# DynamicReliefValve


## Copyright © EDF 2002 - 2026  
## ThermoSysPro Version 4.2  
This component model is documented in Sect. 13.13 of the ThermoSysPro book.   
# Dynamic relief valve   
   
Relief valves are used to limit the pressure in a system.  
When the set pressure level is exceeded, the relief valve opens.  
It closes again once the system pressure reaches the valve reseting pressure.s  

Following assumptions are made:  
- the fluid is subsonic and incompressible.  
- the volume inside the valve is negligible, so that inertia is neglected.  


## Modelica component model  

The equations mentioned below are implemented in the component *DynamicReliefValve*, located in the *WaterSteam.PressureLosses* sub-library.   
This component has 2 connectors:  
- C1: fluid inlet,  
- C2: fluid outlet.  
   
![ThermoSysPro.WaterSteam.PressureLosses.DynamicReliefValve.svg](./icons/ThermoSysPro.WaterSteam.PressureLosses.DynamicReliefValve.svg)  

## Nomenclature  

| Symbol| Description| Unit| Definition| Modelica name |  
| :----------------------------------------------- | :--------------------------------------------------------------------------------------- | :------------------------------- | :------------------------------------- | :----------- |  
| $A_{\mathrm{i}}$| Clapper section at the inlet| $\mathrm{m}^{2}$|| A |  
| $A_{\mathrm{o}}$| Clapper section at the outlet| $\mathrm{m}^{2}$|| A2 |  
| $C_{\mathrm{v}}$| Flow coefficient of the valve| U.S. [USG/min]|| Cvmax |  
| $D$| Valve damping| $-$|| D |  
| $f_{\mathrm{d}}$| Force acting on the clapper due to damping| $\mathrm{N}$|| Fd |  
| $f_{\mathrm{h}}$| Hydraulic force acting on the clapper| $\mathrm{N}$|| Fh |  
| $f_{\mathrm{s}}$| Force acting on the clapper due to the spring| $\mathrm{N}$|| Fr |  
| $f_{\mathrm{t}}$| Total force acting on the clapper| $\mathrm{N}$|| Ft |  
| $f_{\mathrm{w}}$| Force acting on the clapper due to the weight of the clapper| $\mathrm{N}$|| Fp |  
| $h$| Fluid specific enthalpy | $\mathrm{J} / \mathrm{kg}$|| h |  
| $h_{\mathrm{c}}$| Valve height | $\mathrm{m}$| $h_{\mathrm{c}}=z_{\max }-z_{\min }$ | - |  
| $K_{\mathrm{e}}$| Valve spring stiffness| $\mathrm{N} / \mathrm{m}$|| Ke |  
| $\dot{m}$| Fluid mass flow rate| $\mathrm{kg} / \mathrm{s}$|| Q |  
| $M$| Valve mass| $\mathrm{kg}$|| m |  
| $P_{\mathrm{i}}$| Fluid pressure at the inlet| $\mathrm{Pa}$|| C1.P |  
| $P_{\mathrm{o}}$| Fluid pressure at the outlet| $\mathrm{Pa}$|| C2.P |  
| $v$| Clapper velocity| $\mathrm{m} / \mathrm{s}$|| v |  
| $z$| Clapper position| $\mathrm{m}$|| z |  
| $z_{\min }$| Clapper minimum position \(valve fully closed\)| $\mathrm{m}$|| z_min |  
| $z_{\max }$| Clapper maximum position \(valve fully open\)| $\mathrm{m}$|| z_max |  
| $\delta$| Difference between the free spring length and the spring length when the valve is closed | $\mathrm{m}$|| - |  
| $\Delta P$| Pressure loss of the fluid between the valve inlet and outlet| $\mathrm{Pa}$| $P_{\mathrm{i}}-P_{\mathrm{o}}$| deltaP |  
| $\rho$| Fluid density| $\mathrm{kg} / \mathrm{m}^{3}$ ||rho |  
| $\rho_{\text {water, } 60^{\circ} \mathrm{F}}$ | Density of water at $60^{\circ} \mathrm{F}\left(15.5556^{\circ} \mathrm{C}\right) .$| $\mathrm{kg} / \mathrm{m}^{3}$ || rho60F |  
| $\Omega$| Valve position| $-$|| Ouv |  



## Governing equations  

### Static momentum balance equation  


    
    

- Validity domain:   
   
 $\forall \dot{m}$ and $C_{\mathrm{v}} \geq 0$. For $C_{\mathrm{v}}=0, \Delta P$ must be defined.  

- Mathematical formulation:   
   
 $$\Delta P \cdot C_{\mathrm{v}} \cdot\lvert C_{\mathrm{v}}\rvert =1.732189 \times 10^{12} \cdot \frac{\dot{m} \cdot \lvert \dot{m} \rvert }{\rho \cdot \rho_{\text {water, } 60^{\circ} F}}$$  

- Comments:   
   
 This equation is the same as the control valve $C_{\mathrm{v}}=f_{v}(\Omega)$ where $f_{v}$ is the valve characteristic. It is assumed that $\Omega=\frac{z-z_{\min }}{z_{\max }-z_{\min }}$.  


### Clapper equation  

- Validity domain:  

$ z_{\min} \leq z \leq z_{\max} $  

- Mathematical formulation:   

$$   M \cdot \frac{\mathrm{d} 
u}{\mathrm{d}t}=\left\{\begin{array}{l} f_{\mathrm{t}} \text{ if } z_{\min }<z<z_{\max } \   f_{\mathrm{t}} \text{ if } z \leq z_{\min } \text{ and } f_{\mathrm{t}}>0 \   f_{\mathrm{t}} \text{ if } z \geq z_{\max } \text{ and } f_{\mathrm{t}}<0 \   0  \text{ else }\end{array}\right.$$ 
 $$   
u =\left\{\begin{array}{l}\frac{\mathrm{d} z}{\mathrm{d} t}    \text{ when } z_{\min }<z<z_{\max } \   0 \text{ when } z \leq z_{\min } \text{ or } z \geq z_{\max }\end{array}\right.$$ 
 $$   f_{\mathrm{t}} = f_{\mathrm{w}}+f_{\mathrm{s}} +f_{\mathrm{d}}+ f_{\mathrm{h}} \   f_{\mathrm{w}} = -M \cdot g \   f_{\mathrm{d}} = -D \cdot \frac{\mathrm{d} z}{\mathrm{d} t} \   f_{\mathrm{s}} = -K_{\mathrm{e}} \cdot (z - z_{\min} + \delta) \   f_{\mathrm{h}} = P_{\mathrm{i}} \cdot A_{\mathrm{i}} - P_{\mathrm{o}} \cdot A_{\mathrm{o}}$$  

- Comments:  

The velocity  and acceleration are set to zero when the clapper hits the mechanical stops. The equal sign is replaced by $\leq$ or $\geq$ in the transition conditions $z=z_{\min }$ and $z=z_{\max }$ because equal signs are not recognized by solvers to compare real values.  

## References   
   
El Hefni, Baligh and Bouskela, Daniel (2019). [Modeling and Simulation of Thermal Power Plants with ThermoSysPro](https://link.springer.com/book/10.1007/978-3-030-05105-1), sect. 13.1. Springer Nature Switzerland AG.