2.1 Generalized Reynolds Equation Based on the Two-Phase Flow Theory
This paper studies the water-lubricated SGTB with the turbulent cavitating flow film (Figure 1). The classical Reynolds equation is not suitable for describing the bearing lubrication state, a Reynolds equation which contains turbulent and two-phase interfacial effects is needed, so the following basic assumptions are made for the modeling.
-
(1)
The bubbles in the cavitation flow be regarded as spheres.
-
(2)
The flow field is a small perturbation of the turbulent Couette flow [24].
-
(3)
The turbulent shear stress is defined by the law of wall, Reichardt [25] empirical formula is used to define the eddy viscosity coefficient.
-
(4)
The average viscosity along the film thickness is adopted in generalized turbulent Reynolds equation.
The spiral area is an irregular area in the polar coordinate system, a spiral coordinates system \((\eta ,\zeta )\) are adopted to improve numerical calculation accuracy. The polar coordinates \((\theta ,r)\) are converted into the spiral coordinates system \((\eta ,\zeta )\) as follows:
$$\begin{array}{ll}\zeta =r,\\ \eta =\theta -f\left(\zeta \right),\\ f\left(\zeta \right)=\frac{1}{\mathrm{cot}\beta }\mathrm{ln}\left(\frac{\zeta }{{\zeta }_{b}}\right).\end{array}$$
(1)
The spiral patterns before and after the coordinates conversion are shown in Figure 2.
A generalized turbulent Reynolds equation with cavitation interface effects and inertial effects in the \((\eta ,\zeta )\) coordinates is derived based on two-phase flow theory [26]:
$$\begin{array}{ll}\frac{\partial }{\partial \zeta }\left(\frac{{h}^{3}\zeta }{{\overline{\mu }}_{w}{k}_{r}}\frac{\partial {C}_{w}p}{\partial \zeta }\right)+\frac{\partial }{\partial \eta }\left(\left(\frac{{\zeta }^{2}{{f}^{^{\prime}}}^{2}\left(\zeta \right)}{{k}_{r}}+\frac{1}{{k}_{\theta }}\right)\frac{{h}^{3}}{{\overline{\mu }}_{w}\zeta }\frac{\partial {C}_{w}p}{\partial \eta }\right)\\ -\frac{\partial }{\partial \zeta }\left(\frac{{h}^{3}\zeta }{{\overline{\mu }}_{w}{k}_{r}}{f}^{^{\prime}}(\zeta )\frac{\partial {C}_{w}p}{\partial \eta }\right)-\frac{\partial }{\partial \eta }\left(\frac{{h}^{3}\zeta }{{\overline{\mu }}_{w}{k}_{r}}{f}^{^{\prime}}(\zeta )\frac{\partial {C}_{w}p}{\partial \zeta }\right)\\ =\frac{\partial }{\partial \eta }\left(\frac{hU{C}_{w}}{2}\right)+\frac{\partial }{\partial \eta }\left(\frac{{h}^{3}}{{\overline{\mu }}_{w}{k}_{\theta }}M(\eta )\right)\\ \quad+\, \frac{\partial }{\partial \zeta }\left(\frac{{h}^{3}{\rho }_{w}{\overline{u} }^{2}{C}_{w}}{{\overline{\mu }}_{w}{k}_{r}}\right)-\frac{\partial }{\partial \eta }\left({f}^{^{\prime}}(\zeta )\frac{{h}^{3}{\rho }_{w}{\overline{u} }^{2}{C}_{w}}{{\overline{\mu }}_{w}{k}_{r}}\right),\end{array}$$
(2)
where
$$\begin{array}{ll}{\overline{\mu }}_{w}=\frac{1}{h}\int\limits_{0}^{h}\,{\mu }_{w}\,\mathrm{d}z,\\ \overline{u }=\frac{1}{h}\int\limits_{0}^{h}\,u\mathrm{d}z,\end{array}$$
(3)
$$\begin{array}{ll}\frac{1}{{k}_{\theta }}=\int\limits_{0}^{1}\int\limits_{0}^{{z}^{*}}\frac{1}{{f}_{c}}\left(1-\frac{{g}_{c}}{{f}_{c}}\right)\left(\frac{1}{2}-{z}^{*\prime}\right)\mathrm{d}{z}^{*\prime}\mathrm{d}{z}^{*},\\ \frac{1}{{k}_{r}}=\int\limits_{0}^{1}\int\limits_{0}^{{z}^{*}}\frac{1}{{f}_{c}}\left(\frac{1}{2}-{z}^{*\prime}\right)\mathrm{d}{z}^{*\prime}\mathrm{d}{z}^{*},\\ {f}_{c}=0.4\left[{z}^{*}{h}_{c}^{*}-10.7th\left(\frac{{z}^{*}{h}_{c}^{*}}{10.7}\right)\right]+1,\\ {g}_{c}=0.2{z}^{*}{h}_{c}^{*}t{h}^{2}\left(\frac{{z}^{*}{h}_{c}^{*}}{10.7}\right),\\ {z}^{*}=\frac{z}{h},\end{array}$$
(4)
$${h}_{c}^{*}=\frac{h}{{\nu }_{w}}\sqrt{\frac{{\tau }_{c}}{{\rho }_{w}}}.$$
(5)
The following expression can be given for the logarithmic spirals:
$${f}^{\mathrm{^{\prime}}}\left(\zeta \right)=\frac{1}{\zeta \mathrm{cot}\beta }.$$
(6)
The circumferential velocity distribution of the water film is written as
$$u=\frac{U}{2}+\frac{{({h}_{c}^{*})}^{2}}{{R}_{c}}\frac{U}{{h}^{*}}{G}_{1}({z}^{*})-\frac{1}{{C}_{w}}\frac{{h}^{2}}{{\overline{\mu }}_{w}}(\frac{1}{\zeta }\frac{\partial ({C}_{w}p)}{\partial \eta }M(\eta )){G}_{2}({z}^{*}),$$
(7)
where
$$\begin{array}{ll}{G}_{1}\left({z}^{*}\right)=\int\limits_\frac{1}{2}^{{z}^{*}}\frac{1}{{f}_{c}}\mathrm{d}{z}^{*{{\prime}}},\\ {G}_{2}({z}^{*})=\int\limits_{0}^{{z}^{*}}\frac{1}{{f}_{c}}\left(1-\frac{{g}_{c}}{{f}_{c}}\right)\mathrm{d}{z}^{*{{\prime}}},\end{array}$$
(8)
$${C}_{w}=1-{C}_{g},$$
(9)
$${C}_{g}=\frac{\frac{\pi }{6}\int\limits_{0}^{\infty }{\varsigma }^{3}{f}_{eq}(\eta ,\varsigma )\mathrm{d}\varsigma }{1+\frac{\pi }{6}\int\limits_{0}^{\infty }{\varsigma }^{3}{f}_{eq}(\eta ,\varsigma )\mathrm{d}\varsigma },$$
(10)
where \({f}_{eq}(\eta ,\varsigma )\) is the equilibrium distribution function of the cavitation bubble diameter; \({f}_{eq}(\eta ,\varsigma )\mathrm{d}\varsigma\) represents the number of bubbles with diameter between \((\varsigma ,\varsigma +\mathrm{d}\varsigma )\) per unit volume of water at spatial coordinate η under the equilibrium state. The first term on the right-hand side of Eq. (2) is the hydrodynamic effect, the second term on the right side of Eq. (2) is the interface hydrodynamic effect, the third and fourth terms on the right side of Eq. (2) is the inertia hydrodynamic effect. The interface momentum transfer function includes the following three parts:
$$M\left(\eta \right)={M}_{1}\left(\eta \right)+{M}_{2}\left(\eta \right)+{M}_{3}(\eta ),$$
(11)
where \({M}_{1}(\eta )\) is the interface momentum transfer term due to mass transfer, \({M}_{2}(\eta )\) is the interface momentum transfer term due to viscous drag, \({M}_{3}(\eta )\) is the interface momentum transfer term due to surface tension. Because the film thickness is thin, the liquid velocity in \({M}_{1}\left(\eta \right),{M}_{2}(\eta )\) are approximated by the average velocity. \({M}_{1}(\eta ),{M}_{2}(\eta )\) and \({M}_{3}(\eta )\) are written as:
$$\begin{array}{ll}{M}_{1}=-\left[{\rho }_{w}\frac{\pi }{3}\int\limits_{0}^{\infty }{\varsigma }^{2}{f}_{eq}(\eta ,\varsigma )\mathrm{d}\varsigma \right]\overline{u }(\overline{u }-{u}_{b}),\\ {M}_{2}=\left[\frac{\pi }{3}{C}_{Ds}{\rho }_{w}\int\limits_{0}^{\infty }{\varsigma }^{2}{f}_{eq}(\eta ,\varsigma )\mathrm{d}\varsigma \right]{(\overline{u }-{u}_{b})}^{2},\\ {M}_{3}=2\pi \sigma \int\limits_{0}^{\infty }\varsigma {f}_{eq}(\eta ,\varsigma )d\varsigma ,\end{array}$$
(12)
where \({C}_{Ds}\text{=}\frac{24}{{R}_{e}}\).
Assume that the range of bubble diameter is divided into M smaller intervals, the diameter distribution of the bubbles is uniform in a smaller interval. So the equilibrium distribution function of bubble diameter in the kth smaller diameter interval \(({\varsigma }_{k},{\varsigma }_{k+1})\) can be expressed as:
$${f}_{eq}\left(\eta ,\varsigma \right)\approx {n}_{eq}^{k}\left(\eta \right)\delta \left(\varsigma -{x}_{k}\right),$$
(13)
where
$${n}_{eq}^{k}\left(\eta \right)=\int\limits_{{\varsigma }_{k}}^{{\varsigma }_{k+1}}{f}_{eq}(\eta ,\varsigma )\mathrm{d}\varsigma ,$$
(14)
$${x}_{k}=\frac{1}{2}\left({\varsigma }_{k}+{\varsigma }_{k+1}\right),$$
(15)
where \({n}_{eq}^{k}(\eta )\) is the number of bubbles in the kth diameter interval \(({\varsigma }_{k},{\varsigma }_{k+1})\). And the integral in Eq. (12) can be approximated as:
$$\begin{array}{ll}\int\limits_{0}^{\infty }{\varsigma }^{2}\,\,{f}_{eq}(\eta ,\varsigma )d\varsigma \approx \sum\limits_{k=1}^{M}{x}_{k}^{2}{n}_{eq}^{k}(\eta ),\\ \int\limits_{0}^{\infty }\varsigma\,\, {f}_{eq}(\eta ,\varsigma )d\varsigma \approx \sum\limits_{k=1}^{M}{x}_{k}{n}_{eq}^{k}(\eta ).\end{array}$$
(16)
The boundary conditions of Eq. (2) are as follows:
$$\begin{array}{c}{p|}_{\zeta ={\zeta }_{out}}=0,\\ {p|}_{\zeta ={\zeta }_{in}}={p}_{in},\\ p(\eta )=p(\eta +\frac{2\pi }{N}).\end{array}$$
(17)
The pressure at the groove-ridge boundary can be solved by using continuous condition of flow at the groove-ridge boundary. The finite volume at the groove-ridge boundary is shown in Figure 3.
In a finite volume at the groove-ridge boundary, the continuous condition of the flow can be written as
$${\sum }_{i}{Q}_{i}^{\eta }+{\sum }_{j}{Q}_{j}^{\zeta }=0,$$
(18)
where
$${Q}_{i}^{\eta }=\int\limits_{{\zeta }_{1}}^{{\zeta }_{2}}\left(\begin{array}{c}\zeta {f}^{^{\prime}}(\zeta )\left(\frac{{h}^{3}}{{C}_{w}{\bar {\mu }_{w}}}\left(\frac{\partial {C}_{w}p}{\partial \zeta }-\frac{1}{\zeta \mathrm{cot}\beta }\frac{\partial {C}_{w}p}{\partial \eta }\right)-\frac{{h}^{3}}{\bar {\mu }_{w}}\frac{\rho {\overline{u} }^{2}}{\zeta }\right)\frac{1}{{k}_{r}}\\ +\frac{Uh}{2}-\frac{{h}^{3}}{{\bar {\mu }_{w}}}\left(\frac{1}{\zeta {C}_{w}}\frac{\partial {C}_{w}p}{\partial \eta }-\frac{M(\eta )}{{C}_{w}}\right)\frac{1}{{k}_{\theta }}\end{array}\right)\mathrm{d}\zeta,\quad (i=BF,FC,DE,EA).$$
(19)
$${Q}_{j}^{\zeta }=\underset{{\eta }_{1}}{\overset{{\eta }_{2}}{\int }}\zeta \left(-\frac{{h}^{3}}{{C}_{w}{\bar {\mu }_{w}}}\left(\frac{\partial {C}_{w}p}{\partial \zeta }{f}^{^{\prime}}(\zeta )\frac{\partial {C}_{w}p}{\partial \eta }\right)\frac{1}{{k}_{\theta }}\frac{{h}^{3}}{{\bar {\mu }_{w}}}\frac{\rho {\overline{u} }^{2}}{\zeta }\frac{1}{{k}_{r}}\right)\mathrm{d}\eta.\,\, \left(j\text{=}AB,CD\right).$$
(20)
2.2 Turbulent Cavitating Flow Energy Equation
The temperature field of the cavitating flow water film can be obtained by solving the two-phase flow energy equation under high-speed conditions, and the following assumptions are adopted to simplify the energy equation.
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(1)
The circumferential convection term is only considered in the energy equation, because the circumferential velocity is much larger than the radial velocity.
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(2)
Turbulent pulsation heat conduction term is ignored.
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(3)
The viscous dissipation term along the film thickness is only considered in the energy equation.
According to the above assumption, the energy equation in the \((\eta ,\zeta )\) coordinates in the spiral coordinate system can be simplified to
$${\rho }_{w}{c}_{v}\left(\frac{u}{\zeta }\frac{\partial \left({C}_{w}T\right)}{\partial \eta }\right)={C}_{w}{k}_{w}\frac{{\partial }^{2}T}{\partial {z}^{2}}+{C}_{w}\left({\tau }_{t}\left(z\right)\right)\left(\frac{\partial u}{\partial z}\right)+{E}_{int},$$
(21)
where
$$\left({\tau }_{t}\left(z\right)\right)=\frac{{({h}_{c}^{*})}^{2}{\overline{\mu }}_{w}U}{{R}_{c}h}+\frac{1}{{C}_{w}}\left(\frac{1}{\zeta }\frac{\partial ({C}_{w}p)}{\partial \eta }-M(\eta )\right)\left(z-\frac{h}{2}\right),$$
(22)
$$\frac{\partial u}{\partial z}=\frac{{({h}_{c}^{*})}^{2}}{{R}_{c}{h}^{*}}\frac{\partial {G}_{1}({z}^{*})}{\partial {z}^{*}}\frac{U}{h}-\frac{h}{{C}_{w}{\overline{\mu }}_{w}}\left(\frac{1}{\zeta }\frac{\partial \left({C}_{w}p\right)}{\partial \eta }M\left(\eta \right)\right)\frac{\partial {G}_{2}\left({z}^{*}\right)}{\partial {z}^{*}}.$$
(23)
The interface term \({E}_{\mathrm{int}}\) in the energy equation is expressed as
$$\begin{array}{c}{E}_{int}\approx {C}_{Ds}\pi {\left(\overline{u }-{u}_{b}\right)}^{2}\overline{u }\sum_{k=1}^{M}{x}_{k}^{2}{n}_{eq}^{k}\left(\eta \right)\\ -{\rho }_{w}{c}_{v}T(\overline{u }-{u}_{b})\pi \sum_{k=1}^{M}{x}_{k}^{2}{n}_{eq}^{k}(\eta )\\ +2\sigma \overline{u }\pi \sum_{k=1}^{M}{x}_{k}{n}_{eq}^{k}(\eta ).\end{array}$$
(24)
The following boundary conditions are adopted for the energy Eq. (21).
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(1)
At the inlet of the spiral groove
$$T\left(0,z\right)={T}_{0}.$$
(25)
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(2)
On the interface of the water film and the thrust disk
$$T\left(\eta ,h\right)={T}_{d}(\eta ,0).$$
(26)
-
(3)
On the interface of the water film and the stationary ring
$$T\left(\eta ,0\right)={T}_{s}\left(\delta_{s} \right).$$
(27)
The viscosity temperature relationship is written as
$${\mu }_{w}={\mu }_{w0}{e}^{(-b(T-{T}_{0}))}.$$
(28)
The water film temperature field can be obtained solved by simultaneously solving the energy Eq. (21), the heat conduction equation of the stationary ring and the heat conduction equation of the thrust disk.The surface temperature \({T}_{s}(\delta_{s} )\) of the stationary ring and surface temperature \({T}_{d}(\eta ,0)\) of the thrust plate are set as the boundary condition of the energy Eq. (21).
2.3 Heat Conduction Equation in Stationary Ring
The circumferential and radial heat conduction of the stationary ring are ignored, the heat conduction equation is simplified as follows:
$$\frac{{\partial }^{2}{T}_{\mathrm{s}}}{\partial {z}_{s}^{2}}=0 \left(0\le {z}_{s}\le {\delta }_{s}\right).$$
(29)
The boundary conditions of Eq. (29) are given as follows.
-
(1)
At the surface between the stationary ring and the fluid \(\left({z}_{s}={\delta }_{s}\right)\):
$$\begin{array}{c}{-k}_{s}{\left.\frac{\partial {T}_{s}}{\partial {z}_{s}}\right|}_{{z}_{s}={\delta }_{s}}={\alpha }_{w}({\left.{T}_{s}\right|}_{{z}_{s}={\delta }_{s}}-{T}_{m}),\\ {T}_{m}=\frac{1}{h}\int\limits_{0}^{h}T\mathrm{d}z.\end{array}$$
(30)
-
(2)
At the surface between the stationary ring and the ambient \(\left({z}_{s}={0}\right)\):
$${k}_{s}{\left.\frac{\partial {T}_{s}}{\partial {z}_{s}}\right|}_{{z}_{s}=0}={\alpha }_{a}({{T}_{s}|}_{{z}_{s}=0}-{T}_{0}).$$
(31)
The expression of the temperature at the surface of the stationary ring can be obtained from Eq. (29) and boundary conditions (30), (31). The temperature at the surface of the stationary ring is written as
$${{T}_{s}|}_{{z}_{s}={\delta }_{s}}=\frac{{T}_{m}-{T}_{0}}{\frac{{\alpha }_{w}}{{\alpha }_{a}}-\frac{{\alpha }_{w}}{{k}_{s}}{\delta }_{s}-1}+{T}_{m}.$$
(32)
2.4 Heat Conduction Equation in Thrust Disk
For the rotary thrust disk, the heat transfer effects in circumferential and radial directions are also ignored, the heat conduction equation is simplified as follows:
$$\frac{{k}_{d}}{{\rho }_{d}{c}_{d}}\frac{{\partial }^{2}{T}_{d}}{\partial {z}_{d}^{2}}=\omega \frac{\partial {T}_{d}}{\partial \eta }.\begin{array}{cc}& \left(0\le {z}_{d}\le {\delta }_{d}\right).\end{array}$$
(33)
The boundary conditions of Eq. (33) are given as follows:
-
(1)
At the surface between the thrust disk and the fluid \(({z}_{d}=0)\):
$${k}_{d}{\left.\frac{\partial {T}_{d}}{\partial {z}_{d}}\right|}_{{z}_{d}=0}={\alpha }_{w}\left({{T}_{d}|}_{{z}_{d}=0}-{T}_{m}\right).$$
(34)
-
(2)
At the surface between the thrust disk and the ambient \(({z}_{d}={\delta }_{d})\):
$${k}_{d}{\left.\frac{\partial {T}_{d}}{\partial {z}_{d}}\right|}_{{z}_{d}={\delta }_{d}}=-{\alpha }_{a}\left({{T}_{d}|}_{{z}_{d}={\delta }_{d}}-{T}_{0}\right).$$
(35)
2.5 Force Balance Equation of Bubble
The force balance equation of bubbles is expressed as
$${F}_{B}+{F}_{D}+{F}_{G}=0,$$
(36)
where
$${F}_{B}=\langle V\rangle ({\rho }_{w}-{\rho }_{g})g,$$
(37)
$${F}_{G}=({\rho }_{w}-{\rho }_{g})\langle V\rangle \frac{\mathrm{d}u}{\mathrm{d}t},$$
(38)
$$\frac{\mathrm{d}u}{\mathrm{d}t}=-\frac{1}{{\rho }_{w}}\frac{\partial p}{\zeta \partial \eta },$$
(39)
$$\langle V\rangle \approx \frac{\pi }{6}\sum_{k=1}^{M}{x}_{k}^{3}{n}_{eq}^{k}(\eta ),$$
(40)
$${F}_{D}=12\pi {\overline{\mu }}_{w}\langle {R}_{b}\rangle \left({u}_{b}-\overline{u }\right)\frac{1-{C}_{g}^\frac{5}{3}}{{\left(1-{C}_{g}\right)}^{2}},$$
(41)
$$\langle {R}_{b}\rangle \approx \frac{1}{2}\sum_{k=1}^{M}{x}_{k}{n}_{eq}^{k}\left(\eta \right).$$
(42)
The bubble velocity ub in the cavitating flow is calculated by the force balance Eq. (36). Substitute \({F}_{B}\), \({F}_{G}\), \({F}_{D}\) into Eq. (36), the bubble velocity ub can be expressed as [27]
$${u}_{b}=\overline{u }+\frac{{(1-{C}_{g})}^{2}({\rho }_{w}-{\rho }_{g})}{12\pi {\overline{\mu }}_{w}\langle {R}_{b}\rangle (1-{C}_{g}^\frac{5}{3})}\langle V\rangle \left(g+\frac{1}{{\rho }_{w}}\frac{\partial p}{\zeta \partial \eta }\right).$$
(43)
2.6 Population Balance Equation of Bubbles
Interface effect is an important phenomenon for cavitating flow, the momentum, mass and energy transfer occur through the interface between the gas-liquid. The interfacial area per unit volume liquid is positively correlated with the size distribution of the bubble, the bubble size distribution can be described by defining a probability density function \(f({\varvec{r}},\varsigma ,t)\), and the internal coordinates \(\varsigma\) are taken as bubble diameter in the model, and the \(f({\varvec{r}},\varsigma ,t)\mathrm{d}\varsigma\) represents the number of bubbles with between \((\varsigma ,\varsigma +\mathrm{d}\varsigma )\) at per volume liquid. Thus, integration of \(f\) over bubble diameter results in the total number of bubbles per volume liquid.The breakage, coalescence are two main factors affecting bubble size distribution in the cavitating flow, and bubble size distribution is predicted by the model of the breakage and coalescence processes of bubbles, this leads to the so-called population balance equation (PBE).
The PBE is a transport equation, the evolution of function \(f\) can be described by the PBE in the spiral coordinates \((\eta ,\zeta )\), the PBE is written as follows:
$$\begin{array}{ll}\frac{\partial f\left(\eta ,\varsigma ,t\right)}{\partial t}+{u}_{b}\frac{\partial }{\zeta \partial \eta }\left(f\left(\eta ,\varsigma ,t\right)\right)\\= -\,b(\varsigma )f(\eta ,\varsigma ,t)\\ \,\,+\int\limits_{\varsigma }^{{\varsigma }_{\mathrm{max}}}{h}_{b}\left(\xi ,\varsigma \right)b\left(\xi \right)f\left(\eta ,\xi ,t\right)\mathrm{d}\xi \\\,\, -\,f(\eta ,\varsigma )\int\limits_{0}^{{\varsigma }_{\mathrm{max}}}c(\varsigma ,\xi )f(\eta ,\xi ,t)\mathrm{d}\xi \\ \,\,+\,\frac{{\varsigma }^{2}}{2}\int\limits_{0}^{\varsigma }\frac{c\left({\left({\varsigma }^{3}-{\xi }^{3}\right)}^{1/3},\xi \right)}{{\left({\varsigma }^{3}-{\xi }^{3}\right)}^{2/3}}f\left(\eta ,{\left({\varsigma }^{3}-{\xi }^{3}\right)}^{1/3},t\right)f\left(\eta ,\xi ,t\right)\mathrm{d}\xi \\ \,\,+\,{S}_{c}(\varsigma ),\end{array}$$
(44)
where, the first term on the right-hand side of Eq. (44) represents breakup sink term of bubbles with diameter \(\varsigma\) per unit time, the second term on the right-hand side of Eq. (44) represents breakup source term of bubbles with diameter \(\varsigma\) per unit time, the third term on the right-hand side of Eq. (44) represents coalescence sink term of bubbles with diameter \(\varsigma\) per unit time,the fourth term on the right-hand side of Eq. (44) represents coalescence source term of bubbles with diameter \(\varsigma\) per unit time, the fifth term on the right-hand side of Eq. (44) represents bubbles with diameter \(\varsigma\) source term.
The initial condition for Eq. (44) is given
$$f(\eta ,\varsigma ,0)=0.$$
(45)
The breakage frequency \(b(\varsigma )\) is given as [28]:
$$b\left(\varsigma \right)=\frac{{\kappa }_{1}{\varepsilon }^{1/3}}{{\varsigma }^{2/3}}\mathrm{exp}\left(-\frac{\sigma {\kappa }_{2}}{{\rho }_{g}{\varepsilon }^{2/3}{\varsigma }^{5/3}}\right),$$
(46)
$$\varepsilon ={C}_{g}{u}_{b}g.$$
(47)
The daughter bubble size redistribution function \({h}_{b}(\xi ,\varsigma )\) is given as [28]
$${h}_{b}\left(\xi ,\varsigma \right)=\frac{4.8}{\xi }\mathrm{exp}\left(-4.5{\frac{(2\varsigma -\xi )}{{\xi }^{2}}}^{2}\right).$$
(48)
The coalescence closure is given as [28]:
$$\begin{array}{ll}c\left(\varsigma ,\xi \right)=0.05\frac{\pi }{4}{(\varsigma +\xi )}^{2}{\left[{\beta }_{c}{\left(\varepsilon \varsigma \right)}^{2/3}+{\beta }_{c}{(\varepsilon \xi )}^{2/3}\right]}^{1/2}\\ \mathrm{exp}\left(-\frac{{\left({r}_{c}^{3}{\rho }_{w}/16\sigma \right)}^{1/2}{\varepsilon }^{1/3}\mathrm{ln}\left({h}_{i}/{h}_{f}\right)}{{r}_{c}^{2/3}}\right),\end{array}$$
(49)
$${r}_{c}=\frac{1}{4}{\left(\frac{1}{\varsigma }+\frac{1}{\xi }\right)}^{-1}.$$
(50)
The value of parameter \({\beta }_{c}\) is taken as 2.48. The expression of source term \({S}_{c}(\varsigma )\) is
$${S}_{c}\left(\varsigma \right)={C}_{\rho }\sqrt{\left({p}_{v}-p\right)}\chi \left(\varsigma \right),$$
(51)
where
$$\chi \left(\varsigma \right)=\int\limits_{0}^{\varsigma }(\psi (\varsigma ,{\varsigma }_{c}){g}_{c}({\varsigma }_{c}))\mathrm{d}{\varsigma }_{c},$$
(52)
$${C}_{\rho }={\left(\frac{2}{3{\rho }_{w}}\right)}^{1/2}.$$
(53)
The redistribution function of bubble diameter \(\varsigma\) is given as
$$\psi \left({\varsigma }_{c},\varsigma \right)=\frac{{c}_{1}}{{\varsigma }_{c}}\mathrm{exp}\left(-\frac{{c}_{2}{(\varsigma -{\varsigma }_{c})}^{2}}{{\varsigma }_{\mathrm{max}}^{2}}\right).$$
(54)
The diameter distribution density function of the gas nucleus is given by experimental data [29]:
$${g}_{c}\left({\varsigma }_{c}\right)\approx \frac{\left(\alpha +1\right)}{{\varsigma }_{c\mathrm{max}}}{N}_{i}{\left(\frac{{\varsigma }_{c}}{{\varsigma }_{c\mathrm{max}}}\right)}^{\alpha },$$
(55)
where the value of parameter \(\mathrm{\alpha }\) is taken as − 10/3.
