Local Velocity Distribution Research Articles

Radial and azimuthal gradients of the moving groups in Gaia DR3: The slow and fast bar degeneracy problem

The structure and dynamics of the central bar of the Milky Way (MW) are still under debate whilst being fundamental ingredients for the evolution of our Galaxy. The recent Gaia DR3 offers an unprecedented detailed view of the 6D phase space of the MW, allowing for a better understanding of the complex imprints of the bar on the phase space. We aim to identify and characterise the dynamical moving groups across the MW disc, and use their large-scale distribution to help constrain the properties of the Galactic bar. We used 1D wavelet transforms of the azimuthal velocity ($V_ distribution in bins of radial velocity to robustly detect the kinematic substructure in the Gaia DR3 catalogue. We then connected these structures across the disc to measure the azimuthal (phi ) and radial ($R$) gradients of $V_ of the moving groups. We simulated thousands of perturbed distribution functions using backward integration, sweeping a large portion of parameter space of feasible Galaxy models that include a bar, in order to compare them with the data and to explore and quantify the degeneracies. The radial gradient of the Hercules moving group ($ V_ R$ = 28.1pm 2.8 $) cannot be reproduced by our simple models of the Galaxy that show much larger slopes both for a fast and a slow bar. This suggests the need for more complex dynamics (e.g. a different bar potential, spiral arms, a slowing bar, a complex circular velocity curve, external perturbations, etc.). We measured an azimuthal gradient for Hercules of $ V_ = -0.63pm $deg$^ $ and find that it is compatible with both the slow and fast bar models. Our analysis points out that in using this type of analysis, at least two moving groups are needed to start breaking the degeneracies. We conclude that it is not sufficient for a model to replicate the local velocity distribution; it must also capture its larger-scale variations. The accurate quantification of the gradients, especially in the azimuthal direction, will be key for the understanding of the dynamics governing the disc.

Are there any extragalactic high speed dark matter particles in the Solar neighborhood?

We use the APOSTLE suite of cosmological hydrodynamical simulations of the Local Group to examine the high speed tail of the local dark matter velocity distribution in simulated Milky Way analogues. The velocity distribution in the Solar neighborhood is well approximated by a generalized Maxwellian distribution sharply truncated at a well-defined maximum “escape” speed. The truncated generalized Maxwellian distribution accurately models the local dark matter velocity distribution of all our Milky Way analogues, with no evidence for any separate extragalactic high-speed components. The local maximum speed is well approximated by the terminal velocity expected for particles able to reach the Solar neighborhood in a Hubble time from the farthest confines of the Local Group. This timing constraint means that the local dark matter velocity distribution is unlikely to contain any high-speed particles contributed by the Virgo Supercluster “envelope”, as argued in recent work. Particles in the Solar neighborhood with speeds close to the local maximum speed can reach well outside the virial radius of the Galaxy, and, in that sense, belong to the Local Group envelope posited in earlier work. The local manifestation of such envelope is thus not a distinct high-speed component, but rather simply the high-speed tail of the truncated Maxwellian distribution.

4D Neutron Imaging of Solute Transport and Fluid Flow in Sandstone Before and After Mineral Precipitation

AbstractIn many geological systems, the porosity of rock or soil may evolve during mineral precipitation, a process that controls fluid transport properties. Here, we investigate the use of 4D neutron imaging to image flow and transport in Bentheim sandstone core samples before and after in‐situ calcium carbonate precipitation. First, we demonstrate the applicability of neutron imaging to quantify the solute dispersion along the interface between heavy water and a cadmium aqueous solution. Then, we monitor the flow of heavy water within two Bentheim sandstone core samples before and after a step of in‐situ mineral precipitation. The precipitation of calcium carbonate is induced by reactive mixing of two solutions containing CaCl2 and Na2CO3, either by injecting these two fluids one after each other (sequential experiment) or by injecting them in parallel (co‐flow experiment). We use the contrast in neutron attenuation from time‐resolved tomograms to derive three‐dimensional fluid velocity field by using an inversion technique based on the advection‐dispersion equation. Results show mineral precipitation induces a wider distribution of local flow velocities and leads to alterations in the main flow pathways. The flow distribution appears to be independent of the initial distribution in the sequential experiment, while in the co‐flow experiment, we observed that higher initial local fluid velocities tended to increase slightly following precipitation. The outcome of this study contributes to progressing the knowledge in the domain of reactive solute and contaminant transport in the subsurface using the promising technique of neutron imaging.

Hydrodynamics in a randomly packed bed of spheres: A comparison between PR‐CFD simulations and MRI experiments

AbstractMultitubular reactors are commonly used in industry for processes involving highly exothermic chemical reactions. This reactor type consists of individual tubes, with a small diameter compared to the particle size. These slender beds facilitate heat management, but also give rise to flow maldistribution, which decreases the reactor efficiency. The aim of this article is to validate particle‐resolved simulations using Magnetic Resonance Imaging experiments while focusing on the flow maldistribution. The packing structure used in the simulations is reconstructed from the experimental images to facilitate a one‐to‐one comparison. A good match between experiments and simulations is found for the averaged flow profile, probability density function of the velocity in axial direction and even the local velocity distributions. However, a correction of the experimental results for magnetic susceptibility artifacts is necessary to obtain a similar match in the probability density functions and the local profiles.

Implicit Finite Difference Simulation of Hybrid Nanofluid along a Vertical Thin Cylinder with Sinusoidal Wall Heat Flux under the Effects of Magnetic Field

A numerical analysis of magnetohydrodynamic natural convection along a thin vertical cylinder with a sinusoidal heat flux at the wall immersed in copper (Cu) and aluminum-oxide (Al2O3) hybrid nanofluids has been studied. A 2D vertical thin cylinder shape geometry has been considered with a radius of R. The fluid flow is considered laminar and incompressible with the Prandtl number of Pr = 6.2 and 10% concentration of hybrid nanoparticles. The nondimensional governing equations have been solved numerically by using the implicit finite difference method. An in-house FORTRAN 90 code is used for solving this problem and the code is validated with the available benchmark results. Numerical simulations have been performed for a wide range of governing parameters, Hartmann number from Ha = 0 to Ha = 4, nanoparticles volume fractions ϕ = 0.0 to ϕ = 0.1, and the amplitude of the wall heat flux ε = 0.0–0.3. The findings have been illustrated in terms of streamlines, isotherms, local skin friction coefficients, local Nusselt numbers, velocity, and temperature distributions. The flow field and temperature distribution within the boundary layer are deceased by the effects of the wall heat flux amplitudes. It is also noted that the rate of heat transfer increases with particle volume fraction and the amplitude of the wall heat flux. According to the findings, Nu increases by 24.72% as ϕ increases from 0 to 0.1 while ε = 0.3, and 27.66% while ε increases from 0.0 to 0.3 at 5% hybrid nanoparticles. The local skin frictions and Nusselt number diminish with the increment of the Hartman number due to the effects of the Lorenz force. The findings of this study can lead to a better understanding of the fundamental principles regarding the behavior of hybrid nanofluids under complex conditions, such as a vertical thin cylinder with a sinusoidal wall heat flux. Understanding the behavior of hybrid nanofluids in the presence of a magnetic field and a nonuniform wall heat flow can also lead to the development of innovative heat transfer enhancement strategies.

The impact of the Large Magellanic Cloud on dark matter direct detection signals

We study the effect of the Large Magellanic Cloud (LMC) on the dark matter (DM) distribution in the Solar neighborhood, utilizing the Auriga magneto-hydrodynamical simulations of Milky Way (MW) analogues that have an LMC-like system. We extract the local DM velocity distribution at different times during the orbit of the LMC around the MW in the simulations. As found in previous idealized simulations of the MW-LMC system, we find that the DM particles in the Solar neighborhood originating from the LMC analogue dominate the high speed tail of the local DM speed distribution. Furthermore, the native DM particles of the MW in the Solar region are boosted to higher speeds as a result of a response to the LMC's motion.We simulate the signals expected in near future xenon, germanium, and silicon direct detection experiments, considering DM interactions with target nuclei or electrons. We find that the presence of the LMC causes a considerable shift in the expected direct detection exclusion limits towards smaller cross sections and DM masses, with the effect being more prominent for low mass DM. Hence, our study shows, for the first time, that the LMC's influence on the local DM distribution is significant even in fully cosmological MW analogues.

Experimental investigation of a supersonic close-coupled atomizer employing the phase Doppler measurement technique

Along with the growing economic importance of metal additive manufacturing by means of laser powder bed fusion, the demand for high-quality metal powders as the corresponding raw material is also increasing. However, the physics involved in supersonic close-coupled gas atomization, which is often employed for the production of these powders, are not well understood and extensive experimental data is scarce, leading to a lack of reliable predictive modeling capabilities. In this experimental study, local particle size and velocity distributions for the spray produced by a generic supersonic close-coupled atomizer are obtained using the phase Doppler measurement technique. The gas stagnation pressure and the liquid mass flow rate are varied systematically and independently. Three working liquids are considered, investigating the influence of the liquid dynamic viscosity on the atomization result. The particle size is shown to be sensitive to changes in both the gas stagnation pressure and the liquid mass flow rate. Notably, it is not an unambiguous function of the gas-to-liquid ratio. Furthermore, the effect of the liquid dynamic viscosity appears to be negligible. In conclusion, these are important insights for formulating physics-based models for the supersonic close-coupled atomization process.

Scale-up of Interdigitated Design from Small Cell to Large Cell for Aqueous Redox Flow Batteries

Redox flow batteries present attributes for large-scale energy storage. Interdigitated flow configuration is a promising structure due to its higher power density and lower pressure loss. The screening for new electrolyte is usually first tested on small cell (<10 cm2) due to its easy operation and less electrolyte consumption. To test the electrolyte functionality in practical application condition, large-scale (i.e., 780 cm2) testing is required conventionally in laboratory or industry. The twice testing is due to the fact that the testing results in small cell are different from that for large-scale interdigitated cell. In order to reduce the testing cycles and quickly estimate the application of new electrolytes, scale-up of interdigitated design from small cell to large cell is investigated in this work.Numerical simulation on different scales of interdigitated cells (10cm2, 78cm2, 130cm2, 780cm2) is conducted to assess the polarization. The local velocity distribution and redox-active species concentration are extracted to show their dominance on the cell performance. It is shown that with similar velocity and concentration distribution, cell with different scales present similar limiting current density and power density, regardless of active area. It implies that the results in small cells can be directly scaled up for large cells without further testing. It provides an avenue for quick estimation of new electrolyte in application condition.Experimental testing is conducted on vanadium redox flow battery for different scale of interdigitated cell. The experimental results will demonstrate this scale-up from small-scale to middle-scale interdigitated cell.

Tablet coating in lab-scale drum coaters: Combining DEM simulations and spray experiments to predict tablet coating

In pharmaceutical industry the coating of granulates or tablets is an essential production step, which is often realised in a rotating drum moving the tablet bed and a number of spray nozzles installed above this bed. The nozzles atomise the coating liquid through a high-velocity annular air-stream and apply steering air nozzles for producing an oval-shaped spray. The operational conditions determine the produced droplet size spectrum. The collision of the droplets with the solid particles or tablets in the drum and the resulting coating and film thickness strongly influences the product properties of granulates or tablets. In order to be able to predict the coating result, without running time-consuming lab-scale experiments, it is intended to numerically simulate the entire tablet coating process by combining Computational Fluid Dynamics (CFD), the Discrete Element Method (DEM) and the Lagrangian Parcel Concept (LPC) for simulating the spray propagation originating at the nozzle. For doing so, knowledge on the spray droplet properties produced by the two-fluid spray nozzles is indispensable. Therefore, it is necessary to characterise as a first step the local droplet size and velocity distributions in different spray cross-sections, which is the main focus of the present contribution. Such measurements are efficiently done by applying Phase Doppler Anemometry (PDA). It was found that the velocity of the spray emitted from the two-fluid nozzle exceeds almost 200 m/s and the number mean droplet diameter was well below 20 μm. In the initial spray region droplet break-up was observed whereas downstream of 50 mm a slight growth of the mean droplet size indicated further coalescence. Due to the applied pattern air, the spray flux distribution at the normally located tablet bed (i.e., 100–150 mm downstream the nozzle) has an oval shape and the mean droplet velocity still exceeds almost 50 m/s with a mean droplet diameter of about 10 μm. The cross-sectional measurements are used to provide proper inlet conditions for the spray simulations, while the other cross-sections are used for validating the computed spray expansion and dispersion. For demonstrating the effect of a real measured spray flux distribution on the tablet coating quality, additionally DEM simulations were conducted. Here, the measured spray flux data were mapped onto the temporarily evolving and moving tablet surface simulated for a lab-scale test drum by DEM. The uniformity of tablet coating (inter-tablet variability) was judged by calculating the Coefficient Of Variation (COV) for the considered different tablet shapes and operational conditions.

Influence of internal components on the hydrodynamic properties of the scrubbing-cooling chamber

The bubble-breaking plate is a crucial internal component in the scrubbing-cooling chamber. The study of hydrodynamics under its influence is the basis for designing and scaling up industrial installations. Experimental measurements of characteristic bubble parameters and liquid phase structure parameters at local locations were carried out using a conductivity probe and a pitot tube, respectively, to investigate the effects of five types of bubble-breaking plate installation on the hydrodynamic properties of the annular gap bubble bed. The experimental results show that the staggered placement of two plates is the most favorable for increasing the gas holdup, reducing the bubble size, increasing the gas–liquid interface area, improving the liquid velocity distribution and suppressing the turbulence fluctuations. In addition, a dimensionless mathematical model predicted the radial distribution of local gas holdup and axial time-averaged liquid velocity, and the expected results agreed with the experimental values.

Horizontal Stratified Air–Foam–Water Flows: Preliminary Modelling Attempts with OLGA

Water accumulation is a major problem in the flow assurance of gas pipelines. To limit liquid loading issues, deliquification by means of surfactant injection is a promising alternative to the consolidated mechanical methods. However, the macroscopic behavior of foam pipe flow in the presence of other phases has barely been explored. The goal of this work was to propose an approach to simulate air–water–foam flows in horizontal pipes using OLGA by Schlumberger, an industry standard tool for the transient simulation of multiphase flow. The simulation results were compared with experimental data for 60 mm and 30 mm ID (Inner Diameter) horizontal pipelines. Preliminary validation for two-phase air–water flow was carried out, which showed that correct flow pattern recognition is essential to accurately reproduce the experimental data. Then, stratified air–foam–water flows were investigated, assuming different models for the foam local velocity distribution. Foam rheology was considered through the Herschel–Bulkley model with the yield stress varying in time due to foam decay. The results showed good agreement for a uniform velocity profile and fresh foam properties in the case of the 60 mm ID pipeline, whereas for the 30 mm ID, which was characterized by significantly higher velocities, a linear velocity profile and 2000 s foam aging provided the best agreement. In both cases, the pressure gradient was overestimated, and the mean absolute prediction error ranged from about 5% to 30%.

The local dark matter distribution in self-interacting dark matter halos

We study the effects of dark matter self-interactions on the local dark matter distribution in selected Milky Way-like galaxies in the EAGLE hydrodynamical simulations. The simulations were run with two different self-interacting dark matter models, a constant and velocity-dependent self-interaction cross-section. We find that the local dark matter velocity distribution of the Milky Way-like halos in the simulations with dark matter self-interactions and baryons are generally similar to those extracted from cold collisionless dark matter simulations with baryons. In both cases, the local dark matter speed distributions agree well with their best fit Maxwellian distributions. Including baryons in the simulations with or without dark matter self-interactions increases the local dark matter density and shifts the dark matter speed distributions to higher speeds. To study the implications for direct detection, we compute the dark matter halo integrals obtained directly from the simulations and compare them to those obtained from the best fit Maxwellian velocity distribution. We find that a Maxwellian distribution provides a good fit to the halo integrals of most halos, without any significant difference between the results of different dark matter self-interaction models.

Quasi-parallel Whistler Waves and Their Interaction with Resonant Electrons during High-velocity Bulk Flows in the Earth’s Magnetotail

In collisionless space, plasma waves are important channels of energy conversion, affecting the local particle velocity distribution functions through wave–particle interactions. In this paper we present a comparative statistical analysis of the characteristics of quasi-parallel narrowband whistler waves and the properties of resonant electrons interacting with these waves during the intervals of earthward and tailward high-velocity bulk flows produced by the near-Earth X-line and observed by Magnetospheric Multiscale Mission spacecraft. We found that on both sides of the X-line, the suprathermal electrons (≥1 keV) having large pitch angles make the major contribution to the maximal growth rate (γ) of these waves. The whistler waves were observed almost simultaneously with strong enhancements of perpendicular magnetic gradients localized at electron scales near dipolarization fronts associated with the earthward bulk flows, and near flux ropes/magnetic islands embedded into the tailward bulk flows. Betatron energization of electrons due to the appearance of such gradients increases the perpendicular anisotropy of electron distribution, which could be responsible for the whistler wave generation. We found that in the course of electron interactions with the whistler waves the lower-energy resonant electrons can transfer a part of their kinetic energy to the higher-energy electrons, especially in the Central Plasma Sheet. This results in formation/enhancement of energy-dependent perpendicular anisotropy and power-law tails in the high-energy range of electron velocity distribution. We conclude that despite the differences in the magnetic structure of the earthward and tailward bulk flows, the mechanisms of the quasi-parallel whistler wave generation and the properties of resonant electrons are quite similar.

A data-driven machine learning framework for modeling of turbulent mixing flows

A novel computationally efficient machine learning (ML) framework has been developed for constructing the turbulent flow field of single-phase or two-phase particle-liquid flows in a mechanically agitated vessel by feeding a very short-term experimental Lagrangian trajectory. Using a supervised k-nearest neighbors regressor learning algorithm coupled with a Gaussian process, the framework predicts the mean flow and turbulent fluctuations by sharing the statistical features learned from experimental data. The capability of the ML framework is evaluated by comparing the flow dynamics of predicted trajectories to extensive Lagrangian particle tracking measurements under various flow conditions. Local velocity distributions, Lagrangian statistical analysis, solid concentration distributions, and phase flow numbers show very good agreement between ML-predictions and experiments. Being accurate, efficient, and robust, the ML framework is a powerful tool for analyzing and modeling multiphase flow systems using a minimal amount of driver data input, which can equally be provided from any reliable numerical simulation, thus avoiding costly experimental measurements.

Flow at an Ogee Crest Axis for a Wide Range of Head Ratios: Theoretical Model

The discharge coefficient of an ogee crest is a function of the ratio of the effective head to the design head. The purpose of the present study is to derive a theoretical model of this relation, which does not depend on empirical coefficients and whose predictions over a wide range of head ratios are accurate enough for practical use. The developments consider unsubmerged ogee crests without approach flow or lateral contraction effects, heads large enough to enable surface tensions to be neglected, and heads small enough to avoid flow separation. The method is based on potential flow theory, depth integration in a curvilinear reference frame, and critical flow theory. The characteristics of the crest shape are defined by the trajectory of a free jet passing over the crest at the design head. The dimensionless equations show that the position of the critical section is not at the apex of the crest. Nevertheless, they also suggest an approximate equation at the apex of the crest from which the discharge coefficient is derived, together with the local water depth, velocity, and pressure distribution. The results compare well with experimental data for head ratios between 0 and 5, which validates the underlying assumptions of the theoretical model.

Application Of The Phase Doppler Measurement Technique For The Characterization Of Supersonic Gas Atomization

A dual-mode phase Doppler measurement system has been set up for the characterization of the complex spray produced through supersonic gas atomization. The atomizer design has been based on a generic close-coupled configuration operated with water and air, which leads to high local Mach numbers and complex patterns of compression and expansion waves within the gas flow field. As a result, the dense spray is characterized by high particle velocities and a wide range of particle sizes, which proves challenging for experimental investigations. Measurements of local particle size and velocity distributions have been performed in various radial positions within the spray downstream of the atomizer, where the spray is fully developed. The set point of operation, which is defined by the gas stagnation pressure and the liquid mass flow rate, has been varied systematically, covering a wide range. By decoupling both operational parameters, their influence on the atomization result has been investigated independently. Furthermore, the atomizer geometry has been varied in terms of liquid nozzle diameter and liquid nozzle protrusion length. In accordance with data from the literature, the gas stagnation pressure has been observed to lead to decreasing median particle size and dispersion of the particle size, when increased. However, as a novel finding, the atomization result has been shown to be more sensitive to changes in the liquid mass flow rate. Thus, increasing the liquid mass flow rate results in an increase in both the median particle size and the dispersion of the particle size. The relationship between the atomization result and the gas-to-liquid ratio has been clearly shown to be ambiguous, that is, a function of the gas stagnation pressure and the liquid mass flow rate. Neither the liquid nozzle diameter nor the liquid nozzle protrusion length have been found to have an influence on the median particle size. However, the latter clearly influences the dispersion of the particle size in a complicated way, while the effect of the liquid nozzle diameter is more subtle. It has been concluded that the complex coupling between operational parameters and the specific atomizer design evident in supersonic gas atomization requires a deeper understanding of the local gas flow field and its interaction with the liquid flow.

Nonequilibrium effects of reactive flow based on gas kinetic theory* *This work is supported by the National Natural Science Foundation of China (NSFC) under Grant Nos. 51806116 and 11875001.

How to accurately probe chemically reactive flows with essential thermodynamic nonequilibrium effects is an open issue. Via the Chapman–Enskog analysis, the local nonequilibrium particle velocity distribution function is derived from the gas kinetic theory. It is demonstrated theoretically and numerically that the distribution function depends on the physical quantities and derivatives, and is independent of the chemical reactions directly as the chemical time scale is longer than the molecular relaxation time. Based on the simulation results of the discrete Boltzmann model, the departure between equilibrium and nonequilibrium distribution functions is obtained and analyzed around the detonation wave. In addition, it has been verified for the first time that the kinetic moments calculated by summations of the discrete distribution functions are close to those calculated by integrals of their original forms.

Scale-up issues for commercial depth filters in bioprocessing.

Significant increases in cell density and product titer have led to renewed interest in the application of depth filtration for initial clarification of cell culture fluid in antibody production. The performance of these depth filters will depend on the local pressure and velocity distribution within the filter capsule, but these are very difficult to probe experimentally, leading to challenges in both process design and scale-up. We have used a combination of carefully designed experimental studies and computational fluid dynamics (CFD) to examine these issues in both lab-scale (SupracapTM 50) and pilot-scale (StaxTM ) depth filter modules, both employing dual-layer lenticular PDH4 media containing diatomaceous earth. The SupracapTM 50 showed a more rapid increase in transmembrane pressure and a more rapid DNA breakthrough during filtration of a Chinese Hamster Ovary cell culture fluid. These results were explained using CFD calculations which showed very different flow distributions within the modules. CFD predictions were further validated using measurements of the residence time distribution and dye binding in both the lab-scale and pilot-plant modules. These results provide important insights into the factors controlling the performance and scale-up of these commercially important depth filters as well as a framework that can be broadly applied to develop more effective depth filters and depth filtration processes.

Effect of bend of horizontal rectangular duct on Scattering and Breakup of gas-liquid interface

The flow of a liquid film sheared by gas stream in a horizontal rectangular bent duct was investigated using a high-speed camera. Compared to the straight horizontal duct condition, the local gas-phase velocity distribution in the bend and the effects of interactions such as reflections of waves impinging on the wall remain unknown. In this study, we investigate the unique scattering and splitting behavior that occurs at the bend by varying the gas-phase flow velocity and the bend angle of the rectangular duct. Specifically, Bag breakup was quantitatively evaluated using We number and compared with Bag breakup generated from a single droplet using characteristic time t*bk. The results showed that Bag breakups were unevenly distributed due to bend effects, and the We number varied depending on the location of the Bag breakup. It was also found that the possible range of the characteristic time of bag breakup was smaller than that of a single droplet.

Simplified analysis of MHD flow in a porous surrounding bounded by an oscillating vertical cylindrical surface

This study developed the relatively simple mathematical model by adding a radiation effect and transverse magnetic field to the physical model, taking into account heat and mass transfer of a permeable, saturated porous medium, infinite oscillating cylindrical plate. In comparison, magnetohydrodynamics reality flow pattern in most earlier works was ignored in porous media. We have performed a semi-analytical method for numerical inverse Laplace transformation into PDEs of radiation on Newtonian fluids’ heat and mass transfer. The numerical stability method is used to analyze the influence of parameter variation for different values of M and N. We took into account both the local flow velocity distribution and concentration profile with the effect of Sc and time t and the skin friction against time t. There are several variations examined between the heat transfer under the cylinder boundary conditions and the plane boundary conditions. The findings given by empirical and numerical approaches are in good agreement. As seen in the literature, the numerical results on the local flow velocity agree closely with the solutions available. Our observations, meanwhile, show that the rate of heat transfer can be boosted using a magnetic field. The curved surface of the boundary configuration will restrain velocity and the temperature rise, while it has little effect on the mass transfer rate.