Introduction
Importance of droplets on the transmission of infectious diseases
The influenza virus impacts millions annually, with the World Health Organization reporting over 16,455 deaths due to H1N1 (Swine flu) in 2009 To reduce the spread and mortality associated with influenza, further research is essential The Centers for Disease Control and Prevention (CDC) indicates that avian influenza is transmissible among birds such as chickens, ducks, and turkeys, often leading to illness and death in domesticated birds Infected birds can shed the virus through saliva, nasal secretions, and feces, posing a risk of infection to susceptible birds that come into contact with these contaminated materials Although these viruses typically do not affect humans, there is a growing number of confirmed human infections from various avian influenza subtypes.
2009) Therefore, the research on the control of the influenza virus is a matter of great urgency
Coughing and sneezing are significant modes of virus transmission, as they release droplets containing infectious agents into the environment (Weber & Stilianakis, 2008) These droplets, which can carry viruses like the influenza virus, are expelled from the body and can be inhaled by others, facilitating the spread of illness Each cough or sneeze can produce millions of droplets (Duguid, 1946), and their characteristics—such as temperature, salinity, and composition—affect their behavior, including surface tension and evaporation rate Understanding these droplet properties is crucial for effectively reducing airborne transmission of infectious viruses.
There are several ways of transmission of virus and germs: (1) Direct contact with infected individuals; (2) Indirect contact with contaminated objects; (3) Inhalation of virus-laden aerosols;
Inhalation or contact with large droplets, specifically those with a diameter greater than 5μm, is a significant source of infection due to their ability to travel up to 2 meters and contaminate surfaces This research focuses on comparatively large droplets, measuring up to 1mm in diameter, while excluding smaller droplets that are less than 0.5mm.
Effect of droplet properties on evaporation
Viruses are transmitted through droplets released by an infected person during coughing and sneezing As these droplets evaporate, they may remain airborne longer, increasing the risk of infection for those nearby The evaporation process is crucial as it decreases droplet size and influences how long they stay suspended in the air For example, a droplet with a diameter of 0.17mm can fall 2 meters in 3 seconds in dry air, but its descent slows in more humid conditions Research indicates that evaporation affects both the final size of the droplet and its settling time, making it a key factor in influenza transmission This study will examine how the properties of droplets—such as surface tension and salinity—and environmental factors like temperature and humidity impact the evaporation rates of water, saline, and human saliva droplets.
Literature review
Extensive experiments on the evaporation of water droplets, saltwater droplets, and human saliva droplets have been conducted over the past few decades A study by Ranz & Marshall (1952a) focused on the evaporation of distilled water droplets, analyzing energy and mass transfer at low Reynolds numbers The researchers obtained independent correlations for heat and mass transfer rates by measuring drop temperatures with 0.127 mm thermocouples, using droplets with diameters ranging from 0.06 to 0.11 cm Conducted in still dry air at an ambient temperature of 24.9˚C, the droplets were maintained at 9.11˚C Photomicrographs were captured with a feed capillary featuring a 0.127 mm manganin-constantan thermo element junction at its center The findings indicated that a 0.1 cm diameter water droplet would take approximately 780 seconds to evaporate under these conditions, with the experimental data aligning closely with the model predictions.
Moyle et al (2006) conducted a study on the evaporation of small diameter water droplets, ranging from 15 to 50 microns, using an electrodynamics particle trap The research was performed under specific gaseous conditions, with a temperature of approximately -43˚C, a dew point of -33˚C, and a pressure of 59.2 kPa The relative humidity surrounding each droplet was calculated to be 36% using the Magnus-Tetens approximation (Redmond, 2007) To analyze the evaporation process, the mass rate of change of a water droplet in a steady state vapor field, as described by Pruppacher & Klett (1978), was employed to predict changes in diameter over time The findings indicated that the evaporation of small water droplets can be effectively predicted through this numerical method.
In a study by Pyung et al (1968), the evaporation rates of water droplets were measured using two methods: a 0.18mm diameter Fenwal thermistor and a 40-μm diameter glass fiber with a beaded tip, with photographic records taken through a microscope The findings aligned with Maxwell’s quasi-stationary prediction (Fuks, 1959), indicating that the rate of change in the diameter of the evaporating droplet is inversely proportional to its diameter, as expressed in Equation 1.
The dynamics of droplet evaporation can be described by several key parameters: the mass of the droplet (m), time (t), radius (r), the diffusion coefficient of water in air (D v), the saturated vapor concentration at the droplet's surface (C 0), and the ambient vapor concentration (C ).
In 2009, John Redrow developed a numerical model to study the evaporation of human saliva droplets within an air jet, simulating a human cough using FLUENT software The research involved simulations conducted at the Morgantown NIOSH facility, predicting the behavior of multiple consecutive coughs in a controlled environment Small particles, representing saliva droplets, were injected into the room, and their trajectories were meticulously tracked The dispersion of these particles was then compared to experimental results, with a particular focus on the movement of saliva droplets ranging in size from 1.5 micrometers.
The study focused on droplet sizes ranging from 1.5 to 1500 μm, demonstrating that John Redrow's method accurately predicts the properties of binary aqueous solution droplets containing sodium chloride, glucose, or bovine serum albumin However, the predictions for ternary or higher-order solutions did not align with experimental results Additionally, the research simulated the transmission of droplets generated by coughs within a room.
Measurement of Surface Tension
Methodology
Table 2.1 lists the equipment utilized for various measurements, including an electrical balance for measuring mass, a Tencor Alpha-Step 200 for thickness assessment, and a Finnipipette II for liquid volume measurement Additionally, images were captured using a Canon camera with a Tokina lens, while capillaries were employed to measure surface tension.
Table 2 1 Equipments used in the measurement
Tencor Alpha-Step 200 ±160 àm 0.015àm KLA-Tencor, Milpitas,
Finnipipette II 1-5mL 0.01mL Fisherbrand, Pittsburgh,
Canon EOS 450D camera N/A N/A Canon, Ōta, Tokyo, Japan
The experiment involves two types of glass capillaries: one with a red marking, featuring an inside diameter of 1.12 mm and an outside diameter of 1.47 mm, and another with a yellow marking, having an inside diameter of 0.96 mm and an outside diameter of 1.38 mm When these capillaries are submerged in water, they exhibit a phenomenon known as "Capillary Phenomenon," where the liquid level rises against gravity, as illustrated in Figures 2.1 and 2.2 This results in a visible liquid column ascending within the capillary tube.
Figure 2 1 Surface Tension Measurement: Red Tube I.D = 1.12mm, O.D = 1.47mm
Figure 2 2 Surface Tension Measurement: Yellow Tube I.D = 0.96mm, O.D = 1.38mm
Other devices were utilized during the measurements, such as beakers, iron supports, microinjector (1-10μL), etc.
Surface tension plays a crucial role in the evaporation of droplets, as demonstrated by Equation 2.1 To gain a deeper understanding of how surface tension affects evaporation, a variety of liquids were analyzed The liquids tested included distilled water, tap water, and different concentrations of sodium chloride (0.3%, 0.6%, and 0.9%), glucose (0.6% and 0.9%), as well as 5% and 10% milk solutions, brewed coffee, and human saliva.
Surface tension (σ) is measured in newtons per meter (N/m) and is influenced by several factors, including the height (h) of the liquid column in meters (m), the density (ρ) of the liquid in kilograms per cubic meter (kg/m³), the acceleration due to gravity (g) in meters per second squared (m/s²), the inner radius (r) of the tube in meters (m), and the contact angle (θ) Understanding these variables is crucial for applications involving fluid dynamics and capillary action.
To prepare three distinct concentrations of saltwater solutions, we measured the masses of pure solid NaCl and distilled water For example, to create a 0.3% NaCl solution, specific procedures were followed to ensure accurate formulation.
1 Calculate the mass of NaCl and distilled water (Table 2.2);
2 Put a piece of clean dry paper on the pallet of the electrical balance, set the balance to zero, and use the spatulato fill the calculated mass of salt on the paper, and put the salt into a beaker;
3 Lay a clean empty beaker on the pallet, set the balance to zero, and fill it with the calculated mass of distilled water,
4 Pour the water into the beaker which contains the salt, and mix the water and salt with glass rod gently, and
5 Repeat the same procedure to obtain the other two NaCl water solutions.
Table 2 2 Measured mass of water and NaCl for three concentrations of NaCl water solutions
The glucose water solutions are obtained with the same procedure used before but with different concentrations (Table 2.3)
Table 2 3 Measured data of mass of water and glucose for two concentrations of glucose water solutions
Milk water solutions were prepared using 0.02% fat milk, following the previously established procedure The specific measured mass of the milk used in this process is detailed in Table 2.4.
Table 2 4 Measured data of mass of water and milk of the two concentrations of milk water solutions
Density values play a crucial role in calculating surface tension To ensure accurate results, the density of each liquid solution must be measured It is important to note that the density of distilled water is considered equivalent to that of pure water at standard conditions, specifically at 20°C and 1 atm (Efunda, 2011) Throughout the experiments, maintaining a consistent temperature between 20-23°C and a pressure of 1 atm is essential for reliable measurements.
Mass 0.02% Fat Milk [g] Distilled Water [g]
The density of the tap water is obtained as the same temperature and pressure as the distilled water ( SImetric, 2010).
Three methods were employed to determine the densities of NaCl solutions The first method involved calculations, while the second method consisted of measuring the mass and volume of three NaCl solutions through ten repetitions, allowing for the calculation of an average density In this approach, the volume was consistently maintained at 2 mL throughout the procedure.
1 Use Finnipipette II 1-5mL (Fisherbrand, Pittsburgh, Pennsylvania, U.S.) to measure 2 mL of liquids;
2 Lay a beaker onto the pallet of the electrical balance, set the balance to zero;
3 Put the liquid into the beaker, and read the mass
4 Repeat this process 10 times (Table 2.5);
Table 2 5 Measured mass data of three concentrations of NaCl solutions with fixed volume 2mL
The final method is from “The Engineering Toolbox” It offers a relationship between the concentration of NaCl solutions and their densities (Fig 2.3) (EngineeringToolBox, 2009)
Figure 2 3 Density changing with concentration of NaCl
A function of density and concentration is calculated, which is:
Where cis the concentration and ρis the density of the NaCl solution Then the density of any NaCl solution can be obtained as long as the concentration is known.
The CSGNetwork calculator, developed in collaboration with the University of Michigan and NOAA, is designed to calculate the density of NaCl solutions based on specified temperature and concentration values This tool provides accurate density computations, making it a valuable resource for researchers and professionals in various scientific fields.
Table 2.6 shows the results of the three methods.
Table 2 6 Results of densities of NaCl solutions of the three methods
From Table 2.6, the difference of density values of each method is small; since the result of method 2 was obtained by measurement, it was used in this thesis.
The densities of glucose water solutions are obtained by measurement, and the method is the same as used in the NaCl water solutions.
The densities of 0.02% fat milk water solutions were obtained by using equation:
The density of a milk-water solution, represented as ρ mixture, is determined by the densities of its individual components, ρ i, which include 0.02% fat milk and distilled water The mass fraction of each component, ω i, plays a crucial role in this calculation Notably, the density of 0.02% fat milk is measured at 1,033 kg/m³ at a temperature of 20 °C (Jones, 2002).
The density of brewed coffee was measured by the same procedure before, but the fixed volume was set to 5mL.
The density of human saliva was determined using two methods: measurement and calculation Unstimulated whole saliva samples were collected between 9:00 AM and 11:00 AM, following the procedure outlined by Chiappin et al (2007) The results of the measurements are presented in Table 2.7.
Table 2 7 Measured mass and calculated density of human saliva
From Table 2.7, the average value of density of human saliva can be calculated, which is 1001.73 kg/m 3
Human saliva is primarily composed of 99.5% water, along with 0.3% proteins and 0.2% inorganic and trace substances, as established by Schipper et al (2007) These components significantly contribute to the overall density of saliva, which has been analyzed and quantified by various scientists.
In this method, certain assumptions were made, primarily due to the negligible concentration of trace substances in human saliva, which is less than 1/10,000, leading to their exclusion from calculations Additionally, a variety of inorganic ions present in human saliva are detailed in Table 2.8 (Chiappin et al., 2007).
Table 2 8 Mass concentration of inorganic components of human saliva
To accurately calculate the density of human saliva, it is essential to know the density of its individual components Table 2.8 outlines the solvent amounts for each substance, leading to assumptions about the solid components remaining after complete water evaporation Table 2.9 presents the concentrations of NaCl, KCl, MgCl2, and NH4HCO3, derived from the ion concentrations listed in Table 2.8 for 1 mL of human saliva For example, a combination of 5 moles of Na+ with 5 moles of Cl- was considered in this analysis.
5 mol sodium chloride was created.
Table 2 9 Mole number in 1mL and density information of salts in human saliva
Results and Discussion
Collecting all the values needed in Equation 2.1, the surface tension of each liquid can be calculated (Table 2.15)
Table 2 15 Results of surface tension of each kind of liquid
Figure 2.8 shows the surface tension of all the solutions measured Some trends can be observed
The surface tension of NaCl solutions increases with higher concentrations, aligning with Lloyd Trefethen's (1969) findings that inorganic salts raise the surface tension of water Conversely, sugar water solutions exhibit a decrease in surface tension as concentration rises, demonstrating an opposite relationship This trend is supported by earlier research indicating that organic substances lower water's surface tension (Lindfore, 1924; Weissenborn, 2006) Additionally, increased milk concentration in water results in reduced surface tension, while the surface tension of human saliva is similar to that of tap water.
Figure 2 8 Surface tension of each liquid
Measurement of Evaporation Rate
Experimental Set-up and Procedure
Experiments were performed to acquire the characteristics of droplet evaporation The solutions measured included distilled water, tap water, NaCl with different concentrations, and human saliva
NaCl water droplets were prepared from three distinct concentrations: 0.3%, 0.6%, and 0.9% NaCl solutions, following the same method used for surface tension measurements Additionally, the human saliva droplet was sourced from unstimulated human saliva, as referenced by Chiappin et al (2007).
The procedure for measuring the evaporation rate of distilled water droplets involves placing the water sample in a controlled environment for over 30 minutes to achieve thermal equilibrium A calculation provided in Appendix I demonstrates that this duration is sufficient for temperature stabilization Throughout the measurement, the temperature and relative humidity of the surrounding environment are continuously monitored using a hygrometer (Meade Instruments Corp., Irvine, CA, USA) Additionally, wind velocity, which significantly influences the evaporation rate, is measured with a hot wire thermo-anemometer (Extech Instruments Corporation, Waltham).
1 The temperature, relative humidity and the wind velocity were recorded at the beginning of the experiment, and the solutions to be studied would be placed in the experimental environmental for more than half an hour, sometimes, with a cover on top of the beakers;
2 The microinjector (1-10μL) (Figure 3.1) is used to ingest some water;
3 Inject 1μLwater approximately by subtracting the scales on the microinjector;
4 Move the water droplet down to the middle of the needle on the microinjector by slightly shaking, or inject a droplet and lie it on a flat plate;
5 If the droplet is held by the needle, hold it with the iron support vertically;
6 Taking pictures of the droplet every 30 seconds during the whole evaporation process;
7 Record the temperature, relative humidity and wind velocity at the end of the experiment, and if the difference between the initial and final droplet is small then the conditions can be considered as constants;
The 4 d and 5 th step should be done as soon as possible to decrease the amount of water lost by evaporation In order to obtain high quality pictures of the evaporation process, the illumination condition can easily influence the result; there will be a fluctuation of the illumination intensity shown in the pictures even a person just passing by, and afterward the results obtained from Matlab would be influenced Therefore, the illumination intensity should be exactly the same during the entire process In this research, in order to reduce the influence of the fluctuation of illumination intensity, a photo studio (Merax, City of Industry, CA, USA) was used to cover the droplet.
Repeat the same procedure to all the other liquids to obtain the pictures Figure 3.2 shows how the experiment is designed.
Figure 3 2 Photo studio used in the experiment of evaporation rate
Image Analysis
A Matlab code has been developed to analyze previously captured images of evaporation, calculating the droplet's volume and surface area over time The program continuously zooms in on the droplet in the photos and converts the images into binary format, featuring only black and white elements (see Figures 3.3 and 3.4).
Figure 3 3 Pictures of droplet on the needle: (a) Photo of droplet on the needle, (b) Binary photo obtained from program
Figure 3 4 Pictures of droplet on the surface: (a) Photo of droplet on the needle, (b) Binary photo obtained from program
The binary image allows for the clear identification of the droplet's boundary Given that the droplet on the needle is approximately symmetric around its centerline, calculations for its volume and surface area can be derived from the left portion of the droplet.
A r d (3.1) and for a semi-sphere, the surface area is shown in Equation 3.2:
A r d (3.2) where Astands for the surface area and dis the diameter of the droplet.
The program executed a calculation procedure for each image, illustrating the evaporation of a single droplet This process culminated in the creation of a video capturing the entire evaporation sequence Subsequently, the evaporation rate was determined, as demonstrated in Figure 3.5, which presents the experimental results for the evaporation rate of a distilled water droplet at an ambient temperature of 22.6˚C and a relative humidity of 48% The data indicates that the evaporation rate closely follows a linear trend.
Experiment of Distilled Water: D o =1.03mm
Figure 3 5 The experimental result of evaporation rate distilled water droplet: Surface area vs Time
Figure 3.6 shows the Experimental results of 0.3% NaCl water solution droplet, 0.6% NaCl water solution droplet, 0.9% NaCl water solution droplet, and human saliva droplet evaporation under 13.5˚C and relative humidity of 33%;
0.3% NaCl water solution: Do=0.73mm 0.6% NaCl water solution: Do=0.92mm 0.9% NaCl water solution: Do=0.83mm Human saliva: Do=0.80mm
Figure 3 6 Experimental results: saline water and human saliva droplets evaporation:
To ensure the reliability of the experiment, multiple evaporation tests were conducted for each liquid The experimental results, illustrated in Figure 3.10, depict the evaporation of NaCl water solution droplets at a temperature of approximately 34.5˚C and a relative humidity of around 26%.
The experimental results of evaporation, as illustrated in Figures 3.5 and 3.6, show minor fluctuations; however, it is evident that the surface area of the droplet changes linearly over time.
Figure 3.7 shows the experimental evaporation results of droplets of different NaCl water solutions The 0.3% NaCl water solution droplet evaporates slightly faster than the 0.6% and 0.9% NaCl water solution droplets.
0.3% NaCl water solution: Do=0.97mm 0.6% NaCl water solution: Do=1.04mm 0.9% NaCl water solution: Do=0.97mm
Figure 3 7 The experimental result of evaporation of NaCl water solution droplets: Surface area [mm 2 ] vs Time [s]
Figure 3.8 shows the experimental result of evaporation of human saliva droplets under temperature around 23˚C and relative humidity around 30%.
Experiment of human saliva: D o =0.81mm Experiment of human saliva: D o =0.74mm Experiment of human saliva: D o =0.81mm
Figure 3 8 The experimental result of evaporation of human saliva droplets: Surface area
The evaporation rates of various solutions differ significantly, as illustrated in Figures 3.7 and 3.8 However, Figure 3.11 demonstrates that, under identical environmental conditions, different human saliva droplets exhibit a consistent evaporation rate.
The experimental results depicted in Figure 3.9 illustrate the evaporation of human saliva droplets under varying environmental temperatures and relative humidity levels The findings indicate that an increase in temperature leads to a higher evaporation rate, while a decrease in relative humidity also contributes to an increased evaporation rate.
Figure 3 9 Experimental results of semi-sphere Human saliva Droplets evaporation under different environmental temperature and relative humidity: Surface area [mm 2 ] vs Time [s]
Temp# o C, RH0%, D o =0.81mmTemp# o C, RH0%, D o =0.74mmTemp# o C, RH8%, D o =0.90mmTemp6.7 o C, RH%%, D o =0.80mmTemp6.7 o C, RH%%, D o =0.68mm
Mathematical Model of Evaporation
According to Fick’s law, the mass flux at the surface of an evaporating or condensing droplet in a binary mixture is:
(4 1) where nis the coordinate normal to and away from the surface, A is the mass density of species
A, s is the mixture density, D is the diffusion coefficient (Crowe et al., 1997) Then,
(4 2) where A is the mass fraction of species A in the mixture (Crowe et al., 1997).
The mass fraction gradient for a droplet with diameter d is directly proportional to the difference in mass fraction between the droplet's surface and the surrounding free stream, while being inversely proportional to the droplet's diameter.
The mass fraction of species A at the droplet surface is denoted as \( \omega_A^s \), while \( \omega_A^\infty \) represents its value in the free stream The representative density, \( \rho_c \), is defined as the average density between the droplet surface and the free stream, as outlined by Crowe et al (1997).
The droplet continuity equation describes how the mass of a droplet changes, indicating that the rate of mass change is equal to the negative mass efflux across its surface This relationship is mathematically represented as \( s \frac{dm}{dt} = -\rho \omega \), where \( A \) denotes the surface area of the droplet (Crowe et al., 1997).
Thus the rate of change of droplet mass should be proportional to
The constant of proportionality is the Sherwood number so the equation becomes (Crowe et al., 1997).
, , c A A s dm Sh d D dt (4.6) where Sh is Sherwood number (Crowe et al., 1997):
Sh (4.7) where Re r is the Reynolds number based on the relative speed between the droplet and the carrier gas,
Where u and v are the velocity of the continuous and dispersed phases, respectively, ν is the kinetic viscosity.
Sc is the Schmidt number defined by
(4.10) where A , is the vapor density in free stream and A s , is the density at droplet surface.
The mass changing with time can be represented by:
6 2 d d dm d d dd dt dt d dt
(4.13) where d is the droplet density (Pruppacher & Klett, 1978)
From ideal gas law eM
RT (4.14) where Ris the universal gas constant, eis the atmosphere pressure.
(4.15) where e is the vapor pressure in the free stream, e d is the vapor pressure at droplet location, T is the temperature in the free stream and T d is the droplet temperature.
The temperature T d at the surface of a droplet can be derived by analyzing the interplay between heat and mass transfer rates, particularly through the process of heat release (Pruppacher & Klett, 1978).
(4.17) where L v is the latent heat of evaporation, k a ' is the modified thermal conductivity (Crowe et al., 1997):
(4.20) where e sat w , (T ) is the saturated vapor pressure at temperature T
Relative humidity is defined as:
( ) ( ) ( ) exp 1 1 4 / 3 sat w d d d e w w s a s w s sat w sat w d sat w w d s e T e e L M M M M e T e T e T RT RT a a m
The equation (4.22) relates various factors affecting surface tension, including the surface tension coefficient (σ), water density (ρ), the number of ions (ν) a salt molecule dissociates into, the practical osmotic coefficient (Φ), the molecular weight of salt (M), and the mass of salt in the droplet (m), as outlined by Pruppacher & Klett (1978).
ShD M L M M M M rdr RH dt R T RT RT r r m
Equation 4.22 effectively addresses binary solutions; however, to accommodate multiple component solutions, a more general model is required This can be achieved by modifying Equation 4.22 to include an aqueous solution drop that contains a solid insoluble substance, as outlined by Pruppacher & Klett (1978).
(4.24) where m m m s N , m N is the mass of the solid components, r N is the diameter of the nuclei after evaporation
Replacing the term shown in equation (4.22) in equation (4.23):
Diffusion Term Surface Tension Influence Solute Influence y y y w N N y y d N sat w e w w s a d w e T
A Matlab program was developed by using the mathematical model of equation (4.25).
To solve Equation 4.23, the valuables in the equation should be specified
The original equation is only valid for the distances greater than the ‘vapor jump’ length v (∆ ν
The mean free path (λ) refers to the average distance a moving particle travels between successive collisions In the region defined by r ≤ r ≤ r + Δr, it is essential to incorporate gas kinetic expressions to modify the original equation accordingly.
(Pruppacher & Klett, 1978); this equation is used to calculate in the Matlab code;
(Andreas, 2005), and it is used in the Matlab code;
The number of ions into which a salt molecule dissociates: 2;
Liquids used with the model
The numerical solution models pure water, along with sodium chloride (NaCl) solutions at concentrations of 0.3%, 0.6%, and 0.9%, in addition to human saliva The specific constituents of human saliva utilized in this numerical analysis are detailed in Table 4.2 (Redrow, 2009).
Table 4 1 Composition of human saliva in the mathematical solution
Composition of human saliva Composition [mg/mL]
Results and discussion
The experimental results depicted in Figure 5.1 closely align with the numerical data, indicating a strong correlation between the two The slight anomaly observed at the end of the experimental results may be attributed to light reflection and refraction during the photography process This study focuses on the evaporation of distilled water droplets under controlled temperature conditions.
23˚C; the relative humidity is 50% and the wind velocity is zero.
Experiment of Distilled Water Numerical Result of pure water
Figure 5 1 Water droplet evaporation: Surface area vs Time
The results of three different concentrations of NaCl water solution droplets and human saliva droplet are illustrated in Figure 5.2 The environmental temperature is 13.5˚C and relative humidity is 33%.
Figure 5 2 0.3% NaCl water droplet evaporation: Surface area [mm 2 ] vs Time [s}
During the evaporation process of salt solutions, some noise points were eliminated, which were attributed to salt crystallization after water evaporation Ideally, salt should crystallize into a semi-spherical shape; however, it was observed to form randomly on a flat plane instead Figure 5.2 illustrates a strong correlation between numerical and experimental results, indicating that the evaporation rates of salt solution droplets are lower than those of pure water droplets Additionally, the figure highlights that the evaporation rate of human saliva droplets is also slightly slower than that of pure water Overall, the numerical model effectively predicts the experimental outcomes.
Figure 5.3 shows the evaporation of different kinds of droplets at different environmental
0.3% NaCl water solution: Do=0.73mm 0.6% NaCl water solution: Do=0.92mm 0.9% NaCl water solution: Do=0.83mm Human saliva: Do=0.80mm
Symbols are experimental results experimental result; 0.3% means 0.3% NaCl water solution droplet, similarly to 0.6% and 0.9%;
Human saliva (HS) droplets were studied under varying environmental conditions, specifically at temperatures of approximately 34.2°C with 26% relative humidity and 23°C with 30% relative humidity The droplets analyzed included NaCl water solutions at concentrations of 0.3%, 0.6%, and 0.9% The experimental findings, as depicted in Figure 5.2.8, closely align with the numerical results, demonstrating the reliability of the numerical solution for understanding the evaporation of human saliva droplets.
N: 0.3%: T4.2 °C, RH&% Do=0.97mm; E: 0.3%: T4.2 °C, RH&% Do=0.97mm N: 0.6%: T4.2 °C, RH'% Do=1.04mm; E: 0.6%: T4.2 °C, RH'% Do=1.04mm N: 0.9%: T5 °C, RH&% Do=0.97mm; E: 0.9%: T5 °C, RH&% Do=0.97mm N: HS: T# °C, RH0% Do=0.81mm; E: HS: T# °C, RH0% Do=0.81mm N: HS: T# °C, RH0% Do=0.74mm; E: HS: T# °C, RH0% Do=0.74mm N: HS: T# °C, RH0% Do=0.81mm; E: HS: T# °C, RH0% Do=0.81mm.
Figure 5 3 Results of Evaporation of different droplets under various environmental temperatures:
Surface area [mm 2 ] vs Time [s]
Figure 5.4 illustrates the evaporation groups of human saliva droplets subjected to various environmental conditions, including high temperature (36.7°C, 25% relative humidity), medium temperature (23°C, 30% and 38% relative humidity), and low temperatures (13°C and 2°C, 33% relative humidity) In Figure 5.7, "N" denotes numerical results, "E" represents experimental results, and "T" indicates the environmental temperature.
Semi-sphere Human saliva Droplets evaporation
T# o C, RH0%, Do=0.81mm T# o C, RH0%, Do=0.74mm T o C, RH%%, Do=0.90mm T6.7 o C, RH%%, Do=0.80mm T6.7 o C, RH8%, Do=0.68mm T=2 o C, RH%, Do=0.59mm The points are the experimental resutls
Figure 5 4 The evaporation results of human saliva droplets at different temperature and relative humidity; Surface area [mm 2 ] vs Time [s]
In Figure 5.5, it is illustrated that higher environmental temperatures lead to increased evaporation rates, while maintaining a constant temperature and increasing relative humidity results in slower evaporation This indicates that evaporation rates are influenced by varying temperature and relative humidity conditions Furthermore, the numerical results align closely with the experimental findings.
Figure 5 5 Evaporation of human saliva droplets under different environmental temperature, similar relative humidity
S u rf a c e a re a /I n it ia l S u rf a c e A re a
T=0.7 o C, RH5%, Do=0.42mmT# o C, RH0%, Do=0.81mmT6.7 o C, RH(%, Do=0.80mmSymbols are the experimental results
Figure 5.6 illustrates the experimental results and numerical results of 0.9% NaCl water solution droplets and brewed coffee droplet evaporation.
Time / Initial Water Evaporation Time D ro p le t S u rf a c e A re a / I n it ia l D ro p le t S u rf a c e A re a Exp.: Brewed Coffee Droplet
Num.: Brewed Coffee Droplet Exp.: 0.9% NaCl water Droplet Num.: 0.9% NaCl water Droplet
Figure 5 6 Evaporation of coffee and 0.9% NaCl water solution droplet: Temp" ° C,
Table 5.1 is showing the comparison of numerical results from the Matlab program and experimental results from other authors.
Table 5 1 Comparison of numerical results: Matlab vs experiments
Numerical Result [Zhang, 2011] Error Smolik et al. pure water
Table 5.2 shows the experimental results of human saliva droplets evaporation performed in this study and the numerical results under the same experimental conditions.
Table 5 2 Evaporation rate of human saliva droplet: experiment vs numerical results
Numerical Result Error human saliva
Table 5.3 shows the experimental results of NaCl water solution droplets and other droplet evaporation performed in this study and the numerical results under the same experimental conditions.
Table 5 3 Evaporation rate of other kinds droplets: experiment vs numerical results
Figure 5.7 illustrates how the evaporation rate fluctuates with varying temperatures while maintaining constant relative humidity As temperature rises, the evaporation rate increases, whereas a decrease in temperature leads to a reduction in the evaporation rate.
3.5 Numerical results: Semi-sphere human saliva Droplets evaporation (Constant RH), D o =1mm
Figure 5 7 Numerical results of human saliva droplets under different temperature, same relative humidity
As illustrated in Figure 5.8, the evaporation rate fluctuates with changes in relative humidity while maintaining a constant temperature Specifically, an increase in ambient relative humidity leads to a decrease in evaporation rate, whereas a decrease in relative humidity results in an increased evaporation rate.
3.5 Numerical results: Semi-sphere human saliva Droplets evaporation (Constant Temp.), D o =1mm
Figure 5 8 Numerical results of human saliva droplets under different relative humidity, same temperature
To achieve a droplet with an evaporation rate comparable to that of human saliva, numerical results were presented in Figure 5.9, which illustrates various droplets with an initial diameter of 1mm under consistent temperature and relative humidity conditions The data indicates that as salinity increases, the evaporation rate decreases, with pure water droplets exhibiting the highest evaporation rate, while 1.2% NaCl water droplets demonstrate the slowest evaporation rate among the tested samples.
Time / Evaporation Time S u rf a c e a re a / n it ia l d ro p le t s u rf a c e a re a 0.3% NaCl water droplet
0.9% NaCl water droplet Human saliva droplet 1.0% NaCl water droplet Pure water droplet
Figure 5 9 Numerical results of evaporation of different droplets
Figure 5.10 illustrates the evaporation process of artificial human saliva and natural human saliva droplets, both with an initial diameter of 1mm, under identical conditions of 23˚C and 38% relative humidity To create artificial human saliva, the numerical solution code for a 1mm diameter human saliva droplet is executed at 23˚C and 30% relative humidity to determine the complete evaporation time Subsequently, the concentrations of lipid and DNA are set to zero, and the code is rerun with varying concentrations of salt, protein, and carbohydrate to closely replicate the properties of natural human saliva.
Figure 5 10 Numerical solution of evaporation of artificial human saliva evaporation vs
In Figure 5.11, the temperature changed during the evaporation process
3.5 Artificial human saliva vs Human saliva: Temp = 23 o C, RH = 38%
Artificial Human saliva Human saliva
D ro pl et T em pe ra tu re [ o C ]
It can be observed that the temperature falls dramatically at the initial evaporation process to
The temperature initially drops sharply to 10˚C due to the latent heat of evaporation, and then stabilizes At the conclusion of the evaporation process, a further decrease in temperature occurs, attributed to the change in the liquid's density.
Conclusions
This study analyzes the surface tension and evaporation rates of various droplets using Matlab techniques and software The results indicate that for water, a temperature increase of 20˚C leads to a decrease in surface tension by 4.33%.
When the salt concentration increases, the surface tension of the water solution will increase; contrarily, when the glucose concentration increases, the surface tension of the water solution will decrease.
Salt reduces the evaporation rate of water droplets by forming hydrogen bonds with water molecules when dissolved These bonds require additional kinetic energy to break, making it harder for water to transition from liquid to gas As temperature rises, evaporation occurs more rapidly due to increased kinetic energy in water molecules Conversely, higher relative humidity slows evaporation because the increased concentration of water vapor in the air inhibits the escape of water molecules.
The evaporation rate is influenced by temperature and relative humidity As temperature rises, the evaporation rate increases, while a decrease in temperature leads to a lower evaporation rate Conversely, when relative humidity increases, the evaporation rate decreases; however, a decrease in relative humidity results in an increased evaporation rate.
As salinity increases, the evaporation rate of water droplets decreases Among various solutions, pure water droplets exhibit the highest evaporation rate, while droplets of 1.2% NaCl solution demonstrate the slowest evaporation rate This trend highlights the inverse relationship between salinity and evaporation efficiency.
To create artificial human saliva, a numerical solution is employed under specific environmental conditions, including a temperature of #˚C and a relative humidity of 8% The composition of the artificial saliva is detailed in Table 6.1, and it exhibits an evaporation time comparable to that of natural human saliva droplets.
Table 6 1 The concentration of components of artificial human saliva
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This appendix details the calculation of the time required for tap water to reach equilibrium with the surrounding temperature The initial temperature does not impact the outcome The water is held in a beaker, modeled as a cylinder with a volume of 1 x 10^-5 m³, featuring a height and diameter of 0.08 m Notably, convection effects are negligible in this scenario.
Lumped system analysis is valid to solve this problem:
// clear all;close all;clc fns = dir('*.jpg'); fnames = char(fns.name); fdates = datevec(char(fns.date));
[j,k]=size(fnames); ini_crop=[1150 1880 180 410]; vol(j)=0.4664; %Needle Volume for i=1:j
%BW = im2bw(I3,0.5);%graythresh(I3+10)); %to be fixed
BW=(1-bwmorph((1-im2bw(I3,0.56)),'clean')); % 'bwmorph, clean'removes isolated pixels
BW3=double(BW); box = regionprops(BW3,'BoundingBox','centroid'); if i == 1 xw1 = box.BoundingBox(3); yw1 = box.BoundingBox(4); end ctr_x = box.Centroid(1); ctr_y = box.Centroid(2);
% I4=imcrop(I3,[ctr_x-xw1/2-15 ctr_y-yw1/2+120 xw1+60 yw1-150]);
I5=imresize(I3,1.2); %to be fixed imshow(I5)
[m,n] = size(BW3); verlin_x(1:m) = ctr_x; verlin_y = 1:m;
BWT_1=BWT_1(:,1:round(size(BW3,2)/2)+20); % Cut the shadow in the right part
% r_t=round(size(BW3,1)/2)-2:round(size(BW3,1)/2)+2; % the row of the water center (5 length)
The code snippet defines the approximate column and row positions of the water center in a given matrix It calculates the column indices, `c_t`, by rounding half the size of `BW3` and adjusting it with a constant value Additionally, the row indices, `r_t`, are set to the middle of the matrix height, ensuring that all column positions align with this central row This approach effectively centralizes the water center within the specified dimensions of the matrix.
BWT_2=bwselect(BWT_1,c_t,r_t); % select the main body of the water
The code snippet identifies the coordinates of water within a binary water table (BWT_2) by finding where it equals one It then determines the maximum row number of the lowest point and retrieves the index of that point To handle potential duplicates, it selects the middle index if multiple points exist Finally, the corresponding column number of the identified point is obtained.
The code processes an image using the Canny edge detection method to identify the first non-zero pixel in each row, storing the results in the array `zero_pix` It then calculates the adjusted pixel values with a correction factor and uses polynomial fitting to model the relationship between the droplet's vertical position and its radius The surface area of the droplet is computed by integrating the fitted polynomial function over the defined range Time elapsed since the start is calculated, and both the surface area and evaporation rate are displayed as text annotations on the plot, providing insights into the droplet's characteristics over time.
F(i) = getframe; end movie2avi(F,'example.avi','compression','None','quality',100,'fps',2); for i=2:j lambda_e(i-1) = (sur(i)-sur(i-1))*1e-6/(et(i)-et(i-1))/pi lambda_e_mean = mean(lambda_e) end
R = 8.314;%universal gas constant den8.2;%water density kg/m3
RH=0.33; % Relative humidity of the environment
Pi=RH*Pd;%pressure of the environment
%d_0=(6*vol_0/pi)^(1/3);% in meters lamda=4*Dv*M*(Pd-Pi)/(R*den*T);%evaporation constant
%t_evap=d_0^2/lamda;%lifetime or evaporation time sur_0 = sur(1)*1e-6; t_evap=sur_0/lamda/pi;
%n = length(sur); t = 0:t_evap; sur_t = sur_0 - pi*lamda*t; d_0 = (sur_0/pi)^0.5; figure(1) title('Semi-sphere Droplet evaporation');
%plot(et/t_evap,(sur-sur(54))/(sur(1)-sur(54)),'^b',t/t_evap,sur_t/sur_0,'r')
The analysis presents a comparison between experimental and numerical results for the evaporation of distilled water The plots illustrate the relationship between time and the droplet surface area, normalized against the initial surface area The first plot shows the experimental data (marked in blue) against the numerical solution (in red) for pure water, with the x-axis representing time relative to the analytical evaporation time and the y-axis displaying the droplet surface area ratio The second figure further details the surface area of the droplet over time, again comparing experimental findings with numerical predictions Both plots effectively highlight the evaporation dynamics of distilled water, providing valuable insights into the behavior of droplets during evaporation.
64
RH = 0.38; d = (1)*1e-3; %m r=d/2; vol=(1*pi*r^3)/6; vol_semi = vol;
V=0; %wind velocity den_air = 1.1839; % [m3/kg] @25C rho_a = den_air; miu_air = 18.3*1e-6; %[kg/m.s]@25C
%Constituents in droplet kg/m^3 salt=9.00; % If this is the water, it should be 0, pro.9; lip.1; carb.3; dna=0.820;
% salt=9.00; % If this is the water, it should be 0,
% salt=0.001; % If this is the water, it should be 0,
% dna=0; m_pro=pro*vol; m_dna=dna*vol; m_salt=salt*vol;
% Density of water rho_w9.842594 + 6.793952e-2*TC - 9.095290e-3*TC^2 +
1.001685e-4*TC^3 - 1.120083e-6*TC^4 + 6.536332e-9*TC^5; m_w = rho_w * vol;
% droplet total mass m_tot=m_pro+m_lip+m_carb+m_dna+m_salt+m_w;
% mass fractions x_pro=m_pro/m_tot; x_lip=m_lip/m_tot; x_carb=m_carb/m_tot; x_dna=m_dna/m_tot; x_salt=m_salt/m_tot; x_water=m_w/m_tot;
%densities of each component [kg/m3] rho_salt!60; rho_pro62; rho_lip00; rho_carb62; rho_dna50;
%Molecular weight of each component
% specific heat of dry air, const press, T_inf = -40 to 40 C for press near 1atm c_p_d05.60 + 0.017211*T_dC + 0.000392*T_dC^2;
% thermal conductivity of air (W/mC) k_a_O=2.411e-2*(1 + 3.309e-3*T_dC - 1.441e-6*T_dC^2);
Sh = 2+0.6*(miu_air/den_air/D_v)^(1/3)*(d*V*den_air/miu_air)^0.5;
% saturation vapor pressure e_sat=6.1121*(1.0007+(3.46e-6)*P)*exp((17.502*T_dC)/(240.97+T_dC));
% total density of droplet rho_s=(x_pro/rho_pro+x_lip/rho_lip+x_salt/rho_salt+x_dna/rho_dna+ x_carb/rho_carb+x_water/rho_w)^(-1);
% mass of water in the droplet m_w=rho_w*(vol-(m_salt/rho_salt+m_pro/rho_pro+m_lip/rho_lip+ m_dna/rho_dna+m_carb/rho_carb)); delta=T_d/T_inf-1;
% volume of solid part vol_sol=((m_salt/rho_salt + m_pro/rho_pro + m_lip/rho_lip + m_dna/rho_dna + m_carb/rho_carb));
% the radius of the solid part r_sol=(6*vol_sol/pi)^(1/3);
% Surface tension of water sig_w=0.2358*((374.00-TC)/647.15)^1.256*(1-0.625*((374.00-TC)/647.15));
In the dissociation of a salt molecule, the number of ions produced is represented by nu_ion = 2 The mass fraction of salt is calculated as mf_salt = m_salt/m_w, while the mass fractions for carbohydrates and proteins are given by mf_carb = m_carb/m_w and mf_pro = m_pro/m_w, respectively The density of the solution, rho_sol, is determined by the total mass of components including protein, lipid, carbohydrates, DNA, and salt, divided by the volume of the solution.
%% molalities (of binary solution) molal_s=(m_salt/M_s)*1000/m_w; molal_c=(m_carb/M_carb)*1000/m_w; molal_p=(m_pro/M_pro)*1000/m_w;
% practical osmotic coefficients phi_s=0.0055*molal_s^2+0.028*molal_s+0.9128;
% high concentration limiting condition if phi_s > 1.5 phi_s=1.5; end phi_c=0.0233*molal_c+1;
% high concentration limiting condition if phi_c > 1.2 phi_c=1.2; end phi_p".0357*2099560^molal_p-21.718; dRH=((RH)^0.5-0.5916); phi_pU*(((0.70711-0.5916)-dRH)/(0.70711-0.5916))+ phi_p*dRH/(0.70711-0.5916); if dRH