Advanced models of fuel droplet heating and evaporation

Abstract

Recent developments in modelling the heating and evaporation of fuel droplets are reviewed, and unsolved problems are identified. It is noted that modelling transient droplet heating using steady-state correlations for the convective heat transfer coefficient can be misleading. At the initial stage of heating stationary droplets, the well known steady-state result Nu=2 leads to under prediction of the rate of heating, while at the final stage the same result leads to over prediction. The numerical analysis of droplet heating using the effective thermal conductivity model can be based on the analytical solution of the heat conduction equation inside the droplet. This approach was shown to have clear advantages compared with the approach based on the numerical solution of the same equation both from the point of view of accuracy and computer efficiency. When highly accurate calculations are not required, but CPU time economy is essential then the effect of finite thermal conductivity and re-circulation in droplets can be taken into account using the so called parabolic model. For practical applications in computation fluid dynamics (CFD) codes the simplified model for radiative heating, describing the average droplet absorption efficiency factor, appears to be the most useful both from the point of view of accuracy and CPU efficiency. Models describing the effects of multi-component droplets need to be considered when modelling realistic fuel droplet heating and evaporation. However, most of these models are still rather complicated, which limits their wide application in CFD codes. The Distillation Curve Model for multi-component droplets seems to be a reasonable compromise between accuracy and CPU efficiency. The systems of equations describing droplet heating and evaporation and autoignition of fuel vapour/air mixture in individual computational cells are stiff. Establishing hierarchy between these equations, and separate analysis of the equations for fast and slow variables may be a constructive way forward in analysing these systems.

Introduction

The problem of modelling droplet heating and evaporation is not a new one. A discussion of the models developed prior to the early fifties is provided in [1], [2]. A number of widely known monographs and review papers have been published since then, including [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17]. Various aspects of this problem have been covered in numerous review articles, including those published in this journal [18], [19], [20], [21]. In all these monographs and review articles, however, this problem was discussed as an integral part of a wider problem of droplet and spray dynamics. This inevitably limited the depth and the breadth of coverage of the subject. Also, most of the relevant monographs and reviews were published more than 5 years ago, and thus do not include the most recent developments in this area.

In contrast to the articles referenced above, this review will focus on the relatively narrow problem of droplet heating and evaporation. Although the application of the models will be mainly illustrated through examples referring to fuel droplets, most of them could be easily generalised to any liquid droplets if required. Only subcritical heating and evaporation will be considered. Near-critical and supercritical droplet heating and evaporation was covered in the relatively recent reviews published in this journal [22], [23], and in [24]. Analysis of the interaction between droplets, collisions, coalescence, atomization, oscillations (including instabilities of evaporating droplets) and size distribution will also be beyond the scope of this review, although all these processes indirectly influence the processes considered (see [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40]). Neither will the problem of heating and evaporation of droplets on heated surfaces be considered (see [37], [41]). Although the phenomena considered in this review can be an integral part of the more general process of spray combustion, the detailed analysis of the latter will also be beyond the scope of this work (see [42], [43], [44], [45]). Although the problem of radiative heating of droplets is closely linked with the problem of scattering of radiation, the formal modelling of the two processes can be separated. The models of the latter process were reviewed in [46] (see also [47]), and their analysis will be beyond the scope of this paper.

Soret and Dufour effects will be ignored. Soret effect describes the flow of matter caused by a temperature gradient (thermal diffusion), while Dufour effect describes the flow of heat caused by concentration gradients. The two effects occur simultaneously. Both effects are believed to be small in most cases although sometimes their contribution may be significant (see [48], [49], [50], [51], [52]).

In most models of droplet evaporation it is assumed that the ambient gas is ideal. This assumption becomes questionable when the pressures are high enough, as observed in internal combustion engines. The main approaches to taking into account ‘real gas’ effects have been discussed in [53], [54], [55], [56], [57]. The analysis of these effects, however, will be beyond the scope of this review.

This review is intended to be both an introduction to the problem and a comprehensive description of its current status. Most of the review is planned to be a self-sufficient text. On some occasions, however, the reader will be referred to the original papers, without detailed description of the models. Experimental results will be discussed only when they are essential for understanding or validation of the models.

The focus will be on the models suitable or potentially suitable for implementation in computational fluid dynamics (CFD) codes. These are the public domain (e.g. KIVA) or commercial (e.g. PHOENICS, FLUENT, VECTIS, STAR CD) codes. The structures of these codes can vary substantially. However, basic approaches to droplet and spray modelling used in them are rather similar. This will allow us to link the models, described in this review, with any of these codes, without making any specific references.

Following [13] the models of droplet heating can be subdivided into the following groups in order of ascending complexity:

• models based on the assumption that the droplet surface temperature is uniform and does not change with time;

• models based on the assumption that there is no temperature gradient inside droplets (infinite thermal conductivity of liquid);

• models taking into account finite liquid thermal conductivity, but not the re-circulation inside droplets (conduction limit);

• models taking into account both finite liquid thermal conductivity and the re-circulation inside droplets via the introduction of a correction factor to the liquid thermal conductivity (effective conductivity models);

• models describing the re-circulation inside droplets in terms of vortex dynamics (vortex models);

• models based on the full solution of the Navier–Stokes equation.

The first group allows the reduction of the dimension of the system via the complete elimination of the equation for droplet temperature. This appears to be particularly attractive for the analytical studies of droplet evaporation and thermal ignition of fuel vapour/air mixture (see e.g. [58], [59], [60], [61], [62]). This group of models, however, appears to be too simplistic for application in most CFD codes. The groups (5) and (6) have not been used and are not expected to be used in these codes in the foreseeable future due to their complexity. These models are widely used for validation of more basic models of droplet heating, or for in-depth understanding of the underlying physical processes (see, e.g. [13], [63], [64], [65], [66]). The main focus of this review will be on models (2)–(4), as these are the ones which are actually used in CFD codes, or their incorporation in them is feasible.

The review consists of two main parts. First, models of non-evaporating droplets will be reviewed (Section 2). Then evaporation models will be discussed (Section 3). The main results of the review will be summarised in Section 4.