Pipe (TPF)#

The Pipe (TPF) block models the dynamics of a two-phase fluid moving through a pipeline section. The block takes into account the compressibility of the fluid.

Law of Conservation of Mass (1) :

(1)#\[\dot m_A + \dot m_B+ \epsilon=V \cdot (\frac{dp_I}{dt}(\frac{\delta \rho}{\delta p})_u+ \frac{du_I}{dt}(\frac{\delta \rho}{\delta u})_p)\]

where:

\(\dot m_A, \dot m_B\) – mass flow rates at the input and output, [kg/s];

\(V\) – volume of the pipeline section, [m\(^3\)];

\(p\) – pressure, [Pa];

\( ho\) – density of the medium, [kg/m\(^3\)];

\(\epsilon= \frac{M-V/v}{ au}\) correction taking into account the numerical error in smoothing partial derivatives;

\(M\) – mass of the medium in the pipeline section, [kg];

\(v\) – specific volume of the medium in the pipeline section, [m\(^3\)/kg];

\( au\) – phase transition time constant, [s];

\(u\) – specific internal energy, [J/kg].

Mass of the medium in the pipeline section (2):

(2)#\[\dot m_A + \dot m_B=\frac{dM}{dt}\]

Law of conservation of energy (3) :

(3)#\[\phi_A + \phi_B + Q= (\dot m_A + \dot m_B)\cdot u_1 + M\cdot \dot u_1\]

where:

\(\phi_A, \phi_B\) - energy flows at the input and output, [W];

\(Q\) – heat flow through the heat port, [W].

Heat flow (4) :

(4)#\[Q=\alpha \cdot F \cdot (T_w - T_I)\]

where:

\(\alpha\) - heat transfer coefficient, [\(\frac{W}{m^2 \cdot K}\)];

\(F\) - pipeline surface area, [m\(^2\)];

\(T_w\) - pipeline wall temperature, [℃];

\(T_I\) - temperature of the environment inside the pipeline section, [℃].

Calculation of the heat transfer coefficient

For liquid and superheated steam, the heat transfer coefficient is calculated using the following formula (5) :

(5)#\[\begin{split}\alpha_ж=\frac{\lambda_ж \cdot Nu_ж}{D_г} \\ \alpha_{пп}=\frac{\lambda_{пп} \cdot Nu_{пп}}{D_г} \end{split}\]

where:

\(\lambda_ж, \lambda_{пп}\) - thermal conductivity for liquid and superheated steam, respectively, [\(\frac{W}{m \cdot K}\)];

\(Nu_ж, Nu_{пп}\) Nusselt number for liquid and superheated steam, respectively;

\(D_г\) – hydraulic diameter, [m].

For a two-phase mixture (6) :

(6)#\[\alpha_{см}=\frac{\lambda_{нж}^{см} \cdot Nu^{см}}{D_г}\]

\(\lambda_{нж}^{см}\) – thermal conductivity of liquid in the saturation state, [\(\frac{W}{m \cdot K}\)];

\(Nu^{cm}\) – Nusselt number for a two-phase mixture.

Calculation of the Nusselt number

If the flow regime is laminar, the Nusselt number is considered constant and equal to the value of the property Nusselt number for laminar flow heat transfer.

A flow regime is said to be laminar if the Reynolds number is less than the Laminar flow upper Reynolds number limit.

A flow regime is turbulent if the Reynolds number is greater than the value of the property Turbulent flow lower Reynolds number limit.

Cubic interpolation was performed in transition mode.

For the liquid and vapor phase, the following formula is used to calculate the Nusselt number (7) :

(7)#\[Nu=\frac {\frac{f}{8}\cdot (Re - 1000) \cdot Pr_I}{1 + 12.7 \cdot \sqrt{\frac{f}{8}}\cdot (Pr_I^{\frac{2}{3}}-1)}\]

where:

\(Re=\frac{|\dot m_{ср}| \cdot D_г \cdot \nu_I}{F \cdot v_I}\) – Reynolds number;

\(Pr_I\) – Prandtl number for the liquid inside the pipeline;

\( u_I\) – specific volume of liquid inside the pipeline, [m\(^3\)/kg];

\( v_I\) – kinematic viscosity for the liquid inside the pipeline, [m\(^2\)/s];

\(F\) – flow area of the medium, [m\(^2\)];

\(f\) – friction coefficient. (8)

(8)#\[f=(-1.8 \cdot log_{10} [\frac{6.9}{Re}+(\frac{\Delta}{3.7})^{1.11}])^{-2}\]

where \(\Delta\) is the relative roughness of the inner surface of the pipeline section.

For a two-phase mixture, the Nusselt calculation formula is as follows (9) :

(9)#\[Nu_{см}=0.05[(1-x_I + x_I \cdot \sqrt{\frac{v_{нп}}{v_{нж}}})\cdot Re_{нж}]^{0.8}\cdot Pr^{0.33}_{нж}\]

where:

\(x_I\) – degree of steam dryness;

\(v_{нп} ,v_{нж}\) – specific volume of saturated vapor and liquid, respectively;

\(Re_{нж}\) – Reynolds number;(10)

\(Pr_{нж}\) – Prandtl number for a liquid in the saturated state.

(10)#\[Re_{нж}=\frac{|\dot m_{ср}|\cdot D_г \cdot \nu_{нж}}{F \cdot v_{нж}}\]

\( u_{нж}\) – specific volume for liquid in the saturated state, [m\(^3\)/kg];

\(v_{нж}\) – kinematic viscosity for liquid in the saturated state, [m\(^2\)/s].

Pulse Balance (11) :

(11)#\[\begin{split}\begin{cases}p_A - p_I = \frac{\dot m_A \cdot |\dot m_A|}{F^2} \cdot |(\nu_I - \nu_A)|+F_{viscA}\\p_B - p_I = \frac{\dot m_B \cdot |\dot m_B|}{F^2} \cdot |(\nu_I - \nu_B)|+F_{viscB} \end{cases}\end{split}\]

where:

\(p_A,p_B\) – pressure at the inlet and outlet, respectively, [Pa];

\(p_I\) – liquid pressure inside the pipeline section, [Pa];

\(\dot m_A, \dot m_B\) – mass flow at the input and output, respectively, [kg/s];

\( u_A\) – specific volume of liquid at the inlet, [m\(^3\)/kg];

\( u_I\) – specific volume of liquid inside a pipeline section, [m\(^3\)/kg];

\(F_{viscA}, F_{viscB}\) – pressure losses due to viscous friction forces, [Pa].

Calculation of pressure losses due to viscous friction

Laminar mode (12) :

(12)#\[\begin{split}\begin{cases}F_{viscA}^{лам}=\frac{f_{chape} \cdot \dot m_A \cdot L_{eff} \cdot \nu_I \cdot \mu_I}{4 \cdot D_г^2 \cdot F}\\F_{viscB}^{лам}=\frac{f_{chape} \cdot \dot m_B \cdot L_{eff} \cdot \nu_I \cdot \mu_I}{4 \cdot D_г^2 \cdot F} \end{cases}\end{split}\]

where:

\(f_{chape}\) - form factor for calculating the Darcy friction coefficient in the laminar flow regime;

\(L_{eff}\) – the sum of the pipeline length and the total equivalent length to take into account additional pressure losses in local resistances, [m];

\(\mu_I\) – dynamic viscosity of the medium in the pipeline, [Pa \(\cdot\) s].

Turbulent mode (13) :

(13)#\[\begin{split}\begin{cases}F_{viscA}^{турб}=\frac{f_A\cdot \dot m_A \cdot |\dot m_A| \cdot L_{eff} \cdot \nu_I}{4 \cdot D_г \cdot F^2}\\F_{viscB}^{турб}=\frac{f_B\cdot \dot m_B \cdot |\dot m_B| \cdot L_{eff} \cdot \nu_I}{4 \cdot D_г \cdot F^2} \end{cases}\end{split}\]

where \(f_A, f_B\) is the Darcy friction coefficient.

Darcy friction coefficient (14) :

(14)#\[\begin{split}\begin{cases}f_A=(-1.8 \cdot log_{10}[\frac{6.9}{Re_A}+(\frac{\Delta}{3.7})^{1.11}])^{-2}\\f_B=(-1.8 \cdot log_{10}[\frac{6.9}{Re_B}+(\frac{\Delta}{3.7})^{1.11}])^{-2} \end{cases}\end{split}\]

где \(\begin{cases}Re_A=\frac{|\dot m_A| \cdot D_г \cdot \nu_I}{F \cdot v_I}\\Re_B=\frac{|\dot m_B| \cdot D_г \cdot \nu_I}{F \cdot v_I}\end{cases}\)

1. Properties#

Property	Default Value	Description	Units of measurement
Pipe length	3.0	-	m
Cross-sectional area	0.01	-	m\(^2\)
Hydraulic diameter	0.113	-	m
Aggregate equivalent length of local resistance	0.4	-	m
Internal surface absolute roughness	1.5E-5	-	m
Laminar flow upper Reynolds number limit	2000.0	-	-
Turbulent flow lower Reynolds number limit	4000.0	-	-
Laminar friction constant for Darcy friction factor	64.0	-	-
Nusselt number for laminar flow heat transfer	3.66	-	-
Initial fluid energy specification: Temperature;	Temperature	-	-
Initial fluid gauge pressure	0	-	Pa
Initial fluid temperature	20.0	-	°C
Phase change time constant	0.1	-	s

2. Inputs and outputs#

Parameters	Input/Output	Description	Units of measurement	Port type
Input	Input	-	-	Two-phase fluid
Heat port	Output	-	-	Heat exchange
Output	Output	-	-	Two-phase fluid

In order to see the connection capabilities of the block, you need to go to page Description of block ports