Restrictor, Long Orifice

Figure 92: h-s diagram showing the restrictor process

Properties: adiabatic, not isentropic, symmetric, $A_1$ inlet based restrictor

Restrictors are discontinuous geometry changes in gas pipes. The loss factor $\zeta$ can be defined based on the inlet conditions or the outlet conditions. Focusing on the h-s-diagram (entalpy vs. entropy) Figure (92), the inlet conditions are denoted by the subscript 1, the outlet conditions by the subscript 2. The entropy loss from state 1 to state 2 is $s_2-s_1$. The process is assumed to be adiabatic, i.e. $T_{t_1}=T_{t_2}$, and the same relationship applies to the total entalpy $h_t$, denoted by a dashed line in the Figure. $E_1$ denotes the kinetic energy part of the entalpy $v_1^2/2$, the same applies to $E_2$. Now, the loss coefficient $\zeta$ based on the inlet conditions is defined by

$$\zeta_1=\frac{s_2-s_1}{s_{\text{inlet}}-s_1}$$ (131)

and based on the outlet conditions by

$$\zeta_2=\frac{s_2-s_1}{s_{\text{outlet}}-s_2}.$$ (132)

$s_{\text{inlet}}$ is the entropy for zero velocity and isobaric conditions at the inlet, a similar definition applies to ${\text{outlet}}$. So, for $\zeta_1$ the increase in entropy is compared with the maximum entropy increase from state 1 at isobaric conditions. Now we have $s_1=s_A$ and $s_2=s_B4$ consequently,

$$\zeta_1=\frac{s_B-s_A}{s_{\text{inlet}}-s_A}$$ (133)

and based on the outlet conditions by

$$\zeta_2=\frac{s_B-s_A}{s_{\text{outlet}}-s_B}.$$ (134)

Using Equation (60) one obtains:

$$s_2-s_1=r \ln \frac{p_{t_1}}{p_{t_2}},$$ (135)

$$s_{\text{inlet}}-s_1=r \ln \frac{p_{t_1}}{p_{1}},$$ (136)

$$s_{\text{outlet}}-s_2=r \ln \frac{p_{t_2}}{p_{2}},$$ (137)

from which [87]

$$\frac{p_{t_1}}{p_{t_2}} =\left ( {\frac{p_{t_1}}{p_{1}}} \right )... ...\left( 1 + \frac{\kappa -1}{2} M_1^2 \right) ^{\zeta_1 \frac{\kappa}{\kappa-1}}$$ (138)

if $\zeta$ is defined with reference to the first section (e.g. for an enlargement, a bend or an exit) and

$$\frac{p_{t_1}}{p_{t_2}} =\left ({\frac{p_{t_2}}{p_{2}}} \right ) ... ...left( 1 + \frac{\kappa -1}{2} M_2^2 \right) ^{\zeta_2 \frac{\kappa}{\kappa-1}},$$ (139)

if $\zeta$ is defined with reference to the second section (e.g. for a contraction).

Using the general gas equation (46) finally leads to (for $\zeta_1$):

$$\frac{\dot{m} \sqrt{r T_{t_1}}}{A p_{t_1} \sqrt{\kappa}} = \sqrt{... ...} \left(\frac{p_{t_1}}{p_{t_2}}\right)^{-\frac{(\kappa +1)}{2 \zeta_1 \kappa}}.$$ (140)

This equation reaches critical conditions (choking, $M_1=1$) for

$$\frac{p_{t_1}}{p_{t_2}}=\left( \frac{\kappa+1}{2}\right)^{\zeta_1 \frac{\kappa}{\kappa-1}}.$$ (141)

Similar considerations apply to $\zeta_2$.

Restrictors can be applied to incompressible fluids as well by specifying the parameter LIQUID on the *FLUID SECTION card. In that case the pressure losses amount to

$$\Delta_1^2 F = \zeta \frac{\dot{m}^2}{2 g \rho^2 A_1^2 }$$ (142)

and

$$\Delta_1^2 F = \zeta \frac{\dot{m}^2}{2 g \rho^2 A_2^2 },$$ (143)

respectively.

A long orifice is a substantial reduction of the cross section of the pipe over a significant distance (Figure 93).

Figure 93: Geometry of a long orifice restrictor

There are two types: TYPE=RESTRICTOR LONG ORIFICE IDELCHIK with loss coefficients according to [40] and TYPE=RESTRICTOR LONG ORIFICE LICHTAROWICZ with coefficients taken from [53]. In both cases the long orifice is described by the following constants (to be specified in that order on the line beneath the *FLUID SECTION, TYPE=RESTRICTOR LONG ORIFICE IDELCHIK or TYPE=RESTRICTOR LONG ORIFICE LICHTAROWICZ card):

• reduced cross section $A_1$.
• full cross section $A_2$.
• hydraulic diameter $D_h$ defined by $D_h=4 A/P$ where $P$ is the perimeter of the cross section.
• Length $L$ of the orifice.
• oil mass flow in the restrictor (only if the OIL parameter is used to define the kind of oil in the *FLUID SECTION card)
• not used (internally: oil material number)

A restrictor of type long orifice MUST be preceded by a restrictor of type user with $\zeta=0$. This accounts for the reduction of cross section from $A_2$ to $A_1$.

By specifying the parameter LIQUID on the *FLUID SECTION card the loss is calculated for liquids. In the absence of this parameter, compressible losses are calculated.

Example files: restrictor, restrictor-oil.