Showing papers in "Numerical Heat Transfer Part A-applications in 1981"
TL;DR: In this article, a control-volume approach for solving two-dimensional elliptic problems involving fluid flow and heat and mass transfer has been developed based on a power-law formulation for the combined convection-diffusion influence.
Abstract: A calculation method based on the control-volume approach has been developed for solving two-dimensional elliptic problems involving fluid flow and heat and mass transfer. The main features of the method include a power-law formulation for the combined convection-diffusion Influence, an equation-solving scheme that consists of a block-correction method coupled with a line-by-line procedure, and a new algorithm for handling the interlinkage between the momentum and continuity equations. Although the method is described for steady two-dimensional situations, its extension to unsteady flows and three-dimensional problems is very straightforward.
584 citations
TL;DR: In this paper, a modified strongly implicit procedure for solving the system of algebraic equations that arise in the finite-difference or finite-analytic description of field problems is presented.
Abstract: A modified strongly implicit procedure for solving the system of algebraic equations that arise in the finite-difference or finite-analytic description of field problems is presented. The method is derived for a nine-point difference scheme and can readily be applied to the more conventional five-point scheme simply through the use of the five-point scheme coefficients. The method is demonstrated by application to several examples and a comparison is made between the performance of the modified procedure and that of the strongly implicit procedure, the alternating direction implicit method, and successive over-relaxation. In all cases examined the modified strongly implicit procedure offers superior results when the number of iterations required for convergence or the computational cost required for convergence is used as the measure of performance. The method is also less sensitive to control volume aspect ratio, relaxation parameters, and mesh subdivision than other available procedures. Savings in comp...
281 citations
TL;DR: In this article, a second-order accurate quadratic upstream interpolation technique is used for the finite differencing of convection terms in the transport equations, thus reducing numerical diffusion error.
Abstract: Numerical results are reported for thermally driven laminar flow in two-dimensional rectangular geometries with one plane, the aperture plane, removed. Finite-difference expressions are derived from a set of approximated transport equations in which large temperature and density variations are allowed but high-frequency pressure oscillations are not. The approach allows small time step limitations to be removed from the calculation procedure. A second-order accurate quadratic upstream interpolation technique is used for the finite differencing of convection terms in the transport equations, thus reducing numerical diffusion error. Parameters varied in the calculations were cavity aspect ratio and inclination angle with respect to gravity, inside wall temperature, and Grashof number. A value of Prandtl number corresponding to air was fixed (Pr = 0.73). For the conditions studied, flow and temperature fields within the cavity are determined mainly by local heat transfer events.
162 citations
TL;DR: In this article, a numerical analysis is carried out to investigate the local and overall heat transfer between concentric and eccentric horizontal cylinders, based on Stone's strongly implicit method, is extended to the 3 × 3 coupled system of the governing partial differential equations describing the conservation of mass, momentum, and energy.
Abstract: A numerical analysis is carried out to investigate the local and overall heat transfer between concentric and eccentric horizontal cylinders. The numerical procedure, based on Stone's strongly Implicit method, is extended to the 3 × 3 coupled system of the governing partial differential equations describing the conservation of mass, momentum, and energy. This method allows finite-difference solutions of the governing equations without artificial viscosity, and conserves its great stability even for arbitrarily large time steps. The algorithm is written for a numerically generated, body-fitted coordinate system. This procedure allows the solution of the governing equations in arbitrarily shaped physical domains Numerical solutions were obtained for a Raylelgh number In the range 102-103, a Prandtl number of 0.7, and three different eccentric positions of the inner cylinder. The results are discussed in detail and are compared with previous experimental and theoretical results.
120 citations
TL;DR: In this article, the authors used the finite-analytic method to solve heat transfer in cavity flow at high Reynolds number (1000) for Prandtl numbers of 0.1, 1, and 10.
Abstract: Heat transfer in cavity flow is numerically analyzed by a new numerical method called the finite-analytic method. The basic idea of the finite-analytic method is the incorporation of local analytic solutions in the numerical solutions of linear or nonlinear partial differential equations. In the present investigation, the local analytic solutions for temperature, stream function, and vorticity distributions are derived. When the local analytic solution is evaluated at a given nodal point, it gives an algebraic relationship between a nodal value in a subregion and its neighboring nodal points. A system of algebraic equations is solved to provide the numerical solution of the problem. The finite-analytic method is used to solve heat transfer in the cavity flow at high Reynolds number (1000) for Prandtl numbers of 0.1, 1, and 10.
96 citations
TL;DR: In this paper, the Gauss elimination algorithm for solving the tridiagonal system of linear algebraic equations associated with most implicit heat conduction codes is specialized to the inverse problem and the upper limit in additional computation time generally does not exceed 27-36%.
Abstract: A very efficient numerical technique has been developed to solve the one-dimensional Inverse problem of heat conduction. The Gauss elimination algorithm for solving the tridiagonal system of linear algebraic equations associated with most implicit heat conduction codes is specialized to the inverse problem. When compared to the corresponding direct problem, the upper limit in additional computation time generally does not exceed 27-36%. The technique can be adapted to existing one-dimensional implicit heat conduction codes with minimal effort and applied to difference equations obtained from finite-difference, finite-element, finite control volume, or similar techniques, provided the difference equations are tridiagonal in form. It is also applicable to the nonlinear case in- which thermal properties are temperature-dependent and is valid for one-dimensional radial cylindrical and spherical geometries as well as composite bodies. The calculations reported here were done by modifying a one-dimensional impl...
45 citations
TL;DR: In this article, a wide range of Rayleigh numbers and aspect ratios were used to estimate the heat transfer rate of a vertical air layer for both perfectly conducting and adiabatic boundaries at the top and bottom ends of the layer.
Abstract: Finite-difference predictions of natural convection in vertical air layers are reported for a wide range of Rayleigh numbers and aspect ratios. Calculations were carried out for both perfectly conducting and adiabatic boundaries at the top and bottom ends of the layer. Comparisons between the predictions and measurements show that the calculated heat transfer rates are valid only over a limited range of parameters, and that the points of departure correlate closely with predicted points of instability.
40 citations
TL;DR: In this article, the inverse Laplace transform was used to solve the inverse heat conduction problem, which is carried out directly by using a novel technique that is both simple and accurate.
Abstract: The Laplace transform technique is used to solve the inverse heat conduction problem. The inverse Laplace transform is carried out directly by using a novel technique that is both simple and accurate. Exact and noisy data are used to infer the boundary temperature. It is shown that the present method is accurate and also quite insensitive to measurement errors.
35 citations
TL;DR: In this article, a simple and efficient finite-difference technique using the generalized finitedifference (GFD) discretization is presented for two-dimensional heat transfer problems of irregular geometry, where a finite number of nodal points are distributed in the problem domain.
Abstract: A simple and efficient finite-difference technique using the generalized finite-difference (GFD) discretization is presented for two-dimensional heat transfer problems of irregular geometry. A finite number of nodal points are distributed in the problem domain. At every interior node the spatial derivatives of a field equation are approximated by functional values at neighboring nodes after introducing a family of shape functions for the dependent variables. The resulting simultaneous algebraic equations are solved in a usual manner. The results of two examples, a steady-state heat conduction and a steady natural convection problem, are compared with results of the finite-element and conventional finite-difference method, respectively. The present study demonstrates that, if well implemented, this method will become a handy yet efficient tool for solutions to any field problems since its mathematical concept is simple and the problem formulation is straightforward.
34 citations
TL;DR: In this paper, the effect of natural convection on the heat transfer at the interface and on the movement and shape of the interface is found to be significant compared to the results obtained by neglecting NAT in the melt.
Abstract: Two-dimensional solidification in a rectangular enclosure has been analyzed, taking into account the effects of natural convection and considering the convective and radiative boundary conditions at the surface of the mold and at the top of the enclosure. The bottom is taken as insulated. The dimensionless equations governing the unsteady velocity profiles in the melt and the unsteady temperature profiles in the melt, the solid, and the mold are solved by finite-difference methods, using the alternating direction implicit technique for the vorticity and the stream function. The effect of natural convection on the heat transfer at the interface and on the movement and shape of the interface is found to be significant compared to the results obtained by neglecting natural convection in the melt.
29 citations
TL;DR: In this paper, numerical solutions for thermally and hydrodynamically developing laminar flow in a flat plate duct with uniform suction on one wall and uniform temperature or heat flux independently prescribed at each wall were obtained.
Abstract: Numerical solutions are obtained for thermally and hydrodynamically developing laminar flow in a flat plate duct with uniform suction on one wall and uniform temperature or heat flux independently prescribed at each wall. Building-block solutions and linear superposition allow the construction of the solution for any case of uniform boundary conditions. Application of the results to the design of transpired solar air heaters is discussed.
TL;DR: In this paper, the use of the hn method of Richardson and Gaunt for extrapolation of values for only one halving of the grid size was evaluated and it was concluded that this method is applicable if a consistent order of truncation is used in alt of the finite-difference approximations and if the larger grid size is equal to or less than one-fifth of the least dimension.
Abstract: The time requirements for the finite-difference computation of laminar natural convection in enclosures are so great as to preclude convergence by a preemptive reduction in the grid size. Computed results for one- and mo-dimensional natural convection are analyzed in order to evaluate the use of the hn method of Richardson and Gaunt for extrapolation of values for only one halving of the grid size. It is concluded that this method is applicable if a consistent order of truncation is used in alt of the finite-difference approximations and if the larger grid size is equal to or less than one-fifth of the least dimension.
TL;DR: In this article, the low Prandtl number flow of a conducting fluid about a semi-infinite vertical plate in the presence of a strong cross magnetic field was investigated numerically.
Abstract: The low Prandtl number flow of a conducting fluid about a semi-infinite vertical plate in the presence of a strong cross magnetic field is investigated numerically. The range of Prandtl numbers examined extends down to values appropriate to liquid-metal reactor coolants. A numerical scheme is employed that takes advantage of the established limiting similarity states at the leading edge and downstream.
TL;DR: In this article, a numerical analysis about the boundary layer of steady laminar free convection around an isothermal sphere was carried out, and the results for the average heat transfer coefficient for Pr = 0.7, 10, and 100 and Ra = 10-2, 102, and 106 were expressed within an accuracy of 1%.
Abstract: A numerical analysis is carried out about the thick boundary layer of steady laminar free convection around an isothermal sphere by a method quite similar to that of the previous study for a horizontal cylinder. The results for the average heat transfer coefficient for Pr = 0.7, 10, and 100 and Ra = 10-2, 102, and 106 are expressed within an accuracy of 1% as follows: Num = 2 + 0.S07C(Pr) Ram and C(Pr) is a function of Pr. This expression predicts values a little lower than experimental results in the moderate Rayleigh number range. Also discussed are the radial distributions of temperature and velocity components and the azimuthal distribution of local Nusselt number, and the results are compared with previous theoretical and experimental results for the average Nusselt number.
TL;DR: In this article, the authors proposed a solution of the finite-difference equations by using a novel iterative technique that solves simultaneously for vorticity and stream function along lines.
Abstract: In order to predict natural convection in nonrectangular enclosures the equations of motion are written for orthogonal curvilinear coordinates, using stream function and vorticity as dependent variables. Special integration techniques are described that permit an accurate grid to be found. The solution of the finite-difference equations is obtained by using a novel iterative technique that solves simultaneously for vorticity and stream function along lines. This was found to be economical and stable despite the use of a high-order boundary condition on vorticity. These techniques were applied to the problem of laminar two-dimensional natural convection in an air layer bounded above by an isothermal flat plate and below by a higher-temperature vee-corrugated isothermal surface. The dependence of heat transfer on Rayleigh number, aspect ratio and inclination angle is presented. Where possible, comparisons are made with previous predictions and measurements.
TL;DR: In this article, a finite-difference numerical scheme is employed to obtain transient response solutions to the time-dependent partial differential equations describing the flow adjacent to a uniformly heated vertical flat plate.
Abstract: A finite-difference numerical scheme is employed to obtain transient response solutions to the time-dependent partial differential equations describing the flow adjacent to a uniformly heated vertical flat plate. The plate is of finite thickness and has an appreciable thermal capacity. It is assumed to be immersed in an extensive quiescent body of fluid that has a Prandtl number of 7.6, as for water around room temperature. Extensive calculations of the effect of the thermal capacity of the plate on the nature of the transient flow behavior have been carried out. The resulting flow regimes are analyzed in detail.
TL;DR: In this article, the problem of transient heat flow from a disk into a half-space when the disk temperature undergoes a step change is investigated by a separation of variables technique.
Abstract: The problem of transient heat flow from a disk Into a half-space when the disk temperature undergoes a step change is investigated by a separation of variables technique. The solution Is obtained in oblate spheroidal coordinates (η, ξ) defined by t2 =(1 — η2)(l + ξ2) and z = ηξ with 0 < η < 1, 0 < ξ In this system, the boundary conditions are not of mixed type and a separation solution is possible. An artificial boundary is introduced at some value of ξ with a boundary condition chosen to minimize Its impact. The resulting coupled Sturm-Liouville problem has discrete eigenvalues and is treated numerically by using Prufer transformations. Comparisons with existing asymptotic and numerical solutions are made.
TL;DR: In this article, the boundary integral equation (BIE) formulation is developed for predicting the heat transfer in situations where heat is exchanged between fluids separated by a finned interface, which can easily handle problems for which the fins and the supporting surface have different thermal conductivities.
Abstract: In this study the boundary integral equation (BIE) formulation is developed for predicting the heat transfer in situations where heat is exchanged between fluids separated by a finned interface. With the BIE formulation discretization for numerical purposes occurs only on the boundary of the relevant domain and thus a considerable reduction in the computational storage and time requirements is achieved in comparison with other equivalent numerical techniques. The flexibility and conceptual simplicity of the BIE method enable quite general linear boundary conditions to be accommodated without difficulty. In particular, the BIE method can easily handle problems for which the fins and the supporting surface have different thermal conductivities. It is shown in this study that the classical BIE method, which employs a relatively crude piecewise-constant approximation, gives an effective treatment even when comparable finite-difference and finite-element implementations fail to provide a satisfactory solution....
TL;DR: In this paper, a finite-element marching procedure is presented for the calculation of transport processes in three-dimensional parabolic flows as in corresponding finite-difference procedures, equations are solved one after the other.
Abstract: A finite-element marching procedure is presented for the calculation of transport processes in three-dimensional parabolic flows As in corresponding finite-difference procedures, equations are solved one after the other. Similarly, longitudinal and cross-stream pressure gradients are uncoupled. Velocity fields are determined by first calculating intermediate velocity values based on estimated pressure gradient distributions and then obtaining appropriate corrections so as to satisfy the continuity equation Illustrative examples concerning flow in the entrance region of straight ducts demonstrate the accuracy and reliability of the proposed technique.
TL;DR: In this paper, the authors employed an efficient iteration scheme based on a fast elliptic equation solver for the quasi-steady freezing encountered in long cylindrical alloy bars, and gave results showing the two-dimensional temperature fields encountered in solidification processes subject to various input conditions.
Abstract: The material characteristics of directionally solidified superalloy crystals require carefully controlled solidification rates, temperature field distributions, and maximum temperature gradients. Experimentally, these can be determined by vacuum casting of cylindrical rods. Corresponding analytical studies yield nonlinear equation systems due to radiative boundary conditions and latent heat effects, which in the past required lengthy finite-difference or finite-element solutions. For the quasi-steady freezing encountered in long cylindrical alloy bars, we employed an efficient iteration scheme based on a fast elliptic equation solver previously reported in the literature. This paper discusses the principal aspects of the computational scheme and gives results showing the two-dimensional temperature fields encountered in solidification processes subject to various input conditions.
TL;DR: In this article, the authors dealt numerically with natural convective heat transfer in a confined rectangular cavity with a relatively low geometric aspect ratio H/W (ratio of the height to the width of the cavity) and different vertical-wall temperatures.
Abstract: The present paper deals numerically with natural convective heat transfer in a confined rectangular cavity with a relatively low geometric aspect ratio H/W (ratio of the height to the width of the cavity) and different vertical-wall temperatures. The numerical calculations, performed by the upwind finite-difference method, are carried out for a confined rectangular cavity with H/W = 0.03-1, Prandtl number Pr= 1-103, and Rayleigh number Ra = 102-106. The numerical results indicate that the geometric aspect ratio significantly affects the rate of heat transfer through the fluid layer. Moreover, useful and convenient correlations of the heat transfer through the fluid layer are derived from the results.
TL;DR: In this paper, numerical predictions for the flow of a laminar axisymmetric gaseous jet lightly seeded with solid particles of uniform size are presented for the same type.
Abstract: Numerical predictions are presented for the flow of a laminar axisymmetric gaseous jet lightly seeded with solid particles of uniform size. The predictions are in good agreement with the available experimental data. This agreement is possible only when the local effectiveness of the momentum transfer between the particles and the gas is accounted for in the gaseous momentum equations.
TL;DR: Timing and accuracy tests of the GEM (general elliptic marching) codes are described, which solve elliptic and mixed discretized two-dimensional partial differential equations by direct (noniterative) spatial marching methods.
Abstract: Timing and accuracy tests of the GEM (general elliptic marching) codes are described. The GEM codes solve elliptic and mixed discretized two-dimensional partial differential equations by direct (noniterative) spatial marching methods. Both 5-point and 9-point stencils may be solved, with no requirement that the coefficients be separable, and quite general boundary conditions are treated. The basic GEM code depends on problem parameters (primarily a large cell aspect ratio Δ/Δy) to control the instability incurred in marching elliptic equations. For a 5-point operator with non-periodic boundary conditions, repeat solutions are obtained in the time equivalent of two SOR iterations. Stabilizing codes allow an increase of the problem size in the marching direction at some penalty in execution time and core storage.
TL;DR: In this article, a numerical method capable of solving nonlinear ordinary and partial differential equations has been developed, which is slightly similar to, but may be more advantageous or more widely applicable than, this article.
Abstract: A numerical method capable of solving nonlinear ordinary and partial differential equations has been developed. It is slightly similar to, but may be more advantageous or more widely applicable tha...
TL;DR: In this paper, the upper-plate heat transfer response to a lateral offset of the plate from a position of precise alignment with a lower plate was investigated. And the lateral offset was varied parametrically, as was the vertical separation distance between the plates.
Abstract: Numerical solutions have been obtained for the upper-plate heat transfer response to a lateral offset of the plate from a position of precise alignment with a lower plate. Both plates are vertical and are at the same uniform temperature above ambient. The lateral offset was varied parametrically, as was the vertical separation distance between the plates. Relative to an aligned plate, offsetting tends to reduce the local heat flux in the initial portion of the plate and to enhance the flux at larger downstream distances. The extent of the initial-region heat flux reduction is greater for larger vertical separation distances, while the extent of the downstream-region enhancement Is diminished. For short plates and for large vertical separation distances, the surface-integrated heat transfer for an offset plate Is less than that for an aligned plate. Offsetting can lead to an enhancement of the surface-integrated heat transfer for small separation distances and for intermediate and long plates.
TL;DR: In this article, a vertical combustor for refuse particle combustion is analyzed for waste energy recovery and a one-dimensional model is constructed that consists of three components of fuel particles, inert solid particles, and the gaseous mixture.
Abstract: A vertical combustor for refuse particle combustion is analyzed for waste energy recovery. A one-dimensional model is constructed that consists of three components of fuel particles, inert solid particles, and the gaseous mixture. The gaseous component is further divided into six chemical species that evolve in combustion at temperatures below about 1367K (2000°F). The solutions show that such com-bustors may be viable for U.S. refuse because of their high content of volatile matter. Combustion of the relatively small char, however, may not be cost-effective In the present combustor, where the fuel residence time is of the order of 2 s for the combustor height of 6-9 m.
TL;DR: A finite-element procedure is described that utilizes conjugate base functions and a modified form of the secant method for solving the discretized equations and has the simplicity normally associated with explicit time integration and the unconditional stability of an implicit time integrator.
Abstract: A finite-element procedure is described that utilizes conjugate base functions and a modified form of the secant method for solving the discretized equations. Since a variational formulation involving conjugate bases provides a consistent diagonal capacitance matrix, time integration can be performed directly from nodal forces rather than by use of a global conductance matrix. By utilizing an iterative procedure, implicit time integration is achieved with nodal forces computed from information associated with elements connected to a given node. The result is that the algorithm has the simplicity normally associated with explicit time integration and the unconditional stability of an implicit time integrator. Furthermore, computational effort is automatically concentrated in the region where the dependent variable is changing most. The procedure holds considerable promise for large-scale nonlinear heat conduction problem.