Thermostatic Expansion Valve (2P)

Flow control valve that maintains evaporator superheat for use in refrigeration cycles

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Description

The Thermostatic Expansion Valve (2P) block models a valve with a pressure drop that maintains an evaporator superheat in a two-phase fluid network. This valve is typically placed between a condenser and evaporator in a refrigeration system and maintains a specific temperature differential by moderating the flow into the evaporator.

When the superheat, the difference in temperature between the vapor at the evaporator outlet and the fluid evaporating temperature, reaches the Static (minimum) evaporator superheat, the valve closes. This reduces the flow through the evaporator, which reduces the heat transfer in the evaporator and increases the evaporator outlet temperature. When you enable a maximum pressure or temperature limit with the MOP limit for evaporating pressure parameter, the valve closes when the limit is exceeded.

The bulb sensor at port S measures the evaporator outlet temperature. If the valve in your system has external pressure equalization, the evaporator outlet pressure is modeled by a line connection from the evaporator to port E. Otherwise, the pressure at port B is used for internal pressure equalization. The block balances the bulb pressure, which acts to open the valve, with the valve equalization pressure, which acts to close the valve.

Opening Area

The valve operates primarily to control the mass flow rate between a condenser and an evaporator by regulating the effective open area, S_eff. The mass flow rate is calculated as

$$\dot{m}=S_{eff}\sqrt{\frac{2}{v_{in}}}\frac{\Delta p}{{\left(\Delta p^2-\Delta p_{lam}^2\right)}^{0.25}},$$

where:

• v_in is the inlet specific volume, or the fluid volume per unit mass.

• Δp is the pressure differential over the valve, p_A – p_B.

• Δp_lam is the pressure threshold for transitional flow. Below this value, the flow is laminar. It is calculated as:

  $$\Delta p_{lam}=\frac{p_A+p_B}{2}\left(1-B_{lam}\right),$$

  where B_lam is the Laminar flow pressure ratio.

The effective valve area depends on the pressure difference between the measured pressure, p_bulb and the equalization pressure, p_eq:

$$S_{eff}=\beta \left[\left(p_{bulb}-p_{eq}\right)-\left(p_{sat}\left(T_{evap}+\Delta T_{static}\right)-p_{sat}\left(T_{evap}\right)\right)\right],$$

where:

• β is a valve constant determined from nominal operating conditions. See Determining β from Nominal Conditions for more information.

• T_evap is the Nominal evaporating temperature parameter.

• ΔT_static is the Static (minimum) evaporator superheat parameter.

• p_sat(T_evap) is the fluid saturation pressure as a function of T_evap. The block uses the tablelookup function to identify this value.

• p_sat(T_evap+ΔT_static) is the saturation pressure as a function of T_evap+ΔT_static. The block uses the tablelookup function to identify this value.

• p_bulb is the fluid pressure of the bulb. The bulb pressure is the saturation pressure, $$p_{bulb}=p_{sat}(T_{bulb})$$, unless pressure limiting is enabled and the maximum pressure has been reached; see MOP limit for evaporating pressure for more information. T_bulb is the bulb fluid temperature.

• p_eq depends on the valve pressure equalization setting:

  • When you set Pressure equalization to Internal pressure equalization, p_eq is the pressure at port B.

  • When you set Pressure equalization to External pressure equalization, p_eq is the pressure at port E.

The effective valve area has limits. The minimum effective valve area, S_eff,min, is

$$S_{eff,\min }=f_{leak}{S}_{eff,nom},$$

where f_leak is the Closed valve leakage flow as a fraction of nominal flow. The nominal effective valve area, S_eff,nom and maximum effective valve area are discussed in Determining β from Nominal Conditions.

Determining β from Nominal Conditions

β represents the relationship between the nominal evaporator superheat and the nominal evaporator capacity, the rate of heat transfer between the two fluids in the evaporator:

$$\beta =\frac{S_{eff,nom}}{\left[p_{sat}\left(T_{evap}+\Delta T_{nom}\right)-p_{sat}\left(T_{evap}\right)\right]},$$

where p_sat(T_evap+ΔT_nom) is the saturation pressure at the sum of the Nominal evaporating temperature and the Nominal (static + opening) evaporator superheat.

The nominal effective valve area, S_eff,nom, is calculated as a function of the nominal condenser and evaporator thermodynamics:

$$S_{eff,nom}=\frac{\left[\frac{Q_{nom}}{c_{p,evap}\Delta T_{nom}+h_{evap}-h_{cond}+c_{p,cond}\Delta T_{sub}}\right]}{\sqrt{\frac{2}{v_{cond}}\left(p_{sat}\left(T_{cond}\right)-p_{sat}\left(T_{evap}\right)\right)}},$$

where:

• T_cond is the Nominal condensing temperature.

• v_cond is the liquid specific volume at T_cond.

• Q_nom is the Nominal evaporator capacity.

• c_p,evap is the vapor specific heat at T_evap.

• h_evap is the vapor specific enthalpy at T_evap.

• c_p,cond is the liquid specific heat at T_cond.

• h_cond is the liquid specific enthalpy at T_cond.

• ΔT_sub is the Nominal condenser subcooling. Subcooling is the difference in temperature between the condenser outlet and the condensing temperature.

The maximum effective area of the valve is determined in the same way as S_eff,nom, but instead uses Maximum evaporator capacity in the place of the Nominal evaporator capacity.

Pressure Equalization

The equalization pressure is the pressure at the evaporator outlet that governs valve operability. In physical systems with low pressure loss in the evaporator due to viscous friction, pressure equalization can occur internally with the pressure at port B. This is referred to as internal pressure equalization. In systems with larger losses, connect the evaporator outlet port to the valve block at port E.

MOP Limit for Evaporating Pressure

You can limit to the maximum pressure in the evaporator by specifying a maximum pressure or associated temperature with the MOP limit for evaporating pressure parameter. If enabled, the valve closes when the bulb temperature exceeds the temperature associated with maximum bulb pressure, and opens once the pressure reduces. If MOP limit for evaporating pressure is set to Off, or the measured pressure is below the limit, $$p_{bulb}=p_{sat}(T_{bulb})$$. If enabled, when the measurement exceeds the limit, the bulb pressure remains at

$$p_{bulb}=\frac{p_{bulb,MOP}}{T_{bulb,MOP}}{T}_{bulb},$$

where:

• p_bulb,MOP is a function of the Maximum operating pressure, p_eq,MOP, or the pressure associated with the Maximum operating temperature, and the nominal evaporator temperature:

  $$p_{bulb,MOP}=p_{eq,MOP}+p_{sat}\left(T_{evap}+\Delta T_{static}\right)-p_{sat}\left(T_{evap}\right).$$

• T_bulb is the bulb fluid temperature. This is the temperature at port S if Bulb temperature dynamics is set to Off. A first-order delay is applied to the bulb temperature if Bulb temperature dynamics is set to On.

• T_bulb,MOP is the associated temperature at the pressure p_bulb,MOP.

Bulb Temperature Dynamics

You can model the bulb dynamic response to changing temperatures by setting Bulb temperature dynamics to On. This introduces a first-degree lag in the measured temperature:

$$\frac{d{T}_{bulb}}{dt}=\frac{T_S-T_{bulb}}{{\tau }_{bulb}},$$

where:

• T_S is the temperature at port S. If bulb dynamics are not modeled, this is T_bulb.

• τ_bulb is the Bulb thermal time constant.

Fluid Specific Volume Dynamics

When the fluid at the valve inlet is a liquid-vapor mixture, the block calculates the specific volume as:

$$v_{in}=\left(1-x_{dyn}\right)v_{liq}+x_{dyn}{v}_{vap},$$

where:

• x_dyn is the inlet vapor quality. The block applies a first-order lag to the inlet vapor quality of the mixture.

• v_liq is the liquid specific volume of the fluid.

• v_vap is the vapor specific volume of the fluid.

If the inlet fluid is liquid or vapor, v_in is the respective liquid or vapor specific volume.

Vapor Quality Lag

If the inlet vapor quality is a liquid-vapor mixture, the block applies a first-order time lag:

$$\frac{d{x}_{dyn}}{dt}=\frac{x_{in}-x_{dyn}}{\tau },$$

where:

• x_dyn is the dynamic vapor quality.

• x_in is the current inlet vapor quality.

• τ is the Inlet phase change time constant.

If the inlet fluid is a subcooled liquid, x_dyn is equal to x_in.

Conservation Equations

Mass is conserved through the valve:

$${\dot{m}}_A+{\dot{m}}_B=0,$$

where:

• $${\dot{m}}_A$$ is the mass flow rate at port A.

• $${\dot{m}}_B$$ is the mass flow rate at port B.

Reversed flows are numerically supported, however, the valve block is not designed for flows from port B to port A.

Energy flow is also conserved through the valve:

$${\Phi }_A+{\Phi }_B=0,$$

where:

• Φ_A is the energy flow rate at port A.

• Φ_B is the energy flow rate at port B.

Ports

Conserving

A — Fluid inlet

two-phase fluid

Valve inlet port. Connect this port to the outlet of a condenser or liquid receiver within a refrigeration cycle model.

B — Fluid outlet

two-phase fluid

Valve outlet port. Connect this port to the inlet of an evaporator within a refrigeration cycle model.

S — Sensing bulb temperature measurement

two-phase fluid

Temperature measurement port representing a sensing bulb. Connect this port to the outlet port of the evaporator. The valve operation is based on the comparison of the sensed temperature at S to the fluid saturation temperature.

There is no mass or energy flow rate through port S.

E — Equalization line pressure measurement

two-phase fluid

Pressure measurement port representing the pressure equalization line. Connect this port to the outlet port of the evaporator.

There is no mass or energy flow rate through port E.

Dependencies

To enable this port, set Pressure equalization to External pressure equalization.

Parameters

Nominal evaporator capacity — Evaporator heat transfer rate in nominal conditions

15 kW (default) | positive scalar

Rate of heat transfer in the system evaporator under nominal operating conditions. This parameter sets the operational conditions of the thermostatic expansion valve.

Maximum evaporator capacity — Maximum evaporator heat transfer rate

20 kW (default) | positive scalar

Maximum rate of heat transfer in the system evaporator. In most refrigeration cycles, this value is 20% to 50% larger than the Nominal evaporator capacity. This parameter sets the maximum operating conditions for the thermostatic expansion valve.

Nominal evaporating temperature — Evaporator saturation temperature in nominal conditions

283.15 K (default) | positive scalar

Refrigerant saturation temperature in the evaporator under nominal operating conditions.

Static (minimum) evaporator superheat — Minimum allowable temperature differential at the evaporator outlet

5 deltaK (default) | positive scalar

Minimum allowed difference between the superheated vapor temperature at the evaporator outlet and the Nominal evaporating temperature. If the operational difference falls below this value, the valve is closed.

Nominal (static + opening) evaporator superheat — Operational evaporator outlet temperature differential

10 deltaK (default) | positive scalar

Difference between the superheated vapor temperature at the evaporator outlet and the Nominal evaporating temperature under nominal operating conditions. The valve maintains this value by adjusting its open area to allow more or less fluid into the evaporator.

Nominal condensing temperature — Refrigerant saturation temperature in the condenser

313.15 K (default) | positive scalar

Refrigerant saturation temperature in the condenser under nominal operating conditions.

Nominal condenser subcooling — Operational condenser outlet temperature differential

5 deltaK (default) | positive scalar

Difference between the Nominal condensing temperature and the liquid temperature at the condenser outlet under nominal operating conditions.

Pressure equalization — Pressure measurement location

Internal pressure equalization (default) | External pressure equalization

Pressure measurement location. Set this to Internal pressure equalization to measure the evaporator pressure at port B. Set this parameter to External pressure equalization to measure the evaporator pressure at port E, which is connected to the evaporator outlet. This setting depends on the design of the thermostatic expansion valve.

MOP limit for evaporating pressure — Whether to enable pressure or temperature limits

Off (default) | On - Specify maximum operating pressure | On - Specify maximum operating temperature

Whether to enable pressure limits in the evaporator. The options are:

• Off: There is no pressure limit. The valve opens and closes based only on the evaporator superheat.

• On - Specify maximum operating pressure: This setting places an upper limit on the evaporating pressure and temperature. The valve closes when this pressure is reached.

• On - Specify maximum operating temperature: This setting places an upper limit on the evaporating pressure and temperature. The valve closes when this temperature is reached.

Maximum operating pressure — Maximum pressure when evaporator pressure limiting is on

0.5 MPa (default) | positive scalar

Maximum permissible saturation pressure in the evaporator.

Dependencies

To enable this parameter, set MOP limit for evaporating pressure to On - Specify maximum operating pressure.

Maximum operating temperature — Maximum temperature when evaporator temperature limiting is on

288.15 K (default) | positive scalar

Maximum permissible saturation temperature in the evaporator.

Dependencies

To enable this parameter, set MOP limit for evaporating pressure to On - Specify maximum operating temperature.

Bulb temperature dynamics — Whether to model thermal lag

Off (default) | On

Whether to model thermal dynamics in temperature measurement. When set to On, the bulb fluid temperature lags the refrigeration temperature. The Bulb thermal time constant determines the lag response.

Bulb thermal time constant — Thermal lag time constant

20 s (default) | positive scalar

First-order time constant for the bulb fluid temperature delay. The measured temperature is delayed relative to the refrigerant temperature at port S. The time constant is proportional to the thermal mass of the bulb (including any ballast) and inversely proportional to the thermal conductance across the thermal contact surface.

Dependencies

To enable this parameter, set Bulb temperature dynamics to On.

Cross-sectional area at ports A and B — Port area

0.01 m^2 (default) | positive scalar

Cross-sectional area of connecting pipes at ports A and B.

Closed valve leakage flow as a fraction of nominal flow — Minimum flow through the valve at all times

1e-9 (default) | positive scalar

Fraction of leakage to nominal flow through the valve when it is closed. A nonzero value enhances numerical stability in the fluid network.

Laminar flow pressure ratio — Ratio of evaporator pressures at flow regime transition

0.999 (default) | positive scalar

Ratio of the evaporator outlet pressure to evaporator inlet pressure at which the fluid transitions between the laminar and turbulent regimes. The pressure loss corresponds to the mass flow rate linearly in laminar flows and quadratically in turbulent flows.

Smoothing factor — Numerical smoothing factor

0.01 (default) | positive scalar in the range [0,1]

Continuous smoothing factor that introduces a layer of gradual change to the flow response when the valve is in near-open or near-closed positions. Set this value to a nonzero value less than one to increase the stability of your simulation in these regimes.

Inlet phase change time constant — Lag for mixture specific volume

0.1 s (default) | positive scalar

Time lag for liquid-vapor mixtures in computing the fluid specific volume. This parameter does not influence the specific volume when the inlet fluid is a fully supercooled liquid.

Model Examples

Electric Vehicle Thermal Management

Models the thermal management system of a battery electric vehicle. The system consists of two coolant loops, a refrigeration loop, and a cabin HVAC loop. The thermal load are the batteries, powertrain, and cabin.

Open Model

References

[1] Eames, Ian W., Adriano Milazzo, and Graeme G. Maidment. "Modelling Thermostatic Expansion Valves." International Journal of Refrigeration 38 (February 2014): 189-97.

Version History

Introduced in R2020b