Optimal exponent heat balance and refined integral methods applied to Stefan problems

When using a polynomial approximating function the most contentious aspect of the Heat Balance Integral Method is the choice of power of the highest order term. In this paper we employ a method recently developed for thermal problems, where the exponent is determined during the solution process, to analyse Stefan problems. This is achieved by minimising an error function. The solution requires no knowledge of an exact solution and generally produces significantly better results than all previous HBI models. The method is illustrated by first applying it to standard thermal problems. A Stefan problem with an analytical solution is then discussed and results compared to the approximate solution. An ablation problem is also analysed and results compared against a numerical solution. In both examples the agreement is excellent. A Stefan problem where the boundary temperature increases exponentially is analysed. This highlights the difficulties that can be encountered with a time dependent boundary condition. Finally, melting with a time-dependent flux is briefly analysed without applying analytical or numerical results to assess the accuracy.

Application of standard and refined heat balance integral methods to one-dimensional Stefan problems

The work in this paper concerns the study of conventional and refined heat balance integral methods for a number of phase change problems. These include standard test problems, both with one and two phase changes, which have exact solutions to enable us to test the accuracy of the approximate solutions. We also consider situations where no analytical solution is available and compare these to numerical solutions. It is popular to use a quadratic profile as an approximation of the temperature, but we show that a cubic profile, seldom considered in the literature, is far more accurate in most circumstances. In addition, the refined integral method can give greater improvement still and we develop a variation on this method which turns out to be optimal in some cases. We assess which integral method is better for various problems, showing that it is largely dependent on the specified boundary conditions.

Improving the accuracy of heat balance integral methods applied to thermal problems with time dependent boundary conditions

In this paper the two main drawbacks of the heat balance integral methods are examined. Firstly we investigate the choice of approximating function. For a standard polynomial form it is shown that combining the Heat Balance and Refined Integral methods to determine the power of the highest order term will either lead to the same, or more often, greatly improved accuracy on standard methods. Secondly we examine thermal problems with a time-dependent boundary condition. In doing so we develop a logarithmic approximating function. This new function allows us to model moving peaks in the temperature profile, a feature that previous heat balance methods cannot capture. If the boundary temperature varies so that at some time t & 0 it equals the far-field temperature, then standard methods predict that the temperature is everywhere at this constant value. The new method predicts the correct behaviour. It is also shown that this function provides even more accurate results, when coupled with the new CIM, than the polynomial profile. Analysis primarily focuses on a specified constant boundary temperature and is then extended to constant flux, Newton cooling and time dependent boundary conditions.

Influence of summertime mesoscale convective systems on the heat balance and surface mixed layer dynamics of a large Amazonian hydroelectric reservoir

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Heat recovery from wastewater: numerical modelling of sewer systems

Dissertação para obtenção do Grau de Mestre em Engenharia do Ambiente, perfil Engenharia Sanitária

The role of lateral ocean physics in the upper ocean thermal balance of a coupled ocean-atmosphere GCM

The sensitivity of the upper ocean thermal balance of an ocean-atmosphere coupled GCM to lateral ocean physics is assessed. Three 40-year simulations are performed using horizontal mixing, isopycnal mixing, and isopycnal mixing plus eddy induced advection. The thermal adjustment of the coupled system is quite different between the simulations, confirming the major role of ocean mixing on the heat balance of climate. The initial adjustment phase of the upper ocean (SST) is used to diagnose the physical mechanisms involved in each parametrisation. When the lateral ocean physics is modified, significant changes of SST are seen, mainly in the southern ocean. A heat budget of the annual mixed layer (defined as the “bowl”) shows that these changes are due to a modified heat transfer between the bowl and the ocean interior. This modified heat intake of the ocean interior is directly due to the modified lateral ocean physics. In isopycnal diffusion, this heat exchange, especially marked at mid-latitudes, is both due to an increased effective surface of diffusion and to the sign of the isopycnal gradients of temperature at the base of the bowl. As this gradient is proportional to the isopycnal gradient of salinity, this confirms the strong role of salinity in the thermal balance of the coupled system. The eddy induced advection also leads to increased exchanges between the bowl and the ocean interior. This is both due to the shape of the bowl and again to the existence of a salinity structure. The lateral ocean physics is shown to be a significant contributor to the exchanges between the diabatic and the adiabatic parts of the ocean.

A traceable physical calibration of the vertical advection-diffusion equation for modelling ocean heat uptake

The classic vertical advection-diffusion (VAD) balance is a central concept in studying the ocean heat budget, in particular in simple climate models (SCMs). Here we present a new framework to calibrate the parameters of the VAD equation to the vertical ocean heat balance of two fully-coupled climate models that is traceable to the models’ circulation as well as to vertical mixing and diffusion processes. Based on temperature diagnostics, we derive an effective vertical velocity w∗ and turbulent diffusivity k∗ for each individual physical process. In steady-state, we find that the residual vertical velocity and diffusivity change sign in mid-depth, highlighting the different regional contributions of isopycnal and diapycnal diffusion in balancing the models’ residual advection and vertical mixing. We quantify the impacts of the time-evolution of the effective quantities under a transient 1%CO2 simulation and make the link to the parameters of currently employed SCMs.

Study of the equatorial Atlantic Ocean mixing layer using a one-dimensional turbulence model

The General Ocean Turbulence Model (GOTM) is applied to the diagnostic turbulence field of the mixing layer (ML) over the equatorial region of the Atlantic Ocean. Two situations were investigated: rainy and dry seasons, defined, respectively, by the presence of the intertropical convergence zone and by its northward displacement. Simulations were carried out using data from a PIRATA buoy located on the equator at 23º W to compute surface turbulent fluxes and from the NASA/GEWEX Surface Radiation Budget Project to close the surface radiation balance. A data assimilation scheme was used as a surrogate for the physical effects not present in the one-dimensional model. In the rainy season, results show that the ML is shallower due to the weaker surface stress and stronger stable stratification; the maximum ML depth reached during this season is around 15 m, with an averaged diurnal variation of 7 m depth. In the dry season, the stronger surface stress and the enhanced surface heat balance components enable higher mechanical production of turbulent kinetic energy and, at night, the buoyancy acts also enhancing turbulence in the first meters of depth, characterizing a deeper ML, reaching around 60 m and presenting an average diurnal variation of 30 m.

Irrigation requirements and transpiration coupling to the atmosphere of a citrus orchard in Southern Brazil

Crop evapotranspiration (ETc) was measured as evaporative heat flux from an irrigated acid lime orchard (Citrus latifolia Tanaka) using the aerodynamic method. Crop transpiration (T) was determined by a stem heat balance method. The irrigation requirements were determined by comparing the orchard evapotranspiration (ETc) and T with the reference evapotranspiration (ETo) derived from the Penman-Monteith equation, and the irrigation requirements were expressed as ETc/ETo (Kc) and T/ETo (Kcb) ratios. The influence of inter-row vegetation on the ETc was analyzed because the measurements were taken during the summer and winter, which are periods with different regional soil water content. In this study, the average Mc values obtained were 0.65 and 0.24 for the summer and winter, respectively. The strong coupling of citrus trees to the atmosphere and the sensitivity of citrus plants to large vapor pressure deficits and air/leaf temperatures caused variations in the Kcb in relation to the ETo ranges. During the summer, the Kcb value ranged from 0.34 when the ETo exceeded 5 mm d(-1) to 0.46 when the ETo was less than 3 mm d(-1). (C) 2011 Elsevier B.V. All rights reserved.

Dryout phenomena in a three-phase fixed-bed reactor

Understanding the mechanism of liquid-phase evaporation in a three-phase fixed-bed reactor is of practical importance, because the reaction heat is usually 7-10 times the vaporization heat of the liquid components. Evaporation, especially the liquid dryout, can largely influence the reactor performance and even safety. To predict the vanishing condition of the liquid phase, Raoult's law was applied as a preliminary approach, with the liquid vanishing temperature defined based on a liquid flow rate of zero. While providing correct trends, Raoult's law exhibits some limitation in explaining the temperature profile in the reactor. To comprehensively understand the whole process of liquid evaporation, a set of experiments on inlet temperature, catalyst activity, liquid flow rate, gas flow rate, and operation pressure were carried out. A liquid-region length-predicting equation is suggested based on these experiments and the principle of heat balance.

Projecto de entreposto frigorífico a R-744/R-717 para produtos congelados

Pretende-se com este trabalho de Projecto de Mestrado conceber uma câmara frigorífica de 20000 m3 para armazenar produtos congelados e estudar um sistema frigorífico com dois fluidos frigorigénios que serão o CO2 (R-744) e o NH3 (R-717) O trabalho inicia-se com a definição dos objectivos principais para o projecto de um sistema frigorífico deste tipo. Após a definição dos objectivos, o projecto propõe um estudo termodinâmico do CO2 (R-744) como fluido frigorigénio, definindo se o seu historial de utilização, características principais, diagrama pressão-entalpia com a distinção das diversas fases do fluído, comparação em diversos parâmetros com outros fluidos, o porquê da utilização deste fluído, problemas comuns em sistemas com a presença deste fluído, entre outros parâmetros de estudo. De seguida será feito o dimensionamento de uma câmara frigorífica de 20000 m3 para armazenar produtos congelados paletizados através de um balanço térmico manual e um balanço térmico informático através do programa da Centauro comparando no final o resultado dos dois métodos. Será feito uma descrição das características básicas da câmara frigorífica com um esquema simples do edifício. Será abordado de seguida o sistema frigorífico a CO2 (R-744) e a NH3 (R-717) nas suas diversas características, nomeadamente no cálculo de caudais, diagrama pressão-entalpia, dimensionamento de tubagens e finalmente selecção de equipamento. Procede-se assim também ao estudo um sistema apenas a NH3 (R-717) com as suas diversas características nomeadamente no cálculo de caudais, diagrama pressão-entalpia, dimensionamento de tubagens e finalmente selecção dos equipamentos mais importantes, para que desta forma se proceda à comparação dos dois sistemas a nível energético, consumo eléctrico e manutenção de equipamentos. Finalmente proceder-se-á interpretação de resultados com o objectivo final de escolher a melhor solução nos vários parâmetros de comparação, para o esquema em questão.

Estudo, Avaliação e Propostas de melhoria para a Climatização do edifício Biorama do Parque Biológico de Vila Nova de Gaia

A eficiência energética e a preocupação com a sustentabilidade têm vindo a ganhar preponderância na sociedade moderna. Este trabalho é uma contribuição para esta tendência onde se pretendeu avaliar e sugerir alterações ao sistema de climatização do edifício Biorama do Parque Biológico de Vila Nova de Gaia (PBG). Procedeu-se em primeiro lugar a uma caracterização física, química e geográfica dos 5 biomas constituintes do Biorama. Para isso, recorreu-se a documentos fornecidos pelo próprio PBG, visitas ao local e registo de medições de alguns parâmetros (temperatura, humidade relativa, qualidade do ar). Posteriormente foi realizado o balanço térmico dos edifícios, de acordo com a legislação em vigor, recorrendo a expressões e conceitos teóricos. Foram determinados valores dos ganhos térmicos de aquecimento de 15811, 10694, 7939, 9233, e 6621 kWh/ano para Floresta tropical, Mesozoico, Dunas, Savana e Deserto, respetivamente. Foram igualmente determinados valores dos ganhos térmicos no verão de 7093, 4798, 3560, 4144 e 2971 kWh na Floresta tropical, no Mesozoico, nas Dunas, na Savana e no Deserto, respetivamente. As cargas térmicas de aquecimento foram 149, 125, 47, 60 e 51 kW na Floresta tropical, no Mesozoico, nas Dunas, na Savana e no Deserto, respetivamente. As cargas térmicas de arrefecimento foram iguais a 59, 57, 47, 35 e 36 kW na Floresta tropical, no Mesozoico, nas Dunas, na Savana e no Deserto, respetivamente. Algumas soluções são avançadas, bem como alternativas comportamentais de modo a corrigir alguns problemas identificados. Uma proposta é a da instalação de painéis solares e acumuladores de calor, com os quais se estima um ganho médio conjunto de 500 W em cada bioma, e representam um investimento de 1050 euros e terão um retorno de 1 ano. Em relação à humidade é sugerido a utilização mais eficaz dos aspersores existentes e a utilização de esponjas, para fazer subir a humidade relativa para valores superiores a 80%. Em sentido inverso, no inverno, propõem-se a utilização de material higroscópico para fazer baixar a humidade relativa em cerca de 5%. Os custos com os suportes e o material higroscópico rondam os 250 €. Por fim, é sugerido a instalação de um aparelho de ar condicionado de 16 000 BTU no corredor de ligação, pois é a única forma de garantir condições de conforto térmico. Esta proposta de arrefecimento com ar condicionado e ainda o recurso a uma cortina de lâminas de plástico, que servem para efetuar uma separação mais eficiente entre ar frio e ar quente, têm um custo aproximado de 350 €. É ainda sugerida a utilização de lonas ou de uma planta trepadeira com um custo por planta de 5€, nas coberturas dos telhados virados a sul, sendo que a zona do corredor deverá ser totalmente coberta, a fim de evitar a exposição solar direta.

Vibration control with shape-memory alloys in civil engineering structures

Dissertação apresentada à Faculdade de Ciências e Tecnologia da Universidade Nova de Lisboa para obtenção do grau de Doutor em Engenharia Civil

Application of the combined integral method to Stefan problems

In this paper we present a new, accurate form of the heat balance integral method, termed the Combined Integral Method (or CIM). The application of this method to Stefan problems is discussed. For simple test cases the results are compared with exact and asymptotic limits. In particular, it is shown that the CIM is more accurate than the second order, large Stefan number, perturbation solution for a wide range of Stefan numbers. In the initial examples it is shown that the CIM reduces the standard problem, consisting of a PDE defined over a domain specified by an ODE, to the solution of one or two algebraic equations. The latter examples, where the boundary temperature varies with time, reduce to a set of three first order ODEs.

Contact melting of a three-dimensional phase change material on a flat substrate

In this paper a model is developed to describe the three dimensional contact melting process of a cuboid on a heated surface. The mathematical description involves two heat equations (one in the solid and one in the melt), the Navier-Stokes equations for the flow in the melt, a Stefan condition at the phase change interface and a force balance between the weight of the solid and the countering pressure in the melt. In the solid an optimised heat balance integral method is used to approximate the temperature. In the liquid the small aspect ratio allows the Navier-Stokes and heat equations to be simplified considerably so that the liquid pressure may be determined using an igenfunction expansion and finally the problem is reduced to solving three first order ordinary differential equations. Results are presented showing the evolution of the melting process. Further reductions to the system are made to provide simple guidelines concerning the process. Comparison of the solutions with experimental data on the melting of n-octadecane shows excellent agreement.