Direct Substitution Method Research Articles

Singular Finite Elements for the Die-Swell Simulation of Viscoelastic Fluids

In previous work, we have attempted to apply the singular finite element method (SFEM) developed for Newtonian flow problems to the die-swell simulation of viscoelastic (Giesekus) fluids, and reported that SFEM, with less memory size, gave more accurate results than those obtained by the ordinary decoupled FEM. In this paper, a revised version of above SFEM, i.e. SFEM vpE12, has been presented, where the velocity (v) and stress (E) nodes, as well as the pressure (p) node, are removed from a position of singularity; it implies that asymptotic behaviors of field variables near the singular point (i.e., die-lip) are assumed to be such that v - r 0.5 E ∼ r -0.5 and p - r -0.5 , where r denotes a radial distance from the singular point. Furthermore, as for the stress substitution for momentum equation, we attempted two methods, i.e., the rearranged substitution (RS) method and the direct substitution (DS) method. It is found that SFEM vpE12 with RS (i.e., SFEM vpE12-RS) extends the upper limit of the Weissenberg number (We) of the die-swell flow simulation up to 150, whereas SFEM vpE12 with DS (i.e., SFEM vpE12-DS) fails to converge beyond 1.8 of We; however, starting from converged results obtained by SFEM vpE12-RS at 130 of We, SFEM vpE12-DS simulation is able to converge at the same We of 130, and gives more smoothed and accurate results than the original SFEM especially in the very vicinity of singularity.

A critical heat flux approach for square rod bundles using the 1995 Groeneveld CHF table and bundle data of heat transfer research facility

The critical heat flux (CHF) approach using CHF look-up tables has become a widely accepted CHF prediction technique. In these approaches, the CHF tables are developed based mostly on the data bank for flow in circular tubes. A set of correction factors was proposed by Groeneveld et al. [Groeneveld, D.C., Cheng, S.C., Doan, T., 1986. 1986 AECL-UO Critical Heat Flux lookup table. Heat Transf. Eng. 7(1–2), 46] to extend the application of the CHF table to other flow situations including flow in rod bundles. The proposed correction factors are based on a limited amount of data not specified in the original paper. The CHF approach of Groeneveld and co-workers is extensively used in the thermal hydraulic analysis of nuclear reactors. In 1996, Groeneveld et al. proposed a new CHF table to predict CHF in circular tubes [Groeneveld, D.C., et al., 1996. The 1995 look-up table for Critical Heat Flux. Nucl. Eng. Des. 163(1), 23]. In the present study, a set of correction factors is developed to extend the applicability of the new CHF table to flow in rod bundles of square array. The correction factors are developed by minimizing the statistical parameters of the ratio of the measured and predicted bundle CHF data from the Heat Transfer Research Facility. The proposed correction factors include: the hydraulic diameter factor ( K hy), the bundle factor ( K bf), the heated length factor ( K hl), the grid spacer factor ( K sp), the axial flux distribution factors ( K nu), the cold wall factor ( K cw) and the radial power distribution factor ( K rp). The value of constants in these correction factors is different when the heat balance method (HBM) and direct substitution method (DSM) are adopted to predict the experimental results of HTRF. With the 1995 Groeneveld CHF Table and the proposed correction factors, the average relative error is 0.1 and 0.0% for HBM and DSM, respectively, and the root mean square (RMS) error is 31.7% in DSM and 17.7% in HBM for 9852 square array data points of HTRF.

An independent assessment of Groeneveld et al.'s 1995 CHF look-up table

The prediction capability of the 1995 CHF look-up table (Groeneveld D.C., et al., Nucl. Eng. Des. 163 (1996) 1–23) is independently assessed based on the KAIST data base consisting of 10 822 data for uniformly-heated, vertical, round tubes. This confirms the error statistics for the heat balance method reported by Groeneveld et al. and shows overall average and RMS errors of 4.2 and 36.7%, respectively, for the direct substitution method. The new 1995 table shows better prediction capability than the 1986 AECL-UO table ( Groeneveld et al., 1986), especially for the low-pressure, low-flow region. The error analysis indicates the length effect even for significantly long tubes.

Critical heat flux prediction using genetic programming for water flow in vertical round tubes

The genetic programming method is used to develop critical heat flux (CHF) correlations for upward water flow in vertical round tubes under low pressure and low flow conditions. The genetic programming, as a symbolic regression tool, finds both the functional form and fitting coefficients of a correlation without any initial assumptions. Inlet and local condition type correlations are developed based on 414 and 314 CHF data from KAIST CHF data bank, respectively. The inlet condition type correlation shows the rms error of 15.2% and the local condition type one shows the rms errors of 32.7% and 13.2% by the direct substitution method and the heat balance method, respectively. Prediction errors are smaller than or comparable to those for other existing correlations.

A Direct Substitution Method for Multicomponent Solute Transport in Ground Water

AbstractA numerical method was developed using a direct substitution approach for simulating one‐dimensional multicomponent solute transport in ground water. The method has the ability to treat equilibrium reactions of aqueous and surface complexation, and precipitation‐dissolution. The concentrations of aqueous component species, adsorbed component species, and precipitated species are chosen as the principal dependent variables. The substitution takes place after the transport equation is discretized. The resulting system of equations is solved using the Newton‐Raphson method and the Jacobian matrix is computed analytically. This method is computationally more efficient than sequential iteration methods due to its faster convergence rate. It is also computationally more efficient than other direct substitution methods which compute the Jacobian matrix numerically.

Assessment of CHF correlations for internally heated concentric annulus channels

The existing CHF correlations for internally heated concentric annulus channels are assessed using KAIST CHF database for uniformly heated vertical annuli. Five annulus CHF correlations (Jannsen-Kervinen, Barnett, Levitan-Lantsman, Doerffer et al., and Bobkov et al.) are chosen for assessment based on literature survey and Groeneveld et al.'s CHF table for round tube is also assessed for comparison. Among the above correlations, one is inlet-condition type and others local conditions type. To make the comparison meaningful, the local-condition-type correlations are assessed in two ways: direct substitution method (DSM) and heat balance condition method (HBM). Totally 1125 data are classified into 10 groups based on pressure and mass flux conditions and correlations are assessed to each group separately. Prediction capability of each correlation depends on the data group and non shows the best prediction over the entire group. In overall, the correlations by Doerffer et al. and Janssen et al. appear to be the best, but Barnett or Levitan-Lantsman correlations also show reasonable prediction for most groups. However, the low-pressure, low flow CHFs are not well predicted by any correlations. The CHF table for round tubes overpredicts the CHF in annuli at fixed local conditions.

Application of energy balance and direct substitution methods for thermal margins and data evaluation

Thermal limits, such as critical heat flux (CHF), flow excursion (FE) and incipient boiling (IB), are used to determine the maximum operating power for a variety of thermal systems, most importantly nuclear reactors. In doing so, a variety of analytical tools, concepts, and techniques are involved. Concepts and definitions involved include the critical heat flux ratio (CHFR), the critical power ratio (CPR) or departure from nucleate boiling power ratio (DNBPR), the direct substitution method (DSM), the heat balance method (HBM), local correlations, system or global correlations, and upstream effects. Some of these concepts and terms cause confusion and misunderstanding in the technical community as to their real physical meaning and their proper application. Some investigators argue strongly in support of using the CPR for determining safety margins in nuclear reactors rather than the CHFR and still others argue that the use of the HBM for thermal limits data and correlation evaluation gives “consistently better predictions than when using DSM”. Those claims are an indication of a general confusion in understanding and applying these concepts. The intent of this paper is to first define and clarify the issues involved and second to show that either concept, CHFR or CPR, can be used for defining margins. The selection depends on the interest of the investigator and is conditional on understanding what the results mean physically. However, the present author believes that the use of HBM for data evaluation can be misleading and tends to imply incorrectly a closer agreement between the data and the correlation than can be justified. Furthermore, some correlations cannot be used correctly for certain applications, in spite of the fact that they seem to be “statistically superior”.

Evaluation of subcooled critical heat flux correlations for tubes with and without internal twisted tapes

Eleven correlations and models for critical heat flux (CHF) of subcooled flow boiling in water were evaluated. Both a direct substitution method (DSM) and a heat balance condition method (HBM) were compared in the evaluations. The HBM was recommended as a better prediction method in the present study. For straight tubes under uniform heating conditions, the correlations of the Gunther, Knoebel, modified Tong, W-2, and Tong-75, and also the Celata and Weisman-Pei models were confirmed to give reasonably good predictions. Among them, the Celata model was the best with respect to accuracy. For swirl flow under uniform heating conditions. Tong-75-I (involving modification of the water velocity parameter) and Nariai-Inasaka correlations were confirmed to give reasonably good predictions, even though their predictions were too low for the CHF under non-uniform heating conditions.

Consideration of critical heat flux margin prediction by subcooled or low quality critical heat flux correlations

The accurate prediction of the critical heat flux (CHF) margin which is a key design parameter in a variety of cooling and heating systems is of high importance. These margins are, for the low quality region, typically expressed in terms of critical heat flux ratios using the direct substitution method. Using a simple example of a heated tube, it is shown that CHF correlations of a certain type often used to predict CHF margins, expressed in this manner, may yield different results, strongly dependent on the correlation in use. It is argued that the application of the heat balance method to such correlations, which leads to expressing the CHF margins in terms of the critical power ratio, may be more appropriate.

An algorithm for multiphase equilibrium calculations for systems containing polymers

A comprehensive algorithm has been developed for multiphase equilibrium computations in multicomponent systems. The algorithm includes stability analysis and calculation of equilibrium compositions by a combination of a multiphase direct substitution method and Gibbs energy minimization. The algorithm has been specially designed to handle the numerical complexities that are characteristic of systems containing large (polymeric) and small molecules. Such complexities include a very strong composition dependence of equilibrium ratios (K-factors) as well as frequent occurrence of extremely low equilibrium concentrations and fugacity or activity coefficients.

Process simulation in the complex domain

AbstractThe asymptotic behavior of fixed‐point methods in the complex domain is studied. Both direct substitution and Newton's method exhibit stable periodic and aperiodic behavior from real‐ or complex‐valued starting points. Moreover, multiple stable periodic orbits can exist for direct substitution. Traditional trust region (or dogleg) methods, on the other hand, often terminate at singular points, which correspond to nonzero‐valued saddlepoints in the least‐squares function that can be arbitrarily far from a solution. Furthermore, the basins of attraction of these singular points are usually dispersed throughout the basin boundaries in the complex domain, clearly illustrating that singular points (via the dogleg strategy) also attract either real‐ or complex‐valued starting points.In light of this, an extension of the dogleg strategy to the complex domain, based on a simple norm‐reducing, singular point perturbation, is proposed. This extended trust region method removes all forms of nonconvergent behavior and always terminates at a fixed point, even from critical point (worst‐case) initial values. Many numerical results and geometric illustrations using chemical process simulation examples are presented.

Effective numerical methods for elasto-plastic contact problems with friction

Several effective numerical methods for solving the elasto-plastic contact problems with friction are presented. First, a direct substitution method is employed to impose the contact constraint conditions on condensed finite element equations, thus resulting in a reduction by half in the dimension of final governing equations. Second, an algorithm composed of contact condition probes and elasto-plastic iterations is utilized to solve the governing equation, which distinguishes two kinds of nonlinearities, and makes the solution unique. In addition, Positive-Negative Sequence Modification Method is used to condense the finite element equations of each substructure and an analytical integration is introduced to determine the elasto-plastic status after each time step or each iteration, hence the computational efficiency is enhanced to a great extent. Finally, several test and practical examples are presented showing the validity and versatility of these methods and algorithms.

Simple process equations, fixed‐point methods, and chaos

AbstractThe chaotic behavior of fixed‐point methods for steady‐state process simulation is studied. It is shown that direct substitution and Newton's method exhibit all of the rich structure of chaos (period doubling, aperiodicity, fractal basin boundaries, and related properties) on simple process examples. These examples include finding roots to the Soave‐Redlich‐Kwong and Underwood equations, dew point and flash calculations for heterogeneous mixtures, and a simple process flowsheet.For single variable problems, it is shown that direct substitution follows a classical period‐doubling route to chaos. On the other hand, the chaotic behavior of direct substitution and Newton's method on multivariable problems is considerably more complex, and can give the appearance that no organized route to chaos is followed. For example, for the dew point problems, truncated period doubling, odd periodic cycles, and chaotic behavior can be observed, within which are embedded narrow regions of global convergence. Many numerical results and geometric illustrations are presented.

Convergence properties of systems of algebraic equations—explicit loops

AbstractTechniques are developed for using the structural information contained in algebraic equation sets to aid in the study of convergence properties of iterative equation loops.The concept of inverse mapping is applied directly in order to study convergence properties of sequences. For problems in which there exist both a direct and inverse path for the computation of iterative loops, direct substitution techniques will converge via one path and diverge via the other path. When an inverse path existed, the direct substitution method was superior to simultaneous solution techniques for the problems studied. Not all iterative loops have an inverse path. For problems in which inverse paths do not originally exist, rearrangement of the equations of an iterative loop can result in a modified loop for which an inverse path does exist. The digraph representation of an iterative loop aids in deciding whether rearrangements are feasible for a particular problem. When inverse paths cannot conveniently be obtained, direct substitution cannot be successfully used and simultaneous solution techniques such as quasi‐linearization are recommended.

Antishielding and Polarizabilities in Alkali Halide Gases

The ionic model that has been used in an attempt to explain $\frac{\mathrm{eQq}}{h}$ data in ionic gases and solids is examined, extended, and applied to alkali halide gases. It is shown that the antishielding factor, ${\ensuremath{\gamma}}_{\ensuremath{\infty}}$, multiplies the unshielded value of $q$, ($\frac{{\ensuremath{\partial}}^{2}V}{\ensuremath{\partial}{Z}^{2}}$), produced not only by an external charge but by all the external moments, and some of these effects are important in alkali halide gases. In order to estimate the magnitude of the contributions to $q$ of such moments, polarizabilities are calculated that apply to alkali halide internuclear distances. These polarizabilities are often much smaller than those calculated for the free ion. Polarizabilities and ${\ensuremath{\gamma}}_{\ensuremath{\infty}}'\mathrm{s}$ are calculated using both Hartree and Hartree-Fock wave functions and are compared. The values obtained using the Hartree-Fock wave functions are consistently smaller. Some second order calculations of the polarizabilities and antishielding factors are calculated by a direct substitution method. The results indicate that for the alkali halide gas calculations perturbation theory is applicable to the alkali ions but not to the easily deformable halide ions for the situation considered here. This is probably the reason the ionic model has always yielded poor results for the halide ions. When the ionic model is extended, as described in this paper, and applied to the alkali ions, a set of antishielding factors is found for which there is agreement between theory and experiment or would be if Hartree-Fock wave functions were available to calculate ${\ensuremath{\gamma}}_{\ensuremath{\infty}}$. From this model the nuclear quadrupole moment, $Q$, of ${\mathrm{Li}}^{7}$ is found to be -0.016\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}24}$ ${\mathrm{cm}}^{2}$.