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Esta dissertação apresenta uma modelagem matemática do escoamento laminar em tubos de paredes permeáveis aplicada à micro-filtração de suspensões. A modelagem utilizou-se da formulação integral das equações de conservação e de funções pré- estabelecidas para o representar os campos de velocidade e de concentração ao longo do tubo permeável. As equações integrais da quantidade de movimento e da conservação das espécies químicas forneceram duas equações diferenciais ordinárias de primeira ordem para as variáveis funcionais \"n (z)\" e \"m (z)\" presentes nas funções pré-estabelecidas. Para a solução destas equações optou-se pelo método de Runge-Kutta de quarta ordem devido a sua simplicidade e versatilidade conhecida da literatura. No entanto a equação para a conservação da quantidade de movimento apresentou grande instabilidade ao ser submetida à solução numérica, contornada a partir da imposição de diferentes formas de evolução para o campo de velocidade, através do funcional n(z) cujas formas de variação foram impostas segundo uma dependência linear, exponencial e polinomial. Por outro lado, a solução da equação para conservação das espécies foi numericamente convergente. De posse das funções pré-estabelecidas e ajustadas a partir da equação da conservação das espécies na forma integral, obtém-se neste trabalho os valores correspondentes para o adimensional de Sherwood, quantificando o processo de transferência de massa. Com os valores de Sherwood, os resultados desta modelagem foram comparados com os da literatura, Grober et al. (Apud Zeman & Zydney, 1996) e outros, e apresentaram-se de acordo para estudos de casos particulares, no intervalo de Peclet de 104 - 106 . / This dissertation presents a mathematical modeling of the larninar flow in permeable tubes applied to the micro-filtration of suspensions. The modeling uses of integral formulation of the conservation equations and of functions pre-established for to represent the fields of velocity and concentration along the permeable tube. The integral equations of the momentum and of conservation of the chemical species its supplied two differential ordinary equations if first order for the variables functional \"n(z)\" and \"m(z)\" presents in the pre-established functions. For the solution of these equations was opted for the method of Runge-Kutta of fourth order due to its simplicity and well-known versatility of the literature. However the equation for the conservation of the momentum presented great instability to be submitted to the numeric solution, outlined starting from the imposition forms different from evolution for the field of velocity, through the functional \"n(z)\" with lineal, exponential and polynomial dependence. However, the solution of the equation for conservation of the species was convergent numerical. Through of the pre-established functions and adjusted starting from the equation of the conservation of the species in the integral form, it was obtained in this work the corresponding values for the dimensionless of Sherwood, quantifying the process of mass transfer. With the values of Sherwood, the results of this modeling were compared with the one of the literature, Grober et al. (Apud Zeman & Zydney, 1996) and other, and they came of agreement for particular cases in the interval of Peclet of 104 the 106.

Esta dissertação apresenta uma modelagem matemática do escoamento laminar em tubos de paredes permeáveis aplicada à micro-filtração de suspensões. A modelagem utilizou-se da formulação integral das equações de conservação e de funções pré- estabelecidas para o representar os campos de velocidade e de concentração ao longo do tubo permeável. As equações integrais da quantidade de movimento e da conservação das espécies químicas forneceram duas equações diferenciais ordinárias de primeira ordem para as variáveis funcionais \"n (z)\" e \"m (z)\" presentes nas funções pré-estabelecidas. Para a solução destas equações optou-se pelo método de Runge-Kutta de quarta ordem devido a sua simplicidade e versatilidade conhecida da literatura. No entanto a equação para a conservação da quantidade de movimento apresentou grande instabilidade ao ser submetida à solução numérica, contornada a partir da imposição de diferentes formas de evolução para o campo de velocidade, através do funcional n(z) cujas formas de variação foram impostas segundo uma dependência linear, exponencial e polinomial. Por outro lado, a solução da equação para conservação das espécies foi numericamente convergente. De posse das funções pré-estabelecidas e ajustadas a partir da equação da conservação das espécies na forma integral, obtém-se neste trabalho os valores correspondentes para o adimensional de Sherwood, quantificando o processo de transferência de massa. Com os valores de Sherwood, os resultados desta modelagem foram comparados com os da literatura, Grober et al. (Apud Zeman & Zydney, 1996) e outros, e apresentaram-se de acordo para estudos de casos particulares, no intervalo de Peclet de 104 - 106 . / This dissertation presents a mathematical modeling of the larninar flow in permeable tubes applied to the micro-filtration of suspensions. The modeling uses of integral formulation of the conservation equations and of functions pre-established for to represent the fields of velocity and concentration along the permeable tube. The integral equations of the momentum and of conservation of the chemical species its supplied two differential ordinary equations if first order for the variables functional \"n(z)\" and \"m(z)\" presents in the pre-established functions. For the solution of these equations was opted for the method of Runge-Kutta of fourth order due to its simplicity and well-known versatility of the literature. However the equation for the conservation of the momentum presented great instability to be submitted to the numeric solution, outlined starting from the imposition forms different from evolution for the field of velocity, through the functional \"n(z)\" with lineal, exponential and polynomial dependence. However, the solution of the equation for conservation of the species was convergent numerical. Through of the pre-established functions and adjusted starting from the equation of the conservation of the species in the integral form, it was obtained in this work the corresponding values for the dimensionless of Sherwood, quantifying the process of mass transfer. With the values of Sherwood, the results of this modeling were compared with the one of the literature, Grober et al. (Apud Zeman & Zydney, 1996) and other, and they came of agreement for particular cases in the interval of Peclet of 104 the 106.

We studied the effect of a first order irreversible chemical reaction on mass transfer for two-phase flow systems in which the continuous phase is a fluid and the dispersed phase consists in catalystspherical particles. The reactive solute is transported by the fluid flow and penetrates through the particle surface by diffusion. The chemical reaction takes place within the bulk of the particle. Wehandle the problem by coupling mass balance equations for internal-external transfer with two boundary conditions: continuity of concentration and mass flux at the particle surface. We start with the case of a single isolated sphere. We propose a model to predict mass transfer coefficient (`reactive' Sherwood number) accounting for the external convection-diffusion along with internal diffusion-reaction. We validate the model through comparison with fully resolved Direct Numerical Simulations (DNS) performed by means of a boundary-fitted mesh method. For the simulation of multi-particle systems, we implemented a Sharp Interface Method to handle strong concentration gradients. We validate the implementation of the method thoroughly thanks to comparison with existing analytical solutions in case of diffusion, diffusion-reaction and by comparison with previously established correlations for convection-diffusion mass transfer. In case of convectiondiffusion- reaction, we validate the method and we evaluate its accuracy through comparisons with single particle simulations based on the boundary-fitted method. Later, we study the problem of three aligned-interacting spheres with internal chemical reaction. We propose a `reactive' Sherwood number model based on a known non-reactive prediction of mass transfer for each sphere. We validate the model by comparison with direct numerical simulations for a wide range of dimensionless parameters. Then, we study the configuration of a fixed bed of catalyst particles. We model the cup-mixing concentration profile, accounting for chemical reaction within the bed, and the mean surface and volume concentration profiles of the particles. We introduce a model for `reactive' Sherwood number that accounts for the solid volume fraction, in addition to the aforementioned effects. We compare the model to numerical simulations to evaluate its limitations

The convective mass transfer coefficient is determined for evaporation in a horizontal rectangular duct, which forms the test section of the transient moisture transfer (TMT) facility. In the test facility, a short pan is situated in the lower panel of the duct where a hydrodynamically fully developed laminar or turbulent airflow passes over the surface of the water. The measured convective mass transfer coefficients have uncertainties that are typically less than ±10% and are presented for Reynolds numbers (ReD) between 560 and 8,100, Rayleigh numbers (RaD) between 6,100 and 82,500, inverse Graetz numbers (Gz) between 0.003 and 0.037, and operating conditions factors (H*) between -3.6 and -1.4. The measured convective mass transfer coefficients are found to increase as ReD, RaD, Gz and H* increase and these effects are included in the Sherwood number (ShD) correlations presented in this thesis, which summarize the experimental data.<p> An analogy between heat and mass transfer is developed to determine the convective heat transfer coefficients from the experimentally determined ShD correlations. The convective heat transfer coefficient is found to be a function of ShD and the ratio between heat and moisture transfer potentials (S*) between the surface of the water and the airflow in the experiment. The analogy is used in the development of a new method that converts a pure heat transfer NuD (i.e., heat transfer with no mass transfer) and a pure mass transfer ShD (i.e., mass transfer with no heat transfer) into NuD and ShD that are for simultaneous heat and mass transfer. The method is used to convert a pure heat transfer NuD from the literature into the NuD and ShD numbers measured in this thesis. The results of the new method agree within experimental uncertainty bounds, while the results of the traditional method do not, indicating that the new method is more applicable than the traditional analogy between heat and mass transfer during simultaneous heat and mass transfer.<p>A numerical model is developed that simulates convective heat and mass transfer for a vapor permeable Tyvek® membrane placed between an airflow and liquid water. The boundary conditions imposed on the surfaces of the membrane within the model are typical of the conditions that are present within the TMT facility. The convective heat and mass transfer coefficients measured in this thesis are applied in the model to determine the heat and moisture transfer through the membrane. The numerical results show that the membrane responds very quickly to a step change in temperature and relative humidity of the air stream. Since the transients occur over a short period of time (less than 1 minute), it is feasible to use a steady-state model to determine the heat and mass transfer rates through the material for HVAC applications.<p>The TMT facility is also used to measure the heat and moisture transfer through a vapor permeable Tyvek® membrane. The membrane is in contact with a water surface on its underside and air is passed over its top surface with convective boundary conditions. The experimental data are used to verify the numerically determined moisture transfer rate through the Tyvek® membrane. The numerical model is able to determine the mass transfer rates for a range of testing conditions within ±26% of the experimental data. The differences between the experiment and the model could be due to a slightly different mass transfer coefficient for flow over Tyvek® than for flow over a free water surface.

The convective mass transfer coefficient is determined for evaporation in a horizontal rectangular duct, which forms the test section of the transient moisture transfer (TMT) facility. In the test facility, a short pan is situated in the lower panel of the duct where a hydrodynamically fully developed laminar or turbulent airflow passes over the surface of the water. The measured convective mass transfer coefficients have uncertainties that are typically less than ±10% and are presented for Reynolds numbers (ReD) between 560 and 8,100, Rayleigh numbers (RaD) between 6,100 and 82,500, inverse Graetz numbers (Gz) between 0.003 and 0.037, and operating conditions factors (H*) between -3.6 and -1.4. The measured convective mass transfer coefficients are found to increase as ReD, RaD, Gz and H* increase and these effects are included in the Sherwood number (ShD) correlations presented in this thesis, which summarize the experimental data.<p> An analogy between heat and mass transfer is developed to determine the convective heat transfer coefficients from the experimentally determined ShD correlations. The convective heat transfer coefficient is found to be a function of ShD and the ratio between heat and moisture transfer potentials (S*) between the surface of the water and the airflow in the experiment. The analogy is used in the development of a new method that converts a pure heat transfer NuD (i.e., heat transfer with no mass transfer) and a pure mass transfer ShD (i.e., mass transfer with no heat transfer) into NuD and ShD that are for simultaneous heat and mass transfer. The method is used to convert a pure heat transfer NuD from the literature into the NuD and ShD numbers measured in this thesis. The results of the new method agree within experimental uncertainty bounds, while the results of the traditional method do not, indicating that the new method is more applicable than the traditional analogy between heat and mass transfer during simultaneous heat and mass transfer.<p>A numerical model is developed that simulates convective heat and mass transfer for a vapor permeable Tyvek® membrane placed between an airflow and liquid water. The boundary conditions imposed on the surfaces of the membrane within the model are typical of the conditions that are present within the TMT facility. The convective heat and mass transfer coefficients measured in this thesis are applied in the model to determine the heat and moisture transfer through the membrane. The numerical results show that the membrane responds very quickly to a step change in temperature and relative humidity of the air stream. Since the transients occur over a short period of time (less than 1 minute), it is feasible to use a steady-state model to determine the heat and mass transfer rates through the material for HVAC applications.<p>The TMT facility is also used to measure the heat and moisture transfer through a vapor permeable Tyvek® membrane. The membrane is in contact with a water surface on its underside and air is passed over its top surface with convective boundary conditions. The experimental data are used to verify the numerically determined moisture transfer rate through the Tyvek® membrane. The numerical model is able to determine the mass transfer rates for a range of testing conditions within ±26% of the experimental data. The differences between the experiment and the model could be due to a slightly different mass transfer coefficient for flow over Tyvek® than for flow over a free water surface.

This thesis numerically studied mass transfer during fully confined liquid jet impingement on a rotating target disk of finite thickness and radius. The study involved laminar flow with jet Reynolds numbers from 650 to 1500. The nozzle to plate distance ratio was in the range of 0.5 to 2.0, the Schmidt number ranged from 1720 to 2513, and rotational speed was up to 325 rpm. In addition, the jet impingement to a stationary disk was also simulated for the purpose of comparison. The electrochemical fluid used was an electrolyte containing 0.005moles per liter potassium ferricyanide (K3(Fe(CN6)), 0.02moles per liter ferrocyanide (FeCN6?4), and 0.5moles per liter potassium carbonate (K2CO3). The rate of mass transfer of this electrolyte was compared to Sodium Hydroxide (NaOH) and Hydrochloric acid (HCl) electrochemical solutions. The material of the rotating disk was made of 99.98% nickel and 0.02% of chromium, cobalt and aluminum. The rate of mass transfer was also examined for different geometrical shapes of conical, convex, and concave confinement plates over a spinning disk. The results obtained are found to be in agreement with previous experimental and numerical studies. The study of heat transfer involved a microchannel for a composite channel of trapezoidal cross-section fabricated by etching a silicon wafer and bonding it with a slab of gadolinium. Gadolinium is a magnetic material that exhibits high temperature rise during adiabatic magnetization around its transition temperature of 295K. Heat was generated in the substrate by the application of magnetic field. Water, ammonia, and FC-77 were studied as the possible working fluids. Thorough investigation for velocity and temperature distribution was performed by varying channel aspect ratio, Reynolds number, and the magnetic field. The thickness of gadolinium slab, spacing between channels in the heat exchanger, and fluid flow rate were varied. To check the validity of simulation, the results were compared with existing results for single material channels. Results showed that Nusselt number is larger near the inlet and decreases downstream. Also, an increase in Reynolds number increases the total Nusselt number of the system.

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Thesis (D. Min.)--Harding University Graduate School of Religion, 2001. / Includes bibliographical references (leaves 261-285).

Thesis (D. Min.)--Harding University Graduate School of Religion, 2001. / Includes bibliographical references (leaves 261-285).