Compositional controls on melting and dissolving a salt into a ternary melt

We explore theoretically the controls on dissolution of salt A, in an undersaturated brine of salts A and B. We show that, as the concentration of B increases, the dissolution rate of A decreases, for brine of given temperature. We also show that there is a sharper decrease in dissolution rate with increasing concentration, for concentrations of B above a critical value, where B limits the equilibrium concentration. We explore the implications of the predictions for dissolution of KCl or NaCl, by a mixed brine of NaCl and KCl, a common reaction that may arise in dissolution of evaporites. We predict that, with mixed-composition brine, KCl crystals dissolve more rapidly than NaCl crystals, unless the (far-field) brine is nearly saturated in KCl. We also predict that the dissolution rate of these salts is largely independent of fluid temperature and is controlled by compositional diffusion

Diffusion-controlled dissolution of a binary solid into a ternary liquid with partially-molten zone formation

We build a theoretical model of equilibrium dissolution of a homogeneous, solid mixture of two salts A and B, KCl and NaCl being used as the type example, into an aqueous solution of the two salts, with diffusive transport. We find that there are two sharp dissolution fronts, separating fluid, a partially molten zone containing a single solid and mixed solid. The phase change happens almost entirely at the two sharp fronts. In equilibrium, the leading front exhibits a small amount of precipitation of NaCl, simultaneous with complete dissolution of KCl. There is a unique surface in the space of far-field fluid KCl concentration, far-field fluid NaCl concentration and solid composition, dividing conditions where NaCl is the solid in the partially molten zone, from conditions where KCl is the solid in the partially molten zone. The movement rates of the dissolution fronts decrease as the concentration of either salt in the far-field fluid is increased. The movement rates of the dissolution fronts increase as either far-field temperature is increased, but this effect is smaller than that of concentration. In most circumstances, the dissolution front for a given salt moves more slowly, the more of that salt is present in the original solid, although the mass dissolution rate is not greatly affected by the solid composition

Transient ventilation dynamics following a change in strength of a point source of heat

We investigate the transient ventilation flow within a confined ventilated space, with high- and low-level openings, when the strength of a low-level point source of heat is changed instantaneously. The steady-flow regime in the space involves a turbulent buoyant plume, which rises from the point source to a well-mixed warm upper layer. The steady-state height of the interface between this layer and the lower layer of exterior fluid is independent of the heat flux, but the upper layer becomes progressively warmer with heat flux. New analogue laboratory experiments of the transient adjustment between steady states identify that if the heat flux is increased, the continuing plume propagates to the top of the room forming a new, warmer layer. This layer gradually deepens, and as the turbulent plume entrains fluid from the original warm layer, the original layer is gradually depleted and disappears, and a new steady state is established. In contrast, if the source buoyancy flux is decreased, the continuing plume is cooler than the original plume, so that on reaching the interface it is of intermediate density between the original warm layer and the external fluid. The plume supplies a new intermediate layer, which gradually deepens with the continuing flow. In turn, the original upper layer becomes depleted, both as a result of being vented through the upper opening of the space, but also due to some penetrative entrainment of this layer by the plume, as the plume overshoots the interface before falling back to supply the new intermediate layer. We develop quantitative models which are in good accord with our experimental data, by combining classical plume theory with models of the penetrative entrainment for the case of a decrease in heating. Typically, we find that the effect of penetrative entrainment on the density of the intruding layer is relatively weak, provided the change in source strength is sufficiently large. However, penetrative entrainment measurably increases the rate at which the depth of the draining layer decreases. We conclude with a discussion of the importance of these results for the control of naturally ventilated spaces

On the transport of heavy particles through an upward displacement-ventilated space

We explore the transport of heavy particles through an upward displacement-ventilated space. The space incorporates a localised source of buoyancy which generates a turbulent buoyant plume. The plume fluid is contaminated with a small concentration of particles, which are subject to gravitational settling. A constant flow of uncontaminated fluid is supplied at a low level into the space, while an equal amount of fluid is vented from the space at a high level. At steady state, a two-layer density stratification develops associated with the source of buoyancy. New laboratory experiments are conducted to explore how particles are transported by this flow. The experiments identify that the upper layer may either become well-mixed in particles or it may develop a vertical stratification in particle concentration, with the particle concentration decreasing with height. We develop a quantitative model which identifies that such stratification develops for larger particle setting speeds, or smaller ventilation rates. In accord with our experiments, the model predicts that the number of particles extracted from the space through the high-level vent is controlled by the magnitude of the particle stratification in the upper layer, and this in turn depends on the particle settling speed relative to the ventilation speed and also the cross-sectional area and height of the space. We compare the predictions of the model with measurements of the flux of particles vented from the space for a range of operating conditions. We explore the relevance of the model for the removal of airborne contaminants by displacement ventilation in hospital rooms, and we discuss how contamination is propagated in the room as a result of lateral mixing of pathogens in the upper layer.This work has been funded through the BP Institute, EPSRC and Hughes Hall, Cambridge. We gratefully acknowledge the technical assistance of A. Pluck, and the constructive comments of Professor C. Noakes, Dr. C. P. Caulfield and of two anonymous referees.This is the author accepted manuscript. The final version is available via CUP at http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=9694505&fileId=S0022112015002049

On turbulent particle fountains

We describe new experiments in which particle-laden turbulent fountains with source Froude numbers 20>Fr_{0}>6 are produced when particle-laden fresh water is injected upwards into a reservoir filled with fresh water. We find that the ratio $U$ of the particle fall speed to the characteristic speed of the fountain determines whether the flow is analogous to a single-phase fountain ($U\ll 1$) or becomes a fully separated flow ($U\geqslant 1$). In the single-phase limit, a fountain with momentum flux $M$ and buoyancy flux $B$ oscillates about the mean height, $h_{m}=(1.56\pm 0.04)M^{3/4}B^{-1/2}$, as fluid periodically cascades from the maximum height, $h_{t}=h_{m}+{\rm\Delta}h$, to the base of the tank. Experimental measurements of the speed $u$ and radius $r$ of the fountain at the mean height $h_{m}$, combined with the conservation of buoyancy, suggest that $Fr(h_{m})=u(g^{\prime }r)^{-1/2}\approx 1$. Using these values, we find that the classical scaling for the frequency of the oscillations, ${\it\omega}\sim BM^{-1}$, is equivalent to the scaling $u(h_{m})/r(h_{m})$ for a fountain supplied at $z=h_{m}$ with $Fr=1$ (Burridge & Hunt, J. Fluid Mech., vol. 728, 2013, pp. 91–119). This suggests that the oscillations are controlled in the upper part of the fountain where $Fr\leqslant 1$, and that they may be understood in terms of a balance between the upward supply of a growing dense particle cloud, at the height where $Fr=1$, and the downward flow of this cloud. In contrast, in the separated flow regime, we find that particles do not reach the height at which $Fr=1$: instead, they are transported to the level at which the upward speed of the fountain fluid equals their fall speed. The particles then continuously sediment while the particle-free fountain fluid continues to rise slowly above the height of particle fallout, carried by its momentum.This work has been funded through the BP Institute, EPSRC and Hughes Hall, Cambridge. We gratefully acknowledge the technical assistance of A. Pluck, and the constructive comments of three anonymous referees.This is the author accepted manuscript. The final version is available from Cambridge University Press via http://dx.doi.org/10.1017/jfm.2016.16

On the transport of heavy particles through a downward displacement-ventilated space

We investigate the transport of relatively heavy, small particles through a downward displacement-ventilated space. A flux of particles is supplied to the space from a localised source at a high level and forms a turbulent particle-laden plume which descends through the space. A constant flow of ambient fluid which does not contain particles is supplied to the space at a high level, while an equal amount of fluid is vented from the space at a low level. As a result of the entrainment of ambient fluid into the particle plume, a return flow is produced in the ambient fluid surrounding the plume in the lower part of the space. At steady state, particles are suspended by this return flow. An interface is formed which separates the ambient fluid in the lower part of the space, which contains particles, from the particle-free ambient fluid in the upper part of the space. New laboratory experiments show that the concentration of particles in the ambient fluid below the interface is larger than the average concentration of particles in the plume fluid at the level of the interface. Hence, as the plume fluid crosses the interface and descends through the particle-laden fluid underneath, it becomes relatively buoyant and forms a momentum-driven fountain. If the fountain fluid impinges on the floor, it then spreads radially over the surface until lifting off. We develop a quantitative model which can predict the height of the interface, the concentration of particles in the lower layer, and the partitioning of the particle flux between the fraction which sediments over the floor and that which is ventilated out of the space. We generalise the model to show that when particles and negatively buoyant fluid are supplied at the top of the space, a three-layer stratification develops in the space at steady state: the upper layer contains relatively low-density ambient fluid in which no particles are suspended; the central layer contains a mixture of ambient and plume fluid in which no particles are suspended; and the lower layer contains a suspension of particles in the same mixture of ambient and plume fluid. We quantify the heights of the two interfaces which separate the three layers in the space and the concentration of particles in suspension in the ambient fluid in the lower layer. We then discuss the relevance of the results for the control of airborne infections in buildings. Our experiments show that the three-layer stratification is subject to intermittent large-scale instabilities when the concentration of particles in the plume at the source is sufficiently small, or the rate of ventilation of the space is sufficiently large: we describe the transient concentration of particles in the space during one of these instabilities.This work has been funded through the BP Institute, EPSRC and Hughes Hall, Cambridge. We gratefully acknowledge the technical assistance of A. Pluck, and the constructive comments of Professor C. Noakes, Dr. C. P. Caulfield and of three anonymous referees.This is the author accepted manuscript. The final version is available from Cambridge University Press via http://dx.doi.org/10.1017/jfm.2015.24

Buoyancy-driven flow in a confined aquifer with a vertical gradient of permeability

We examine the injection of fluid of one viscosity and density into a horizontal permeable aquifer initially saturated with a second fluid of different viscosity and density. The novel feature of the analysis is that we allow the permeability to vary vertically across the aquifer. This leads to recognition that the interface may evolve as either a rarefaction wave that spreads at a rate proportional to$t$, a shock-like front of fixed length or a mixture of shock-like regions and rarefaction-wave-type regions. The classical solutions in which there is no viscosity ratio between the fluids and in which the formation has constant permeability lead to an interface that spreads laterally at a rate proportional to$t^{1/2}$. However, these solutions are unstable to cross-layer variations in the permeability owing to the vertical shear which develops in the flow, causing the structure of the interface to evolve to the rarefaction wave or shock-like structure. In the case that the viscosities of the two fluids are different, it is possible that the solution involves a mixture of shock-like and rarefaction-type structures as a function of the distance above the lower boundary. Using the theory of characteristics, we develop a regime diagram to delineate the different situations. We consider the implications of such heterogeneity for the prediction of front locations during$\text{CO}_{2}$sequestration. If we neglect the permeability fluctuations, the model always predicts rarefaction-type solutions, while even modest changes in the permeability across a layer can introduce shocks. This difference may be very significant since it leads to the$\text{CO}_{2}$plume occupying a greater fraction of the pore space between the injector and the leading edge of the$\text{CO}_{2}$front in a layer of the same mean permeability. This has important implications for estimates of the fraction of the pore space that the$\text{CO}_{2}$may access.</jats:p

Exsolved volatiles in magma reservoirs

We review our understanding of the exsolved volatile phase co-existing with magmas during pre-eruptive storage at the pressures and temperatures corresponding to crustal magma reservoirs. We explore the consequences and implications of such a volatile phase for magma and ore body petrogenesis and the fluid dynamics of magma reservoirs. We outline the geochemical constraints on the size and composition of the exsolved volatile phase that may co-exist with magmas in the crust. We distinguish between decompression-driven and crystallization-driven exsolution, and describe the implications of the volatiles for the dynamics of the magma reservoir, using key natural examples and case studies. We discuss eruptions triggered by second boiling, and the various regimes of magma mixing and magma overturn that may be induced by second boiling in a layered reservoir. We also explore the control of the volatile content of the magma on the mass erupted during an eruption episode, and compare our models to eruption datasets. We then turn to the mechanisms for magma-volatile separation, noting that in crystal-poor melts convective separation of exsolved volatiles may dominate while in crystal-rich melts, volatiles may generate channels or permeable-flow pathways through the crystal mush, thereby separating from the parent magma. We discuss the implications of the accumulation of the exsolved volatile phase at the roof zones of crystal-rich reservoirs for the large gas emissions observed during explosive eruptions, and for the development of metal-rich porphyry deposits

Linear estimation of flux sensitivity to uncertainty in porous media

We derive an integral expression for the flux of a single-phase fluid through a porous medium with prescribed boundary conditions. Taking variations with respect to the parameters of a given permeability model yields an integral expression for the sensitivity of the flux. We then extend the method to consider linear changes in permeability. This yields a linearised flux expression which is independent of changes in the pressure field that result from the changes in the permeability. For demonstration purposes, we first consider an idealised layered porous medium with a point source and point sink. We show how the effects of changes in permeability are affected by the position of the source and sink relative to the layered structure as well as the layer height and orientation of the layered structure. The results demonstrate that, even in a simple porous system, flux estimates are sensitive to the way in which the permeability is represented. We derive relationships between the statistical moments of the flux and of the permeability parameters which are modelled as random variables. This allows us to estimate the number of permeability parameters that should be varied in a fully nonlinear calculation to determine the variance of the flux. We demonstrate application of the methods to permeability fields generated through fast Fourier transform and kriging methods. We show that the linear estimates for the variability in flux show good agreement with fully nonlinear calculations for sufficiently small standard deviations in the underlying permeability.The support of an Industrial CASE studentship from EPSRC and BP for A. J. Evans entitled "Optimal Reservoir Modelling" is gratefully acknowledged.This is the accepted manuscript. The final version is available from CUP at http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=9603056&fileId=S0022112015001020

The mixing of airborne contaminants by the repeated passage of people along a corridor

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