Optimal Control of Nonlocal Thermistor Equations

We are concerned with the optimal control problem of the well known nonlocal thermistor problem, i.e., in studying the heat transfer in the resistor device whose electrical conductivity is strongly dependent on the temperature. Existence of an optimal control is proved. The optimality system consisting of the state system coupled with adjoint equations is derived, together with a characterization of the optimal control. Uniqueness of solution to the optimality system, and therefore the uniqueness of the optimal control, is established. The last part is devoted to numerical simulations.Comment: Submitted 21-March-2012; revised 11-June-2012; accepted 13-June-2012; for publication in the International Journal of Contro

Optimal Control of the Thermistor Problem in Three Spatial Dimensions

This paper is concerned with the state-constrained optimal control of the three-dimensional thermistor problem, a fully quasilinear coupled system of a parabolic and elliptic PDE with mixed boundary conditions. This system models the heating of a conducting material by means of direct current. Local existence, uniqueness and continuity for the state system are derived by employing maximal parabolic regularity in the fundamental theorem of Pr\"uss. Global solutions are addressed, which includes analysis of the linearized state system via maximal parabolic regularity, and existence of optimal controls is shown if the temperature gradient is under control. The adjoint system involving measures is investigated using a duality argument. These results allow to derive first-order necessary conditions for the optimal control problem in form of a qualified optimality system. The theoretical findings are illustrated by numerical results

Nucleation and growth control in protein crystallization

The five topics summarized in this final report are as follows: (1) a technique for the expedient, semi-automated determination of protein solubilities as a function of temperature and application of this technique to proteins other than lysozyme; (2) a small solution cell with adjustable temperature gradients for the growth of proteins at a predetermined location through temperature programming; (3) a microscopy system with image storage and processing capability for high resolution optical studies of temperature controlled protein growth and etching kinetics; (4) growth experiments with lysozyme in thermosyphon flow ; and (5) a mathematical model for the evolution of evaporation/diffusion induced concentration gradients in the hanging drop protein crystallization technique

A Study of Heat Transfer and Heat Flow Asymmetry through Water in the presence of the Density Maximum

This thesis is concerned with heat flow asymmetry through composites of water and aqueous solutions. Central to this exploration are the behaviour of heat transfer in the vicinity of the density maximum and the behaviour of the temperature of maximum density of aqueous solutions. Both of these topics are investigated by cooling a rectangular enclosure of the test fluid in a quasi-steady state manner. During cooling, the temperature at select points within the liquid is monitored and the flow of heat at both isothermal walls is measured. As the liquid cools through its density maximum the normal single-cell convection that occurs in the presence of a horizontal temperature gradient changes to a double cell configuration in the vicinity of the density maximum. This transition manifests itself in changes in the horizontal temperature profile across the cavity and in the rate of cooling of the fluid. A measurement technique to study the behaviour of the temperature of maximum density of aqueous solutions is described in this thesis that relies on these changes. These changes are investigated experimentally and numerically. The study of the behaviour of heat transfer and the temperature of maximum density of aqueous solutions revealed that heat transfer is reduced in the vicinity of the density maximum and that the temperature of maximum density of aqueous solutions depends on the nature and concentration of the solute. Both of these results are exploited in the study of heat flow asymmetry through a device that consists of two cubic enclosures side by side; one enclosure contains water with a density maximum at 4oC and the other enclosure contains a saline solution with a density maximum at 2oC. A temperature gradient, which spans both of these temperatures of maximum density, is applied horizontally across the composite system, resulting in different rates of heat transfer through the device depending on the gradient direction. Experiments performed with a 12cm x 6cm x 6cm container yield heat transfer rates of 0.55W and 0.19W depending on the direction of the temperature gradient, resulting in a rectification factor of 65.4%. Asymmetrical heat transfer rates are also found in composite systems of water and solids when the temperature gradient spans the temperature of maximum density of the water. Results from computational fluid dynamics confirm the experimental results, and are used to investigate the influence of such parameters as temperature gradient and container aspect ratio on the rectification factor

An investigation of the effects of surfactant monolayers on natural convection heat transfer and evaporative mass transfer

Laboratory experiments are presented that reveal the effects of surfactant monolayers on natural convection heat transfer and evaporation from bodies of water; more specifically, the situation is studied where the bulk temperature of the water is greater than the temperature of the air so that evaporative convection occurs. Four sets of experiments were performed in a laboratory environment on insulated tanks of different widths and depths for the following surface conditions: 1) clean surface, 2) oleyl alcohol covered surface, 3) stearic acid covered surface, and 4) stearyl alcohol covered surface. An infrared (IR) camera was used to verify the existence of each of these conditions, and to measure the surface temperature during these experiments. The Nusselt-Rayleigh (Nu-Ra) and Sherwood-Rayleigh (Sh-Ra) dimensionless power law parameterizations were used to characterize the efficiency of convective heat transfer and evaporative mass transfer, respectively. This dissertation research presents the first such parameterizations that definitively show how these heat and mass transport processes are affected by surfactant monolayers. All three of the surfactants reduced Nu by nearly one order of magnitude compared to clean surface conditions at equivalent Ra. This is attributed to the ability of the surfactants to change the hydrodynamic boundary condition at the interface from shear-free for the case of the clean surface to one that supports shear. The convective motion of the water is consequently damped in the presence of the surfactants, and the efficiency of convective heat transfer decreases. Despite this fact, all four surface conditions share essentially the same Nu-Ra power law exponent of 0.360. This result is different from typical Rayleigh-Benard Nu-Ra studies which find a power law exponent close to 1/3. The present results indicate that the rate at which Nu increases with Raw is greater for a free surface condition than for the solid boundary condition of Rayleigh-Benard type studies. For all four surface conditions, the Sh-Ra parameterizations are found to be very close to obeying a 1/3 exponent which indicates that the mass transfer coefficient is independent of the horizontal extent of the surface W. The oleyl alcohol and stearic acid surface conditions both give the same Sh-Ra result as the clean surface condition; this surprising result reveals that the efficiency of evaporation is unaffected by these two surfactants at equivalent Ra. Stearyl alcohol, however, decreased Sh by approximately 50% compared to all other conditions. This is attributed to the ability of stearyl alcohol to block the transport of water vapor through the surfactant film. An effective relative humidity is defined which allows for the density of water vapor at the surface to be less than the equilibrium saturation value; this newly defined parameter allows for the stearyl alcohol Sh-Ra data to collapse with the other cases

Comparative analysis of different methods of modeling the thermal effect of circulating blood flow during RF cardiac ablation

Our aim was to compare the different methods of modeling the effect of circulating blood flow on the thermal lesion dimensions created by radio frequency (RF) cardiac ablation and on the maximum blood temperature. Computational models were built to study the temperature distributions and lesion dimensions created by a nonirrigated electrode by two RF energy delivery protocols (constant voltage and constant temperature) under high and low blood flow conditions. Four methods of modeling the effect of circulating blood flowon lesion dimensions and temperature distribution were compared. Three of them considered convective coefficients at the electrode-blood and tissue-blood interfaces to model blood flow: 1) without including blood as a part of the domain; 2) constant electrical conductivity of blood; and 3) temperaturedependent electrical conductivity of blood (+2%/°C). Method 4) included blood motion andwas considered to be a reference method for comparison purposes. Only Method 4 provided a realistic blood temperature distribution.The other three methods predicted lesion depth values similar to those of the reference method (differences smaller than 1 mm), regardless of ablation mode and blood flow conditions. Considering the aspects of lesion size and maximum temperature reached in blood and tissue, Method 2 seems to be the most suitable alternative to Method 4 in order to reduce the computational complexity. Our findings could have an important implication in future studies of RF cardiac ablation, in particular, in choosing the most suitable method to model the thermal effect of circulating blood

Doctor of Philosophy

dissertationPeriodic temperature measurements in the DOI/GTN-P Deep Borehole Array on the western Arctic Slope of Alaska have shown a strong near-surface permafrost warming over the last 40 years, particularly since ∼ 1990. Due to the manner in which these deep wells were drilled, the portion of the observed permafrost warming caused by climate change has remained unclear. Other factors that have strongly influenced temperatures near the wellbores include the heat deposited into permafrost during drilling and local-landscape changes associated with drilling operations (creation of reserve pits and drill pads). Multidimensional heat-transfer models capable of assessing the magnitude of the drilling and local-landscape disturbances near the wellbores have not been available. For the western Arctic Slope, such models must be capable of simulating heat-transfer processes in layered fine-grained mudrocks whose thermal properties are highly nonlinear due to the occurrence of unfrozen water at temperatures well below 0°C. An assessment of the drilling and landscape-change effects also requires knowledge of the specific thermophysical properties occurring at the well sites. Little information has been available about these properties on the western Arctic Slope. To establish the portion of the observed permafrost warming related to drilling and landscape-change effects, multidimensional (2-D cylindrical, 3-D cartesian) numerical heat-transfer models were created that simulate heat flow in layered heterogenous materials surrounding a wellbore, phase changes, and the unfrozen water properties of a wide range of fine-grained sediments. Using these models in conjunction with the borehole temperature measurements, the mean thermophysical properties of permafrost rock units on the western Arctic Slope were determined using an optimization process. Incorporation of local meteorological information into the optimization allows a more refined estimate of the thermal properties to be determined at a well site. Applying this methodology to the East Simpson #1 well on the Beaufort Sea coast (70°55.046'N, 154°37.286'W), the freezing point of permafrost is found to be -1.05°C at this site and thermal diffusivities range 0.22-0.40 × 10 -6 m2 s-1. Accounting for the drilling and landscape-change effects, tundra adjacent to East Simpson is found to have warmed 5.1 K since the mid-1880s. Of this, 3.1 K (60%) of the warming has occurred since 1970

Natural Convection Above Heated Inclined Surfaces.

Motivated by a need to assess pollutant transport by upslope flows, an investigation has been conducted into the fundamentals of natural convection flow over inclined surfaces. Particular attention was focused on the influence of ambient fluid stability. Field studies were performed using tracer gas releases into the upslope flow over a Southern California mountain range. The field studies served to reveal the presence of a split slope flow recirculation and demonstrated the impact of this recirculation on the transport of pollutants from a valley. In order to pursue a controlled investigation of the phenomena found in the field work, a laboratory model was developed using water as a working fluid. Extensive dye studies demonstrated the presence of this recirculation to varying degrees in nearly every configuration with a stable layer present. Heat transfer experiments were conducted with the laboratory model to refine and validate the experimental techniques used. Comparisons are made with existing theoretical and empirical predictions where available. Existing correlations for inclined surfaces are extended two orders of magnitude lower in Rayleigh number. A modification to vertical theory based on simply replacing g with gcos$\theta$ is shown to be useful for inclinations down to 75\sp\circ from vertical. Transition ranges and empirical correlations are expressed for inclinations of 0\sp\circ, 15\sp\circ, 30\sp\circ, 45\sp\circ, 60\sp\circ, 75\sp\circ, and 90\sp\circ. Overall correlations are also reported with apparently far less scatter of data than for any previously reported research with inclined surfaces. Experimentation with stratified ambient fluid resulted in the observation that, with turbulent flow, stratification could apparently be disregarded and heat transfer simply calculated from local conditions. The observed heat transfer coefficients are essentially independent of position along the slope suggesting that an approximate analytical model of upslope flows developed by L. Prandtl in 1942 is applicable. However, quantitative laboratory results showed that Prandtl\u27s one dimensional theory underpredicted the observed boundary layer depths. The results are in good agreement with observed characteristics of atmospheric slope flows