Short-term behavior of classical analytic solutions for the design of ground-source heat pumps

Over the years several methods have been proposed to simulate and design the earth heat exchanger for a ground-source heat pump (GSHP) system. Some of these methods are based on numerical techniques while others rely on analytic solutions. Among the latter, two classical solutions have been extensively used over the years, the infinite line source (ILS) solution and the infinite cylindrical source (ICS). These solutions were known to overestimate the fluid temperature when the time scale is important and are valid only in a time range between a minimum and a maximum value which are often adequate for must design applications. It is usually accepted that for small Fourier numbers, the ICS solution should be used instead of the ILS. This paper revisits the short-term behavior of these solutions and we arrive at different conclusions than those usually accepted in the literature if the Fourier number is based on the borehole radius, which is normally the case. The reasons for these discrepancies are discussed and several options are proposed.     

A study of fundamental solution in orthotropic thermodiffusive elastic media

The present investigation is to study the fundamental solution for two dimensional problem in orthotropic thermodiffusive elastic medium. After applying the dimensionless quantities, we have derived the most general expression for displacements, concentration, temperature, normal and tangential stress. The general solution for a point heat source on the surface of a semi infinite orthotropic thermodiffusive plane has been obtained. A special case of interest is also deduced from the general expression in the absence of diffusion effect. The amplitude of surface displacements, temperature change and concentration are computed and presented graphically to depict the effect of diffusion.     

Slow solidification of a cylinder with constant heat efflux

An asymptotic solution for the slow solidification of an infinite cylinder with constant surface heat flux is presented. The solidification of a 10 cm. diameter aluminum cylinder in 30 minutes corresponds to a value of 0.00901 for the small parameter. The predictions of the solidification front position based on the first three terms in the asymptotic expansion are presented. The extension to steady-state continuous casting is discussed.     

Unsteady mhd flow and heat transfer on a rotating disk in an ambient fluid

An unsteady flow and heat transfer of a viscous incompressible electrically conducting fluid over a rotating infinite disk in an otherwise ambient fluid are studied. The unsteadiness in the flow field is caused by the angular velocity of the disk which varies with time. The magnetic field is applied normal to the disk surface. The new self-similar solution of the Navier–Stokes and energy equations is obtained numerically. The solution obtained here is not only the solution of the Navier–Stokes equations, but also of the boundary layer equations. Also, for a simple scaling factor, it represents the solution of the flow and heat transfer in the forward stagnation-point region of a rotating sphere or over a rotating cone. The asymptotic behaviour of the solution for a large magnetic field or for a large independent variable is also examined. The surface shear stresses in the radial and tangential directions and the surface heat transfer increase as the acceleration parameter increases. Also the surface shear stress in the radial direction and the surface heat transfer decrease with increasing magnetic field, but the surface shear stress in the tangential direction increases.     

Estimation of front surface temperature and heat flux of a locally heated plate from distributed sensor data on the back surface

We present a new method of solving the three-dimensional inverse heat conduction (3D IHC) problem with the special geometry of a thin sheet. The 3D heat equation is first simplified to a 1D equation through modal expansions. Through a Laplace transform, algebraic relationships are obtained that express the front surface temperature and heat flux in terms of those same thermal quantities on the back surface. We expand the transfer functions as infinite products of simple polynomials using the Hadamard Factorization Theorem. The straightforward inverse Laplace transforms of these simple polynomials lead to relationships for each mode in the time domain. The time domain operations are implemented through iterative procedures to calculate the front surface quantities from the data on the back surface. The iterative procedures require numerical differentiation of noisy sensor data, which is accomplished by the Savitzky–Golay method. To handle the case when part of the back surface is not accessible to sensors, we used the least squares fit to obtain the modal temperature from the sensor data. The results from the proposed method are compared with an analytical solution and with the numerical solution of a 3D heat conduction problem with a constant net heat flux distribution on the front surface.     

Green’s function for a line heat source acting on the surface of a coated isotropic thermoelastic material

Two-dimensional Green’s function, for a line heat source acting on the surface of a coated isotropic thermoelastic material, is investigated in this paper to improve the understanding of interface mechanisms of coating/substrate system. The coating and substrate are modeled as infinite layer and semi-infinite substrate respectively. They are perfectly bonded or are in smooth contact at the interface. Based on the two-dimensional general solution of isotropic thermoelastic materials expressed by harmonic functions, the corresponding harmonic functions with undetermined constants for coating and semi-infinite substrate are constructed, respectively. The thermoelastic field can be obtained by substituting the harmonic function into the general solution. The constants can be determined by the free surface boundary conditions and interface continuous conditions between the coating and the semi-infinite substrate. Numerical results are exhibited in the form of contours and some valuable conclusions for interface effect, interface shear debonding and coating tensile failure are presented.     

Temperature fields of the extruded pipe under conditions of co-current cooling

This article examines the temperature fields of the thick wall pipe while cooling it under the process of extrusion. We solved the Fourier–Kirchhoff equation by Fourier method in the form of an infinite row and with the help of Bessel’s functions. The equations were transformed into dimensionless forms and a solution of during heat we got as the function of Biot’s and Fourier’s number, dimensionless inner radius and thermal capacitance ratio of contact phases.     

A numerical solution of developing temperature for laminar developing flow in eccentric annular ducts

The energy equation expressed in bipolar coordinates is used to determine the temperature distribution in the thermal entrance region of an eccentric annular duct. An implicit alternating-direction method is used in the numerical solution. The analysis of the hydrodynamic entrance region, which provides the velocity distributions needed for the thermal solution, was obtained from a published solution by the present authors. A published Graetz solution for an eccentric annulus and a published combined thermal and hydrodynamic entrance region solution for the circular tube are used in the verification of the present solution. In the present analysis 17 combinations of fundamental thermal boundary conditions, Prandtl number, and annular geometry are considered. The annular geometry with equal relative eccentricity and radius ratio of 0.5 is used to study the effects of eccentricity and Prandtl number on the fluid temperature and surface heat flux distributions.     

Effects of free convection and mass transfer on the flow past an impulsively moving infinite vertical circular cylinder

The effects of free convection and mass transfer are taken into account for the Stokes' problem of the flow near an impulsively moving infinite vertical circular cylinder. Expressions of the velocity, temperature, concentration and skin friction of the fluid in closed form are obtained by the Laplace transform technique. The results based on various values of the parameters Gr (Grashof number), Gm (modified Grashof number), Sc (Schmidt number) and Pr (Prandtl number) are given in graphical form. It will be seen that there is a rise in the velocity due to the presence of a foreign mass. But higher Sc yields the lower velocity and skin friction. As the radius of the circular cylinder approaches to infinite, the results presented in this paper agree with those of V.M. Soundalgekar's and C.K. Chen's etc. for the flow past an impulsively moving infinite vertical plate.     

Analysis of Mode-III fracture problem with multiple cracks by boundary element alternating method

An efficient boundary element alternating method was developed in this present study for the analysis of a Mode-III fracture problem with single or multiple cracks in a finite sheet. Firstly, an analytical solution for a crack in an infinite domain, subjected to an arbitrary longitudinal shear loading across the crack surface, is developed herein. The solution can then be obtained through the iterative superposition of this analytical solution and the boundary element of a finite uncracked sheet. Several Mode-III fracture problems in a finite sheet, with single and multiple cracks under various boundary conditions, are discussed for confirming the validity of this work. Excellent agreement can be observed. The interaction effects among cracks and influence of the boundaries are also considered.     

The Deceleration of two hollow concentric infinite cylinders submerged in three fluids with different viscosities

In this paper, an analytical solution is presented for the angular velocities of two infinite, hollow and concentric cylinders having initial angular velocities in viscous fluid with time. The momentum equation is solved for the three regions, neglecting the inertia force because of the difficulty of the deduced equations in this case. The resulting torque affected on each cylinder surface is calculated, and the equations of motion for the two cylinders are solved using the analysis of the eigenvalue problem.     

KIRCHHOFF'S TYPE FORMULA IN THERMOELASTICITY WITH FINITE WAVE SPEEDS

Kirchhoff's type formula for a solution of a central system of field equations of thermoelasticity with finite wave speeds is established in the paper. The solution of the system (φ, θ), where φ and θ denote the thermoelastic displacement potential and temperature respectively, is expressed by surface integrals over the boundary of a thermoelastic solid whose kernels have the form of infinite series involving Bessel functions. Application of the formula to reduce the solution of an initial-boundary problem to that of a vector Fredholm integral equation of the first kind is outlined.     

Analysis of rewetting of an infinite tube by numerical fourier inversion

A semi—analytical model for the two—dimensional quasi—steady conduction equation, governing conduction controlled rewetting of an infinite tube, has been suggested. The solution yields the temperature field as a function of various input model parameters such as Peclet number, Biot number and radius ratio of the tube. Unlike earlier investigations, the present semi-analytical model predicts the temperature field for the entire domain of a tube, employing the Wiener—Hopf technique and by inverse discrete Fourier transform (IDFT) algorithm.     

THERMAL SHOCK FRACTURE IN A W-Cu DIVERTOR PLATE WITH A FUNCTIONALLY GRADED NONHOMOGENEOUS INTERFACE

The plane elasticity solution is presented in this article for the crack problem of a W-Cu divertor plate under thermal shock. The material is made of a graded layer with exponentially varying thermomechanical properties bonded between a homogeneous substrate and a homogeneous coating and is subjected to a cycle of heating and cooling on the coating surface of the material. The surface layer contains an embedded or a surface crack perpendicular to the boundaries. Using superposition the problem is reduced to a perturbation problem in which the crack surface tractions are only external forces. The dimensions, geometry, and loading conditions of the original problem are such that the perturbation problem may be approximated by a plane strain mode I crack problem for an infinite divertor plate. Fourier transforms are used to formulate the crack problem in terms of a singular integral equation. After giving some sample results regarding the distribution of thermal stresses, stress intensity factors for embedded and surface cracks are presented. Also included are the results for a crack/contact problem in a divertor plate that is under compression near and at the surface and tension in the interior region.     

An analysis of surface-microstructures effects on heterogeneous nucleation in pool boiling

The interaction of surface microstructures and wettability effects on heterogeneous nucleation in pool boiling is analyzed in this paper based on the changes of free energy and availability. It is shown that the bubble is most easily formed on a concave surface in comparison with a convex surface or a plane surface at the same wettability and the same wall temperature. It is found that the effect of microstructures greatly enhances nucleation of bubbles when the curvature radius of these microstructures is in the range of 5–100 times less than the bubble radius. Larger than this limit, the surface roughness effect is negligible and the wettability effect predominates. Closed form analytical solutions for the critical radius and change in availability are obtained for the special case of homogeneous nucleation where no wall temperature gradient exists on surfaces with microstructures. Under this simplified assumption, it is found that the microstructures have no effect on critical nucleation radius and their effect on the change in availability is underestimated.     

反射式高倍聚光光伏“光漏斗”匀光技术理论与实验研究

在聚光器和接收器之间加入“光漏斗”匀光器来实现砷化镓(GaAs)电池表面能流密度的均匀分布。根据蒙特卡洛射线法和照射在接收器表面能流密度相同的原理,设计“光漏斗”匀光器。聚光器开口半径为1600 mm,焦距为1700 mm,Tracepro仿真结果显示接收器表面能流密度的均匀度为98.51%,实验显示均匀度为95.05%,仿真值与实验值相近。根据GaAs电池输出电量最大化来确定电池芯片的布置范围,仿真结果得到在半径为36 mm内布置电池芯片,与实验值39 mm相近。     

SOLUTION OF DISPLACEMENT BOUNDARY VALUE PROBLEM UNDER UNIFORM HEAT FLUX

The general solution of the displacement boundary value problem is obtained for an infinite plate with an arbitrary shaped hole under uniform heat flux in any direction. The complex stress functions, the dislocation method, and a rational mapping function are used and the closed solution is obtained. An infinite plate with a circular hole and a slit is analyzed under the condition of the constrained displacements. The singularity at the tip of the slit of the constrained displacement is investigated     

Modeling a fire whirl generated over a 5-cm-diameter methanol pool fire

We conducted a series of laboratory-scale fire whirl experiments spinning 5-cm-diameter methanol pool fires and observed elongated flame height compared with the pool fire without spin. A simple scaling analysis was conducted to obtain dependency of the axial flame height on the momentum-controlled circulation and the effect of buoyancy. To obtain a specific functional relationship for the parameters obtained by the scaling analysis, we developed an analytical model consisting of coupled species and energy equations and Burgers vortex for circulation generated by a fire whirl. The solution of the coupling equations shows that the average rate of heat transfer from the flame to the fuel surface is a function of the vortex core radius; a smaller vortex core radius provides more heat to the fuel surface enhancing evaporation thereby producing the longer flame height. This new model predicts both flame height and flame shape. The flame height prediction compare favorably with results from the scaling analysis and experiment.     

The line-spring model for embedded crack and slightly emergent surface crack

Based on the solutions of through-cracked strip, the constitutive relations of line-springs for embedded crack and slightly emergent surface crack are formulated for the first time. The line-spring model has been improved with the aid of a three-dimensional solution of embedded elliptical crack in an infinite body to include the curvature effects of surface crack front, i.e. the interaction between springs. The calculated results from the improved model are very consistent with those from the alternative iteration reported previously (Shah, R. C. & Kobayashi, A. S. J. Pres. Vessel Technol., 1 (1974) 47–54); the improved model has higher precision and a larger effective range than the unimproved model.     

Thermal propagation in solids due to surface laser pulsation and oscillation

Most studies of heat transfer pertaining to a planar medium subjected to time-varying and spatially-decaying laser incidence along with external surface cooling are based on the diffusion theory (parabolic heat conduction equation), an approximation that implies a non-physical infinite speed of energy transport. In this study, temperature distributions within one-dimensional plates subjected to the aforementioned heating and cooling conditions, but with thermal energy propagation at a finite speed, are presented. Incident energy that is partially absorbed at the external surface is transferred through the plate by conduction, while the remaining energy is convectively cooled to the environment. The present investigation will examine three different time characteristics of the incident heat sources which include a continuously operating, a pulsed and an oscillatory laser source. The temperature results were obtained by using a simple and concise finite-difference algorithm based on the Godunov scheme, a method developed for the solution of resulting characteristic equations that govern thermal wave propagations within the solid.