1 Introduction

Selective catalytic reduction (SCR) is a process to reduce the emission of nitrogen oxide in combustion engines, whose relevance in the automotive sector has strongly increased during the last years. In this process, urea-water-solution (UWS, 32.5 wt% urea in water) is injected into the hot exhaust gas. It decomposes into ammonia and reacts in the presence of a suitable catalyst with nitrogen oxides to form molecular nitrogen. The underlying physical-chemical processes of the formation of ammonia include the evaporation of liquids, diffusion in gas and liquid phase, convective gas flow and chemical reactions, all of which are coupled. Thus, there is a need for efficient and accurate models which can reliably describe the complex multiphase flow as well as the chemical kinetics of the SCR process. At present, the model for the decomposition of urea is commonly separated into three main steps [16, 35]:

Evaporation of water from a UWS droplet:

$$ \text{CO}(\text{NH}_{2})_{2}\text{ (aq)} \to \text{CO}(\text{NH}_{2})_{2}\text{ (s or l)} + \text{H}_{2}\text{O (g)}. $$
(1)

Thermolysis of urea into ammonia (NH3) and isocyanic acid (HNCO):

$$ \text{CO}(\text{NH}_{2})_{2}\text{ (s or l)} \to \text{NH}_{3}\text{ (g)} + \text{HNCO}\text{ (g)}. $$
(2)

Hydrolysis of isocyanic acid:

$$ \text{HNCO (g)} + \text{H}_{2}\text{O (g)} \to \text{NH}_{3}\text{ (g)} + \text{CO}_{2}\text{ (g)}. $$
(3)

Due to the different properties of water and urea, the evaporation of a UWS droplet can be separated into two main phases of water evaporation and urea decomposition where each phase is preceded by a short heating phase of the droplet [32]. Experimental results show that both main phases are close to a D2 law progression as expected from an evaporation process [21, 22, 31, 32]. Beyond this similarity, the quantitative results of the experimental studies vary strongly which is most likely caused by differences in the experimental conditions [25]. For instance, different types of support fibres or natural convection inside the experimental vessel influence the droplet evaporation [6]. There are several different approaches to model the evaporation of UWS and to describe the decomposition of urea. van Helden et al. [30] added ammonia in a post-processing step where the concentration was estimated based on the water vapour concentration. Wurzenberger and Wanker [33] modelled the thermal decomposition of urea as a fast evaporation of UWS to water vapour and gaseous urea in a homogeneous system. Other studies assume [7, 10] that urea decomposition happens directly during the evaporation. Kim et al. [15] presented a simple model where the release of isocyanic acid is neglected. Within the context of selective non-catalytic reduction of NO x , which usually happens at a higher temperature, a fast decomposition of urea is assumed to happen after the water has evaporated [8]. Altogether, there is the choice between an evaporation model and a model based on an Arrhenius expression to describe the urea decomposition [5]. Abu-Ramadan et al. [1] simulated a single droplet considering the distribution of urea within the droplet and comparing both models. The best fit with the experimental data of Wang et al. [31] was obtained using an evaporation model. The direct decomposition using an Arrhenius equation leads to high droplet temperatures [4] that do not appear during urea decomposition in experiments [32]. Evaporation models also have an advantage because a vapour pressure curve to model the urea decomposition can be easily measured, fitted and implemented in CFD simulations [18].

For these reasons, mostly evaporation models have been used during the recent years. There is some uncertainty about the amount of gaseous urea emitted from a droplet or film and while the majority of publications assume gaseous urea to be unstable and only present in very low concentrations, some measurements depict a much higher amount [3]. Birkhold [4] described two different cases how an evaporation model can be implemented into a simulation. The major difference is that in one case, gaseous urea exists and decomposes in the gas phase and in the other case urea directly decomposes into ammonia and isocyanic acid on the interface so that no gaseous urea exists. If urea decomposes in the gas phase, the enthalpy of reaction can be applied at the evaporation of urea or during the following reaction. He assumed that both give comparable results. Furthermore, due to the use of droplet evaporation models in large-scale CFD simulations of the exhaust system, the gas phase close to the droplet surface cannot be resolved in detail without massively increasing the computational time and a simplification like a film model based on the 1/3 rule is frequently used [27].

The results of several numerical and experimental studies conducted for typical conditions found in the exhaust system can be used to simplify the evaporation model used in this work. Droplets in a UWS spray usually have a radius of only around 100 μm [5]. The terminal velocity for water droplets of this size is approximately 1 m/s which gives Reynolds numbers in air of around 1 [2, 13]. The density of UWS droplets is only approximately 10% higher compared to pure water so the results for water are applicable to UWS droplets. Using a typical relation for the Nusselt number and Sherwood number [12], only a limited increase of the heat and mass transfer of the droplet is expected and the convective flow around the droplet can be neglected in this study. In addition, it is shown that the D2 curves are similar for both a diffusion model in the liquid phase and a rapid mixing model, where no concentration gradients within the droplet are assumed [1]. Despite this, a diffusion model predicts significant concentration gradients within the droplet [25]. The gradients may lead to the formation of bubbles of gaseous water within the droplet [31] or the formation of a crust at the surface of the droplet [17]. Kontin et al. [17] also developed a theoretical model to estimate the influence of urea oversaturation and crystallization on the droplet life time and show that despite the crust having a massive influence at 298 K, it only affects the droplet lifetime by less than 2% above 473 K, so this effect can also be neglected.

The focus of the present work, however, is to compare the different evaporation models using detailed one-dimensional simulations of the gas phase and liquid phase instead of a film model. Given the aforementioned results and to keep the simulation as simple as possible, the influence of a convective flow around the droplet is neglected and no oversaturation or crystallization of urea is considered. The convective flow away from the droplet due to the evaporation flux, however, is not neglected. The major differences expected from analysing the underlying equations are presented and then compared with simulation results under varying ambient conditions. According to this analysis, it is shown which model gives the best results depending on the required data from the simulations, for instance, droplet lifetime and temperature or accurate prediction of ammonia production. In a second step, chemical kinetics for the HNCO hydrolysis and NO oxidation are added in the gas phase and their influence on the evaporation model is determined.

2 Mathematical Modelling

The influence of the evaporation model, the chemical kinetics and the ambient conditions on the decomposition of UWS is investigated using detailed simulations of a single droplet in ambient gas at a constant uniform pressure. The droplet is assumed to be spherical during the whole process, as shown by Pruppacher and Beard [23] for water, which allows the formulation of a one dimensional model [29]. The resulting conservation equations can be found in [19]. Due to the difficulties integrating the convective terms [19], the gas phase equations are transformed into modified Lagrangian coordinates

$$ f(r,t)\to f(\psi,t), $$
(4)
$$ \psi(r,t)={\int}_{r_{D}}^{r}\rho(r,t)r^{2}dr. $$
(5)

After the transformation, the equations in the gas-phase read [29]

$$ \left( \frac{\partial r}{\partial \psi}\right)_{t}=\frac{1}{\rho r^{2}}, $$
(6)
$$ \frac{\partial w_{i}}{\partial t}+z\frac{\partial w_{i}}{\partial\psi}+\frac{\partial}{\partial\psi}(r^{2} j_{i})=\frac{\dot{\omega}_{i} M_{i}}{\rho}, $$
(7)
$$\begin{array}{@{}rcl@{}} 0 &=&\frac{\partial T}{\partial t}-\frac{1}{\rho c_{p}}\frac{\partial p}{\partial t}+z\frac{\partial T}{\partial\psi}-\frac{1}{c_{p}}\frac{\partial}{\partial\psi}\left( \rho r^{4} \lambda\frac{\partial T}{\partial\psi}\right)\\ &&-\frac{r^{2}}{c_{p}}\sum\limits_{i=1}^{n_{S}}j_{i} c_{p,i}\frac{\partial T}{\partial\psi} +\frac{1}{\rho c_{p}}\sum\limits_{i=1}^{n_{S}}\dot{\omega}_{i} h_{i} M_{i}, \end{array} $$
(8)
$$ \rho=\frac{p\bar{M}}{RT}, $$
(9)
$$ z=v(\psi_{0})\rho(\psi_{0}){r_{D}^{2}}. $$
(10)

The equation system for the liquid phase is transformed into Lagrangian coordinates analogous to the gas phase. As the mass of the droplet changes over the time a second transformation is added

$$ f(r,t)\to f(\psi,t)\to f(\eta,t), $$
(11)
$$ \eta=\frac{\psi}{{\psi_{D}^{0}}(t)}, $$
(12)
$$ {\psi_{D}^{0}}(t)=\frac{m_{D}(t)}{4\pi}. $$
(13)

The transformed equations of the liquid phase read

$$ \left( \frac{\partial r}{\partial \eta}\right)_{t}=\frac{{\psi_{D}^{0}}}{\rho r^{2}}, $$
(14)
$$ \frac{\partial w_{i}}{\partial t}-\frac{1}{{\psi_{D}^{0}}}(\eta\vartheta^{0})\frac{\partial w_{i}}{\partial\eta}+\frac{1}{{\psi_{D}^{0}}}\frac{\partial}{\partial\eta}(r^{2} j_{i})=0, $$
(15)
$$\begin{array}{@{}rcl@{}} 0&=&\frac{\partial T}{\partial t}-\frac{1}{{\psi_{D}^{0}}}(\eta\vartheta^{0})\frac{\partial T}{\partial\eta} -\frac{1}{c_{p}{\psi_{D}^{0}}}\frac{\partial}{\partial\eta}\left( \rho r^{4}\lambda \frac{1}{{\psi_{D}^{0}}}\frac{\partial T}{\partial\eta}\right)\\ &&-\frac{r^{2}}{c_{p}}\sum\limits_{i=1}^{n_{S}}j_{i} c_{p,i}\frac{1}{{\psi_{D}^{0}}}\frac{\partial T}{\partial\eta}, \end{array} $$
(16)
$$ \rho=\rho(w_{1},\dots,w_{n_{S}},p,T), $$
(17)
$$ \vartheta^{0}(t)=\frac{d{\psi_{D}^{0}}}{dt}=-\phi_{vap}{r_{D}^{2}}. $$
(18)

In the gas phase, the transport processes are modelled in detail. Fourier’s law is used to determine the heat fluxes. The diffusion coefficients are calculated using the approximation by Hirschfelder and Curtiss [14]. Convection is neglected due to the small droplet size which results in a limited effect on heat and mass transfer as shown in Section 1. The liquid phase is modelled as an ideal mixture of liquids. This assumption introduces only a small error as UWS does not deviate strongly from an ideal mixture [26]. Urea is assumed to be liquid due to the limited influence of crystallization and oversaturation on the general evaporation process [17] and the short period of time required to increase the droplet temperature above the urea melting point after most of the water is evaporated [4, 25, 32]. Liquid property data are taken from Yaws et al. [34]. The diffusion coefficient for solid urea in water is used during the phase of water evaporation. Afterwards, diffusion within the droplet is negligible as the amount of water becomes small. For consistency, the diffusion coefficient of water in liquid urea is approximated with correlations taken from Reid et al. [24].

2.1 Numerical Solution

The system of partial differential equations is integrated by using the method of lines. The spatial coordinates ψ for the gas phase and η for the liquid phase are discretized with finite differences on a nonequidstant adaptive grid. To resolve the changing steep gradients close to the droplet during the evaporation process and to keep the amount of grid points at a fixed number, a regridding procedure based on a grid function is applied [20]. In the liquid phase, the grid is refined towards the droplet surface because the steepest gradients are also expected at this position [28]. At the outer boundary, Neumann boundary conditions with zero gradients are applied to approximate an adiabatic vessel. It is easier to analyse the overall mass and reaction rate in the gas phase after the droplet evaporation without mass flowing through the boundary. To prevent any interaction between the outer boundary and the evaporation process, the numerical domain extends to at least 100 droplet diameters. In the centre of the droplet, a symmetry boundary condition is used. The resulting differential and algebraic equations of both liquid and gas phase are integrated by the linearly implicit extrapolation method LIMEX [9].

2.2 Interface

The original multi-component evaporation model developed by Stauch [29] encounters difficulties handling conditions where one species is near the vapour-liquid equilibrium while another species is still evaporating. These are typical conditions found during UWS evaporation as there is a high water content in the exhaust gas. The evaporation model was modified to allow each species to condensate or evaporate independently of the direction of the overall mass flow. Similar to the original model, a local phase equilibrium is assumed at the surface of the droplet along a continuous temperature profile T g = T l. Raoult’s law is used to determine the molar fraction of the evaporating species directly above the surface. Transformed for mass fractions the equation for all evaporating species becomes

$$ {w_{i}^{g}}=\frac{M_{i}}{\bar{M}_{v}^{g}}\frac{p_{vap,i}}{p_{amb}}{x_{i}^{l}}, $$
(19)

with

$$ {x_{i}^{l}}={w_{i}^{l}}\frac{\bar{M}_{v}^{l}}{M_{i}}, $$
(20)

At the surface, the mass flux each species reads

$$ \phi_{vap,i}={\rho_{i}^{g}}v^{g}+{j_{i}^{g}}={\rho_{i}^{g}}v^{l}+{j_{i}^{l}}, $$
(21)

which can be reformulated using

$$ \phi_{vap}=\rho^{g}v^{g}=\rho^{g}v^{l}, $$
(22)

and

$$ \rho_{i}=w_{i}\rho, $$
(23)

to give

$$ \phi_{vap,i}={w_{i}^{g}}\phi_{vap}+{j_{i}^{g}}={w_{i}^{l}}\phi_{vap}+{j_{i}^{l}}. $$
(24)

Adding Eq. 24 over all evaporating species results in the overall vaporization rate which is used in Eq. 12. For the non-evaporating species, the boundary condition is then given as

$$ {w_{i}^{g}}=-\frac{{j_{i}^{g}}}{\phi_{vap}}. $$
(25)

Equation 24 is also applied to the liquid side to close the system. Both liquid and gas with the interface conditions are solved in a fully coupled way. The simplified energy conservation for the interface is [29]

$$ 0=\sum\limits_{i}\phi_{vap,i}\Delta h_{vap,i}+j_{q,c}^{g}-j_{q,c}^{l}. $$
(26)

The resulting system of partial differential algebraic equations consists of Eqs. 6–10 for the gas phase and Eqs. 14–18 for the liquid phase which are closed by Eq. 19 and Eqs. 24–26.

2.3 Evaporation Model

In Section 1, two ways to handle urea decomposition based on an evaporation approach have been presented. The major difference between the two cases is whether urea is directly decomposed at the droplet surface or first evaporated into the gas phase and then decomposed using a chemical reaction. In addition, the enthalpy change due to the endothermic urea decomposition can be applied at different places. Evaporation is coupled with diffusion in both the liquid phase and in the gas phase and with gas phase kinetics. Varying ambient conditions, like temperature and gas phase composition, strongly affect the physical and chemical processes. The objective of this study is to determine how the decomposition of urea is influenced by the evaporation model and its coupled processes under different ambient conditions and if both variants of the evaporation model lead to comparable results. For better analysis, the two main cases were separated into four cases as shown in Fig. 1.

Fig. 1
figure 1

Four cases of the evaporation model from left to right: No decomposition (ND), decomposition in gas phase with enthalpy applied at evaporation (DG1) and during the reaction (DG2) and decomposition at the interface (DI)

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The simplest case includes no decomposition (ND). Only gaseous urea is evaporated from the droplet without any decomposition reaction or other chemistry. To preserve the energy balance compared to the other cases, the enthalpy change of the endothermic urea decomposition is added to the evaporation and melting enthalpy to form a new combined evaporation enthalpy.

Two cases with decomposition in the gas phase (DG1 and DG2) are built on the basis of case ND and add a decomposition reaction for gaseous urea (Eq. 27) analogous to the reaction given for liquid or solid urea in Eq. 2. As gaseous urea is not expected to appear in large quantities, the reaction rate of this artificial reaction is set sufficiently high to make sure all urea is decomposed within a distance of half a droplet diameter. The difference between both cases is where the decomposition enthalpy is applied. In case DG1, it is part of the evaporation enthalpy like in case ND while in case DG2, it is applied directly during the reaction in the gas phase.

$$ \text{CO}(\text{NH}_{2})_{2}\text{ (g)} \to \text{NH}_{3}\text{ (g)} + \text{HNCO}\text{ (g)}. $$
(27)

The last case (DI) assumes that liquid urea decomposes at the interface into gaseous NH3 and HNCO [4]. The interface equations have to be modified for this case. The molar fraction and mass flux of urea in the liquid phase has to be coupled with the mass fractions and mass fluxes of NH3 and HNCO in the gas phase. The decomposition reaction (Eq. 2) shows that urea decomposes into the same amount of NH3 and HNCO. Based on this, it is assumed that NH3 and HNCO share the same molar fraction determined by the vapour pressure in the gas phase directly above the surface of the droplet [4]. The following changes to Eqs. 19 and 24 are then implemented

$$ w_{\text{NH}_{3}}^{g}=\frac{M_{\text{NH}_{3}}}{\bar{M}_{v}^{g}}\frac{p_{vap,\text{CO}(\text{NH}_{2})_{2}}}{p_{amb}}x_{\text{CO}(\text{NH}_{2})_{2}}^{l}, $$
(28)
$$ w_{\text{HNCO}}^{g}=\frac{M_{\text{HNCO}}}{\bar{M}_{v}^{g}}\frac{p_{vap,\text{CO}(\text{NH}_{2})_{2}}}{p_{amb}}x_{\text{CO}(\text{NH}_{2})_{2}}^{l}, $$
(29)
$$\begin{array}{@{}rcl@{}} \phi_{vap,\text{CO}(\text{NH}_{2})_{2}}&=&w_{\text{CO}(\text{NH}_{2})_{2}}^{l}\phi_{vap}+j_{\text{CO}(\text{NH}_{2})_{2}}^{l}\\ &=&w_{\text{NH}_{3}}^{g}\phi_{vap}+j_{\text{NH}_{3}}^{g}\\ &&+w_{\text{HNCO}}^{g}\phi_{vap}+j_{\text{HNCO}}^{g}. \end{array} $$
(30)

In the energy conservation, only the mass flux of urea is used for case DI and the evaporation enthalpy is combined with the decomposition enthalpy as in case ND and DG1. The vapour pressure curve, evaporation enthalpy and decomposition enthalpy are the same for all cases and are taken from Birkhold [4].

Case ND is used for reference. There is no chemistry involved and only urea evaporates into air. The interaction between water content or temperature of the ambient air and the evaporation process can be easily identified. Cases DG1 and DG2 show the coupling of gas phase chemistry close to the droplet surface when compared with case ND. The different position where the decomposition enthalpy of urea is applied between DG1 and DG2 highlights if the energy transport close to the droplet is affecting the evaporation of water or urea. Case DI compared to case ND shows if different species properties, especially concerning the diffusion, affect the evaporation. Due to the direct interaction of HNCO with the gas-liquid equilibrium at the droplet surface, the effect of slower reactions like the HNCO hydrolysis can also be shown.

3 Results and Discussion

Simulations were done for typical exhaust conditions of diesel engines but also for potential uses at higher ambient temperatures [5]. The initial composition of the UWS contains 32.5% urea by weight. Table 1 gives an overview of the range of conditions considered in this work. The lowest temperature was chosen as 473 K so that there is always a significant rate of urea evaporation [31].

Table 1 Range of conditions for droplet simulations
Full size table

Two different compositions for the ambient gas were used: dry air for reference purposes and a model exhaust gas with 10% O2, 10% CO2, 5% H2O, and 250 ppm of NO and NO2 where the remaining part is N2. The chemical kinetics are modelled by a one-step reaction for the HNCO hydrolysis [35] and a detailed mechanism (25 species, 132 reactions) for NO oxidation and reduction without a catalyst [11]. The reaction rate of the urea decomposition in the gas phase for cases DG1 and DG2 (Eq. 27) is determined using an Arrhenius expression that was fitted to fulfil the conditions given in Section 2.3:

$$ r_{dec}=c_{\text{CO}(\text{NH}_{2})_{2}}\cdot 3\cdot 10^{10}\frac{1}{s}e^{\frac{-23000J/mol}{RT}}. $$
(31)

Unless otherwise specified, simulations were done at 1 bar in model exhaust gas without HNCO hydrolysis and without NO oxidation and reduction in the gas phase using a droplet with an initial diameter of 200 μm.

3.1 Four Phases of UWS Droplet Evaporation

The evaporation of a UWS droplet does not follow exactly the four separate phases consisting of water evaporation and urea evaporation, each phase preceded by a heating period. Figure 2 shows a typical D2 ratio curve, droplet surface temperature curve and mass flow curve during the evaporation of a droplet based on reference case ND. The intermediate heating period is not very distinct and water evaporation continues well into the phase of urea evaporation, especially at higher ambient temperatures. The reason is that the diffusion velocity of water is limited. Water evaporates fast at the droplet surface resulting in a high urea concentration. Due to the lower vapour pressure of urea, the local gas-liquid equilibrium moves to higher temperatures and leads to an increased evaporation of urea. There is still water left inside the droplet which diffuses slowly outwards and evaporates when reaching the surface. At higher ambient temperatures, this process is faster and more water is still left inside the droplet when urea evaporation starts. Such a behaviour is in line with experimental results and previous theoretical works as presented in Section 1. It highlights not only the need to resolve and calculate mass transport inside the droplet but also that D2 ratio curves are not sensitive to these processes.

Fig. 2
figure 2

D2 ratio, droplet surface temperature and species mass flow, case ND, \(T_{\infty } = 673\) K, D 0 = 200 μm

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3.2 Evaporation Model

Figures 3 and 4 compare the D2 ratio curves and the droplet surface temperature of the four cases. Based on the results, the studied cases can be separated into two groups. Case DG1 and DG2 show nearly identical results and case ND and DI differ only by a maximum of a few %. DG1 and DG2 predict significantly greater evaporation rates at lower ambient temperatures. This difference decreases at higher temperature until all cases are close to each other. The variation of the droplet surface temperature between the two groups is independent of the ambient temperature and cases DG1 and DG2 predict a surface temperature 20–30 K lower than cases ND and DI.

Fig. 3
figure 3

D2 ratio curves for all cases, model exhaust, D 0 = 200 μm

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Fig. 4
figure 4

Droplet surface temperature for all cases, model exhaust, D 0 = 200 μm

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As shown in Eqs. 24 and 26, the evaporation mass flux is determined by the diffusion flux of the evaporating species while the energy flux is determined by the heat transfer of the whole gas phase consisting to more than 70% of nitrogen. Thus, the heat flux depends mostly on the temperature gradient and it is nearly independent of the properties and gradients the evaporating species unless the droplet surface temperature is close to the boiling point. During evaporation, both processes are in equilibrium. In cases DG1 and DG2, the urea decomposition reaction reduces the molar fraction of urea relatively close to the droplet surface. The concentration gradient increases and so does the diffusion flux of gaseous urea away from the droplet. The resulting higher evaporation rate requires more heat and the droplet cools down. This in turn leads to an increased heat transfer and a lower urea concentration with a decreased diffusion flux. The local gas-liquid equilibrium at the droplet surface moves to a lower temperature.

At growing ambient temperatures above 700 K, the droplet surface temperature increases only slightly during water and urea evaporation. The difference of the surface temperature between both evaporation phases becomes small compared to the difference to the ambient temperature. Thus, the temperature gradient and the resulting heat flux are nearly the same for both phases. In addition, the droplet temperature is closer to the boiling point. Due to the exponential vapour pressure curve, even a small change in temperature results in a large difference in the resulting diffusion flux while the heat flux changes minimally. This explains why the D2 ratio curves of all four cases are similar at 873 K in comparison to 473 K. Furthermore, the combined enthalpy of the evaporation and decomposition of urea is only slightly larger than the evaporation enthalpy of water. It is the most likely reason why the gradient of the D2 ratio of water at 873 K is only marginally steeper than that of urea (Fig. 3).

In case DG2, the increase of temperature before urea begins to evaporate happens slightly earlier compared to all other cases. It is the only case where the decomposition enthalpy is not applied at the evaporation. The gas phase decomposition reaction is not instantaneous so some urea can evaporate based on the lower effective evaporation enthalpy leading to a short heating period. When urea starts to decompose in the gas phase, the energy used for the process is again identical to DG1 and the temperature at the droplet surface becomes nearly the same. The differences between case ND and DI are small compared to the differences between these two cases and DG1 and DG2. As both ND and DI do not include a urea decomposition reaction and are solely based on diffusion, the main factor of influence is the property data of gaseous urea, NH3 and HNCO. NH3 and HNCO have a greater effective diffusion coefficient compared to gaseous urea. Thus for case DI, the evaporation rate of urea is slightly increased.

There is no urea decomposition in case ND so the amount of NH3 and HNCO produced by a droplet can only be compared between cases DG1, DG2 and DI in Fig. 5. For cases DG1 and DG2, the gas phase decomposition without further reactions results in a near equimolar distribution of NH3 and HNCO as expected by the reaction shown in Eq. 27. It is clearly visible that case DI overestimates the NH3 production from urea. The mass of material evaporated from the droplet is still identical in all cases. The changed mass distribution in case DI (see Fig. 6) is also independent of the ambient temperature. The different properties NH3 and HNCO, especially diffusion coefficients, lead to different diffusion fluxes of both species towards the interface. While the overall mass conservation is guaranteed due to both species mass flux being added in Eq. 24 to form the mass flux of urea in the droplet the species conservation is not explicitly enforced.

Fig. 5
figure 5

Amount of NH3 and HNCO in the gas phase, model exhaust, cases DG1, DG2 and DI, \(T_{\infty }\) = 573 K, D 0 = 200 μm

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Fig. 6
figure 6

Mass of NH3 and HNCO in the gas phase, model exhaust, cases DG1, DG2 and DI, \(T_{\infty } = 573\) K, D 0 = 200 μm

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For this reason, if an accurate prediction of the ammonia production rate is required, only cases DG1 and DG2 provide good results. Case DG2 is preferable due to the more realistic distribution of the evaporation and reaction enthalpies. An improved fit for both the urea vapour pressure and the reaction rate of the urea decomposition could be used to compensate for the increased evaporation rate of case DG2 at lower ambient temperatures. In case the prediction of the droplet lifetime and temperature is important and some deviation in the amount of ammonia is acceptable, case DI can also be used. It has the additional advantage of not requiring gaseous urea and therefore no decomposition reaction very close to the droplet which would also require a very fine grid. Thus in the following sections, only cases DG2 and DI are considered.

3.3 Influence of Pressure, Initial Droplet Diameter and Gas Composition

The behaviour of cases DG2 and DI is analysed under varying ambient pressure. In Fig. 7, the evaporation mass flow of urea and H2O is shown for 1 bar and 5 bar. The first heating period is longer but the overall droplet lifetime is only slightly increased and the mass flow is nearly identical when corrected for the different starting time.

Fig. 7
figure 7

Evaporation mass flow at 1 bar and 5 bar, model exhaust, case DI, \(T_{\infty }\) = 773 K, D 0 = 200 μm

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The initial droplet diameter was varied from 50 to 200 μ m. As expected for an evaporation process, the D2 curves over the reduced time \(\mathrm {t}/\mathrm {D}_{0}^{2}\) are nearly independent of the initial diameter for both cases. Only in case DG2, the droplet temperature is affected by the changing droplet diameter as shown in Fig. 8. The differences are relatively small and most likely caused by the decomposition reaction. Close to the droplet, the temperature is within a limited range and thus the Arrhenius parameter of the reaction, which depends on the temperature, is within the same range. The second factor is the urea concentration which is higher at larger droplets due to geometrical reasons. A higher reaction rate is reached and combined with the higher urea concentration at the surface it leads to increased urea concentration gradients which result in a lower temperature due to the mechanism presented in Section 3.2.

Fig. 8
figure 8

Droplet surface temperature at 1 bar, model exhaust, cases DG2 and DI, \(T_{\infty }\) = 573 K, D 0 = 50–200 μm

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Changes in the composition of the ambient gas also show the expected effect. The most relevant value is the water content which should slow the evaporation of water at increasing values. This effect is small but still visible in Fig. 9 equally for both case DG2 and DI. The evaporation of urea starts a bit earlier in dry air but is otherwise unchanged. The difference is better visible when looking at the droplet surface temperature in Fig. 10 where the increased evaporation rate of water in dry air leads to a lower temperature. This effect is more distinct at lower ambient temperatures and barely visible at 873 or 973 K.

Fig. 9
figure 9

Evaporation mass flow at 1 bar, model exhaust and dry air, case DG2, \(T_{\infty }~=~473\) K, D 0 = 200 μm

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Fig. 10
figure 10

Droplet surface temperature at 1 bar, model exhaust and dry air, cases DG2 and DI, \(T_{\infty }~=~473\) K, D 0 = 200 μm

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3.4 Gas Phase Chemistry

Besides the ambient conditions the effect of gas phase chemistry on UWS droplet evaporation has to be investigated. Thus, for the results presented in this section, the HNCO hydrolysis and NO reduction were added. If the influence of these reactions is low, the evaporation and decomposition of a UWS droplet can be separated from the gas phase chemistry. The hydrolysis reaction attains a significant reaction rate only above 600 K so that only a small amount of HNCO is present in the gas phase at any time. This is shown in Fig. 11 and can be compared to simulations without the reaction where the HNCO mass is much higher (Fig. 6). There is still some HNCO left after the droplet has fully evaporated, and thus the NH3 mass continues to increase for a short period of time. The following decrease of NH3 mass is caused by the NO reduction. This reaction is much slower even at 973 K and only becomes relevant after the droplet has evaporated and there is no more HNCO left.

Fig. 11
figure 11

Mass of NH3 and HNCO in the gas phase, model exhaust, cases DG2 and DI with complete gas phase chemistry, \(T_{\infty }~=~973\) K, D 0 = 200 μm

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Over the whole parametric range, the change in droplet evaporation rate, temperature and other data is less than 1% compared to simulations without HNCO hydrolysis and NO reduction. Therefore, these reactions can be considered uncoupled from the evaporation process at least for the conditions investigated in this work. It was already shown in Section 3.2 that case DI leads to an unequal production of NH3 and HNCO from urea. As there is less HNCO present in the gas phase in case DI the amount of NH3 produced from the hydrolysis reaction is also lower. Despite this mechanism, the overestimation of the NH3 production cannot be completely corrected but the error is reduced to around 10% as shown in Fig. 11.

4 Conclusion

In this study, several variants of evaporation models for urea decomposition from UWS are analysed and compared. An improved multi-component evaporation model that can cover both variants was developed for an existing simulation tool for one-dimensional droplet evaporation. The effects of changing ambient conditions and gas phase chemistry on the droplet surface temperature and NH3 production rates predicted by the different models were determined. The evaporation and decomposition of UWS is usually separated into two heating phases where the first is followed by the evaporation of water and second by the decomposition of urea. Recent numerical studies predicted a certain amount of water left inside the droplet when there is nearly none left at the surface, which was also found in this study. Subsequently, the leftover water slowly diffuses to the surface and evaporates. At this point, urea is already decomposing and so the three phases of water evaporation, second heating and urea decomposition overlap. It was also found that D2 ratio time histories are often very similar for varying ambient conditions and changing evaporation model variants despite significant differences in droplet surface temperature and actual evaporation mass flow.

The different evaporation models were separated into four cases for better analysis of the different phenomena and coupled processes. Two main variants of the urea evaporation model were determined: direct decomposition of urea at the surface (case DI) and decomposition by a fast gas phase reaction after evaporation (case DG2). For direct decomposition at the surface, the NH3 production is overestimated because of the different property data of NH3 and HNCO, the model assumption of both gases having the same molar fraction at the surface and the strongly coupled processes of diffusion and evaporation. The variant using a gas phase decomposition reaction does not show any discrepancy in the NH3 and HNCO production from the UWS droplet. Due to the coupling between the droplet evaporation and the fast decomposition reaction in this variant, the evaporation rate is significantly increased at ambient temperatures below 700 K. A refined fit for both the vapour pressure and the decomposition reaction should be able to solve this issue. Such a fit is best based on experimental data of evaporating UWS droplets. As the existing experimental data is limited and shows strong variations, further experiments under well-defined conditions can help to considerably improve current evaporation models. Additionally, the gas-phase chemistry for HNCO hydrolysis and NO reduction was considered. At the studied conditions neither HNCO hydrolysis nor NO reduction couple with the evaporation process and the urea decomposition.

Overall, if an accurate prediction of the ammonia production rate is required and if a good fit for the urea vapour pressure and decomposition reaction is available using an evaporation model with a gas phase decomposition reaction can be recommended. Alternatively, direct decomposition at the surface can be considered if the focus of the simulation is the droplet lifetime and temperature or if the inclusion of gaseous urea and its decomposition reaction is difficult. A different approach using chemical kinetics within the liquid phase instead of an evaporation model might also be feasible and can be considered for further numerical investigations.