The first section of the model generates the initial droplet distribution used for the simulation for a custom set of operational conditions. Part of this section overlaps with the description in the companion paper by . Measurements of air-blast-generated droplet distributions suggest that droplets emerge according to a log-normal diameter distribution: 1pdp=1dpσ2πexp⁡-ln⁡dp-μ22σ2, where p(dp) is the normalized probability distribution function of droplets of size dp. Log-normal distributions are controlled by the parameters μ and σ, which are the distribution mode and width, respectively, in logarithmic space. A related metric is the Sauter mean diameter (SMD), D32, which represents the mean ratio of volume to area of the droplets. Given an assumed log-normal distribution, the SMD is related to the log-normal parameter μ via : 2D32=exp⁡μ+32σ2.

Investigations in the companion paper by lay out the procedure for experimental measurements of the incoming spray particle size distribution and the final results on particle number distribution downstream of the nebulizer. The study obtained measurements of the mean incoming droplet size distribution using a combination of phase Doppler particle analysis (PDPA) and aerosol aerodynamic classification (AAC). Scaling of the measurements for solvent fluids with different properties relies on previous measurements of the Sauter mean diameter (SMD) D32 for similar air blast nebulizers by . Log-normal distributions measured by for twin-fluid concentric pneumatic nebulizers similar to the current one were used as a validated baseline for a range of compounds whose physical properties are well within the range of industrially relevant solvents. The particular values of μ and σ for the validation case were determined experimentally in the companion study . The same authors offered an equation linking the D32 of the initial droplet distribution to the solvent properties as follows: 3D32DL=C11+mrbgDL1/2ρLρgRebg1/41WeDL⋅1+C2DLbg1/6RebgρL/ρg1/12WeDL1/6OhDL2/3, where DL is the liquid orifice diameter; mr is the ratio of liquid to gas flow; bg is taken as the thickness of the atomizer gas annulus ; and ρL and ρg are densities of the liquid and gas, respectively. The non-dimensional numbers Rebg are based on the atomizer gas annulus and properties, We_DL is the Weber number based on the liquid properties and nozzle diameter, and OhDL is the Ohnesorge number based on liquid properties and jet diameter. The constants C1 and C2 were experimentally determined in the same study to be 1.734 and 1.0, respectively.

Once D32 is obtained for the particular fluid via Eq. (), the inlet probability distribution function (pdf) is obtained via Eq. (). The final droplet number concentration N is obtained via conservation of liquid mass flow rate for the droplets: 4QL=QgN∫0∞p(dp)π6dp3d(dp).

The initial velocity of the droplets at the outlet of the atomizer is assumed to be the same as that of the surrounding gas, which is the local speed of sound in the choked flow, with a velocity of ug = 318 m s⁻¹ at the relevant nebulizer operating conditions. Particles start slowing down immediately due to drag owing to the surrounding gas. The value for ug in the atomization chamber was obtained from CFD simulations (see Sect. ) across the whole ELSD from atomizer tip to optical detector and was taken to be represented by the value of the velocity at the centreline of the cross-section.

The equation of motion for a spherical particle of diameter dp moving along the duct with a velocity of up = dxpdt and surrounded by gas at a velocity ug is given as follows: 5dxpdt=up6dupdt=34Cddρgρpug-upug-up, where the drag coefficient Cd = Fd12ρgup-ug2=Cd*Cc is extracted from theory and experiments as 7Cd*=24Re,forRe≤124Re1+Re2/36,for1≥Re≥10000.424,forRe≥1000. Here the Reynolds number is based on the diameter of the particle, dp, and the relative velocity between particle and gas, up-ug. The relative velocity of the gas varies along the different sections, from very high at the inlet where droplets emerge to zero near the final evaporator section when small particles move at the same velocity as the gas.

The coefficient Cc is unity for particles much larger than the mean free path of the gas in the flow, λ, which is typically of the order of tens of nanometres. The correction is based on the Knudsen number, Kn = 2λ/d, 8Kn=2λd, and is given as 9CcKn=1+Kn1.257+0.4exp⁡-1.1Kn.

The mean free path can be calculated as 10λ=kBT2pξ, where kB is the Boltzmann constant, T is the gas temperature, p is the pressure, and ξ is the effective cross-sectional area of the gas particle. For nitrogen at 1 bar, λ = 34.5 nm.

There are two sections where droplets of sufficient inertia may hit the walls and cease to be transported with the flow, namely the ELSD Y piece or diffuser cartridge. The transmission efficiency of the particles across an obstacle is expressed as a function of the ability of the flow to provide sufficient drag relative to particle inertia to allow the particle to follow the flow as it travels across the regions where the flow is forced through a corner. This is expressed as the Stokes number, Stk, which is the ratio of the characteristic particle stopping time to that of the flow time . A solution of Eq. () for Re < 1 yields a characteristic stopping time τs = CCρpdp218μg, leading to a Stokes number equal to 11Stk=CCρpd218μgugL, where τg = L/ug is the characteristic flow time. Particles with motion characterized by Stk lower than unity follow the flow streamlines around obstructions, whereas they otherwise deviate from the gas flow streamlines and may impact obstacles. For particles with a Reynolds number larger than unity, there is no analytical solution to the stopping time, but the approximate correlations below based on Stokes number are still routinely used, with appropriate validation.

A collection efficiency Ceff(Stk) is defined as the fraction of total particle concentration of a given diameter (and thus Stokes number) in a flow lost to impaction onto an obstacle. For each impaction element, the ratio of the transmitted particle number concentration fraction to the incoming fraction is thus given as follows: 12NoutNin=1-Ceff(Stk). The collection efficiency is often represented using the following expression : 13Ceff(Stk)=12erfWStk-Stk50+1, where erf is the error function, W is a non-dimensional parameter characterizing the width of the function in Stokes number space, and Stk₅₀ is the Stk value which gives a collection efficiency of 50 %. The values of these two parameters depend on the type of obstacle and are discussed further on. An example of the change in the particle concentrations and efficiency as a function of droplet diameter is shown in Fig. .

In this study, collection efficiency curves were determined through a detailed investigation via CFD simulations of droplets in flows (see Sect. ) and were fitted according to Eq. (). Simulations were carried out for scenarios involving the Y piece and the diffuser cartridge, both of which act as impaction surfaces within the ELSD.

The output from Module 2 described in the previous section is used as the input distribution for the evaporator tube section of the ELSD in Module 3. The model solves the equations for evaporation for a given diameter bin of a given size. The calculations assume that the original number concentration of particles remains constant during the evaporation process as the corresponding droplet diameter decreases over the convective time. Number concentrations can change by collision (which is neglected for the presently dilute mixture) or impaction (which is active only across the obstructions described in the previous section) and are considered to be negligible in the evaporation tube.

The following assumptions are made: (a) the temperature along the duct is constant. This is justified as the mixture is dilute, the tube walls operate with a feedback loop for constant temperature, and evaporation does not consume sufficient energy to change the temperature of the bath gas significantly. (b) The rate of evaporation is sufficiently low so that the vapour pressure of the evaporating droplet does not change the vapour pressure of the bulk mixture. The latter is assumed to correspond to its upper bound throughout the domain, corresponding to the molar fraction attained if all droplets did vapourize completely. Since the mixture is very dilute, this means there is very little change in bulk vapour pressure. (c) The vapour pressure at the surface of the evaporating material is assumed to be associated with the temperature of the droplet, which is assumed to remain constant during the process.

Once the evaporation process is concluded, the final reconstructed evaporated aerosol distribution n(dp) is used in the final Module to determine the signal for light scattering by convolving the number distribution with the calculated intensity of scattered light for spherical particles of a given diameter.

describe the equations for the fraction of monochromatic coherent light intensity scattered by a single spherical particle as a function of scattering angle as follows: 34I(θ,dp)I0=λ28π2R2i1(d,θ)+i2(dp,θ), where I(θ) is the scattered intensity as a function of angle from the incoming beam, I0 is the incident intensity, R is the distance of the laser source to the scattering particle, and dp is the particle diameter. The functions i1 and i2 are the intensities of vertically and horizontally scattered light, respectively, and involve complex expressions ; they are coded in MATLAB in . The expressions also require the wavelength λ and the index of refraction nr of the particle material.

The final convolution of the distributions of diameter with intensity yields the expected signal as a function of angle, given as follows: 35S(θ)=N∫0∞I(θ,dp)p(dp)d(dp), where, again, N is the total number of particles, p(d) is the probability distribution function at the detector location, and θ is the collection angle. The signal is assumed to be proportional (except for a calibration constant) to the signal obtained at the detector.

The model was implemented in MATLAB using the ode15s method for the integration of governing Eqs. ()–(). As explained briefly above in Sect. , Eq. () is solved until the total droplet mass is lower than or equal to the mass of the solute, (md ≤ msol); this is done by setting an event function which stops the integration at this point. From this point on, the system is solved for the vaporization of the analyte only.

Details of the experimental measurements are provided in the companion study and are summarized below. An ELSD (Agilent Technologies; 1290 Infinity II ELSD) recorded 10 µL injections of analytes with a total volumetric gas and liquid flow rate of 1.9 L min⁻¹ and 0.5 mL min⁻¹, respectively. The gas used was dry nitrogen gas (BOC Ltd.), and the liquid was fed via the pump in a GPC unit (Agilent Technologies; 1260 Infinity II GPC). ELSD experiments were conducted with evaporator temperatures set to 25, 40, 60, 80, and 100 °C. One set of experiments was conducted using water as a solvent, with various analyte species, including ethylene glycol (Alfa Aesar, Thermo Fisher Scientific), glycerol (Scientific Laboratory Supplies Ltd.) and citric acid (Alfa Aesar, Thermo Fisher Scientific), and four different analyte concentrations (0.125, 0.25, 0.5 and 1 g L⁻¹). Another set of experiments kept the analyte species (caffeine) and concentration constant at 0.25 g L⁻¹ but varied the solvent choice (water (HPLC-grade, > 99.9 %), acetone (> 99.8 %), iso-propyl alcohol (> 99.5 %) and methanol (> 99.5 %) (Merck Life Science UK Ltd.). After mixing the analytes in their respective solvents, all solutions were sonicated in a water bath held at 25 °C for 10 min to ensure total dissolution.

In order to compare simulated final particle size distributions with experimental values, a scanning mobility particle sizer (SPMS, TSI Instruments Ltd.) was used to sample particles downstream of the ELSD outlet. The SMPS was run with a negative polarity such that positively charged particles were selected, with an aerosol flow of 1.5 L min⁻¹ and a sheath flow 15 L min⁻¹. Particle sizes were measured over 90 s scans across the explored size range of 8 to 232 nm. In this study, we investigate the comparison between theory and experiment for one particular analyte, namely for 1 g L⁻¹ aqueous citric acid. The choice was made because, among the analytes considered (dextran, caffeine, citric acid), citric acid has a known bulk density (1680 kg m⁻³, ) and also produces dried particles of spherical shape. These conditions reduce the uncertainty in the comparisons of particle aerodynamic and mobility diameters and in the relationship between mass and volume, which are used in the model.

Figure a shows CFD results of the velocity magnitude across the central cross-section of the flow path. Velocities are highest near the sonic atomizer, and the diffuser restriction along the evaporator shows reasonably uniform values across the radius except near the constrictions. Path lines show the circulation pattern, with most of the particles following through to the outlet (Fig. b). Additional details of simulations are shown in Appendix H.

The gas-phase solutions were used to solve for droplet trajectories and impingement fraction as a function of droplet diameter. Non-evaporating droplet spheres were introduced using the DPM model for monodisperse particle distributions, with the explicit purpose of determining the collection efficiency of the Y piece, which serves as a large droplet trap. Figure  illustrates path lines followed by particles 1 µm in diameter, coloured by particle residence times, where flow is from right to left. Particles start from the injector and impinge on the Y piece. Their lifetime in the flow path is also shortened by the labyrinth diffuser until they reach the straight section of the evaporator and eventually make their way to the outlet.

Simulations using the DPM were performed for a range of particle diameters, from 400 nm to 20 µm. These results allowed for the construction of a collection efficiency curve, defined as the fraction of particles removed by collision with the walls (Fig. ). The symbols in Fig.  show calculated collection efficiencies for the Y piece (blue) and labyrinth diffuser (red). The collection efficiency increases with droplet diameter as droplets with higher inertia fail to follow the bend in the flow and impinge onto the obstructions. The results can be represented as a function of the square root of the Stokes number in Fig. b, which shows that both obstructions review similar behaviour. Nevertheless, there is significant scatter in the simulation results for smaller particles at low Stk. This is a result of incomplete convergence of the statistical results for the small particles, which can be trapped in a variety of recirculation zones. The physics of the problem dictates that sufficiently small particles should follow the flow, and so we neglect CFD DPM calculations for values of Stk below 1 µm and use an error function fit suggested in . The fit results show a universal curve for either obstruction, with parameter W = 1 and Stk50 = 0.49.

Final particle size distributions for different initial concentrations are shown in Fig. a, showing how the final size depends on the original concentrations, approximately as expected from conservation of mass for a fixed number of particles: the larger the concentration, the larger the remaining mass of analyte and, thus, the larger the diameter expected after drying. The differential scattering signal is shown for different initial analyte concentrations: the signal scales with a large power (of the order 3–4) of the diameter of the particle. Therefore, most of the scattering signal originates from the larger particle diameters, as explained in Sect. . Since larger concentrations lead to larger mean diameters, the overall signal to the total scattered signal increases with the concentration of analyte, first slowly and then rising with concentration, as shown in Fig. b.

Experimental results obtained in provide both validation for the input droplet distributions and also the corresponding measured total number and size distribution of dried particles emerging from the outlet of the ELSD. These values can be compared to the simulated results emerging from applying the simulations in the previous sections.

Figure  shows measured and simulated particle size distributions under normalized and non-normalized conditions. The model is able to capture the large decrease in particle diameter from around micrometre size (Fig. ) down to 100 nm particles, where the mode of the dried particles is 40 nm, and the peak number is about 30 % lower than the predicted value. The higher particle number suggests that impingement losses may be underestimated, and the higher predicted diameter suggests that the rate of evaporation is underestimated. The predicted overall shape and width of the distribution, however, appear to be in very good agreement with the prediction once the peak concentration is normalized Fig.  (right).

In spite of the discrepancy in absolute values, the key outcome in the present work is to test the model in its ability to reproduce the sensitivities of the ELSD signal to changing variables. This is addressed in the next section.

The experimental ELSD signal response results are compared against simulations using solutes with low, middle and high degrees of volatility in water. Measurement results for the detector signal are shown in the top row of Fig. , along with normalized simulations in the bottom row, for three different analytes and five different evaporator temperatures. The simulations are able to correctly capture dependence of the signal on analyte concentration and volatility, predicting negligible signal for the most volatile analyte considered (ethylene glycol), an increasing signal with increasing analyte concentration for both semi-volatile (glycerol) and non-volatile (citric acid) analytes. However, the model predicts no dependence of the signal on temperature for the latter case, whilst the experiments show a change in signal with temperature in the opposite direction as compared to the semi-volatile glycerol.

In order to further challenge the model, experiments were produced using solvents of different volatility while keeping the analyte (caffeine) and its concentration unchanged. The comparison (Fig. ) shows that the normalized signals are well captured for the different fluids with respect to the relevant atomization properties.