Liquid Mixing in Evaporation of the Droplet

It is accepted that the liquid evaporates only when it is exposed to the surface. Therefore, mixing within the liquid droplet has become an important subject of research. In the literature two kinds of limiting cases for the transport of the mass and heat within the liquid are found:

(1) Well mixed model (Infinite diffusivity model) (2) Diffusion limited model

2.4.5.1 Well Mixed Model (Infinite Diffusivity Model)

This model is applicable for low viscosity mixtures in which the internal circulation within the liquid is fast enough so that the droplet concentrations and temperature remain uniform, but varying with time. In this model, the evaporation of the entire droplet is controlled by the relative volatilities of the components. In other words, the more volatile components are continuously brought to the surface where they evaporate in order of their volatility, leaving the less volatile components in the droplet. This kind of evaporation is similar to batch distillation. The motion within the liquid droplet can be caused by the buoyant motion of the air or evaporated vapour in the surrounding [55].

Law [55] presented this kind of well mixed model (spatially uniform droplet properties but temporarily varying). A binary mixture of octane and heptane was used. In the results two separate slopes were observed for the prediction of d2 -law, Dominant evaporating species evaporates in order of their relative volatility. Both regions follow the d2 law behaviour and showed the preferential type of evaporation. Law et al.[63] developed a model by assuming a uniform liquid temperature. Their results showed that the intensity of mixing increases as the viscosity decreases. As noted earlier, Hallett & Ricard [61] assumed a well-mixed droplet and developed a model for the droplet evaporation and ignition using

seven different components to accurately model the distillation curve. They pointed out that the distillation curve is not a good indicator of the ignition behaviour but overall ignition is governed by the chemical nature and boiling point of the most volatile species in the mixture.

In 1995, Tamim & Hallett [18] developed a new approach for the evaporation of multicomponent fuels using a Probability Density Function (PDF) for the fuel composition. This approach is called the theory of continuous thermodynamics and will be described in great detail in the next section. They assumed a well-mixed droplet behaviour in their approach, their result showed good agreement with the experimental observations. Moreover, Law [58] has pointed out that the theoretical analysis showed the diffusion limited type behaviour of the multicomponent droplet but experimental evidence supported the batch type distillation behaviour. Baert [14] pointed out that HFO evaporation process is different from multicomponent droplet evaporation. Opinions of different researchers about evaporation behaviour of heavy fuel oil droplet are given in Chapter 4.

2.4.5.2 Diffusion Limited Model

This model is relevant for the high viscosity mixtures where internal circulation within the liquid does not exist and transport within the liquid phase is only governed by molecular diffusion. Molecular diffusion is extremely slow compared to the liquid surface regression rate. The concentration changes at the droplet surface very rapidly but the core of the droplet remains unaffected. Therefore, the relative volatility of the liquid does not affect the overall process of the evaporation [18, 55].

Landis & Mills [64] studied the effect of internal diffusional resistance on the evaporation of binary droplets. Their result showed that internal concentration approaches a constant profile after a short initial transient period and it remains there till the end of droplet lifetime. They concluded that the liquid phase diffusion is a slow process and it controls the surface composition.

Law & Law [58] presented the d2-law behaviour of a single component and multicomponent fuel. They assumed the gas phase as quasi-steady and Lewis number as unity in the gas phase. Their derived equations showed that multicomponent equations are closely related to single component equations; except their values were replaced by mixture mass fraction weighted values. The droplet surface temperature in a single component model can be derived from the liquid boiling point but in the case of multicomponent mixture it needs an iteration procedure to obtain it (refer Law & Law [58] for detailed expressions). Their result showed that due to the extremely slow rate of diffusion, an approximately constant concentration profile exists for a significant portion of the droplet lifetime, with a sharp concentration gradient near the droplet surface. They also suggested that the quasi-steady gas phase assumption can be used to simplify the complex multicomponent modelling.

Mawid & Aggarwal [60] studied the transient combustion of a multicomponent fuel mixture droplet and they showed that at the surface of the droplet a boundary layer exists. Sirignano [62] studied both cases of the internal mixing (well mixed model and diffusion limited model) of the droplet and They result shows the considerable amount of difference with both models.

Jin & Borman [65] proposed a model with internal circulation by means of an effective molecular diffusivity. This model includes the effect of high pressure to calculate the vapour-liquid equilibrium at the droplet surface by means of the Redlich-Kwong equation of state. It also utilises an intermediate approach between the well-mixed and the diffusion limited model. The model of Jin & Borman [65] showed a linear slope of the d2-law for the steady evaporation beyond the initial droplet heating. Preferential evaporation was observed to a lesser extent, when the ambient pressure is close to critical pressure. Abraham & Magi [66] developed a model for the multicomponent droplet evaporation in sprays. They weighted the molar vapour fraction at the droplet surface by the molecular diffusivity of each component. This model was limited to theoretical study only, no experimental results were found.

Kneer et al.[56] studied the importance of variable properties in diffusion controlled evaporation of a multicomponent fuel. Their result showed that the diffusion resistance of the droplet can be altered by varying the properties of liquids (density, viscosity, binary

diffusion coefficient and thermal conductivity) which are dependent on liquid concentrations and temperature. It was also shown that the sharp increase in vapour concentration at the initial stage of droplet lifetime caused by high volatile components is important for ignition.

2.4.5.3 Droplet with Internal Circulation

Sirignano et al. (Lara-Urneja & Sirignano [67], Prakash & Sirignano [68, 69], Sirignano [54], Tong & Sirignano [70]) have conducted detailed studies on the effect of circulation in the convective environment which is summarised in ref [62]. Parkash and Sirignano [68, 69] developed a model for the temperature distribution of a droplet. Later, Lara-Urneja & Sirignano [67] extended that model to mass transfer of a multicomponent droplet. Furthermore, Tong & Sirignano [70] simplified the complex model developed by earlier researchers and implemented a so called vortex model for spray calculations. Their model accounted for liquid phase internal circulation, transient droplet diffusion, and asymmetric gas phase convection. However, they noted that due to large scattering in the results with experimental comparison, a concrete conclusion was not drawn. Abdel-Qader & Hallett [71, 72] investigated the role of internal mixing in the evaporation of droplets of mixture containing many components using continuous thermodynamics. Their results showed that internal mixing has a smaller influence on the droplet containing many components than the binary component droplet.

In general, many papers are found in the literature concerning heat and mass transfer models of a multicomponent droplet. The models have varying levels of complexity in terms of computational cost. Sirignano has categorised liquid phase models into different levels of complexity in his book [62]. In increasing order of the complexity these models are summarised in Table 2-2.

Table 2-2: Levels of liquid phase modelling complexity (Sirignano [62]).

Level Liquid Phase Modelling

1 Constant droplet temperature model (d2 law)

2 Infinite liquid conductivity model (uniform droplet temperature but varying with time)

3 Finite liquid conductivity, droplet divided into 2 regions (shell or skin and core region)

4 Spherically symmetric droplet heating (conduction limited or finite conductivity)

5 Vortex model for droplet heating (using Hill’s vortex in the modelling) 6 Navier-Stokes solution of the droplet internal flow