IntechOpen contains original content and research findings.
\n\nWe are honored to be among such prestigious publishers and we hope to continue to spearhead that growth in our quest to promote Open Access as a true pioneer in OA book publishing.
\n\n\n\n
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\r\n\tLiquid atomization, sprays, and fuel injection are at the heart of many chemical-reaction based propulsion and electric energy production such as internal combustion reciprocation, gas turbine and liquid rocket engines. It is known that both performance and emission of pollutants from such systems are heavily influenced by details of the fuel sprays generated from the fuel nozzles or injectors. For these reasons there has been an increasing number of R&D works targeted towards generation of fuel sprays with a desired characteristics to “match” the environment into which they are injected and subsequently burned. Such research efforts span from injector designs to spray characterization and dynamics using both experimental and modeling/simulation approaches. Therefore, the main goal of this book is to cover important advances in this field for the past fifteen years.
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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. Whether that be identifying an exceptional author and proposing an editorship collaboration, or contacting researchers who would like the opportunity to work with IntechOpen, I establish and help manage author and editor acquisition and contact."}},relatedBooks:[{type:"book",id:"1591",title:"Infrared Spectroscopy",subtitle:"Materials Science, Engineering and Technology",isOpenForSubmission:!1,hash:"99b4b7b71a8caeb693ed762b40b017f4",slug:"infrared-spectroscopy-materials-science-engineering-and-technology",bookSignature:"Theophile Theophanides",coverURL:"https://cdn.intechopen.com/books/images_new/1591.jpg",editedByType:"Edited by",editors:[{id:"37194",title:"Dr.",name:"Theophanides",surname:"Theophile",slug:"theophanides-theophile",fullName:"Theophanides Theophile"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"3161",title:"Frontiers in Guided Wave Optics and Optoelectronics",subtitle:null,isOpenForSubmission:!1,hash:"deb44e9c99f82bbce1083abea743146c",slug:"frontiers-in-guided-wave-optics-and-optoelectronics",bookSignature:"Bishnu Pal",coverURL:"https://cdn.intechopen.com/books/images_new/3161.jpg",editedByType:"Edited by",editors:[{id:"4782",title:"Prof.",name:"Bishnu",surname:"Pal",slug:"bishnu-pal",fullName:"Bishnu Pal"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"3092",title:"Anopheles mosquitoes",subtitle:"New insights into malaria vectors",isOpenForSubmission:!1,hash:"c9e622485316d5e296288bf24d2b0d64",slug:"anopheles-mosquitoes-new-insights-into-malaria-vectors",bookSignature:"Sylvie Manguin",coverURL:"https://cdn.intechopen.com/books/images_new/3092.jpg",editedByType:"Edited by",editors:[{id:"50017",title:"Prof.",name:"Sylvie",surname:"Manguin",slug:"sylvie-manguin",fullName:"Sylvie Manguin"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"371",title:"Abiotic Stress in Plants",subtitle:"Mechanisms and Adaptations",isOpenForSubmission:!1,hash:"588466f487e307619849d72389178a74",slug:"abiotic-stress-in-plants-mechanisms-and-adaptations",bookSignature:"Arun Shanker and B. 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"}},{type:"book",id:"72",title:"Ionic Liquids",subtitle:"Theory, Properties, New Approaches",isOpenForSubmission:!1,hash:"d94ffa3cfa10505e3b1d676d46fcd3f5",slug:"ionic-liquids-theory-properties-new-approaches",bookSignature:"Alexander Kokorin",coverURL:"https://cdn.intechopen.com/books/images_new/72.jpg",editedByType:"Edited by",editors:[{id:"19816",title:"Prof.",name:"Alexander",surname:"Kokorin",slug:"alexander-kokorin",fullName:"Alexander Kokorin"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"314",title:"Regenerative Medicine and Tissue Engineering",subtitle:"Cells and Biomaterials",isOpenForSubmission:!1,hash:"bb67e80e480c86bb8315458012d65686",slug:"regenerative-medicine-and-tissue-engineering-cells-and-biomaterials",bookSignature:"Daniel Eberli",coverURL:"https://cdn.intechopen.com/books/images_new/314.jpg",editedByType:"Edited by",editors:[{id:"6495",title:"Dr.",name:"Daniel",surname:"Eberli",slug:"daniel-eberli",fullName:"Daniel Eberli"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"57",title:"Physics and Applications of Graphene",subtitle:"Experiments",isOpenForSubmission:!1,hash:"0e6622a71cf4f02f45bfdd5691e1189a",slug:"physics-and-applications-of-graphene-experiments",bookSignature:"Sergey Mikhailov",coverURL:"https://cdn.intechopen.com/books/images_new/57.jpg",editedByType:"Edited by",editors:[{id:"16042",title:"Dr.",name:"Sergey",surname:"Mikhailov",slug:"sergey-mikhailov",fullName:"Sergey Mikhailov"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"1373",title:"Ionic Liquids",subtitle:"Applications and Perspectives",isOpenForSubmission:!1,hash:"5e9ae5ae9167cde4b344e499a792c41c",slug:"ionic-liquids-applications-and-perspectives",bookSignature:"Alexander Kokorin",coverURL:"https://cdn.intechopen.com/books/images_new/1373.jpg",editedByType:"Edited by",editors:[{id:"19816",title:"Prof.",name:"Alexander",surname:"Kokorin",slug:"alexander-kokorin",fullName:"Alexander Kokorin"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"2270",title:"Fourier Transform",subtitle:"Materials Analysis",isOpenForSubmission:!1,hash:"5e094b066da527193e878e160b4772af",slug:"fourier-transform-materials-analysis",bookSignature:"Salih Mohammed Salih",coverURL:"https://cdn.intechopen.com/books/images_new/2270.jpg",editedByType:"Edited by",editors:[{id:"111691",title:"Dr.Ing.",name:"Salih",surname:"Salih",slug:"salih-salih",fullName:"Salih Salih"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"872",title:"Organic Pollutants Ten Years After the Stockholm Convention",subtitle:"Environmental and Analytical Update",isOpenForSubmission:!1,hash:"f01dc7077e1d23f3d8f5454985cafa0a",slug:"organic-pollutants-ten-years-after-the-stockholm-convention-environmental-and-analytical-update",bookSignature:"Tomasz Puzyn and Aleksandra Mostrag-Szlichtyng",coverURL:"https://cdn.intechopen.com/books/images_new/872.jpg",editedByType:"Edited by",editors:[{id:"84887",title:"Dr.",name:"Tomasz",surname:"Puzyn",slug:"tomasz-puzyn",fullName:"Tomasz Puzyn"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}]},chapter:{item:{type:"chapter",id:"67121",title:"Interface Treatment for Conjugate Conditions in the Lattice Boltzmann Method for the Convection Diffusion Equation",doi:"10.5772/intechopen.86252",slug:"interface-treatment-for-conjugate-conditions-in-the-lattice-boltzmann-method-for-the-convection-diff",body:'Heat and mass transfer between multi-phases or different materials with interfacial conjugate conditions is frequently encountered in fundamental sciences and numerous engineering applications involving fluid dynamics, thermal transport, materials sciences, and chemical reactions. Examples are cooling of turbine blades, heat exchangers and electronic devices, thermal insulation on heat pipes and chemical reactors, heat conduction in composite materials, and heat and mass transfer between solid particles and their surrounding fluids [1, 2, 3, 4, 5, 6, 7, 8], to name a few. The most well-known conjugate conditions include the continuity of both the temperature (concentration) and the heat (mass) flux at the interface. Other conjugate conditions, such as with temperature (concentration) jumps and/or flux discontinuities [9], and Henry’s law relationship [10], are also noticed at fluid-solid interfaces or interfaces of two solids or fluids of different thermal (mass diffusion) properties.
The non-smoothness or discontinuities/jumps in the physical or transport properties, and consequently in the distribution of the temperature (concentration) field across the interface, pose a great challenge to any numerical method applied to solve the interface problems. Development of accurate and efficient numerical schemes to treat the interface conditions has attracted much attention in the literature, such as the immersed boundary method (IBM) [11, 12], the immersed interface method (IIM) [13, 14], the ghost fluid method (GFM) [15, 16], the sharp interface Cartesian grid method [17, 18], and the matched interface and boundary (MIB) method [19]. Most of these methods are formulated in the finite-difference, finite-volume or finite-element frameworks.
When applying those traditional numerical methods, a popular approach to implement the conjugate conditions is to employ iterative schemes, in which a Dirichlet interface condition is imposed for one phase or material and a Neumann interface condition for the other. The heat and mass transfer in each phase is separately solved, and the continuity or prescribed jump condition at the interface could be satisfied after multiple iterations. For conjugate transport with complex interface geometry, the iterative schemes would become difficult to implement and they normally necessitate a considerable amount of computational effort.
The lattice Boltzmann method (LBM), which has emerged as an attractive alternative numerical method for modeling fluid flows and heat mass transfer (see [20, 21, 22] and Refs. therein), has been demonstrated to be an effective and efficient numerical approach for conjugate interface conditions in tandem with the convection diffusion equation (CDE) [9, 23]. In this chapter, we present a critical review of the various interface schemes proposed in the literature, with a focus on the comparison of numerical accuracy.
The well-known features of the LBM method include its explicit algorithm, ease in implementation, capability to treat complex geometry, and compatibility with parallel computing [20, 21]. Boundary condition treatment is essential to the integrity of LBM since the kinetic theory-based method deals directly with the microscopic distribution functions (DFs) rather than the macroscopic conservation equations. Earlier LB models treat the collision effects with a single-relaxation-time (SRT) approximation, commonly referred to as the Bhatnagar-Gross-Krook (BGK) model [24, 25, 26]. However, the SRT model is limited such that it can only describe isotropic diffusion [20]. In recent years, models such as the two-relaxation-time (TRT) [27, 28] and multiple-relaxation-time (MRT) [20, 29, 30] LB models have been proposed that can handle anisotropic diffusion. Representative LB models proposed in the literature include the general BGK model by Shi and Guo [31] for the nonlinear CDE, the D3Q7/D2Q5 MRT models by Yoshida and Nagaoka [20] for the general convection anisotropic diffusion equation, and the D1Q3/D2Q9/D3Q19 MRT models by Chai and Zhao [30] for the general nonlinear convection anisotropic diffusion equation, to name a few. The MRT models have improved numerical accuracy and stability compared to the SRT models [20, 28, 30]. The D3Q7/D2Q5 model proposed in [20] is used for this review, as it preserves second order spatial accuracy when recovering the general CDE following an asymptotic analysis. Based on the D3Q7/D2Q5 LB models, Li et al. [21] proposed second-order accurate boundary treatments for both the Dirichlet and Neumann conditions; they have also established a general framework for heat and mass transfer simulations with direct extension to curved boundary situations. In their framework, explicit analytical expressions were developed to relate the macroscopic quantities, such as boundary temperature (concentration) and their fluxes, and interior temperature (concentration) gradients, to the microscopic DFs in the LB model.
The first work that explicitly addressed the fluid-solid interface condition in LBM was conducted by Wang et al. [6]. They proposed a simple “half lattice division” (HLD) treatment in which no special treatment is required and the temperature and flux continuity condition at the interface was automatically satisfied for steady cases. Improvement of this HLD based scheme was conducted by several groups (see a review in Ref. [23]) for unsteady cases. Importantly, with the interface treated as a shared boundary between the adjacent domains, the boundary conditions by Li et al. [21] were applied to interface conditions and particular interface schemes were proposed and verified in [23] for standard conjugate conditions, and in [9] for conjugate heat and mass transfer with interfacial jump conditions. This idea of developing analytical relationships for the DFs to satisfy the conjugate conditions was also extended to handle general interface conditions in [32, 33]. In all the previous schemes in [9, 23, 32, 33], the local geometry was taken into account and the second-order accuracy of the LBM solution can be preserved.
There is another category of interface schemes that has attracted interest in the LBM community. In those schemes, additional source terms [34], alternative LBE formulations [35, 36], or modified equilibrium DFs [37, 38], were proposed to handle the conjugate conditions. The main motivation for those schemes is to avoid the consideration of the interface geometry or topology, which can be a challenge in complex systems such as porous media. As pointed out in [33], however, these schemes usually suffer from degraded numerical accuracy and/or convergence orders. This perspective will be illustrated in detail in this chapter.
The rest of this chapter is organized as follows. In Section 2, the different types of conjugate conditions are presented. The LB model for the general CDE is briefly described in Section 3. In Section 4, the representative interface schemes are summarized. Discussion on the numerical accuracy of the selected interface schemes is provided in Section 5 with representative numerical examples. Conclusions and outlook are given in Section 6.
In order to define the conjugate conditions, we begin by defining two domains 1 and 2, as shown in Figure 1. The conjugate interface conditions, from a heat and mass transfer perspective, can be defined as:
Illustration of the local geometry of an interface in the lattice (filled circles: lattice nodes in domain 1, filled squares: interface nodes, and open circles: lattice nodes in domain 2). With permission from [
where
Some examples of jump conditions can be found in cases such as concentration jumps (Henry’s law [10]) or temperature jumps at the interface [9]. Eqs. (1a)–(1c) reduce to the standard conjugate conditions in [23] with no jumps and zero normal velocity; they can also be extended to yield two general relationships between interfacial scalar values and their fluxes as in [32, 33].
The governing heat and mass transfer equation within each domain can be written as a general convection diffusion equation as:
where
For fluid flow simulations, the D2Q9/D3Q19 LB models are the most popular selections due to their accuracy and robustness [39]. While for the scalar CDE (2), the D2Q5/D3Q7 LB models are most widely used [20, 40]. To recover the CDE to second-order accuracy, the evolution equation follows
where the microscopic distribution function,
The equilibrium distribution function can be defined as [20, 40]
where
When using the multiple-relaxation-time (MRT) collision operator, and the collision-streaming process for efficient computations, Eq. (3) is split into two steps:
In the above,
One unique feature of the LBM method is that both the Dirichlet-type boundary value and the Neumann-type boundary flux, i.e., temperature/concentration gradient, can be obtained from a simple moment of the distribution functions with appropriate boundary schemes [20, 21]. It eliminates finite-difference type approximation schemes for the flux. This idea can also be applied to construct interface schemes by treating the interface as a shared boundary between the two adjacent domains [9, 23, 32, 33]. We consider the basic situation with zero convective flux (
with σ = 1,
Depending on whether the local interface geometry is considered, the interface schemes fall into two different categories, as discussed in Sections 4.1 and 4.2.
According to the second-order interpolation-based boundary schemes developed in [21], the relationships between the distribution functions and the interfacial
Similarly, for the Neumann condition treatment,
where
When
The coefficients in Eqs. (11a) and (11b) are now determined by the geometry fraction Δ and the property ratio σ. It is worth noting that there is an adjustable parameter in those coefficients since the second-order Dirichlet boundary scheme allows one adjustable parameter, as shown in [21], where three particular Dirichlet schemes were also presented:
The corresponding interface schemes, can thus also be obtained. Those schemes will be numerically verified in Section 5.2 for a test case including interfacial jump conditions.
When the straight interface is located “halfway” between the lattice nodes (∆ = 0.5), the unknown DFs can be calculated by only knowing two single-node post-collision DFs, i.e., without interpolation:
Furthermore, for the most simplified case of ∆ = 0.5 and σ = 1, Eqs. (13a) and (13b) reduce to:
For straight interfaces where Δ ≠ 0.5 and for curved interfaces, the complete interface conditions can be found in [9, 23].
It should be noted that second-order accurate boundary schemes can also be obtained using only the single lattice node next to the boundary, as demonstrated in [32], instead of using interpolations in Eqs. (9) and (10). However, such boundary schemes were constructed with complex coefficients that are related not only to geometry-related Δ, but also to the LB model-related relaxation coefficient. Interested readers are referred to [32] for details and such interface schemes are not discussed in this chapter.
The second-order interpolation based interface scheme developed in [23] has attracted much interest. In the last 5 years, there have been various modified interface schemes proposed with the objective of simplifying the original scheme, as it becomes complex and computationally expensive when applied to curved or irregular interfaces. The applicability of those modified schemes was demonstrated in those publications while their accuracy and convergence order have not been fully investigated. In this section, we present three groups of those modified schemes that do not account for the local interface geometry. Most of those schemes were formulated for conjugate heat transfer problems, and conjugate mass transfer can be similarly handled. Comparison of their numerical accuracy with the original interpolation based scheme will be presented in Section 5.1.
In the first group, additional source terms were introduced to the lattice nodes in the domains next to the interface. For example, the following additional source was given in [34]:
In LBM, the total flux in the second bracket can be conveniently obtained from the moment of the nonequilibrium DFs [20, 21, 34]. In [34], the gradient of the heat capacity-related term in Eq. (15) was computed from a first-order one-sided finite-difference (FD) scheme:
It should be noted that the above introduction of the source term and the calculation of the heat capacity-related gradient cause two issues: first, for adjacent domains with distinct
Another area of interest is found in enthalpic formulations for the LBE. With the definition of an “enthalpic term”
Comparing Eq. (16) with Eq. (2), the last two terms in Eq. (16) need to be included in the source term implementation in LBM simulations. Clearly, those additional terms also have the heat capacity-related gradients and thus the discontinuity effect. In [36], the gradient was approximated from
In this group, modified equilibrium DFs were introduced. The heat capacity is typically involved in the modified equilibrium DFs, such as [37, 38]
When using the MRT D2Q5 model the equilibrium moments are calculated to be
The temperature is solved from
In order to verify the applicability of the different interface schemes to simulate conjugate heat and mass transfer and compare their accuracy, we consider two benchmark cases with analytical solutions: (i) 2D convection-diffusion in a channel with two-layered fluids, and (ii) 2D diffusion within a circular domain of two solids with interfacial jump conditions. The computational domains are depicted in Figures 2 and 9, respectively. Those two cases have been widely employed in [9, 23, 32, 33, 34] to verify the various interface schemes.
Schematic layout of the lattice on a 2D channel containing two fluids in domain 1 (0
For straight interfaces, the following relative L2 norm errors are defined to check the numerical accuracy and convergence orders following [23]:
where
The computational domain is depicted in Figure 2. The two fluids are assumed immiscible and both have the same velocity
When considering isotropic diffusion, the governing CDE can be expressed as
We consider only the steady case with sinusoidal boundary conditions on the horizontal walls
The interpolation-based interface scheme in Section 4.1 and the modified schemes in Groups 1–3 in Section 4.2 are implemented with the D2Q5 MRT-LB model. For all cases presented,
Figures 3a and b show the scalar value and flux profiles at selected locations along the
Profiles of the (a) scalar variable, and (b) flux values at selected vertical lines in the channel.
Comparison of (a) interfacial scalar value, and (b) interfacial fluxes.
Figures 5–7 show the
Relative L2 norm error, E2, for the interior scalar versus the grid resolution, 1/H, for steady convection-diffusion in the channel at Δ = 0.5.
Relative L2 norm error, E2, ϕint, for the interfacial scalar versus the grid resolution, 1/H, for steady convection-diffusion in the channel at Δ = 0.5.
Relative L2 norm error, E2, qint, for the interfacial flux versus the grid resolution, 1/H, for steady convection-diffusion in the channel at Δ = 0.5.
Figure 5 clearly shows that the original scheme in [23] and Group 3 are able to preserve the second-order accuracy in LBM. However, Groups 1 and 2 show only a linear convergence at low resolution, and it reduces further towards zeroth-order convergence at high resolution. Similar observations can be found in the errors for the interfacial scalar values in Figure 6. For the interfacial flux errors shown in Figure 7, Group 2 always exhibits zeroth-order accuracy, while Group 1 exhibits linear convergence in one domain and zeroth-order in the other. As previously mentioned, a discontinuity approximation is present in the development of interface schemes in Groups 1 and 2. This is the direct cause for the degradation of the order of accuracy and the much higher error magnitude in these two groups.
To further demonstrate the necessity of taking into account the local geometry to preserve second-order accuracy, Figure 8 presents the
Relative L2 norm error, E2, for the interior scalar versus the grid resolution, 1/H, for steady convection-diffusion in the channel at different Δ values.
It is clear in Figure 8 that only the interpolation-based scheme considering the local geometry is able to preserve second-order accuracy at different Δ values. The error behavior for Groups 1 and 2 is similar to that in Figure 5, the near identical errors at high resolution for different Δ values confirm the error caused by the same discontinuity; moreover, while Group 3 presents second-order convergence for Δ = 0.5, the convergence order drops to first-order for Δ ≠ 0.5. This is similar to the behavior of the “half-lattice division” scheme discussed in [23]. It is evident that when the local interface geometry is not considered for cases Δ ≠ 0.5, the inherent second-order accuracy in LBM computation can be lost.
This test is applied to demonstrate the applicability and accuracy of the interpolation-based interface scheme to simulate transport in complex geometry and with interfacial jump conditions. Figure 9 schematically presents the lattice layout and computational domain for 2D diffusion within a circular domain. On the outer circle with radius,
Schematic layout of the computational domain for circular diffusion. With permission from [
and
the analytical solution can be found [9]. For the numerical LBM computation, the parameters are
Figures 10 and 11a, b provide the respective relative
Relative L2 norm error, E2, for the interior temperature versus the grid resolution for 2D diffusion in a circular domain. With permission from [
Relative L2 norm errors (a) E2_tint for the interfacial temperature, and (b) E2_qint for the interfacial flux, versus the grid resolution for 2D diffusion in a circular domain. With permission from [
The results in Figures 10 and 11 demonstrate the first-order accuracy at high resolution for all cases. It should also be noted that the temperature and flux jump conditions at the interface do not affect the order of convergence. The decrease of the convergence order from second-order in Section 5.1 to first-order in this test is due to the implementation of the Cartesian decomposition method [21] that was used to convert the normal fluxes into those in the discrete velocity directions. It is expected that the modified geometry-ignored interface schemes in Groups 1–3 would result in much higher error magnitude for curved interfaces, and Groups 1 and 2 would yield only zeroth-order convergence for all the three quantities of interest. This will be presented in future publications.
The chapter presents a brief review of the interface schemes within the scope of the lattice Boltzmann method (LBM) for conjugate transport between multiphases or different materials. Compared to the interface schemes developed to satisfy the macroscopic conjugate conditions using traditional CFD methods, the LBM method deals with the microscopic distribution functions (DFs); the physical conjugate conditions can be converted to those for the DFs, and they are satisfied in each time-marching step without iterations. In the last decade, a number of interface schemes have been proposed. The interpolation-based schemes [9, 23, 33] taking into account the local interfacial geometry are able to preserve the second-order accuracy in LBM for straight interfaces; while those “modified” geometry-ignored schemes [34, 35, 36, 37, 38] have at most first-order accuracy in general, and with the introduction of heat capacity-related discontinuity in those schemes (e.g., Groups 1 and 2), the order of accuracy becomes essentially zeroth order.
Furthermore, it is verified that when using the interpolation-based schemes, the interfacial jump conditions can be conveniently modeled with no effect on the order of accuracy of the LBM solutions.
Curved geometry also has a substantial effect and it reduces the order of accuracy of LBM solutions in modeling conjugate heat and mass transfer problems. In addition, the interpolation-based schemes would demand for a higher computational cost than those modified schemes. The readers are thus recommended to take into account both numerical accuracy and computational cost when selecting effective interface schemes for curved geometries.
LL acknowledges the support and start-up fund from the Department of Mechanical Engineering at Mississippi State University.
Endotoxins, a type of pyrogen, are natural compounds found in the outer cell membrane of Gram-negative bacteria and can impact over 30 biological activities. Endotoxin can lead to cell death by initiating complement activation. The Limulus amebocyte lysate (LAL) test was commercially introduced in the 1970s. LAL is derived from the blood cells, or amebocytes, of the horseshoe crab,
Limulus amebocyte lysate test is an aqueous extract of blood cells (amoebocytes) which obtain from the horseshoe crab (
Activation of inflammation in body [
Horseshoe crab [
Among the most well-known and important applications of the LAL test are the ones related to the pharmaceutical industry. It can be said that the most common pyrogens in pharmaceutical products are endotoxins, which is why the pyrogen tests on rabbits have been replaced by the LAL test according to the recommendations of the international pharmacopeia. One of the reasons that has made the LAL test prevail in the pharmaceutical industry is the careful avoidance by the LAL manufacturers of bringing harm to live animals during both production and testing. It is important to clarify that the crabs, from which part of the hemolymph used for the LAL test was extracted, are returned to alive to their natural habitat with no lasting problems after the extraction.
Add volume of lysate to a volume of product dilution. Incubating the reaction mixture at 37.5°C. Endotoxin in the reaction would activate the LAL reagent. Cleave small chromogenic peptides and liberates pNA. pNA, color is yellow and absorbs light at 405 nm. For samples that absorb at 405–410 nm, Diazo-coupling agent modification may be used. In this method, pNA reacted with nitrite in hydrochloric acid, ammonium sulfamate and N-(1-naphthyl)-ethylenediamine (NEDA). Absorbs at a range between 540 and 550 nm. A standard curve is used to establish concentrations in product specimens.
10 × 75 mm fully depyrogenated borosilicate glass culture tubes (Associates of Cape Cod, Inc. catalog numbers TB050).
Optical reader is capable of reading at 405 nm, or at 540–550 nm for the diazo method. Incubator is able to maintaining 37 ± 1°C. A water bath can be used for the endpoint test tube method. Both devices should have a uniform heat distribution. Test tube racks to hold the tubes and/or incubate dilution and reaction tubes. Micropipettes or disposable pipette tips free of interfering endotoxins and glucans are recommended. Vortex-type mixer, Para film (American National Can™) and hot-air oven with the capacity to heat to at least 250°C for depyrogenation of glassware.
Limulus amebocyte lysate (LAL), LAL reconstitution buffer, control standard endotoxins (CSE), solution 1 (nitrite), solution 1A (0.1 N hydrochloric acid), solution 2 (ammonium sulfamate), solution 3 (N-(1-naphthyl)-ethylenediamine (NEDA)), LRW.
The endotoxins limit for USP/BP sterile WFI is only 0.25 EU/ml; therefore, sterile WFI may contain detectable endotoxins and be unsuitable for use. Use certified LRW to make dilutions of standards, and to prepare positive controls.
Collect aseptically containers that are free of detectable endotoxins in depyrogenated glassware apparatus.
The pH must be 6–8. Adjust the pH of the product specimen with dilute HCl, NaOH, or buffer (free of endotoxins). Dilute concentrated HCl or NaOH with LRW. Use a volume that will not lead to significant dilution of the test specimen. Dilution (LRW) alone can overcome the issue sometimes.
Gently tap the vial of lysate. Loose material fall to the bottom. Break the vacuum by lifting the gray stopper. Do not contaminate the mouth of the vial. Remove and discard the stopper. Start the reconstituted lysate with 3.2 ml buffer. Avoid vigorous mixing that may cause excessive foaming and a loss of sensitivity. Wrap the vials with parafilm and store in a cold place (2–8°C) when not in use and use within 8 h of reconstitution.
This is relatively well stable and, if stored properly, will retain full activity through the expiration date on the label. Store the product at 2–8°C. Excess temperature over 37°C cause rapid deterioration, loss of sensitivity and distinct yellowing.
Each vial of control standard endotoxins (CSE) contains 10 ng of endotoxins. Reconstitute CSE with the volume mentioned on the Certificate of Analysis (CA, which gives the potency of the CSE). Gently knocks the vial of control standard endotoxins (CSE) to cause loose material to fall to the bottom. Break the vacuum by lifting the gray stopper. Do not contaminate the mouth of the vial. Remove the stopper and place it in a cold place aseptically for reuse.
Reconstitute CSE with the volume specified on the Certificate of Analysis (CA, which gives the potency of the CSE) and as directed in the package insert. Place the stopper. Vortex the vial for 40–60 s to form a homogenous mixture. Discard solution if not used immediately, vortex the vial for 30 s prior to use.
Read the tubes UV/visible spectrophotometers (Table 1).
CSE + lysate | Incubation time (min) |
---|---|
50 μl of 0.50 EU/ml + 50 μl | 30 |
50 μl of 0.250 EU/ml + 50 μl | 30 |
50 μl of 0.125 EU/ml + 50 μl | 30 |
50 μl of 0.0625 EU/ml + 50 μl | 30 |
Dilution mixing and incubation time.
Sample + lysate | Incubation (min) |
---|---|
50 μl of sample + 50 μl | 30 |
Stop the reaction by adding 50% acetic acid. Add 0.025 ml (25 μl) read the optical density (OD) at 405 nm read the test.
Reconstitute vial 1 with entire contents of vial, reconstitute vial 2 with 4 ml of water, reconstitute vial 3 with 4 ml of water. Add 0.05 ml (50 μl) of solution 1 (sodium nitrite reconstituted with dilute HCL). Add 0.05 ml (50 μl) of solution 2 (ammonium sulfamate). Add 0.05 ml (50 μl) of solution 3 (NEDA) use new pipette tip agitate the plate to mix. Full color (magenta) should develop immediately. Read the test at 540–550 nm.
Positive control must be included to verify that it is appropriate to use the parameters of a previous (archived) standard curve to calculate endotoxin concentrations.
LRW negative controls should be included with each test
1: Equation of straight line (results)
y = mx + c
m = slop
x = endotoxin concentration,
c = y-intercept and
y = mean absorbance
X = y-c/m
Example calculation
Prepare sample solutions by dissolving or diluting drugs (pH 6.0–8.0). The pH may be adjusted by the use of acid, base, or suitable buffers as recommended. Do not exceed the MVD or MCV while making dilutions and adjusting the pH.
MVD = (endotoxin limit × concentration of sample solution)/(λ)
Endotoxin limit given in USP, concentration of a sample of the label, λ: the labeled lysate sensitivity in the gel-clot technique (IU/ml) or the lowest concentration used in the standard curve for the turbidimetric or chromogenic techniques.
MVC = λ/endotoxin limit
λ: the labeled lysate sensitivity in the gel-clot technique (IU/ml) or the lowest concentration used in the standard curve for the turbidimetric or chromogenic techniques.
Sample 1
Endotoxin limit: 0.5 EU/ml
Concentration of sample: 100 mg/ml
λ: 0.06 EU/ml
MVD = 0.5 EU/ml × 100 mg/ml/0.06 EU/ml
MVD = 833
Add 1 ml of sample 1 in to 832 ml of LRW. Prepare sample 2 in using the same method.
Using 10-fold and 2-fold dilution methods prepare the following dilutions of control standard endotoxins (CSE)
0.5 EU/ml
0.25 EU/ml
0.125 EU/ml
0.0625 EU/ml
Reconstitute the lysate with 3.2 ml of buffer provided with it. Follow the standard procedure for reconstitution.
Stop reaction.
For sample 1 and sample 2:
Stop the reaction by adding 50% acetic acid. Add 0.025 ml (25 μl) (Tables 2 and 3).
CSE + lysate | Incubation (min) |
---|---|
50 μl of 0.50 EU/ml + 50 μl | 30 |
50 μl of 0.250 EU/ml + 50 μl | 30 |
50 μl of 0.125 EU/ml + 50 μl | 30 |
50 μl of 0.0625 EU/ml + 50 μl | 30 |
50 μl of sample 1 + 50 μl | 30 |
50 μl of sample 2 + 50 μl | 30 |
Different dilution of CSE and lysate.
Make two replicates of each CSE and sample preparation to reduce any errors.
Use Microsoft word for further calculations and results. Make standard curve and endotoxin concentration (Figure 3).
Validation of standard curve.
R2 = coefficient of determination
R = correlation coefficient
R ≥ 0.98
R2 = 0.99
R = √R2 = 0.99
Equation of straight line
y = mx + c
m = slop
x = endotoxin concentration
c = y intercept
y = mean absorbance
Equation of straight line
Y = 1.019X − 0.026
X = Y + 0.026/1.019
m = slop = 1.019,
C = y intercept = 0.026,
Y = mean absorbance,
X = endotoxin concentration
X = Y + 0.026/1.019
Y = 0.300, X = 0.300 + 0.026/1.019, X = 0.319EU/ml
X = Y + 0.026/1.019
Y = 0.335, X = 0.335 + 0.026/1.019, X = 0.354 EU/ml
Gel Clot LAL provides a simple positive/negative result and is most often mentioned in pharmacopeial monographs as the official referee test.
This is very easy to perform.
This is not time consuming.
Accuracy is 100 percent.
The LAL Gel-Clot assay, gives a more quantitative measurement of endotoxin over a range of concentrations.
Gel Clot lysate for 20 test, Gel Clot standard 0.5 EU/Vial, LAL reagent water (LRW 50 ml).
Lysate: add 2 mL LRW and mix it slowly. Do not shake and avoid foaming. Transfer 0.1 ml in 20 test tubes. Store it at –degree (in freezer).
Standard: Add 2 mL of LRW in the vial and mix it well for 15 min. Store the vial at 2–8°C. Storage life is 15 days.
Take three test tubes and mark them as test, positive control and negative control [1].
Add your sample in test tube marked as sample. Add standard in test tube marked as Positive control. Add LRW in test tube marked as negative control. Incubate the test tubes at 37 + 2°C for 60 min. After an incubation, check for the gel by inverting the test tube. If the material remains firm in the bottom of the test tube, it means gel has formed. This positive if the material gets the flow down, it means gel has not formed. This means negative.
Take similarly three test tubes as above and add water for injection (WFI) in test tube marked as sample. And proceed as above. The results should be as follows (Table 4):
Sample | Positive control | Negative control | Result |
---|---|---|---|
−ve (gel not formed) | +ve | −ve | Pass |
Sample | Positive control | Negative control | Result |
---|---|---|---|
+ve (gel formed) | +ve | −ve | Fail |
Results shown sample pass or not.
We have to make dilution.
Example: If the product endotoxin limit is 1 EU/ml, then we have to make the dilution as follows:
Since we are using 0.25 EU/ml, this is called lambda. Divide the endotoxin limit of product with lambda
1/0.25 = 1:4
As per USP, we have to test 3 test as follows:
One test tube | 1:3 | The result should be positive |
Second | 1:4 | The result may be positive or negative |
Third | 1:5 | The result should be negative |
This means the product is passed.
Chromogenic lysate [2],
Respective endotoxin standard,
Diazo coupling reagent (set of four bottles).
Note: All reagents must be stored in refrigerator at 2–8°C.
Dissolve 45.6 ml of acetic acid in 1 liter of distilled water. The final concentration of acetic acid is 0.8 M. This solution can be stored for 3 months.
Remove the plastic cover. Wipe off with 70% alcohol around the rubber cap and top portion of every vial. Remove the aluminum cap with sterile and pyrogen free forceps and then cover with depyrogenated aluminum foil to avoid any Endotoxin contamination. (2.8 ml LAL water vial is provided with Endotoxin vial, concentration is mentioned on the label). Pour whole quantity of LAL water into the ET vial and cover with foil. Mix vigorously for at least 10 s by vortexer. During stirring solution must not touch the foil.
Note: Stir every time vigorously before use.
Toxicolor lysate
(Buffer vial 0.35 ml and LAL water are provided with Lysate. Sensitivity is mentioned on the certificate). After taking from the refrigerator, pour whole quantity of buffer and 0.35 ml LAL water into the lysate vial as soon as possible, covers with foil. Then quickly stir to dissolve. Avoid air bubbling during stirring. Place the vial in ice water bath for 2–3 min before use.
Note: Be sure that the reagent is completely dissolved. This reagent must be reconstituted just before use. The reagent is extremely sensitive and must be consumed at one time. Storage should be avoided, but can be stored at −20°C in 0.1 ml dispensed quantities in small test tubes. Use stored lysate if the color is not changed. Reconstituted lysate may only be deep frozen once.
Four bottles are provided with one set, marked as 7, 8, 9 and 10s respectively. Transfer whole quantity of bottle no. 7 s into bottle no. 8 s. Then add 12 ml distilled water into each of bottle no. 9 s and 10s. Ultimately, we will have three bottles 8, 9, and 10 s, which are used stepwise to block the reaction.
The pH of the sample is adjusted by pyrogen free 0.1 N NaOH or 0.1 N HCl. The pH of the sample should be between 6.0 and 8.0.
Arrange test tubes in two stands as under; stand 1—test tubes for sample and standard dilutions; stand 2—test tubes for reaction.
Take 0.05 ml well-mixed sample into small test tubes. If required, make 1/10 dilution of the sample with Pyrogen free water as Below, Take 4.5 ml of pyrogen free water in the test tube. Then add 0.5 ml of well-mixed sample. Vortex mixing for a few seconds.
Take 0.1 ml into a small test tube for further process.
Make a dilution of the endotoxin (concentration 0.470 EU/ml) according to the product limit. For making 0.235 EU/ml (if the product limit is 0.25) proceed as follows;
Take 0.05 ml of the reconstituted endotoxin in the test tube after stirring. Add 0.05 ml of pyrogen free water and vortex to mix. Now the final dilution is 0.235 EU/ml.
Take 0.05 ml of step 2 into a small test tube for further process.
Pour 0.05 ml of pyrogen free water (being used in the test) in small test tube as a blank for further process.
Lysate addition
Place the tube stand for small test tubes (containing the tubes of blank, standard and diluted samples) in ice water bath or suitable ice water container. Add 0.05 ml of lysate to all of the tubes as soon as possible. Stir the contents of every tube soon after the addition of lysate for a few seconds. Avoid foaming.
Soon after the addition of lysate, place the test tube rack in the incubator set at 32.5 + 2.5°C for 30 min. The tube rack can be placed in the water container placed in the incubator.
After completion of the incubation period, place tube rack in ice water bath, then blocks the reaction immediately from one of the two methods mentioned below:
By acetic acid
Add 0.4 ml of 0.8 M acetic acid into each tube and stir to mix.
By diazo coupling reagent
Three bottles of the reagent are used as under;
Add 0.5 ml from bottle no 8 s to each tube and stir to mix.
Add 0.5 ml from bottle no 9 s to each tube and stir to mix.
Add 0.5 ml from bottle no 10s to each tube and stir to mix.
Absorbance reading (using spectrophotometer) measurement at 405 nm
If 0.8 M acetic acid is used to block the reaction, then absorbance reading is taken at 405 nm.
Note: The readings. Glass photocell is used for reading at 405 nm. Because the volume of the tube content is not sufficient, the distilled water is added to each tube and is stirred to mix.
When Diazo coupling reagent is used for blockage of the reaction then the reading is taken at 545 nm. Note all the readings.
Note: Distilled water is used for reference in both cases.
All the absorbance readings are fed in the “Software reader for window version 1.51” to collect the results.
Following Formula is used to calculate the results
Calculations of MVC, MVD and ELC.
Where the lowest sensitivity of lysate, M is the maximum dose/kg body weight and K is constant having value equal to 5.
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