Numerical simulation of dynamics of premixed flames: flame instability and vortex–flame interaction

The characteristics of cellular flames generated by intrinsic instability has been studied using two-dimensional (2-D) and three-dimensional (3-D) unsteady calculations of reactive flows, based on the compressible Navier–Stokes equation. Three basic types of phenomena, responsible for the intrinsic instability of premixed flames, are examined here, i.e. hydrodynamic, body-force and diffusive-thermal effects. Cellular flames are generated by these effects, and their characteristics—cell size, cell depth, flame-surface area, and flame velocity—depend on the adiabatic flame temperature, acceleration, and Lewis number. As intrinsic instability becomes stronger, the flame–surface area and flame velocity of cellular flames increase, and the behavior of cellular-flame fronts becomes unstable. The increment in the flame–surface area and the flame velocity of 3-D cellular flames is about twice that of 2-D cellular flames. This is due to the difference in the disposition of cells between 2-D and 3-D flames. Moreover, the flame velocity of cellular flames depends strongly on the length of computational domain in the direction tangential to the flame surface. As the length of computational domain becomes larger, the flame velocity increases. This is because the long-wavelength components of disturbances play an important role in the shape of cellular flames, i.e. the flame–surface area. Next, colliding interaction of a vortex pair with a premixed flame has been numerically studied in order to understand how the vortex affects the flame and how the flame affects the vortex. Three types of interacting behavior appear, depending on the ratio of the maximum circumferential velocity of the vortex to the burning velocity of the flame. The temporal evolution of the curvature, the strain rate, the stretch rate at the stagnation point, and that of the flame surface area and the global burning velocity are also analyzed, with different Lewis numbers and vortex strengths. Flame propagation along a vortex core, i.e. vortex bursting, is also numerically studied in order to understand how the vortex affects the flame propagation and how the flame affects the vortex. It is shown that flame evolution along a fine vortex tube is related to the formation of the Azimuthal component of vorticity, which is produced by convection and stretch effects, and that the density ratio of the flame and the Reynolds number of the vortex affect the propagation velocity. Flame propagation in a rotating cylinder has been also reviewed.     

Laminar burning velocities and flame instabilities of diluted H2/CO/air mixtures under different hydrogen fractions

An experimental study was conducted using outwardly propagating flame to evaluate the laminar burning velocity and flame intrinsic instability of diluted H₂/CO/air mixtures. The laminar burning velocity of H₂/CO/air mixtures diluted with CO₂ and N₂ was measured at lean equivalence ratios with different dilution fractions and hydrogen fractions at 0.1 MPa; two fitting formulas are proposed to express the laminar burning velocity in our experimental scope. The flame instability was evaluated for diluted H₂/CO/air mixtures under different hydrogen fractions at 0.3 MPa and room temperature. As the H₂ fraction in H₂/CO mixtures was more than 50%, the flame became more unstable with the decrease in equivalence ratio; however, the flame became more stable with the decrease in equivalence ratio when the hydrogen fraction was low. The flame instability of 70%H₂/30%CO premixed flames hardly changed with increasing dilution fraction. However, the flames became more stable with increasing dilution fraction for 30%H₂/70%CO premixed flames. The variation in cellular instability was analyzed, and the effects of hydrogen fraction, equivalence ratio, and dilution fraction on diffusive-thermal and hydrodynamic instabilities were discussed.     

Onset of cellular instability in adiabatic H2/O2/N2 premixed flames anchored to a flat-flame heat-flux burner

The onset of cellular instability in adiabatic H₂/O₂/N₂ premixed flames anchored to a heat-flux burner is investigated numerically. Both hydrodynamic instability and diffusional-thermal instability are shown to play an important role in the onset of cellular flames. The burner can effectively suppress cellular instability when the flames are close to the burner, otherwise the burner can suppress the instabilities only at large wavenumbers. Because of differential diffusion, local extinction can occur in lean H₂/O₂/N₂ flames. When the flames develop to take on cellular shapes, the surface length, the overall heat release rate and the mean burning velocity are all increased. For near stoichiometric fuel-rich flames the mean burning velocity can increase by as much as 20%–30%. For lean flames with an equivalence ratio of 0.56, the mean burning velocity can be 2–3 times of the burning velocity of the corresponding planar flame.     

Diffusive-thermal instability of premixed tubular flames

A systematic study of the diffusive-thermal instability of premixed tubular flames is carried out. The problem becomes amenable to a complete analysis in the frame of the diffusive-thermal approximation when a simplified flow field and the flame-sheet combustion model are used. The dispersion relation determining the growth rate of instability is obtained in an analytical form which therefore analyzed numerically. Stability diagrams showing stable and unstable states, in particular those manifesting the cellular flames, are presented.     

Physical Insight and Modelling for Lewis Number Effects on Turbulent Heat and Mass Transport in Turbulent Premixed Flames

The effects of global Lewis number on the behavior of Reynolds heat and mass fluxes in turbulent premixed flames are studied based on three-dimensional direct numerical simulation (DNS) of a number of statistically planar turbulent premixed flames with a global Lewis number ranging from Le = 0.34 to 1.2. For the same values of initial turbulent flow field parameters and duration of flame-turbulence interaction, it has been found that both Reynolds heat and mass fluxes may exhibit countergradient transport for flames with a Lewis number significantly smaller than unity; whereas predominantly gradient-type transport is obtained for flames with a Lewis number closer to unity. It is demonstrated that strong flame normal acceleration due to greater heat release in the low Lewis number flames acts to promote countergradient transport, and that the magnitude of the flame normal acceleration decreases with increasing Lewis number. Algebraic models for Reynolds heat and mass fluxes are proposed in which the effects of the Lewis number on flame normal acceleration are explicitly taken into account. The predictions of the new models are compared with DNS data, and the models are found to capture the influence of the Lewis number on turbulent scalar flux in a satisfactory manner for all the flames considered in this study.     

Effects of the external turbulence on centrally-ignited spherical unstable Ch4/H2/air flames in the constant-volume combustion bomb

In the present study, we conducted experiments to investigate the effects of external turbulence on the development of spherical H₂/CH₄/air unstable flames developments at two different equivalence ratios associated with different turbulent intensities using a spherical constant-volume turbulent combustion bomb and high speed schlieren photography technology. Flame front morphology and acceleration process were recorded and different effects of weak external turbulent flow field and intrinsic flame instability on the unstable flame propagation were compared. Results showed the external turbulence has a great influence on the unstable flame propagation under rich fuel conditions. For fuel-lean premixed flames, however, the effects of external turbulence on the morphology of the cellular structure on the flame front was not that obvious. Critical radius decreased firstly and then kept almost unchanged with the augment of the turbulence intensity. This indicated the dominating inhibiting effect of flame stretch on the turbulent premixed flame at the initial stage of the flame front development. Beyond the critical radius, the acceleration exponent was found increasing with the enhancement of initial turbulence intensity for fuel-lean premixed flames. For fuel-rich conditions, however, the initial turbulence intensity had little effect on acceleration exponent. In order to evaluate the important impact of the intrinsic flame instability and external turbulent flow field for spherical propagating premixed flames, intrinsic flame instability scale and average diameter of vortex tube were calculated. Intrinsic flame instability scale decreased greatly and then stayed unchanged with the propagation of the flame front. The comparison between intrinsic flame instability scale and average diameter of vortex tube demonstrated that the external turbulent flow filed will be more important for the evolution of wrinkle structure in the final stage of the flame propagation, when the turbulence intensity was more than 0.404 m/s.     

Effect of the initial pressures on evolution of intrinsically unstable hydrogen/air premixed flame fronts

Laminar premixed flame front may be wrinkled due to the hydrodynamic and diffusive-thermal instabilities. This may lead to the occurrence of the cellular structure and the self-acceleration. The lean unstable hydrogen/air premixed flame at various initial pressures are studied to clarify the effect of the initial pressure on the evolution of the unstable laminar flame. Linear and nonlinear development stages of the unstable flame are simulated and investigated separately. In the linear stage, the initial sinusoidal wave disturbance on the flame front will still keep its initial configuration. The growth rate increases firstly and then decreases with the increase of the wavenumbers. The effect of the self-acceleration on the unstable flame front will be stronger in the linear stage at the higher initial pressure, since there are larger thermal expansion and constant Lewis number for hydrogen/air premixed flame at higher pressure. There are little discrepancies for the calculated growth rates with those predicted by the revised dispersion relation. The nonlinear stage of the unstable flame propagation could be divided into two stages, the transitional and the stable nonlinear stages. In the transitional stage, the flame front cells splits, merges and moves all the time and the initial wavenumber has a great influence on the cell evolution process. With the evolution of the cell on the flame front, the cellular structure on the flame front will not change greatly with the initial wavenumbers in the stable nonlinear stage. The effect of self-acceleration due to the wrinkling of the flame front at this stage is weakened with the increase of the initial pressure. At the higher pressure, more wrinkled structures with smaller mean curvature are distributed on the flame front. At last, results show that the flame front will propagate faster for the larger computation domain. Based on the fractal theory, the fractal dimension of lean hydrogen/air premixed flame with the equivalence ratio of 0.6 at 0.5 MPa in the 2D domain is obtained and around 1.26.     

Research on cellular instabilities in outwardly propagating spherical hydrogen-air flames

Experiments were conducted in a closed vessel using Schlieren system to study the cellular instabilities of hydrogen-air premixed flames at different equivalence ratios (from 0.6 to 2.5), initial temperatures (from 300 K to 450 K), and initial pressures (from 0.1 MPa to 0.3 MPa). The cellular instabilities of hydrogen-air flames were interpreted and evaluated in the viewpoint of the diffusive-thermal and hydrodynamic instabilities. Also, critical flame radius and critical Peclet number were measured. The results showed that for lean hydrogen-air flames, the cellular instabilities are dominantly influenced by the diffusive-thermal instability; for stoichiometric and rich hydrogen-air flames, the cellular instabilities are just influenced by the hydrodynamic instability. Critical flame radius is increased with the increase of equivalence ratio and/or initial temperature, and is decreased with the increase of initial pressure. Critical Peclet number is increased with the increase of equivalence ratio, and is insensitive to initial temperature and initial pressure.     

Asymptotic structure of premixed flames for a simple chain-branching chemistry model with finite activation energy near the flammability limit

The structure of freely propagating, planar and adiabatic premixed flames is investigated for a simple two-step chain-branching model with finite activation energy within the diffusive-thermal pproximation. This article is focused on the flame structure near the flammability limit where the phenomenon of appreciable fuel leakage takes place.The asymptotic analysis was carried out with the help of the method of matched asymptotic expansions using a large parameter inversely proportional to the square of the flame propagation velocity S_L. Two terms of the corresponding expansions were obtained. The leading term provided the flammability limit condition in analytical form. The existence of multiple solutions for flames with fuel Lewis number larger than one was demonstrated analytically. The results of the asymptotic analysis were successfully compared with numerical computations.     

Extinction of a premixed flame by a large variation in axial velocity

Unsteady flame propagation in a tube is examined by introducing a mean velocity variation larger than the burning velocity to a stabilized flame for a period longer than the reaction time scale. In our previous work, stabilized propane-air flames were classified as either one-dimensional or two-dimensional flames. The eventual extinction during the velocity increase was categorized as either acoustic extinction or boundary layer extinction. In this work, the effects of a nonunity Lewis number were estimated through experiments with a methane-air flame; the eventual extinction during the velocity decrease was investigated in more detail; and the growth of the extinction boundary layer was analyzed with a transient one-dimensional model of the flame stretch. In our experiments, the Lewis number did not affect the existence or characteristics of the critical velocity and the characteristic time for boundary layer extinction. An additional critical velocity was found, however, for acoustic extinction when the Lewis number was smaller than unity. In the transient one-dimensional model, the velocity transition along the flame was calculated with a continuity equation and an axial momentum equation. The spatial gradient of the burning velocity and the extinction criterion were simplified with the experimental results and some theoretical studies. The analysis shows that the unsteady flame stretch at the flame edge during a large axial velocity variation is the prevailing cause of the growth of the extinction boundary layer. These results provide some evidence that flame stretch affects the behavior of the flame edge; they also suggest the cause of the finger flame. The findings help explain the unsteady behavior of premixed flames near a flammability limit.     

Burning rates and surface characteristics of hydrogen-enriched turbulent lean premixed methane–air flames

The burning rates and surface characteristics of hydrogen-enriched turbulent lean premixed methane–air flames were experimentally studied by laser tomography visualization method using a V-shaped flame configuration. Turbulent burning velocity was measured and the variation of flame surface characteristics due to hydrogen addition was analyzed. The results show that hydrogen addition causes an increase in turbulent burning velocity for lean premixed CH₄–air mixtures when turbulent level in unburned mixture is not changed. Moreover, the increase of turbulent burning velocity is faster than that of the corresponding laminar burning velocity at constant equivalence ratio, suggesting that the kinetics effect is not the sole factor that results in the increase in turbulent burning velocity when hydrogen is added. The further analysis of flame surface characteristics and brush thickness indicates that hydrogen addition slightly decreases local flame surface density, but increases total flame surface area because of the increased flame brush thickness. The increase in flame brush thickness that results in the increase in total surface area may contribute to the faster increase in turbulent burning velocity, when hydrogen is added. Besides, the stretched local laminar burning velocity may be enhanced with the addition of hydrogen, which may also contribute to the faster increase rate of turbulent burning velocity. Both the variation in flame brush thickness and the enhancement in stretched local laminar burning velocity are due to the decreased fuel Lewis number when hydrogen is added. Therefore, the effects of fuel Lewis number and stretch should be taken into account in correlating burning velocity of turbulent premixed flames.     

Nonlinear dynamical behaviour of intrinsic thermal-diffusive oscillations of laminar flames with varying premixedness

This study focuses on the thermal-diffusive oscillations of the full spectrum of co-flow laminar flames—from premixed to non-premixed through partially premixed flames for Lewis numbers sufficiently greater than unity. A premixedness parameter is identified using the analytical solution to the mixing field downstream of a splitter plate. Different premixedness levels ranging from zero (fully non-premixed) to unity (fully premixed) are then used as Dirichlet data for inlet concentrations of the reactants to the flame. The initial value problem is numerically solved by adopting a thermo-diffusive model and single-step finite rate chemical reaction and using higher-order compact schemes. A thumb-shaped region in the Damköhler number – premixedness parameter space is identified as corresponding to limit cycle oscillations of the flame. This fills the gap between what is reported for the pulsating instability of non-premixed edge flames recently and fully premixed flames earlier. The thumb-shaped region enlarges with increase in the Lewis number in the range of 1.5–1.7, with marked increase in the oscillatory amplitude of the total heat release. A subcritical Hopf bifurcation is seen to occur at the boundary of this region in the above parametric space. The mechanism of oscillations is studied: hysteresis is found between the upstream and downstream propagation of the flame due to the thermal-diffusive imbalance. The oscillations are plotted in the phase space of the domain-integrated mass and heat diffusive fluxes of and to the reactants. The area of the limit cycle peaks at a particular Damköhler number and premixedness parameter and drops to zero on both sides in this parametric space.     

Experimental study on premixed combustion of spherically propagating methanol-air-nitrogen flames

The outward propagation and development of surface instability of the spark-ignited spherical premixed flames for methanol-air-nitrogen mixtures were experimentally studied by using a constant volume combustion chamber and a high-speed schlieren photography system. The laminar burning velocities, the mass burning fluxes, and the Markstein lengths were obtained at different equivalence ratios, dilution ratios, initial temperatures, and pressures. The laminar burning velocities and the mass burning fluxes give a similar curve versus the equivalence ratios. They increase with the increase of initial temperature and decrease with the increase of dilution ratio. The laminar burning velocity decreases with elevating the initial pressure, while the mass burning flux increases with the increase of the initial pressure. Markstein length decreases slightly with the increase of initial temperature for the rich mixtures. High initial pressure corresponds to low Markstein length. Markstein length increases with the increase of dilution ratio, which is more obvious when the mixture becomes leaner. Equivalence ratio has a slight impact on the development of the diffusive-thermal cellular structure at elevated initial pressures. The initial pressure has a significant influence on the occurrence of the flame front cellular structure. At the elevated pressures, the cracks on the flame surface branch and develop into the cell structure. These cells are bounded by cracks emitting a bright light, which may indicate soot formation. For very lean mixture combustion, the buoyancy effect and cooling effect from the spark electrodes have a significant impact on the flame propagation. The hydrodynamic instability, inhibited with the increase of initial temperature around the stoichiometric equivalence ratio, is enhanced with the increase of initial pressure and suppressed by mixture dilution.     

Experimental study on premixed combustion of spherically propagating methanol-air-nitrogen flames

The outward propagation and development of surface instability of the spark-ignited spherical premixed flames for methanol-air-nitrogen mixtures were experimentally studied by using a constant volume combustion chamber and a high-speed schlieren photography system. The laminar burning velocities, the mass burning fluxes, and the Markstein lengths were obtained at different equivalence ratios, dilution ratios, initial temperatures, and pressures. The laminar burning velocities and the mass burning fluxes give a similar curve versus the equivalence ratios. They increase with the increase of initial temperature and decrease with the increase of dilution ratio. The laminar burning velocity decreases with elevating the initial pressure, while the mass burning flux increases with the increase of the initial pressure. Markstein length decreases slightly with the increase of initial temperature for the rich mixtures. High initial pressure corresponds to low Markstein length. Markstein length increases with the increase of dilution ratio, which is more obvious when the mixture becomes leaner. Equivalence ratio has a slight impact on the development of the diffusive-thermal cellular structure at elevated initial pressures. The initial pressure has a significant influence on the occurrence of the flame front cellular structure. At the elevated pressures, the cracks on the flame surface branch and develop into the cell structure. These cells are bounded by cracks emitting a bright light, which may indicate soot formation. For very lean mixture combustion, the buoyancy effect and cooling effect from the spark electrodes have a significant impact on the flame propagation. The hydrodynamic instability, inhibited with the increase of initial temperature around the stoichiometric equivalence ratio, is enhanced with the increase of initial pressure and suppressed by mixture dilution.     

A parametric study of sound generation by premixed laminar flame annihilation

A numerical and theoretical study of sound generation by planar, axisymmetric and spherically symmetric premixed laminar flame annihilation is presented in this paper. The compressible Navier–Stokes, energy, and progress variable equations are solved using Direct Numerical Simulation with one-step chemistry. A theory is developed to relate the pressure amplitude to the flame’s propagation velocity and consumption speed, which were identified as key parameters in the generation of sound in our previous study (Talei et al., 2011[38]). The results show that by obtaining the propagation velocity of the flame from the simulations, the sound generation can be predicted accurately for unity Lewis number. Use of Markstein’s linear theory relating flame speed and curvature is also investigated as a more practical way of applying this theory. It is shown that this leads to under-prediction of the radiated sound in the spherical and axisymmetric configurations, but it is able to capture the qualitative trends.The effects of laminar flame speed, temperature ratio and Zel’dovich number are then investigated. These are interpreted in terms of a scaling argument put forward in our previous study, and also in terms of the flame’s propagation velocity and consumption speed. Finally, cases involving non-unity Lewis number are investigated. As the flames approach annihilation, either flame acceleration or deceleration is observed depending on the Lewis number, and this is shown to affect the radiated sound significantly. For Lewis numbers greater than unity, the annihilation of the flame results in a local increase in consumption speed at annihilation, leading to more sound production compared with that of unity Lewis number. Lewis numbers less than unity exhibit the opposite behaviour.     

Pulsating and hydrodynamic instabilities at large Lewis numbers

The dynamic behavior of freely propagating premixed flames with large Lewis numbers was computationally simulated using a sixth-order central difference scheme and nonreflective boundary conditions. Results in the linear stage of the instability growth show that the growth rate dramatically decreases with increasing Lewis number and that the large activation energy excites the pulsating instability and increases the growth rate of the hydrodynamic instability. In the nonlinear growth stage, there exist regimes of stable cell propagation, periodic pulsating cellular flames, and irregular pulsating cellular flames as the activation energy is increased. Characteristics of these regimes were further studied for the effects of Lewis number on the flame front structure in the regime of stable cell propagation, the effects of flame pulsation on the flow and flame cell structure in the regime of periodic pulsating cellular flame, and the complex pattern formation in the regime of irregular pulsating cellular flame. It is further demonstrated that unsteady pulsating flames can propagate faster than the adiabatic flame when the local stretch rate is positive, implying that models based on quasi-steady flame propagation may not correctly predict the behavior of unsteady flames with large Lewis numbers.     

Lewis number effect on the propagation of premixed flames in narrow adiabatic channels: Symmetric and non-symmetric flames and their linear stability analysis

The propagation of premixed flames with different Lewis numbers in a planar channel subject to a Poiseuille flow is considered within the diffusive-thermal model for steady and time-dependent cases. It was found that, depending on the Lewis number and the flow rate, symmetric and non-symmetric flames are possible. The existence of multiple steady solutions in cases of the low Lewis number is demonstrated. The time-dependent simulations carried out for high Lewis number flames also showed the symmetric and non-symmetric oscillatory solutions.Linear stability analysis of two-dimensional steady-states was performed using a practical method developed in the paper and applied to calculate the main eigenvalue. It was shown that for symmetric flames with a low Lewis number the increase in the flow rate leads to a loss of stability with subsequent formation of non-symmetric solutions. For flames with a high Lewis number the Poiseuille flow produces a stabilization effect. The results of the stability analysis were successfully compared with the results of direct numerical simulations.     

The asymptotic structure of premixed tubular flames

The steady isobaric combustion of premixed tubular flames undergoing a direct one-step irreversible Arrhenius-type exothermic global reaction with a constant but general Lewis number is studied in the physically interesting limit of large activation energy. This analysis applies the combustion approximation and differs from previous asymptotic analyses of tubular flames by applying the Hirschfelder boundary condition at the burner exit for chemical species and the application of the delta-function closure scheme. The analysis yields a solution for flame sheet position, flame temperature, heat loss rate to the burner, temperature in the burned region, and stabilization limits as functions of the mass flow rate supplied to the burner and temperature of the burner surface in the near equidiffusional flame limit. Results predict the existence of dual flame behavior consistent with other investigations on the planar and cylindrical burner-stabilized flame. Two stabilization limits are identified, one for approaching flames, consistent with previous studies, and one for receding flames that has not been reported to date. Flame temperature profiles predict a nonmonotonic response, unique to the tubular flame. Consistent with the excess enthalpy of the tubular flame, the results demonstrate a strong dependence of a Lewis number differing from unity. Previous asymptotic analyses of tubular flames with a plug flow boundary condition for mass fraction are reanalyzed with the application of the delta-function closure scheme. Unlike previous results, the analysis predicts an interesting dual response for the temperature in the burned region.     

The influence of radiation on the flame propagation through micro organic dust particles with non-unity Lewis number

In this paper, the analytical study of effects of radiation and non-unity Lewis number on the laminar premixed flames of organic dust clouds has been done. The research is focused on a combustion model for premixed flames and the flame structure is composed of preheat-vaporization, narrow reaction and finally the post-flame zone. The normalized governing equations with help of boundary and matching conditions are solved by perturbation method. The results show that increasing equivalence ratio and decreasing Lewis number are resulted in the increase of flame temperature and burning velocity. For the sake of this model validation, fuel conversion is compared by published experimental data and shows an acceptable agreement.     

Heat release rate variations in high hydrogen content premixed syngas flames at elevated pressures: Effect of equivalence ratio

Three-dimensional direct numerical simulations with detailed chemistry were performed to investigate the effect of equivalence ratio on spatial variations of the heat release rate and flame markers of hydrogen/carbon monoxide syngas expanding spherical premixed flames under turbulent conditions at elevated pressures. The flame structures and the heat release rate were analysed and compared between fuel-lean, stoichiometric and fuel-rich centrally ignited spherical flames. The equivalence ratio changes the balance among thermo-diffusive effects, Darrieus–Landau instability and turbulence, leading to different flame dynamics and the heat release rate distribution, despite exhibiting similar cellular and wrinkling flames. The Darrieus–Landau instability is relatively insensitive to the equivalence ratio while the thermo-diffusive process is strongly affected by the equivalence ratio. As the thermo-diffusive effect increases as the equivalence ratio decreases, the fuel-lean flame is more unstable than the fuel-rich flame with the stoichiometric flame in between, under the joint effects of the thermo-diffusive instability and the Darrieus–Landau instability. The local heat release rate and curvature display a positive correlation for the lean flame, no correlation for the stoichiometric flame, and negative correlation for the rich flame. Furthermore, for the fuel-lean flame, the low and high heat release rate values are found in the negative and positive curvature zones, respectively, while for the fuel-rich flame, the opposite trends are found. It is found that heat release rate markers based on species concentrations vary strongly with changing equivalence ratio. The results suggest that the HCO, HO₂ concentrations and product of OH and CH₂O concentrations show good correlation with the local heat release rate for H₂/CO premixed syngas-air stoichiometric flame under turbulent conditions at elevated pressures.