Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro / Este trabalho apresenta uma modelagem matemática para o processo de aquecimento de um corpo exposto a uma fonte pontual de radiação térmica. O resultado original que permite a solução exata de uma equação diferencial parcial não linear a partir de uma seqüência de problemas lineares também é apresentado. Gráficos gerados com resultados obtidos pelo método de diferenças finitas ilustram a solução do problema proposto. / This work presents a mathematical model for the heating process on a body exposed to a punctual source of thermal radiation. An original result, that allows the construction of the exact solution for a non-linear partial differential equation by solving a sequence of linear problems, is also presented. Graphic images generated from the results obtained through the Finite Difference Method illustrate the solution of the proposed problem.

A radiação térmica é frequentemente considerada um mecanismo de transferência de calor muito importante em processos de combustão em alta pressão, devido à presença de meios participantes e às altas temperaturas envolvidas. Resolver a radiação térmica em meios participantes é um problema complexo devido à natureza integro-diferencial da equação governante e à dependência espectral altamente irregular das propriedades de radiação. Atualmente, o método mais preciso para resolver a integração espectral é o método linha-porlinha (LBL), que possui um custo computacional muito elevado. Para contornar essa dificuldade, o problema espectral é geralmente resolvido usando modelos espectrais e, consequentemente, a equação da transferência radiativa (RTE) é simplificada. Um destes modelos é o da soma-ponderada-de-gases-cinza (WSGG), que substitui o comportamento espectral altamente irregular do coeficiente de absorção, por bandas de coeficientes de absorção uniforme e tem mostrado um bom desempenho em diversas aplicações, mesmo sendo um modelo bastante simplificado. Entretanto, recentemente alguns autores não obtiveram bons resultados ao tentar aplicar o WSGG a problemas de combustão em alta pressão. Este artigo desenvolve um modelo WSGG para CO2 e H2O em condições de alta pressão. Para validar o modelo, a emitância total é calculada usando os coeficientes WSGG e comparada à solução do LBL obtida usando o banco de dados espectrais HITEMP 2010. Os resultados mostraram grande convergência entre os valores de emitância de ambos os métodos, mesmo para valores de alta pressão, tanto para o CO2 quanto para H2O, provando que o método WSGG é aplicável a condições de alta pressão. O modelo também foi validado pelo cálculo do fluxo de calor e termo fonte radiativo, e comparando-os com os obtidos através do método LBL. O H2O teve melhores resultados para baixas pressões, enquanto o CO2 apresentou melhores resultados para pressões mais altas. O efeito da pressão total sobre a solução de LBL foi maior para o H2O, o que pode ser um dos motivos pelo qual os desvios foram maiores para os casos de alta pressão. / Thermal radiation is often a very important heat transfer mechanism in high pressure combustion processes due to the presence of participating media and the high temperatures involved. Solving thermal radiation in participating media is a tough problem due to the integro-differential governing equation and the complex spectral dependence of radiation properties. Currently, the most accurate method to solve the spectral integration is the line-byline (LBL) method, which has a very high computational cost. In order to avoid this drawback the spectral problem is usually solved using spectral models, and as a consequence the radiative transfer equation (RTE) is simplified. One of the models is the weighted-sum-ofgray- gases (WSGG) which replaces the highly irregular spectral behavior of the absorption coefficient by bands of uniform absorption coefficients, and has shown great performance a lot of applications even though it is a very simple model. However, recently some authors didn’t have good results when trying to apply the WSGG to high pressure combustion problems. This thesis develops a WSGG model for both CO2 and H2O on high pressure conditions. In order to validate the model the total emittance is calculated using the WSGG coefficients and compared to the LBL solution which was obtained using the HITEMP 2010 spectral emissivity database. The results showed that the emittance values from both methods were very close even for high pressure values for both CO2 and H2O proving that the WSGG method is applicable to high pressure conditions. The model was also validated by calculating the radiative heat flux and source, and comparing them with the LBL method. H2O had better results for low pressures while CO2 had better results for higher pressures. The effect of total pressure on the LBL solution was higher for H2O, which might be the reason why deviations were higher at high pressure values.

This thesis is devoted to the investigation of passivesafety and inherent features of subcritical nucleartransmutation systems - accelerator-driven systems. The generalobjective of this research has been to improve the safetyperformance and avoid elevated coolant temperatures inworst-case scenarios like unprotected loss-of-ow accidents,loss-of-heat-sink accidents, and a combination of both theseaccident initiators. The specific topics covered are emergencydecay heat removal by reactor vessel auxiliary cooling systems,beam shut-off by a melt-rupture disc, safety aspects fromlocating heat-exchangers in the riser of a pool-type reactorsystem, and reduction of pressure resistance in the primarycircuit by employing bypass routes. The initial part of the research was focused on reactorvessel auxiliary cooling systems. It was shown that an 80 MWthPb/Bi-cooled accelerator-driven system of 8 m height and 6 mdiameter vessel can be well cooled in the case of loss-of-owaccidents in which the accelerator proton beam is not switchedoff. After a loss-of-heat-sink accident the proton beam has tobe interrupted within 40 minutes in order to avoid fast creepof the vessel. If a melt-rupture disc is included in the wallof the beam pipe, which breaks at 150 K above the normal coreoutlet temperature, the grace period until the beam has to beshut off is increased to 6 hours. For the same vessel geometry,but an operating power of 250 MWth the structural materials canstill avoid fast creep in case the proton beam is shut offimmediately. If beam shut-off is delayed, additional coolingmethods are needed to increase the heat removal. Investigationswere made on the filling of the gap between the guard and thereactor vessel with liquid metal coolant and using water spraycooling on the guard vessel surface. The second part of the thesis presents examinationsregarding an accelerator-driven system also cooled with Pb/Bibut with heat-exchangers located in the risers of the reactorvessel. For a pool type design, this approach has advantages inthe case of heat-exchanger tube failures, particularly if wateris used as the secondary uid. This is because a leakage ofwater from the secondary circuit into the Pb/Bi-cooled primarycircuit leads to upward sweeping of steam bubbles, which wouldcollect in the gas plenum. In the case of heatexchangers in thedowncomer steam bubbles may be dragged into the ADS core andadd reactivity. Bypass routes are employed to increase the owspeed in loss-of-ow events for this design. It is shown thatthe 200 MWth accelerator-driven system with heat-exchangers inthe riser copes reasonably well with both a loss-of-ow accidentwith the beam on and an unprotected loss-of-heat-sink accident.For a total-loss-of-power (station blackout) and an immediatebeam-stop the core outlet temperature peaks at 680 K. After acombined loss-of-ow and loss-of-heat-sink accident the beamshould be shut off within 4 minutes to avoid exceeding the ASMElevel D of 977 K, and within 8 minutes to avoid fast creep.Assuming the same core inlet temperature, both the reactordesign with heat-exchanger in the risers and the downcomershave similar temperature evolutions after a total-loss-ofpoweraccident. A large accelerator-driven system of 800 MWth with a 17 mtall vessel may eventually become a standard size. For thishigher power ADS, the location of the heat-exchangers hasgreater impact on the natural convection capability. This isdue to that larger heatexchangers have more inuence on thedistance between the thermal centers during a lossof- owaccident. The design with heat-exchangers in the downcomers,the long-term vessel temperature peaks at 996 K during aloss-of-ow accident with the beam on. This does not pose athreat of creep rupture for the vessel. However, the locationof the heat-exchangers in the downcomers will probably requiresecondary coolant other than water, like for example oil (fortemperatures not higher than 673 K) or Pb/Bi coolant.

<p>This thesis is devoted to the investigation of passivesafety and inherent features of subcritical nucleartransmutation systems - accelerator-driven systems. The generalobjective of this research has been to improve the safetyperformance and avoid elevated coolant temperatures inworst-case scenarios like unprotected loss-of-ow accidents,loss-of-heat-sink accidents, and a combination of both theseaccident initiators. The specific topics covered are emergencydecay heat removal by reactor vessel auxiliary cooling systems,beam shut-off by a melt-rupture disc, safety aspects fromlocating heat-exchangers in the riser of a pool-type reactorsystem, and reduction of pressure resistance in the primarycircuit by employing bypass routes.</p><p>The initial part of the research was focused on reactorvessel auxiliary cooling systems. It was shown that an 80 MWthPb/Bi-cooled accelerator-driven system of 8 m height and 6 mdiameter vessel can be well cooled in the case of loss-of-owaccidents in which the accelerator proton beam is not switchedoff. After a loss-of-heat-sink accident the proton beam has tobe interrupted within 40 minutes in order to avoid fast creepof the vessel. If a melt-rupture disc is included in the wallof the beam pipe, which breaks at 150 K above the normal coreoutlet temperature, the grace period until the beam has to beshut off is increased to 6 hours. For the same vessel geometry,but an operating power of 250 MWth the structural materials canstill avoid fast creep in case the proton beam is shut offimmediately. If beam shut-off is delayed, additional coolingmethods are needed to increase the heat removal. Investigationswere made on the filling of the gap between the guard and thereactor vessel with liquid metal coolant and using water spraycooling on the guard vessel surface.</p><p>The second part of the thesis presents examinationsregarding an accelerator-driven system also cooled with Pb/Bibut with heat-exchangers located in the risers of the reactorvessel. For a pool type design, this approach has advantages inthe case of heat-exchanger tube failures, particularly if wateris used as the secondary uid. This is because a leakage ofwater from the secondary circuit into the Pb/Bi-cooled primarycircuit leads to upward sweeping of steam bubbles, which wouldcollect in the gas plenum. In the case of heatexchangers in thedowncomer steam bubbles may be dragged into the ADS core andadd reactivity. Bypass routes are employed to increase the owspeed in loss-of-ow events for this design. It is shown thatthe 200 MWth accelerator-driven system with heat-exchangers inthe riser copes reasonably well with both a loss-of-ow accidentwith the beam on and an unprotected loss-of-heat-sink accident.For a total-loss-of-power (station blackout) and an immediatebeam-stop the core outlet temperature peaks at 680 K. After acombined loss-of-ow and loss-of-heat-sink accident the beamshould be shut off within 4 minutes to avoid exceeding the ASMElevel D of 977 K, and within 8 minutes to avoid fast creep.Assuming the same core inlet temperature, both the reactordesign with heat-exchanger in the risers and the downcomershave similar temperature evolutions after a total-loss-ofpoweraccident.</p><p>A large accelerator-driven system of 800 MWth with a 17 mtall vessel may eventually become a standard size. For thishigher power ADS, the location of the heat-exchangers hasgreater impact on the natural convection capability. This isdue to that larger heatexchangers have more inuence on thedistance between the thermal centers during a lossof- owaccident. The design with heat-exchangers in the downcomers,the long-term vessel temperature peaks at 996 K during aloss-of-ow accident with the beam on. This does not pose athreat of creep rupture for the vessel. However, the locationof the heat-exchangers in the downcomers will probably requiresecondary coolant other than water, like for example oil (fortemperatures not higher than 673 K) or Pb/Bi coolant.</p>

It has been noted by several authors that General Relativistic frame dragging in rotating neutron stars is a first order effect which has to be included in a self-consistent model of pulsar magnetospheric structure and associated radiation and transport processes. To this end, I undertook the present study with the aim of investigating the effect of General Relativity (GR) on millisecond pulsar (MSP) visibility. I developed a numerical code for simulating a pulsar magnetosphere, incorporating the GR-corrected expressions for the electric potential and field. I included curvature radiation (CR) due to primary electrons accelerated above the stellar surface, as well as inverse Compton scattering (ICS) of thermal X-ray photons by these electrons. I then applied the model to PSR J0437-4715, a prime candidate for testing the GR-Electrodynamic theory, and examined its visibility for the H.E.S.S. telescope. I also considered the question of whether magnetic photon absorption would take place for this particular pulsar. In addition, I developed a classical model for comparison with the GR results. I found that the typical electron energies and associated CR photon energies are functions of position above the polar cap (PC). These energies are also quite smaller in the GR case than in the classical case due to the different functional forms of the GR and classical electric fields. I found the CR energy cut-off to be ~ 4 GeV compared to the well-known classical value of ~ 100 GeV. Since the H.E.S.S. energy threshold is ~ 100 GeV, it seems as though the CR component will not be visible, contrary to wide-held opinion. However, the ICS component seems to be well in excess of the H.E.S.S. energy threshold and is expected to be visible. I also found that no pair production will take place for PSR J0437-4715. Hopefully, forthcoming H.E.S.S. observations will provide validation of these results. KEY WORDS: General relativistic frame dragging, GR electrodynamics, millisecond pulsar visibility, non-thermal radiation processes, pair production, H.E.S.S., individual pulsars: PSR J0437-4715. / Thesis (M.Sc. (Physics))--North-West University, Potchefstroom Campus, 2004.

It has been noted by several authors that General Relativistic frame dragging in rotating neutron stars is a first order effect which has to be included in a self-consistent model of pulsar magnetospheric structure and associated radiation and transport processes. To this end, I undertook the present study with the aim of investigating the effect of General Relativity (GR) on millisecond pulsar (MSP) visibility. I developed a numerical code for simulating a pulsar magnetosphere, incorporating the GR-corrected expressions for the electric potential and field. I included curvature radiation (CR) due to primary electrons accelerated above the stellar surface, as well as inverse Compton scattering (ICS) of thermal X-ray photons by these electrons. I then applied the model to PSR J0437-4715, a prime candidate for testing the GR-Electrodynamic theory, and examined its visibility for the H.E.S.S. telescope. I also considered the question of whether magnetic photon absorption would take place for this particular pulsar. In addition, I developed a classical model for comparison with the GR results. I found that the typical electron energies and associated CR photon energies are functions of position above the polar cap (PC). These energies are also quite smaller in the GR case than in the classical case due to the different functional forms of the GR and classical electric fields. I found the CR energy cut-off to be ~ 4 GeV compared to the well-known classical value of ~ 100 GeV. Since the H.E.S.S. energy threshold is ~ 100 GeV, it seems as though the CR component will not be visible, contrary to wide-held opinion. However, the ICS component seems to be well in excess of the H.E.S.S. energy threshold and is expected to be visible. I also found that no pair production will take place for PSR J0437-4715. Hopefully, forthcoming H.E.S.S. observations will provide validation of these results. KEY WORDS: General relativistic frame dragging, GR electrodynamics, millisecond pulsar visibility, non-thermal radiation processes, pair production, H.E.S.S., individual pulsars: PSR J0437-4715. / Thesis (M.Sc. (Physics))--North-West University, Potchefstroom Campus, 2004.

Neste trabalho são obtidos os coeficientes do modelo da Soma-Ponderada-dos-gases-cinza (WSGG) a partir do banco de dados HITEMP 2010, permitindo o uso do modelo com os dados mais precisos disponíveis atualmente. Neste trabalho também se faz uma comparação dos valores de emitância total obtidos a partir do modelo WSGG com valores benchmark, obtidos nesse trabalho, mostrando uma excelente concordância. Com os coeficientes obtidos, problemas unidimensionais de transferência de calor radiante são resolvidos de modo a comparar a solução obtida pelo modelo WSGG com a solução obtida pela integração LBL (solução benchmark). Nas comparações, diferentes perfis de temperatura, comprimentos de trajeto, gradientes de temperatura e concentrações de espécies são utilizadas. Em todos os casos é possível verificar uma boa concordância entre os resultados WSGG e LBL. Para comparações com perfil de temperatura parabólico, verifica-se erros locais abaixo de 8%. Para perfis de temperatura cossenoidais, é possível observar erros de até 18% para alguns casos, porém com erros médios menores que 1,6%. / In this work the coefficients for the Weighted Sum-of-Gray-Gases model (WSGG) are determined from HITEMP 2010 database, allowing the use of the model with the most accurate data available nowadays. This study also makes a comparison of the total emittance values obtained from the model with benchmark values, obtained in this work, showing an excellent agreement. With the obtained coefficients, one-dimensional radiant heat transfer problems are solved in order to compare the solution obtained by the WSGG model with the solution obtained by the LBL integration (benchmark solution). In the comparisons, different temperature profiles, path lengths, temperature gradients and species concentrations are used. In all cases it is possible to verify the good agreement of the WSGG and LBL results. For comparisons with parabolic temperature profile, the local error is below 8%. For cosine temperature profile, the local error is about 18% for some cases, but with average errors less than 1,6%.

A radiação térmica é o principal mecanismo de transferência de calor em fenômenos que envolvem meios participantes em temperaturas elevadas, tais como em processos de combustão. A dependência fortemente irregular do coeficiente de absorção em relação ao número de onda torna desafiador o estudo de situações em que a radiação é apenas parte de um problema mais complexo. A exatidão do cálculo da radiação fica condicionada à solução da equação da transferência radiativa (RTE) por meio da integração linha-por-linha (LBL), sendo, muitas vezes, impraticável, em virtude do esforço computacional requerido para contabilizar as centenas de milhares ou milhões de linhas espectrais do coeficiente de absorção. Alternativamente, modelos espectrais, como a soma-ponderada-de-gases-cinza (WSGG), têm sido empregados de maneira eficaz na obtenção de resultados em substituição à integração LBL. Nessa dissertação, o modelo WSGG é aplicado na solução da transferência de calor radiativa em um sistema unidimensional, formado por duas placas planas paralelas infinitas e preenchido por uma mistura homogênea de dióxido de carbono e vapor de água, considerando-se perfis distintos de temperatura. Diferentemente da maioria dos estudos da literatura que empregam a mesma geometria, mas com paredes negras, o presente trabalho supõe superfícies cinzas e não cinzas. O objetivo central é, portanto, avaliar o erro em se assumir fronteiras negras quando estas não apresentam esse comportamento. Os resultados para o modelo WSGG aplicado a superfícies não cinzas, cinzas e negras são comparados com a solução linha-por-linha para paredes não cinzas. As análises dos desvios entre as soluções pelo modelo da soma-ponderada-de-gases-cinza e pela integração LBL mostram que a suposição de paredes negras, para casos em que as superfícies deveriam ser consideradas não cinzas, pode levar a erros de até 50% nos resultados para o fluxo de calor e para o termo fonte radiativo. / Thermal radiation is the main heat transfer mechanism in phenomena that involves high temperatures, such as in combustion processes. The strongly irregular dependence of the absorption coefficient on the wavenumber makes challenger the study of situations in which the radiation is only part of a more complex problem. The accuracy of the calculation of the radiation is conditioned to the solution of the radiative transfer equation (RTE) by line-by-line (LBL) integration, being frequently impracticable, due to the computational effort required to account for the hundreds of thousands or millions spectral lines of the absorption coefficient. Alternatively, spectral models, such as the weighted-sum-of-gray-gases (WSGG) model, have been used with success to obtain results in comparison to LBL integration. In this study, the WSGG model is applied to solve the radiative heat transfer in a one-dimensional system, formed by two infinite flat parallel plates and filled by a homogeneous mixture of carbon dioxide and water vapor, for different temperature profiles. Unlike most studies of the literature that employ the same geometry, but with black walls, the present work supposes gray and non-gray surfaces. The central objective is, therefore, to evaluate the error in assuming black boundaries when they do not present this behavior. The results for the WSGG model applied to non-gray, gray and black surfaces are compared with the line-by-line solution for non-gray walls. Analyzes of the deviations between the solutions by the weighted-sum-of-gray-gases model and the LBL integration show that the assumption of black walls, for cases where the surfaces should be considered as non-gray, may lead to errors of up to 50% in results for the heat flux and the radiative source term.

A radiação térmica é frequentemente considerada um mecanismo de transferência de calor muito importante em processos de combustão em alta pressão, devido à presença de meios participantes e às altas temperaturas envolvidas. Resolver a radiação térmica em meios participantes é um problema complexo devido à natureza integro-diferencial da equação governante e à dependência espectral altamente irregular das propriedades de radiação. Atualmente, o método mais preciso para resolver a integração espectral é o método linha-porlinha (LBL), que possui um custo computacional muito elevado. Para contornar essa dificuldade, o problema espectral é geralmente resolvido usando modelos espectrais e, consequentemente, a equação da transferência radiativa (RTE) é simplificada. Um destes modelos é o da soma-ponderada-de-gases-cinza (WSGG), que substitui o comportamento espectral altamente irregular do coeficiente de absorção, por bandas de coeficientes de absorção uniforme e tem mostrado um bom desempenho em diversas aplicações, mesmo sendo um modelo bastante simplificado. Entretanto, recentemente alguns autores não obtiveram bons resultados ao tentar aplicar o WSGG a problemas de combustão em alta pressão. Este artigo desenvolve um modelo WSGG para CO2 e H2O em condições de alta pressão. Para validar o modelo, a emitância total é calculada usando os coeficientes WSGG e comparada à solução do LBL obtida usando o banco de dados espectrais HITEMP 2010. Os resultados mostraram grande convergência entre os valores de emitância de ambos os métodos, mesmo para valores de alta pressão, tanto para o CO2 quanto para H2O, provando que o método WSGG é aplicável a condições de alta pressão. O modelo também foi validado pelo cálculo do fluxo de calor e termo fonte radiativo, e comparando-os com os obtidos através do método LBL. O H2O teve melhores resultados para baixas pressões, enquanto o CO2 apresentou melhores resultados para pressões mais altas. O efeito da pressão total sobre a solução de LBL foi maior para o H2O, o que pode ser um dos motivos pelo qual os desvios foram maiores para os casos de alta pressão. / Thermal radiation is often a very important heat transfer mechanism in high pressure combustion processes due to the presence of participating media and the high temperatures involved. Solving thermal radiation in participating media is a tough problem due to the integro-differential governing equation and the complex spectral dependence of radiation properties. Currently, the most accurate method to solve the spectral integration is the line-byline (LBL) method, which has a very high computational cost. In order to avoid this drawback the spectral problem is usually solved using spectral models, and as a consequence the radiative transfer equation (RTE) is simplified. One of the models is the weighted-sum-ofgray- gases (WSGG) which replaces the highly irregular spectral behavior of the absorption coefficient by bands of uniform absorption coefficients, and has shown great performance a lot of applications even though it is a very simple model. However, recently some authors didn’t have good results when trying to apply the WSGG to high pressure combustion problems. This thesis develops a WSGG model for both CO2 and H2O on high pressure conditions. In order to validate the model the total emittance is calculated using the WSGG coefficients and compared to the LBL solution which was obtained using the HITEMP 2010 spectral emissivity database. The results showed that the emittance values from both methods were very close even for high pressure values for both CO2 and H2O proving that the WSGG method is applicable to high pressure conditions. The model was also validated by calculating the radiative heat flux and source, and comparing them with the LBL method. H2O had better results for low pressures while CO2 had better results for higher pressures. The effect of total pressure on the LBL solution was higher for H2O, which might be the reason why deviations were higher at high pressure values.

Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro / Este trabalho apresenta uma modelagem matemática para o processo de aquecimento de um corpo exposto a uma fonte pontual de radiação térmica. O resultado original que permite a solução exata de uma equação diferencial parcial não linear a partir de uma seqüência de problemas lineares também é apresentado. Gráficos gerados com resultados obtidos pelo método de diferenças finitas ilustram a solução do problema proposto. / This work presents a mathematical model for the heating process on a body exposed to a punctual source of thermal radiation. An original result, that allows the construction of the exact solution for a non-linear partial differential equation by solving a sequence of linear problems, is also presented. Graphic images generated from the results obtained through the Finite Difference Method illustrate the solution of the proposed problem.