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

$\displaystyle \zeta_1=\frac{s_2-s_1}{s_{\text{inlet}}-s_1}$ (118)

and based on the outlet conditions by

$\displaystyle \zeta_2=\frac{s_2-s_1}{s_{\text{outlet}}-s_2}.$ (119)

$ 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,

$\displaystyle \zeta_1=\frac{s_B-s_A}{s_{\text{inlet}}-s_A}$ (120)

and based on the outlet conditions by

$\displaystyle \zeta_2=\frac{s_B-s_A}{s_{\text{outlet}}-s_B}.$ (121)

Using Equation (51) one obtains:

$\displaystyle s_2-s_1=r \ln \frac{p_{t_1}}{p_{t_2}},$ (122)

$\displaystyle s_{\text{inlet}}-s_1=r \ln \frac{p_{t_1}}{p_{1}},$ (123)

$\displaystyle s_{\text{outlet}}-s_2=r \ln \frac{p_{t_2}}{p_{2}},$ (124)

from which [75]

$\displaystyle \frac{p_{t_1}}{p_{t_2}} =\left ( {\frac{p_{t_1}}{p_{1}}} \right )...
...\left( 1 + \frac{\kappa -1}{2} M_1^2 \right) ^{\zeta_1 \frac{\kappa}{\kappa-1}}$ (125)

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

$\displaystyle \frac{p_{t_1}}{p_{t_2}} =\left ({\frac{p_{t_2}}{p_{2}}} \right ) ...
...left( 1 + \frac{\kappa -1}{2} M_2^2 \right) ^{\zeta_2 \frac{\kappa}{\kappa-1}},$ (126)

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

Using the general gas equation (37) finally leads to (for $ \zeta_1$):

$\displaystyle \frac{\dot{m} \sqrt{r T_{t_1}}}{A p_{t_1} \sqrt{\kappa}} = \sqrt{...
...} \left(\frac{p_{t_1}}{p_{t_2}}\right)^{-\frac{(\kappa +1)}{2 \zeta_1 \kappa}}.$ (127)

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

$\displaystyle \frac{p_{t_1}}{p_{t_2}}=\left( \frac{\kappa+1}{2}\right)^{\zeta_1 \frac{\kappa}{\kappa-1}}.$ (128)

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

$\displaystyle \Delta_1^2 F = \zeta \frac{\dot{m}^2}{2 g \rho^2 A_1^2 }$ (129)


$\displaystyle \Delta_1^2 F = \zeta \frac{\dot{m}^2}{2 g \rho^2 A_2^2 },$ (130)


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 [34] and TYPE=RESTRICTOR LONG ORIFICE LICHTAROWICZ with coefficients taken from [44]. 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):

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.