Heat exchange of an evaporating water droplet in a high-temperature environment

Highlights

• Three approaches were used: free-falling droplet, fixed in a hot air flow or muffle furnace.

• Quasi-steady temperature field of droplet and its heating rates have been determined.

• With gas temperature of 1100 °C, droplet evaporation rate may reach 0.18 kg/(m²s).

• When Re < 100, Ranz and Marshall's correlation adequately describes experimental data.

• When T_a > 500 °C, Nu(Re,Pr) requires an adjustment coefficient, (1-T_s/T_a)^Ts/Tb.

Abstract

In this research, we present the results of experiments capturing the characteristics of high-temperature heating and evaporation of water droplets. The experimental parameters are as follows: initial droplet radius 1–2.5 mm, temperature 20–1100 °C, air flow velocity 0–5 m/s. We use two schemes of water droplet fixation in the heated air flow. In the experiments, we record the heating and evaporation of free-falling droplets so that the holder does not interfere with the experimental results. Typical water droplet heating rates range from 0.4 to 92.4 °C/s and evaporation rates from 0.003 to 0.178 kg/(m²s). Criterial processing of our experimental data is based on the classical equations formulated by Ranz and Marshall and equations proposed by a set of research groups and scientists. We determine the variation ranges of the Reynolds numbers, in which one can only use a limited set of Nu(Re,Pr) correlations. Adjustment coefficients are proposed to take into account the droplet surface temperatures and evaporation rates as functions of the gas medium temperature. Finally, we hypothesize as to how modern mathematical models can be modified to bring the simulation results closer to the experimental data in droplet heating and evaporation rates during fast heat supply.

Introduction

The intensity and typical rates of liquid droplet evaporation are defined by the vapor concentration gradient in the thin layer composed of the ambient gas and vapors of the evaporating liquid [[1], [2], [3], [4], [5], [6]]. It is nearly impossible to determine these concentrations in the regions that are typically much smaller than 1 mm [[7], [8], [9], [10]]. The evaporation of liquid drops is the main process behind many advanced technologies of various industries, such as chemical, petrochemical, oil-refining, and energy sectors [[11], [12], [13], [14], [15]]. Thus, in order to choose the right process conditions, we primarily need to know the liquid evaporation rates (W_e) and droplet heating rates (W_h). The latter is basically the rate, at which an almost quasi-steady temperature field is generated.

It is usually difficult, if at all possible, to experimentally measure the W_h and W_e in the actual production process [[12], [13], [14], [15], [16], [17], [18], [19], [20]]. Therefore, when calculating the droplet heating and evaporation rates in a gas [21], the researchers often assume that all the energy supplied to the liquid droplet surface is spent on the phase transition. The heat flux to the droplet surface can be calculated using criterial equations obtained for a solid spherical particle. When using this approach, it is hard to consider the actual physical properties of the processes occurring in a vapor-gas mixture near the droplet surface.

With the recent boom in technology, it has become necessary to reliably describe liquid evaporation from the heated droplet surface. The heating temperatures in this case range from 500 °C to 1500 °C, sometimes even reaching 2000 °C, for instance, in flame liquid treatment. Empirical equations for evaporation rates were obtained for a limited temperature range (under 500 °C) in an assumption that water vapor concentration may be linked to the Reynolds and Prandtl numbers obtained in the experimental data processing [22,23]. In other words, the heated gas flow around a droplet entrains vapor molecules. Their concentration in the near-wall layer only depends on the inertia and viscosity in this layer. Under such conditions, it is extremely difficult to predict the phase transition rates using models that only consider convective heat exchange. Thus, traditional assumptions and models should be improved to reproduce the evaporation conditions and characteristics more adequately.

Conventional approaches to the experimental research of evaporation assume that the temperature distributions in a liquid droplet are homogeneous and all the energy supplied to a droplet is spent on the phase transition. Many renowned scientists including O. Knake, I.N. Stranskiy, N. Fuchs, D. Spalding, M. Yen, L. Chen and M. Renksizbulut, S.S. Kutateladze, D.V. Labuntsov, V.E. Nakoryakov, V.I. Terekhov, A.A. Avdeev, and S.Ya. Misyura developed models to simulate these processes. However, both the modeling and experimental approaches are limited to the temperature range of 300–500 °C, in which the evaporation characteristics are in satisfactory agreement with the experimental data (with deviations of less than 10–15%). For gas temperatures of over 500 °C, there are no adequate models so far that would predict phase transition rates with deviations of no more than 10–20% [24]. The main reason of this state of things is the lack of reliable experimental data. Experimental and simulated results by Chen [25], Yuen and Renksizbulut [26,27], building on the ideas by Fuchs [28], Ranz and Marshall [29], are widely known. For many years, there have been discussions over the feasibility of the approach considering the so-called thermal balance at the droplet surface. It is based on the equality of heat spent on the liquid evaporation and the energy supplied to the droplet. Criterial processing under such conditions is reduced to the modification of classical correlations like Nu = f(Re,Pr) using additional factors and effects, for instance, Refs. [14,[30], [31], [32]]. In their monograph [33], Terekhov and Pakhomov give a number of the most widespread equations of the form Nu = f(Re,Pr). According to the research findings from Refs. [33,34], it is important which procedure is used to record the main characteristics of high-temperature heating and evaporation of droplets. There are plenty of high-potential applications for high temperatures in the liquid droplet – gas medium systems [[22], [23], [24]]: direct-contact heat exchangers, fuel ignition in combustion chambers, heat carriers based on flue gases, water vapor and droplets. At the same time, solving the performance problems of both current and emerging heat-power equipment is a key to the further development of the whole global heat and power industry. Therefore, high-temperature heating and evaporation of droplets is attracting more and more interest each year.

In the mathematical modeling of the heat exchange between a gas medium and a liquid droplet, a major challenge is adequately describing the temperature fields of the latter. The optical methods of measuring droplet temperature fields (e.g., Refs. [[7], [8], [9], [10],[35], [36], [37], [38], [39]]) show their high inhomogeneity. Two stages are distinguished in the process of heating. At the first one, the temperature in all the droplet sections grows rapidly. At the second stage, this growth slows down, and the temperature field becomes quasi-steady. Under rapid phase transformations, the duration of these stages changes considerably. Significant temperature gradients are recorded from the droplet surface to its deep layers almost over the whole droplet lifetime. Such processes greatly affect the temperature distribution in aerosol flows and jets [35,38].

The aim of this research is to experimentally study the high-temperature heating and evaporation of free-falling water droplets or those fixed on holders using well-known recording procedures. We employ typical Nu = f(Re,Pr) correlations to determine the approaches that would allow us to use the latter to reliably predict the values of key heating and evaporation parameters of various liquids.