1. Introduction
Recently, there has been a rising interest in the application of thermal phase change processes to nanofluids, particularly flow boiling of nanofluids [
1,
2,
3]. Flow boiling is an efficient cooling technique because it removes latent heat of vaporization while pushing the flow owing to the density difference. Furthermore, nanofluids can be used to improve heat transfer in industrial applications, which was proved to be a very successful technique by Choi and Eastman [
4] for the first time. Following that, several researchers began to investigate various aspects of employing nanofluids in cooling applications both experimentally [
5,
6,
7,
8,
9,
10,
11] and numerically [
12,
13,
14,
15,
16,
17,
18,
19].
Phase change research was generally experimental; however, experiments in this field are restricted, and even a little mistake might result in a huge error [
20]. As a result, it is necessary to numerically model phase change processes in order to fully comprehend the available experimental data and to accurately characterize the underlying physics. The ability to examine complex geometries and various conditions is one of the most important benefits of numerical simulation [
21,
22].
While phase transition has been researched for more than half a century, its complicated physical nature, which involves simultaneous heat, mass, and momentum transfer, makes numerical predictions difficult. Locating the interface position is a significant difficulty in CFD modeling of two-phase flows. The issue is how to construct a sharp interface by analyzing the curvature of the interface and forecasting physical parameters that change dramatically across the interface. As seen in
Figure 1, the gas–liquid interface can be represented using interface tracking and capturing techniques. The interface tracking techniques are primarily divided into three categories: front tracking methods, marker methods, and arbitrary Lagrangian–Eulerian methods. The primary advantage of this technique is its ability to appropriately analyze the interface profile; however, since the computational domain requires meshing updates when the interface deforms, it will be prohibitively difficult to precisely monitor the interface while maintaining a reasonable degree of re-meshing repetition. This is related to the production of an overwhelming mesh, which renders interface tracking approaches outdated.
As an alternative to interface tracking techniques, interface capturing techniques can be discussed, which specify the interface location using a scalar function. The volume-of-fluid (VOF) and level-set (LS) functions are the most well-known scalar functions for describing the gas–liquid interface. The VOF method, which was pioneered by [
23], has evolved into a norm for both open-source and commercial CFD software. A volume fraction (
) is employed in the VOF technique to distinguish between the fluid 1 and fluid 2 inside the domain. The cell is filled with fluid 1 when
, and with fluid 2 when
. Each phase’s volume fraction is tracked across the domain in this model, which uses a single momentum equation to address the fluids.
OpenFOAM’s phase change solvers mostly employ VOF to describe interfaces, as well as research into simulating flow boiling of nanofluids, and it has been applied in a broad variety of phase change circumstances, some of which are listed in the
Table 1. The volume fraction gradient is used as the normal vector field perpendicular to the gas–liquid interface in the VOF technique. Because the volume fraction is not a smooth function, the direction of the normal vector is chaotic. The curvature of the unit normal vector is equal to its divergence. Additionally, due to the smeared interface, a finite gradient of curvature develops in the normal direction. These impacts lead to an incorrect curvature calculation, which results in a larger numerical error for phase change calculations [
24].
Using other interface capture techniques, most notably the level-set approach, results in a more accurate estimation of interface curvature. The level-set approach, presented by [
30], calculates a distance function for the entire computational domain’s cells. The zero distance denotes the interface between gases and liquids. The cells with a positive distance are believed to be in the continuous phase (liquid), whereas those with a negative distance are thought to be in the scattered phase (gas). Dhir’s group has extensively employed the level-set technique [
31,
32,
33,
34] to investigate a variety of phase change events, including nucleate boiling, film boiling from a horizontal cylinder, and subcooled pool boiling. Apart from Dhir’s group, ref. [
35] investigated film boiling using the level-set approach, and [
36] simulated the evaporation of a moving and deforming droplet. The level-set technique, on the other hand, is a non-conservative methodology that does not ensure mass conservation in phase transition calculations [
37]. The primary drawback of the level-set approach is the fluid mass loss. Ref. [
38] presented a combination approach of level-set and VOF termed CLSVOF, which was extensively adopted by others [
39,
40,
41,
42]. It generates a precise, sharp interface and compensates for the level-set method’s drawback. The volume fractions of liquid and gas are calculated using VOF, and level-set determines geometrical interface properties such as the curvature and normal vector in each cell.
According to the best knowledge of the writers, prior research on nanofluid flow boiling has used VOF to capture the interface; however, this technique produces less accurate gas–liquid curvature calculations, which in turn yields in less accurate results. Due to the fact that numerous studies in the literature have identified level-set as an interface description method for calculating curvature, level-set is used for the first time in this paper for nanofluids’ thermally driven phase change modeling, and the obtained results are compared to VOF’s utilizing benchmark scenarios to demonstrate their respective strengths and limitations in different thermal phase change circumstances. Benchmark instances are
Evaporation by heat conduction,
Condensation by heat conduction,
Film condensation on a vertical plate,
2D film boiling.
We picked these four benchmark cases because of their relevance in a wide range of processes: one-dimensional boiling, one-dimensional condensation, two-dimensional condensation, and two-dimensional boiling. To ensure the solver’s accuracy, simulation results were validated using analytical solutions.
2. Numerical Formulation
The mass, momentum, and energy conservation equations as well as the advection interface description equations are expressed as follows for two incompressible and immersible fluids
where
represents the rate of exchanged volumetric mass flux, which was determined by applying the Tanasawa model ([
43]). In this model, the interface superheat temperature (
) is used to determine the interface mass flux:
where
is the evaporation coefficient (here ir is equal to 1),
is the fluid’s molecular weight, and
R the universal gas constant.
The domain is created for a mixture, using the
function to differentiate the liquid and gas. In this study, the interface description is provided via the use of two separate techniques for interface capture, VOF and CLSVOF. The VOF approach represents the interface and separate phases using a liquid (nanofluid) volume fraction function (
).
where
denotes the nanofluid’s volume fraction,
denotes the volume of nanofluid contained inside a cell, and
V is the total cell volume. The volume fraction function of liquid (
) is also used to determine the density of a mixture (
), the viscosity of a mixture (
), the constant pressure specific heat of a mixture (
), and the thermal conductivity of a mixture (
).
We integrated the CLSVOF technique with VOF into an OpenFOAM solver for thermally induced phase changes. The CLSVOF method makes use of the VOF method’s mass conservatism and the LS method’s interface sharpness. A new scalar field termed level-set function (
) is defined in the CLSVOF method instead of
, which is used in VOF method. The reader is directed to Omar et al’s [
24] paper for more thorough details on the CLSVOF method’s implementation and governing equations.
2.1. Governing Equations Used to Calculate Nanofluids Thermophysical Properties
This paper investigates nanofluids thermally driven phase change cases. Nanofluids are assumed to be homogeneous, and their properties can be determined experimentally or theoretically. The density, specific heat capacity, viscosity, thermal conductivity, and surface tension of a nanofluid are calculated using [
44,
45,
46,
47]:
where
is the volumetric concentration of nanoparticles. The number
n specifies the empirical shape factor, which is equal to three for spherical nanoparticles. The
,
, and
, respectively, indicate the characteristics of nanoparticles, base fluid, and obtained nanofluid. The experimental values (a and b) are
and
[
47]. Spherical Al
2O
3 nanoparticles with
are added to the base fluid.
2.2. Discretization Schemes for Solution Algorithm
OpenFOAM has a number of built-in numerical schemes for discretizing the terms in each conservation equation. As seen in
Table 2, second-order schemes are used in different cases for all OpenFOAM calculations in this study.