Analysis of Real-Gas and Matrix-Conduction Effects in Cyclic Cryogenic Regenerators

1973 ◽  
Vol 95 (2) ◽  
pp. 199-205 ◽  
Author(s):  
M. F. Modest ◽  
C. L. Tien
Keyword(s):  
Numerical Solutions ◽  
Operating Conditions ◽  
Physical Parameters ◽  
Governing Equations ◽  
Real Gas ◽  
The Matrix ◽  
Compression And Expansion ◽  
Temperature Levels ◽  
The One ◽  
Gas Conduction

The one-dimensional governing equations for the thermal performance of cryogenic regenerators are developed and simplified by neglecting gas conduction and pressure drop along the matrix. The present formulation includes the effects of matrix conduction and real-gas behavior, which can be quite important in actual situations but were neglected in all previous analyses. The time dependence of the governing equations is eliminated by integration over the compression and expansion periods. Numerical solutions of the resulting time-independent equations are presented for various values of physical parameters and temperature levels. Comparison with the corresponding cases neglecting real-gas and matrix-conduction effects demonstrates the significant nature of these effects for many operating conditions.

Consequences of viscous anisotropy in a deforming, two-phase aggregate. Part 1. Governing equations and linearized analysis

2013 ◽  
Vol 734 ◽  
pp. 424-455 ◽  
Author(s):  
Yasuko Takei ◽  
Richard F. Katz
Keyword(s):  
Shear Stress ◽  
Numerical Solutions ◽  
Shear Plane ◽  
High Porosity ◽  
Governing Equations ◽  
Two Phase ◽  
The Matrix ◽  
Continuum Scale ◽  
Torsional Flow ◽  
Matrix Shear

AbstractIn partially molten regions of Earth, rock and magma coexist as a two-phase aggregate in which the solid grains of rock form a viscously deformable framework or matrix. Liquid magma resides within the permeable network of pores between grains. Deviatoric stress causes the distribution of contact area between solid grains to become anisotropic; in turn, this causes anisotropy of the matrix viscosity at the continuum scale. In this two-paper set, we predict the consequences of viscous anisotropy for flow of two-phase aggregates in three configurations: simple shear, Poiseuille, and torsional flow. Part 1 presents the governing equations and an analysis of their linearized form. Part 2 (Katz & Takei, J. Fluid Mech., vol. 734, 2013, pp. 456–485) presents numerical solutions of the full, nonlinear model. In our theory, the anisotropic viscosity tensor couples shear and volumetric components of the matrix stress/strain rate. This coupling, acting over a gradient in shear stress, causes segregation of liquid and solid. Liquid typically migrates toward higher shear stress, but under specific conditions, the opposite can occur. Furthermore, it is known that in a two-phase aggregate with a porosity-weakening viscosity, matrix shear causes porosity perturbations to grow into a banded or sheeted structure. We show that viscous anisotropy reduces the angle between these emergent high-porosity features and the shear plane. Laboratory experiments produce similar, high-porosity features. We hypothesize that the low angle of porosity bands in such experiments is the result of viscous anisotropy. We therefore predict that experiments incorporating a gradient in shear stress will develop sample-wide liquid–solid segregation due to viscous anisotropy.

scholarly journals A Generalized Analytical Model for Joule Heating of Segmented Wires

2018 ◽  
Vol 140 (7) ◽  
Author(s):  
Vivekananthan Balakrishnan ◽  
Toan Dinh ◽  
Hoang-Phuong Phan ◽  
Dzung Viet Dao ◽  
Nam-Trung Nguyen
Keyword(s):  
Temperature Distribution ◽  
Analytical Solution ◽  
Joule Heating ◽  
Numerical Solutions ◽  
Analytical Data ◽  
Experimental Studies ◽  
Governing Equations ◽  
Steady State Temperature ◽  
The One ◽  
Good Agreement

This paper presents an analytical solution for the Joule heating problem of a segmented wire made of two materials with different properties and suspended as a bridge across two fixed ends. The paper first establishes the one-dimensional (1D) governing equations of the steady-state temperature distribution along the wire with the consideration of heat conduction and free-heat convection phenomena. The temperature coefficient of resistance of the constructing materials and the dimension of the each segmented wires were also taken into account to obtain analytical solution of the temperature. COMSOL numerical solutions were also obtained for initial validation. Experimental studies were carried out using copper and nichrome wires, where the temperature distribution was monitored using an IR thermal camera. The data showed a good agreement between experimental data and the analytical data, validating our model for the design and development of thermal sensors based on multisegmented structures.

Prediction of Ingress Through Turbine Rim Seals: Part 2—Combined Ingress

2010 ◽  
Author(s):  
J. Michael Owen ◽  
Oliver Pountney ◽  
Gary Lock
Keyword(s):  
Experimental Data ◽  
Rotational Speed ◽  
Numerical Solutions ◽  
External Flow ◽  
Operating Conditions ◽  
Flow Rates ◽  
Wide Range ◽  
Experimental Values ◽  
The One ◽  
Empirical Constants

In Part1 of this two-part paper, the orifice equations were solved for the case of externally-induced ingress, where the effects of rotational speed are negligible. In Part 2, the equations are solved, analytically and numerically, for combined ingress (CI) where the effects of both rotational speed and external flow are significant. For the CI case, the orifice model requires the calculation of three empirical constants, including Cd,e,RI and Cd,e,EI, the discharge coefficients for rotationally-induced (RI) and externally-induced (EI) ingress. For the analytical solutions, the external distribution of pressure is approximated by a linear saw-tooth model; for the numerical solutions, a fit to the measured pressures is used. It is shown that, although the values of the empirical constants depend on the shape of the pressure distribution used in the model, the theoretical variation of Cw,min (the minimum nondimensional sealing flow rate needed to prevent ingress) depends principally on the magnitude of the peak-to-trough pressure difference in the external annulus. The solutions of the orifice model for Cw,min are compared with published measurements, which were made over a wide range of rotational speeds and external flow rates. As predicted by the model, the experimental values of Cw,min could be collapsed onto a single curve, which connects the asymptotes for RI and EI ingress at the respective smaller and larger external flow rates. At the smaller flow rates, the experimental data exhibit a minimum value of Cw,min, which undershoots the RI asymptote. Using an empirical correlation for Cd,e, the model is able to predict this undershoot, albeit smaller in magnitude than the one exhibited by the experimental data. The limit of the EI asymptote is quantified, and it is suggested how the orifice model could be used to extrapolate effectiveness data obtained from an experimental rig to engine-operating conditions.

The Representation of Regenerator Fluid Carryover by Bypass Flows

1984 ◽  
Vol 106 (1) ◽  
pp. 216-220 ◽  
Author(s):  
P. J. Banks
Keyword(s):  
Numerical Solutions ◽  
Fluid Flows ◽  
Porous Matrix ◽  
Heat Transfer Process ◽  
The Other ◽  
Sensible Heat ◽  
Governing Equations ◽  
Fluid Stream ◽  
The Matrix ◽  
Two Fluid

A regenerator transfers sensible heat between two fluid streams by means of a porous matrix through which the streams are passed alternately. The fluid contained in the matrix passages is carried over from one fluid stream to the other, and contributes to the heat transfer process. It has been suggested that this contribution may be predicted by treating the fluid carryover as fluid flows bypassing the matrix. The validity of this representation is explained, and its accuracy is explored in a case for which numerical solutions of the governing equations are available. A published analysis of the representation is discussed and completed.

ANALYTICAL AND NUMERICAL SOLUTIONS OF PERISTALTIC FLOW OF WILLIAMSON FLUID MODEL IN AN ENDOSCOPE

2011 ◽  
Vol 11 (04) ◽  
pp. 941-957 ◽  
Author(s):  
NOREEN SHER AKBAR ◽  
S. NADEEM
Keyword(s):  
Numerical Solutions ◽  
Fluid Model ◽  
Pressure Rise ◽  
Peristaltic Flow ◽  
Physical Parameters ◽  
Peristaltic Motion ◽  
Governing Equations ◽  
Williamson Fluid ◽  
Long Wavelength ◽  
Good Agreement

The present studies deal with the peristaltic motion of an incompressible Williamson fluid model in an endoscope. The governing equations of Williamson fluid model are first simplify using the assumptions of long wavelength and low Reynolds number. The four types of solutions have been presented for velocity profile named (i) exact solution, (ii) perturbation solution, (iii) HAM solution, and (iv) numerical solutions. The comparisons of four solutions have been found a very good agreement between all the solutions. In addition, the expressions for pressure rise and velocity against various physical parameters are discussed through graphs.

scholarly journals The Numerical Approximation of the Bioheat Equation of Space-Fractional Type Using Shifted Fractional Legendre Polynomials

2020 ◽  
pp. 875-889
Author(s):  
Firas A. Al-Saadawi ◽  
Hameeda Oda Al-Humedi
Keyword(s):  
Legendre Polynomials ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
Dimensional Space ◽  
Bioheat Equation ◽  
One Dimensional ◽  
Good Convergence ◽  
The Matrix ◽  
The One ◽  
Fractional Type

The aim of this paper is to employ the fractional shifted Legendre polynomials (FSLPs) in the matrix form to approximate the fractional derivatives and find the numerical solutions of the one-dimensional space-fractional bioheat equation (SFBHE). The Caputo formula was utilized to approximate the fractional derivative. The proposed methodology applied for two examples showed its usefulness and efficiency. The numerical results showed that the utilized technique is very efficacious with high accuracy and good convergence.

Prediction of Ingress Through Turbine Rim Seals—Part II: Combined Ingress

2011 ◽  
Vol 134 (3) ◽  
Author(s):  
J. Michael Owen ◽  
Oliver Pountney ◽  
Gary Lock
Keyword(s):  
Experimental Data ◽  
Rotational Speed ◽  
Numerical Solutions ◽  
External Flow ◽  
Operating Conditions ◽  
Flow Rates ◽  
Wide Range ◽  
Experimental Values ◽  
The One ◽  
Empirical Constants

In Part I of this two-part paper, the orifice equations were solved for the case of externally induced (EI) ingress, where the effects of rotational speed are negligible. In Part II, the equations are solved, analytically and numerically, for combined ingress (CI), where the effects of both rotational speed and external flow are significant. For the CI case, the orifice model requires the calculation of three empirical constants, including Cd,e,RI and Cd,e,EI, the discharge coefficients for rotationally induced (RI) and EI ingress. For the analytical solutions, the external distribution of pressure is approximated by a linear saw-tooth model; for the numerical solutions, a fit to the measured pressures is used. It is shown that although the values of the empirical constants depend on the shape of the pressure distribution used in the model, the theoretical variation of Cw,min (the minimum nondimensional sealing flow rate needed to prevent ingress) depends principally on the magnitude of the peak-to-trough pressure difference in the external annulus. The solutions of the orifice model for Cw,min are compared with published measurements, which were made over a wide range of rotational speeds and external flow rates. As predicted by the model, the experimental values of Cw,min could be collapsed onto a single curve, which connects the asymptotes for RI and EI ingress at the respective smaller and larger external flow rates. At the smaller flow rates, the experimental data exhibit a minimum value of Cw,min, which undershoots the RI asymptote. Using an empirical correlation for Cd,e, the model is able to predict this undershoot, albeit smaller in magnitude than the one exhibited by the experimental data. The limit of the EI asymptote is quantified, and it is suggested how the orifice model could be used to extrapolate the effectiveness data obtained from an experimental rig to engine-operating conditions.

Approximate solutions to the one-particle Dirac equation: numerical results

1986 ◽  
Vol 64 (3) ◽  
pp. 297-302 ◽  
Author(s):  
R. A. Moore ◽  
T. C. Scott
Keyword(s):  
Dirac Equation ◽  
Numerical Solutions ◽  
Structure Constant ◽  
Approximate Solutions ◽  
Fine Structure Constant ◽  
Matrix Elements ◽  
Atomic Systems ◽  
The Matrix ◽  
The One ◽  
Significant Figures

The zero-, first-, and second-order differential equations in a previously defined hierarchy of equations giving approximate solutions to the one-particle Dirac equation and the corresponding eigenvalue contributions are each written as power series in α, the fine structure constant, for an arbitrary, spherically symmetric potential. These equations are solved numerically for the hydrogen-atom potential to obtain wave functions to order α2 and eigenvalues to order α4 for all states with n = 1–4, inclusive. The numerical solutions are then used to evaluate a number of matrix elements to order α2. A comparison with the exact expressions shows that the numerical values for the coefficients of the different powers of α have at least six significant figures in the eigenfunctions and eigenvalues and five in the matrix elements. Thus, the procedure is validated and can be applied with confidence to other atomic systems.

Stirling Engines With a Chemically Reactive Working Fluid—Some Thermodynamic Effects

1977 ◽  
Vol 99 (2) ◽  
pp. 284-287 ◽  
Author(s):  
M. M. Metwally ◽  
G. Walker
Keyword(s):  
Heat Pumps ◽  
Substantial Improvement ◽  
Regenerative Process ◽  
Operating Conditions ◽  
Working Fluid ◽  
Nitrogen Tetroxide ◽  
Specific Output ◽  
Stirling Engines ◽  
Compression And Expansion ◽  
Temperature Levels

Stirling engines operate on closed regenerative thermodynamic cycles with compression and expansion of the working fluids at different temperature levels. They may be used as prime movers, refrigerating machines, heat pumps, or pressure generators. Conventional machines use a gaseous working fluid, but substantial improvement in specific output may be gained with a partially reactive, condensing working fluid. The working fluid then consists of an inert gaseous carrier with a chemically reactive, condensing working fluid such as nitrogen tetroxide (N2O4). This may be liquid in the cold compression space and then evaporates and dissociates in the regenerative process to be in the elemental gaseous phase in the hot expansion space. The change of state of one component reduces the required compression work and has the effect of increasing the engine volume compression ratio with consequent benefit to the specific output. The results obtained using idealized theory show that an improvement may be gained in net cycle work of twice the output with a simple gaseous working fluid with no penalties in size, weight, or cost of the engine. The degree of improvement depends on the design and operating conditions of the engine. The effects of variation of some of these parameters are explored.

scholarly journals Numerical solutions of one-dimensional parabolic convection-diffusion problems arising in biology by the Laguerre collocation method

BIOMATH ◽  
2017 ◽  
Vol 6 (1) ◽  
pp. 1706047 ◽  
Author(s):  
Burcu Gürbüz ◽  
Mehmet Sezer
Keyword(s):  
Collocation Method ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Equation System ◽  
Diffusion Problem ◽  
One Dimensional ◽  
Convection Diffusion ◽  
Laguerre Series ◽  
The Matrix ◽  
The One

In this work, we present a numerical scheme for the approximate solutions of the one-dimensional parabolic convection-diffusion model problems. Diffusion models form a reasonable basis for studying insect and animal dispersal and invasion, which arise from the question of persistence of endangered species, biodiversity, disease dynamics, multi-species competition so on. Convection diffusion problem is also a form of heat and mass transfer in biological models. The presented method is based on the Laguerre collocation method used for these problems of differential equations.In fact, the approximate solution of the problem in the truncated Laguerre series form is obtained by this method. By substituting truncated Laguerre series solution into the problem and by using the matrix operations and the collocation points, the suggested scheme reduces the problem to a linear algebraic equation system. By solving this equation system, the unknown Laguerre coecients can be computed. The accuracy and the efficiency of the method is showed by numerical examples and the comparisons by the other methods.

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