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Numerical analysis of subcritical evaporation and transcritical mixing of droplet using a tabulated multicomponent vapor-liquid equilibrium model

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In this paper, a fully compressible real-fluid and homogeneous equilibrium model (HEM) has been developed, in which the two-phase characteristics are obtained using a tabulated multicomponent vapor-liquid equilibrium approach. This classical HEM model consists of four balance equations posed in terms of mass density, partial species density, momentum, and internal energy (e). The thermodynamic properties of the mixture are calculated as a function of temperature, pressure and species composition (z_i) based on the Peng-Robinson equation of state. Most importantly, a bijective look-up table linking (ρ, e) and (T, P) is constructed using a computationally efficient isothermal isobaric (TPn) flash. This look-up table also includes various thermodynamic derivatives such as sound speed, heat capacity as well as the transport properties. During the simulation, all thermal and transport properties are linearly interpolated using (T, P, z_i). This tabulation approach has been successfully applied to the investigation of subcritical evaporation and transcritical mixing characteristics of spherical n-dodecane droplets in a nitrogen ambient. Primarily, an isolated droplet with uniform initial temperature is put into a moderate ambient condition (P_(amb.)=62 bar,T_(amb.)=700 K), in which it undergoes a classical evaporation process with the continuous diameter reduction. Then, the droplet is injected into a high temperature and pressure condition (P_(amb.) = 102 bar,T_(amb.) = 1200 K), in which the droplet firstly remains spherical for a while, and then deforms to an olive shape. The predicted results are shown to be in good agreement with the recent experimental findings. The thermodynamic analysis also demonstrates that the droplet has entered the two-phase regime with a diffused interface in which vapor and liquid coexist. This proves that the experimentally observed clouds around the droplet at (P_(amb.)=102 bar,T_(amb.)=1200 K) is still mainly generated by evaporation, and not due to diffusive mixing, even though the initial ambient gas is significantly above the n-dodecane critical point (P_c== 18.1 bar). The transition from the subcritical classical evaporation to the supercritical mixing regimes is also discussed in this work based on thermodynamic arguments.
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... Several contributions based on pre-tabulated thermodynamic closures have been presented in the literature for two-phase flow simulations. Yi et al. [59] investigated n-dodecane droplets evaporation in transcritical conditions using a three-dimensional (3D) uniform tabulation approach based on the (VLE) solver developed by [60,57]. Besides, Koukouvinis et al. [32] employed a tabulated thermodynamic approach based on (log 10 P −T ) tables to investigate the high pressure/temperature injection of n-dodecane in ECN spray A condition [1]. ...
... However, these approaches are still under investigation and their efficiency for multi-component problems is not evaluated. Therefore, the current work adopts an efficient pre-tabulation approach as initially proposed by Yi et al. [59], for binary systems. In this work, the tabulation approach is further developed to handle ternary mixtures as those encountered in dual-fuel engines, in contrast to the previous research limited to binary mixtures tabulation. ...
... This behavior is due to the multi-component mixture critical point variation compared to that of the pure component. For instance, Yi et al. [59] have demonstrated that the transition from subcritical evaporation to diffusive mixing regime, is based on the mixture's critical point for the multi-species problem, not the pure-fuel critical point. It also implies that considering the vapor-liquid equilibrium theory is essential for the correctness of the modeling as subcritical and supercritical states may exist simultaneously based on the local temperature, pressure, and species composition [24,57,35]. ...
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... Indeed, there are previous studies that have proposed various methods for tabulation, interpolation, and look-up of data [46][47][48][49][50][51][52]. Azimian et al. [48] introduced an artificial neural network model for generating the water/steam thermodynamic tables to reduce the computational expenses of solving flash calculations. ...
... In this study, a fully compressible multi-component two-phase real fluid model (RFM) [8,9,46] considering vapor liquid equilibrium (VLE) is proposed using a generalized three-dimensional (3D) tabulation method. In this method, an in-house thermodynamic library IFPEN-Carnot is used to generate the 3-D table (with T-P-Y as the axis for a binary mixture) based on various real fluid EoS. ...
... The uniform look-up table is generated based on an isothermal-isobaric flash (TPn flash) [53]. This tabulation method also improves the efficiency of the tabulation approach previously developed by Yi et al. [46] based only on the PR-EoS [8,9]. The RFM model along with the generalized 3D tabulation method was implemented in the CONVERGE solver [54]. ...
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... To reduce the computational effort, different researchers have proposed to calculate the fluid properties before the simulation[78,[87][88][89][90][91][92]. Then, these properties are stored inside a table,and an interpolation of the updated values is carried out during the simulation. ...
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In this thesis, a fully compressible real-fluid model has been developed, in which the two-phase characteristics are obtained using a tabulated vapor- liquid equilibrium (VLE) approach. This tabulated multicomponent real-fluid model (RFM) is proposed to overcome most limitations and make real-fluid simulations affordable. Basically, the RFM model consists of four balance equations: mass density, partial species density, momentum, and energy. The thermodynamic properties of the mixture are calculated as a function of temperature (T), pressure (P), and compositions (Y) based on different equations of state (EoS). This is carried out using the IFPEN-Carnot thermodynamic library which generates a 3D-table with (T,P,Y) as inputs. This look-up table is generated using a computationally efficient isothermal-isobaric (TPn)-flash, thereby avoiding the costlier iterative isochoric-isoenergetic (UVn)-flash employed in previous works. It specifically includes different thermodynamic outputs such as sound speed, heat capacity, and transport properties. The RFM model, along with the 3D tabulation method, has been implemented in the CONVERGE CFD solver. All thermal and transport properties are linearly interpolated using the updated (T,P,Y) during the simulation. First, various studies have been done for the refinement, and grid in-dependency of the thermodynamic tables, especially near the thermodynamic phase boundary using uniform and nonuniform grids. These studies have demonstrated that nonuniform grids, like octree and quadtree, is costly compared to the uniform approach. Therefore, uniform tabulation coupled with IFPEN's shared memory technique proved to be the most appropriate approach for tabulation, for the targeted industrial studies. Next, the present work has also investigated the robustness and accuracy of the proposed RFM model and the tabulation methodologies in conjunction with two different modified numerical schemes, a modified PISO and modified SIMPLE algorithms, adapted for the current real fluid modeling approach. Then, the proposed RFM model has been successfully applied to different academic and industrial applications to investigate subcritical classical evaporation/condensation and transcritical mixing characteristics. Among them, two industrially important test cases for which recent experimental results are available have been simulated and analyzed to validate the RFM model. 1- Simulation of a conventional cryogenic injection of liquid nitrogen coaxially with a hot hydrogen jet was performed using thermodynamic tables generated by two different equations of state: Peng-Robinson (PR) and Soave-Redlich-Kwong (SRK). 2- Simulation of the interaction between phase transition and turbulent fluid dynamics for subcritical and supercritical multi-species jets using different turbulence models including large-eddy simulations (LES Sigma and Smagorinsky) models as well as a Reynolds Averaged Navier-Stokes (RANS K-ε). The numerical results were found to be in good agreement with the available experimental data and published numerical studies, which also showed the relevance of the LES approach associated with the Sigma model for these very complex two-phase flows. Finally, numerical results showed that the tabulation method improves the liquid-vapor equilibrium (VLE) efficiency for real fluid modeling and provides a means to study and understand the structure of subcritical and transcritical liquid-gas interfaces revealing the hydro-thermodynamic characteristics of the multicomponent jet mixture.
... We found that more than 70% of the CPU time is consumed in the thermodynamic equilibrium computation. A possible solution to address this issue is using tabulation method [70], [71]. Indeed, all the thermal properties including phase composition, speed of sound, internal energy…etc, from phase equilibrium calculation could be stored in a table prepared before the simulation starts. ...
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