Simultaneous Nonlinear Differential Equations Research Articles

Performance of a packed bed absorber for aqua ammonia absorption refrigeration system

A mathematical model of a packed bed absorber for aqua-ammonia absorption refrigeration system is presented. The model is used to predict the performance of the bed at various design and operating conditions. The governing equations and the boundary conditions are derived to predict the bed performance. A numerical integral method and an iteration scheme are used to solve the governing one dimensional, non-linear simultaneous differential equations which are subjected to three point boundary value problem. A computer program is prepared and carefully debugged to solve the governing equations with the help of some supporting equations to describe the properties of the working fluids and the heat and mass transfer coefficients in the bed. The analysis show that the absorption process is affected by the following parameters: the volumetric heat rejection model, bed height, vapor and solution flow rates to the bed and the inlet conditions; and packing material type. The effect of changing each of those parameters on the performance of the bed is studied after suggesting a model for the volumetric heat rejection from the bed. The results showed that changing the bed pressure and/or the vapor inlet temperature have negligible effect on the performance of the bed. Changing other parameters are found to affect the performance of the bed by different degrees. Also, the results show that within the present range of parameters, a bed height less than 0.7 m guarantees an absorption efficiency better than 91%.

Dirac Equation in the Spatially Flat Friedmann Model

The unitary transformation which diagonalizesthe field-free Dirac Hamiltonian in the spatially flatFriedmann-Robertson-Walker metric is analyzed, and apair of simultaneous first-order nonlinear differential equations is derived for the two parameters(two angles) that characterize the transformation. Theequations are solved approximately for a test particlewhose kinetic energy is small compared to its mass energy, and minimum-uncertainty wave packetsare constructcd from the solutions. It is found thatgeneral relativity limits the quantum mechanical spreadof the wave packets, but forces then to expand with the expanding space, as if they were embeddedin it. The massless Dirac equation is solved exactly forthe two-component neutrino spinor, and yieldsgeneralized nonspreading wave packets which display no quantum mechanical spread at all, but areconstrained to expand with the expanding space as theyfollow null geodesics.

New Analytical Solution for Pyle-Popovich's Peritoneal Dialysis Model.

Continuous Ambulatory Peritoneal Dialysis (CAPD) is one of the standard treatments for kidney disease patients. A washing solution, called dialysate, is put into the peritoneal cavity to remove waste products and excess amounts of water in CAPD. The dialysate is exchanged four to five times a day by the patient. However, it is not easy to prescribe CAPD therapy, which may have precluded popularization of CAPD therapy. Popovich et al. constructed a mathematical model (P-P model) that applies to the prescription of the treatment schedule. It requires, however, a number of iterative calculations to obtain an exact numerical solution because the model is a set of nonlinear simultaneous ordinary differential equations. In this paper, the authors derived a new approximated analytical solution by employing a time-discrete technique, assuming all the parameters to be constant within each piecewise period of time for the P-P model. We have also described an algorithm of a numerical calculation with the new solution for determining a set of unknown parameters in the P-P model. We compare the validation of the new solution for clinical use with another analytical solution (Vonesh’s solution).The new analytical solution consists of a forward solution (FW solution), that is the solution for the plasma and dialysate concentrations from ti to ti+1 (ti ti+1), and a backward solution (BW solution) from ti to ti–1 (ti–1 ti). The unknown parameters were determined by employing the Newton–Raphson method, a trial-and-error method and the modified Powell method in combination with FW and BW solutions.The new analytical solution show an excellent agreement with the exact numerical solution for entire dwelling time. Moreover, optimized parameters with the new analytical solution show much smaller discrepancy than those with Vonesh’s solution.Although the proposed method requires a slightly longer calculation time than Vonesh’s, it can simulate concentrations in plasma and dialysate for an entire single exchange in CAPD using the clinical data measured at arbitrary time. The proposed method may be useful for determining unknown parameters as well as for prescribing CAPD treatment.

Simulation of pervaporation process for ethanol dehydration by using pilot test results

This paper focuses on providing a pervaporation simulation method for ethanol dehydration from a practical point of view. The simulation procedure is performed by setting up simulation equations which describe the pervaporation process, obtaining the necessary data from pervaporation batch mode pilot tests, verifying the simulation tool through simulations of continuous mode pilot tests, and comparing the simulation results with the real pilot test results. We considered the mass and energy balances that describe separating an ethanol/water mixture by a pervaporation membrane. The simulation equations were mathematically expressed into simultaneous non-linear differential equations based on these balances. The necessary data for simulation consist of the thermophysical properties for the ethanol-water mixture and the characteristic data of a PVA composite membrane. The membrane characteristic data are permeation flux and membrane selectivity, which are functions of feed composition and operating conditions. These data were experimentally determined by a batch mode pilot test. The continuous mode pilot tests were simulated and the simulation results were compared to the real test results. The resulte were fairly good.

Numerical simulation of active suppression of rotating stall in axial compression systems

In the present paper, a theoretical model is proposed to analyze the transient behavior of suppression of rotating stall in axial compression systems through the use of an additional disturbance. The governing equations of the model are a set of simultaneous nonlinear first order partial differential equations, and for numerical calculations, a simple explicit time marching method can be used. The influence of system parameters on the suppression effectiveness and the interaction between rotating stall and surge have been discussed initially. The analysis of the influence of system parameters presents that both theB parameter and axisymmetric compressor characteristic have significant effect on the stabilization effectiveness of a control strategy. The effectiveness decreases as the value ofB and the number of stages or stage loading of the compressor increase. It has been found that the onset flow rate of rotating stall and surge in a compression system may be different, and there is a strong interaction between these two kinds of instabilities. The onset flow rate of pure one dimensional surge depends on the value ofB and axisymmetric compressor characteristic, besides the slope of the compressor characteristic. In some cases, when rotating stall which is the natural mode of instability in a compression system is suppressed, one dimensional surge can occur. It often limits the effectiveness of a control strategy to suppress rotating stall. But when surge is initiated by rotating stall, it is also possible to inhibit the occurrence of surge by suppressing rotating stall in a compression system.

Nonlinear standing waves: Theory and experiments

The theoretical treatments of nonlinear standing waves had been hampered for years by the simultaneous nonlinear partial differential equations of Riemann. It is suggested that the situation may become encouraging if attention is concentrated on the motion of a particle at a point fixed in space, or on the waveform of the standing wave with fixed alternating values of the particle velocities. It is shown that the motion of a particle at a point fixed on the waveform of a standing wave is readily soluble from the Riemann equations, and an exact, stable solution of the nonlinear standing wave is obtained, which differs from that in linear acoustics only in containing an additional term proportional to the particle displacements in the arguments of the time and space functions. These implicit functions are readily developed into explicit power series, and the field parameters of the standing wave result in terms representing the harmonics. The properties of the nonlinear standing waves thus obtained have been confirmed by experiments. A standing-wave tube has been built and equipped with a high power loudspeaker driver as the source of excitation. The gradual spectral variation of the standing wave with increasing excitation has been observed. Experimental results agree well with the theoretical predictions and further uncover a number of interesting phenomena. Under certain conditions, saturation and bifurcation appear at a high sound level.

BEST-KIT; Development of Biochemical Engineering System Analyzing Tool-KIT

The final goal of this study is the implementation of an efficient and user-friendly “biochemical engineering system analyzing tool (BEST-KIT)” integrated the following key modules: 1) mathematical modeling and editing of reaction scheme, 2) aulomatic derivation of simultaneous nonlinear differential equations of reaction scheme, 3)numcrical simulation of reaction scheme, 4)visualizalion of the numerical results, S)cvaluation of kinetic parameters and estimation of unknown parameters, 6)estimation of control variable for artificial control of reactor system. This paper mainly deals with the development of integrated interface (UNIX version) of these key modules and with the algorithm of automatic derivation of simultaneous nonlinear differential equations.

A MULTIVARIABLE NEURAL NETWORK SHIP MATHEMATICAL MODEL

The classical approach to modelling the dynamic behaviour of rigid bodies is to express their behaviour as a set of simultaneous linear and non-linear differential equations, and to obtain a solution for various input simuli. An alternative approach is that of system identification whereby a given input such as a sinusoid or pseudo- random binary sequence is applied to the real system and from a set of input/output measurements a mathematical model may be obtained. This paper investigates the generation of a state variable representation of a ship in three degrees of freedom by the application of an artificial neural network.

One zone modeling of irregular variability of stellar convective envelope

One zone modeling of the irregular variability of red super-giants is intended with regard to the nonlinear coupling of finite amplitude pulsation with convection. The nonlocal mixing length is employed for the evaluation of the convective flux, the turbulent pressure and the turbulent power of temperature fluctuations. The radial pulsation and the Boussinesq convection are assumed for simplicity. The one zone is defined as the layer having the entropy maximum and the minimum at the bottom and at the top, respectively. The quasi-adiabatic approximation is consistent with this definition in fixing the zone to the same mass range. The spatial derivatives are evaluated under the assumption of homologous changes with the equilibrium homologous parameters. Then, a set of 6 simultaneous first order nonlinear ordinary differential equations are obtained as the one zone representation of the irregular variability of the convective envelope.

Recent VLSI neural networks in Japan

In this article, recent research activities on the development of electronic neural networks in Japan are reviewed. Most of the largest Japanese electronic companies have developed VLSI neural chips using analog, digital or optoelectronic circuits. They have run various neural networks on them. Recently, in Japan, digital approach becomes active. Several fully-digital VLSI chips for on-chip BP learning have been developed, and 2.3 GCUPS (Giga Connection Updates per Second) learning speed has already been attained. Although the numbers of neurons and synapses containable in single digital chips are small, a large neural network can be developed by cascading the chips. By cascading 72 chips, a fully interconnected PDM (Pulse Density Modulating) digital neural network system has been developed. The behavior of the system follows simultaneous nonlinear differential equations and the processing speed amounts to 12 GCPS (Giga Connections per Second). Intensive researches on analog and optoelectronic approaches have also been carried out in Japan. An analog VLSI neural chip attains 28 GCUPS on-chip learning speed and 1 TCPS (Tera Connections per Second) processing speed for Boltzmann machine with 1 bit digital output. For the optoelectronic approach, although the network size is small, 640 MCUPS BP learning speed has been attained.

One Zone Modeling of Irregular Variability of Stellar Convective Envelope

AbstractOne zone modeling of the irregular variability of red super-giants is intended with regard to the nonlinear coupling of finite amplitude pulsation with convection. The nonlocal mixing length is employed for the evaluation of the convective flux, the turbulent pressure and the turbulent power of temperature fluctuations. The radial pulsation and the Boussinesq convection are assumed for simplicity. The one zone is defined as the layer having the entropy maximum and the minimum at the bottom and at the top, respectively. The quasi-adiabatic approximation is consistent with this definition in fixing the zone to the same mass range. The spatial derivatives are evaluated under the assumption of homologous changes with the equilibrium homologous parameters. Then, a set of 6 simultaneous first order nonlinear ordinary differential equations are obtained as the one zone representation of the irregular variability of the convective envelope.

Nonlinear Oscillation of a Magnetically Levitated Body. Unsteady Bouncing and Pitching Oscillations.

A pair of nonlinear simultaneous differential equations which govern the unsteady bouncing and pitching oscillations of a simple levitated body, are solved by using the method of multiple scales, for the special case in which the fluctuation of the magnetic lift force changes harmonically in the neighborhood of twice the natural frequency of the bouncing oscillation Ωz, and in which the natural frequency of the pitching oscillation Ωo is equal to Ωz/2. As a result, it is shown that there are three types of oscillations ; periodic, quasi-periodic and complicated. Furthermore these three types of oscillations are observed with a simple experimental apparatus.

Numerical simulation of a tropical Storm boundary layer

A model has been designed to study the surface boundary layer of a tropical storm. The numerical method consists of solving a two point boundary value problem for two systems of simultaneous non-linear differential equations by finite differences. A Stoke's stream function suitable to represent the flow both in interior and exterior regions of a tropical storm boundary layer has been developed. The advantage of the method is that the, boundary layer of the tropical storm can be studied starting from the outer region to the centre of the storm without neglecting non-linear terms. In addition, there IS no need for assumptions on the vertical profiles for tangential and radial velocities. The method is stable and converges within a few iterations. The flow above the friction layer is represented by a steady axisymmetric vortex in gradient balance. To investigate the effect of turbulence- on boundary layer characteristics, turbulence has been represented by four different variations of the eddy coefficient of viscosity with no slip boundary conditions. Computations have been performed 1aking 40-grid points in the vertical direction. It is observed that, if the eddy coefficient of viscosity is assumed to vary with the superimposed flow above the boundary layer, the solutions compare favourably well with observations. The solution also shows an outflow from the Inner core of the boundary layer which is necessary for creation of an eye of the storm.

An existence and uniqueness theorem for the transfer of mass and heat in a rectangular cavity

We demonstrate an existence and uniqueness theorem for the solutions of a system of simultaneous nonlinear partial differential equations which models the transfer of mass and heat in an upright rectangular cavity. To facilitate numerical simulations, all constants which appear in the estimates are given explicitly in terms of the physical parameters

Microstructural Model of a Fibrous Composite Fracture

Abstract A new explicitly solvable model of a unidirectional composite with nonlinear deformable fibers is introduced in this paper. This model generalizes the shear-lag model by including the nonlinear behavior of fibers. The equations of the model allow a Lax representation. This establishes the existence of an infinite number of invariants. The present theory is unique in that the analytical solution of the boundary value problem for simultaneous nonlinear differential equations is reduced to simultaneous nonlinear equations in terms of the boundary values of unknown functions. The two-dimensional multifilament failure problems of unidirectional fiber composites are analyzed. Example numerical solutions that focus specifically on the stress concentration factors of fibers adjacent to the local imperfections and the cracks are discussed.

A simple model for prediction of oxygen transport rates by flowing blood in large capillaries

A simple model has been developed for simulation of oxygen transport to and from blood flowing in conduits of the diameter of arterioles and larger (⩾ 20 μm). The basis is the large capillary model (LCAP) of P. K. Nair, et al. 1989 which has been validated experimentally. Detailed calculations of the oxygen concentration distribution reveal that the dominant resistance to oxygen transport is distributed in the plasma. Relatively little resistance is present within or in the immediate vicinity of the red cells. On the basis of these findings, LCAP was simplified from four simultaneous nonlinear partial differential equations (PDEs) to one PDE by (1) assuming chemical equilibrium within the red blood cells, (2) neglecting intracellular and extracellular boundary layer resistances, and (3) incorporating transport in the cell-free region adjacent to the capillary wall into the boundary conditions. The simplified model is much easier to apply mathematically to new situations. A comparison between LCAP and the simpler model shows that they give virtually the same predictions, and the predictions agree well with experimental measurements. The model is predictive in that all the parameters are determined from the literature or from in-independent measurements. Thus it should be useful in studies of physiological significance, as well as in design and analysis of extracorporeal blood oxygenators.

Two-dimensional convection and radiation with scattering from a Poiseuille flow

Two-dimensional combined convection and radiation heat transfer from a gray scattering fluid in a reflecting channel is considered. The model, represented by a set of simultaneous nonlinear integro-partial differential equations, is solved numerically. The effects of aspect ratio, conduction-radiation parameter, scattering albedo, and wall emissivity, are systematically investigated. It is found that these parameters have a significant influence on the temperature field and alter the radiative and convective fluxes at the hot and cold walls. In particular, when radiation effects are considerable, the heat-transfer characteristics of the fluid at the hot and cold walls are very different.

Use of Adair four-step kinetics in mathematical simulation of oxygen transport in the microcirculation.

The Adair four-step kinetic model for the reactions of haemoglobin and oxygen recognizes five haemoglobin species, corresponding to deoxyhaemoglobin and one species for each level of oxygenation of the four haem groups. Thus, an oxygen transport problem involves a system of five simultaneous non-linear partial differential equations for diffusion with chemical reaction. This mathematical complexity has impeded application of the Adair model despite its theoretical advantages over the one-step model often used in practice. The Adair kinetic model has been incorporated into a simulation of microcirculatory oxygen transport. The results show that the usual one-step kinetic model is inaccurate in comparison with the Adair model. However, an empirical modification can be made to the one-step model to ensure compatibility with the equilibrium curve. This modified one-step kinetic model (the VRC model) is much more tractable mathematically than the Adair model. In the physiological range of fluxes, the VRC kinetic model appears to be of sufficient accuracy for most purposes, and the mathematical complexity of the Adair model is not required.

Similarity representation of wave propagation in a nonlinear viscoelastic rod on a group theoretic basis

Similarity representation of wave propagation in a nonlinear viscoelastic rod, subjected to a velocity impact is constructed. By the use of a multiparameter dimensional group of transformations, under which the system of basic equations and auxiliary conditions are invariant, similarity transformations are obtained which are used to construct the similarity representation in the form of an ordinary nonlinear boundary value problem of the original system, consisting of a set of simultaneous nonlinear partial differential equations and the association boundary and initial conditions.

A Theory of Post-Stall Transients in Axial Compression Systems: Part I—Development of Equations

An approximate theory is presented for post-stall transients in multistage axial compression systems. The theory leads to a set of three simultaneous nonlinear third-order partial differential equations for pressure rise, and average and disturbed values of flow coefficient, as functions of time and angle around the compressor. By a Galerkin procedure, angular dependence is averaged, and the equations become first order in time. These final equations are capable of describing the growth and possible decay of a rotating-stall cell during a compressor mass-flow transient. It is shown how rotating-stall-like and surgelike motions are coupled through these equations, and also how the instantaneous compressor pumping characteristic changes during the transient stall process.