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The D^{2}-Law of Droplet Evaporation When Calculating the Droplet Evaporation Process of Liquid Containing Solid State Catalyst Particles

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## Abstract

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^{2}-law of droplet evaporation, which is used to describe the spraying process involving the evaporation of droplets. This law, the subject of numerous publications, can be successfully applied to describe the droplet evaporation process under various conditions, including the calculations of the process of feeding the boiler with a liquid that contains catalyst particles. To date, not a lot of work has been devoted to this issue. The paper is a continuation of previous research concerning the spraying of liquids with a catalyst, which improves the efficiency of the process. The conducted analysis showed that the experimental data from previously published work are very compatible with the data obtained from the D

^{2}-law of droplet evaporation. At the standard speed of about 20 m/s of an aerosol flowing through a dust duct, droplets in the stream should be observed up to a distance of 1 m from the outlet of the apparatus supplying the system. Under such flow conditions, a droplet’s lifetime must be above 0.05 s. The dependence between a droplet’s lifetime and its diameter and temperature was determined. The obtained results confirmed that the effective droplet diameter is above 30 µm. Such droplets must be generated and then fed to the boiler for the catalyst to work properly. This law is an engineering approach to the problem, which uses relatively simple model equations in order to determine the evaporation time of a droplet.

## 1. Introduction

^{2}-law during n-decane droplet vaporization in a microgravity environment. The experimental results show that the D

^{2}-law is always valid in normal gravity within the range of the ambient temperatures explored in the study. The authors of article [18] proposed a revision of the D

^{2}-law that is capable of accurately determining the droplet evaporation rate in dilute conditions by properly estimating the asymptotic properties of a droplet. The authors of paper [18] tested the proposed model against data from direct numerical simulations, and found an excellent agreement for the predicted droplet evaporation time in dilute turbulent jet-sprays.

^{2}-law was modified to take into account droplet heating. Comparison with numerical solutions of the transient problem for moving droplets shows the applicability of this approximation to modelling the heating and evaporation processes of droplets. The simplicity of the model makes it particularly convenient for implementation into multidimensional CFD codes to replace the widely used model of isothermal droplets.

^{2}model of the evaporation of droplets enables the diameter of the evaporating droplet to be calculated without the need to use complicated, cost-consuming and time-consuming numerical calculations. However, it does not give a broad picture of the process [23,24,25].

^{2}-law of droplet evaporation, it is difficult to find any in which this law was used to describe the droplet evaporation process during the feeding of the boiler with a liquid containing catalyst particles. The main goal of this study was to present the literature data for the D

^{2}-law of droplet evaporation, and to attempt to perform calculations for the process of dosing the catalyst into a combustion chamber. The obtained results were analyzed, and confirmed its practical (i.e., engineering) approach to the problem, which uses relatively simple model equations that allow for the determination of the droplets’ evaporation time.

## 2. Analysis of the Evaporation Process

_{t}(W/m

^{2}) is generated, which in turn causes droplets to heat and evaporate:

_{w}—heat flux heating a droplet (W/m

^{2}), L

_{par}—latent heat of evaporation (J/kg), ${{\dot{m}}^{\prime}}_{p}$—mass flow rate (kg/(m

^{2}× s)).

^{2}-law of droplet evaporation is obtained:

_{0}, D—initial and final (current) diameter of a droplet, K

_{par}—evaporation constant, t—time.

_{p}—thermal conductivity coefficient, B—exchange number, (C

_{p})

_{p}—isobaric specific heat of gas, ρ

_{c}—liquid density.

_{par}—latent heat of evaporation of the liquid (fuel) at temperature T

_{s}, which is the surface temperature of the droplet. The calculation of K

_{par}requires knowledge of B, where B is a function of the generally unknown temperature T

_{s}. It is possible to determine this temperature graphically or numerically, but it is very difficult. When simplifying this, however, it can be assumed that temperature T

_{s}is equal to the boiling point of the liquid. It was also shown that the B value is close to zero at temperatures close to room temperature.

_{par})

_{e}was introduced as:

_{p}is the mass transfer coefficient in the gas phase (mol/N/s), β

_{p}is the mass transfer coefficient (kg/m

^{2}× s), D

_{d}is the dynamic diffusion coefficient (kg/m × s), and D is the dynamic molecular diffusion coefficient expressed in (m

^{2}/s). The gas constant in Equation (12) is equal to R = 8314 (J/mol/K). The speed of droplets was defined as the average velocity of the mixture in the outlet channel.

_{m}of the mass transfer, defined as:

_{m}by about 3%. The process conditions were mapped on the basis of real data for a dust duct supplying coal dust to an existing coal dust boiler at a combined heat and power (CHP) plant in Poland [24].

^{3}n/h (at a speed of 26.8 m/s), the amount of supplied dust is 1.2 t/h, and its grain size ranges from 20 to 200 μm. The liquid with the catalyst is fed at a volumetric flow rate of 300 mL/h. Droplets of liquid containing the catalyst with diameters from 0.1 to 100 µm were analyzed. The diameter of the generated droplets should be >30 μm [24,25]. The analysis of the data presented in Figure 2 shows that the apparatus feeding a liquid containing the catalyst works in the laminar flow regime. It should be noted that the data presented in [30] show the values of the Colburn modulus for a Reynolds number from 1 to 1000, and the values of the Reynolds number 1–2 are consistent with the values shown in Figure 2.

_{m}with regard to the droplet’s diameter. An exemplary graph is shown in Figure 3. The analysis of the obtained dependence showed that the value of the Colburn modulus decreases with an increase in the droplet diameter.

^{2}/s).

- -
- the physical properties of the surrounding gas, i.e., temperature, pressure, thermal conductivity, specific heat, and viscosity,
- -
- the droplet’s speed in relation to the environment,
- -
- the properties of the liquid and its vapor, i.e., density, vapor pressure, thermal conductivity, and specific heat,
- -
- the properties of the droplet in its initial state, especially its diameter and temperature.

^{2}-law of droplet evaporation. The concept of the effective evaporation constant, which is described by Equation (11), is very convenient from a practical point of view. The graph of the dependence between the effective evaporation constant and the temperature, which was obtained by extrapolating the data presented in the paper of Chin and Lefebvre [31], is shown in Figure 4.

## 3. Conclusions

^{2}-law of droplet vaporization, which is used in the liquid spraying process. The aspect related to the evaporation of the liquid that carries catalyst particles is particularly interesting, because it is of a great practical importance—especially from the point of view of energy. As a result of the conducted analysis, earlier predictions regarding the size of a droplet, which were presented in articles [24,25], were confirmed. The obtained results confirmed that the effective diameter of a droplet is above 30 µm, which must be generated and then fed to the boiler for the catalyst to work properly. The use of the D

^{2}-law of droplet evaporation allows for a relatively simple calculation of a droplet’s diameter during the conducted process without the need for complicated, cost-intensive and time-consuming CFD calculations. It could be an alternative to process simulations. However, it should be borne in mind that this method does not give as broad a picture of the process as the one that can be obtained using CFD simulations.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Dependence between the kinematic diffusion coefficient and the temperature for the water-to-air vapor system.

**Figure 2.**Dependence between the dimensionless Colburn modulus j

_{m}of the mass exchange and the Reynolds number for the tested process.

**Figure 3.**Dependence between the dimensionless Colburn modulus j

_{m}of the mass transfer and the droplet diameter at the temperature of aerosol of T

_{s}= 308 K.

**Figure 4.**Diagram of the dependence between the effective evaporation constant and the temperature of the surrounding gas (the assumed boiling point of the liquid T

_{wrz}= 373 K).

**Figure 5.**Dependence between the evaporation time of a droplet and its diameter for selected temperatures.

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**MDPI and ACS Style**

Ochowiak, M.; Bielecki, Z.; Bielecki, M.; Włodarczak, S.; Krupińska, A.; Matuszak, M.; Choiński, D.; Lewtak, R.; Pavlenko, I. The *D*^{2}-Law of Droplet Evaporation When Calculating the Droplet Evaporation Process of Liquid Containing Solid State Catalyst Particles. *Energies* **2022**, *15*, 7642.
https://doi.org/10.3390/en15207642

**AMA Style**

Ochowiak M, Bielecki Z, Bielecki M, Włodarczak S, Krupińska A, Matuszak M, Choiński D, Lewtak R, Pavlenko I. The *D*^{2}-Law of Droplet Evaporation When Calculating the Droplet Evaporation Process of Liquid Containing Solid State Catalyst Particles. *Energies*. 2022; 15(20):7642.
https://doi.org/10.3390/en15207642

**Chicago/Turabian Style**

Ochowiak, Marek, Zdzisław Bielecki, Michał Bielecki, Sylwia Włodarczak, Andżelika Krupińska, Magdalena Matuszak, Dariusz Choiński, Robert Lewtak, and Ivan Pavlenko. 2022. "The *D*^{2}-Law of Droplet Evaporation When Calculating the Droplet Evaporation Process of Liquid Containing Solid State Catalyst Particles" *Energies* 15, no. 20: 7642.
https://doi.org/10.3390/en15207642