A quasi-discrete model for droplet heating and evaporation: Application to Diesel and gasoline fuels

Fuel 97 (2012) 685–694

Contents lists available at SciVerse ScienceDirect

Fuel journal homepage: www.elsevier.com/locate/fuel

A quasi-discrete model for droplet heating and evaporation: Application to Diesel and gasoline fuels A.E. Elwardany, S.S. Sazhin ⇑ Sir Harry Ricardo Laboratories, Centre for Automotive Engineering, School of Computing, Engineering and Mathematics, Faculty of Science and Engineering, University of Brighton, Brighton BN2 4GJ, UK

a r t i c l e

i n f o

Article history: Received 15 November 2011 Received in revised form 26 January 2012 Accepted 31 January 2012 Available online 16 February 2012 Keywords: Diesel fuel Gasoline fuel Multi-component droplets Evaporation Modelling

a b s t r a c t The previously suggested quasi-discrete model for heating and evaporation of complex multi-component hydrocarbon fuel droplets is generalised to take into account the dependence of density, viscosity, heat capacity and thermal conductivity of the liquid components on carbon numbers and temperature. This model is applied to the modelling of heating and evaporation of Diesel and gasoline fuel droplets. In agreement with the prediction of the previously reported simplified version of this model in which density, viscosity, heat capacity and thermal conductivity of all liquid components were assumed to be the same as for n-dodecane, it is pointed out that Diesel fuel droplet surface temperatures and radii, predicted by a rigorous model taking into account the effect of all 20 quasi-components, are almost the same as those predicted by the model using five quasi-components. For the Effective Thermal Conductivity/Effective Diffusivity (ETC/ED) model, the number of quasi-components used can be reduced to three. In the case of gasoline fuel, with the maximal number of quasi-components equal to 13, a good approximation for the ETC/ED model can be achieved based on the analysis of just three components. The difference in predictions of the 13 and 1 component models appears to be particularly important when droplets evaporate in gas at a relatively low temperature (450 K) and low pressure (0.3 MPa). In this case the evaporation time predicted by the one component model is less than half of the time predicted by the 13 component model. The surface mass fraction of the lightest quasi-component in gasoline fuel monotonically decreases with time, while the surface mass fraction of the heaviest component monotonically increases with time. Surface mass fractions of intermediate components initially increase with time, but at later times they rapidly decrease with time. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Most of the practically important fuels, including those used in automotive engines, are multi-component [1]. These fuels are supplied to combustion chambers in the form of sprays. Droplets in these sprays are heated and evaporated and this eventually leads to the ignition of the air/fuel vapour mixture [2]. This paper focuses on modelling of the first two stages of this process only. Two main approaches to modelling multi-component droplet heating and evaporation have been suggested: those based on the analysis of individual components (Discrete Component Model (DCM)) [3–11], applicable in the case when a small number of components needs to be taken into account, and those based on the probabilistic analysis of a large number of components (e.g. Continuous Thermodynamics approach [12–19] and the Distillation Curve Model [20]) (see [21] for details). In the second family of models a number of additional simplifying assumptions have ⇑ Corresponding author. E-mail address: [email protected] (S.S. Sazhin). 0016-2361/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.fuel.2012.01.068

been used, including the assumption that species inside droplets mix infinitely quickly or do not mix at all. A model containing features of both these groups of models has been suggested in [22–24]. In [22] the mixtures of multi-component (Diesel, gasoline and biodiesel fuels) and mono-component substances were studied based on the combination of the Continuous Thermodynamics approach and the Discrete Component Model. As in the case of the classical Continuous Thermodynamics approach, it was assumed that the mixing processes inside droplets are infinitely fast both for species and temperature. The analysis of [23,24] was based on the application of the Quadrature Method of Moments (QMoMs), originally developed in [25]. This method allows one to use two or three pseudo-components for each group of components instead of dozens of real components for the whole mixture. The normal boiling point of each pseudo-component was allowed to change during the vaporisation process. As in the case of the conventional Continuous Thermodynamics approach, it was assumed that the droplets are well mixed. As follows from our analysis of heating and evaporation of bi-component droplets [10], this assumption appears to be questionable.

686

A.E. Elwardany, S.S. Sazhin / Fuel 97 (2012) 685–694

In [26] the Discrete Component Model was applied for studying the mixtures of substances containing many components (Diesel and gasoline fuels) by approximating these fuels with relatively small numbers of physical species (six species for Diesel fuel and seven species for gasoline fuel). They took into account the effects of temperature gradient inside droplets, but assumed that the mass transfer processes inside droplets are infinitely fast. As already mentioned, this assumption is not at all obvious. A new approach to modelling heating and evaporation of multicomponent droplets, suitable for the case when a large number of components is present in the droplets, is suggested in [27]. As in [23,24], this approach is based on the introduction of pseudocomponents, but these pseudo-components have been introduced in a way which differs from the one described in [23,24]. In contrast to the previously suggested models, designed for large numbers of components, the new model takes into account the diffusion of liquid species and thermal diffusion as in the classical Discrete Component Models. This model was called the ‘quasi-discrete model’. At the same time the model described in [27] was based on a number of simplistic assumptions. All physical properties of the components, except saturation vapour pressure and latent heat of evaporation, were assumed to be the same. This model was applied only to Diesel fuel. In this paper, the model, originally developed in [27], is generalised to take into account the differences in liquid density, viscosity, specific heat capacity, and thermal conductivity for different liquid components. The generalised version of the quasi-discrete model is applied to the analysis of heating and evaporation of two automotive fuels: Diesel and gasoline fuels. The generalised version of the quasi-discrete model is described in Section 2. The results of the application of the new model to Diesel and gasoline fuel droplets are presented and discussed in Section 3. The main results of the paper are summarised in Section 4. 2. Quasi-discrete model As in the case of the Continuous Thermodynamics approach, the quasi-discrete model is based on the introduction of the distribution function fm(I) such that:

Z

I2

fm ðIÞ dI ¼ 1;

ð1Þ

I1

where I is the property of the component, fm characterises the relative contribution of the components having this property in the vicinity of I, I1 and I2 are limiting values of this property. An obvious limitation of this approach is that it is applicable only in the case when all properties of the components depend on only one parameter I. Although molar mass M is almost universally used to describe the property I, this choice is certainly far from being a unique one. Remembering that most practically important hydrocarbon fuels consist mainly of molecules of the type CnH2n+2 (alkanes), where n P 1 in the general case or n P 5 for liquid fuels, it is more practical to write the distribution function fm as a function of the carbon number n rather than M [17]. These two parameters are linked by the following equation:

M ¼ 14n þ 2;

ð2Þ

where M is measured in kg/kmole. As in [27] we assume that fm(n) can be approximated as:

fm ðnÞ ¼ C m ðn0 ; nf Þ

ðMðnÞ  cÞa1 ba CðaÞ

   MðnÞ  c exp  ; b

C m ðn0 ; nf Þ ¼

(Z

nf

n0

   )1 ðMðnÞ  cÞa1 MðnÞ  c exp  : dn b ba CðaÞ

ð4Þ

This choice of Cm assures that

Z

nf

fm ðnÞ dn ¼ 1:

n0

Note that real life automotive fuels, including Diesel and gasoline fuels, apart from alkanes, contain significant amounts of alkenes, alkynes, naphthenes and aromatics. The contribution of these elements is not taken into account at this stage, and this is a serious limitation of our model. As in [27] we use the following approximation for the dependence of the saturation vapour pressure (in MPa) on n:

 psat ðnÞ ¼ exp AðnÞ 

 BðnÞ ; T  CðnÞ

ð5Þ

where

AðnÞ ¼ 6:318n0:05091 ; BðnÞ ¼ 1178n0:4652 ; CðnÞ ¼ 9:467n0:9143 ; T is in K. The above approximations for A(n), B(n), C(n) were derived for 4 < n < 17, but we will assume that they can be applied for n P 17 as well if the contribution of hydrocarbon fuels with these n is relatively small. From the Clausius–Clapeyron equation it follows that [17]

L¼

Ru d ln psat ðnÞ ; MðnÞ dð1=TÞ

ð6Þ

where Ru is the universal gas constant. Remembering (5), formula (6) can be rewritten as:

L¼

Ru BðnÞT 2 MðnÞðT  CðnÞÞ2

:

ð7Þ

Following [27], this formula will be used in our analysis. The results predicted by Eq. (5) for n = 10 and n = 12 in the temperature range 300–500 K differed from those reported in [28] by not more than 6.05% and 5.62% respectively. Also, the results predicted by Eqs. (6) and (7) for n = 10 and n = 12 in the temperature range 300–500 K differed from those reported in [28] by not more than 4.82% and 3.52% respectively. In [27] the dependence of liquid density, viscosity, specific heat capacity and thermal conductivity on n was ignored. In our analysis this dependence is taken into account. The approximations of the dependence of these parameters on n and temperature are presented and discussed in Appendix A. As follows from (5) and (7) and the results presented in Appendix A, the transport and thermodynamic properties of the fuels under consideration are relatively weak functions of n. In this case, following [27], it would be sensible to assume that the properties of hydrocarbons in a certain narrow range of n are close, and replace the continuous distribution (3) with a discrete one, consisting of Nf quasi-components with carbon numbers

R nj n

 j ¼ R j1 n nj

nfm ðnÞ dn

f ðnÞ nj1 m

dn

;

ð8Þ

the corresponding molar fractions

Xj ¼

Z

nj

fm ðnÞ dn;

ð9Þ

nj1

ð3Þ

where n0 6 n 6 nf, subscripts 0 and f stand for initial and final, C(a) is the Gamma function, a and b are parameters that determine the shape of the distribution, c determines the original shift,

and mass fractions

 j ÞX j Mðn Y j ¼ Pj¼N ; f  j¼1 ½Mðnj ÞX j 

ð10Þ

687

A.E. Elwardany, S.S. Sazhin / Fuel 97 (2012) 685–694

where j is an integer in the range 1 6 j 6 Nf. Note that

X

0.3

j¼N f

Xj ¼

j¼1

X

Y j ¼ 1:

ð11Þ

The choice of nj can be arbitrary. In our model we assume that all nj  nj1 are equal, i.e. all quasi-components have the same range of values of n. For the case when Nf = 1 this approach reduces the analysis of multi-component droplets to mono-component ones. These new quasi-components are not the actual physical  j are not integers in the general case). hydrocarbon components (n Hence we call this model a quasi-discrete model. These quasi-components are treated as actual components in the conventional DCM, including taking into account diffusion of liquid species in droplets. This model is expected to be particularly useful when Nf is much less than the number of actual species in the hydrocarbon mixture. The mixtures are treated as ideal (Raoult’s law is assumed to be valid). In this case, partial pressures of individual quasi-components can be estimated as:

 j Þ ¼ X lsi ðn  j Þpsat ðn  j Þ; pv ðn

Diesel Diesel fuel

0.24

j¼1

gasoline gasoline fuel 0.18

fm(n)

j¼N f

0.12

0.06

0 5

10

15

20

25

n Fig. 1. Plots of fm(n) versus n as predicted by Eq. (3) for Diesel (thin solid) and gasoline (thick solid) fuels with the values of parameters given in Table 1.

800

ð12Þ

12

Diesel fuel

where Xlsi is the molar fraction of liquid quasi-components at the  j Þ is determined by Eq. (5). surface of the droplet, psat ðn  j we obtain the required valHaving replaced n in Eq. (7) with n ues of L for all quasi-components.

10

700

8

Ts (K)

3. Application to Diesel and gasoline fuel droplets

6

Rd (µm)

600

500

Following [17] we assume the values of parameters for the distribution function (3) for Diesel and gasoline fuels shown in Table 1. Plots of fm versus n for Diesel and gasoline fuels for the values of parameters given in Table 1 are shown in Fig. 1. The curve referring to Diesel fuel is a reproduction of the corresponding plot given in Fig. 3 of Sazhin et al. [27]. As follows from this figure the forms of the plots of fm versus n for Diesel and gasoline fuels appear to be rather different. The values n for which fm is maximal are equal to 12.4 and 5 for Diesel and gasoline fuels respectively. The average  Þ for these fuels are 12.56 and 7.05 respectively. The values of nðn analysis of droplet heating and evaporation for both fuels will be performed separately below. 3.1. Diesel fuel As in [27], we assume that the initial droplet temperature is equal to 300 K, and is homogeneous throughout its volume. Gas temperature is assumed to be equal to 880 K and gas pressure is assumed to be equal to 3 MPa. The initial composition of droplets is described by distribution function (3) with the values of parameters for Diesel fuel given in Table 1, as shown in Fig. 1. Plots of droplet surface temperature Ts and droplet radius Rd versus time for the initial droplet radius equal to 10 lm and velocity 1 m/s are shown in Fig. 2. The droplet velocity is assumed to be constant during the whole process. The calculations were performed for the case of Nf = 1 (one quasi-component droplet,  ¼ 12:56) and Nf = 20 (20 quasi-component droplet), using the n Table 1

Fuel

a

b (kg/kmole)

c (kg/kmole)

n0

nf

Diesel Gasoline

18.5 5.7

10 15

0 0

5 5

25 18

4 One quasi-component-ETC/ED Twenty quasi-components-ETC/ED Twenty quasi-components-ITC/ID Twenty quasi-components-ETC/ED as in [27]

400

2

300

0 0

0.4

0.8

1.2

1.6

t (ms) Fig. 2. Plots of Ts and Rd, predicted by four models, versus time. The initial droplet radius and temperature are assumed to be equal to 10 lm and 300 K respectively; the droplet velocity is assumed to be equal to 1 m/s and its changes during the heating and evaporation process are ignored; gas temperature is assumed equal to 880 K. These are the models used for calculations: Effective Thermal Conductivity (ETC)/Effective Diffusivity (ED) model using one quasi-component; ETC/ED model using 20 quasi-components and Infinite Thermal Conductivity (ITC)/Infinite Diffusivity (ID) model using 20 quasi-components; and the approximations for density, viscosity, heat capacity and thermal conductivity of liquid components given in Appendix A. The results are shown by thick solid, thin solid and thin dashed curves. The fourth model is the ETC/ED model using 20 quasi-components with the density, viscosity, heat capacity and thermal conductivity of the liquid components assumed to be equal to those on n-dodecane (thick dashed).

ETC/ED and ITC/ID models. In the same figure, the plots of Ts and Rd versus time for Nf = 20, using the ETC/ED model, but assuming that the density, viscosity, heat capacity and thermal conductivity of all components are the same and equal to those of n-dodecane, are presented. As one can see from this figure, the droplet radii Rd and and surface temperature Ts, predicted by the ETC/ED model, using 1 and 20 quasi-components are noticeably different, especially at the final stages of droplet heating and evaporation. The model, using 20 quasi-components predicts higher surface temperatures and longer evaporation time compared with the model using one quasicomponent. This can be related to the fact that at the final stages of droplet evaporation the species with large n become the dominant, as will be demonstrated later. These species evaporate more slowly than the species with lower n and have higher wet bulb temperatures. One can see from this figure that the results

688

A.E. Elwardany, S.S. Sazhin / Fuel 97 (2012) 685–694

predicted by a simplified model used in [27] are noticeably different from those predicted by the more rigorous model used in this paper. This shows the limitations of the earlier used simplified model for the density, viscosity, heat capacity and thermal conductivity of the liquid components. Also, there are noticeable differences in predictions of the ETC/ ED and ITC/ID models, using 20 quasi-components, especially in the case of the surface temperature at the initial stage of droplet heating and evaporation. The accurate prediction of this temperature is particularly important for the prediction of the auto-ignition timing in Diesel engines [29]. This questions the reliability of the models for heating and evaporation of multi-component droplets, based on the ITC/ID approximations. As mentioned in Section 1, these models are almost universally used for modelling these processes, especially when large numbers of components are involved in the analysis. Plots of Ts and Rd at time equal to 0.5 ms versus the number of quasi-components Nf, predicted by the ETC/ED and ITC/ID models, are shown in Fig. 3 for the same conditions as in Fig. 2. Symbols refer to those Nf for which calculations were performed. As follows from this figure, for Nf P 10 the predicted Ts and Rd no longer depend on Nf. In fact the difference between the values for surface temperature and radius, predicted for Nf = 5 and Nf = 20, can be considered negligible compared with the difference between the values for temperature, predicted by the ETC/ED and ITC/ID models. Hence, heating and evaporation of Diesel fuel droplets can be safely modelled using just five quasi-components, in agreement with our earlier results [27], obtained for time equal to 0.25 ms using a simplified version of the quasi-discrete model. The errors due to the ITC/ID approximation for Nf P 3 are signifi-

(a) t = 0.5 ms

543

cantly larger than those due to the choice of a small number of quasi-components, especially for the surface temperature. These errors cannot be ignored in most engineering applications, and this questions the applicability of the models using the ITC/ID approximation, including the widely used Continuous Thermodynamics models. Plots similar to those shown in Fig. 3 but at time equal to 1 ms are shown in Fig. 4. As one can see from this figure, both droplet surface temperature and radius can be well predicted if only five quasi-components are used. Note that the sensitivity of the values of Ts and Rd to the number of components at Nf < 5 is far greater in the case shown in Fig. 4 than in the case shown in Fig. 3. The closeness of the temperatures predicted by ETC/ED and ITC/ ID models at the later stages of droplet heating and evaporation can be related to the fact that at this stage the droplet temperature becomes almost homogeneous (see Fig. 9 of [27]) and the effects of the temperature gradient inside droplets can be ignored. In agreement with [27], smaller droplet radii are predicted by the ITC/ID model, compared with the ETC/ED model, at the final stages of droplet heating and evaporation. Comparing Figs. 3 and 4 one can see that at early stages of droplet heating and evaporation (t = 0.5 ms), the predicted droplet radius reduces slightly with the increase in the number of quasicomponents used, while at a later stage (t = 1 ms) the opposite effect is observed, in agreement with the results reported in [27]. This could be related to the fact that at the early stages, droplet evaporation is controlled by the most volatile quasi-components, while at the later stages it is controlled by less volatile quasi-components. When the number of quasi-components increases then the volatility of the most volatile component increases and that of the least volatile decreases.

(a) 630

ETC, ED models ETC/ED model ITC, ID models ITC/ID model

Ts (K)

541

Ts (K)

620

539

610

ETC/ED model ETC/ED model ETC, ED models t = 1 ms ITC/ID model ITC, ID models ITC/ID model

537 0

4

8

12

16

20

Number of quasi-components

600 0

(b)

4

9.7

8

12

16

20

Number of quasi-components

t = 0.5 ms

(b)

ETC, ETC/ED ED models model

7.7

ITC, ID models ITC/ID model

9.65

Rd (µm)

Rd (µm)

7.5

9.6

7.3

ETC, ED models ETC/ED model

7.1

t =1 ms ITC, ID models ITC/ID model

9.55 0

4

8

12

16

20

Number of quasi-components

6.9 0

Fig. 3. Plots of Ts (a) and Rd (b) versus the number of quasi-components Nf for the same conditions as in Fig. 2 at time 0.5 ms as predicted by the ETC/ED (squares) and ITC/ID (triangles) models.

4

8

12

Number of quasi-components Fig. 4. The same as Fig. 3 but at time 1 ms.

16

20

689

A.E. Elwardany, S.S. Sazhin / Fuel 97 (2012) 685–694

(a)

12

800

Onequasi-component-ETC/ED Thirteen quasi-components-ETC/ED Thirteen quasi-components-ITC/ID

700

10

8

6

Rd (µm)

Ts (K)

600

500 4 400

2

gasoline fuel 300 0

0.25

0.5

0.75

1

0 1.25

t (ms) Fig. 5. The same as Fig. 2 for the first three curves but for the gasoline fuel with the maximal number of quasi-components Nf = 13. All plots are based on the approximations for density, viscosity, heat capacity and thermal conductivity of liquid components given in Appendix A.

430

t = 0.2 ms

ETC/ED model ETC, ED models ITC, ID models ITC/ID model

426

Ts (K)

Plots similar to those shown in Fig. 2, but for gasoline fuel, are presented in Fig. 5. The maximal number of quasi-components for gasoline fuel is 13. The initial conditions are assumed to be the same as in the case of Diesel fuel droplets to enable us to perform the direct comparison between heating and evaporation of Diesel and gasoline fuel droplets in identical conditions. As in the case shown in Fig. 2, the droplet velocity is assumed to be constant during the whole process. The calculations were performed for the  ¼ 7:05) and Nf = 13 case of Nf = 1 (one quasi-component droplet, n (13 quasi-component droplet), using the ETC/ED and ITC/ID models. The density, viscosity, heat capacity and thermal conductivity of all liquid components are described in Appendix A. As in the case of Diesel fuel droplets, the droplet radii Rd and surface temperatures Ts, predicted by the ETC/ED models, using 1 and 13 quasi-components are noticeably different, especially at the final stages of droplet heating and evaporation. The model, using 13 quasi-components predicts higher surface temperatures and longer evaporation time compared with the model using one quasi-component. As in the case of Diesel fuel droplets, this can be related to the fact that at the final stages of droplet evaporation the species with large n become the dominant, as will be demonstrated later. These species evaporate more slowly than the species with lower n and have higher wet bulb temperatures. The differences in predictions of the ETC/ED and ITC/ID models, using 13 quasi-components, are more noticeable than in the case of Diesel fuel droplets. This difference can be seen not only at the initial stage of droplet heating and evaporation, but also at the later stages of these processes. This provides additional support for our questioning of the reliability of the models for heating and evaporation of multi-component droplets, based on the ITC/ID approximations. Plots of Ts and Rd at time equal to 0.2 ms versus the number of quasi-components Nf, predicted by the ETC/ED and ITC/ID models for gasoline fuel droplets, are shown in Fig. 6 for the same conditions as in Fig. 5. As follows from this figure, for Nf P 6 the predicted Ts and Rd no longer depend on Nf. In fact the difference between the values for temperature and radius, predicted for Nf = 3 and Nf = 13, can be considered negligible compared with the difference between the values for temperature and radius, predicted by the ETC/ED and ITC/ID models. Hence, heating and evaporation of gasoline fuel droplets can be safely modelled using just

422

418

414 0

3

6

9

12

Number of quasi-components

(b) 9.89 9.87

t = 0.2 ms

ETC, ED models ETC/ED model ITC, ID models ITC/ID model

9.85

Rd (µm)

3.2. Gasoline fuel

9.83

9.81

9.79

9.77 0

3

6

9

12

Number of quasi-components Fig. 6. Plots of Ts (a) and Rd (b) versus the number of quasi-components Nf for the same conditions as in Fig. 5 at time 0.2 ms as predicted by the ETC/ED (squares) and ITC/ID (triangles) models.

three quasi-components. As in the case of Diesel fuel droplets, the errors due to the ITC/ID approximation for Nf P 3 are significantly larger than those due to the choice of a small number of quasicomponents, especially for the surface temperature. Plots similar to those shown in Fig. 6 but at time equal to 0.75 ms are shown in Fig. 7. As one can see from this figure, both droplet surface temperature and radius can be well predicted by the ETC/ED model if only three quasi-components are used. Comparing Figs. 6 and 7 one can see that at the early stages of droplet heating and evaporation (t = 0.2 ms), the predicted droplet radius reduces slightly with the increase in the number of quasicomponents used, while at a later stage (t = 0.75 ms) the opposite effect is observed, in agreement with the results reported in [27], and those shown in Figs. 3 and 4. Plots similar to those shown in Fig. 5, but for more realistic conditions in gasoline engines, are presented in Fig. 8. Following [30], we assume that gas temperature is equal to 450 K, gas pressure is equal to 0.3 MPa and droplet velocity is equal to 10 m/s. As in the case shown in Fig. 5, we assume that the initial droplet temperature is equal to 300 K, and is homogeneous throughout its volume, while the droplet initial radius is equal to 10 lm. Comparing Figs. 5 and 8, one can see that in the latter case the difference between the predicted temperatures and droplet radii for 1 and 13 quasi-components is much more visible than in the former. The same conclusion applies to the predictions of the ETC/ED and ITC/ID models. This can be attributed to much slower evaporation for the case shown in Fig. 8, compared with the case shown in Fig. 5. The general trends of the curves shown in Fig. 8 are similar to the ones shown in Fig. 5. In the case when 13 qua-

690

A.E. Elwardany, S.S. Sazhin / Fuel 97 (2012) 685–694

(a)

(a)

350

560

t = 0.5 ms

ITC, ID models ITC/ID model

345 540

Ts (K)

Ts (K)

ETC, ED models ETC/ED model

520

ETC, ED models ETC/ED model

340

335

t = 0.75 ms ITC, ID models ITC/ID model 500 0

3

6

9

330

12

0

Number of quasi-components

(b)

3

6

9

12

Number of quasi-components

7

(b)

6.6

9.5

t = 0.5 ms

9.45

ETC/ED model ETC, ED models ITC, ID models ITC/ID model

Rd (µm)

Rd (µm)

6.2

5.8

9.4

9.35

ETC/ED model ETC, ED models t = 0.75 ms

5.4

ITC, ID models ITC/ID model

9.3

5 0

3

6

9

12

9.25 0

Number of quasi-components

3

6

9

12

Number of quasi-components Fig. 7. The same as Fig. 6 but at time 0.75 ms.

500

Fig. 9. Plots of Ts (a) and Rd (b) versus the number of quasi-components Nf for the same conditions as in Fig. 8 at time 0.5 ms as predicted by the ETC/ED (squares) and ITC/ID (triangles) models.

12

One quasi-component-ETC/ED model Thirteen quasi-components-ETC/ED model Ts_n=10 Thirteen quasi-components-ITC/ID model Ts-n=20,ITC

450

10

400

6

gasoline fuel

Rd (µm)

Ts (K)

8

4 350 2

300

0 0

2

4

6

8

t (ms) Fig. 8. The same as Fig. 5 but for droplet velocity equal to 10 m/s, gas temperature equal to 450 K and pressure equal to 0.3 MPa.

si-components are considered, at the end of the evaporation process, mainly the heavier components in the droplets remain. These can reach higher temperatures and evaporate more slowly compared with the light and middle-range components. Plots of Ts and Rd at time equal to 0.5 ms versus the number of quasi-components Nf, predicted by the ETC/ED and ITC/ID models for gasoline fuel droplets, are shown in Fig. 9 for the same conditions as in Fig. 8. For Nf P 6 the predicted Ts and Rd no longer depend on Nf (cf. the cases shown in Figs. 3, 4, 6 and 7). This range can be extended to Nf P 3 at least for the ETC/ED model. In contrast to the cases shown in Figs. 3, 4, 6 and 7, the temperatures and radii, predicted by the ETC/ED and ITC/ID models, appear to be very close

for small numbers of quasi-components. This can be related to the fact that in this case the temperature reaches saturation level by 0.5 ms, when one or two components are considered. Plots similar to those shown in Fig. 9 but at time equal to 2 ms are shown in Fig. 10. As in the case shown in Fig. 9, both droplet surface temperatures and radii can be well predicted by the ETC/ ED model if only three quasi-components are used. In contrast to the case shown in Fig. 9, the temperatures and radii, predicted by the ETC/ED and ITC/ID models, appear to be very close only for the case when one quasi-component is used. Plots of Ysi versus time for the four quasi-components for the same case as shown in Fig. 8, are presented in Fig. 11. The results presented in this figure are consistent with those shown in Fig. 8 of Sazhin et al. [27] for Diesel fuel using a simplistic approach to approximate density, viscosity, heat capacity and thermal conductivity of the liquid components. The values of Ys1 monotonically decrease with time, while those of Ys4 monotonically increase with time. The values of Ys2 and Ys3 initially increase with time, but at later times they rapidly decrease with time. At times close to the moment when the droplet completely evaporates, only the quasi-component Ys4 remains. Since this quasi-component is the most slowly evaporating one and has the highest wet bulb temperature, the model based on four quasi-components is expected to predict longer evaporation times and larger droplet surface temperatures at the final stages of droplet evaporation, compared with the model using one quasi-component. This result can be generalised to the case when the number of quasi-components is greater than four. It is consistent with results shown in Figs. 2, 5 and 8.

A.E. Elwardany, S.S. Sazhin / Fuel 97 (2012) 685–694

(a) 400

Ts (K)

380

360

ETC, ED models ETC/ED model t = 2 ms ITC, ID models ITC/ID model 340 0

3

6

9

12

Number of quasi-components

(b)

7 6.8

Rd (µm)

6.6 6.4 6.2 6

ETC, ED models ETC/ED model t = 2 ms

5.8

ITC, ID models ITC/ID model

5.6 0

3

6

9

12

Number of quasi-components Fig. 10. The same as Fig. 9 but at time 2 ms.

1

0.8

Ys1 y1 Ys2 y2

Ys

0.6

Ys3 y3 y4 Ys4

0.4

0.2

0 0

2

4

6

t (ms) Fig. 11. Plots of Ysi versus time for four quasi-components (i = 1, 2, 3, 4) for the same case as shown in Figs. 9 and 10.

4. Conclusions The previously suggested quasi-discrete model for heating and evaporation of complex multi-component hydrocarbon fuel droplets has been generalised to take into account the dependence of density, viscosity, heat capacity and thermal conductivity of liquid components on carbon numbers and temperatures. This model is applied to the modelling of the Diesel and gasoline fuel droplet heating and evaporation. The model is based upon the assumption that properties of the components vary relatively slowly from one

691

component to another and depend on a single parameter: number of carbon atoms in the components (carbon number: n). The components with relatively close n are replaced by quasi-components with properties calculated as average properties of the a priori defined groups of actual components. Thus the analysis of the heating and evaporation of droplets consisting of many components is replaced with the analysis of the heating and evaporation of droplets consisting of relatively few quasi-components. In contrast to previously suggested approaches to modelling the heating and evaporation of droplets consisting of many components, the effects of temperature gradient and quasi-component diffusion inside droplets are taken into account. In agreement with the previously reported results referring to simplified version of this model in which density, viscosity, heat capacity and thermal conductivity of all liquid components were assumed to be the same as for n-dodecane (see [27]), it has been pointed out that Diesel fuel droplet surface temperatures and radii, predicted by a rigorous model taking into account the effect of all 20 quasi-components, are close to those predicted by the model using five quasi-components. For the Effective Thermal Conductivity/Effective Diffusivity (ETC/ED) model, the number of quasi-components used can be reduced to three. The droplet surface temperature, and evaporation time predicted by the simplified model, described in [27], and the model developed in this paper are noticeably different. The evaporation time predicted by the simplified model is about 10% shorter compared with the model used in this paper. This confirms the need to take into account the dependence of density, viscosity, heat capacity and thermal conductivity of liquid components on carbon numbers and temperatures, as has been done in this paper. Also, in agreement with [27], errors due to the assumptions that the droplet thermal conductivity and species diffusivities are infinitely large cannot be ignored in the general case. These errors are particularly important when the droplet surface temperature at the initial stage of heating is predicted. It is pointed out that in the case of gasoline fuel, with the maximal number of quasi-components equal to 13, a good approximation for the case of the ETC/ED model can be achieved based on the analysis of just three components. The difference in predictions of the 13 and 1 component models appears to be particularly important in the case when droplets evaporate in gas at a relatively low temperature (450 K) and low pressure (0.3 MPa). In this case the evaporation time predicted by the one component model is less than half of the evaporation time predicted by the 13 component model. The surface mass fraction of the lightest quasi-component in gasoline fuel is shown to monotonically decrease with time, while the mass fraction of the heaviest component is shown to monotonically increase with time. Mass fractions of intermediate components initially increase with time, but at later times they rapidly decrease with time. Acknowledgments The authors are grateful to the European Regional Development Fund Franco-British INTERREG IVA (Project C5, Reference 4005) and EPSRC (Grant EP/F058276/1) for financial support of the work on this project. Appendix A. Approximations for alkane fuel properties In what follows the new approximations for the temperature dependencies of density, viscosity, heat capacity and thermal conductivity for liquid alkanes (CnH2n+2) with 5 6 n 6 25, as inferred from published approximations and data, are described.

692

A.E. Elwardany, S.S. Sazhin / Fuel 97 (2012) 685–694

where

A.1. Liquid density Following [31], the temperature dependence of the density of liquid alkanes for 5 6 n 6 25 is approximated as: Cq 1TT cr

ð

ql ðTÞ ¼ 1000Aq Bq

Þ

ð13Þ

;

where Tcr are critical temperatures (approximations for Tcr for 5 6 n 6 25 are shown in Fig. 2 of Sazhin et al. [27]), the numerical values of Aq, Bq and Cq for individual values of n are given in [31]. These values have been approximated by the following expressions:

8 > < Aq ¼ 0:00006196104  n þ 0:234362 Bq ¼ 0:00004715697  n2  0:00237693  n þ 0:2768741 > : C q ¼ 0:000597039  n þ 0:2816916 ð14Þ The range of applicability of Eq. (13) depends on the values of n. For n = 5 this range was determined as 143.42–469.65 K; for n = 10 this range was determined as 243.49–618.45 K; for n = 25 this range was determined as 315.15–850.13 K [31] (the upper limits are critical temperatures of the components). Remembering that the contribution of alkanes with n close to 25 is relatively small, we will assume that Eqs. (13) and (14) are valid in the whole range from room temperature until close to the critical temperature. Plots of ql versus n for T = 300 K and T = 450 K, as inferred from Eq. (13) with coefficients Aq, Bq and Cq given by Yaws [31] (filled squares j and filled triangles N), and approximated by Eqs. (14) (solid and dashed curves) are shown in Fig. A1. As follows from this figure, the agreement between the values of liquid density predicted by approximation (13), with the values of the coefficients given in [31] and approximated by Eq. (14), looks almost ideal. For temperatures of 300 and 450 K, the values of density inferred from Eq. (13) with coefficients given by Eq. (14) differ by less than 0.51% and 0.76% respectively from the values of density inferred from Eq. (13) with coefficients given by Yaws [31]. A.2. Liquid viscosity Following [32], the temperature dependence of the dynamic viscosity of liquid alkanes for 4 6 n 6 44 is approximated as:

ll ðn; TÞ ¼ 103 ½10½100ð0:01 TÞ

bðnÞ



 0:8;

ð15Þ

bðnÞ ¼ 5:745 þ 0:616 lnðnÞ  40:468 n1:5 :

ð16Þ

The temperature range of the applicability of approximations (15) and (16) was not explicitly specified in [32], but the author of this paper demonstrated good agreement between the predictions of these approximations and experimental data in the range of temperatures from 10 °C to 100 °C. Plots of ll versus n for T = 300 K and T = 450 K, as inferred from Eqs. (15) and (16) (solid and dashed curves), and the corresponding values of ll in the range 5 6 n 6 12, inferred from Ref. [33] (filled squares j (T = 300 K) and filled triangles N (T = 450 K)), are shown in Fig. A2. As follows from this figure, the agreement between the values of liquid viscosity predicted by approximations (15) and (16) and the results presented on the NIST website [33] looks almost ideal. The differences between the results predicted by Eq. (15) and NIST data for 300 K and 450 K were found to be less than 5.13% and 40.16% respectively. Large errors in the latter case are linked with small values of l. Note that the values of dynamic viscosity affect droplet heating and evaporation only via the corrections to the values of thermal conductivity and diffusivity in the Effective Thermal Conductivity and Effective Diffusivity (ETC/ED) models. In most practically important cases, the influence of viscosity on the final results is expected to be very weak. A.3. Heat capacity Following [34], the temperature dependence of the heat capacity of liquid alkanes for 2 6 n 6 26 is approximated as:

cl ðn; TÞ ¼ 1000

  43:9 þ 13:99ðn  1Þ þ 0:0543ðn  1ÞT ; MðnÞ

ð17Þ

where M(n) = 14n + 2 is the molar mass of alkanes. The temperature range of applicability of Eq. (17) was not clearly identified in [34] for all n, except to say that this approximation is not valid at temperatures close to the temperature of fusion. For n = 16 and n = 17 these ranges were identified as 340–400 K and 335–400 respectively. In the case of n = 16 and n = 25 the temperatures of fusion are equal to 295.1 K and 329.25 K respectively. However, remembering that the contribution of the alkanes with n > 16 is very small, it will be assumed that approximation (17) is valid in the whole temperature range from room temperature onwards.

900

0.014

800

0.012

0.01

µ l (Pa.s)

ρl (kg/m3)

700

600

0.008

0.006

500 0.004 400

0.002

300

0 5

10

15

20

25

n Fig. A1. Plots of ql versus n for T = 300 K and T = 450 K, as inferred from Eq. (13) with coefficients Aq, Bq and Cq given by Yaws [31] (filled squares j for T = 300 K and filled triangles N for T = 450 K), and approximated by Eq. (14) (solid curve for T = 300 K and dashed curve for T = 450 K).

5

10

15

20

25

n Fig. A2. Plots of ll versus n for T = 300 K and T = 450 K, as inferred from Eqs. (15) and (16) (solid (T = 300 K) and dashed (T = 450 K) curves), and the corresponding values of ll in the range 5 6 n 6 12, inferred from Ref. [33] (filled squares j (T = 300 K) and filled triangles N (T = 450 K)).

693

A.E. Elwardany, S.S. Sazhin / Fuel 97 (2012) 685–694 0.2 0.18 0.16

kl (W/m.K)

Plots of cl versus n for T = 300 K and T = 450 K, as inferred from Eq. (17) (solid and dashed curves), and the corresponding experimental values of cl for T = 300 K in the range 5 6 n 6 18, inferred from Refs. [33] (filled squares j) and [34] (filled circles ), are shown in Fig. A3. As follows from this figure, the agreement between the values of the liquid heat capacity predicted by approximation (17) and the experimental results for T = 300 K looks almost ideal. The difference between the results predicted by Eq. (17) and data reported in [33,34] for 300 K was found to be less than 0.91%. We are not aware of experimental data for T = 450 K.

0.14 0.12 0.1 0.08

A.4. Thermal conductivity

0.06

Following [35], the temperature dependence of thermal conductivity of liquid alkanes for 5 6 n 6 20 is approximated as:

0.04

  T 2=7 kl ðn; TÞ ¼ 10 Ak þBk ð1T cr Þ ;

ð18Þ

where Tcr are critical temperatures as in Eq. (13), the numerical values of Ak and Bk for individual values of n are given in [35]. These values have been approximated by the following expressions:

(

Ak ¼ 0:002911  n2  0:071339  n  1:319595 Bk ¼ 0:002498  n2 þ 0:058720  n þ 0:710698

ð19Þ

Although approximations (18) and (19) have been derived for 5 6 n 6 20, they are used in the whole range 5 6 n 6 25. Possible errors imposed by these approximations in the range 21 6 n 6 25 are expected to have a very small effect on the final results as the mass fractions of alkanes in this range of n are very small in Diesel fuel, and negligible in gasoline fuel. The range of applicability of Eq. (18) depends on the values of n. For n = 5 this range was determined as 143–446 K; for n = 10 this range was determined as 243–588 K; for n = 20 this range was determined as 310–729 K [35]. Remembering that the contribution of alkanes with n P 20 is relatively small, we have assumed that Eqs. (18) and (19) are valid in the whole range from room temperature until close to the critical temperature, as in the case of approximations (13) and (14). Plots of kl versus n for T = 300 K and T = 450 K, as inferred from Eq. (18) with coefficients Ak and Bk given by Yaws [35] (filled squares j (T = 300 K) and filled triangles N (T = 450 K)) and approximated by Eq. (19) (solid and dashed curves) are shown in Fig. A4. In the same figure we have shown the values of kl inferred from Ref. [33] (squares h (T = 300 K) and triangles M (T = 450 K)). As follows from this figure, the agreement between the values of

2800

cl (J/kg.K)

2600

2400

2200

2000 5

10

15

20

25

n Fig. A3. Plots of cl versus n for T = 300 K and T = 450 K, as inferred from Eq. (17) (solid (T = 300 K) and dashed (T = 450 K) curves), and the corresponding experimental values of cl for T = 300 K in the range 5 6 n 6 18, inferred from Refs. [33] (filled squares j) and [34] (filled circles ).

5

10

15

20

25

n Fig. A4. Plots of kl versus n for T = 300 K and T = 450 K, as inferred from Eq. (18) with coefficients Ak and Bk given by Yaws [35] (filled squares j (T = 300 K) and filled triangles N (T = 450 K)) and approximated by Eq. (19) (solid (T = 300 K) and dashed (T = 450 K) curves); the values of kl inferred from Ref. [33] (squares h (T = 300 K) and triangles M (T = 450 K)).

thermal conductivity predicted by approximation (18) with the values of the coefficients given in [35] and approximated by Eq. (19), looks almost ideal. Both these values agree well with the data reported in [33]. For temperatures 300 and 450 K, the values of thermal conductivity inferred from Eq. (18) with coefficients given by Eq. (19) differ by less than 2.46% and 7.80% respectively from the values of thermal conductivity inferred from Eq. (18) with coefficients given by Yaws [35]. Note that, during calculations, a small number of lighter components inside droplets could have temperatures exceeding their critical temperatures. In this case, the values of saturation pressure, latent heat of evaporation, density, viscosity, heat capacity and thermal conductivity were assumed equal to those at T = Tcr. This assumption allows us to avoid the analysis of heat and mass transfer in supercritical conditions, without imposing significant errors in our analysis due to the fact that the number of components affected by this assumption is very small.

References [1] Heywood JB. Internal combustion engines fundamentals. New York: McGrawHill Book Company; 1988. [2] Sazhina EM, Sazhin SS, Heikal MR, Babushok VI, Johns R. A detailed modelling of the spray ignition process in Diesel engines. Combust Sci Technol 2000;160:317–44. [3] Faeth GM. Evaporation and combustion of sprays. Prog Energy Combust Sci 1983;9:1–76. [4] Tong AY, Sirignano WA. Multicomponent transient droplet vaporization with internal circulation: integral equation formulation. Numer Heat Transfer 1986;10:253–78. [5] Continillo G, Sirignano WA. Unsteady, spherically-symmetric flame propagation through multicomponent fuel spray clouds. In: Angelino G, De Luca L, Sirignano WA, editors. Modern research topics in aerospace propulsion. Springer-Verlag; 1991. p. 173–98. [6] Klingsporn M, Renz U. Vaporization of a binary unsteady spray at high temperature and high pressure. Int J Heat Mass Transfer 1994;37(Suppl. 1):265–72. [7] Lage PLC, Hackenberg CM, Rangel RH. Nonideal vaporization of dilating binary droplets with radiation absorption. Combust Flame 1995;101:36–44. [8] Torres DJ, O’Rourke PJ, Amsden AA. Efficient multi-component fuel algorithm. Combust Theory Model 2003;7:67–86. [9] Maqua C, Castanet G, Lemoine F. Bi-component droplet evaporation: temperature measurements and modelling. Fuel 2008;87:2932–42. [10] Sazhin SS, Elwardany A, Krutitskii PA, Castanet G, Lemoine F, Sazhina EM, et al. A simplified model for bi-component droplet heating and evaporation. Int J Heat Mass Transfer 2010;53:4495–505. [11] Sazhin SS, Elwardany A, Krutitskii PA, Deprédurand V, Castanet G, Lemoine F, et al. Multi-component droplet heating and evaporation: numerical simulation versus experimental data, Int. J Therm Sci 2011;50:1164–80. [12] Tamim J, Hallett WLH. Continuous thermodynamics model for multicomponent vaporization. Chem Eng Sci 1995;50:2933–42.

694

A.E. Elwardany, S.S. Sazhin / Fuel 97 (2012) 685–694

[13] Lippert AM, Reitz RD. Modelling of multicomponent fuels using continuous distributions with application to droplet evaporation and sprays. SAE Technical Paper 972882; 1997. [14] Hallett WLH. A simple model for the vaporization of droplets with large numbers of components. Combust Flame 2000;121:334–44. [15] Zhu G-S, Reitz RD. A model for high-pressure vaporization of droplets of complex liquid mixture using continuous thermodynamics. Int J Heat Mass Transfer 2002;45:495–507. [16] Pagel S, Stiesch G, Merker GP. Modelling of evaporation of a multicomponent fuel. In Taine J, editor. Proceedings of the twelfth international heat transfer conference, vol. 1. Grenoble (August 18–23, 2002). Paris: Editions Scientifique et Medicale Elsevier SAS; 2002. CD ROM. [17] Arias-Zugasti M, Rosner DE. Multicomponent fuel droplet vaporization and combustion using spectral theory for a continuous mixture. Combust Flame 2003;135:271–84. [18] Abdel-Qader Z, Hallett WLH. The role of liquid mixing in evaporation of complex multicomponent mixtures: modelling using continuous thermodynamics. Chem Eng Sci 2005;60:1629–40. [19] Rivard E, Brüggemann D. Numerical investigation of semi-continuous mixture droplet vaporization. Chem Eng Sci 2010;65:5137–45. [20] Burger M, Schmehl R, Prommersberger K, Schäfer O, Koch R, Wittig S. Droplet evaporation modelling by the distillation curve model: accounting for kerosene fuel and elevated pressures. Int J Heat Mass Transfer 2003;46:4403–12. [21] Sazhin SS. Advanced models of fuel droplet heating and evaporation. Prog Energy Combust Sci 2006;32:162–214. [22] Zhang L, Kong S-C. Vaporization modeling of petrolium-biofuel drops using a hybrid multi-component approach. Combust Flame 2010;157:2165–74. [23] Laurent C. Dévelopment et validation de modèles d’évaporation multicomposant. Thèse de l’Institut Supériour de l’Aéronautique et de l’Espace de Toulouse; 2008.

[24] Laurent C, Lavergne G, Villedieu P. Continuous thermodynamics for droplet vaporization: comparison between gamma-PDF model and QMoM. Comptes Rendus Mechanique 2009;337:449–57. [25] Lage PLC. The quadrature method of moments for continuous thermodynamics. Comput Chem Eng 2007;31:782–99. [26] Ra Y, Reitz RD. A vaporization model for discrete multi-component fuel sprays. Int J Multiphase Flow 2009;35:101–17. [27] Sazhin SS, Elwardany A, Sazhina EM, Heikal MR. A quasi-discrete model for heating and evaporation of complex multicomponent hydrocarbon fuel droplets. Int J Heat Mass Transfer 2011;54:4325–32. [28] Abramzon B, Sazhin SS. Convective vaporization of fuel droplets with thermal radiation absorption. Fuel 2006;85:32–46. [29] Sazhin SS, Abdelghaffar WA, Sazhina EM, Heikal MR. Models for droplet transient heating: effects on droplet evaporation, ignition, and break-up. Int J Therm Sci 2005;44:610–22. [30] Basshuysen RV. Gasoline engine with direct injection: processes, systems, development, potential, first ed. GWV Fachverlage GmbH, Wiesbaden (9788348-0670-3); 2009. [31] Yaws CL, editor. Thermophysical properties of chemicals and hydrocarbons. William Andrew Inc.; 2008. [32] Mehrotra AK. Correlation and prediction of the viscosity of pure hydrocarbon. Can J Chem Eng 1994;72:554–7. [33] http://www.webbook.nist.gov/chemistry/. [34] van Miltenburg JC. Fitting the heat capacity of liquid n-alkanes: new measurements of n-heptadecane and n-octadecane. Thermochim Acta 2000;343:57–62. [35] Yaws CL. Handbook of thermal conductivity. Vol. 2 (Organic compounds, C5– C7) and Vol. 3 (Organic compounds, C8–C28). Houston, London, Paris, Zurich, Tokyo: Gulf Publishing Company; 1995.