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Pressurized thermal liquid container with compressed gas charge

Thermal Liquid/Tanks & Accumulators

The Gas-Charged Accumulator (TL) block models a pressurized thermal liquid container with a compressed gas charge. The accumulator consists of thermal liquid and gas chambers separated by a hermetic and insulated diaphragm.

**Accumulator Schematic**

If the inlet pressure is greater than the gas charge pressure, the liquid chamber volume increases, compressing the gas chamber. If the inlet pressure is lower than the gas charge pressure, the liquid chamber volume decreases, decompressing the gas chamber.

Hard stops limit the diaphragm motion when the liquid chamber is at capacity and when the liquid chamber is empty. The hard-stop compliance is modeled through spring and damper forces. If the specified spring stiffness is low, the liquid volume can momentarily fall below zero or rise above capacity.

The liquid chamber volume is the difference between the total accumulator volume and the gas chamber volume:

$${V}_{L}={V}_{T}-{V}_{G},$$

where:

*V*_{L}is the liquid chamber volume.*V*_{T}is the total accumulator volume.*V*_{G}is the gas chamber volume.

**Chamber Volumes**

The liquid chamber capacity is the difference between the total accumulator volume and the dead volume of gas in the accumulator at full capacity:

$${V}_{C}={V}_{T}-{V}_{Dead},$$

where:

*V*_{C}is the liquid chamber capacity.*V*_{Dead}is the dead volume of gas at full capacity.

The gas chamber pressure and volume follow from the precharge states as described by the polytropic equation

$${p}_{G}{V}_{G}^{k}={p}_{pr}{V}_{T}^{k},$$

where:

*p*_{G}is the gas chamber pressure at a given time step.*V*_{G}is the gas chamber volume at a given time step.*p*_{pr}is the precharge gas chamber pressure when the liquid chamber is empty.*V*_{T}is the total liquid chamber volume.*k*is the polytropic index.

The mass conservation equation in the liquid chamber is

$${V}_{L}{\rho}_{L}\left(\frac{1}{{\beta}_{L}}\frac{d{p}_{L}}{dt}-{\alpha}_{L}\frac{d{T}_{L}}{dt}\right)+{\rho}_{L}\frac{d{V}_{L}}{dt}={\dot{m}}_{A},$$

where:

*ρ*_{L}is the thermal liquid density.*β*_{L}is the isothermal bulk modulus.*α*_{L}is the isobaric thermal expansion coefficient.*p*_{L}is the thermal liquid pressure.*T*_{L}is the thermal liquid temperature.$${\dot{m}}_{A}$$ is the thermal liquid mass flow rate into the accumulator through port A.

The time variation of the liquid chamber volume is given by the conditional equation

$$\frac{d{V}_{L}}{dt}=\{\begin{array}{ll}\frac{{\dot{p}}_{L}}{k{p}_{pr}{V}_{T}^{k}{V}_{G}^{(-k-1)}+{K}_{s}+{K}_{d}{\dot{m}}^{+}/{\rho}_{L}},\hfill & {V}_{L}\ge {V}_{C}\hfill \\ \frac{{\dot{p}}_{L}}{k{p}_{pr}{V}_{T}^{k}{V}_{G}^{(-k-1)}+{K}_{s}-{K}_{d}{\dot{m}}^{-}/{\rho}_{L}},\hfill & {V}_{L}\le 0\hfill \\ \frac{{\dot{p}}_{L}}{k{p}_{pr}{V}_{T}^{k}{V}_{G}^{(-k-1)}},\hfill & \text{Else}\hfill \end{array},$$

where:

*K*_{s}is the hard-stop stiffness coefficient.*K*_{d}is the hard-stop damping coefficient.$${\dot{m}}^{+}$$ is the mass flow rate into the liquid chamber when the accumulator diaphragm contacts the top hard stop:

$${\dot{m}}^{+}=\{\begin{array}{ll}\dot{m},\hfill & \dot{m}>0\hfill \\ 0,\hfill & \text{Else}\hfill \end{array},$$

$${\dot{m}}^{-}$$ is the mass flow rate from the liquid chamber when the accumulator diaphragm contacts the bottom hard stop:

$${\dot{m}}^{-}=\{\begin{array}{ll}\dot{m},\hfill & \dot{m}<0\hfill \\ 0,\hfill & \text{Else}\hfill \end{array},$$

The momentum conservation equation in the accumulator volume is

$${p}_{L}={p}_{G}+{p}_{HS},$$

where:

*p*_{HS}is the hard-stop contact pressure:$${p}_{HS}=\{\begin{array}{ll}\left({V}_{L}-{V}_{C}\right)\left({K}_{s}+{K}_{d}{\dot{m}}^{+}/\rho \right),\hfill & {V}_{L}\ge {V}_{C}\hfill \\ {V}_{L}\left({K}_{s}-{K}_{d}{\dot{m}}^{-}/\rho \right),\hfill & {V}_{L}\le 0\hfill \\ 0,\hfill & \text{Else}\hfill \end{array},$$

The energy conservation equation in the liquid chamber volume is

$$\frac{d}{dt}\left({\rho}_{L}{u}_{L}{V}_{L}\right)={\varphi}_{A}+{\varphi}_{H},$$

where:

*u*_{L}is the thermal liquid specific internal energy.*Φ*_{A}is the energy flow rate into the liquid chamber through the accumulator inlet.*Φ*_{H}is the thermal energy flow rate into the liquid chamber through the accumulator wall.

Gas chamber compression is treated as a polytropic process.

Diaphragm loading is ignored.

Fluid inertia is ignored.

**Total accumulator volume**Combined liquid and gas volume in the accumulator. The default value is

`8e-3`

m^3.**Minimum gas volume**Remnant gas volume in the accumulator in a completely filled state. The default value is

`4e-5`

m^3.**Precharge pressure**Initial gas charge pressure of the empty accumulator. Fluid enters the accumulator if the inlet pressure is equal to or greater than the precharge pressure. The default value is

`0`

MPa gauge pressure.**Specific heat ratio**Ratio of the gas specific heat at constant pressure to that at constant volume. The default value is

`1.4`

.**Hard-stop stiffness coefficient**Stiffness coefficient of the top and bottom accumulator hard stops. The hard stops restrict diaphragm motion between zero and the maximum liquid chamber level. The stiffness coefficient accounts for the restorative portion of the hard-stop contact forces. The default value is

`1e4`

MPa/m^3.**Hard-stop damping coefficient**Damping coefficients of the top and bottom accumulator hard stops. The hard stops restrict diaphragm motion between zero and the maximum liquid chamber level. The damping coefficients account for the dissipative portion of the hard-stop contact forces. The default value is

`1e4`

s*MPa/m^6.**Cross-sectional area at port A**Flow cross-sectional area at the accumulator inlet. The default value is

`0.01`

m^2.

**Volume of liquid**Volume of thermal liquid in the accumulator at the start of simulation. The default value is

`0.005`

m^3.**Mass of liquid**Mass of thermal liquid in the accumulator at the start of simulation. The default value is

`5`

kg.**Pressure of liquid volume**Pressure in the thermal liquid chamber at the start of simulation. The default value is

`0.101325`

MPa.**Temperature of liquid volume**Temperature in the thermal liquid chamber at the start of simulation. The default value is

`293.15`

K.

A — Thermal liquid port representing the accumulator inlet

H — Thermal port representing heat transfer between the liquid and the environment through the accumulator wall