The Effect of Biot Number on a One-Dimensional General Conduction Solution

2021 ◽  
Author(s):  
Robert L. McMasters ◽  
Filippo de Monte ◽  
James V. Beck
Keyword(s):  
Boundary Condition ◽  
Analytical Solution ◽  
Boundary Conditions ◽  
Biot Number ◽  
Analytical Solutions ◽  
Heat Fluxes ◽  
Graphical Form ◽  
Large Temperature ◽  
Convection Coefficient ◽  
General Analytical Solution

Abstract Analytical solutions for thermal conduction problems are extremely important, particularly for verification of numerical codes. Temperatures and heat fluxes can be calculated very precisely, normally to eight or ten significant figures, even in situations involving large temperature gradients. It can be convenient to have a general analytical solution for a transient conduction problem in rectangular coordinates. The general solution is based on the principle that the three primary types of boundary conditions (prescribed temperature, prescribed heat flux, and convective) can all be handled using convective boundary conditions. A large convection coefficient closely approximates a prescribed temperature boundary condition and a very low convection coefficient closely approximates an insulated boundary condition. Since a dimensionless solution is used in this research, the effect of various values of dimensionless convection coefficients, or Biot number, are explored. An understandable concern with a general analytical solution is the effect of the choice of convection coefficients on the precision of the solution, since the primary motivation for using analytical solutions is the precision offered. An investigation is made in this study to determine the effects of the choices of large and small convection coefficients on the precision of the analytical solutions generated by the general convective formulation. Results are provided, in tablular and graphical form, to illustrate the effects of the choices of convection coefficients on the precision of the general analytical solution.

A General Analytical Solution for the Concentration Profiles of Ion-Exchanged Planar Waveguides

1991 ◽  
Vol 244 ◽  
Author(s):  
Xiaoming Li ◽  
Paul F. Johnson
Keyword(s):  
Ion Exchange ◽  
Analytical Solution ◽  
Boundary Conditions ◽  
Surface Concentration ◽  
Concentration Profiles ◽  
Concentration Boundary ◽  
Glass Waveguides ◽  
General Analytical Solution ◽  
Exchange Processes ◽  
One Step

ABSTRACTDuring the recent years, a great variety of ion-exchange processes, including one-step or two-step electric field assisted ion-exchange processes, have been developed to fabricate different kinds of passive planar glass waveguides, e.g., surface waveguides, which correspond to surface maximum concentration, or buried waveguides, which correspond to inside maximum concentration [1,2,3]. Theoretical calculation of ionic concentration distribution has been of interest since refractive index is generally a linear function of concentration. A general analytical solution to calculate both surface and buried concentration distributions from different ion-exchange processes, however, has not yet been presented. In addition, traditional ion-exchange has been carried out only with constant surface concentration boundary conditions. Little attention has been paid, either experimentally or theoretically, to ion-exchange processes with variable boundary conditions. In fact, the time-dependent surface concentration is experimentally observed for the ion-exchange of GRIN glass in molten salt bath [4]. Very recently, a novel one-step technique [5,6] involving electric field assisted ion-exchange of Na+ in glass by Ag+ from molten AgNO3 bath with decaying silver concentration has been developed to produce buried Ag+ concentration profiles in glass. As the accurate and reproducible processes are very important for fabricating ion-exchanged glass waveguides, theoretical modeling and analysis on the new process are needed.In this paper, the one-dimensional field-assisted linear diffusion equation has been analytically solved by Laplace transformation to theoretically calculate concentration profiles produced by field enhanced ion-exchange process with exponentially decaying surface concentration boundary conditions. The applications of the solution to a variety of ion-exchange processes with different boundary or processing conditions for optical waveguide fabrication have been discussed. The theoretical results prove that the solution is a general analytical solution which can be used to calculate either surface concentration profiles or buried concentration profiles.

Boundary Conditions and Initial Value Lines for Unsteady Homentropic Flow Calculations

1970 ◽  
Vol 21 (2) ◽  
pp. 145-162 ◽  
Author(s):  
W. A. Woods ◽  
H. Daneshyar
Keyword(s):  
Boundary Condition ◽  
Analytical Solution ◽  
Boundary Conditions ◽  
Good Accuracy ◽  
Numerical Calculations ◽  
Time Dependent ◽  
Straight Pipe ◽  
Initial Value ◽  
Use Of Time ◽  
The Difference

SummaryA detailed discussion on the difference between an initial value line and a line characterised by a boundary condition has been presented. Two types of boundaries are described and illustrated. To examine each boundary, several different calculations have been performed for a straight pipe. The results of the numerical calculations are compared with an analytical solution. It is shown that known pressure and velocity at the pipe ends give the most accurate results. Comparisons are also made between several practical types of calculations which give similar findings. The use of time-dependent boundaries can lead to errors as large as 40 per cent in derived results. It is shown that good accuracy can be restored by converting the boundaries into initial value lines. It is concluded that in general no more than one time-dependent boundary should be used in any calculation. Finally it is demonstrated that errors are not revealed by means of pressure diagrams alone.

Effects of thermal boundary conditions on rotating Rayleigh-Bénard convection with implications on geophysical and astrophysical systems

2021 ◽  
Author(s):  
Janet Peifer ◽  
Onno Bokhove ◽  
Steve Tobias
Keyword(s):  
Boundary Condition ◽  
Heat Flux ◽  
Boundary Conditions ◽  
Biot Number ◽  
Thermal Flux ◽  
Thermal Boundary Condition ◽  
Actual Condition ◽  
Fixed Temperature ◽  
Thermal Boundary Conditions ◽  
Core Dynamics

<p>Rayleigh-Bénard convection (RBC) is a fluid phenomenon that has been studied for over a century because of its utility in simplifying very complex physical systems. Many geophysical and astrophysical systems, including planetary core dynamics and components of weather prediction, are modeled by including rotational forcing in classic RBC. Our understanding of these systems is confined by experimental and numerical limits, as well as theoretical assumptions. </p><p>The role of thermal boundary condition choice on experimental studies of geophysical and astrophysical systems has been often been overlooked, which could account for some lack of agreement between experimental and numerical models as well as the actual flows. The typical thermal boundary conditions prescribed at the top and the bottom of a convection system are fixed temperature conditions, despite few real geophysical systems being bounded with a fixed temperature. A constant heat flux is generally more applicable for real large-scale geophysical systems. However, when this condition is applied in numerical systems, the lack of fixed temperature can cause a temperature drift. In this study, we seek to minimize temperature drifting by applying a fixed temperature condition on one boundary and a fixed thermal flux on the other.</p><p>Experimental boundary conditions are also often assumed to be a fixed temperature. However, the actual condition is determined by the ratio of the height and thermal conductivity of the boundary material to that of the contained fluid, known as the Biot number. The relationship between the Biot number and thermal boundary condition behavior is defined by the Robin, or 'thin-lid', boundary condition such that low Biot number boundaries are essentially fixed thermal flux and high Biot number boundaries are essentially fixed temperature. </p><p>This study seeks to strengthen the link between numerical and experimental models and geophysical flows by investigating the effects of thermal boundary conditions and their relationship to real-world processes. Both fixed temperature and fixed flux boundary conditions are considered. In addition, the Robin boundary condition is studied at a range of Biot numbers spanning from fixed temperature to fixed flux, allowing intermediate conditions to be investigated. Each system is studied at increasingly rapid rotation rates, corresponding to decreasing Ekman numbers as low as Ek=10<sup>-5</sup> Heat transport is analyzed using the Nusselt number, Nu, and the form of the solution is described by the number of convection rolls and time-dependency. Further investigations will analyze Nu and fluid movement within a system with heterogeneous heat flux condition on the  sidewall boundary conditions, which is useful in the study of planetary core dynamics. The results of this study have implications for improvements in modeling geophysical systems both experimentally and numerically. </p>

Analytical Solution of the Problem of Conjugate Heat Transfer between a Gasdynamic Boundary Layer and Anisotropic Strip

2020 ◽  
pp. 44-59
Author(s):  
V.F. Formalev ◽  
S.A. Kolesnik ◽  
B.A. Garibyan
Keyword(s):  
Heat Transfer ◽  
Thermal Conductivity ◽  
Boundary Layer ◽  
Heat Conduction ◽  
Analytical Solution ◽  
Boundary Conditions ◽  
Conjugate Heat Transfer ◽  
Heat Fluxes ◽  
The Body ◽  
Conductivity Tensor

The paper focuses on the problem of conjugate heat transfer between the thermal-gas-dynamic boundary layer and the anisotropic strip in conditions of aerodynamic heating of aircraft. Under the assumption of an incompressible flow which takes place in the shock layer behind the direct part of the shock wave, we found a new analytical solution for the components of the velocity vector, temperature distribution, and heat fluxes in the boundary layer. The obtained heat fluxes at the interface between the gas and the body are included as boundary conditions in the problem of anisotropic heat conduction in the body. The study introduces an analytical solution to the second initial-boundary value problem of heat conduction in an anisotropic strip with arbitrary boundary conditions at the interfaces, with heat fluxes which are obtained by solving the problem of a thermal boundary layer used at the interface. An analytical solution to the conjugate problem of heat transfer between a boundary layer and an anisotropic body can be effectively used to control, e.g. to reduce, heat fluxes from the gas to the body if the strip material chosen is such that the longitudinal component of the thermal conductivity tensor is many times larger than the transverse component of the thermal conductivity tensor. Such adjustment is possible due to an increase in body temperature in the longitudinal direction, and, consequently, a decrease in the heat flow from the gas to the body, as well as due to a favorable change in the physical characteristics of the gas. Results of numerical experiments are obtained and analyzed

An Analytical Method for Dielectrophoresis and Traveling Wave Dielectrophoresis Generated by an n-Phase Interdigitated Parallel Electrode Array

2008 ◽  
Vol 130 (8) ◽  
Author(s):  
Hongjun Song ◽  
Dawn J. Bennett
Keyword(s):  
Boundary Condition ◽  
Analytical Solution ◽  
Boundary Conditions ◽  
Analytical Method ◽  
Traveling Wave ◽  
Electrode Array ◽  
Accurate Analysis ◽  
Exact Boundary ◽  
Latex Beads ◽  
Parallel Electrode

In this paper, we present an analytical method for solving the electric potential equation with the exact boundary condition. We analyze the dielectrophoresis (DEP) force with an n-phase ac electric field periodically applied on an interdigitated parallel electrode array. We compare our analytical solution with the numerical results obtained using the commercial software CFD-ACE. This software verifies that our analytical method is correct for solving the problem. In addition, we compare the analytical solutions obtained using the exact boundary conditions and the approximate boundary conditions. The comparison shows that the analytical solution with the exact boundary condition gives a more accurate analysis for DEP and traveling wave DEP forces. The DEP forces of latex beads are also investigated with different phase arrays for (n=2,3,4,5,6).

An Analytical Method for Dielectrophoresis and Traveling Wave Dielectrophoresis Generated by an Interdigitated Parallel Electrode Array

2007 ◽  
Author(s):  
Hongjun Song ◽  
Dawn J. Bennett
Keyword(s):  
Boundary Condition ◽  
Analytical Solution ◽  
Analytical Solutions ◽  
Electric Potential ◽  
Analytical Method ◽  
Traveling Wave ◽  
Mixed Type ◽  
Dielectrophoretic Force ◽  
Potential Equation ◽  
Exact Boundary

As a new technique, dielectrophoresis has been proven to be successful in the separation, transportation, entrapment and manipulation of cells, DNA molecules, and viruses. One typical design uses an array with interdigitated parallel electrodes to manipulate and separate particles using traveling wave and conventional dielectrophoresis. In order to obtain an analytical solution for the dielectrophoretic force or traveling wave dielectrophoretic force, the electric potential equation needs to be solved. Unfortunately, the mixed type of boundary condition (Dirichet and Neumann) for the electric potential equation poses a large challenge for obtaining an analytical solution. Although some analytical solutions have been achieved using an approximate single type of boundary condition instead of the exact boundary condition, this leads to inaccurate results especially in the zone near the electrodes which cannot be neglected. In this paper, we present an analytical method for solving the electric potential equation with the mixed type of boundary condition. We compare our analytical solution with the numerical results obtained using the Computational Fluid Dynamics Research Corporation, CFDRC, code which verifies our analytical method is correct for solving this problem. In addition, comparisons are made between the analytical solutions with approximate boundary conditions and those with exact boundary conditions. The comparison shows our analytical solution gives a more accurate analysis for the conventional and traveling wave dielectrophoretic forces.

scholarly journals Eshelby's problem of a spherical inclusion eccentrically embedded in a finite spherical body

2017 ◽  
Vol 473 (2198) ◽  
pp. 20160808 ◽  
Author(s):  
W.-N. Zou ◽  
Q.-C. He
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Analytical Solutions ◽  
Superposition Principle ◽  
Auxiliary Problem ◽  
Spherical Inclusion ◽  
Spherical Body ◽  
Physical Parameters ◽  
Tensor Fields ◽  
Spherical Domain

Resorting to the superposition principle, the solution of Eshelby's problem of a spherical inclusion located eccentrically inside a finite spherical domain is obtained in two steps: (i) the solution to the problem of a spherical inclusion in an infinite space; (ii) the solution to the auxiliary problem of the corresponding finite spherical domain subjected to appropriate boundary conditions. Moreover, a set of functions called the sectional and harmonic deviators are proposed and developed to work out the auxiliary solution in a series form, including the displacement and Eshelby tensor fields. The analytical solutions are explicitly obtained and illustrated when the geometric and physical parameters and the boundary condition are specified.

scholarly journals Mathematical modeling of belt and rigid conveyer’s drum contact with non-affect Coulomb friction

2019 ◽  
Vol 109 ◽  
pp. 00050
Author(s):  
Hryhorii Larionov ◽  
Mykola Larionov ◽  
Artem Pazynich
Keyword(s):  
Mathematical Modeling ◽  
Boundary Condition ◽  
Analytical Solution ◽  
Boundary Conditions ◽  
Opposite Direction ◽  
Coulomb Friction ◽  
Surface Boundary ◽  
Arc Length ◽  
Cylindrical Coordinates ◽  
Surface Boundary Condition

Conveyor’s belt with the drum contact task is paid a great researcher attention. It had solved in stresses with the mix boundary condition and Coulomb friction low action applies. Obtained solutions, as usual, do not satisfy to the displacement on drum surface boundary condition. The attempt to solve the system of Lamb equations in cylindrical coordinates in condition h/R<<1 values where h the belt thin and R drums radius is done. The Prandtl‘s technique and substitution methods are used to analytical solution obtain. As a result, the displacement expressions for rest and slide drum arcs are obtained. In search of rest arc length, radial and peripheral displacement equalities are used. Applied rest arc efforts are equal in values and have an opposite direction. The equality means the fact that traction efforts do not transfer under rest drum arc. The obtained results are coincided with the ones obtained by Zhukovskiy N.E. for flexible thread model. The fact to simultaneously satisfy to boundary conditions on inner and outer belt sides have to do the conclusion for boundary belt seam existing. Expressions for rest drum arc length without applied friction law are obtained. The displacement and corresponding stresses graphics are demonstrated.

scholarly journals ANALYTICAL SOLUTION OF A 2D TRANSIENT HEAT CONDUCTION PROBLEM USING GREEN´S FUNCTIONS

2020 ◽  
Vol 19 (1) ◽  
pp. 66
Author(s):  
J. R. F. Oliveira ◽  
J. A. dos Santos Jr. ◽  
J. G. do Nascimento ◽  
S. S. Ribeiro ◽  
G. C. Oliveira ◽  
...  
Keyword(s):  
Temperature Field ◽  
Heat Conduction ◽  
Analytical Solution ◽  
Boundary Conditions ◽  
Heat Conduction Problem ◽  
Transient Heat Conduction ◽  
Heat Fluxes ◽  
Two Dimensional ◽  
One Dimensional ◽  
Transient Heat Conduction Problem

Through the present work the authors determined the analytical solution of a transient two-dimensional heat conduction problem using Green’s Functions (GF). This method is very useful for solving cases where heat conduction is transient and whose boundary conditions vary with time. Boundary conditions of the problem in question, with rectangular geometry, are of the prescribed temperature type - prescribed flow in the direction x and prescribed flow - prescribed flow in the direction y, implying in the corresponding GF given by GX21Y22. The initial temperature of the space domain is assumed to be different from the prescribed temperature occurring at one of the boundaries along x. The temperature field solution of the two-dimensional problem was determined. The intrinsic verification of this solution was made by comparing the solution of a 1D problem. This was to consider the incident heat fluxes at y = 0 and y = 2b tending to zero, thus making the problem one-dimensional, with corresponding GF given by GX21. When comparing the results obtained in both cases, for a time of t = 1 s, it was seen that the temperature field of both was very similar, which validates the solution obtained for the 2D problem.

Steady Heat Conduction With Generalized Boundary Conditions

2016 ◽  
Author(s):  
Kevin D. Cole ◽  
Filippo de Monte ◽  
Robert L. McMasters ◽  
Keith A. Woodbury ◽  
A. Haji-Sheikh ◽  
...  
Keyword(s):  
Heat Transfer ◽  
Boundary Condition ◽  
Boundary Conditions ◽  
Biot Number ◽  
Internal Heat ◽  
Computer Code ◽  
Number System ◽  
Bioheat Equation ◽  
The Relationship ◽  
Steady Heat

Heat transfer in solids provides an opportunity for students to learn of several boundary conditions: the first kind for specified temperature, the second kind for specified heat flux, and the third kind for specified convection. In this paper we explore the relationship among these types of boundary conditions in steady heat transfer. Specifically, the normalized third kind of boundary condition (convection) produces the first kind condition (specified temperature) for large Biot number, and it produces the second kind condition (specified flux) for small Biot number. By employing a generalized boundary condition, one expression provides the temperature for several combinations of boundary conditions. This combined expression is presented for several simple geometries (slabs, cylinders, spheres) with and without internal heat generation. The bioheat equation is also treated. Further, a number system is discussed for each combination to identify the type of boundary conditions present, which side is heated, and whether internal generation is present. Computer code for obtaining numerical values from the several expressions is available, along with plots and tables of numerical values, at a web site called the Exact Analytical Conduction Toolbox. Classroom strategies are discussed regarding student learning of these issues: the relationship among boundary conditions; a number system to identify the several components of a boundary value problem; and, the utility of a web-based resource for analytical heat-transfer solutions.

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