# Bend


## Copyright © EDF 2002 - 2026  
## ThermoSysPro Version 4.2  
This component model is documented in Sect. 13.6 of the ThermoSysPro book.   

# Bend   
   
The bend models the singular pressure loss of a fluid circulating inside a bend pipe. For water/steam, the flow regime can be single-phase or homogeneous two-phase flow.   

Following assumptions are made:  
- The flow inside pressure losses is adiabatic. Non-adiabatic pipes must be  
modeled by connecting pressure losses to volumes.  
- The specific enthalpy inside the components is equal to the specific enthalpy at the inlet.  
- The properties of the fluid are computed for the average pressure..  
- Inertia due to momentum inside pipes is neglected.  

The friction pressure loss coefficient is calculated using the geometry of the bend. The bend thus completes the [singular pressure loss component](ThermoSysPro.WaterSteam.PressureLosses.SingularPressureLoss), in which the pressure loss coefficient is a single parameter.  

## Modelica component model  

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

## Nomenclature  

| Symbol| Description| Unit| Definition| Modelica name |  
| :------------------ | :------------------------------------------------ | :------------------------------- | :-------------------------------------------------------------- | :----------- |  
| $A_{1}$| Factor for the singular pressure loss coefficient | $-$|| yA1 |  
| $B_{1}$| Factor for the singular pressure loss coefficient | $-$|| yB1 |  
| $C_{1}$| Factor for the singular pressure loss coefficient | $-$|| yC1 |  
| $D$| Internal diameter of the bend| $\mathrm{m}$|| D |  
| $h$| Fluid specific enthalpy at the inlet| $\mathrm{J} / \mathrm{kg}$|| h |  
| $K_{\varepsilon}$ | Roughness factor| $-$|| kdelta |  
| $\dot{m}$| Fluid mass flow rate| $\mathrm{kg} / \mathrm{s}$|| Q |  
| $P_{\mathrm{i}}$| Fluid pressure at the inlet| $\mathrm{Pa}$|| C1.P |  
| $P_{\mathrm{o}}$| Fluid pressure at the outlet| $\mathrm{Pa}$|| C2.P |  
| $R$| Bend radius| $\mathrm{m}$|| R0 |  
| $Re$| Reynolds number| $-$| $\frac{4 \cdot \lvert \dot{m} \rvert}{\pi \cdot D \cdot \mu}$ | Re |  
| $Re_{\lim }$| limiting Reynolds number| $-$|| Relim |  
| $\delta$| Bend angle| $\circ$|| delta |  
| $\Delta P$| Pressure loss between the inlet and the outlet| $\mathrm{Pa}$| $P_{\mathrm{i}}-P_{\mathrm{o}}$| deltaP |  
| $\varepsilon$| Pipe roughness| $\mathrm{m}$|| rugosrel |  
| $\zeta$| Hydraulic pressure loss coefficient| $-$|| khi |  
| $\zeta_{m}$| singular pressure loss coefficient| $-$|| khim |  
| $\zeta_{f}$| Friction pressure loss coefficient| $-$|| khif |  
| $\lambda$| Friction pressure loss coefficient| $-$|| lambda |  
| $\mu$| Fluid dynamic viscosity| $\mathrm{Pa} \mathrm{s}$|| mu |  
| $\rho$| Fluid density| $\mathrm{kg} / \mathrm{m}^{3}$ || rho |  



## Governing equations  

### Static momentum balance equation  


    
    

- Validity domain:   
   
 $\forall \dot{m}$  

- Mathematical formulation:   
   
 $$\Delta P=8 \cdot \zeta \cdot \frac{\dot{m} \cdot \lvert \dot{m} \rvert}{\pi^{2} \cdot D^{4} \cdot \rho}$$  

- Comments:   
   

### Hydraulic pressure loss coefficient  

- Validity domain:  

$\forall \dot{m}$ with $Re>Re_{\lim }$  

- Mathematical formulation:   

$$\zeta=K_{\varepsilon} \cdot \zeta_{m}+\zeta_{f}$$ 
 $$   K_{\varepsilon} = \left\{ \begin{array}{l}   2 \; \text{if} \; \varepsilon \geq 10^{-3} \   1 + 10^3 \cdot \varepsilon \; \text{if} \; \varepsilon<10^{-3} \; \text{and} \; \frac{R}{D}<1.5 \   1 + 10^6 \cdot \varepsilon^2 \; \text{if} \; \varepsilon<10^{-3} \; \text{and} \; \frac{R}{D} \geq 1.5   \end{array} \right.$$ 
 $$\zeta_{f}=0.0175 \cdot \lambda \cdot \frac{R}{D} \cdot \delta$$ 
 $$\zeta_{m}=A_{1} \cdot B_{1} \cdot C_{1}$$  

- Comments:  

The limiting Reynolds number is given by:  

$$   Re_{\lim } = \left\{ \begin{array}{l}   2 \times 10^{5} \; \text{if} \; \varepsilon<5 \times 10^{-5} \   \max \left(\frac{560}{\varepsilon}, 2 \times 10^{5}\right) \; \text{if} \; \varepsilon \geq 5 \times 10^{-5}   \end{array} \right.$$  

## 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. 13.6. Springer Nature Switzerland AG.