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

**Library:**Simscape / Fluids / Two-Phase Fluid / Valves & Orifices / Flow Control Valves

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.

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}:

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

where:

*β*is a valve constant determined from nominal operating conditions. See Determining β from Nominal Conditions for more information.*p*_{sat}is the fluid saturation pressure, which is a function of temperature and evaluated at the indicated temperatures.*p*is the saturation pressure at_{sat}(T_{evap}+ΔT_{static})*T*._{evap}+ΔT_{static}*T*_{evap}is the**Nominal evaporating temperature**.*ΔT*_{static}is the**Static (minimum) evaporator superheat**.*p*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._{bulb}*T*is the bulb fluid temperature._{bulb}*p*_{eq}depends on the valve pressure equalization setting:If

**Pressure equalization**is set to`Internal pressure equalization`

,*p*_{eq}is the pressure at port**B**.If

**Pressure equalization**is set 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

*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

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*is the_{cond}**Nominal condensing temperature**.*v*is the liquid specific volume at_{cond}*T*._{cond}*Q*is the_{nom}**Nominal evaporator capacity**.*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**.*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

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**.

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}.

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**.

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.

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**Inlet phase change time constant**.

If the inlet fluid is a subcooled liquid,
*x _{dyn}* is equal to

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**.

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