Boundary Conditions Of Third Kind Research Articles

A gradient based numerical algorithm to solve inverse dynamic viscoelastic problems of multi-variable identification

A gradient based numerical method is put forward to solve the inverse dynamic viscoelastic problem with multi-variable identification for the constitutive and TKBC (third kind boundary conditions) related parameters. A three parameter model is employed to describe the time dependent TKBC concerned with structural interaction, and is formulated in the forward and inverse dynamic viscoelastic analysis. In addition to the reliable platform provided by TPAA-SBFEM to gain the solution of forward problems, a piecewise recursive adaptive algorithm is developed to obtain the derivatives required in the sensitivity analysis, which is able to provide a stable temporal solution accuracy for different step sizes. SBFEM (Scaled Boundary Finite Element Method) is stressed in the structural analysis with its own advantages, such as the efficient and convenient treatment of stress singularity problem and flexibility of mesh generation with polygonal elements and image based quadtree gridding. The inverse problem of identification is solved by the Levenberg-Marquardt method, and the regional inhomogeneity, crack, initial guess, location/number of measurement points, material contrast ratios, and noisy data are taken into account. Numerical examples are provided to verify the proposed approach, and the satisfactory results are achieved.

Quantitative thermal-wave depth profiles of solids with spatially variant cooling coefficients imaged using lock-in thermography

A generalized treatment of the thermal-wave boundary-value problem in a solid slab strip with large sidewalls exposed to the ambient and third-kind boundary conditions over the extended sidewall surfaces was presented. Unlike conventional heat conduction theoretical practice, it was found that the cooling coefficient h is not constant and depends on the thermal gradient variation with depth/length. The exponential damping of the TW amplitude with depth presented an ideal condition to reveal the depth dependence of h which under normal steady heat transfer conditions might appear approximately constant in a non-steeply changing thermal field. The observed depth dependence of the heat loss coefficient can be used to establish the physically expected constancy of thermal diffusivity at different modulation frequencies in juxtaposition to some published reports. It is expected that this work can be used to optimize other thermophysical parameters for materials subject to periodic heating.

Analytical Inverse Analysis Methodological Approach for Thermo-Physical Parameters Estimation of Multilayered Medium Terrain with Homogenized Sampled Measurements

The current paper presents results of the inverse theory approach utilized for the analytical estimation of thermo-physical properties for a multi-layered medium terrain with homogenized experimental measurements. We demonstrate the derivation steps of the exact solution for the heat transfer problem with third-kind boundary conditions due to natural convection on the outlets posed for the considered experimental domain. There are received analytical expressions. Initially, we illustrate the homogenization of the boundary conditions. We then discuss the process of derivation for the analytical solution of the posed problem with the help of key elements of the Fourier method. We provide an algorithm for applying the contact condition to extend received expressions for multiple layers. After that we demonstrate the major steps for the construction of nonlinear systems of equations to be solved in order to obtain exact values of key thermo-physical and geometrical parameters of the investigated medium with the help of received exact analytical expressions. Along with analytical procedures, we present a posed experimental design and discuss an algorithm of numerical exploitation for a suggested method, outlining its advantages and possible limitations in terms of initial approximations.

Boundary optimal control of time-space SIR model with nonlinear Robin boundary condition.

A boundary optimal control problem arising in time-space SIR epidemic models is treated. In this work we aim with the control of the flux of infected individuals crossing part of boundary. On the other side of the domain, we suppose a nonlinear boundary condition of third kind: nonlinear Robin boundary condition, this condition models immersing individual crossing this part of the boundary of the domain of study. We give the existence and uniqueness of the solution of both state and optimal control problem ending some numerical tests throughout a simple example.

Impact of boundary conditions of third kind on nanoliquid flow and Radiative heat transfer through asymmetrical channel

The present study is devoted to analyze the influence of thermal radiation and chemical reaction on a transient incompressible natural convective flow of nanoliquid saturated porous medium through two rigid, parallel asymmetric plates with the negligible thickness and capable to transfer the heat with nano-liquid by convection. It is assumed that nano-liquids are made up with pouring various types of nanoparticles into the water based fluid. Single phase model is adopted to determine the thermo-physical properties of the nano-liquid and simple Darcy model is utilized to predict the flow through porous medium. The transport equations are solved to analyze the important characteristics of momentum, energy and concentration by Differential Transform Method. To compare these semi-analytical results, Crank-Nicolson finite difference scheme is used and it was in good agreement. The study reveals that Biot number influences appreciably on the velocity, temperature and concentration distributions. The significant variation in the concentration profile is found by imposing chemical reaction in the case of large Biot number.

Transient Convective Heating Transport of the Micropolar Fluid Flow Between Asymmetric Channel with Activation Energy

Here modelling and computations are presented to introduce the concept of activation energy involved in the micropolar fluid between the porous vertical channel with the boundary conditions of third kind. The fluid is considered to be gray, absorbing-emitting but non scattering medium. The resulting system of equations are solved numerically by Crank-Nicolson implicit finite difference method. The effect of various physical parameter such as activation energy, micropolar parameter, reaction rate, Reynolds number, Schmidt number, heat and mass transfer Biot numbers on the velocity, temperature and concentration field are discussed by means of pictorial representation.

The effects of exterior boundary conditions on a internally heated tumor tissue with a thermoporoelastic model

Modeling flow field in tumor regions interstitial space is of primary importance, because of the importance of advection in macromolecule drug delivery. Its deformation has also to be taken into account because of the forces caused by the fluid; if the tumor region is not isothermal, this deformation can be also strongly affected by temperature fields.In this paper, the effects of thermal boundary conditions on a tumor region periphery with an internal heat source are investigated. The tumor region is modeled as a deformable sphere, in which two phases can be distinguished. The fluid phase is the interstitial fluid, while the rest of the tumor is modeled as the solid phase, including also capillaries and tissues. Transient-state governing equations for mass, momentum and energy are written for both phases, by also considering tumor deformation under the linear elastic material assumption. A situation of Tumor Blood Flow (TBF) rapid decay, in which vascular pressure rapidly approaches to zero, is considered, while the heat source is modeled as a fourth-grade radial-decay function. Boundary conditions for the energy equation are varied on the external surface of the sphere, in order to appreciate the effects of the surrounding on flow and temperature fields inside the tumor. After scaling equations, a finite-element scheme is employed for the numerical solution. Comparisons with analytical solutions from literature show a good agreement. Results are shown for different dimensionless parameters that are referred to temperature, volumetric strain, pressure and velocity, showing in which case external boundary conditions strongly affect tumor region flow fields and a third-kind boundary condition is needed.

Theoretical studies of filtration movement of moisture in capillaries of woody plants

The article presents theoretical studies of filtration transport of moisture through capillaries of woody plants that is determined by humidity gradients. The emerging humidity fields and the magnitude of the gradients significantly affect the growth and development of woody plants biomass. The aim of theoretical research was to use mathematical tools of operational calculus that provide an opportunity to consider the filtration of moisture through the capillaries of wood materials using a third-kind boundary condition, when the substance is exchanged with the environment at the considered boundary. As a result of the use of Laplace transform a mathematical model for studying the movement of moisture in the capillaries of woody vegetation through filtration transport has been obtained. Modelling of a moisture transport process in wood materials is an important tool in theoretical and experimental studies. This is particularly important for conducting long-term experimental studies of the site quality of stand of trees from the technogenic impact of vehicles on roadside ecological systems.

Steady natural convection Couette flow with wall conduction and thermal boundary condition of third kind

AbstractThis study theoretically investigates the effects of viscous dissipation and boundary plate thickness on an incompressible heat generating/absorbing fluid with non‐uniform internal temperature unlike lumped heat capacitance assumption. The flow, which is laminar is induced by free convection caused by asymmetric heating of the channel boundaries and internally generated energy caused by viscous dissipation. In addition, convection through the boundary plates is also considered in the flow formation. One of the plates channel moves along the flow direction while the other is stationary. Due to the non‐linear and coupling nature of the governing flow equations, homotopy perturbation method (HPM) has been adopted to analytically find the approximate solution to the problem. The effects of thermodynamic and hydrodynamic parameters are depicted in graphs and tables. It is discovered from the investigation that, both the velocity and temperature profiles increase with increase in viscous dissipation. Velocity distribution decreases with increase in Biot number while the temperature distribution near the heated plate increases with increase in Biot number. The increase in boundary plate thickness d causes a boost in the fluid flow across the medium and reduces the temperature of the fluid near the heated plate. It is further discovered that the rate of heat transfer on both plates increase with increase in Biot number while they drastically dropped when the boundary plate thickness d is increased. The shear stresses on the surface of both plates increase with increase in heat generation while reverse cases were observed with increase in heat absorption . It is further observed that the volume flow rate within the channel increases with increase in viscous dissipation .

Numerical investigation of laminar forced convection for a non-Newtonian nanofluids flowing inside an elliptical duct under convective boundary condition

PurposeThe purpose of this study is to investigate the thermal response of the laminar non-Newtonian fluid flow in elliptical duct subjected to a third-kind boundary condition with a particular interest to a non-Newtonian nanofluid case. The effects of Biot number, aspect ratio and fluid flow behavior index on the heat transfer have been examined carefully.Design/methodology/approachFirst, the mathematical problem has been formulated in dimensionless form, and then the curvilinear elliptical coordinates transform is applied to transform the original elliptical shape of the duct to an equivalent rectangular numerical domain. This transformation has been adopted to overcome the inherent mathematical deficiency due to the dependence of the ellipsis contour on the variables x and y. The yielded problem has been successfully solved using the dynamic alternating direction implicit method. With the available temperature field, several parameters have been computed for the analysis purpose such as bulk temperature, Nusselt number and heat transfer coefficient.FindingsThe results showed that the use of elliptical duct enhances significantly the heat transfer coefficient and reduces the duct’s length needed to achieve the thermal equilibrium. For some cases, the reduction in the duct’s length can reach almost 50 per cent compared to the circular pipe. In addition, the analysis of the non-Newtonian nanofluid case showed that the addition of nanoparticles to the base fluid improves the heat transfer coefficient up to 25 per cent. The combination of using an elliptical duct and the addition of nanoparticles has a spectacular effect on the overall heat transfer coefficient with an enhancement of 50-70 per cent. From the engineering applications view, the results demonstrate the potential of elliptical duct in building light-weighted compact shell-and-tube heat exchangers.Originality/valueA complete investigation of the heat transfer of a fully developed laminar flow of power law fluids in elliptical ducts subject to the convective boundary condition with application to non-Newtonian nanofluids is addressed.

Analysis of Crack Resistance of an Oil Trunk Pipeline Considering the Varying Failure Viscosity in the Neighborhood of a Welded Joint

This article presents a development of the method of analysis of crack resistance of an oil trunk pipeline in the zone of the transverse welded joint during operation. The nonlinear nonstationary heat conductivity problem was solved by the finite difference method using the boundary conditions of third kind. Modeling of the kinetics of conversion of austenite to ferrite and bainite under non-isothermal conditions during welding was carried out based on the theory of isokinetic reactions. The calculation of the residual welding stress is performed by solving the problem of thermoelastic plasticity using the finite element method for material with a nonstationary structure. The calculation of the crack resistance is based on Irwin’s force failure criterion, and the dependence of the failure viscosity on the structural composition is considered.

Unsteady-state moisture behavior calculation for multilayer enclosing structure made of capillary-porous materials

The paper studies unsteady-state moisture behavior of multilayer enclosing structure using discrete-continuous method. Modification of Gagarin’s differential moisture transfer equation is proposed. The proposed moisture transfer equation within one layer of a multilayer enclosing structure is a second-order differential parabolic equation with constant coefficients. Third-kind boundary conditions of moisture exchange are set for enclosing structure boundaries. Fourth-kind boundary conditions represented by moisture potential flows equality are set at the joint of two different materials. An analytical expression for moisture potential calculation in any enclosing structure section, at any time, under continuous control for temperature distribution, has been derived using discrete-continuous method. Calculations for multilayer enclosing structure consisting of clay brick base and lime brick lining have been made. The obtained results have been compared to calculation results obtained by well-known Gagarin’s unsteady-state method and Kozlov’s engineering method. It has been shown that the discrete-continuous method allows calculating unsteady-state moisture behavior of multilayer enclosing structures by analytical expression, and obtaining moisture distribution similar to Gagarin’s unsteady-state method, thus it can be used by design engineers in practice.

Modelling the third kind boundary condition in scaled boundary finite element method based numerical analysis

Numerical models to deal with the third kind boundary condition (TKBC) are presented under the framework of Scaled Boundary Finite Element Method (SBFEM) for the 2-D static analysis. A gradient based algorithm is presented to tackle with nonlinear TKBC, and a temporally-piecewise adaptive algorithm is developed for a kind of time dependent TKBC. In addition, a proof that the appended stiffness matrix is block-circulant is given if the TKBC is cyclically symmetric, resulting in a reduction of computational expense of SBFEM based numerical analysis. Numerical examples are provided to verify proposed approaches, and satisfactory results are obtained.

Critical Conditions of Thermal Explosion Considering Space-Temporal Nonlocality

The equation for locally non-equilibrium heat exchange with nonlinear internal heat source was obtained by using Fourier’s law formula with the relaxation of heat flux and temperature gradient. The thermal conductivity problem for the infinite plate was studied by the numerical method at nonsymmetrical boundary conditions of the third kind. The conclusion was made that the thermal explosion delay time considerably increased as compared to the case of neglecting the space-temporal nonlocality.

Moisture behavior calculation of single-layer enclosing structure by means of discrete-continuous method

The paper proposes mathematical model for unsteady-state moisture behaviour calculation of single-layer enclosing structures by means of moisture potential and discrete-continuous method. A formula for moisture potential value calculation for single-layer enclosing structure in any enclosing structure section, at any moment of time, under continuous control for temperature distribution, has been derived. Moisture transfer between the enclosing structure and ambient air is taken into account by means of third-kind boundary conditions. Temperature distribution is taken as constant during a month. The proposed method allows for moisture potential determination according to the proposed formula for every month. Calculation results obtained by the proposed method, well-known unsteady-state method and engineering method developed by V.G. Gagarin and V.V.Kozlov have been compared for single-layer enclosing structure made of aerated concrete. It was shown that average value calculation results disagreement does not exceed 7% for the proposed method and unsteady-state method. However, the proposed approach allows for moisture distribution determination using analytical expression, which is convenient for use in engineering practice.

Convective drying of osmo-dehydrated apple slices: kinetics and spatial behavior of effective mass diffusivity and moisture content

The main objective of this study is the presentation of a numerical model of liquid diffusion for the description of the convective drying of apple slices submitted to pretreatment of osmotic dehydration able of predicting the spatial distribution of effective mass diffusivity values in apple slabs. Two models that use numerical solutions of the two-dimensional diffusion equation in Cartesian coordinates with the boundary condition of third kind were proposed to describe drying. The first one does not consider the shrinkage of the product and assumes that the process parameters remain constant along the convective drying. The second one considers the shrinkage of the product and assumes that the effective mass diffusivity of water varies according to the local value of the water content in the apple samples. Process parameters were estimated from experimental data through an optimizer coupled to the numerical solutions. The osmotic pretreatment did not reduce the drying time in relation to the fresh fruits when the drying temperature was equal to 40 °C. The use of the temperature of 60 °C led to a reduction in the drying time. The model that considers the variations in the dimensions of the product and the variation in the effective mass diffusivity proved to be more adequate to describe the process.

Inverse problem on the identification of temperature and thermal stresses in an FGM hollow cylinder by the surface displacements

ABSTRACTA problem on the identification of time-dependent temperature on one of the limiting surfaces of a radially inhomogeneous hollow cylinder is formulated and solved under the temperature and radial displacement given on the other limiting surface. The analysis of temperature and thermal stress distribution in the cylinder is performed. The solution has been constructed by the reduction to an inverse thermoelasticity problem. By making use of the finite difference method, a stable solution algorithm is suggested for the analysis of inverse problem. The solution technique is verified numerically by making use of the solution to a relevant direct problem. It is shown that the proposed technique can be efficiently used for the identification of a heat flux or unknown parameters (the surrounding temperature or the heat-exchange coefficient) in the third-kind boundary conditions.

Numerical Solution and it’s Analysis during Solar Drying of Green Peas

A mathematical model is developed for solar drying of green peas (Botanical name: Pisum Sativum). The problem is solved assuming the shape of the green peas is spherical. The governing transient mass transfer equation is discretized into finite difference scheme. The time marching is performed by implicit scheme. The governing equations and boundary conditions are non-dimensionalized to get generic results. The product in the chamber is in contact with air which is heated by solar energy, so the boundary conditions of third kind (convective boundary conditions) are considered. By space and time discretization a set of algebraic equations are generated and these algebraic equations are solved by tridiagonal matrix algorithm. A computer code is developed in MATLAB in order to compute the transient moisture content distribution inside the product. Center point, boundary and mean moisture of green peas are estimated at different temperatures and drying time. Present numerical result is compared with experimental result from literature and it was found that there is a good agreement of results. The drying time is predicted for how quickly the mean moisture of green peas is reached to 50, 40, 30, 20 and 10% of its initial moisture corresponding to different temperatures.

Flow and heat transfer of couple stress fluid in a vertical channel in the presence of heat source/sink

PurposeThe purpose of this paper is to consider the problem of fully developed laminar mixed convection flow of a couple stress fluid in a vertical channel with the third-kind boundary conditions in the presence or absence of heat source/sink effect.Design/methodology/approachThrough proper choice of dimensionless variables, the governing equations are developed. These governing equations are solved analytically by the differential transform method and numerically by the Runge–Kutta shooting method. Analytical solutions for the velocity and temperature profiles for heat generation and absorption of the problem are reported.FindingsThe mass flow rate and Nusselt numbers at both the left and right channel walls on mixed convection parameter, Brinkman number, couple stress parameter and heat generation/absorption parameter for equal and unequal Biot numbers are presented. Favorable comparisons of special cases with previously published work are obtained. It is found that velocity, temperature, mass flow rate and Nusselt number decrease with couple stress parameter and increase with mixed convection parameter and Brinkman number.Originality/valueThe work done in this paper is not done earlier to the authors’ knowledge. This is the first paper in which the sixth-order differential equation is solved using the semi-numerical method, which is a differential method.

Non-linear drag induced entropy generation analysis in a microporous channel: The effect of conjugate heat transfer

The entropy generation analysis in a viscous dissipative flow of a Newtonian fluid through a hyper-porous microchannel formed between two heated parallel plates is conferred. Employing an analytical method, which is consistent with the perturbation analysis, the transport equations governing the thermo-hydrodynamics are studied. The effects of nonlinear Forchheimer drag and conjugate heat transfer on the thermal transport characteristics of heat are considered, while the thermal boundary conditions of third kind have been employed at the outer boundaries of channel for the conjugate heat transfer analysis. The explicit alterations are made in the thermal transport of heat in the system as attributable to the effect of dissipative heat generated due to the non-linear effect Forchheimer drag, Darcy frictional effect and the viscous shearing stress in the flow field. To account these effects, the explicit variation of Forchhiemer constant, Darcy number, porosity, thickness and conductivity of upper wall and Biot number of upper wall are carried out to shows the changes in the available energy of the system. Also, it is shown that the effect of non-linear drag mainly stemming from the presence of complex porous structure in the flow field and its interaction with the conjugate transport of heat alters the heat transfer rate in the system non-trivially, which, in turn, gives rise to the entropy generation in the system. The individual contribution of two different effects viz., the heat transfer and viscous dissipation on the system entropy generation rate for different cases are studied. It is believed that the implications of the present analysis may have direct bearing on the design of micro devices/systems typically used in electronic cooling, micro-heat pipes.