# Gas-Charged Accumulator (TL)

Gas-charged accumulator in a thermal liquid network

**Libraries:**

Simscape /
Fluids /
Thermal Liquid /
Tanks & Accumulators

## Description

The Gas-Charged Accumulator (TL) block represents 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. If the specified spring stiffness is too low, the liquid
volume can momentarily fall below zero or rise above capacity. Use the **Hard
stop stiffness coefficient** parameter to correct this.

### Mass Balance

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 block computes the time variation of the liquid chamber volume using 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*is the_{s}**Hard-stop stiffness coefficient**parameter.*K*is the_{d}**Hard-stop damping coefficient**parameter.$${\dot{m}}^{+}$$ 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:

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

### Momentum Balance

The momentum conservation equation in the accumulator volume is

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

where *p _{HS}* is the
hard-stop contact pressure:

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

### Energy Balance

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.

### Chamber Volumes

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.

### Initial Targets

To set the priority and initial target values for the block variables prior to simulation, use
the **Initial Targets** section in the block dialog box or
Property Inspector. For more information, see Set Priority and Initial Target for Block Variables.

Nominal values provide a way to specify the expected magnitude of a variable in a model.
Using system scaling based on nominal values increases the simulation robustness. Nominal
values can come from different sources, one of which is the **Nominal
Values** section in the block dialog box or Property Inspector. For more
information, see Modify Nominal Values for a Block Variable.

### Assumptions and Limitations

The block treats the gas chamber compression as a polytropic process.

The block ignores diaphragm loading.

The block ignores the effects of fluid inertia.

## Ports

### Conserving

## Parameters

## Extended Capabilities

## Version History

**Introduced in R2016a**