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Numerical Heat Transfer, Part A: ApplicationsAn International Journal of Computation and Methodology

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Evaporation of pure and blended droplets ofdiesel and alcohols (C2–C9) under diesel engineconditions

Ping Yi, Ming Jia, Wuqiang Long, Li Qiao, Tianhao Yang & Liyan Feng

To cite this article: Ping Yi, Ming Jia, Wuqiang Long, Li Qiao, Tianhao Yang & Liyan Feng (2017):Evaporation of pure and blended droplets of diesel and alcohols (C2–C9) under diesel engineconditions, Numerical Heat Transfer, Part A: Applications, DOI: 10.1080/10407782.2016.1264749

To link to this article: http://dx.doi.org/10.1080/10407782.2016.1264749

Published online: 13 Feb 2017.

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NUMERICAL HEAT TRANSFER, PART A http://dx.doi.org/10.1080/10407782.2016.1264749

Evaporation of pure and blended droplets of diesel and alcohols (C2–C9) under diesel engine conditions Ping Yia, Ming Jiaa, Wuqiang Longa, Li Qiaob, Tianhao Yanga, and Liyan Fenga

aKey Laboratory of Ocean Energy Utilization and Energy Conservation of Ministry of Education, Dalian University of Technology, PR China; bSchool of Aeronautics and Astronautics, Purdue University, West Lafayette, Indiana, 47907, USA

ABSTRACT An improved multicomponent evaporation model was developed to study the evaporation characteristics of the pure diesel and alcohol (C2–C9) as well as their blended droplets. In this model, the Predictive Soave–Redlich–Kwong (PSRK) equation of state (EOS) was employed to evaluate the vapor–liquid equilibrium for pure and blended droplets. The results indicated that the model predictions agree very well with the experimental measurements. The evaporation characteristics of different pure alcohol droplets were analyzed, and the influence of the addition of alcohols in diesel on the evaporation behavior of droplets under various conditions was discussed.

ARTICLE HISTORY Received 6 August 2016 Accepted 4 November 2016

1. Introduction

Due to their pollution-reducing and octane-enhancing capabilities, many oxygenated additives have been added into practical fuels such as diesel and gasoline to reduce emissions and meet legislative requirements [1]. Blending alcohols with diesel allows the use of energy from renewable sources without seriously changing the evaporation and combustion characteristics in the engines.

The limitation, however, is the miscibility of hydrocarbons and alcohols. Based on previous works [2, 3], short-chain alcohols such as methanol and ethanol are miscible with n-paraffins with carbon numbers less than 13 at room temperature. For long-chain alkanes, the blends can be made miscible by increasing the temperature of the fuel or adding a higher molecular weight alcohol as a stabilizer. The long-chain alcohols, unlike short-chain alcohols, show good blending stability with diesel. Studies have shown that the concentration of alcohols in the blends with diesel can be as high as 50% without significantly changing the key physical properties [4, 5].

The combustion process in an engine is influenced not only by the type of the injected fuel, but also by the distribution of the fuel vapor inside the cylinder. The latter is largely determined by the evaporation behavior of the multicomponent fuel and the subsequent mixing process in the gas phase [6]. An in-depth understanding of the impact of the difference in the vaporability of alcohols and its blends with diesel is necessary to reap the benefits of alcohols’ utilization in practical diesel engines. Most of the previous studies related to alcohols as alternative fuels for internal combustion engines were focused on the spray performance of short-chain alcohols, mainly meth-anol, and ethanol [7–9]. Yeh et al. [10] found that the alcohols of C9–C12 have better performance in particle matter (PM) reduction. Nevertheless, few studies concentrated on the detailed evaporation characteristics of long-chain alcohols. Motivated by this, the present paper focuses on the evaporation

CONTACT Wuqiang Long yiping@mail.dlut.edu.cn Institute of International combustion Engine, Dalian University of Technology, Dalian, 116024, PR China. Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/unht. © 2017 Taylor & Francis

characteristics of the diesel droplets blended with short- (i.e., C2–C5) and long-chain (C9) alcohols, and compares their differences in the evaporation behaviors.

To understand the evaporation behaviors of the fuel blends, a multicomponent evaporation model is essential to gain insight into the evaporation characteristics of each component and the complex interactions among each other. Hallett et al. [5] measured and calculated the distillation curves for the blends of ethanol and diesel, which can help understand in depth the blends’ evaporation beha-viors. However, most of the previous numerical studies on alcohols assumed uniform temperature and mass fraction in the liquid phase [4, 8]. This assumption becomes questionable when the fuel droplet evaporation proceeds at high-temperature and -pressure conditions as encountered in diesel engines [11, 12]. As a result, the finite heat conductivity and mass diffusivity in the liquid phase must be considered in the evaporation model.

On the other hand, the influence of ambient pressure on the evaporation behavior should be taken into account [8, 13]. Most previous works [5, 14] only focused on the evaporation characteristics of the blended droplet of diesel and alcohols under the atmospheric pressure. Hallet and Clark [15] used a continuous form of Raoult’s law with the Clausius–Clapeyron equation to describe the vapor–liquid equilibrium. However, according to the work of Ref. [5], Raoult’s law is incapable of predicting the equilibrium behavior for the blends of diesel and alcohols. Indeed, the Peng–Robinson equation of state and the van der Waals mixing rules are commonly employed in the literature to calculate the vapor–liquid equilibrium for some hydrocarbons [16, 17]. This is because the van der Waals mixing rules are generally limited to nonpolar or slightly polar mixtures [18]. It was found that the UNIFAC (UNIQUAC Functional-group Activity Coefficients) method [19–21] coupled with the Antoine equation can give satisfactory predictions on the vapor–liquid equilibria for the blended droplet with oxygen additives at low pressures [1, 4]. To investigate the effects of high pressure on the evaporation behaviors, the Predictive Soave–Redlich–Kwong (PSRK) method, which combines the Soave–Redlich–Kwong equation of state (SRK EOS) and the UNIFAC method [22, 23], was applied to predict the vapor–liquid equilibrium for the blended droplet of diesel and alcohols in this study.

In the literature, most numerical works were carried out only for the blended droplet of diesel with short-chain alcohols undergoing the evaporation process at the atmospheric pressure. The detailed effects of ambient pressure, as well as the evaporation characteristics of the long-chain alcohol droplets have not been discussed. Motivated by these problems, the present study aims at analyzing the evaporation characteristics of the pure and blended droplets of diesel and various alcohols at a wide range of ambient conditions using an improved multicomponent evaporation model. This evap-oration model was developed based on the previous works [11, 12], and the PSRK EOS was applied to evaluate the vapor–liquid equilibrium for the blended droplet. The predictions of this improved

Nomenclature

Di diffusion coefficient Dl droplet diameter hi specific enthalpy Hlatent latent heat of evaporation _ml mass evaporation rate

N;Nl number of species Nu Nusselt number P pressure T temperature u velocity vi, g diffusion velocity x mole fraction Y mass fraction ρ density λ thermal conductivity εi normalized fraction

ϕi fugacity coefficient γi activity coefficient

Subscripts i, j species g gas l liquid p pressure re reduced v vapor 0 initial values

Superscripts a average c corrected s surface

2 P. YI ET AL.

multicomponent evaporation model were validated by the available measurements for the pure diesel and alcohols, as well as their blended droplets. Then, the validated evaporation model was used to analyze the detailed evaporation characteristics of the pure diesel, alcohols (C2–C9), and their blended droplets under various ambient conditions.

2. Numerical model

An improved multicomponent evaporation model was used herein. Most basic equations and approx-imations used in the present model are essentially the same as those used in our previous works [11, 12], and they are only briefly summarized below. The description of the PSRK EOS, which was not used in Refs. [11, 12], is represented in detail. The following assumptions are introduced for the present evaporation model: (1) the droplet is assumed to be a spherical symmetry; (2) the effects of Dufour and Soret are negligible; (3) both the liquid and gas phases at the droplet surface follow the thermodynamic equilibrium; and (4) there is no internal circumferential movement in the liquid droplet.

2.1. Droplet evaporation rate

Based on the previous works [11, 12], the droplet mass evaporation rate is controlled by the mass diffusion rate in the gas phase and is estimated as

_ml ¼XNl

i¼1_mi;l ¼

XNl

i¼14pR2

l qgysi;l u �

XN

i¼1ys

i;gvsi;g þ uc

� �ð1Þ

where Rl is the droplet radius; ρg is the density of ambient gas; u is the ambient velocity; uc is a correction velocity estimated based on Hirschfelder’s law [24]; yi, g and vi, g are the mass fraction and diffusion velocity of species i, respectively; and Nl and N are the number of species in the liquid and gas phase, respectively.

2.2. Species concentration and temperature distributions within a droplet

The species concentration and temperature distributions within the droplet are described as

qTl Rð Þqt

¼kl

clql

q2Tl Rð ÞqR2 þ

2RqTl Rð ÞqR

� �

ð2Þ

qyi;l Rð Þqt

¼ Di;lq2yi;l Rð ÞqR2 þ

2Rqyi;l Rð ÞqR

� �

ð3Þ

where Tl(R) and yi,l(R) are the droplet temperature and mass fraction distribution within the droplet, respectively; λl is the liquid thermal conductivity; and Di, l is the mass diffusion coefficient of species i. The boundary conditions for the droplet are described as

� Di;lqyi;l Rð Þ

RlqR¼

_ml

4pqlR2l

ei � ysi;l

� �; R ¼ Rl ð4Þ

klqTl Rð ÞRlqR

¼ _qliquid; R ¼ Rl ð5Þ

where εi is the normalized fraction of fuel vapor and _qliquid is the heat flux absorbed by the droplet and described as

_qliquid ¼ 4pR2l kgNu Tg � Ts� �

= 2Rlð Þ � qg

XN

i¼1hi ys

i;gvsi;g

� �h i� _mlHlatent ð6Þ

where λg is the thermal conductivity of gas phase; hi is the specific enthalpy of species i; Tg and Ts are the ambient and droplet surface temperature, respectively; _ml is the mass evaporation rate; Hlatent is

NUMERICAL HEAT TRANSFER, PART A 3

the latent heat of evaporation; and Nu is the Nusselt number, which is calculated according to the work of Ref. [25] as

Nu ¼ 2:0009þ 0:514 max Re;max Gr; 0ð Þ1=2

� �h i1=2Pr1=3 ð7Þ

where Re and Gr are the Reynolds number and the Grashof number of the gas mixture, respectively; and Pr is the Prandtl number. They are all calculated as those in Ref. [25].

The liquid mass diffusivity and thermal conductivity have significant effects on the temperature and mass distribution within the droplet [12], which are calculated based on the method of Ref. [26] as

Di;m g0:8m ¼

XNl

j¼1j6¼i

xj;lDij;b g0:8j ð8Þ

where Di, m is the liquid mass diffusion coefficient of species i in cm2/s; ηm and ηj are the liquid viscosity of mixture and species j in cP, respectively; and xj,l is the mole fraction of species of j. The binary diffusion coefficient is estimated using the method of Ref. [27] as follows:

Dij;b ¼ 8:93� 10� 8 Vi=Vj� �1=6 Pj;p=Pi;p

� �0:6 Tg=gj;l

� �ð9Þ

where Vi and Vi are the molar volume of the species i and j at the normal boiling temperature in cm3/mol, respectively [28]; and Pi,p and Pj,p are parachors for the species i and j, which can be captured from addi-tive group contributions in Ref. [27].

The liquid thermal conductivity is estimated using the method of Latini et al. [29] as

ki;l ¼ A�i Taii;b=Mbi

i Tcii;c

� �1 � Ti;r� �0:38

=T1=6i;r ð10Þ

where Ti, b, Ti,c, and Ti,r are the boiling temperature, critical temperature, and reduced temperature of species i, respectively; Mi is the molecular weight; and A�i , αi, βi, and γi are correlation parameters of Latini et al. [29]. A�i ¼ 0:00339, αi ¼ 1.2, βi ¼ 0.5, and γi ¼ 0.167 for alcohols, A�i ¼ 0:0035, αi ¼ 1.2, βi ¼ 0.5, and γi ¼ 0.167 for saturated hydrocarbons, and A�i ¼ 0:0346, αi ¼ 1.2, βi ¼ 1.0, and γi ¼ 0.167 for aromatics. The other physical properties are calculated based on the works of Refs. [11, 28, 30].

2.3. Vapor–liquid interface

The species mass fraction on the droplet surface is estimated from the thermodynamic equilibrium equation at the vapor–liquid interface. For the blends of diesel and alcohols [21], the phase equilibrium at the vapor–liquid interface is calculated by PSRK EOS, which combines the SRK EOS and the UNIFAC method. The SRK EOS is expressed as

P ¼RgTs

V � B�

AV V þ Bð Þ

ð11Þ

where Rg is the gas constant; V is the mole volume; A and B are EOS parameters, which are determined by the critical state of the pure component as

Ai ¼ 0:42748 R2gT2

i;cai Tð Þ=Pi;c ð12ÞBi ¼ 0:08664 RgTi;c=Pi;c ð13Þ

ai Tð Þ ¼ 1:0þ ci;1 1:0 �ffiffiffiffiffiffiffiTi;c

p� �þ ci;2 1:0 �

ffiffiffiffiffiffiffiTi;c

p� �2þ ci;3 1:0 �

ffiffiffiffiffiffiffiTi;c

p� �2h ið14Þ

B ¼Xi¼N

i¼1

Xj¼N

j¼1xi;lxj;l Bij ð15Þ

A ¼ BRgTs�C� 1A

Xi¼N

i¼1xi;l ln ci þ C� 1

A

Xi¼N

i¼1xi;lB=Bi þ

Xi¼N

i¼1xi;lAi= BiRgTs� ��

ð16Þ

4 P. YI ET AL.

Bij ¼ 0:5 B0:5i þ B0:5

j

� �ð17Þ

where CA is a constant of 0.64663. The fugacity coefficient of each component in the mixture is evaluated as

ln /i ¼ Bi Z � 1ð Þ=B � ln Z 1 � B=Vð Þ½ � � b ln 1þ B=Vð Þ ð18Þb ¼ C� 1

A ln ci þ ln B=Bi þ B=Bi � 1:0ð Þ þ Ai= BiRgTs� �ð19Þ

where γi is the activity coefficient, which is calculated by the UNIFAC method as the sum of a combinatorial and a residual term

ln ci ¼ ln cci þ ln cr

i ð20Þln cc

i ¼ 1 � V 0i þ ln V 0i � 5qi 1 � Vi=Fi þ ln Vi=Fið Þð Þ ð21Þ

ln cri ¼

Xk¼N

k¼1vk;i ln Ck � ln Ck;i� �

ð22Þ

The parameters in Eqs. (21) and (22) are calculated as

V 0i ¼ r3=4i =

Xj¼N

j¼1xs

j; g r3=4j

� �ð23Þ

Vi ¼ ri=Xj¼N

j¼1xs

j; grj

� �ð24Þ

ri ¼Xj¼N

k¼1vk; i Rk ð25Þ

Fi ¼ qi=Xj¼N

j¼1xs

j; gqi

� �ð26Þ

qi ¼Xk¼N

k¼1vk; iQk ð27Þ

ln Ck ¼ Qk 1 � lnXm¼N

m¼1hmwmk �

Xm¼N

m¼1hmwkm=

Xn¼N

n¼1hnwnm

� �� �ð28Þ

where θm is the group area fraction and ψnm is the temperature-dependent parameter, which are calculated as

hm ¼ QmXm=XN

n¼1QnXn

� �ð29Þ

wnm ¼ exp � anm þ bnmT þ cnmT2� �=Ts� �

ð30Þ

where parameters anm, bnm, and cnm are binary coefficients depending on the molecular structure, which can be obtained from Ref. [28].

Therefore, the vapor–liquid equilibrium is expressed as

xsi;g=xs

i;l ¼ /i;l=/i;g ð31Þ

where xsi;g and xs

i;l are the mole fraction fuel vapor and liquid species on the droplet surface, respectively.

3. Results and discussion

The above-improved multicomponent evaporation model was validated by the experimental data for the evaporation of droplets composed of pure diesel and alcohols, as well as their mixtures. The evaporation behavior of pure diesel, alcohols (C2–C9), as well as their blended droplets under various ambient conditions is discussed in this section.

The temporal behavior of the blended droplet can be divided into four distinct stages as shown in Figure 1. First, the droplet experiences the first heating period and the evaporation of more-volatile components. Then, the droplet goes through the second heating period, and followed by the less-volatile components evaporating. During the first heating period, characterized by the thermal expansion, most

NUMERICAL HEAT TRANSFER, PART A 5

heat flux absorbed from the ambient gas is used to raise the droplet temperature. In contrast, during the evaporation period, the heat flux is attributed to the enthalpy of evaporation. The transient evaporation constant is defined as the slope of the droplet size curve during the evaporation period (ktv ¼ ΔD2/Δt), and the constant evaporation rate is estimated as Kv ¼

R t2t1

kvdt. The droplet lifetime herein is defined as the time period from the beginning to the moment when the instantaneous squared diameter of the droplet becomes 5% of the initial squared value, which is widely used as the termination criteria of the simulation.

3.1. Validation of pure diesel and alcohol droplets

The multicomponent evaporation model developed in this study was validated by comparing the model predictions with the experimental measurements of Ref. [5] for pure diesel and alcohol droplets. This experiment [5] was performed in a furnace with nitrogen as the ambient gas to prevent combustion and allow for pure evaporation. A fuel droplet with the diameter in the range of 1.4–1.86 mm was suspended at a quartz fiber. The ambient temperature varies from 688 to 1023 K, and the droplet initial temperature was set as 300 K. Based on the compositions and major physical properties (e.g., density, oxygen content, hydrogen content, distillation curve, etc.) of the test diesel [5], five discrete components were selected to surrogate the practical diesel in the predictions. The composition and distillation curve of the modeled diesel are shown in Figure 2. It can be observed that the predicted distillation curve is in good agreement with the measurement [5]. Hence, the selected compositions herein can represent the practical diesel.

Figure 3a and b shows comparisons of the droplet size history for pure diesel and alcohol droplets, respectively. The modeled alcohol herein is a blend with 85.5% ethanol and 14.5% methanol by mass based on Ref. [5] It can be observed that the predictions of the present model are in good agreement with the measurements [5] for both pure diesel and alcohol droplets under various ambient tem-peratures. Although the model slightly underpredicts the evaporation rate in the late evaporation stage for both fuel droplets, the variation of droplet diameters during the droplet lifetime is well predicted. In addition, the variation of transient evaporation rate is satisfactorily captured for diesel because the fuel components are changing as the more-volatile components in diesel evaporate first. The deviation between the predictions and measurements in Figure 3 might be caused by two reasons. The first one is that the suspended fiber used in the experiment can enhance the evaporation rate [31], which is not considered in the present model. Another one is that the uncertain factors in the experi-mental measurements and the numerical model, such as the uncertain droplet initial temperature and the predictions of physical properties, all of them together affect the derivations between the experimental measurements and the model predictions.

Figure 1. Definition of the evaporation process for blended droplets.

6 P. YI ET AL.

3.2. Effects of the vapor–liquid equilibrium model on the evaporation behavior of the diesel/ethanol droplet

Because the vapor–liquid equilibrium model has a major impact on the predictions of evaporation for fuel droplets, two models, i.e., the PSRK method [22] and Raoult’s law [32], are tested in this section for the blended droplet of diesel and ethanol with different mixing ratios. Figure 4 shows the temporal evolution of droplet size history under different ambient temperatures, and the predictions of the present model with PSRK EOS and Raoult’s law are compared with the measurements of Ref. [5]. In the experiment, the stagnant blended droplet of diesel and ethanol with the concentration of ethanol in the range of 10–40% by mass was exposed to the ambient temperature of 703 K. The droplet diameter was in the range of 1.4–1.8 mm, and the initial temperature was set as 300 K.

The observations in Figure 4 present that both the predictions by PSRK EOS and Raoult’s law show a little difference with the experimental data. The reasons are the same as that discussed in Figure 3. It is worth noting that the transient evaporation constants, especially the inflection point of ethanol distilled out, predicted by the PSRK EOS are in better agreement with the measurements. However, the predictions by Raoult’s law cannot show the changes in droplet size history when the ethanol is consumed out. This is because the activity coefficient of ethanol varies with its con-centration, which subsequently affects the evaporation rate of each component in the mixture. As the mole fraction of ethanol decreases, the activity coefficient rises sharply, and reaches a maximum at infinite dilution [5]. This behavior compensates for the decreasing concentration of ethanol and results in a relatively high evaporation constant.

3.3. Validation of blended droplets of diesel and various alcohols

The evaporation process of the diesel droplet blended with various alcohols (C2–C5) was simulated using the present multicomponent evaporation model. Figure 5 shows the measured [4] and predicted distillation curve for the droplet of diesel blended with various alcohols (20% by volume). As can be

Figure 3. Comparison between the measured [5] and modeled droplet size history for the pure diesel and alcohol droplets.

Figure 2. Diesel compositions [5] and comparison between the measured [5] and modeled distillation curves.

NUMERICAL HEAT TRANSFER, PART A 7

observed, although the use of five discrete surrogates to model the ultralow sulfur diesel may result in erroneous prediction for the evaporation behavior of ultralow sulfur diesel used in Ref. [4], the predicted distillation characteristics of the fuel are in good agreement with the measured data [4]. Hence, the present model is employed to analyze the evaporation behaviors of diesel droplets blended with various alcohols in the following sections.

Because of the large difference in the boiling temperatures of diesel and alcohols [4], the alcohol in the blended droplet evaporates first, followed by a sudden rise in boiling temperature close to that of more-volatile components in diesel, as shown in Figure 5. Among these test alcohols, the boiling temperature increases from ethanol to pentanol. As a result, the starting temperature of the distil-lation curve increases from Figure 5a to d. Additionally, the sudden rise in the boiling temperature becomes less severe. It can also be seen from Figure 5 that the initial latent heat of evaporation

Figure 5. Comparison between the measured [4] and modeled boiling temperature and latent heat of evaporation versus time for the blended droplets of 80% diesel and 20% alcohol at 0.1 MPa.

Figure 4. Comparison between the measured [5] and modeled droplet size history for the blended droplets of diesel and ethanol at different ambient temperatures.

8 P. YI ET AL.

decreases obviously from ethanol to pentanol. It has been well known that the alcohols have lower boiling temperature but higher latent heat of evaporation than diesel components [6]. The lower boiling point indicates the higher volatility, whereas the higher latent heat of evaporation decreases the evaporation rate. With these two factors competing with each other, the pure and blended droplets of diesel and alcohols show distinct evaporation characteristics, and the details are discussed in the following section.

3.4. Evaporation characteristics of pure and blended droplets of diesel and alcohols

3.4.1. Evaporation of pure diesel and alcohol droplets at the atmospheric pressure Evaporation of different pure fuel droplets (e.g., diesel, ethanol, propanol, butanol, pentanol, and nonanol) was simulated using the present model. In order to model the practical conditions relevant to diesel engines, the droplet size and temperature were fixed at 0.2 mm and 300 K, respectively [33]. To understand the effect of ambient temperature on the evaporation behavior, the ambient tem-perature was in the range of 573–873 K.

Figure 6 shows the evaporation characteristics of different pure fuel droplets at the atmospheric pressure. It is observed that the pure alcohol droplets show distinct evaporation behaviors with the diesel droplet. During the heating period, the diesel droplet shows an obvious thermal expansion characterized by the increased droplet diameter. On the contrary, due to the low boiling temperature of alcohols, the thermal expansion is less noticeable than that of diesel, especially for the ethanol droplet, its surface regresses immediately at the beginning as presented in Figure 6a and c. However, the thermal expansion of the pure diesel droplet in Figure 6 is a little more apparent than that in Figure 3. This is because the test droplet diameter is 0.2 mm in Figure 6, which is smaller than that used in the experiment in Figure 3 (Dl,0 ¼ 1.83 mm). Because the temperature of the small droplet increases quickly, higher droplet temperature results in lower liquid density, and subsequently causes more obvious thermal expansion. Additionally, since the heating period depends on both the heat

Figure 6. Variation of droplet size, constant evaporation rate, and heating period for different pure fuel droplets at atmospheric pressure.

NUMERICAL HEAT TRANSFER, PART A 9

capacity and boiling temperature of the liquid [34], the heating period of alcohols increases with the increase of carbon number, as shown in Figure 6b and d.

The diesel droplet shows different evaporation rates with alcohol droplets, including the transient evaporation rate (ktv) and constant evaporation rate (Kv), which are defined in Figure 1. For the diesel droplet, the transient evaporation rate continuously varies with time due to preferential evaporation of the more-volatile components (see Figure 6a and c) as can be seen in other works in the literature [11, 35]. Note, however, the droplet size of pure alcohols varies in the manner of the well-known D2-law, which is characterized by the fact that the transient evaporation rate is a constant.

The boiling temperature and latent heat of evaporation both substantially influence the constant evaporation rate of fuel droplets. The lower boiling point indicates the higher volatility, and the lower latent heat of evaporation improves the evaporation rate. Under the low ambient temperature of 573 K, the diesel droplet with higher boiling point has lower constant evaporation rate than the alcohol droplets, as expected. However, the alcohol droplet with higher boiling temperature shows a higher constant evaporation rate than that with a lower boiling temperature, which is different from that of hydrocarbons [36, 37]. For instance, the nonanol droplet shows the highest constant evapor-ation rate, followed by the pentanol one, and the ethanol droplet shows the lowest value as shown in Figure 6b. This is attributed to the fact that the short-chain alcohols have a large latent heat of evaporation, which decreases the droplet surface temperature and reduces the evaporation rate. Under the high ambient temperature of 873 K, the diesel droplet shows a higher constant evaporation rate than that of short-chain alcohol droplets (ethanol, propanol, butanol, and pentanol) as shown in Figure 6d. This phenomenon indicates that the higher latent heat of alcohols has a more prominent influence on the evaporation rate at high-temperature conditions.

Considering the effect of both the heating and evaporating periods at the atmospheric pressure, the lifetime of alcohol droplets decreases with the increase of carbon number. For instance, the ethanol droplet shows the longest lifetime, and the nonanol droplet shows the shortest one. Note, however, the lifetime of alcohols is closer to or even longer than diesel droplets at the high-temperature condition of 873 K, which is different from the phenomenon at the low-temperature condition. This is mainly because the diesel droplet has a higher evaporation rate than the alcohol droplets at high-temperature conditions.

3.4.2. Evaporation of the pure diesel and alcohol droplets at the high ambient pressure The ambient pressure has significant effects on the phase equilibrium and droplet surface temperature [17]. From the phase equilibrium, the high ambient pressure improves the boiling temperature, which decreases the mass transfer number and thus reduces the droplet evaporation rate. On the contrary, the high ambient pressure increases the droplet surface temperature due to the higher conduction heat flux, subsequently reducing the latent heat of evaporation and improving the droplet evaporation rate. These two factors complete each other and the net result determines the effects of ambient press-ure on the evaporation rate of fuel droplets [11, 38]. To investigate the influence of ambient pressure on the evaporation behavior of pure alcohol droplets, the calculations were performed under the ambient pressure of 2.0 MPa, and the other conditions were the same as those in Figure 6. From the comparison between Figures 6 and 7, it is clear to see that the higher ambient pressure results in more pronounced thermal expansion and a longer heating period. This is because the higher ambient pressure increases the boiling temperature of the fuel components, and the droplet needs to be heated up to a high temperature to evaporate [38]. Additionally, the variation of heating period for different fuel droplets under the ambient pressure of 2.0 MPa in Figure 7b and d is similar to that under the atmospheric pressure.

The ambient temperature affects the way the evaporation rate of pure fuel droplets varies with ambient pressure. For the diesel droplet, the higher ambient pressure decreases the constant evaporation rate under both ambient temperatures as shown in Figure 7b and d. Nevertheless, the ambient pressure shows distinct effects on the evaporation behavior of alcohol droplets. Under the low-temperature condition of 573 K in Figure 7b, the constant evaporation rate of alcohol droplets

10 P. YI ET AL.

decreases with the increase of ambient pressure, especially for the long-chain alcohols (nonanol). On the contrary, under the high-temperature condition of 873 K in Figure 7d, the constant evaporation rate of all alcohol droplets increases with the ambient pressure, and the increased extent decreases with the increase of carbon number of alcohols. The explanation for this phenomenon is discussed above. The high ambient pressure reduces the mass transfer number and thus decreases the evapor-ation rate [38]. On the contrary, the high ambient pressure improves the droplet surface temperature and enhances the evaporation rate [11]. Note that the effect of pressure on droplet surface tempera-ture acts dominantly at high-temperature conditions [17]. Accordingly, the higher ambient pressure is more likely to improve the constant evaporation rate at high-temperature conditions, and exhibits a constant evaporation rate at low-temperature conditions. Based on the work of Ref. [11], this thresh-old temperature is different for different fuel components, and the value for diesel droplet is about 923 K. Comparing the results for diesel and alcohol droplets, the constant evaporation rate of alcohols is more likely to increase with ambient pressure than that of diesel.

3.4.3. Evaporation of blended droplets of diesel and alcohols at the atmospheric pressure The improved multicomponent evaporation model was used here to investigate the evaporation char-acteristics of the blended droplets of diesel and alcohols (Et 10: 10% mass fraction of ethanol; Et 20: 20% mass fraction of ethanol; Et 50: 50% mass fraction of ethanol; No 10: 10% mass fraction of nonanol; No 20: 20% mass fraction of nonanol; No 50: 50% mass fraction of nonanol). The calcu-lation was performed at the ambient condition of 0.1 MPa and 573 K. The temporal evolutions of droplet size, temperature, and vapor mass fraction on droplet surface for the specified fuel composi-tions are presented in Figure 8. It is worth noting that there is significant slope change in the droplet size evolution history for the blended droplet cases, especially for diesel and ethanol droplets, which is also observed in a similar work in the literature [5]. The extent of slope change increases with the increase of alcohol concentration. This is caused by the difference in the vaporability and the relative proportion of alcohols. The slope change is more pronounced in the blended droplet of diesel and

Figure 7. Variation of droplet size, constant evaporation rate, and heating period for different pure fuel droplets at high ambient pressure.

NUMERICAL HEAT TRANSFER, PART A 11

ethanol than that of diesel and nonanol as compared in Figure 8a and b. This is because the vaporability of nonanol is closer to the value of diesel.

The time evolution of surface temperature for the diesel droplet blended with ethanol and nonanol is shown in Figure 8c and d. Qualitative differences are noticed among different fuel droplets. For the pure ethanol and nonanol droplets, the wet bulb temperature is maintained after a short period, and the wet bulb temperature for either droplet is lower than its boiling temperature [39]. For the blended droplet, once the majority of alcohol fraction evaporates first and the latent heat of evaporation is larger than that of diesel, the droplet surface temperature during the alcohol evaporation period is very low. Nearly the end of alcohols distilled out, the droplet goes into the second heating period, and the droplet surface temperature increases sharply. The slope change in the evolution of surface temperature history is more apparent for the blended droplet with a larger concentration of alcohols. During the diesel components evaporation period, the surface temperature of the blended droplets is similar to that of pure diesel droplets. Since the latent heat of evaporation of nonanol is less than that of ethanol, the slope change in the evolution of surface temperature in the blends of diesel and nonanol is less pronounced than that in the blends of diesel and ethanol.

Figure 8. Variation of droplet size, temperature as well as surface vapor mass fraction versus time for the blended droplets of diesel and ethanol.

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Figure 8e and f shows the vapor mass faction on the droplet surface for blended droplets of Et 50 (50% mass fraction of ethanol) and No 50 (50% mass fraction of nonanol). It can be observed that for the droplet of Et 50, only ethanol vapor shows up on the droplet surface in the initial period. When the ethanol is nearly exhausted, the more-volatile components in diesel begin to evaporate. In comparison, the vapor mass fraction of more-volatile components in diesel (C8H18 and C10H22) increases synchro-nously with nonanol as shown in Figure 8f. The total vapor mass fraction on the droplet surface during the initial period includes not only nonanol, but also C8H18, C10H22, and C10H14. Hence, it can be con-cluded that under atmospheric pressure, the vaporability of short-chain alcohols is much higher than that of diesel components, whereas the long-chain alcohol of nonanol shows comparable vaporability with the more-volatile components of C8H18 and C10H22 in diesel. However, the droplet lifetime of pure ethanol is shorter than that of the pure nonanol droplet; this is mainly caused by the large latent heat of ethanol, which reduces the droplet temperature and thus decreases the droplet constant evaporation rate.

3.4.4. Evaporation of blended droplets of diesel and alcohols at the high ambient pressure To investigate the influence of the ambient pressure on the blended droplets of diesel and alcohols, the ambient condition was fixed at 873 K and 2.0 MPa, and the other initial conditions were the same as those in Figure 8. Figure 9 shows the evaporation behaviors of diesel droplets blended with ethanol and nonanol, namely the temporal evolution of droplet size and vapor mass fraction on the droplet surface. The addition of ethanol weakens the extent of thermal expansion for the diesel droplet. A noticeable slope change is also observed in the evolution of droplet size history for the blended drop-let of diesel and ethanol in Figure 9a, which is very similar to that at the atmospheric pressure in Figure 8a. The ethanol is expected to evaporate faster than the diesel components, and its vapor mass fraction shows up on the droplet surface initially, as shown in Figure 9c.

Note, unlike ethanol, the blended droplet of diesel and nonanol shows distinct evaporation beha-vior under different ambient pressures. As shown in Figure 9b, under the high-pressure condition, the

Figure 9. Variation of droplet size and surface vapor mass fraction versus time for the blended droplets of diesel and ethanol at high ambient pressure.

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evolution of droplet size history for the blended droplet of diesel and nonanol is very close to that of the pure diesel droplet, and there is no slope change because of the little difference in volatility of diesel and nonanol. The details are shown in Figure 9d. The vapor of nonanol and the more-volatile components in diesel simultaneously increases on the droplet surface. Compared with the results at the atmospheric pressure (Figure 8f), the difference in the volatilities of the nonanol and diesel com-ponents is decreased. Hence, it can be concluded that the evaporation characteristics of the long-chain alcohol of nonanol are more close to the diesel components under high-pressure conditions.

4. Conclusions

An improved multicomponent evaporation model was developed to simulate the evaporation process of the pure and blended droplets of diesel and different alcohols under various ambient conditions. This model considers a multicomponent diffusion submodel in the gas phase, the finite heat conductivity and mass diffusivity in the liquid phase, as well as the PSRK EOS on the vapor–liquid equilibrium. Based on the modeling results, the following conclusions are drawn. 1. Comparison between the model predictions and experimental measurements shows that the

improved evaporation model is capable of predicting accurately the evaporation characteristics of pure diesel, alcohols, and their blended droplets. The PSRK EOS is more suitable for predicting the evaporation behavior of the blended droplets of diesel and alcohols than Raoult’s law.

2. The pure diesel droplet shows a longer heating period than either alcohol droplet (C2–C9) because of the higher boiling temperature. For the pure alcohol droplets, the heating period increases with the increase of carbon number under all ambient conditions. Hence, the nonanol droplet shows the longest heating period, and the ethanol shows the shortest one among all the test alcohols.

3. Under low ambient temperatures, the evaporation rate of alcohol droplets decreases with the increase of ambient pressure, contrary to the high-temperature case. This finding reveals that the effect of pressure on the evaporation rate of alcohol droplets depends on the ambient tem-perature. The effect of pressure on reducing the mass transfer and decreasing the evaporation rate acts dominantly under low temperatures. On the contrary, improving the surface tempera-ture and increasing the evaporation rate caused by the high pressure have primary effects on the evaporation of alcohol droplets under high temperatures.

4. Under atmospheric pressure, the constant evaporation rate of alcohols significantly increases with the increase of carbon number, whereas this increasing extent decreases under the high-pressure condition. This phenomenon reveals that large difference in the latent heat of evaporation has primary effects on the evaporation rate of alcohol droplets under the low-pressure condition, and this effect for alcohol droplets decreases with the increase of ambient pressure.

5. The addition of short-chain alcohols in the diesel droplets causes significant slope change in the evolution of droplet size and temperature history under all ambient conditions. On the contrary, since there is a little difference in vaporability between the diesel components and nonanol, the addition of nonanol in diesel droplet under the high-temperature and -pressure condition has little effects on the evolution of droplet size history. This phenomenon indicates that the evaporation behavior of long-chain alcohol of nonanol is close to the diesel components under high-temperature and -pressure conditions.

Funding

This work was supported by the National Natural Science Foundation of China (Grant Nos. 51379034 and 51476020) and the China Scholarship Council.

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