The steady performance of a porous and compliant aerostatic thrust bearing

Recent developments in aerostatic thrust bearings have included: (a) the porous aerostatic thrust bearing containing a porous pad and (b) the inherently compensated compliant surface aerostatic thrust bearing containing a thin elastomer layer. Both these developments have been reported to improve the bearing load capacity compared to conventional aerostatic thrust bearings with rigid surfaces. This development is carried one stage further in a porous and compliant aerostatic thrust bearing incorporating both a porous pad and an opposing compliant surface. The thin elastomer layer forming the compliant surface is bonded to a rigid backing and is of a soft rubber like material. Such a bearing is studied experimentally and theoretically under steady state operating conditions. A mathematical model is presented to predict the bearing performance. In this model is a simplified solution to the elasticity equations for deflections of the compliant surface. Account is also taken of deflections in the porous pad due to the pressure difference across its thickness. The lubrication equations for flow in the porous pad and bearing clearance are solved by numerical finite difference methods. An iteration procedure is used to couple deflections of the compliant surface and porous pad with solutions to the lubrication equations. Comparisons between experimental results and theoretically predicted bearing performance are in good agreement. However these results show that the porous and compliant aerostatic thrust bearing performance is lower than that of a porous aerostatic thrust bearing with a rigid surface in place of the compliant surface. This discovery is accounted to the recess formed in the bearing clearance by deflections of the compliant surface and its effect on flow through the porous pad.

Comments on multiple oscillatory solutions in systems with time-delay feedback

A complex Ginzburg-Landau equation subjected to local and global time-delay feedback terms is considered. In particular, multiple oscillatory solutions and their properties are studied. We present novel results regarding the disappearance of limit cycle solutions, derive analytical criteria for frequency degeneration, amplitude degeneration, frequency extrema. Furthermore, we discuss the influence of the phase shift parameter and show analytically that the stabilization of the steady state and the decay of all oscillations (amplitude death) cannot happen for global feedback only. Finally, we explain the onset of traveling wave patterns close to the regime of amplitude death.

Particle and gaseous emissions from compressed natural gas and ultralow sulphur diesel-fuelled buses at four steady engine loads

Exhaust emissions from thirteen compressed natural gas (CNG) and nine ultralow sulphur diesel in-service transport buses were monitored on a chassis dynamometer. Measurements were carried out at idle and at three steady engine loads of 25%, 50% and 100% of maximum power at a fixed speed of 60 kmph. Emission factors were estimated for particle mass and number, carbon dioxide and oxides of nitrogen for two types of CNG buses (Scania and MAN, compatible with Euro 2 and 3 emission standards, respectively) and two types of diesel buses (Volvo Pre-Euro/Euro1 and Mercedez OC500 Euro3). All emission factors increased with load. The median particle mass emission factor for the CNG buses was less than 1% of that from the diesel buses at all loads. However, the particle number emission factors did not show a statistically significant difference between buses operating on the two types of fuel. In this paper, for the very first time, particle number emission factors are presented at four steady state engine loads for CNG buses. Median values ranged from the order of 1012 particles min-1 at idle to 1015 particles km-1 at full power. Most of the particles observed in the CNG emissions were in the nanoparticle size range and likely to be composed of volatile organic compounds The CO2 emission factors were about 20% to 30% greater for the diesel buses over the CNG buses, while the oxides of nitrogen emission factors did not show any difference due to the large variation between buses.

Wavelet based approaches and high resolution methods for complex process models

Many industrial processes and systems can be modelled mathematically by a set of Partial Differential Equations (PDEs). Finding a solution to such a PDF model is essential for system design, simulation, and process control purpose. However, major difficulties appear when solving PDEs with singularity. Traditional numerical methods, such as finite difference, finite element, and polynomial based orthogonal collocation, not only have limitations to fully capture the process dynamics but also demand enormous computation power due to the large number of elements or mesh points for accommodation of sharp variations. To tackle this challenging problem, wavelet based approaches and high resolution methods have been recently developed with successful applications to a fixedbed adsorption column model. Our investigation has shown that recent advances in wavelet based approaches and high resolution methods have the potential to be adopted for solving more complicated dynamic system models. This chapter will highlight the successful applications of these new methods in solving complex models of simulated-moving-bed (SMB) chromatographic processes. A SMB process is a distributed parameter system and can be mathematically described by a set of partial/ordinary differential equations and algebraic equations. These equations are highly coupled; experience wave propagations with steep front, and require significant numerical effort to solve. To demonstrate the numerical computing power of the wavelet based approaches and high resolution methods, a single column chromatographic process modelled by a Transport-Dispersive-Equilibrium linear model is investigated first. Numerical solutions from the upwind-1 finite difference, wavelet-collocation, and high resolution methods are evaluated by quantitative comparisons with the analytical solution for a range of Peclet numbers. After that, the advantages of the wavelet based approaches and high resolution methods are further demonstrated through applications to a dynamic SMB model for an enantiomers separation process. This research has revealed that for a PDE system with a low Peclet number, all existing numerical methods work well, but the upwind finite difference method consumes the most time for the same degree of accuracy of the numerical solution. The high resolution method provides an accurate numerical solution for a PDE system with a medium Peclet number. The wavelet collocation method is capable of catching up steep changes in the solution, and thus can be used for solving PDE models with high singularity. For the complex SMB system models under consideration, both the wavelet based approaches and high resolution methods are good candidates in terms of computation demand and prediction accuracy on the steep front. The high resolution methods have shown better stability in achieving steady state in the specific case studied in this Chapter.

Use of IEEE 1588-2008 for a sampled value process bus in transmission substations

IEC Technical Committee 57 (TC57) published a series of standards and technical reports for “Communication networks and systems for power utility automation” as the IEC 61850 series. Sampled value (SV) process buses allow for the removal of potentially lethal voltages and damaging currents inside substation control rooms and marshalling kiosks, reduce the amount of cabling required in substations, and facilitate the adoption of non-conventional instrument transformers. IEC 61850-9-2 provides an inter-operable solution to support multi-vendor process bus solutions. A time synchronisation system is required for a SV process bus, however the details are not defined in IEC 61850-9-2. IEEE Std 1588-2008, Precision Time Protocol version 2 (PTPv2), provides the greatest accuracy of network based time transfer systems, with timing errors of less than 100 ns achievable. PTPv2 is proposed by the IEC Smart Grid Strategy Group to synchronise IEC 61850 based substation automation systems. IEC 61850-9-2, PTPv2 and Ethernet are three complementary protocols that together define the future of sampled value digital process connections in substations. The suitability of PTPv2 for use with SV is evaluated, with preliminary results indicating that steady state performance is acceptable (jitter < 300 ns), and that extremely stable grandmaster oscillators are required to ensure SV timing requirements are met when recovering from loss of external synchronisation (such as GPS).

An analytical method to solve a general class of nonlinear reactive transport models

Three recent papers published in Chemical Engineering Journal studied the solution of a model of diffusion and nonlinear reaction using three different methods. Two of these studies obtained series solutions using specialized mathematical methods, known as the Adomian decomposition method and the homotopy analysis method. Subsequently it was shown that the solution of the same particular model could be written in terms of a transcendental function called Gauss’ hypergeometric function. These three previous approaches focused on one particular reactive transport model. This particular model ignored advective transport and considered one specific reaction term only. Here we generalize these previous approaches and develop an exact analytical solution for a general class of steady state reactive transport models that incorporate (i) combined advective and diffusive transport, and (ii) any sufficiently differentiable reaction term R(C). The new solution is a convergent Maclaurin series. The Maclaurin series solution can be derived without any specialized mathematical methods nor does it necessarily involve the computation of any transcendental function. Applying the Maclaurin series solution to certain case studies shows that the previously published solutions are particular cases of the more general solution outlined here. We also demonstrate the accuracy of the Maclaurin series solution by comparing with numerical solutions for particular cases.

Oxygen transport in soil and the vertical distribution of roots

An analytical solution for steady-state oxygen transport in soils including 2 sink terms, viz roots and microbes with the corresponding vertical distribution scaling lengths forming a ratio p, showed p governed the critical air-filled porosity, θ_c, needed by most plants. For low temperature and p, θ_c was <0.1 but at higher temperatures and p = 1, θ_c was >0.15 m³/m³. When root length density at the surface was 10⁴ m/m³ and p > 3, θ_c was 0.25 m³/m³, more than half the pore space. Few combinations of soil and climate regularly meet this condition. However, for sandy soils and seasonally warm, arid regions, the theory is consistent with observation, in that plants may have some deep roots. Critical θ_c values are used to formulate theoretical solutions in a forward mode, so different levels of oxygen uptake by roots may be compared to microbial activity. The proportion of respiration by plant roots increases rapidly with p up to p ≈2.

Scaling analysis of the unsteady natural convection boundary layer adjacent to an inclined plate for Pr > 1 following instantaneous heating

The unsteady natural convection boundary layer adjacent to an instantaneously heated inclined plate is investigated using an improved scaling analysis and direct numerical simulations. The development of the unsteady natural convection boundary layer following instantaneous heating may be classified into three distinct stages including a start-up stage, a transitional stage and a steady state stage, which can be clearly identified in the analytical and numerical results. Major scaling relations of the velocity and thicknesses and the flow development time of the natural convection boundary layer are obtained using triple-layer integral solutions and verified by direct numerical simulations over a wide range of flow parameters.

An analytical solution for diffusion and nonlinear uptake of oxygen in a spherical cell

We develop a new analytical solution for a reactive transport model that describes the steady-state distribution of oxygen subject to diffusive transport and nonlinear uptake in a sphere. This model was originally reported by Lin (Journal of Theoretical Biology, 1976 v60, pp449–457) to represent the distribution of oxygen inside a cell and has since been studied extensively by both the numerical analysis and formal analysis communities. Here we extend these previous studies by deriving an analytical solution to a generalized reaction-diffusion equation that encompasses Lin’s model as a particular case. We evaluate the solution for the parameter combinations presented by Lin and show that the new solutions are identical to a grid-independent numerical approximation.

Solution methods for advection-diffusion-reaction equations on growing domains and subdomains, with application to modelling skin substitutes

Problems involving the solution of advection-diffusion-reaction equations on domains and subdomains whose growth affects and is affected by these equations, commonly arise in developmental biology. Here, a mathematical framework for these situations, together with methods for obtaining spatio-temporal solutions and steady states of models built from this framework, is presented. The framework and methods are applied to a recently published model of epidermal skin substitutes. Despite the use of Eulerian schemes, excellent agreement is obtained between the numerical spatio-temporal, numerical steady state, and analytical solutions of the model.

Multiscale coupling and multiphysics approaches in earth sciences : applications

Geoscientists are confronted with the challenge of assessing nonlinear phenomena that result from multiphysics coupling across multiple scales from the quantum level to the scale of the earth and from femtoseconds to the 4.5 Ga of history of our planet. We neglect in this review electromagnetic modelling of the processes in the Earth’s core, and focus on four types of couplings that underpin fundamental instabilities in the Earth. These are thermal (T), hydraulic (H), mechanical (M) and chemical (C) processes which are driven and controlled by the transfer of heat to the Earth’s surface. Instabilities appear as faults, folds, compaction bands, shear/fault zones, plate boundaries and convective patterns. Convective patterns emerge from buoyancy overcoming viscous drag at a critical Rayleigh number. All other processes emerge from non-conservative thermodynamic forces with a critical critical dissipative source term, which can be characterised by the modified Gruntfest number Gr. These dissipative processes reach a quasi-steady state when, at maximum dissipation, THMC diffusion (Fourier, Darcy, Biot, Fick) balance the source term. The emerging steady state dissipative patterns are defined by the respective diffusion length scales. These length scales provide a fundamental thermodynamic yardstick for measuring instabilities in the Earth. The implementation of a fully coupled THMC multiscale theoretical framework into an applied workflow is still in its early stages. This is largely owing to the four fundamentally different lengths of the THMC diffusion yardsticks spanning micro-metre to tens of kilometres compounded by the additional necessity to consider microstructure information in the formulation of enriched continua for THMC feedback simulations (i.e., micro-structure enriched continuum formulation). Another challenge is to consider the important factor time which implies that the geomaterial often is very far away from initial yield and flowing on a time scale that cannot be accessed in the laboratory. This leads to the requirement of adopting a thermodynamic framework in conjunction with flow theories of plasticity. This framework allows, unlike consistency plasticity, the description of both solid mechanical and fluid dynamic instabilities. In the applications we show the similarity of THMC feedback patterns across scales such as brittle and ductile folds and faults. A particular interesting case is discussed in detail, where out of the fluid dynamic solution, ductile compaction bands appear which are akin and can be confused with their brittle siblings. The main difference is that they require the factor time and also a much lower driving forces to emerge. These low stress solutions cannot be obtained on short laboratory time scales and they are therefore much more likely to appear in nature than in the laboratory. We finish with a multiscale description of a seminal structure in the Swiss Alps, the Glarus thrust, which puzzled geologists for more than 100 years. Along the Glarus thrust, a km-scale package of rocks (nappe) has been pushed 40 km over its footwall as a solid rock body. The thrust itself is a m-wide ductile shear zone, while in turn the centre of the thrust shows a mm-cm wide central slip zone experiencing periodic extreme deformation akin to a stick-slip event. The m-wide creeping zone is consistent with the THM feedback length scale of solid mechanics, while the ultralocalised central slip zones is most likely a fluid dynamic instability.

Unsteady Compressible Boundary-Layer Flow On The Stagnation Line Of An Infinite Swept Cylinder

Numerical solutions of flow and heat transfer process on the unsteady flow of a compressible viscous fluid with variable gas properties in the vicinity of the stagnation line of an infinite swept cylinder are presented. Results are given for the case where the unsteady temperature field is produced by (i) a sudden change in the wall temperature (enthalpy) as the impulsive motion is started and (ii) a sudden change in the free-stream velocity. Solutions for the simultaneous development of the thermal and momentum boundary layers are obtained by using quasilinearization technique with an implicit finite difference scheme. Attention is given to the transient phenomenon from the initial flow to the final steady-state distribution. Results are presented for the skin friction and heat transfer coefficients as well as for the velocity and enthalpy profiles. The effects of wail enthalpy parameter, sweep parameter, fluid properties and transpiration cooling on the heat transfer and skin friction are considered.

On the nature of power flow equations in security analysis

The power system network is assumed to be in steady-state even during low frequency transients. However, depending on generator dynamics, and toad and control characteristics, the system model and the nature of power flow equations can vary The nature of power flow equations describing the system during a contingency is investigated in detail. It is shown that under some mild assumptions on load-voltage characteristics, the power flow equations can be decoupled in an exact manner. When the generator dynamics are considered, the solutions for the load voltages are exact if load nodes are not directly connected to each other

State space formulation of ammonia reactor dynamics

The specific objective of this paper is to develop a state space model of a tubular ammonia reactor which is the heart of an ammonia plant in a fertiliser complex. A ninth order model with three control inputs and two disturbance inputs is generated from the nonlinear distributed model using linearization and lumping approximations. The lumped model is chosen such that the steady state temperature at the exit of the catalyst bed computed from the simplified state space model is close enough to the one computed from the nonlinear steady state model. The model developed in this paper is very useful for the design of continuous/discrete versions of single variable/multivariable control algorithms.

A Nonlinear System Under Combined Periodic and Random Excitation

The anharmonic oscillator under combined sinusoidal and white noise excitation is studied using the Gaussian closure approximation. The mean response and the steady-state variance of the system is obtained by the WKBJ approximation and also by the Fokker Planck equation. The multiple steadystate solutions are obtained and their stability analysis is presented. Numerical results are obtained for a particular set of system parameters. The theoretical results are compared with a digital simulation study to bring out the usefulness of the present approximate theory.