Fluid Density Research Articles

Double-diffusive convection in flow of Carreau fluid with variable density inside converging and diverging channels of rectilinear walls

In this article, the effects of double-diffusive convection have been seen on the flow of Carreau fluid of variable density, maintained inside straight and converging (diverging) channels. Note that the channel walls exhibit variable porosity and undergo stretching or shrinking at non-uniform velocities. In the flow domain, we have taken into account the convective transport of both heat and species mass concentrations. The pseudo-similarity solutions of previous investigation for such type flows are not used here; however, a new set of similarity transformations has been formed, which reduced the governing system (PDEs) into an exact set of ordinary differential equations (ODEs), whereas the longstanding issues of semi-similarity solutions have been successfully resolved in this particular case and in general for all situations of Carreau fluid flow. In a nutshell, a unified mathematical approach has been devised in this paper. We strictly emphasized the simulation of flow of Carreau fluid and double diffusive convection in flow inside a channel with multiple dynamical behaviours and structures (both Jeffery Hammel and Poissuielle flow types) of walls. The boundary layer approximation and stream function assumptions have not been used in the study's execution. The shear thinning and shear thickening features of Carreau fluids with variable density cause them to exhibit lower axial velocity in the centre of a horizontal channel than Newtonian fluids due to variable wall configurations and complex flow conditions. By enhancing buoyancy driven convection, which promotes effective heat dispersion, increasing the solutal and thermal Grashof numbers (GrS=GrT>0) lowers the temperature field. It has been found that in channel with diverging walls (a1=2.0), increasing the modified Reynolds numbers (σ1,σ2>0) increases concentration profiles, i.e. ϕ(η), while decreasing them in channel with rectilinear walls (a1=0.0).

Free movement of a near-wall particle in fluid of comparable density

The fundamental problem of nonlinear interaction between a freely moving particle and surrounding fluid flow is investigated for a density ratio of order unity, with potential applications to biomedical, environmental, and aerodynamic configurations. The interaction takes place near a fixed wall, with the particle being relatively thin. A mathematical model is presented, showing the fluid pressure forces to be dominant over the mass-acceleration effects in the particle motion (in contrast with previous analyses). The added mass due to the fluid motion, thus, greatly exceeds the body mass. Numerical simulations and asymptotic analysis reveal a range of possible particle motions. The main properties emerging are: (a) the distinction between collisions with the wall and fly-away responses; (b) the time scales involved in such behaviors; (c) the pressures and velocities induced at collision; (d) the occurrence of flow reversal in certain cases; and (e) the results being independent of the particle mass and moment of inertia as well as independent of the density ratio provided that the ratio is of order unity (0–7, say) or only slightly larger (8–30, say).

Viscous-Layer Compressibility Correction for Two-Equation Reynolds-Averaged Navier–Stokes Models

The baseline two-equation Reynolds-averaged Navier–Stokes (RANS) models include fluid density but lack calibration for compressible flows, making them inadequate for high Mach numbers. Various compressibility corrections have been proposed to integrate the van Driest transformation or the semilocal transformation, i.e., a given compressible law of the wall, into the RANS formulation. Prior work has focused mainly on the logarithmic layer, but upon evaluating these log-layer compressibility corrections, we find that they do not significantly improve skin friction estimates. To overcome this challenge, we develop viscous-layer compressibility corrections. We do that by altering the dissipation terms. These corrections conform the RANS model to the semilocal scaling, resulting in more accurate predictions of mean velocity and temperature in a posteriori tests. Although not the primary focus of this study, we also find that the baseline one-equation Spalart–Allmaras model, which was proposed before the concept of semilocal scaling, produces results consistent with the semilocal scaling in a posteriori tests without requiring any compressibility correction.

Mathematical theory and CFD solutions for internal convective cooling of gas turbine blades highlighting the effects of rotation

A quasi-one-dimensional mathematical theory is developed for determining the effects of rotation on the fluid flow and heat transfer in a cooling channel within a rotating gas turbine blade. The three-dimensional intricacies in the solutions of the same problem are captured through accurate computational fluid dynamic simulations. The derived analytical closed-form solutions predict the amount by which the relative total temperature of the coolant changes as a result of rotation. It is shown that such changes due to rotation are significant proportion of the change of coolant temperature due to heat transfer. The computed spatial evolutions of the primary velocity field, secondary velocity field and the force fields are presented as the coolant progresses from the inlet to the outlet so that the complexities and subtleties are exposed to a high degree of visualizability. Detailed considerations to various components of the centrifugal, Coriolis and pressure forces are given. The flow field is shown to depend strongly on whether the coolant moves in the spanwise outward or inward direction. The three-dimensionality of the flow field is reflected in complex circumferential variations of the blade temperature and the internal heat transfer coefficient. When the effects of the x–component of the Coriolis force interacts non-linearly with the effects of the z–component of the centrifugal force and a drastic x-gradient in fluid density due to a thin thermal boundary layer, then the primary velocity profile may exhibit great complexity, including flow separation. The present paper thus captures the strong, non-linear, two-way coupling between the fluid dynamics and heat transfer in the internal convective cooling occurring in a channel within a rotating gas turbine blade. Based on about 100 three-dimensional computational fluid dynamics simulations, new heat transfer correlations are developed for average Nusselt number for a rotating channel that will be of practical use.

Drilling Fluid Density Optimization for Narrow Density Window Fractured Formation

Abstract Fractures are prevalent in the Leikoupo Formation in western Sichuan, with a narrow safe density window of 0.06-0.12 g/cm3, leading to frequent downhole accidents like kicks and losses. Using Pengzhou X well as a case study, this research employed numerical simulation to analyze the impact of various drilling fluid densities within this window on displacement rate and bottomhole pressure. It established an optimal drilling fluid density model for fractured formations with narrow density windows. Findings reveal that as drilling fluid density increases, annulus up-return velocity decreases, potentially trapping fracture gas in the upper annulus. Optimal drilling fluid density lies within this narrow window, maintaining a dynamic balance between preventing fast gas return and fluid accumulation in fractured formations. The model applied to Pengzhou Y well yields a drilling density correlation of 0.000001036, consistent with optimized model results. This approach optimizes drilling fluid density within narrow density windows, mitigating downhole gushing and leakage accidents.

Influential mechanism of operational parameters on internal resonance of drill strings by numerical simulations

Abstract Drill string vibration is highly conductive to drill string fatigue and component failure. In this study, a finite element model of a drill string with bottom hole assembly (BHA) is constructed to investigate the effect of different operational parameters, such as different string length, density, pressure, and velocity of drilling fluid on drill string natural frequencies, and thus internal resonance by considering fluid-structure interaction (FSI) in ANSYS. Numerical simulations demonstrate that the natural frequencies decrease as the length of the drill string increases, which is especially true for the axial mode. The condition of 1:1 internal resonance between the axial and torsional modes only occurs when the length of drill string is sufficiently high. Nevertheless, this resonance fades away in the presence of fluid. The increment of fluid density slightly increases the natural frequency in lateral mode, whereas decreases that in axial mode. On the other hand, this effect of density is negligible in the torsional mode. The natural frequencies for axial, torsional and lateral modes decrease as the fluid velocity increases, and the impact of fluid pressure on natural frequency is almost not observable. The numerical results for the case without drill fluid are in qualitative agreement with analytical solutions. when considing the effect of drill length on natural frequency.

Investigating surface and internal waves and viscosity effects in two-layer fluids with a streamlined moving body using fully nonlinear simulation and experiment

Waves generated by a moving submerged moving body were analyzed in a two-layer density fluid. The experiments were performed in a two-dimensional towing tank and compared to the numerical results from a two-dimensional, fully nonlinear numerical towing tank (NTT) based on potential flow. The experimental conditions were determined by the relationship between the depth-dependent Froude number and the critical Froude number in baroclinic mode. The study was performed with various body velocities, changing the upper and lower fluid depth while maintaining the total water depth. An artificial damping coefficient was applied to the entire free surface and interface boundary condition to simulate the viscosity effect of the fluid interface. When the baroclinic mode was dominant, internal precursor soliton waves and depression waves were similar to the experimental results with strong nonlinearity. In contrast, the trailing waves were significantly different from the experimental results without applying an artificial damping scheme. A comparison with the experimental data showed that the effects of the fluid viscosity can be simulated with an appropriate damping coefficient. As the body velocity increases, the depressed or rising waves associated with the baroclinic mode are mixed with the trailing waves of the barotropic mode. As the depth of the upper fluid decreases, the characteristics of the baroclinic mode decrease, and the barotropic mode becomes dominant under the same velocity conditions.

Cryogenic thermosiphon used for indirect cooling of superconducting magnets

A thermosiphon is a thermodynamic phenomenon that facilitates the circulation of cryogen within a cooling system, relying solely on gravitational forces and phase change. This mechanism leverages the variations in the density of the cryogenic fluid throughout the entire cooling loop, creating a pressure gradient. This gradient serves as the primary driving force for the circulation of the cryogen. To negate the necessity of a circulation pump, it is crucial to determine the geometry of the cooling loop, the configuration of the thermosiphon, its height, and the vertical placement of the cryogen phase separator. This paper introduces a simplified computational model and the geometric calculations of the cryogenic thermosiphon for two distinct configurations of the indirect cooling loop for superconducting magnets.

Study on the Equivalent Density Tool and Depressurisation Mechanism of Suction-Type Depressurisation Cycle

In order to further regulate equivalent circulating density (ECD), a novel downhole apparatus for reducing circulating pressure in high-temperature and high-pressure wells, the suction-type ECD reduction tool, was devised. The utilisation of this tool enables the bottomhole pressure of the equivalent circulating density to be attained in close proximity to its hydrostatic pressure, thereby facilitating the attainment of deeper drilling depths. The tool is composed primarily of a screw motor, scroll blades, annular seals, universal joints, and drilling columns. The tool operates by utilising the suction effect and hydraulic energy extracted from the circulating fluid by the screw motor, which is then converted into mechanical energy to create suction and enhance the flow energy of the drilling fluid within the annulus at the bottom of the well, thereby reducing the equivalent circulating density. Furthermore, based on ANSYS-FLUENT analysis simulations, the alteration of pressure drop characteristics in response to varying drilling fluid densities, displacements, and tool sizes was modelled. The simulation results demonstrate that the pressure drop effect is 1.0 MPa when the drilling fluid density is 1.2 g/cm3, 1.7 MPa when the drilling fluid density is 1.5 g/cm3, and 1.9 MPa when the drilling fluid density is 1.8 g/cm3. A pressure drop of approximately 2.3 MPa was observed when the drilling fluid density was 2.0 g/cm3. The maximum pressure drop is achievable with a flow rate ranging from 1500 to 2500 L/min. A maximum pressure drop of 2.3 MPa is observed when the flow rate is within the range of 1500 to 2500 L/min. Two distinct viscosity values (0.02 and 0.06 kg/(m·s)) were employed to assess the impact of viscosity on pressure drop characteristics in a suction-type ECD tool. The results demonstrated that the pressure drop remained largely unaltered, indicating that viscosity had minimal influence on this parameter. The flow rate emerged as the primary factor affecting pressure drop, with viscosity exerting a relatively minor effect.

Dissipation Effects in the Tea Leaf Paradox

The Tea Leaf Paradox (TLP) describes unsteady fluid motions which help entrain and deposit suspended particles at the center of rotation. Various applications depend on the TLP for particle separations—spanning orders of magnitude in length scales—making it an important problem in fluid mechanics. Despite papers describing the phenomenon, the efficacy of particle separation using the TLP remains unclear as to the relative importance of, for example, hydrostatics, particle-fluid density ratio, wall friction, liquid bath aspect ratio and the rotation speed. The dynamics involved are notably complex and require a careful tuning of each variable. In this study, we have investigated the role of the limit of the aggregation dynamics in rotational flows within 3D-printed vessels of various sizes in tandem with particle imaging to probe the dissipation effects on the particle motions. We have found that the liquid bath aspect ratio limits how much aggregation may occur for a particle-fluid density ratio greater than unity (e.g., ρp/ρf>1), where ρp is the density of the particle and ρf is the ambient fluid density.

Exploring Natural Convection and Entropy Generation in a Closed Water Tank Utilizing Nanofluid: a Computational Study

Theoretical framework: This study presents a comprehensive investigation into three-dimensional natural convection within a confined cavity filled with an alumina-water nanofluid. Objective: Utilizing numerical simulations, we explore the influence of nanoparticles' volume fraction and Rayleigh number () on entropy generation, heat transfer efficiency, and fluid behavior within the water tank. Method: The study delves into the conservation equations governing mass, momentum, and energy within the context of three-dimensional laminar natural convection. It focuses on steady, incompressible flow, employing the Boussinesq approximation to account for variations in fluid density due to temperature gradients. Results and conclusion: Our findings indicate that increasing Rayleigh numbers lead to heightened entropy generation, with values ranging from to, primarily driven by intensified buoyant flow. However, the presence of nanoparticles significantly mitigates entropy generation, enhancing the overall thermal performance of the system. Moreover, nanofluids demonstrate superior heat transfer capabilities compared to pure fluids, with higher nanoparticle concentrations resulting in increased heat transfer rates, ranging from to alumina. Research implications: As a result of such thorough research, qualitative examinations of velocity fields and entropy generation patterns further highlight the role of nanofluids in improving heat transfer efficiency while reducing irreversible processes within the cavity.

Simulating the Rising Air Bubbles Stream through Heavy Fuel Oil Residue Bubble Column Reactor using COMSOL

Several industrial operations rely on the rise of air bubbles through heavy fuel oil residue (HFOr), including oil recovery and wastewater treatment. In order to increase efficiency and decrease expenses, it is necessary to understand the peculiarities of this process. With the help of COMSOL Multiphysics, we numerically simulate the ascent of a single air bubble via HFOr in this article. The dimensions of the container used in the simulations were (100) mm in diameter and (200) mm in height, with an air bubble diameter of 4.5 mm. For an air bubble ascending through an HFOr column, the predicted numerical outcomes are contrasted with nine experimental versions. As the bubble rose higher, the region with the greatest number of vortices was located on its side. The remaining vortex is contained inside the bubble. In addition, since drag decreases the flow, a lighter fluid usually exhibits a region of strong vortex and low pressure. Because of the oscillation effect, the results reveal that the rising velocity initially rises and then gradually falls between 0.6 and 0.65 s. The air bubbles' wavy motion is caused by the high fluid density, which becomes worse as the air velocity gets higher. The values of the drag coefficient rise sharply with decreasing air extrusion speed. It turns out that the height of the fluid has an inverse relationship with the pressure.

Artificial neural network prediction of wellbore stability in offshore shallow formations

Wellbore instability is one of the most critical challenges during drilling, which may result in complex problems such as stuck pipe, high torque and mud loss, impeding the drilling progress and increasing the cost of drilling operations. Offshore shallow formations are characterized by weak consolidation and low strength, causing serious wellbore collapse frequently. The traditional physics-based wellbore instability models attempt to predict the risk of failure of the wellbore for given drilling fluid densities and provide optimal fluid density for safe drilling. However, these models generally involve quite a few empirical coefficients, determination of which heavily rely on the field experiences of the engineers, and thus the results may vary greatly from person to person. In this study, an artificial neural network (ANN) model has been established for wellbore stability prediction, aiming to reduce the subjectivity while mapping the drilling performance of neighboring wells into the predicted results. Eleven factors that may affect wellbore stability have been identified from the prevailing physics-based model and used as the input of the ANN model, while the wellbore enlargement rate (WER) is used as the output to quantify the wellbore stability performance. The model has been trained using data from 5 wells drilled in the offshore shallow formations of the Bohai Bay Basin in east China and then used to predict the WER of another well in the same region. The results show that the mean absolute percentage error is 6.113%, and at most well depths the predicted wellbore diameter deviate from the measured values not more than 5%. Comparison of the ANN model and some other machine learning models, including random forest model, decision trees model, linear regression model, was conducted, which demonstrated the best performance of the ANN model in terms of the predicted wellbore diameter profile. Finally, the potential application of the ANN model in optimizing mud weight has been illustrated through the predicted wellbore diameter profiles for different mud weights. It is worth noting that the method presented in this paper is of physics and data-driven nature and provides a new research insight for wellbore stability analysis.

Determination of acoustic properties of paraffin oil mixed with activated coal nanoparticles or SPAN80 using only BAW time delay measurement

An important aspect of controlling the properties of liquids relates to their use in modern engines powered by composite materials. In this context, controlling the properties of industrial fluids containing micro- and nanoscale particles is essential. Understanding the temperature-dependent behavior of fluid density and elasticity is crucial for proper engine operation. Seven physical parameters of pure paraffin oil, paraffin oil with varying concentrations of activated coal nanoparticles or sorbitane monooleate (SPAN 80) were simultaneously determined using only BAW time delay measurement in the single experimental run. This technique relies on propagating LBAW through the test fluid and measuring the time delay and amplitude of these waves at different temperatures. By calculating temperature coefficients for velocity, density, time delay, and expansion using well-established formulas the physical properties of fluids under study were determined. The frequency of the LBAW used in the experiment was 13 MHz. The length of the liquid sample in the propagation direction of LBAW was ∼ 5 mm. The experiments were carried out within a temperature range of −20 °C to +90 °C. The volume of the test sample was around 1 milliliter. The comparison of the physical properties of different suspensions under these conditions allows us to demonstrate the dependence of the measured properties on the type and composition of the medium tested.

A novel formulation of an eco-friendly calcium nitrate-based heavy completion fluid

Calcium-nitrate-based transparent completion fluids are widely used in the oil and gas industry for well completion and stimulation operations in carbonate reservoirs. These fluids have many advantages, such as medium density, low corrosion, good temperature stability, low clay swelling, and high wettability. However, the density of calcium nitrate brine is limited by its solubility, which can be increased by the addition of alcohols. This study investigated the effects of adding different types of alcohols (G1, G2, and G3) to calcium nitrate brine on the properties and performance of the completion fluid for carbonate reservoirs. The fluid properties and performance were evaluated through a series of laboratory tests, including density measurement, corrosion test, viscosity measurement, rheology test, temperature stability test, clay swelling test, wettability test, and compatibility test. The results showed that the addition of alcohols to brine can improve or reduce the fluid density, viscosity, corrosion, temperature stability, clay swelling, wettability, and compatibility, depending on the type and amount of alcohols. The optimum fluids have been selected based on their highest density, lowest viscosity, lowest corrosion, highest temperature stability, lowest clay swelling, highest wettability change, and highest compatibility with formation fluids. The densities of the fluids CN4, CNG14, CNG24, and CNG34 were respectively 96, 101, 101.5, and 100 pounds per cubic foot (pcf). Their rates of corrosion on L80 steel were respectively 0.829, 0.589, 0.720, and 0.599 mils per year (mpy). Their apparent viscosities at 62.4 °F were respectively around 15, 120, 100, and 60 centipoise (cp). Their apparent viscosities at 176 °F were all around 10 to 25 mpy. The fluids remained clear with no evidence of suspended solid particles at 20 °F, indicating their resilience to low-temperature conditions and suitability for use in cold weather operations. The fluid CNG24 becomes cloudy at 285 °F, and the fluid CNG34 becomes cloudy from 212 °F onward, but the CN4 fluid remains clear and transparent at all temperatures. In the tested high temperatures, the effect on their pH was around the same. The fluids CN4, CNG24, and CNG34 had a lower swelling index than 5 ml per 2 g of bentonite clay. The contact angles of oil and carbonate-type thin sections after their wettabilities were affected by the fluids were also 99.5, 36.54, and 46.03°, respectively. Finally, they’re all relatively compatible with formation fluids. The best completion fluid for carbonate reservoirs was CNG24, which contained calcium nitrate and G2 alcohol. This fluid had the best overall performance and safety of the fluids tested.

An approach simulating interacting solid, liquid, and gas domains for dynamic seal applications

AbstractIn seal applications a solid elastic body (the seal) ensures the separation of a liquid (e.g., hydraulic oil in one chamber) from a gas (e.g., air in another chamber). In the case of a dynamic seal, the rod is moving and transports a thin liquid film from the liquid chamber into the air chamber. The seal gap consists of a converging gap, a minimum distance between seal and rod and a diverging gap, where transported liquid detaches from the seal. Between the three states of matter (gas, liquid, gas) appear the following contact conditions: liquid‐to‐solid, gas‐to‐liquid, and gas‐to‐solid. A classical fluid‐structure‐interaction is technically possible with limited effort, but is not sufficient to distinguish between the liquid and the gas. A solution can make use of the fact, that the liquid has significant dimensions only in the liquid chamber and is pressurized only there. There is a thin liquid film on the rod in the air chamber, but it has dimensions clearly lower than the air chamber and has the same pressure as the air. Therefore a simulation is possible with the “variable viscosity approach”: Liquid and gas are modeled in one single fluid domain. Differences between liquid and gas are included by pressure‐dependent fluid parameters, especially viscosity and density. The resulting violation of the mass balance is very small as the low mass flow through the seal lip is low. The presented approach allows to apply a fluid‐structure simulation with special fluid properties to model a system with three states of matter. Advantages are a less complex model with less complex contact conditions and a reduced simulation time.

Entropy of strongly coupled Yukawa fluids.

The entropy of strongly coupled Yukawa fluids is discussed from several perspectives. First, it is demonstrated that a vibrational paradigm of atomic dynamics in dense fluids can be used to obtain a simple and accurate estimate of the entropy without any adjustable parameters. Second, it is explained why a quasiuniversal value of the excess entropy of simple fluids at the freezing point should be expected, and it is demonstrated that a remaining very weak dependence of the freezing point entropy on the screening parameter in the Yukawa fluid can be described by a simple linear function. Third, a scaling of the excess entropy with the freezing temperature is examined, a modified form of the Rosenfeld-Tarazona scaling is put forward, and some consequences are briefly discussed. Fourth, the location of the Frenkel line on the phase diagram of Yukawa systems is discussed in terms of the excess entropy and compared with some predictions made in the literature. Fifth, the excess entropy scaling of the transport coefficients (self-diffusion, viscosity, and thermal conductivity) is reexamined using the contemporary datasets for the transport properties of Yukawa fluids. The results could be of particular interest in the context of complex (dusty) plasmas, colloidal suspensions, electrolytes, and other related systems with soft pairwise interactions.

Numerical Modelling of Cylindrical Fluid Filled Tank

The demand for drinking and service water storage is rising with changing climate conditions and increasing life expectancy. The tanks are commonly used to store large volumes of liquids and materials in various fields of the economy. This paper presents the model of the numerical simulation for the steel tank filled with fluid, using the finite element method. The results of the tank filled with water are presented, by the results: the pressure of the fluid the effective stress, and the maximum deformation of the tank solid domain. The correctness of the pressure values was verified by the simple calculation of the fluid pressure. Finally, the paper documents the results for various fluid fillings with a considered range of fluid densities. The influence of the fluid filling height on the behavior of the solid domain of the fluid filling container loaded by the static loading as well as the effect of the width of the tank on the behavior of the solid domain of the fluid filling container.

Prediction and Application of Drilling-Induced Fracture Occurrences under Different Stress Regimes

Identifying and categorizing drilling-induced fractures is pivotal for understanding the mechanisms underlying wellbore instability, drilling fluid loss, and assessing reservoirs using imaging logging data. This study employs a linear elastic stress model around the wellbore, coupled with a tensile failure criterion, to establish a predictive framework for the orientation of drilling-induced fractures. It investigates how engineering parameters like wellbore trajectory and bottomhole pressure influence the distribution of principal stresses around the wellbore, as well as the angle and orientation of drilling-induced fractures relative to the wellbore axis, across various faulting scenarios. The results indicate that drilling-induced fractures exhibit structured arrangements and consistent patterns, often appearing at approximately 180° symmetric intervals and descending in similar orientations. This provides a theoretical basis for their systematic identification and classification. Under different stress conditions, these fractures can manifest as feather-like shapes, “J”-shaped, or transitional states between feather-like and “J”-shaped orientations, as well as “V”-shaped or “M”-shaped orientations. Accurate detection and classification of these fractures are essential for interpreting effective fractures, conducting thorough reservoir evaluations, and predicting appropriate drilling fluid densities to mitigate the wellbore failure risk. Moreover, this knowledge aids in effectively determining the magnitude and direction of in situ stress inversion.

Solubility of anhydrite in supercritical water from 380 ˚C to 625 ˚C and 220 bar to 270 bar

The solubility of anhydrite in deionized water has been determined experimentally from 380 ˚C to 625 ˚C and 220 bar to 270 bar. The experiments were performed using a unique flow-through reactor capable of reaching supercritical conditions for pure water. The results cover the approximate range of temperature and pressure expected to be found in deep geothermal systems where supercritical conditions could be expected. Anhydrite solubility has not been previously well-defined in this region.The new experimental data are used to define empirical parameters for a new thermodynamic model based on Dolejš and Manning (2010), for anhydrite dissolution in silica-bearing aqueous fluids at elevated temperatures and pressures:-RTlnm=a+b+elnρwwhere m is the molar concentration mol∙kg−1, R is the gas constant in J mol−1 K−1, and ρw is the density of pure water in kg m−3. These parameters only apply in solutions containing silica and at fluid densities above 200 kg m−3:a = 14921.46; b = 369.46; e = -48.81.T is temperature in K.