Pressure-dependent viscosity and free volume of atactic and syndiotactic polystyrene

The effect of pressure on viscosity is an important but often overlooked aspect of the flow properties of polymeric materials. In this work, two polymers (an atactic and a syndiotactic Polystyrene) were characterized to determine the effect of pressure on viscosity. In particular, a device was adopted to increase the exit pressure of a standard capillary rheometer, thus obtaining data of viscosity under high pressure and high shear rates. The Simha-Somcynsky equation of state was applied to the pressure–volume–temperature experimental data of both materials to obtain the dependence of free volume on temperature and pressure. The Doolittle equation was eventually employed to verify the dependence of viscosity on free volume. It was found that, for both materials, a linear relationship holds between the logarithm of zero-shear-rate viscosity (at several temperatures and pressures) and the inverse of free volume.     

Use of Magnetic Resonance Imaging as a Tool for the Study of Foamy Oil Behavior for an Extra‐Heavy Crude Oil. T2 /Viscosity Correlation with Respect to Pressure

The purpose of this work is to present the results of the phase behavior study for a live heavy oil during a pressure depletion process using magnetic resonance imaging (MRI) as a tool to characterize foamy oil phenomena. The experiments were carried out in the pressure range of 13.1 to 1.4MPa. Signal intensity images were obtained at each pressure and with respect to time, that is, approximately for a total time of 3h after each pressure change. It is possible to see a variation in intensity across the sample. These changes can be associated with changes in mobility as well as segregation of the oil. It was also possible to observe that what we trust is the formation of gas channels at the last two pressure values, as it comes out of solution. A correlation between the transverse relaxation time T2 and temperature was established with the aim of producing one between T2 and viscosity. In this way viscosity maps for the live oil were obtained as a function of pressure and time. It was observed that above the bubble point, the viscosity maps varied from low to high to low with respect to time for the same pressure. Below the bubble point the situation is reverse. The viscosity map changes from high to low to high with respect to time for the same pressure. The study shows the potential use of MRI to follow viscosity changes during pressure depletion test in a PVT MRI cell.     

Effect of fluids properties on non-equilibrium capillarity effects: Dynamic pore-network modeling

We have developed a Dynamic Pore-network model for Simulating Two-phase flow in porous media (DYPOSIT). The model is applicable to both drainage and imbibition processes. Employing improved numerical and geometrical features in the model facilitate a physically-based pore-scale simulator. This computational tool is employed to perform several numerical experiments (primary and main drainage, main imbibition) to investigate the current capillarity theory. Traditional two-phase flow formulations state that the pressure difference between the two phase is equal to the capillary pressure, which is assumed to be a function of saturation only. Many theoretical and experimental studies have shown that this assumption is invalid and the pressure difference between the two fluids is not only equal to the capillary pressure but is also related to the variation of saturation with time in the domain; this is referred to as the non-equilibrium capillarity effect. To date, non-equilibrium capillarity effect has been investigated mainly under drainage. In this study, we analyze the non-equilibrium capillarity theory under drainage and imbibition as a function of saturation, viscosity ratio, and effective viscosity. Other aspects of the dynamics of two-phase flow such as trapping and saturation profile are also studied.     

Pore Network Modeling of the Effects of Viscosity Ratio and Pressure Gradient on Steady-State Incompressible Two-Phase Flow in Porous Media

We perform steady-state simulations with a dynamic pore network model, corresponding to a large span in viscosity ratios and capillary numbers. From these simulations, dimensionless steady-state time-averaged quantities such as relative permeabilities, residual saturations, mobility ratios and fractional flows are computed. These quantities are found to depend on three dimensionless variables, the wetting fluid saturation, the viscosity ratio and a dimensionless pressure gradient. Relative permeabilities and residual saturations show many of the same qualitative features observed in other experimental and modeling studies. The relative permeabilities do not approach straight lines at high capillary numbers for viscosity ratios different from 1. Our conclusion is that this is because the fluids are not in the highly miscible near-critical region. Instead they have a viscosity disparity and intermix rather than forming decoupled, similar flow channels. Ratios of average mobility to their high capillary number limit values are also considered. Roughly, these vary between 0 and 1, although values larger than 1 are also observed. For a given saturation, the mobilities are not always monotonically increasing with the pressure gradient. While increasing the pressure gradient mobilizes more fluid and activates more flow paths, when the mobilized fluid is more viscous, a reduction in average mobility may occur.

     

On the Mathematical Modelling of a Compressible Viscoelastic Fluid

Thermodynamical considerations have largely been avoided in the modelling of complex fluids by invoking the assumption of incompressibility. This approximation allows pressure to be defined as a Lagrange multiplier, and therefore its natural connection with other thermodynamic variables such as density and temperature is irretrievably lost. Relaxing this condition to allow more realistic modelling involves much more than prescribing an equation of state. Even for a simple isothermal viscoelastic model, as explored in this paper, the transition to a compressible model is non-trivial. This paper shows that pressure enters the governing equations in a non-intuitive way. Furthermore, a fluid volume element, which is no longer constant, radically changes the way the basic element of the constitutive equations is viewed—stress is no longer the fundamental constitutive link between the momentum equations and velocity. The importance of geometry in fluid modelling is emphasised through the use of the Lie derivative, which is of a more fundamental character than the “upper” and “lower” convected derivatives prevalent in the literature and which are found to be almost redundant for a compressible fluid. There is now a strong body of non-equilibrium thermodynamics theory for flowing systems, which proves indispensible for this development. These fundamental principles are described herein using methodology and examples, that are sometimes conflicting, from the literature. The main conflict arises from the relationship between thermodynamic pressure and the trace of Cauchy stress, where the current preferred choice is (up to a constant) to set them equal—this is shown to be incorrect. Other issues such as the dependence of viscosity on density, bulk viscosity, integral modelling, the principle of objectivity and convected derivatives, are also clarified and resolved.     

Computational analysis of techniques to determine extensional viscosity from entrance flows

Recent computational analysis of entrance flows (Mitsoulis et al. 1998) suggests that the entrance pressure drop is insensitive to large changes in steady extensional viscosity-a result that directly contradicts a large body of experimental work in this area. A re-examination of entrance flows using numerical simulations is presented in this work which shows that entrance pressure drops do depend on the steady extensional viscosity, provided the extension rate in the entrance flow is large enough. Numerical simulations are presented using both the strain thinning and thickening versions of the Phan-Thien–Tanner (PTT) constitutive model. Several techniques for extracting extensional viscosity from entrance pressure are applied to the results of these simulations. The resulting predictions of extensional viscosity are compared to the steady extensional viscosity curves predicted by the PTT constitutive model used to generate the simulated pressure drop curves. The analytical techniques examined here are shown to provide reasonably accurate estimates of the steady extensional viscosity. This work also clearly demonstrates the advantage of using variable power-law coefficients for the rheological properties, used as inputs to the analyses, to capture the extensional behavior at deformation rates below the power law region more accurately. Received: 23 July 1999/Accepted: 24 November 1999     

Relative Permeability Analysis of Tube Bundle Models

The analytical solution for calculating two-phase immiscible flow through a bundle of parallel capillary tubes of uniform diametral probability distribution is developed and employed to calculate the relative permeabilities of both phases. Also, expressions for calculating two-phase flow through bundles of serial tubes (tubes in which the diameter varies along the direction of flow) are obtained and utilized to study relative permeability characteristics using a lognormal tube diameter distribution. The effect of viscosity ratio on conventional relative permeability was investigated and it was found to have a significant effect for both the parallel and serial tube models. General agreement was observed between trends of relative permeability ratios found in this work and those from experimental results of Singhal et al. (1976) using porous media consisting of mixtures of Teflon powder and glass beads. It was concluded that neglecting the difference between the average pressure of the non-wetting phase and the average pressure of the wetting phase (the macro-scale capillary pressure) – a necessary assumption underlying the popular analysis methods of Johnson et al. (1959) and Jones and Roszelle (1978) – was responsible for the disparity in the relative permeability curves for various viscosity ratios. The methods therefore do not account for non-local viscous effects when applied to tube bundle models. It was contended that average pressure differences between two immiscible phases can arise from either capillary interfaces (micro-scale capillary pressures) or due to disparate pressure gradients that are maintained for a flow of two fluids of viscosity ratio that is different from unity.     

Role of pressure dependent viscosity in measurements with falling cylinder viscometer

The falling cylinder viscometer is frequently used in measuring the dependence of the viscosity on the pressure. The viscosity is calculated using an indirect procedure, namely by appealing to the linear relation between the time taken for the fall and the viscosity. Under certain assumptions, the coefficient of proportionality can be derived analytically, and one gets the classical formula for the viscosity as a function of geometric parameters of the device, density of the fluid and the sinker, gravitational acceleration, and the distance and the time of the fall. Although the classical formula is valid only for fluids with constant viscosity, it is indiscriminately used even for fluids with pressure dependent viscosity. We investigate the role of variable viscosity, and we derive a heuristic correction to the classical formula for the case of fluids with pressure dependent viscosities. The systematic error introduced by the unwarranted application of the classical formula for fluids with pressure dependent viscosity is analysed, and it is shown it is measurable and it can in some cases significantly influence the experimental results.     

Influence of pressure and temperature on the flow behaviour of heavy fuel oils

Transportation and consumption of petroleum products around the world have created a potential risk for oil spills in the environment. Knowledge of high-pressure rheological behaviour of heavy crude oil fractions, which are usually transported in oil tankers, is very important to design deep recovering operations of the oil remaining in the tanks after an accident. The effect of pressure on the viscosity of these materials is not well understood, this is mainly due to experimental constraints involving high-pressure rheology measurements at low shear rates. Consequently, the overall objective of this work is to model the temperature–pressure–viscosity dependence of a selected heavy fuel oil in a wide range of pressure and temperature. With this aim, viscous flow tests at different temperatures and differential pressures and modulated differential scanning calorimetry tests were carried out on the heavy fuel oil selected. A temperature–pressure–viscosity model (FMT model) fits fairly well the experimental results obtained in the whole differential pressure range studied. However, viscosity values at temperatures lower than 10°C cannot be predicted due to microstructural changes associated with the solidification process of the heaviest components of the fuel oil tested.Paper presented at the European Rheology Conference (AERC) held in Grenoble, France, April 21–23, 2005.     

Crossflow of viscoplastic materials through tube bundles

The flow of viscoplastic materials through staggered arrays of tubes is analyzed. The mechanical behavior of the materials is assumed to obey the generalized Newtonian liquid (GNL) model, with a viscosity function given by the biviscosity law. The governing equations of this flow are solved numerically using a finite-volume method with a non-orthogonal mesh. For a representative range of the relevant parameters, results are presented in the form of velocity, pressure and viscosity fields. The pressure drop is also given as a function of rheological and geometric parameters.     

Disappearance of criticality for reactive third-grade fluid with Reynold's model viscosity in a flat channel

In this paper, we consider how a model for the motion with exponential viscosity of a third-grade fluid flowing between parallel plates, under the action of externally imposed pressure gradient, affects the fully developed and laminar reactive flow. The non-dimensional form of the coupled equations are solved numerically. The temperature distribution is presented for various viscosity variational parameters and the number is related to the Frank-Kamenetskii parameter. The criticality and disappearance of criticality (transition) of physical parameters and the competing effects of viscous dissipation, viscosity variational number, heat generation and non-Newtonian are discussed.     

A numerical study of unsteady gas-solid flow between parallel porous plates submitted to a magnetic field

This paper studies unsteady laminar flow of dusty conducting fluid between parallel porous plates with temperature dependent viscosity and the Network Simulation Method (NSM) is used to solve the governing nonlinear partial differential equations. The fluid is acted upon by a constant pressure gradient and an external uniform magnetic field is applied perpendicular to the plates that are assumed to be porous. The NSM is applied to solve the steady-state and transient problems of flow and heat transfer for both the fluid and dust particles. With this method, only discretization of the spatial co-ordinates is necessary, while time remains as a real continuous variable. The velocity and temperature are studied for different values of the viscosity and magnetic field parameters.     

Influence of pressure on polyamide 66 shear viscosity: a case study towards polar polymers behavior

In the present work, we consider the dependence of the viscosity of polyamide 66 (PA66) as a function of both temperature and pressure. Measurements were carried out using a standard capillary rheometer to evaluate the effect of temperature and the corresponding shift factor of viscosity data. The pressure influence was accounted for by employing an at-line rheometer equipped with a counter-pressure cell, the so-called enhanced exit pressure method. Time-temperature-pressure superposition was applied to obtain the viscosity master curve. Interestingly, PA66 displays an unusual increase of the pressure coefficient as a function of temperature. On the contrary, previous studies on nonpolar liquids or polymers for which polar interactions are relatively weak indicated that the sensitivity to pressure, measured by the so-called Barus parameter, is a decreasing function of the temperature. Thus, we attribute the peculiar behavior of PA66 to the strong polar interactions which are present and which may dominate the viscosity at relatively low temperatures and would be less dominant at higher temperatures.     

Effective property models for homogeneous two-phase flows

Using an analogy between thermal conductivity of porous media and viscosity in two-phase flow, new definitions for two-phase viscosity are proposed. These new definitions satisfy the following two conditions: namely (i) the two-phase viscosity is equal to the liquid viscosity at the mass quality = 0% and (ii) the two-phase viscosity is equal to the gas viscosity at the mass quality = 100%. These new definitions can be used to compute the two-phase frictional pressure gradient using the homogeneous modeling approach. These new models are assessed using published experimental data of two-phase frictional pressure gradient in circular pipes, minichannels and microchannels in the form of Fanning friction factor (f_m) versus Reynolds number (Re_m). The published data include different working fluids such as R-12, R-22, argon (R740), R717, R134a, R410A and propane (R290) at different diameters and different saturation temperatures. Models are assessed on the basis minimizing the root mean square error (eRMS). It is shown that these new definitions of two-phase viscosity can be used to analyze the experimental data of two-phase frictional pressure gradient in circular pipes, minichannels and microchannels using simple friction models.     

Numerical simulations of Boger fluids through different contraction configurations for the development of a measuring system for extensional viscosity

This paper reports the flow behaviour of Newtonian and Boger fluids through various axisymmetric contraction configurations by means of numerical predictions. A principal aim has been to evaluate the geometrical design choice of the hyperbolic contraction flow. The FENE-CR model has been used to reflect the behaviour of Boger fluids, with constant shear viscosity, finite (yet large) extensional viscosity and less than quadratic first normal stress difference. Numerical calculations have been performed on six different contraction configurations to evaluate an optimized geometry for measuring extensional viscosity in uniaxial extensional flow. The influence of a sharp or rounded recess-corner on the nozzle has also been investigated. Few commercial measuring systems are currently available for measurement of the extensional rheology of medium-viscosity fluids, such as foods and other biological systems. In this context, a technique based on the hyperbolic contraction flow would be a suitable alternative. The pressure drop, the velocity field, the first normal stress difference and the strain rate across the geometry have each been evaluated for Newtonian and Boger fluids. This numerical study has shown that the hyperbolic configuration is superior to the other geometry choices in achieving a constant extension rate. In this hyperbolic configuration, no vortices are formed, the measuring range is broader and the strain rate is constant throughout the geometric domain, unlike in the alternative configurations tested. The difference between sharp and rounded recess-corner configurations proved to be negligible and a rise in excess pressure drop (epd) for increasing deformation rates has been observed.     

A numerical study of fluids with pressure‐dependent viscosity flowing through a rigid porous medium

Much of the work on flow through porous media, especially with regard to studies on the flow of oil, are based on ‘Darcy's law’ or modifications to it, such as Darcy–Forchheimer or Brinkman models. While many theoretical and numerical studies concerning flow through porous media have taken into account the inhomogeneity and anisotropy of the porous solid, they have not taken into account the fact that the viscosity of the fluid and drag coefficient could depend on the pressure in applications, such as enhanced oil recovery (EOR). Experiments clearly indicate that the viscosity varies exponentially with respect to the pressure and the viscosity can change, in some applications, by several orders of magnitude. The fact that the viscosity depends on pressure immediately implies that the ‘drag coefficient’ would also depend on the pressure. In this paper we consider modifications to Darcy's equation wherein the drag coefficient is a function of pressure, which is a realistic model for technological applications, such as EOR and geological carbon sequestration. We first outline the approximations behind Darcy's equation and the modifications that we propose to Darcy's equation, and derive the governing equations through a systematic approach using mixture theory. We then propose a stabilized mixed finite element formulation for the modified Darcy's equation. To solve the resulting nonlinear equations we present a solution procedure based on the consistent Newton–Raphson method. We solve representative test problems to illustrate the performance of the proposed stabilized formulation. One of the objectives of this paper is to show that the dependence of viscosity on the pressure can have a significant effect both on the qualitative and quantitative nature of the solution. Copyright © 2010 John Wiley & Sons, Ltd.     

On stagnation pressure increases in calorically perfect,ideal gases

When stagnation pressure rises in a natural or numerically simulated flow it is frequently a cause for concern, as one usually expects viscosity and turbulence to cause stagnation pressure to decrease. In fact, if stagnation pressure increases, one may suspect measurement or numerical errors. However, this need not be the case, as the laws of nature do not require that stagnation pressure continually decreases. In order to help clarify matters, the objective of this work is to understand the conditions under which stagnation pressure will rise in the unsteady/steady flows of compressible, viscous, calorically perfect, ideal gases. Furthermore, at a more practical level, the goal is to understand the conditions under which stagnation pressure will increase in flows simulated with the Reynolds averaged Navier–Stokes equations and eddy-viscosity turbulence models. In order to provide an improved understanding of increases in stagnation pressure for both these scenarios, transport equations are derived that govern its behavior in the unaveraged and Reynolds averaged settings. These equations are utilized to precisely determine the relationship between changes in stagnation pressure and zeroth, first, and second derivatives of fundamental flow quantities. Furthermore, these equations are utilized to demonstrate the relationship between changes in stagnation pressure and fundamental non-dimensional quantities that govern the conductivity, viscosity, and compressibility of the flow. In addition, based on an analysis of the Reynolds averaged equation (for eddy-viscosity turbulence models), it is shown that stagnation pressure is particularly likely to experience a spurious rise at the outer edges of shear layers that are undergoing convex curvature. Thereafter, numerical experiments are performed which confirm the primary aspects of the theoretical analysis.     

Flow characteristics of liquid with pressure-dependent viscosities in microtubes

It is obvious that the pressure gradient along the axial direction in a pipe flow keeps constant according to the Hagen-Poiseuille equation. However, recent experiments indicated that the distribution of the pressure seemed no longer linear for liquid flows in microtubes driven by high pressure (1-30MPa). Based on H-P equation with slip boundary condition and Bridgman's relation of viscosity vs. static pressure, the nonlinear distribution of pressure along the axial direction is analyzed in this paper. The revised standard Poiseuille number with the effect of pressure-dependent viscosity taken into account agrees well with the experimental results. Therefore, the dependence of the viscosity on the pressure is one of the dominating factors under high driven pressure, and is represented by an important property coefficient α of the liquid. The project supported by the Chinese Academy of Sciences Major Innovation Project (KJCX2-SW-L2) and the National Natural Science Foundation of China (10272107). The English text was polished by Yunming Chen.     

Comparative analysis of CFD models for jetting fluidized beds: Effect of particle-phase viscosity

Under the Eulerian–Eulerian framework of simulating gas–solid two-phase flow, the accuracy of the hydrodynamic prediction is strongly affected by the selection of rheology of the particulate phase, for which a detailed assessment is still absent. Using a jetting fluidized bed as an example, this work investigates the influence of solid rheology on the hydrodynamic behavior by employing different particle-phase viscosity models. Both constant particle-phase viscosity model (CVM) with different viscosity values and a simple two-fluid model without particle-phase viscosity (NVM) are incorporated into the classical two-fluid model and compared with the experimental measurements. Qualitative and quantitative results show that the jet penetration depth, jet frequency and averaged bed pressure drop are not a strong function of the particle-phase viscosity. Compared to CVM, the NVM exhibits better predictions on the jet behaviors, which is more suitable for investigating the hydrodynamics of gas–solid fluidized bed with a central jet.     

Relative Permeability Analysis of Tube Bundle Models,Including Capillary Pressure

The analytical equations for calculating two-phase flow, including local capillary pressures, are developed for the bundle of parallel capillary tubes model. The flow equations that are derived were used to calculate dynamic immiscible displacements of oil by water under the constraint of a constant overall pressure drop across the tube bundle. Expressions for averaged fluid pressure gradients and total flow rates are developed, and relative permeabilities are calculated directly from the two-phase form of Darcy's law. The effects of pressure drop and viscosity ratio on the relative permeabilities are discussed. Capillary pressure as a function of water saturation was delineated for several cases and compared to a steady-state mercury-injection drainage type of capillary pressure profile. The bundle of serial tubes model (a model containing tubes whose diameters change randomly at periodic intervals along the direction of flow), including local Young-Laplace capillary pressures, was analyzed with respect to obtaining relative permeabilities and macroscopic capillary pressures. Relative permeabilities for the bundle of parallel tubes model were seen to be significantly affected by altering the overall pressure drop and the viscosity ratio; relative permeabilities for the bundle of serial tubes were seen to be relatively insensitive to viscosity ratio and pressure, and were consistently X-like in profile. This work also considers the standard Leverett (1941) type of capillary pressure versus saturation profile, where drainage of a wetting phase is completed in a step-wise steady fashion; it was delineated for both tube bundle models. Although the expected increase in capillary pressure at low wetting-phase saturation was produced, comparison of the primary-drainage capillary pressure curves with the pseudo-capillary pressure profiles, that are computed directly using the averaged pressures during the displacements, revealed inconsistencies between the two definitions of capillary pressure.