The piston pump provides liquid supply from the working chamber to the gas cap via connecting pipe 2 . When performing the calculations, we assume that the reduction in the delivered liquid volume due to leakage, underfilling of the working chamber and compressibility is negligible. Then, the liquid delivered to the lid can be determined as:

In most practical situations, ${\mathrm{S}}_{\mathrm{H}1}={\mathrm{S}}_{\mathrm{H}2}={\mathrm{S}}_{\mathrm{H}I}$, ${I}_{1}={I}_{2}={\mathsf{I}}_{I}$, ${\mathrm{Phi}}_{2}=F\left({\mathrm{Phi}}_{1}\right)$; ${\mathrm{Phi}}_{I}=F\left({\mathrm{Phi}}_{1}\right)$.

#### 2.1. Mathematical model of the gas cap working process

We consider the most general case where there is no separating element between the gas and liquid phases in the gas cap.

The calculation of the gas phase thermodynamic parameters of the gas cap is similar to the calculation of the changes in thermodynamic parameters during the compression and expansion processes of a liquid piston reciprocating compressor.

$$\mathrm{of}=\mathrm{dQ}\u2013\mathrm{dL}+{I}_{pHAd}\mathrm{d}{\mathrm{medium\; size}}_{pHAd}\u2013{I}_{pH\mathrm{oh}}\mathrm{d}{\mathrm{medium\; size}}_{pH\mathrm{oh}}+{I}_{Gr}\mathrm{d}{\mathrm{medium\; size}}_{Gr}\u2013{I}_{\mathrm{oh}}\mathrm{d}{\mathrm{medium\; size}}_{G\mathrm{oh}}$$

$$\mathrm{dQ}=A{F}_{1}\left({\stackrel{\uffe3}{\mathrm{time}}}_{\mathrm{dimension}}\u2013\mathrm{time}\right)\mathrm{d}t+A{F}_{2}\left({\mathrm{time}}_{\mathrm{w}}\u2013\mathrm{time}\right)\mathrm{d}t$$

Where ${F}_{1}=\frac{\mathrm{PI}{d}_{C}^{2}}{4}+\mathrm{PI}{d}_{C}{I}_{r}$(${F}_{2}=\frac{\mathrm{PI}{d}_{C}^{2}}{4}$); ${\stackrel{\uffe3}{\mathrm{time}}}_{wI}=\frac{{{\displaystyle \int}}_{{F}_{1}}^{}{\mathrm{time}}_{wI}\left(F\right)dF}{{F}_{1}}$ is the average temperature of the inner wall surface of the cap (${\mathrm{time}}_{wI}$ It is determined through experiments [29]).

$${\mathrm{dM}}_{\mathrm{w}}=\Sigma {\mathrm{dM}}_{\mathrm{Well}I}\u2013{\mathrm{dM}}_{\mathrm{oh}}\u2013{\mathrm{dM}}_{\mathrm{Pad}}+{\mathrm{dM}}_{\mathrm{Pho}}\u2013{\mathrm{dM}}_{\mathrm{grid}}+{\mathrm{dM}}_{\mathrm{go}}$$

Where $\Sigma {\mathrm{dM}}_{\mathrm{Well}I}={\mathsf{r}}_{\mathrm{w}}\xb7\mathrm{d}\mathsf{t}\xb7\Sigma {\mathrm{ask}}_{I}$ It is the mass of liquid entering the gas cap from the pump cylinder.

$${v}_{w}=\frac{{\mathrm{dM}}_{\mathrm{w}}}{{F}_{2}\mathrm{d}\mathsf{t}\xb7{\mathsf{r}}_{\mathrm{w}}}$$

$$\mathrm{dV}=\u2013{\mathrm{v}}_{\mathrm{w}}{\mathrm{F}}_{2}\mathrm{d}\mathsf{t}$$

In-cap gas phase mass: Changes in the gas phase mass within the cap through leakage through the cap without external leakage are due to condensation or evaporation of the working fluid (first-order phase change). In this case, mass transfer occurs through concentration diffusion, thermal diffusion, and pressure diffusion. Given the accepted assumption that pressure and temperature are constant throughout the gas cap, there will be no thermal or pressure diffusion.

$${\mathrm{d}\mathrm{medium\; size}}_{\mathrm{pH}}={\mathsf{Second}}_{\mathrm{pH}}{\mathrm{F}}_{2}\left({\mathrm{C}}_{\mathrm{w}}\u2013{\mathrm{C}}_{2}\right)\mathrm{d}\mathsf{t}$$

$${\mathrm{C}}_{\mathrm{w}}={\mathrm{p}}_{eIst}/\left({\mathrm{right}}_{\mathrm{s}}{\mathrm{time}}_{\mathrm{w}}\right)$$

Where ${\mathrm{p}}_{eIst}$ is the vapor elastic pressure at the liquid surface (a function of liquid temperature and interface curvature, determined by the Clausius-Cleperon equation [31]).

$${\mathrm{Second}}_{\mathrm{pH}}=\frac{\mathsf{A}}{{\mathrm{C}}_{\mathrm{p}}\mathsf{r}{\mathsf{A}}_{\mathrm{time}}/\mathrm{D}}$$

It should be noted that the solubility of gases in liquids is generally described by Henry’s equation and increases with increasing pressure.

$$\mathrm{dM}={\mathrm{d}\mathrm{medium\; size}}_{\mathrm{Pad}}\u2013{\mathrm{d}\mathrm{medium\; size}}_{\mathrm{Pho}}+{\mathrm{d}\mathrm{medium\; size}}_{\mathrm{grid}}\u2013{\mathrm{d}\mathrm{medium\; size}}_{\mathrm{go}}$$

$$\mathrm{of}=\mathrm{d}\left({\mathrm{MC}}_{\mathrm{v}}\mathrm{time}\right)={\mathrm{C}}_{\mathrm{v}}\mathrm{time\; domain\; management}+{\mathrm{C}}_{\mathrm{v}}\mathrm{time\; domain\; management}$$

If the gas cap pressure is greater than 10 MPa, you need to introduce a compressibility factor in the equation of state and use one of the existing equations of state for ideal gases: van der Waals, Berthelot, Dupre, Clausius or Vukalovich-Kirilin. It must be remembered that for real gases, $\mathrm{you}=F\left(v,\mathrm{time}\right)$ and $I=F\left(p,\mathrm{time}\right)$Where $v=\frac{1}{r}$ It’s specificon.

$$\left\{\begin{array}{c}\mathrm{d}\mathrm{U}=\mathrm{d}\mathrm{ask}\u2013\mathrm{p}\mathrm{d}\mathrm{V}\\ dV=\u2013\left({\mathsf{S}\mathrm{dM}}_{\mathrm{northwest}I}\u2013{\mathrm{dM}}_{\mathrm{SW}}\right)/{\mathsf{r}}_{\mathrm{w}}\\ \mathrm{p}=\left(\mathrm{k}\u20131\right)\mathrm{U}/\mathrm{V}\\ \mathrm{time}=\mathrm{Voltage}/\left(\mathrm{gentlemen}\right)\end{array}\right.$$

$$Dp=\frac{{\mathrm{watt}}_{\mathrm{this}}}{{\mathrm{F}}_{2}}=\frac{{\mathrm{C}}_{\mathrm{Stef}}\left({I}_{\mathrm{w}}\u2013{I}_{\mathrm{w}0}\right)}{{\mathrm{F}}_{2}}$$

If there are no split elements then ${\mathrm{p}}_{\mathrm{w}}=\mathrm{p}$.

To determine the temperature of the liquid in the lid, without the dividing element we use the law of conservation of energy.

$${\mathrm{of}}_{\mathrm{w}}={\mathrm{dQ}}_{\mathrm{w}}\u2013{\mathrm{VAT}}_{\mathrm{bathroom}}+\Sigma {I}_{AdwI}{\mathrm{dM}}_{\mathrm{wireless}}\u2013{I}_{\mathrm{oh}w}{\mathrm{dM}}_{0\mathrm{w}}+{I}_{pHAd}{\mathrm{dM}}_{\mathrm{Pad}}\u2013{I}_{pH\mathrm{oh}}{\mathrm{dM}}_{\mathrm{Pho}}+{I}_{Gr}{\mathrm{dM}}_{\mathrm{grid}}\u2013{I}_{\mathrm{oh}}{\mathrm{dM}}_{\mathrm{go}}$$

$${\mathrm{of}}_{\mathrm{w}}=\mathrm{d}({\mathrm{you}}_{\mathrm{w}}\xb7{\mathrm{medium\; size}}_{\mathrm{w}})={\mathrm{medium\; size}}_{\mathrm{w}}{\mathrm{of}}_{\mathrm{w}}+{\mathrm{you}}_{\mathrm{w}}{\mathrm{dM}}_{\mathrm{w}}$$

$${\mathrm{of}}_{\mathrm{w}}={\left(\frac{\partial {\mathrm{you}}_{\mathrm{w}}}{\partial {v}_{\mathrm{w}}}\right)}_{\mathrm{time}}\mathrm{d}{v}_{\mathrm{w}}+{\left(\frac{\partial {\mathrm{you}}_{\mathrm{w}}}{\partial {\mathrm{time}}_{\mathrm{w}}}\right)}_{{\mathrm{v}}_{\mathrm{w}}}{\mathrm{d}\mathrm{time}}_{\mathrm{w}}$$

$${\mathrm{of}}_{\mathrm{w}}={\mathrm{C}}_{\mathrm{w}}{\mathrm{medium\; size}}_{\mathrm{w}}{\mathrm{d}\mathrm{time}}_{\mathrm{w}}+{\mathrm{C}}_{\mathrm{w}}{\mathrm{time}}_{\mathrm{w}}{\mathrm{d}\mathrm{medium\; size}}_{\mathrm{w}}$$

Where ${\left(\frac{\partial {\mathrm{you}}_{\mathrm{w}}}{\partial {\mathrm{time}}_{\mathrm{w}}}\right)}_{{\mathrm{v}}_{\mathrm{w}}}={\mathrm{C}}_{w}$——Specific heat capacity of liquid.

$${\mathrm{d}\mathrm{time}}_{\mathrm{w}}=\frac{1}{{\mathrm{C}}_{\mathrm{w}}{\mathrm{medium\; size}}_{\mathrm{w}}}$$

$${\mathrm{d}\mathrm{T}}_{\mathrm{w}}=\frac{1}{{\mathrm{C}}_{\mathrm{w}}{\mathrm{M}}_{\mathrm{w}}}\left({\mathrm{dQ}}_{\mathrm{w}}+\sum {i}_{\mathrm{adw}i}{\mathrm{dM}}_{\mathrm{wi}}-{i}_{0w}{\mathrm{dM}}_{0\mathrm{w}}\right)$$

$${i}_{adwi}={\mathrm{C}}_{w}{T}_{adi};{i}_{0w}={\mathrm{C}}_{w}{T}_{w}$$

$${\mathrm{dQ}}_{\mathrm{w}}={\overline{\alpha}}_{w}{F}_{w}\left({\overline{T}}_{wl}-{T}_{w}\right)d\tau $$

$${\stackrel{\uffe3}{A}}_{w}=\frac{{A}_{w}}{{d}_{C}}$$

and for turbulent flow mode:

$${\overline{\alpha}}_{w}=\frac{{\alpha}_{w}}{{d}_{c}}\left[0.021R{e}_{w}^{0.8}P{r}_{w}^{0.43}{\left(P{r}_{w}/P{r}_{wl}\right)}^{0.25}\right]$$

where $R{e}_{w}=\frac{{v}_{w}{d}_{\mathrm{K}}}{{\mu}_{w}/{\rho}_{w}}$ is the Reynolds number; $P{r}_{w}=\frac{{\mu}_{w}{C}_{w}}{{\alpha}_{w}}$ is the Prandtl number; and $P{r}_{wl}$ is the Prandtl number at wall temperature.

$${F}_{w}={l}_{w}\pi {d}_{c}+\frac{\pi {d}_{c}^{2}}{2}$$

$${\mathrm{dM}}_{\mathrm{w}}=\sum {\mathrm{dM}}_{\mathrm{adw}i}-{\mathrm{dM}}_{o\mathrm{w}}$$

#### 2.2. Mathematical Model of Liquid Flow in the Pipeline from the Gas Cap

Currently, various models are used to describe the flow of liquid in a pipeline, ranging from the simplest ones based on the energy conservation equation (Bernoulli), both without taking into account inertial pressure losses and taking them into account, to complex ones using two-parameter turbulence models: k-ε, k-ω, SST and others.

When developing a mathematical model of fluid flow in a connecting pipeline, we use the principle of hierarchy and consider the calculation of the flow based on the Bernoulli equation and the unsteady one-dimensional flow of a viscous incompressible fluid.

$${{\displaystyle \int}}_{{I}_{pp2}}^{}\frac{\partial}{\partial I}\left(z+\frac{p}{j}+\frac{{v}_{pp2}^{2}}{2G}\right)dI+\frac{1}{y}{{\displaystyle \int}}_{{I}_{pp2}}^{}\frac{\partial {v}_{pp2}}{\partial t}dI+\mathsf{S}D{H}_{I}=0$$

Where $\mathsf{S}D{H}_{I}=\left(\mathsf{S}{X}_{I}+{I}_{pp2}\frac{{I}_{pp2}}{{d}_{pp2}}\right)\frac{{v}_{pp2}^{2}}{2G}$— Head loss caused by local resistance along the length and hydraulic resistance.

$${v}_{pp2}=\sqrt{\frac{2G\left[\left({z}_{1pp2}+\frac{p}{{\rho}_{w}g}\right)-\left({z}_{2pp2}+\frac{{P}_{d}}{{\rho}_{w}g}\right)\right]}{\left({\lambda}_{pp2}\frac{{l}_{pp2}}{{d}_{pp2}}+\mathsf{\Sigma}{\xi}_{i}\right)}}$$

$$\begin{array}{c}{r}_{w}\frac{\partial {\mathrm{ask}}_{wpp2}}{\partial t}+{F}_{pp2}\frac{\partial p}{\partial X}+\frac{{I}_{pp2}{r}_{w}}{2{d}_{pp}{F}_{pp2}}{\mathrm{ask}}_{wpp\mathrm{time}\mathrm{phosphorus}2}\left|{\mathrm{ask}}_{w\mathrm{time}\mathrm{phosphorus}2}\right|=0\\ \end{array}$$

$$\frac{{r}_{w}{A}^{2}}{{F}_{pp2}}\frac{\partial p}{\partial X}+\frac{\partial p}{\partial t}=0$$

We propose a “feature” approach to this system.

The boundary conditions at the end of the pipe are adjacent to the gas cap on the one hand and the consumption end of the liquid in the form of pressure on the other hand: in the gas cap –p_{w}; at the user of the liquid –p_{d}.

$${\mathrm{dM}}_{oh\mathrm{w}}={r}_{w}{v}_{pp2}{F}_{pp2}dt={\mathrm{ask}}_{wpp2}{r}_{w}dt$$