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An analytical solution for wall film heating and evaporation
International Communications in Heat and Mass Transfer 87 (2017) 125–131
Contents lists available at ScienceDirect
International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt
An analytical solution for wall film heating and evaporation
MARK
⁎
Hong Liu, Yan'an Yan, Ming Jia , Yanzhi Zhang, Maozhao Xie, Hongchao Yin Key Laboratory of Ocean Energy Utilization and Energy Conservation of Ministry of Education, School of Energy and Power Engineering, Dalian University of Technology, Dalian 116024, PR China
A R T I C L E I N F O
A B S T R A C T
Keywords: Wall film Evaporation Analytical solution Internal combustion engine
The evaporation of the fuel wall film considerably affects the performance and exhaust emissions of internal combustion engines. An analytical model for wall film heating and evaporation was developed and applied in this paper for predicting the temporal and spatial temperature distributions of the liquid film. The effects of the heat conduction between the fuel film and the wall, the convection between the film surface and the surrounding gas, and the film evaporation were taken into account in the present analytical model. This analytical solution was validated by the predictions from the discrete numerical vaporization model, and it is found that accurate predictions can be obtained by the present model. In order to understand the evolution behavior of the wall film, the influence of the wall temperature, the ambient pressure, and the initial film thickness were investigated. The results indicate that the evolution of the wall film evaporation can be divided into two distinct stages, i.e., an initial rapid heating stage and a slow cooling stage. The lifetime of the wall film can be shortened by increasing the wall temperature, decreasing the ambient pressure and the initial wall film thickness. The purpose of this work is to reproduce the transient behaviors of the wall film heating and evaporation with an analytical solution, which is easy to setup and solve.
1. Introduction Spray-wall interaction is considered as a common and important phenomenon in internal combustion engines (IC) [1]. The wall film considerably affects engine performance and exhaust emissions, because the evaporation process of the fuel films attached on the piston and the cylinder walls is much slower than that of the airborne droplets, and produces a large amount of hydro‑carbons (HC) and particulate matter (PM) [2]. Thus, accurate prediction of the heating up and evaporation of the wall film is essential for understanding the wall film dynamics and evaporation characteristics. Several experiments [3–5] have been conducted to measure the thickness of the wall-wetted fuel film by using the laser induced fluorescence (LIF) after spray/wall impingement. The thickness of the adhered fuel film [6], the adhered fuel mass ratio [7], and the distribution of the liquid fuel and vapor [8] after impingement were measured. In addition, the influences of injection pressure [9], injection duration [6], injection angle [10,11], wall temperature [4,12], air flow [4], and injector configuration [13] on the wall film after spray/wall impingement were discussed in detail. However, to the best of our knowledge, no experimental data are available to describe the temperature and evaporation characteristics of the fuel film adhered on the wall, especially under high ambient
⁎
temperature, which is relevant to the operating conditions of practical engines, because the piston temperature is around 500 K [14], and the cylinder wall temperature is generally within 400–500 K [15]. This makes it a challenge to measure the instantaneous evolution of the fuel film thickness, since the evaporation of the wall film is in very small scales of time (several milliseconds) and space (several micrometers in thickness). Sazhin [16–19] proposed several evaporation models for fuel droplets based on the analytical method. It was found that the model was capable of providing accurate solutions by validating the analytical solutions against the available experimental data on droplet temperature. In this paper, a new wall film evaporation model is derived using the unsteady one-dimensional analytical method. The focus of this paper is on the convective heating of the wall film, as well as the wall heat flux considering constant wall temperature. The effects of the wall temperature, the ambient pressure, and the initial wall film thickness on the wall film heating up and evaporation are investigated. 2. Methodology Several simplifications are made in this study for the derivation of
Corresponding author. E-mail addresses: [email protected] (H. Liu), [email protected] (Y. Yan), [email protected] (M. Jia), [email protected] (Y. Zhang), [email protected] (M. Xie), [email protected] (H. Yin). http://dx.doi.org/10.1016/j.icheatmasstransfer.2017.07.009
0735-1933/ © 2017 Elsevier Ltd. All rights reserved.
International Communications in Heat and Mass Transfer 87 (2017) 125–131
H. Liu et al.
Nomenclature a A B BM c f F Fo h L m ṁ M N p t T x Y
Greek symbols
thermal diffusivity (m2 s− 1) parameter introduced in Eq. (16) parameter introduced in Eq. (16) Spalding number specific heat capacity (J kg− 1 K− 1) function defined by Eq. (20) function defined by Eq. (23) Fourier number convective heat transfer coefficient (W m− 2 K− 1) latent heat of evaporation (kJ kg− 1) the number of values in Eq. (23) evaporation rate (g m− 2 s− 1) molar mass (g mol− 1) coefficient defined in Eq. (23) pressure (MPa) time (s) temperature (K) the distance to wall (m) mass fraction
β δ δ̇ Δt θ λ ρ Subscripts a eff g l p s v w 0 ∞
the wall film evaporation model. First, the temperature of the wall is less than the boiling point temperature of fuel. This is consistent with the conditions in typical diesel engines, where the in-cylinder pressure is very high, and the temperature of the wall is usually less than the boiling temperature of the fuel. Second, the wall film is assumed to be stationary based on the fact that the film velocity and surface waves can be ignored since the film velocity is much smaller than the air velocity above the film in the cylinder. Third, it is assumed that the liquid film directly contacts with the wall, and the convective heat transfer between the film and the turbulent air, and the conduction between the film and the wall are taken into account. Fourth, the inertial and gravitational forces on the wall film are negligible. Fifth, only heat transfer along the film height direction is considered. According to above assumptions, the basic mechanism considered for the simplified wall film model is shown in Fig. 1. These models and analytical solutions are presented and discussed below.
∂T ∂x
x=δ
δ̇ =
h ln(1 + BM) ρ ls c pg
(5)
and the evaporation rate (ṁ ) can be expressed as
ṁ =
h ln(1 + BM) c pg
(6)
where cpg is the specific heat capacity of gas, BM is the Spalding mass transfer number defined as [17]
BM =
Yvs − Yv ∞ 1 − Yvs
(7)
where Yvs and Yv ∞ are the mass fractions of the fuel vapor near the film surface [17] and at ambient conditions, respectively. −1
⎛ p ⎞ Ma ⎤ Yvs = ⎡ ⎢1 + ⎜ p − 1⎟ Mf ⎥ vs ⎝ ⎠ ⎣ ⎦
(1)
(8)
where p and pvs are the ambient pressure and the pressure of the saturated fuel vapor near the surface of wall film, respectively. Ma and Mf are the molecular weight of air and fuel, respectively. The newly obtained thickness of the wall film (δnew) can be expressed as
where t is time; x is the distance from solid wall; al = λl/(ρlcl) is thermal diffusivity, where λl, ρl and cl are the liquid thermal conductivity, density, and specific heat capacity, respectively. Assuming that the initial wall film temperature (T0) is uniform as (2)
The boundary condition at x = 0 is considered as a constant wall temperature (Tw) as
T (0, t ) = Tw
(4)
where h is the convection heat transfer coefficient; Tg is the gas ambient temperature; Ts is the film surface temperature; L is specific latent heat of evaporation, and δ is the thickness of wall film. Based on the following equation from Ref. [20], the variation of the liquid film thickness (δ )̇ is taken into account during the evaporation process, which is estimated as
According to the above assumptions, the model of the wall film evaporation can be derived based on the unsteady one-dimensional equations including the liquid heating and film evaporation. The transient conduction equation for a stationary liquid film with the temperature T = T (x, t) can be written as
T (x , 0) = T0
air effective ambient gas liquid constant pressure surface vapor wall initial ambient conditions
h (Tg − Ts) = ρ ls Lδ ̇ + λl
2.1. Energy equations for wall film
∂T ∂ 2T = al 2 ∂t ∂x
parameter introduced in Eq. (23) film thickness (m) reduction rate of wall film thickness (m s− 1) time step (ms) excess temperature (K) thermal conductivity (W m− 1 K− 1) density (kg m− 3)
(3)
Considering that the film surface contacted with the surrounding gas is heated by convection, and cooled down due to the evaporation fuel, another boundary condition at x = 0 can be obtained according to the energy balance equation as [19]
Fig. 1. Sketch of the wall film model.
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International Communications in Heat and Mass Transfer 87 (2017) 125–131
H. Liu et al.
̇ t δnew = δold − δ ∆
The final analytical solution of the temperature distribution within the wall film can be presented in the following form
(9)
where δold is the previous thickness of the film, and Δt is the time step. Based on the unsteady heat conduction differential equations in Eqs. (1)–(4), an analytical solution of the temperature distribution within the film can be obtained. Introducing several new variables
ρ ls Lδ ̇ h
(10)
θ = T (x , t ) − Teff
(11)
Teff = Tg −
T (x , t ) = F (x , t ) −
(12)
θ (x , 0) = T0 − Teff
(13)
θ (0, t ) = Tw − Teff
(14)
∂θ ∂x
(15)
h θ=0 λl
+ x=δ
1. Initialize δ0, Tw, Tg, p, h and T0. 2. Calculate the thermodynamic properties of the liquid fuel and the gases based on the formulas in Ref. [22]. 3. Calculate the evaporation rate of the fuel film on the surface using Eq. (6). 4. Calculate the effective temperature using Eq. (10). 5. Calculate F(x, t) using Eq. (23). Because the term exp(− βm2Fo) rapidly reduces to zero with the increase of m. So m = 5 is adopted in this paper, which is based on a compromise of computational accuracy and efficiency. 6. The temperature distribution in the wall film is obtained based on Eq. (26). The temperature on the surface (Ts) is very important in the calculation process, because it direct impacts the prediction of evaporation rate. 7. Calculate the film thickness using Eq. (9). Assuming that the evaporation process of the liquid film is considered to be finished when the instantaneous thickness of the wall film (δnew) is less than 1 μm. 8. Return to Step 2 and repeat the calculations for the next time step.
where Teff is the effective temperature [21], and θ is the excess temperature. In order to solve the above equations using the method of separating variables, the boundary condition (Eq. (14)) must be converted to a homogeneous equation by introducing
F (x , t ) = θ (x , t ) + Ax + B
(16)
where A and B are coefficients to be solved. When F(x, t) is substituted to the boundary conditions Eqs. (14)–(15), the expressions of A and B can be obtained as
A=
3. Results and discussion
h (Tw − Teff ) λl + hδ
(17)
The above model is applied to study the heating and evaporation of wall film under the conditions relevant to typical diesel engines in this section. As a representative diesel surrogate, dodecane (C12H26) is taken in this study. The variations of thermodynamic properties with temperature are taken into account [22]. If there is no special statement, the input parameters are taken as: the initial thickness of wall film δ0 = 20 μm, the ambient temperature Tg = 1000 K, the wall temperature Tw = 450 K, the initial film temperature T0 = 400 K, the ambient pressure p = 1 MPa, and the convective heat transfer coefficient h = 2000 W m− 2 K− 1. Meanwhile, according the operation conditions in practical engines, the range of the initial parameters used in the calculation are listed in Table 1.
(18)
B = Teff − Tw
Then substituting Eq. (16) into Eqs. (12)–(15), a new set of equations can be shown as
∂F ∂ 2F = al 2 ∂t ∂x
(19)
F (x , 0) = T0 − Tb +
h (Tw − Teff ) x = f (x ) λl + hδ
(20)
F (0, t ) = 0 ∂F ∂x
+ x=δ
(21)
h F=0 λl
(22)
3.1. Validation of analytical method and the discrete numerical method
Finally, the solution of F(x, t) is deduced by using the method of variable separation as ∞
F (x , t ) =
βm ⎞ 1 x exp(−βm2 Fo) N (βm ) ⎝ δ ⎠
∑ sin ⎛
⎜
m=1
⎟
∫0
δ
To verify the present analytical model, the analytical solutions are compared with the predictions of the discrete numerical model [25] for wall film evaporation under the conditions with the wall temperature Tw = 400 K. Comparisons of the evolution of the film surface temperature and the film thickness are shown in Fig. 2. As can be seen, the wall film thickness almost keeps constant at the very early stage, and then rapidly decreases in most period of its lifetime. Meanwhile, the film surface temperature rapidly increases at the early stage, and gradually approaches to the wall temperature after its peak. It can be found from Fig. 2 that the predicted film thickness and the film surface temperature from the analytical model agree with the numerical solutions from the discrete numerical model rather well.
β f (x ) sin ⎛ m x ⎞ dx ⎝ δ ⎠ ⎜
⎟
(23) where Fo = alt/δ is Fourier number, βm is a set of positive values numbered in ascending order (m = 1, 2, 3, …), which meets the condition λβ cos β + hδ sin β = 0. By solving the transcendental equation, the infinitely many characteristic functions sin(βmx/δ) (m = 1, 2, 3, …) can be obtained, which compose an orthogonal function system, and has following property 2
∫0
δ
0, β β sin ⎛ m x ⎞ sin ⎛ n x ⎞ dx = ⎧ ⎨ ⎝ δ ⎠ ⎝δ ⎠ ⎩ N (βm ), ⎜
⎟
⎜
⎟
m≠n m=n
(24)
Table 1 The range of the initial parameters.
where N(βm) is the norm of the above-mentioned orthogonal function system. In this paper, it can be written as
N (βm ) =
∫0
δ
(26)
2.2. Major steps for solving the analytical model
Eqs. (1)–(4) can be rearranged as
∂θ ∂ 2θ = al 2 ∂t ∂x
h (Tw − Teff ) x + Tw λl + hδ
β δ (βm − sin βm cos βm ) sin2 ⎛ m x ⎞ dx = δ 2βm ⎝ ⎠ ⎜
⎟
(25) 127
δ0 (μm) [1,5]
Tg (K) [23]
Tw (K) [14]
T0 (K)
p (MPa) [24]
h (W m− 2 K− 1) [22,24]
10–40
800–1100
400–550
300–450
0.5–4.0
1000–3000
International Communications in Heat and Mass Transfer 87 (2017) 125–131
H. Liu et al.
Fig. 2. Comparisons of the analytical solutions and the discrete numerical solutions for the evolution of the wall film thickness and the surface temperature.
3.2. Wall film heat and evaporation characteristics Based on the analytical evaporation model, the wall film temperature and thickness versus time can be obtained in detail. From the temperature distribution inside the wall film shown in Fig. 3, the nonlinear evaporation characteristics of the wall film can be well illustrated. As can be seen, the overall trend of the temperature distribution is similar for the cases with different wall temperatures. The evolution of the wall film can be divided into two distinct stages based on the variation of the film surface temperature, i.e., an initial rapid heating stage and a slow cooling stage. The wall film surface temperature increases rapidly at the initial heating stage, because it is continuously heated by the surrounding gas due to the temperature difference between the film surface and the surrounding gas. Thereafter, the surface temperature approaches to the wall temperature gradually because of the cooling effects from the wall conduction and the film evaporation. During this stage, the heating and evaporation of the liquid film coexist. In addition, the temperature profiles inside the wall film at different time are shown in Fig. 4. At the condition with the wall temperature of 450 K, it can be seen that the temperatures on both the two boundaries rise immediately at the very initial stage since both the wall temperature and the ambient temperature are higher than the initial wall film temperature. Because the conductive heat flux slowly transfers from the two boundaries to the film inside, the temperatures inside the film are still relatively low. Then the average temperature of the overall film increases gradually, and the wall film thickness becomes thinner with the extension of time due to the film evaporation. At the final stage, as the whole film temperature is higher than the wall temperature, heat only transfers from the film surface contacting the surrounding gas to the wall through the film. Meanwhile, because the fuel film is very thin and the heat flux is relatively steady, the temperature profile inside the film tends to be a linear distribution.
Fig. 3. Temperature distributions inside the wall film at different wall temperatures.
temperature resulting in a larger evaporation rate. Consequently, the evaporation process requires more heat energy from the wall film. Meanwhile, another interesting phenomenon is that the surface temperatures reach the peak at almost the same time under different wall temperature conditions, which can be explained as follows. At the conditions with the same initial film temperature and ambient temperature, the heat flux from the film surface is balanced with the convective heat energy exchanged between the film and the surrounding gas, thus the heat flux is almost the same at different wall temperatures. Although the wall temperature affects the temperature profile inside the film at this stage, its influence on the heat diffusion from the film surface to the film inside is insignificant. Meanwhile, the heat diffusion rate is mainly controlled by the thermal diffusivity, which is almost identical under the conditions with different wall temperatures. As a
3.3. Influence of various parameters on the wall film evaporation 3.3.1. Effects of wall temperature The effects of wall temperature on the film evaporation process are illustrated in Figs. 5–7. It can be observed that the wall temperature significantly affects the film heating up and evaporation. As shown in Fig. 5, a higher wall temperature results in a more rapid increase of the surface temperature at the initial stage, leading to a higher maximum of surface temperature. The reason is that a higher wall temperature provides larger heat flux for the fuel film to heat up. In addition, it is interesting to note that the differences in the maximums of surface temperatures among the cases with various wall temperatures are much smaller than the differences in the wall temperatures. It is mainly because a higher wall temperature leads to a higher wall film surface 128
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H. Liu et al.
Fig. 4. Temperature distribution inside the wall film at different time.
Fig. 7. Wall film thickness versus time at different wall temperatures.
Fig. 5. Surface temperature versus time at different wall temperatures. Fig. 8. Temporal surface temperature versus film thickness at different ambient pressures.
Fig. 6. Evaporation rate versus time at different wall temperatures. Fig. 9. Surface temperature versus time at different ambient pressures.
result, the maximums surface temperatures at different wall temperatures appear nearly simultaneously. As can be found from Fig. 6, the trends of the evaporation rates are closely related to the evolutions of the surface temperatures. With the increase of wall temperature, the differences in the peaks of evaporation rates tend to become more significant. This phenomenon also indicates that a higher wall temperature would lead to a much larger evaporation rate.
Fig. 7 shows the film thickness versus time at different wall temperatures. It demonstrates that the lifetime of the liquid wall film reduces remarkably by raising the wall temperature. In addition, with increasing wall temperature, the differences in the lifetimes of the film among various cases decrease. In other word, the evaporation of the wall film can be strengthened by increasing the wall temperature, and
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H. Liu et al.
Fig. 13. Evaporation rate versus time at different initial film thicknesses.
Fig. 10. Evaporation rate versus time at different ambient pressures.
from the surrounding gas to the wall film is used to heat up the wall film rather than for evaporation, since the temperature of the wall film is much lower than the saturation temperature at the local pressure. Thus, although the convective heat transfer between the air and the film is quite strong, the evaporation of the wall film cannot occur observably. With the time extension, the surface temperature rises much slower than that at the beginning, and the wall film thickness decreases obviously, especially when the surface temperature is around the peak of the curve shown in Fig. 8. It can be seen from Fig. 9 that with the increase of ambient pressure, the maximum surface temperature increases markedly. However, the instantaneous evaporation rate decreases with the increasing ambient pressure, as shown in Fig. 10. That means high ambient pressure condition would restrain the evaporation of fuel, because the saturation temperature is high when its corresponding ambient pressure is large. Meanwhile, the heat flux absorbed from the environment is mainly used to heat up the fuel film rather than for evaporation. The evolution of the wall film thickness at different ambient pressures is shown in Fig. 11, which reveals that higher ambient pressure prolongs the lifetime of the wall film.
Fig. 11. Wall film thickness versus time at different ambient pressures.
3.3.3. Effects of the initial wall film thickness The effects of the initial wall film thickness on the film evaporation are illustrated in Figs. 12 and 13. At the first stage, the case with a thinner wall film exhibits a faster heating trend on the surface. This is because less heat energy of evaporation is required for thinner films, so the heat flux from the wall and the environment can heat up the thinner fuel films more quickly. However, when the wall film is thinner, the cooling effect of the wall is more significant, so the peak surface temperature of the thinner film is lower than that of the thicker one, as shown in Fig. 12. Fig. 13 shows the evolution of the evaporation rate under different initial wall film thicknesses. It is obvious that the wall film with the thinner initial thickness vanishes sooner owing to the more intense evaporation process. From above results, it can be also found that the liquid wall film will exist for several cycles after the formation of the liquid fuel film in engines, especially for the case with thick fuel film. The slow evaporation rate of the liquid fuel on the in-cylinder surface could cause high pollution emissions.
Fig. 12. Surface temperature versus time at different initial film thicknesses.
its effect is more evident when the wall temperature is relatively low. 4. Conclusions 3.3.2. Effects of the ambient pressure Fig. 8 shows the evolution of the surface temperature with the wall film thickness at different ambient pressures. It can be seen that, at the initial heating up stage, the surface temperature rises quickly, but the thickness of the wall film nearly keeps constant without evident evaporation. This phenomenon indicates that most of the heat transfer
A new analytical solution was proposed for description of the transient heating and evaporation of the liquid wall film under the conditions with the constant wall temperature. In the analytical solution, the effects of convection from the surrounding hot gas and the heat conduction from the contacted wall are considered. Based on the 130
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model, the characteristics of the evaporation of the wall film under engine-relevant conditions were investigated. The main conclusions are summarized as follows.
[4] Y.S. Cheng, K. Deng, T. Li, The coupling influence of airflow and temperature on the wall-wetted fuel film distribution, Exp. Thermal Fluid Sci. 34 (2) (2010) 227–233. [5] J. Senda, M. Ohnishi, T. Takahashi, Measurement and modeling on wall wetted fuel film profile and mixture preparation in intake port of SI engine, SAE Paper, 199901-0798, 1999. [6] M. Behnia, B.E. Milton, Fundamentals of fuel film formation and motion in SI engine induction systems, Energy Convers. Manag. 42 (1) (2001) 1751–1768. [7] H. Cho, K. Min, H. Cho, Measurement of liquid fuel film distribution on the cylinder liner of a spark ignition engine using the laser-induced fluorescence technique, Meas. Sci. Technol. 14 (7) (2003) 975–982. [8] W. Mathews, C.F. Lee, J.E. Peters, Experimental investigations of spray/wall impingement, Atomization Sprays 13 (2003) 223–242. [9] M. Alonso, P.J. Kay, P.J. Bowen, Quantification of transient fuel films under elevated ambient pressure environments, Atomization Sprays 22 (1) (2012) 79–95. [10] K. Muramatsu, K. Yamamoto, N. Jinno, Measurement of fuel liquid film under the different injection pressure, SAE Paper, 2013-32-9167, 2013. [11] M.R. Wei, F. Liu, H. Wen, Numerical simulation and experimental research of DME spray wall impingement, Chin. Intern. Combust. Engine Eng. 2 (2006) 10–15. [12] V.S. Costanzo, Effect of In-cylinder Liquid Fuel Films on Engine-out Unburned Hydrocarbon Emissions for SI Engines (PhD thesis), Massachusetts Institute of Technology, Boston, United States, 2011. [13] H. Cho, M. Kim, K. Min, The effect of liquid fuel on the cylinder liner on engine-out hydrocarbon emissions in SI engines, SAE Paper, 2001-01-3489, 2001. [14] X. Lu, Q. Li, W. Zhang, Thermal analysis on piston of marine diesel engine, Appl. Therm. Eng. 50 (1) (2013) 168–176. [15] C.D. Rakopoulos, E.G. Giakoumis, D.C. Rakopoulos, Cylinder wall temperature effects on the transient performance of a turbocharged diesel engine, Energy Convers. Manag. 45 (17) (2004) 2627–2638. [16] S.S. Sazhin, W. Abdelghaffar, P. Krutitskii, New approaches to numerical modelling of droplet transient heating and evaporation, Int. J. Heat Mass Transf. 48 (1) (2005) 4215–4228. [17] A.P. Kryukov, V.Y. Levashov, S.S. Sazhin, Evaporation of diesel fuel droplets: kinetic versus hydrodynamic models, Int. J. Heat Mass Transf. 47 (12) (2004) 2541–2549. [18] S.S. Sazhin, A.E. Elwardany, P.A. Krutitskii, Multicomponent droplet heating and evaporation: numerical simulation versus experimental data, Int. J. Therm. Sci. 50 (7) (2011) 1164–1180. [19] S.S. Sazhin, P.A. Krutitskii, W.A. Abdelghaffar, Transient heating of diesel fuel droplets, Int. J. Heat Mass Transf. 47 (14) (2004) 3327–3340. [20] Y.A. Yan, H. Liu, M. Jia, A one-dimensional unsteady wall film evaporation model, Int. J. Heat Mass Transf. 88 (2015) 138–148. [21] S.S. Sazhin, Advanced models of fuel droplet heating and evaporation, Prog. Energy Combust. Sci. 32 (2) (2006) 162–214. [22] B. Abramzon, S.S. Sazhin, Convective vaporization of a fuel droplet with thermal radiation absorption, Fuel 85 (1) (2006) 32–46. [23] R. Finesso, E. Spessa, A real time zero-dimensional diagnostic model for the calculation of in-cylinder temperatures, HRR and nitrogen oxides in diesel engines, Energy Convers. Manag. 79 (2) (2014) 498–510. [24] C.D. Rakopoulos, G.C. Mavropoulos, Experimental evaluation of local instantaneous heat transfer characteristics in the combustion chamber of air-cooled direct injection diesel engine, Energy 33 (7) (2008) 1084–1099. [25] Y.Z. Zhang, M. Jia, H. Liu, Development of an improved liquid film model for spray/ wall interaction under engine-relevant conditions, Int. J. Multiphase Flow 79 (2016) 74–87.
(1) The predictions of the analytical model are well consistent with the results of the discrete numerical model. The new analytical model provides an accurate and effective solution for the wall film heating up and evaporation. (2) According to the variation of the surface temperature of the liquid film, the evolution behavior of the wall film can be divided into an initial rapid heating stage and a slow cooling stage. At the initial heating stage, because huge temperature difference exists between the wall film and the surrounding gas, the surface temperature of fuel film rise significantly and the heat flux transfers to the film inside gradually. Thereafter, the surface temperature approaches to the wall temperature gradually because of the cooling effects from the wall conduction and the film evaporation. (3) The wall temperature, the ambient pressure, and the initial film thickness have important effects on the heating and evaporation characteristics of the wall film. (4) The vaporizing time decreases with the increasing wall temperature, and the decreasing ambient pressure and wall film thickness. (5) An increase in the ambient pressure results in a higher surface temperature of the film, but a lower evaporation rate. (6) At the final stage of the evaporation process, the temperature of the fuel film approaches to the wall temperature until the film is evaporated completely. Acknowledgments The authors are grateful to the National Natural Science Foundation of China (Grant No. 51376038, 51476020) and the Fundamental Research Funds for the Central Universities (Grant No. DUT15JJ(G)04) for the financial supports of this project. References [1] J. Senda, T. Kanda, M. Al-Roub, Modeling spray impingement considering fuel film formation on the wall, SAE Paper, 970047, 1997. [2] T. Alger, Y. Huang, M. Hall, Liquid film evaporation off the piston of a direct injection gasoline engine, SAE Paper, 2001-01-1204, 2001. [3] Y.S. Cheng, K. Deng, T. Li, Measurement and simulation of wall-wetted fuel film thickness, Int. J. Therm. Sci. 49 (4) (2010) 733–739.
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An analytical solution for wall film heating and evaporation
MARK
⁎
Hong Liu, Yan'an Yan, Ming Jia , Yanzhi Zhang, Maozhao Xie, Hongchao Yin Key Laboratory of Ocean Energy Utilization and Energy Conservation of Ministry of Education, School of Energy and Power Engineering, Dalian University of Technology, Dalian 116024, PR China
A R T I C L E I N F O
A B S T R A C T
Keywords: Wall film Evaporation Analytical solution Internal combustion engine
The evaporation of the fuel wall film considerably affects the performance and exhaust emissions of internal combustion engines. An analytical model for wall film heating and evaporation was developed and applied in this paper for predicting the temporal and spatial temperature distributions of the liquid film. The effects of the heat conduction between the fuel film and the wall, the convection between the film surface and the surrounding gas, and the film evaporation were taken into account in the present analytical model. This analytical solution was validated by the predictions from the discrete numerical vaporization model, and it is found that accurate predictions can be obtained by the present model. In order to understand the evolution behavior of the wall film, the influence of the wall temperature, the ambient pressure, and the initial film thickness were investigated. The results indicate that the evolution of the wall film evaporation can be divided into two distinct stages, i.e., an initial rapid heating stage and a slow cooling stage. The lifetime of the wall film can be shortened by increasing the wall temperature, decreasing the ambient pressure and the initial wall film thickness. The purpose of this work is to reproduce the transient behaviors of the wall film heating and evaporation with an analytical solution, which is easy to setup and solve.
1. Introduction Spray-wall interaction is considered as a common and important phenomenon in internal combustion engines (IC) [1]. The wall film considerably affects engine performance and exhaust emissions, because the evaporation process of the fuel films attached on the piston and the cylinder walls is much slower than that of the airborne droplets, and produces a large amount of hydro‑carbons (HC) and particulate matter (PM) [2]. Thus, accurate prediction of the heating up and evaporation of the wall film is essential for understanding the wall film dynamics and evaporation characteristics. Several experiments [3–5] have been conducted to measure the thickness of the wall-wetted fuel film by using the laser induced fluorescence (LIF) after spray/wall impingement. The thickness of the adhered fuel film [6], the adhered fuel mass ratio [7], and the distribution of the liquid fuel and vapor [8] after impingement were measured. In addition, the influences of injection pressure [9], injection duration [6], injection angle [10,11], wall temperature [4,12], air flow [4], and injector configuration [13] on the wall film after spray/wall impingement were discussed in detail. However, to the best of our knowledge, no experimental data are available to describe the temperature and evaporation characteristics of the fuel film adhered on the wall, especially under high ambient
⁎
temperature, which is relevant to the operating conditions of practical engines, because the piston temperature is around 500 K [14], and the cylinder wall temperature is generally within 400–500 K [15]. This makes it a challenge to measure the instantaneous evolution of the fuel film thickness, since the evaporation of the wall film is in very small scales of time (several milliseconds) and space (several micrometers in thickness). Sazhin [16–19] proposed several evaporation models for fuel droplets based on the analytical method. It was found that the model was capable of providing accurate solutions by validating the analytical solutions against the available experimental data on droplet temperature. In this paper, a new wall film evaporation model is derived using the unsteady one-dimensional analytical method. The focus of this paper is on the convective heating of the wall film, as well as the wall heat flux considering constant wall temperature. The effects of the wall temperature, the ambient pressure, and the initial wall film thickness on the wall film heating up and evaporation are investigated. 2. Methodology Several simplifications are made in this study for the derivation of
Corresponding author. E-mail addresses: [email protected] (H. Liu), [email protected] (Y. Yan), [email protected] (M. Jia), [email protected] (Y. Zhang), [email protected] (M. Xie), [email protected] (H. Yin). http://dx.doi.org/10.1016/j.icheatmasstransfer.2017.07.009
0735-1933/ © 2017 Elsevier Ltd. All rights reserved.
International Communications in Heat and Mass Transfer 87 (2017) 125–131
H. Liu et al.
Nomenclature a A B BM c f F Fo h L m ṁ M N p t T x Y
Greek symbols
thermal diffusivity (m2 s− 1) parameter introduced in Eq. (16) parameter introduced in Eq. (16) Spalding number specific heat capacity (J kg− 1 K− 1) function defined by Eq. (20) function defined by Eq. (23) Fourier number convective heat transfer coefficient (W m− 2 K− 1) latent heat of evaporation (kJ kg− 1) the number of values in Eq. (23) evaporation rate (g m− 2 s− 1) molar mass (g mol− 1) coefficient defined in Eq. (23) pressure (MPa) time (s) temperature (K) the distance to wall (m) mass fraction
β δ δ̇ Δt θ λ ρ Subscripts a eff g l p s v w 0 ∞
the wall film evaporation model. First, the temperature of the wall is less than the boiling point temperature of fuel. This is consistent with the conditions in typical diesel engines, where the in-cylinder pressure is very high, and the temperature of the wall is usually less than the boiling temperature of the fuel. Second, the wall film is assumed to be stationary based on the fact that the film velocity and surface waves can be ignored since the film velocity is much smaller than the air velocity above the film in the cylinder. Third, it is assumed that the liquid film directly contacts with the wall, and the convective heat transfer between the film and the turbulent air, and the conduction between the film and the wall are taken into account. Fourth, the inertial and gravitational forces on the wall film are negligible. Fifth, only heat transfer along the film height direction is considered. According to above assumptions, the basic mechanism considered for the simplified wall film model is shown in Fig. 1. These models and analytical solutions are presented and discussed below.
∂T ∂x
x=δ
δ̇ =
h ln(1 + BM) ρ ls c pg
(5)
and the evaporation rate (ṁ ) can be expressed as
ṁ =
h ln(1 + BM) c pg
(6)
where cpg is the specific heat capacity of gas, BM is the Spalding mass transfer number defined as [17]
BM =
Yvs − Yv ∞ 1 − Yvs
(7)
where Yvs and Yv ∞ are the mass fractions of the fuel vapor near the film surface [17] and at ambient conditions, respectively. −1
⎛ p ⎞ Ma ⎤ Yvs = ⎡ ⎢1 + ⎜ p − 1⎟ Mf ⎥ vs ⎝ ⎠ ⎣ ⎦
(1)
(8)
where p and pvs are the ambient pressure and the pressure of the saturated fuel vapor near the surface of wall film, respectively. Ma and Mf are the molecular weight of air and fuel, respectively. The newly obtained thickness of the wall film (δnew) can be expressed as
where t is time; x is the distance from solid wall; al = λl/(ρlcl) is thermal diffusivity, where λl, ρl and cl are the liquid thermal conductivity, density, and specific heat capacity, respectively. Assuming that the initial wall film temperature (T0) is uniform as (2)
The boundary condition at x = 0 is considered as a constant wall temperature (Tw) as
T (0, t ) = Tw
(4)
where h is the convection heat transfer coefficient; Tg is the gas ambient temperature; Ts is the film surface temperature; L is specific latent heat of evaporation, and δ is the thickness of wall film. Based on the following equation from Ref. [20], the variation of the liquid film thickness (δ )̇ is taken into account during the evaporation process, which is estimated as
According to the above assumptions, the model of the wall film evaporation can be derived based on the unsteady one-dimensional equations including the liquid heating and film evaporation. The transient conduction equation for a stationary liquid film with the temperature T = T (x, t) can be written as
T (x , 0) = T0
air effective ambient gas liquid constant pressure surface vapor wall initial ambient conditions
h (Tg − Ts) = ρ ls Lδ ̇ + λl
2.1. Energy equations for wall film
∂T ∂ 2T = al 2 ∂t ∂x
parameter introduced in Eq. (23) film thickness (m) reduction rate of wall film thickness (m s− 1) time step (ms) excess temperature (K) thermal conductivity (W m− 1 K− 1) density (kg m− 3)
(3)
Considering that the film surface contacted with the surrounding gas is heated by convection, and cooled down due to the evaporation fuel, another boundary condition at x = 0 can be obtained according to the energy balance equation as [19]
Fig. 1. Sketch of the wall film model.
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̇ t δnew = δold − δ ∆
The final analytical solution of the temperature distribution within the wall film can be presented in the following form
(9)
where δold is the previous thickness of the film, and Δt is the time step. Based on the unsteady heat conduction differential equations in Eqs. (1)–(4), an analytical solution of the temperature distribution within the film can be obtained. Introducing several new variables
ρ ls Lδ ̇ h
(10)
θ = T (x , t ) − Teff
(11)
Teff = Tg −
T (x , t ) = F (x , t ) −
(12)
θ (x , 0) = T0 − Teff
(13)
θ (0, t ) = Tw − Teff
(14)
∂θ ∂x
(15)
h θ=0 λl
+ x=δ
1. Initialize δ0, Tw, Tg, p, h and T0. 2. Calculate the thermodynamic properties of the liquid fuel and the gases based on the formulas in Ref. [22]. 3. Calculate the evaporation rate of the fuel film on the surface using Eq. (6). 4. Calculate the effective temperature using Eq. (10). 5. Calculate F(x, t) using Eq. (23). Because the term exp(− βm2Fo) rapidly reduces to zero with the increase of m. So m = 5 is adopted in this paper, which is based on a compromise of computational accuracy and efficiency. 6. The temperature distribution in the wall film is obtained based on Eq. (26). The temperature on the surface (Ts) is very important in the calculation process, because it direct impacts the prediction of evaporation rate. 7. Calculate the film thickness using Eq. (9). Assuming that the evaporation process of the liquid film is considered to be finished when the instantaneous thickness of the wall film (δnew) is less than 1 μm. 8. Return to Step 2 and repeat the calculations for the next time step.
where Teff is the effective temperature [21], and θ is the excess temperature. In order to solve the above equations using the method of separating variables, the boundary condition (Eq. (14)) must be converted to a homogeneous equation by introducing
F (x , t ) = θ (x , t ) + Ax + B
(16)
where A and B are coefficients to be solved. When F(x, t) is substituted to the boundary conditions Eqs. (14)–(15), the expressions of A and B can be obtained as
A=
3. Results and discussion
h (Tw − Teff ) λl + hδ
(17)
The above model is applied to study the heating and evaporation of wall film under the conditions relevant to typical diesel engines in this section. As a representative diesel surrogate, dodecane (C12H26) is taken in this study. The variations of thermodynamic properties with temperature are taken into account [22]. If there is no special statement, the input parameters are taken as: the initial thickness of wall film δ0 = 20 μm, the ambient temperature Tg = 1000 K, the wall temperature Tw = 450 K, the initial film temperature T0 = 400 K, the ambient pressure p = 1 MPa, and the convective heat transfer coefficient h = 2000 W m− 2 K− 1. Meanwhile, according the operation conditions in practical engines, the range of the initial parameters used in the calculation are listed in Table 1.
(18)
B = Teff − Tw
Then substituting Eq. (16) into Eqs. (12)–(15), a new set of equations can be shown as
∂F ∂ 2F = al 2 ∂t ∂x
(19)
F (x , 0) = T0 − Tb +
h (Tw − Teff ) x = f (x ) λl + hδ
(20)
F (0, t ) = 0 ∂F ∂x
+ x=δ
(21)
h F=0 λl
(22)
3.1. Validation of analytical method and the discrete numerical method
Finally, the solution of F(x, t) is deduced by using the method of variable separation as ∞
F (x , t ) =
βm ⎞ 1 x exp(−βm2 Fo) N (βm ) ⎝ δ ⎠
∑ sin ⎛
⎜
m=1
⎟
∫0
δ
To verify the present analytical model, the analytical solutions are compared with the predictions of the discrete numerical model [25] for wall film evaporation under the conditions with the wall temperature Tw = 400 K. Comparisons of the evolution of the film surface temperature and the film thickness are shown in Fig. 2. As can be seen, the wall film thickness almost keeps constant at the very early stage, and then rapidly decreases in most period of its lifetime. Meanwhile, the film surface temperature rapidly increases at the early stage, and gradually approaches to the wall temperature after its peak. It can be found from Fig. 2 that the predicted film thickness and the film surface temperature from the analytical model agree with the numerical solutions from the discrete numerical model rather well.
β f (x ) sin ⎛ m x ⎞ dx ⎝ δ ⎠ ⎜
⎟
(23) where Fo = alt/δ is Fourier number, βm is a set of positive values numbered in ascending order (m = 1, 2, 3, …), which meets the condition λβ cos β + hδ sin β = 0. By solving the transcendental equation, the infinitely many characteristic functions sin(βmx/δ) (m = 1, 2, 3, …) can be obtained, which compose an orthogonal function system, and has following property 2
∫0
δ
0, β β sin ⎛ m x ⎞ sin ⎛ n x ⎞ dx = ⎧ ⎨ ⎝ δ ⎠ ⎝δ ⎠ ⎩ N (βm ), ⎜
⎟
⎜
⎟
m≠n m=n
(24)
Table 1 The range of the initial parameters.
where N(βm) is the norm of the above-mentioned orthogonal function system. In this paper, it can be written as
N (βm ) =
∫0
δ
(26)
2.2. Major steps for solving the analytical model
Eqs. (1)–(4) can be rearranged as
∂θ ∂ 2θ = al 2 ∂t ∂x
h (Tw − Teff ) x + Tw λl + hδ
β δ (βm − sin βm cos βm ) sin2 ⎛ m x ⎞ dx = δ 2βm ⎝ ⎠ ⎜
⎟
(25) 127
δ0 (μm) [1,5]
Tg (K) [23]
Tw (K) [14]
T0 (K)
p (MPa) [24]
h (W m− 2 K− 1) [22,24]
10–40
800–1100
400–550
300–450
0.5–4.0
1000–3000
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Fig. 2. Comparisons of the analytical solutions and the discrete numerical solutions for the evolution of the wall film thickness and the surface temperature.
3.2. Wall film heat and evaporation characteristics Based on the analytical evaporation model, the wall film temperature and thickness versus time can be obtained in detail. From the temperature distribution inside the wall film shown in Fig. 3, the nonlinear evaporation characteristics of the wall film can be well illustrated. As can be seen, the overall trend of the temperature distribution is similar for the cases with different wall temperatures. The evolution of the wall film can be divided into two distinct stages based on the variation of the film surface temperature, i.e., an initial rapid heating stage and a slow cooling stage. The wall film surface temperature increases rapidly at the initial heating stage, because it is continuously heated by the surrounding gas due to the temperature difference between the film surface and the surrounding gas. Thereafter, the surface temperature approaches to the wall temperature gradually because of the cooling effects from the wall conduction and the film evaporation. During this stage, the heating and evaporation of the liquid film coexist. In addition, the temperature profiles inside the wall film at different time are shown in Fig. 4. At the condition with the wall temperature of 450 K, it can be seen that the temperatures on both the two boundaries rise immediately at the very initial stage since both the wall temperature and the ambient temperature are higher than the initial wall film temperature. Because the conductive heat flux slowly transfers from the two boundaries to the film inside, the temperatures inside the film are still relatively low. Then the average temperature of the overall film increases gradually, and the wall film thickness becomes thinner with the extension of time due to the film evaporation. At the final stage, as the whole film temperature is higher than the wall temperature, heat only transfers from the film surface contacting the surrounding gas to the wall through the film. Meanwhile, because the fuel film is very thin and the heat flux is relatively steady, the temperature profile inside the film tends to be a linear distribution.
Fig. 3. Temperature distributions inside the wall film at different wall temperatures.
temperature resulting in a larger evaporation rate. Consequently, the evaporation process requires more heat energy from the wall film. Meanwhile, another interesting phenomenon is that the surface temperatures reach the peak at almost the same time under different wall temperature conditions, which can be explained as follows. At the conditions with the same initial film temperature and ambient temperature, the heat flux from the film surface is balanced with the convective heat energy exchanged between the film and the surrounding gas, thus the heat flux is almost the same at different wall temperatures. Although the wall temperature affects the temperature profile inside the film at this stage, its influence on the heat diffusion from the film surface to the film inside is insignificant. Meanwhile, the heat diffusion rate is mainly controlled by the thermal diffusivity, which is almost identical under the conditions with different wall temperatures. As a
3.3. Influence of various parameters on the wall film evaporation 3.3.1. Effects of wall temperature The effects of wall temperature on the film evaporation process are illustrated in Figs. 5–7. It can be observed that the wall temperature significantly affects the film heating up and evaporation. As shown in Fig. 5, a higher wall temperature results in a more rapid increase of the surface temperature at the initial stage, leading to a higher maximum of surface temperature. The reason is that a higher wall temperature provides larger heat flux for the fuel film to heat up. In addition, it is interesting to note that the differences in the maximums of surface temperatures among the cases with various wall temperatures are much smaller than the differences in the wall temperatures. It is mainly because a higher wall temperature leads to a higher wall film surface 128
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Fig. 4. Temperature distribution inside the wall film at different time.
Fig. 7. Wall film thickness versus time at different wall temperatures.
Fig. 5. Surface temperature versus time at different wall temperatures. Fig. 8. Temporal surface temperature versus film thickness at different ambient pressures.
Fig. 6. Evaporation rate versus time at different wall temperatures. Fig. 9. Surface temperature versus time at different ambient pressures.
result, the maximums surface temperatures at different wall temperatures appear nearly simultaneously. As can be found from Fig. 6, the trends of the evaporation rates are closely related to the evolutions of the surface temperatures. With the increase of wall temperature, the differences in the peaks of evaporation rates tend to become more significant. This phenomenon also indicates that a higher wall temperature would lead to a much larger evaporation rate.
Fig. 7 shows the film thickness versus time at different wall temperatures. It demonstrates that the lifetime of the liquid wall film reduces remarkably by raising the wall temperature. In addition, with increasing wall temperature, the differences in the lifetimes of the film among various cases decrease. In other word, the evaporation of the wall film can be strengthened by increasing the wall temperature, and
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Fig. 13. Evaporation rate versus time at different initial film thicknesses.
Fig. 10. Evaporation rate versus time at different ambient pressures.
from the surrounding gas to the wall film is used to heat up the wall film rather than for evaporation, since the temperature of the wall film is much lower than the saturation temperature at the local pressure. Thus, although the convective heat transfer between the air and the film is quite strong, the evaporation of the wall film cannot occur observably. With the time extension, the surface temperature rises much slower than that at the beginning, and the wall film thickness decreases obviously, especially when the surface temperature is around the peak of the curve shown in Fig. 8. It can be seen from Fig. 9 that with the increase of ambient pressure, the maximum surface temperature increases markedly. However, the instantaneous evaporation rate decreases with the increasing ambient pressure, as shown in Fig. 10. That means high ambient pressure condition would restrain the evaporation of fuel, because the saturation temperature is high when its corresponding ambient pressure is large. Meanwhile, the heat flux absorbed from the environment is mainly used to heat up the fuel film rather than for evaporation. The evolution of the wall film thickness at different ambient pressures is shown in Fig. 11, which reveals that higher ambient pressure prolongs the lifetime of the wall film.
Fig. 11. Wall film thickness versus time at different ambient pressures.
3.3.3. Effects of the initial wall film thickness The effects of the initial wall film thickness on the film evaporation are illustrated in Figs. 12 and 13. At the first stage, the case with a thinner wall film exhibits a faster heating trend on the surface. This is because less heat energy of evaporation is required for thinner films, so the heat flux from the wall and the environment can heat up the thinner fuel films more quickly. However, when the wall film is thinner, the cooling effect of the wall is more significant, so the peak surface temperature of the thinner film is lower than that of the thicker one, as shown in Fig. 12. Fig. 13 shows the evolution of the evaporation rate under different initial wall film thicknesses. It is obvious that the wall film with the thinner initial thickness vanishes sooner owing to the more intense evaporation process. From above results, it can be also found that the liquid wall film will exist for several cycles after the formation of the liquid fuel film in engines, especially for the case with thick fuel film. The slow evaporation rate of the liquid fuel on the in-cylinder surface could cause high pollution emissions.
Fig. 12. Surface temperature versus time at different initial film thicknesses.
its effect is more evident when the wall temperature is relatively low. 4. Conclusions 3.3.2. Effects of the ambient pressure Fig. 8 shows the evolution of the surface temperature with the wall film thickness at different ambient pressures. It can be seen that, at the initial heating up stage, the surface temperature rises quickly, but the thickness of the wall film nearly keeps constant without evident evaporation. This phenomenon indicates that most of the heat transfer
A new analytical solution was proposed for description of the transient heating and evaporation of the liquid wall film under the conditions with the constant wall temperature. In the analytical solution, the effects of convection from the surrounding hot gas and the heat conduction from the contacted wall are considered. Based on the 130
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model, the characteristics of the evaporation of the wall film under engine-relevant conditions were investigated. The main conclusions are summarized as follows.
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(1) The predictions of the analytical model are well consistent with the results of the discrete numerical model. The new analytical model provides an accurate and effective solution for the wall film heating up and evaporation. (2) According to the variation of the surface temperature of the liquid film, the evolution behavior of the wall film can be divided into an initial rapid heating stage and a slow cooling stage. At the initial heating stage, because huge temperature difference exists between the wall film and the surrounding gas, the surface temperature of fuel film rise significantly and the heat flux transfers to the film inside gradually. Thereafter, the surface temperature approaches to the wall temperature gradually because of the cooling effects from the wall conduction and the film evaporation. (3) The wall temperature, the ambient pressure, and the initial film thickness have important effects on the heating and evaporation characteristics of the wall film. (4) The vaporizing time decreases with the increasing wall temperature, and the decreasing ambient pressure and wall film thickness. (5) An increase in the ambient pressure results in a higher surface temperature of the film, but a lower evaporation rate. (6) At the final stage of the evaporation process, the temperature of the fuel film approaches to the wall temperature until the film is evaporated completely. Acknowledgments The authors are grateful to the National Natural Science Foundation of China (Grant No. 51376038, 51476020) and the Fundamental Research Funds for the Central Universities (Grant No. DUT15JJ(G)04) for the financial supports of this project. References [1] J. Senda, T. Kanda, M. Al-Roub, Modeling spray impingement considering fuel film formation on the wall, SAE Paper, 970047, 1997. [2] T. Alger, Y. Huang, M. Hall, Liquid film evaporation off the piston of a direct injection gasoline engine, SAE Paper, 2001-01-1204, 2001. [3] Y.S. Cheng, K. Deng, T. Li, Measurement and simulation of wall-wetted fuel film thickness, Int. J. Therm. Sci. 49 (4) (2010) 733–739.
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