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# Thermostatic Expansion Valve (2P)

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

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• Simscape / Fluids / Two-Phase Fluid / Valves & Orifices / Flow Control Valves

## 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, Seff. The mass flow rate is calculated as

`$\stackrel{˙}{m}={S}_{eff}\sqrt{\frac{2}{{v}_{in}}}\frac{\Delta p}{{\left(\Delta {p}^{2}-\Delta {p}_{lam}^{2}\right)}^{0.25}},$`

where:

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

• Δp is the pressure differential over the valve, pApB.

• Δplam 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 Blam is the Laminar flow pressure ratio.

The effective valve area depends on the pressure difference between the measured pressure, pbulb and the equalization pressure, peq:

`${S}_{eff}=\beta \left[\left({p}_{bulb}-{p}_{eq}\right)-\left({p}_{sat}\left({T}_{evap}+\Delta {T}_{static}\right)-{p}_{sat}\left({T}_{evap}\right)\right)\right],$`

where:

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

• Tevap is the Nominal evaporating temperature parameter.

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

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

• psat(Tevap+ΔTstatic) is the saturation pressure as a function of Tevap+ΔTstatic. The block uses the `tablelookup` function to identify this value.

• pbulb is the fluid pressure of the bulb. The bulb pressure is the saturation pressure, ${p}_{bulb}={p}_{sat}\left({T}_{bulb}\right)$, unless pressure limiting is enabled and the maximum pressure has been reached; see MOP limit for evaporating pressure for more information. Tbulb is the bulb fluid temperature.

• peq depends on the valve pressure equalization setting:

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

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

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

`${S}_{eff,\mathrm{min}}={f}_{leak}{S}_{eff,nom},$`

where fleak is the Closed valve leakage flow as a fraction of nominal flow. The nominal effective valve area, Seff,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 psat(Tevap+ΔTnom) is the saturation pressure at the sum of the Nominal evaporating temperature and the Nominal (static + opening) evaporator superheat.

The nominal effective valve area, Seff,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:

• Tcond is the Nominal condensing temperature.

• vcond is the liquid specific volume at Tcond.

• Qnom is the Nominal evaporator capacity.

• cp,evap is the vapor specific heat at Tevap.

• hevap is the vapor specific enthalpy at Tevap.

• cp,cond is the liquid specific heat at Tcond.

• hcond is the liquid specific enthalpy at Tcond.

• ΔTsub 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 Seff,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, ${p}_{bulb}={p}_{sat}\left({T}_{bulb}\right)$. 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:

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

`${p}_{bulb,MOP}={p}_{eq,MOP}+{p}_{sat}\left({T}_{evap}+\Delta {T}_{static}\right)-{p}_{sat}\left({T}_{evap}\right).$`

• Tbulb 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`.

• Tbulb,MOP is the associated temperature at the pressure pbulb,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:

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

• τbulb 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:

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

• vliq is the liquid specific volume of the fluid.

• vvap is the vapor specific volume of the fluid.

If the inlet fluid is liquid or vapor, vin 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:

• xdyn is the dynamic vapor quality.

• xin is the current inlet vapor quality.

• τ is the Inlet phase change time constant.

If the inlet fluid is a subcooled liquid, xdyn is equal to xin.

### Conservation Equations

Mass is conserved through the valve:

`${\stackrel{˙}{m}}_{A}+{\stackrel{˙}{m}}_{B}=0,$`

where:

• ${\stackrel{˙}{m}}_{A}$ is the mass flow rate at port A.

• ${\stackrel{˙}{m}}_{B}$ 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:

• ΦA is the energy flow rate at port A.

• ΦB is the energy flow rate at port B.

## Ports

### Conserving

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

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

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.

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

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

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.

Refrigerant saturation temperature in the evaporator under nominal operating conditions.

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.

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.

Refrigerant saturation temperature in the condenser under nominal operating conditions.

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

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.

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 permissible saturation pressure in the evaporator.

#### Dependencies

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

Maximum permissible saturation temperature in the evaporator.

#### Dependencies

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

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.

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 of connecting pipes at ports A and B.

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

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.

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.

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.

## 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