Abstract The aim of this work is to improve the overall robustness in equation-oriented chemical engineering simulation work. Because the performance of locally convergent solving methods is strongly dependent on a favourable initial guess, bounded homotopy methods were investigated as a way to enlarge the domain of convergence. Bounded homotopies make it possible to keep the homotopy path inside a feasible problem domain. Thus the fatal errors possibly caused by unfeasible variable values in thermodynamic subroutines can be avoided. To enable the utilization of a narrow bounding zone, modifications were proposed for bounded homotopies. The performance of the modifications was studied with simple test problems and several types of distillation systems in the MATLAB environment. The findings illustrate that modified bounded homotopies with variables mapping make it possible to bound the homotopy path strictly to run inside a feasible problem domain. The homotopy path can be tracked accurately and flexibly also inside a narrow bounding zone. It was also noticed that by utilizing the concept of bounding the homotopy path with respect to the homotopy parameter, the possibility of approaching starting point and solution multiplicities is increased in cases where the traditional problem-independent homotopy method fails. The concept aims to connect separate homotopy path branches thus offering a trackable path with real space arithmetic. Even though the modified bounded homotopies were found to overcome several challenges often encountered with traditional problem-independent homotopy continuation methods, alone they are not enough to guarantee that the solution is approached from an arbitrary starting point. Therefore, problem-tailored solving procedures were implemented in the consideration of complex column configurations. Problem-tailored solving procedures aim to offer feasible consecutive sub-problems and thus direct the solving towards the state distribution that fulfils exact product purity specifications. As a whole, the modified bounded homotopies and problem-tailored solving procedures were found to improve the overall robustness of an equation-oriented solving approach. Thus the threshold for designing and implementing complex process systems such as complex distillation configurations for practical use could be lowered.

No description available.

In school algebra, using different methods including factorization to solve quadratic equations is one common teaching and learning topic at upper secondary school level. This study is about analyzing the algebra content related to solving quadratic equations and the method of factorization as presented in Swedish mathematics textbooks with subject matter content knowledge (CK) and pedagogical content knowledge (PCK) as analytical tools. Mathematics textbooks as educational resources and artefacts are widely used in classroom teaching and learning. What is presented in a textbook is often taught by teachers in the classroom. Similarly, what is missing from the textbook may not be presented by the teacher. The study is based on an assumption that pedagogical content knowledge is embedded in the subject content presented in textbooks. Textbooks contain both subject content knowledge and pedagogical content knowledge. The primary aim of the study is to explore what pedagogical content knowledge regarding solving quadratic equations that is embedded in mathematics textbooks. The secondary aim is to analyze the algebra content related to solving quadratic equations from the perspective of mathematics as a discipline in relation to algebra history. It is about what one can find in the textbook rather than how the textbook is used in the classroom. The study concerns a teaching perspective and is intended to contribute to the understanding of the conditions of teaching solving quadratic equations. The theoretical framework is based on Shulman’s concept pedagogical content knowledge and Mishra and Koehler’s concept content knowledge. The general theoretical perspective is based on Wartofsky’s artifact theory. The empirical material used in this study includes twelve mathematics textbooks in the mathematics B course at Swedish upper secondary schools. The study contains four rounds of analyses. The results of the first three rounds have set up a basis for a deep analysis of one selected textbook. The results show that the analyzed Swedish mathematics textbooks reflect the Swedish mathematics syllabus of algebra. It is found that the algebra content related to solving quadratic equations is similar in every investigated textbook. There is an accumulative relationship among all the algebra content with a final goal of presenting how to solve quadratic equations by quadratic formula, which implies that classroom teaching may focus on quadratic formula. Factorization method is presented for solving simple quadratic equations but not the general-formed quadratic equations. The study finds that the presentation of the algebra content related to quadratic equations in the selected textbook is organized by four geometrical models that can be traced back to the history of algebra. These four geometrical models are applied for illustrating algebra rules and construct an overall embedded teaching trajectory with five sub-trajectories. The historically related pedagogy and application of mathematics in both real world and pure mathematics contexts are the pedagogical content knowledge related to quadratic equations.

Abstract The aim of this work was to formulate separation process models with both predictive capabilities and robust solution strategies. Although all separation process models should have predictive capabilities, the current literature still has multiple applications in which predictive models having the combination of a clear phenomenon base and robust solving strategy are not available. The separation process models investigated in this work were liquid-liquid phase separation and membrane separation models. The robust solving of a liquid-liquid phase separation model typically demands the solution of a phase stability analysis problem. In addition, predicting the liquid-liquid phase compositions reliably depends on robust phase stability analysis. A phase stability analysis problem has multiple feasible solutions, all of which have to be sought to ensure both the robust solving of the model and predictive process model. Finding all the solutions with a local solving method is difficult and generally inexact. Therefore, the modified bounded homotopy methods, a global solving method, were further developed to solve the problem robustly. Robust solving demanded the application of both variables and homotopy parameter bounding features and the usage of the trivial solution in the solving strategy. This was shown in multiple liquid-liquid equilibrium cases. In the context of membrane separation models, predictive capabilities are achieved with the application of a Maxwell-Stefan based model. With the Maxwell-Stefan approach, multicomponent separation can be predicted based on pure component permeation data alone. On the other hand, the solving of the model demands a robust solving strategy with application-dependent knowledge. These issues were illustrated in the separation of a H2/CO2 mixture with a high-silica MFI zeolite membrane at high pressure and low temperature. Similarly, the prediction of mixture adsorption based on pure component adsorption data alone was successfully demonstrated. In the context of membrane separation models, predictive capabilities are achieved with the application of a Maxwell-Stefan based model. With the Maxwell-Stefan approach, multicomponent separation can be predicted based on pure component permeation data alone. On the other hand, the solving of the model demands a robust solving strategy with application-dependent knowledge. These issues were illustrated in the separation of a H2/CO2 mixture with a high-silica MFI zeolite membrane at high pressure and low temperature. Similarly, the prediction of mixture adsorption based on pure component adsorption data alone was successfully demonstrated. / Tiivistelmä Työn tavoitteena oli muotoilla prosessin käyttäytymisen ennustamiseen kykeneviä erotusprosessimalleja ja niiden ratkaisuun käytettäviä luotettavia strategioita. Vaikka kaikkien erotusprosessimallien tulisi olla ennustavia, on tällä hetkellä useita kohteita, joissa prosessin käyttäytymistä ei voida ennustaa siten, että käytettävissä olisi sekä ilmiöpohjainen malli että ratkaisuun soveltuva luotettava strategia. Tässä työssä erotusprosessimalleista kohteina tarkasteltiin neste-neste-erotuksen ja membraanierotuksen kuvaukseen käytettäviä malleja. Neste-neste-erotusmallien luotettava ratkaisu vaatii yleensä faasistabiilisuusongelman ratkaisua. Lisäksi faasien koostumusten luotettava ennustaminen pohjautuu faasistabiilisuusanalyysiin. Faasistabiilisuusongelmalla on useita mahdollisia ratkaisuja, jotka kaikki tulee löytää, jotta voitaisiin varmistaa luotettava mallin ratkaisu sekä prosessimallin ennustuskyvyn säilyminen. Kaikkien ratkaisujen löytäminen on sekä vaikeaa että epätarkkaa paikallisesti konvergoituvilla ratkaisumenetelmillä. Tämän vuoksi globaaleihin ratkaisumenetelmiin kuuluvia modifioituja rajoitettuja homotopiamenetelmiä kehitettiin edelleen, jotta faasistabiilisuusongelma saataisiin ratkaistua luotettavasti. Ratkaisun luotettavuus vaati sekä muuttujien että homotopiaparametrin rajoittamista ja ongelman triviaalin ratkaisun käyttöä ratkaisustrategiassa. Tämä käyttäytyminen todennettiin useissa neste-nestetasa-painoa kuvaavissa esimerkeissä. Membraanierotusta tarkasteltaessa ennustava malli voidaan muotoilla käyttämällä Maxwell-Stefan pohjaista mallia. Maxwell-Stefan lähestymistavalla voidaan ennustaa monikomponenttiseosten erotusta perustuen puhtaiden komponenttien membraanin läpäisystä saatuun mittausaineistoon. Toisaalta mallin ratkaisu vaatii luotettavan ratkaisustrategian, jossa hyötykäytetään kohteesta riippuvaa tietoa. Näitä kysymyksiä havainnollistettiin H2/CO2 seoksen erotuksessa MFI-zeoliitti-membraanilla korkeassa paineessa. Samoin seosten adsorboitumiskäyttäytymistä ennustettiin onnistuneesti pelkästään puhtaiden komponenttien adsorptiodatan pohjalta. Kokonaisuutena voidaan todeta, että tarkasteltujen erotusprosessimallien ennustavuutta voidaan parantaa yhdistämällä malli, jolla on selkeä ilmiöpohja ja luotettava ratkaisustrategia. Lisäksi mallien käytettävyys erotusprosessien suunnittelussa on parantunut työn tulosten pohjalta.

This Master's thesis discusses the Problem - based learning as one of the constructive teaching concept methods, it's position in the system of teaching methods and the possible uses in primary school regard to the key competencies of solving the problems. This theme is very actual because of the influence in the real life. The objectives of the Master's thesis is to define the basic concept of the Problem-based learning (problem, problematic situations, types of the problematic tasks and the way of their solving), key competencies and appropriate organizational forms of the teaching. The objective of the practical level is to define the principles of the effective use of this method in teaching, to verify the possibilities of uses of the Problem-based learning and focus on the way of fulfilling of the key competencies of solving the problems using various organizational forms of teaching. The thesis contains the empirical and theoretical part. The empirical part comes from the analysis and the comparation of the study of the vocational literature. This part focuses on the research as the method of achieving the goals. The research uses two basic methods; the method of the observation of the pupils during their work and method of the collection and analysis of the outcome of the pupils work. The...

<p>  </p> <p>The dissertation study presented here explores what higher education instructors believe about problem-solving. Beliefs about problem-solving pedagogy and the influences that change pedagogical approaches in the post-secondary realm of physics education require more robust exploration. The level of change that occurs through the day-to-day teaching cycle and the support that garners improvement are essential aspects of teaching in higher education that need robust understanding.</p> <p>Insight into higher education could illuminate the transitional experience of students between high school and college-level physics. This study explores the beliefs of Indiana college and high school educators, all of whom teach college-level physics content, and probed how those beliefs shaped higher education instructional strategies and teaching philosophies. The study was conducted using a Constructivist Grounded Theory approach.</p> <p>The findings show that physics educators in college and high school learning environments lacked support explicitly geared toward them and physics. All the educators included in the study taught college-level physics. Four of the six participants were the only ones teaching physics in their schools. Despite the isolation, all participants noted the importance of peer-to-peer learning for themselves and their students, noting interactions with exterior training opportunities (e.g., educational conferences or online educator communities). However, the most crucial source of change in their teaching beliefs and approaches that the participants noted was the feedback they received from students.</p>

Objective: Cardiovascular diseases are the leading cause of death. The gold standard for their diagnosis and treatment are angiographic procedures. Clinicians rely on dedicated and specialized equipment for these interventions, e.g. angiography systems. The speed of the associated development is important as better technology enables progress in treatment methods and clinical outcomes. The goal of this article is to show how to optimize the innovation and development process such that it takes minimal time. Methods: 672 data sets on 302 topics were collected over 47 months during a long-term observation of the innovation and development process of angiographic systems. The total data collected is equivalent to efforts worth 30 man-years. This input was used to calculate key process parameters, analyse key process roles, evaluate the use of problem-solving methods and identify key technologies. We also developed a process model comprising the primary innovation sources, important input providers and key processes. This model is characterized by a continuous loop for the innovation and development process. Results: The conducted literature research identifies this closed loop process model as being unique in comparison to the well-established models proposed by Brockhoff, Cooper, Crawford, Durfee, Ebert, Eppinger, Hughes, Pleschak, Thom, Ulrich, Vahs and Witt. According to the best knowledge of the authors no comparable data collection has been performed and presented anywhere else yet. When analysing our 672 data sets, we found that the median process time ( in this data pool (n=672) was to be 10 weeks (p<0,05). The median number of task owners (xPA) per task across all topics was 2. Our data revealed that the number of task owners had a direct impact on the process time. For data sets with up to eight task owners the relationship between process time and task owners can be described as tPd=3.6*xPA^1.4. The median time of owning a topic was determined for Sales (7 weeks), Service (11 weeks), Customer Relationship Management (6 weeks), Product Lifecycle Management (10 weeks) and Research & Development (11 weeks). Main input providers were Sales (53%) and customers (28%). Sales (42%) and PLM (37%) are significant connectors. Problem solvers are PLM (35%), CRM (27%) and R&D (27%). The problem-solving methods were analysed and it was found that clarification (77%) as well as dialog and variation method (both 50%) were used most often. We found that changes to the application software (33%), mechanics, device interfaces and user interface (all 21%) are the four out of six components that were involved in most often. In the analysed datasets a potential of an up to 20% shorter process time was identified. Conclusion: This article proposes a new model for the innovation and development process. Based on our data, we recommend to apply a continuous loop process in the context of innovation and development of medical devices. Our results can, for example, be used for Activity Based Costing Approach or be applied to bring new or upgraded angiography systems faster to market benefitting patient outcome due to improved diagnosis and treatment of cardiovascular diseases.