Views: 329 Author: Site Editor Publish Time: 2024-07-12 Origin: Site

**1. Microbubble formation process**

The dissolved air water in the dissolved air tank is in a high-pressure state after being pressurized by the air compressor. The oxygen and nitrogen in the air are dissolved in the water by gap filling and hydration to form supersaturated dissolved air water.

After the releaser releases the high-pressure supersaturated dissolved air water, it changes from a high-pressure state to a normal pressure. Due to the sudden pressure drop, the solubility of the gas in the water also decreases significantly. At this time, many bubbles are precipitated from the water.

It generally takes three processes for the gas to precipitate from the water to form stable microbubbles:

(1) As the pressure of dissolved air water decreases, gas molecules dissolved in water by hydration and gap filling are

continuously precipitated from the water and aggregate with neighboring gas molecules to form gas molecule clusters;

(2) Gas molecule clusters aggregate with each other to form gas nuclei;

(3) Gas nuclei expand and become stable microbubbles. The nucleation process of bubbles during the decompression of dissolved air water is shown in Figure 1-1.

a. Free gas molecules

b. Gas molecules aggregate

c. Agglomeration to form gas cores

d. The gas core expands to form stable microbubbles

Figure 1-1 Bubble nucleation process during decompression of dissolved air water

2. Nucleation mechanism of microbubbles

According to the properties of the solution, the bubble nucleation region and the supersaturation, the nucleation mechanism of microbubbles can be divided into:

(1) classical homogeneous nucleation; (2) classical heterogeneous nucleation; (3) quasi-classical nucleation; (4) non-classical nucleation.

The relationship between the saturated solubility of a solution and the amount of dissolved gas is called supersaturation. Formula 2-1 is the definition of solution supersaturation:

（2-1）

In the formula: Xb is the mole fraction of gas in the supersaturated solution; Xi is the mole fraction of gas in the saturated solution.

This nucleation mechanism assumes that: in a homogeneous solution without air cavities before supersaturation, a very high supersaturation is required to form bubbles, even more than 100 or higher.

Once a bubble is generated, it will immediately rise to the surface of the liquid and new bubbles will not be generated at the same position. Figure 2-2 shows the classic homogeneous nucleation process.

Figure 2-2 Classical homogeneous nucleation

The mechanism of classical heterogeneous nucleation is very similar to that of classical homogeneous nucleation, and they both require a considerable level of supersaturation.

A sudden decrease in system pressure causes bubbles to form on pits on the container wall, smooth surfaces, or particles in the solution, and then the bubbles grow and separate, leaving behind a portion of the gas. As shown in Figure 2-3.

Figure 2-3 Classical heterogeneous nucleation

The nucleation mechanism believes that: before the solution is saturated, there is an air cavity with a radius smaller than the critical radius of bubble formation inside it. Therefore, bubbles can only be generated by overcoming the nucleation energy barrier, such as the generation of bubbles with a radius of R1 as shown in Figure 2-4.

This form of nucleation mechanism believes that before the solution is saturated, there is an air cavity with a radius more significant than the critical radius of bubble generation, so there is no need to overcome the nucleation energy barrier. Classical heterogeneous nucleation or quasi-classical nucleation occurs first before non-classical nucleation occurs.

The critical radius of bubble generation is inversely proportional to the supersaturation, that is, when the liquid supersaturation decreases, the critical curvature radius of bubble nucleation will increase. When this radius is equal to the air cavity's radius, the bubbles' generation will stop. The generation of bubbles with a radius of R2 is shown in Figure 2-4.

Figure 2-4 Quasiclassical nucleation and nonclassical nucleation

In the figure, R1 and R2 represent the radii of the air cavities in the solution described above. R1 is smaller than the critical nucleation radius, indicating quasiclassical nucleation; R2 is larger than the critical nucleation radius, indicating nonclassical nucleation.

The formation of bubbles is also affected by the properties of the interface. The generation of a single bubble starts with a gas nucleus. Once the gas nucleus is formed, the bubble will continue to grow until it detaches from the matrix.

Many factors, such as surface tension, inertia of the solution, pressure, and buoyancy, also restrict the growth of bubbles.

This article introduces the study of the size and number of microbubbles by dissolving gas under high pressure and releasing the dissolved gas water into clean water at standard pressure through a releaser. Assuming that the wall is smooth, there are no impurities in the water, and there is no air cavity before the dissolved gas water is saturated, it can be considered that the generation of bubbles conforms to the classical homogeneous nucleation mechanism in this study.

As shown in Figure 2-1, the transformation of free gas molecules into stable microbubbles requires three thermodynamic processes: (1) free gas molecules aggregate into gas molecule clusters; (2) gas clusters aggregate to form gas nuclei; and (3) gas nuclei expand to form stable microbubbles.

Assuming that the free energy changes of the three processes are ΔG1, ΔG2, and ΔG3 respectively, the total free energy change of the microbubble formation process is formula 2-2:

ΔG = ΔG1 + ΔG2 + ΔG3 （2-2）

Assume that n gas molecules form a gas molecule cluster, and the volume of the gas molecule cluster is the sum of the volumes of the n gas molecules (Formula 2-3), then the effective radius of the gas molecule cluster is Formula 2-4

（2-3）

（2-4）

Therefore, the surface energy of gas molecules agglomeration is given by formula 3-5：

（2-5）

Assuming that the volume of a single gas molecule in the free state is V0, and the process of gas molecules agglomerating into gas agglomerates is an isothermal process, the chemical potential change of this process is formula 2-6:

（2-6）

Integrating Formula 2-6 gives Formula 2-7:

ΔF2 = F2 - F1 = -(P* - P0)nV0 (2-7)

From formulas 2-5 and 2-7, we can get that the total free energy change ΔG1 in the process of free gas molecules agglomerating into gas molecule clusters is formula (2-8):

（2-8）

In the formula: Rn is the effective radius of the gas molecule cluster; σ is the surface tension of the liquid; p* is the saturation pressure; p0 is the average pressure; V0 is the volume of the free gas molecules.

(2) Gas clusters gather to form gas cores

Under the driving force p* - p0, assuming that the gas molecules aggregate into gas cores and expand instantly that the expansion process obtains enough space to turn them into a gas state, and that the process is a thermodynamic equilibrium process, then:

ΔG2 = 0 (2-9)

In this process, the gas core expands, the volume increases, and the surface area increases. The change in the surface energy of the system can be expressed by formula 2-10:

ΔF3 = 4πR2 ● σ - 4πR2n ● σ (2-10)

The bubble expands and does work as shown in Formula 2-11:

（2-11）

Therefore, the total free energy change ΔG3 in the process of gas core expansion to form stable microbubbles can be expressed by formula 2-12:

（2-12）

In summary, by substituting formulas 2-8, 2-9, and 2-12 into formula 2-2, we can obtain the total free energy change of pressurized dissolved air water from pressure release to the generation of microbubbles as formula 2-13:

（2-13）

The internal pressure of the bubble in the formula, substituting it into formula 2-13, we get formula 2-14:

（2-14）

It can be seen from formula 2-14 that the change in free energy during bubble generation is related to the surface tension σ of the liquid, the pressure difference Δp when dissolved gas and water are released, the size R0 of the gas molecules dissolved in the liquid, the size R of the generated bubbles, and the number n of generated bubbles.

The size and rising rate of bubbles greatly influence the efficiency of flotation water purification, so it is necessary to study the motion characteristics of bubbles during their rise.

When the bubble diameter is too large, its rise is almost unaffected by the properties of the liquid;

But when the bubble diameter is less than 150μm, its floating process in the liquid is affected by five forces in the Y-axis direction: buoyancy, gravity, liquid thrust, drag and virtual mass force.

Bubbles will quickly reach a stable state during movement, and we can infer the change in bubble diameter by analyzing the force on a single bubble.

The following is an analysis of the rise of a single tiny bubble in still water.

The vertical forces acting on the bubble are shown in Figure 2-5.

Figure 2-5 Analysis of the force on the bubble in the vertical direction

Assuming that the mass of a single bubble is m, the gravity acting on the bubble can be expressed by equation 2-15:

G = -m ● g = - pgvgg (2-15)

The negative sign in the formula indicates that the direction of gravity is vertically downward, ρg is the gas density in the bubble, Vg is the volume of the bubble, and g is the acceleration due to gravity, g = 9.8m/s2.

According to the buoyancy principle proposed by Archimedes, the buoyancy of water on an object equals the weight of the water displaced by the object. The buoyancy of a bubble in water is obtained as formula 2-16:

*F _{f }= ρ_{l} gV_{g }*

_{}

In the formula: ρ_{l} is the density of the liquid.

When bubbles rise in a liquid, there is a relative motion between the bubble itself and the liquid. At this time, the bubble will be affected by the resistance of the liquid. This force is called drag. And as long as there is a speed difference (magnitude and direction) between the bubble and the fluid, the bubble will be affected by drag. Drag can usually be expressed by formula 2-17:

_{}

_{ （2-17）}

_{}

In the formula, *u _{l}*

When the bubble in the fluid accelerates upward relative to the fluid, it will drive part of the fluid around the bubble to accelerate together, equivalent to the bubble having an additional mass, called virtual mass. The extra force generated by this extra mass is called virtual mass force, and the extra mass will increase the effective inertial mass of the bubble. Formula 2-18 is the expression of virtual mass force:

(2-18)

Suppose the liquid is not stationary and has a velocity in the same direction as the bubble. The velocity of the liquid is more incredible than the velocity of the bubble rising. In that case, the liquid will exert a thrust on the rising bubble. If the liquid is stationary during the bubble rising process, the liquid has no thrust on the bubble, that is, *F _{p}* is zero.

By analyzing the force acting on a single bubble and based on Newton’s second law, the force balance equation for the bubble in a static liquid can be obtained by combining equations 2-15 to 2-18. After sorting, the calculation equation for the bubble volume (2-19) and the calculation equation for the bubble diameter (2-20) can be obtained.

(2-19)

(2-20)

In the formula, *a _{g}* is the acceleration of the bubble when it rises in still water, and

It can be seen from formula 2-20 that the diameter of the bubble is related to its rising speed *u _{g}*, acceleration

The drag coefficient varies with the Reynolds number and the properties of the liquid phase. Peebles Garber proposes the most representative drag coefficient calculation formula. When a single bubble rises in water:

Reynolds number less than 2: （2-21）

Reynolds number greater than 2: （2-22）

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