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

A diaphragm separates the gas sealed gas chamber from the liquid chamber with a port. There are hard stops near the top and bottom of the accumulator.

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

$V_{L} ρ_{L} (\frac{1}{β_{L}} \frac{d p_{L}}{d t} - α_{L} \frac{d T_{L}}{d t}) + ρ_{L} \frac{d V_{L}}{d t} = {\dot{m}}_{A},$

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

$\frac{d V_{L}}{d t} = {\begin{array}{l} \frac{{\dot{p}}_{L}}{k p_{p r} V_{T}^{k} V_{G}^{(- k - 1)} + K_{s} + K_{d} {\dot{m}}^{+} / ρ_{L}}, & V_{L} \geq V_{C} \\ \frac{{\dot{p}}_{L}}{k p_{p r} V_{T}^{k} V_{G}^{(- k - 1)} + K_{s} - K_{d} {\dot{m}}^{-} / ρ_{L}}, & V_{L} \leq 0 \\ \frac{{\dot{p}}_{L}}{k p_{p r} V_{T}^{k} V_{G}^{(- k - 1)}}, & Else \end{array},$

where:

K_s is the Hard-stop stiffness coefficient parameter.
K_d is the 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:

${\dot{m}}^{+} = {\begin{array}{l} \dot{m}, & \dot{m} > 0 \\ 0, & Else \end{array},$
${\dot{m}}^{-}$ is the mass flow rate from the liquid chamber when the accumulator diaphragm contacts the bottom hard stop:

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

Momentum Balance

The momentum conservation equation in the accumulator volume is

$p_{L} = p_{G} + p_{H S},$

where p_HS is the hard-stop contact pressure:

$p_{H S} = {\begin{array}{l} (V_{L} - V_{C}) (K_{s} + K_{d} {\dot{m}}^{+} / ρ), & V_{L} \geq V_{C} \\ V_{L} (K_{s} - K_{d} {\dot{m}}^{-} / ρ), & V_{L} \leq 0 \\ 0, & Else \end{array},$

Energy Balance

The energy conservation equation in the liquid chamber volume is

$\frac{d}{d t} (ρ_{L} u_{L} V_{L}) = ϕ_{A} + ϕ_{H},$

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 figure shows a diagram of the accumulator volume measurements.

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

$V_{C} = V_{T} - V_{D e a d},$

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

$p_{G} V_{G}^{k} = p_{p r} V_{T}^{k},$

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.

Examples

Engine Cooling System

Model an engine cooling system with an oil cooling circuit.

Open Model

House Heating System

Model a simple house heating system. The model contains a heater, a controller, and a house structure with four radiators and four rooms. Each room exchanges heat with the environment through its exterior walls, roof, and windows. Each path is simulated as a combination of a thermal convection, thermal conduction, and the thermal mass. It is assumed that heat is not transferred internally between rooms. The heater consists of a furnace, a boiler, an accumulator, and a pump to circulate hot water in the system. The controller starts admitting fuel into the furnace if the overall average temperature of rooms falls below 21 degree C and it stops if the temperature exceeds 25 degree C. The simulation calculates the heating cost and indoor temperatures.

Open Model

Ports

Conserving

expand all

A — Liquid port
thermal liquid

Thermal liquid conserving port associated with the accumulator inlet. The flow rate is positive if liquid flows into the accumulator.

H — Heat port
thermal

Thermal conserving port associated with the thermal mass of the liquid volume.

Parameters

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Total accumulator volume — Accumulator volume
`8e-3` `m^3` (default) | positive scalar

Total volume of the accumulator, including the liquid chamber and the gas chamber. This is the sum of the liquid chamber capacity and the minimum gas volume.

Minimum gas volume — Gas chamber dead volume
`4e-5` `m^3` (default) | positive scalar

Gas chamber dead volume. This is the amount of remaining gas when the liquid chamber is full. A nonzero volume is necessary so that the gas pressure does not become infinite when the liquid chamber is full.

Precharge pressure (gauge) — Gas chamber pressure
`0` `MPa` (default) | nonnegative scalar

Initial gauge pressure of the empty accumulator chamber. Fluid enters the accumulator if the inlet pressure is equal to or greater than the precharge pressure.

Specific heat ratio — Specific heat ratio
`1.4` (default) | positive scalar

Specific heat ratio for the gas in the chamber. This value is typically 1.4 for air at 20°C.

Cross-sectional area at port A — Inlet cross-sectional area
`0.01` `m^2`positive scalar (default)

Flow cross-sectional area at the accumulator inlet.

Hard stop stiffness coefficient — Stiffness coefficient
`1e4` `MPa/m^3` (default) | positive scalar

Stiffness coefficient of the top and bottom accumulator hard stops. The hard stops restrict diaphragm motion between zero and the maximum liquid chamber level.

Hard-stop damping coefficient — Damping coefficient
`1e4` `MPa*s/m^6` (default) | positive scalar

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.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2016a

Gas-Charged Accumulator (TL)

Description

Mass Balance

Momentum Balance

Energy Balance

Chamber Volumes

Initial Targets

Assumptions and Limitations

Examples

Engine Cooling System

House Heating System

Ports

Conserving

A — Liquid port thermal liquid

H — Heat port thermal

Parameters

Total accumulator volume — Accumulator volume 8e-3 m^3 (default) | positive scalar

Minimum gas volume — Gas chamber dead volume 4e-5 m^3 (default) | positive scalar

Precharge pressure (gauge) — Gas chamber pressure 0 MPa (default) | nonnegative scalar

Specific heat ratio — Specific heat ratio 1.4 (default) | positive scalar

Cross-sectional area at port A — Inlet cross-sectional area 0.01 m^2positive scalar (default)

Hard stop stiffness coefficient — Stiffness coefficient 1e4 MPa/m^3 (default) | positive scalar

Hard-stop damping coefficient — Damping coefficient 1e4 MPa*s/m^6 (default) | positive scalar

Extended Capabilities

C/C++ Code Generation Generate C and C++ code using Simulink® Coder™.

Version History

See Also

A — Liquid port
thermal liquid

H — Heat port
thermal

Total accumulator volume — Accumulator volume
`8e-3` `m^3` (default) | positive scalar

Minimum gas volume — Gas chamber dead volume
`4e-5` `m^3` (default) | positive scalar

Precharge pressure (gauge) — Gas chamber pressure
`0` `MPa` (default) | nonnegative scalar

Specific heat ratio — Specific heat ratio
`1.4` (default) | positive scalar

Cross-sectional area at port A — Inlet cross-sectional area
`0.01` `m^2`positive scalar (default)

Hard stop stiffness coefficient — Stiffness coefficient
`1e4` `MPa/m^3` (default) | positive scalar

Hard-stop damping coefficient — Damping coefficient
`1e4` `MPa*s/m^6` (default) | positive scalar

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.