Local restriction (TPF)#

The Local restriction (TPF) block models a local reduction in the flow cross-sectional area. The block can operate in a constant-area mode or in a controlled mode, where the cross-sectional area is determined by an external signal.

Law of Conservation of Mass (1) :

(1)#\[\dot m_A + \dot m_B=0\]

where:

\(\dot m_A\) – mass flow rate at the input, [kg/s];

\(\dot m_B\) – mass flow rate at the outlet, [kg/s].

Law of conservation of energy (2), (3), (4) :

(2)#\[\phi_A + \phi_B = 0\]

where:

\(\phi_A\) - energy flow at the input, [W];

\(\phi_B\) - energy flow at the output, [W].

(3)#\[u_A+p_A\cdot v_A + \frac{w^2_A}{2}=u_R + p_R \cdot v_R + \frac{w^2_R}{2}\]

(4)#\[u_B+p_B\cdot v_B + \frac{w^2_B}{2}=u_R + p_R \cdot v_R + \frac{w^2_R}{2}\]

where:

\(u_A, u_B, u_R\) – specific internal energies of the medium at the entrance, exit and in the constriction, [J/kg];

\(p_A, p_B, p_R\) – pressure of the medium at the inlet, outlet and in the constriction, [Pa];

\(v_A, v_B, v_R\) – specific volumes of the medium at the inlet, outlet and in the constriction, [m\(^3\)/kg];

\(w_A, w_B, w_R\) – ideal flow velocities at the inlet, outlet and constriction.

Ideal speeds are calculated according to the following equation (5) :

(5)#\[w=\frac{\dot m_{ideal}\cdot v}{S}\]

where \(\dot m_{ideal}=\frac{\dot m}{c_d}\) is the ideal mass flow rate, [kg/s].

where:

\(\dot m\) – mass flow rate of the medium, [kg/s];

\(С_d\) - flow coefficient.

Pressure drop

The block implements several pressure loss modes. The Bernoulli mode assumes a constant density at the inlet and outlet of the resistance. This mode is more stable.

The Control Volume mode is more accurate, but less reliable in terms of solution convergence

For Bernoulli mode (6) :

(6)#\[\dot m = C_d \cdot S_R \cdot \frac{\Delta p}{(\Delta p^2+ \Delta p^2_{кр})^{1/4}}\cdot \sqrt{\frac{2}{v_{вх}\cdot K \cdot (1-(\frac{S_R}{S})^2)}}\]

where:

\(S_R\) – cross-sectional area of the hole, [m\(^2\)];

\(S\) – cross-sectional area of ports, [m\(^2\)];

\(v_{in}\) – specific volume of the medium at the inlet, [m\(^3\)/kg];

\(\Delta p\) – pressure drop across the block, [Pa];

\(\Delta p_{кр}\) – pressure drop across the block, [Pa];

\(K\) - pressure loss coefficient. (1) :

(7)#\[K=\frac{\sqrt{1-(\frac{S_R}{S})^2 \cdot (1-C_d^2)}-C_d \cdot \frac{S_R}{S}}{\sqrt{1-(\frac{S_R}{S}^2) \cdot (1-C_d^2)}+C_d \cdot \frac{S_R}{S}}\]

For the Control Volume mode

Mass flow rate for turbulent flow regime (8) :

(8)#\[\dot m_A = C_d \cdot S_R \cdot \Delta p \cdot \sqrt{\frac{2}{v_R \cdot K_T \cdot |\Delta p|}}\]

где \(K_T= (1+ \frac{S_R}{S}) \cdot (1 - \frac{v_{вх}}{v_R} \cdot \frac{S_R}{S})-2 \cdot \frac{S_R}{S}\cdot (1 - \frac{v_{вых}}{v_R} \cdot \frac{S_R}{S})\)

Mass flow rate for laminar flow regime (9) :

(9)#\[\dot m_A = C_d \cdot S_R \cdot \Delta p \cdot \sqrt{\frac{2}{v_R \cdot \Delta p_{лам} \cdot(1-\frac{S_R}{S})^2}}\]

where \(\Delta p_{lam}=\frac{p_A+p_B}{2}\cdot (1-B_{lam})\),

where \(B_{lam}\) is the ratio of the transition pressures between the laminar and turbulent regimes.

1. Properties#

Property	Default Value	Description	Units of measurement
Restriction type: Fixed; Variable	Fixed	-	-
Restriction area	0.001	-	m\(^2\)
Pressure loss model	Bernoulli	-	-
Ports cross-sectional area	0.01	-	m\(^2\)
Discharge coefficient	0.64	-	-
Laminar flow pressure ratio	0.999	-	-

2. Inputs and outputs#

Parameters	Input/Output	Description	Units of measurement	Port type
Input	Input	-	-	Two-phase fluid
Restriction area	Input	Available when `Restriction type` - `Variable`	m\(^2\)	Automation
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