# Thermostatic Expansion Valve (2P)

Flow control valve that maintains evaporator superheat

**Library:**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. You typically place this valve between a condenser and an evaporator in a refrigeration system to maintain a specific temperature differential by moderating the flow into the evaporator.

The valve behavior depends on the *superheat*, which is the difference in
temperature between the vapor at the evaporator outlet and the fluid evaporating
temperature. The valve opens if the superheat increases to let more flow through, and
the valve closes if the superheat decreases to let less flow through. When the superheat
drops to or below the value of the **Static (minimum) evaporator
superheat** parameter, then the valve is fully closed. The closed valve
reduces the flow through the evaporator, which reduces the heat transfer and increases
the outlet temperature. When you use the **MOP limit** parameter to
enable a maximum pressure or temperature limit, the valve closes when the temperature or
pressure exceeds the limit.

The bulb sensor at port **S** measures the evaporator outlet temperature.
If you set the **Pressure equalization** parameter to
`External pressure equalization`

, the block uses data from
the evaporator at port **E** for internal pressure equalization.
Otherwise, the block uses the pressure at port **B** 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.

### Analytical Parameterization

When you set **Valve parameterization** to ```
Nominal capacity,
superheat, and operating conditions
```

, the block uses an analytical
model. In this setting, the block fits the analytical model to the performance data
so that when the refrigeration system operates at the specified nominal evaporating
temperature, condensing temperature, and condenser subcooling, the block meters
enough flow into the evaporator to maintain the evaporator superheat while the block
transfers heat at the evaporator capacity. Subcooling is the difference in
temperature between the condenser outlet and the condensing temperature.

For this parameterization, the block assumes a straight-charged bulb and assumes that the
sensing bulb contains the same fluid as the refrigerant. To model a cross-charged
bulb, set **Valve parameterization** to ```
Tabulated data
- quadrant diagram
```

.

**Opening Area**

The valve operates to control the mass flow rate between a condenser and an evaporator by
regulating the effective opening area,
*S _{eff}*. The mass flow rate is

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

where:

*v*is the inlet specific volume, or the fluid volume per unit mass._{in}*Δp*is the pressure differential over the valve,*p*–_{A}*p*._{B}*Δp*is the pressure threshold for transitional flow. Below this value, the flow is laminar_{lam}$$\Delta {p}_{lam}=\frac{{p}_{A}+{p}_{B}}{2}\left(1-{B}_{lam}\right),$$

where

*B*_{lam}is the value of the**Laminar flow pressure ratio**parameter.

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 the nominal operating conditions. See Determining β from Nominal Conditions.*T*is the evaporating temperature._{evap}When

**Nominal pressure specification**is`Pressure at specified saturation temperature`

,*T*is the value of the_{evap}**Nominal evaporating (saturation) temperature**parameter.When

**Nominal pressure specification**is`Specified pressure`

,*T*is the saturation temperature that corresponds to the value of the_{evap}**Nominal evaporator outlet pressure**parameter.*ΔT*is the_{static}**Static (minimum) evaporator superheat**parameter.*p*is the fluid saturation pressure as a function of_{sat}(T_{evap})*T*. The block uses the_{evap}`tablelookup`

function to identify this value.*p*is the saturation pressure as a function of_{sat}(T_{evap}+ΔT_{static})*T*. The block uses the_{evap}+ΔT_{static}`tablelookup`

function to identify this value.*p*is the fluid pressure of the bulb. The bulb pressure is the saturation pressure, $${p}_{bulb}={p}_{sat}({T}_{bulb})$$, unless you set_{bulb}**MOP limit**to`On - Specify maximum operating pressure`

and the pressure reaches the maximum pressure. See Maximum Outlet Pressure Limit for more information.*T*is the bulb fluid temperature._{bulb}*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

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

where *f _{leak}* is the
value of the

**Closed valve leakage as a fraction of nominal flow**parameter. To see how the block calculates the nominal effective valve area,

*S*and maximum effective valve area, see Determining β from Nominal Conditions.

_{eff,nom}**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 evaporator outlet temperature and
the

**Nominal (static + opening) evaporator superheat**parameter.

The block calculates the nominal effective valve area,
*S*_{eff,nom}, as a function of the
nominal condenser and evaporator thermodynamics. When **Capacity
specification** is ```
Evaporator heat
transfer
```

, the nominal effective valve area is

$${S}_{eff,nom}=\frac{\left[\frac{{Q}_{nom}}{{c}_{p,evap}\Delta {T}_{nom}+{h}_{evap}-{h}_{cond}+{c}_{p,cond}\Delta {T}_{sub}}\right]}{\sqrt{\frac{2}{{v}_{cond}}\left({p}_{sat}\left({T}_{cond}\right)-{p}_{sat}\left({T}_{evap}\right)\right)}}.$$

When **Capacity specification** is ```
Mass flow
rate
```

, the nominal effective valve area is

$${S}_{eff,nom}=\frac{{\dot{m}}_{nom}}{\sqrt{\frac{2}{{v}_{cond}}\left({p}_{sat}\left({T}_{cond}\right)-{p}_{sat}\left({T}_{evap}\right)\right)}},$$

where:

*T*is the condensing saturation temperature._{cond}When

**Nominal pressure specification**is`Pressure at specified saturation temperature`

,*T*is the value of the_{cond}**Nominal condensing (saturation) temperature**parameter.When

**Nominal pressure specification**is`Specified pressure`

,*T*is the saturation temperature that corresponds to the value of the_{cond}**Nominal condenser outlet pressure**parameter.*v*is the liquid specific volume at_{cond}*T*._{cond}*Q*is the_{nom}**Nominal evaporator heat transfer**parameter.*c*is the vapor specific heat at_{p,evap}*T*._{evap}*h*is the vapor specific enthalpy at_{evap}*T*._{evap}*c*is the liquid specific heat at_{p,cond}*T*._{cond}*h*is the liquid specific enthalpy at_{cond}*T*._{cond}*ΔT*is the_{sub}**Nominal condenser subcooling**parameter.$${\dot{m}}_{nom}$$ is the

**Nominal mass flow rate**parameter.

When **Capacity specification** is
`Evaporator heat transfer`

, the block determines
the maximum effective area of the valve in the same way as
*S _{eff,nom}*, but uses the value
of

**Maximum evaporator heat transfer**parameter instead of the

**Nominal evaporator heat transfer**parameter. When

**Capacity specification**is

```
Mass flow
rate
```

, the block uses the value of the **Maximum mass flow rate**parameter instead of the

**Nominal mass flow rate**parameter.

**Maximum Outlet Pressure Limit**

You can limit the maximum outlet pressure (MOP) in the evaporator by setting the
**MOP limit** parameter to ```
On - Specify
maximum operating pressure
```

or```
On - Specify
maximum operating temperature
```

. When you use one of these
settings, the valve closes when the bulb temperature or pressure exceeds the
temperature or pressure associated with maximum bulb pressure, and opens once
the pressure reduces. If you set **MOP limit** to
`Off`

, or the measured pressure is below the limit, $${p}_{bulb}={p}_{sat}({T}_{bulb})$$. Otherwise, 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 evaporator outlet pressure**parameter,*p*_{eq,MOP}, or the pressure specified by the**Maximum evaporating (saturation) temperature**parameter, 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 value is the temperature at port**S**if**Bulb temperature dynamics**is set to`Off`

. The block applies a first-order delay 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}.

### Four-Quadrant Diagram

To visualize the four-quadrant diagram, right-click the block and select **Fluids** > **Plot 4-Quadrant Diagram**.

When **Valve parameterization** is ```
Tabulated data - quadrant
diagram
```

, the block uses user-provided data to plot in all
quadrants. See Tabulated Data Parameterization.

When **Valve parameterization** is ```
Nominal capacity,
superheat, and operating conditions
```

, the block calculates the data
in all quadrants by using the analytical valve model, described in Analytical Parameterization. Because the
analytical model assumes a straight-charged bulb, the diagram plots the two curves
in quadrant 1 on top of each other.

The diagram shows the four quadrants:

**Quadrant 1:**Plot of the fluid pressure of the sensing bulb versus the evaporator outlet temperature when the sensing bulb is attached to the evaporator outlet. In this image, the graph includes a second line that represents the vapor saturation curve of the refrigerant in the cycle, which is a plot of saturation pressure versus saturation temperature.**Quadrant 2:**Plot of the valve lift versus the evaporator outlet pressure. The valve lift is the position of the needle of the valve. As the evaporator outlet pressure decreases, the valve opens and valve lift increases.When

**Valve parameterization**is`Tabulated data - quadrant diagram`

, the plot holds the evaporator outlet temperature constant at the**Reference evaporator outlet temperature**parameter value.**Quadrant 3:**Plot of the mass flow rate through the valve versus valve lift.When

**Valve parameterization**is`Tabulated data - quadrant diagram`

, the plot holds the inlet pressure of the valve constant at the**Reference condenser outlet pressure**parameter, the inlet temperature constant at the**Reference condenser subcooling**parameter, and the outlet pressure constant at the**Reference evaporator outlet pressure parameter**.**Quadrant 4:**Plot of the mass flow rate through the valve versus evaporator outlet temperature.When

**Valve parameterization**is`Tabulated data - quadrant diagram`

, the plot holds the inlet pressure of the valve constant at the**Reference condenser outlet pressure**parameter, the inlet temperature constant at the**Reference condenser subcooling**parameter, and the outlet pressure constant at the**Reference evaporator outlet pressure parameter**.

### Tabulated Data Parameterization

When you set **Valve parameterization** to ```
Tabulated data -
quadrant diagram
```

, the block uses tabulated data. Three of the four
curves in the thermostatic expansion valve quadrant diagram specify the valve
characteristics. The supplier typically provides this diagram.

To determine block characteristics, the block performs interpolation on the data you
provide in the parameters **Quadrant 1 - Evaporator outlet temperature
vector**, **Quadrant 1 - Bulb pressure vector**,
**Quadrant 2 - Valve lift vector**, **Quadrant 2 -
Evaporator outlet pressure vector**, **Quadrant 3 - Valve lift
vector**, and **Quadrant 3 - Mass flow rate vector**.

The block determines the mass flow rate through the valve from the inlet pressure, outlet pressure, equalization pressure, and sensing bulb temperature by using the data from quadrant 1 to quadrant 3 and a series of interpolations. The block first calculates the quadrant 2 evaporator outlet pressure as

$${p}_{evap,Q2}={p}_{eq}-{p}_{bulb}\left({T}_{bulb}\right)+{p}_{bulb}\left({T}_{evap,ref}\right)$$

where:

*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**.

*p*is the bulb fluid pressure interpolated at the bulb fluid temperature_{bulb}(T_{bulb})*T*. You specify the bulb fluid pressure data in the parameter_{bulb}**Quadrant 1 – Bulb pressure vector**.*p*is the bulb fluid pressure interpolated at the reference evaporator outlet temperature_{bulb}(T_{evap,ref})*T*._{evap,ref}

The block uses the valve lift data, *L*, from the **Quadrant 2 –
Valve lift vector** parameter to interpolate the valve lift at the
values in **Quadrant 2 – Evaporator outlet pressure vector**, which
gives *L = L(p _{evap,Q2})*. The block then uses
the reference mass flow rate, ${\dot{m}}_{ref}$, from the

**Quadrant 3 – Mass flow rate vector**, to interpolate the mass flow rate at the valve lift values $${\dot{m}}_{ref}={\dot{m}}_{ref}(L)$$. The block scales the reference mass flow rate to the actual mass flow rate with

$$\dot{m}={\dot{m}}_{ref}\frac{\sqrt{\frac{2}{{v}_{in}}}\frac{\Delta p}{{(\Delta {p}^{2}-\Delta {p}_{lam}^{2})}^{0.25}}}{\sqrt{\frac{2}{{v}_{ref}}({p}_{cond,ref}-{p}_{evap,ref})}}$$

where:

*p*is the_{cond,ref}**Reference condenser outlet pressure**parameter.*p*is the_{evap,ref}**Reference evaporator outlet pressure**parameter.*v*is the specific volume that corresponds to the_{ref}**Reference condenser outlet pressure**and**Reference condenser subcooling**parameters.*v*is the inlet specific volume, or the fluid volume per unit mass._{in}*Δp*is the pressure differential over the valve,*p*–_{A}*p*._{B}*Δp*is the pressure threshold for transitional flow. Below this value, the flow is laminar. The block calculates this value as:_{lam}$$\Delta {p}_{lam}=\frac{{p}_{A}+{p}_{B}}{2}\left(1-{B}_{lam}\right),$$

where

*B*_{lam}is the value of the**Laminar flow pressure ratio**parameter.

### 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 *internal pressure
equalization*. In systems with larger losses, connect the evaporator
outlet port to the valve block at port **E**.

### 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 you do not model bulb dynamics, this value is*T*_{bulb}.*τ*_{bulb}is the value of the**Bulb thermal time constant**parameter.

### 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*is the inlet vapor quality. The block applies a first-order lag to the inlet vapor quality of the mixture._{dyn}*v*is the liquid specific volume of the fluid._{liq}*v*is the vapor specific volume of the fluid._{vap}

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*is the dynamic vapor quality._{dyn}*x*is the current inlet vapor quality._{in}*τ*is the value of the**Inlet phase change time constant**parameter.

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**.

The block supports reversed flows numerically, however, the valve
block is not designed for flows from port **B** to port
**A**.

The block conserves energy flow 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

## Parameters

## References

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

## Extended Capabilities

## Version History

**Introduced in R2020b**