319 ACUERDOS MINISTERIALES
4. DESCRIPCIÓN DEL PROYECTO
4.1 Características técnicas del proyecto :
The high-pressure evaporation models are different from low-pressure models on a number of levels. These differences are summarised as [47, 73, 74];
• Lewis number is not unity.
• Ambient gas may dissolve in the liquid at the droplet surface.
• Raoult’s law cannot be assumed since neither the liquid nor the gas are ideal.
• The transient heating time is longer than the low-pressure model due to high pseudo wet bulb temperature.
Furthermore, to include the effect of high pressure the thermodynamics and transport properties of the liquid and vapour phase need to be adjusted. Droplets may reach the critical temperature at high pressures which does not occur at low pressures. Therefore, some of the transport properties which need to be adjusted at high pressure are given as [47, 73, 74];
• The evaporation enthalpy decreases with an increase in the temperature and at the critical temperature it completely vanishes [75]. Moreover, it also diminishes with increase in pressure, and is can be represented by ideal latent heat of evaporation.
• The surface tension of the liquid also decreases as the temperature approaches the critical temperature and it completely vanishes at the critical point. In order to allow the effect of high pressure by allowing the ambient gas to dissolve into liquid surface, the surface tension needs to be taken into consideration during the modelling.
• The specific heat is a function of pressure and needs to be corrected at high pressure.
• The viscosity of a mixture is a function of temperature, which may enhanced at critical point therefore it needs to be adjusted for high pressure calculations.
The vapour flux from the evaporating droplet can be written as [74];
( ) ( ) F F F F N FN G dy W y W W D dr
ρ
= + −& & & (11)
Where W&F =m&F / (4
π
R2) is the mass flux of fuel species, subscript N stands for surrounding gas nitrogen and (ρ
DFN)G is the product of the gas phase density and the binary diffusion coefficient of the fuel vapour in nitrogen. W&N is the rate of gas diffusion into the liquid. For slow evaporation and no gas diffusion into the liquid, the above equation can be simplified to;, 1 ( ) 1 F F R FN G R F dy W D dr y
ρ
= − & (12)For high rates of surface regression and diffusion of many species, the above equation can be modified to; , 1 ( ) 1 F F R FN G R G F F L dy W D dr x y
ρ
ρ
ρ
= − − & (13)Where, xand y stand for the liquid and vapour phase mass fractions. Kodota & Hiroyasu [76] included the surface regression effect on convectional evaporation rate which can be given as; * * , , 2 , ( ) ( ) , 4 1 (1 ) o F R F F G M F R y y m D Sh B wh B R R y
ρ
ξ
π
ς
∞ − = ⋅ = − + & (14)Where, Sho is the Sherwood number without evaporation effect,
ξ
M is the correction factor for evaporation (in low pressure this correction is given as ln(1+BM) /BM),* B is the modified transfer number, which includes the surface correction due to pressure as;
, (1 ) 3 G F R F L N L y W R d dR W dt dt
ρ
ς
ρ
ρ
− = = − + & & (15)Similar to the mass transfer correction factor the correction to the heat transfer also is found in Kodota & Hiroyasu’s paper [76]. They included the non-ideality of liquid and gas by using the Redlich-Kwong EOS. Their results showed the difference between low- pressure and high-pressure models and also the importance of high pressure model.
Givler & Abraham [47] presented a very good paper on the evaporation and the combustion of droplets at supercritical conditions. They used many numerical and experimental studies and therefore it is worthwhile to note their findings here. Their results which are summarised in the form of figures are shown in Figure 2-8. As shown in Figure 2-8(a), if the reduced pressure and temperature exceeds 2, pure evaporation and unsteady evaporation exists throughout the droplet lifetime. The evaporation rate increases with increase in the pressure. Moreover, it increases strongly for the subcritical pressures but for the supercritical condition it may decrease. The droplet temperature may be transient for the droplet lifetime but evaporation rate can follow quasi-steady behaviour.
Figure 2-8: (a) Qualitative representation for quasi-steady and non-quasi-steady evaporation as a function of experimental reduced temperature and pressures. (b) Qualitative representation of the droplet temperature over a range of ambient temperatures and pressures (from Givler and Abraham [47]).
Furthermore, as shown in Figure 2-8 (b), Givler & Abraham also defined three regions for droplet temperature, namely pseudo wet bulb temperature, transient temperature and critical temperature region. The first region (pseudo wet bulb temperature) is the steady temperature the droplet reaches under higher pressure and temperature. The second region is the transient temperature which is observed throughout the droplet lifetime. The third region is the critical temperature region, when the reduced pressure and temperature are greater than 2 then the droplet may reach critical state in the liquid phase.
Most recently, Yan & Aggarwal [77] developed a high-pressure droplet model for spray simulations of a single component fuel. They compared the model for quasi-steady and transient effects. Their result showed good engagement between the quasi-steady and the transient model for the wide range of pressures at low ambient temperature, and also for the high temperature but at pressures only up to the critical pressure of the liquid.
In the literature, many papers are found that dealt with the evaporation of multicomponent fuels at high pressure and use the equation of state to describe the vapour-liquid equilibrium, and they are summarised in the present section. Pedersen et al.[78-80] published three papers which described the thermodynamics of the petroleum mixtures containing heavy hydrocarbons. Pedersen et al.[79] showed that the critical temperature and pressure of each of the carbon number fractions heavier than the C6 should be
determined by Cavett and Lee-Kesler relations (refer Pedersen et al.[79] for details of
these relations). They reported that the most suitable method to determine the liquid phase density is Alani-Kennedy equation and Standing-Katz method [80].
Zhu & Aggarwal [75, 81, 82] reported a numerical investigation of droplet evaporation in a supercritical environment. The physical-numerical model was developed to simulate the trans-critical and supercritical evaporation based on time dependent conservation equations for liquid and vapour phases and pressure dependent thermo-physical properties. They have used three different equations of state; Redlich-Kwong (RK), Soave-Redlich- Kwong (SRK) and Peng-Robinson (PR). Their result showed variations with all three EOS. Moreover, their results also indicated that at low to moderate temperature, droplet lifetime first increases then decreases with increase in pressure. On the other hand, at high temperatures droplet lifetime continuously decreases with pressure.
Zhang [83] developed a numerical model and studied the evaporation of a suspended droplet in forced convective high-pressure conditions. The model included liquid phase internal circulation, real gas effect, gas-liquid transient effect and solubility of gas into liquid phase. The results showed good agreement with the microgravity experimental results. Further, the result also showed that droplet lifetime decreases with increase in pressure and temperature. This result confirms the findings of Zhu & Aggarwal [75, 81, 82] described in the previous paragraph. Zhang [83] also noted that solubility of nitrogen in liquid (n-heptane) can be neglected at low ambient pressure but it cannot be neglected at higher pressure.
Kim & Sung [84] studied the effect of ambient pressure on the evaporation of a single droplet and the spray. The developed model considered the fugacities of the liquid and gas phases for the calculation of VLE. Their result showed that the droplet lifetime decreases with increase in the ambient pressure. The evaporation of the spray was enhanced at high pressure and temperature. At high pressure, the atomisation and the evaporation rate from a single droplet was increased compared to low pressure.
In the above section, the literature based on the evaporation of single and multicomponent liquids at high-pressure are reviewed. In the following section literature based on the theory of continuous thermodynamics used for high pressure evaporation are reviewed. In 1985, Cotterman & Prausnitz [85] introduced the equation of state to calculate the phase
equilibrium of a mixture containing many components. Around the same time, they [86] also introduced the equation of state for the flash calculation (calculation of bubble points and dew points) of the semi-continuous mixture. In that paper they introduced two different methods; (1) method of moment and (2) quadrature method to determine the flash calculations. This paper is found to be very useful in later studies where continuous thermodynamics is applied to complex hydrocarbons. Based on Cotterman & Prausnitz’s work [86], Baer et al.[87] developed a method for the prediction of vapour-liquid equilibrium of a complex multicomponent mixture based on a two parameter equation of state using the method of continuous thermodynamics. They used a continuous version of the Wong-Sandler mixing rule and RK EOS. Their result showed good agreement with the experimental data.
Zhu & Reitz [75, 88] developed a comprehensive model for the transient evaporation process of real engine fuels at high pressure using continuous thermodynamics. They derived transport equations for a semi-continuous mixture for both the gas phase and also the liquid phase. Moreover, their approach to the equation of state was generic; hence vapour-liquid equilibrium can be obtained for any EOS. Their result showed that at high pressure, light species evaporation is inhibited compared to heavy species. Their result of the comparison of multi-component with the single-component droplets also emphasizes the importance of considering multi-component fuels.
In the literature a paper [89] is found where researchers used the continuous thermodynamics for modelling of practical diesel engine spray combustion at high- pressures. Yi et al.[89] developed a model for the spray combustion of diesel at high pressures by assuming a well-mixed droplet composition and temperature. The model predicted the liquid length in an evaporating spray and showed a good match with experimental results. Yi et al.[89] noted that the high pressure model provides better mixing in the spray region because it considers the effect of temperature and pressure in the calculation of physical properties. The comparison of Sauter Mean Diameter (SMD) as a function of axial distance from the nozzle exit by low-pressure and high-pressure model showed similar results.