.ThermoSysPro.WaterSteam.PressureLosses.Bend

Information

## 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](modelica://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.  
   
![modelica://ThermoSysPro/UsersGuide/Documentation/ThermoSysPro.WaterSteam.PressureLosses.Bend.svg](modelica://ThermoSysPro/UsersGuide/Documentation/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.  
    

Revisions