Restrictor, Long Orifice

Figure: h-s diagram showing the restrictor process
\begin{figure}\epsfig{file=hsdiagram.eps,width=11cm}\end{figure}

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 ([*]), 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} (119)

and based on the outlet conditions by

\zeta_2=\frac{s_2-s_1}{s_{\text{outlet}}-s_2}. (120)

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} (121)

and based on the outlet conditions by

\zeta_2=\frac{s_B-s_A}{s_{\text{outlet}}-s_B}. (122)

Using Equation ([*]) one obtains:

s_2-s_1=r \ln \frac{p_{t_1}}{p_{t_2}}, (123)

s_{\text{inlet}}-s_1=r \ln \frac{p_{t_1}}{p_{1}}, (124)

s_{\text{outlet}}-s_2=r \ln \frac{p_{t_2}}{p_{2}}, (125)

from which [73]

\frac{p_{t_1}}{p_{t_2}} =\left ( {\frac{p_{t_1}}{p_{1}}} \right ) ^{\zeta_1} = \left( 1 + \frac{\kappa -1}{2} M_1^2 \right) ^{\zeta_1 \frac{\kappa}{\kappa-1}} (126)

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 ) ^{\zeta_2} = \left( 1 + \frac{\kappa -1}{2} M_2^2 \right) ^{\zeta_2 \frac{\kappa}{\kappa-1}}, (127)

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

Using the general gas equation ([*]) finally leads to (for \zeta_1):

\frac{\dot{m} \sqrt{r T_{t_1}}}{A p_{t_1} \sqrt{\kappa}}=\sqrt{\frac{2}{\kappa-1}\left( \left( \frac{p_{t_1}}{p_{t_2}}\right)^{\frac{\kappa-1}{\zeta_1 \kappa}} -1\right)} \left(\frac{p_{t_1}}{p_{t_2}}\right)^{-\frac{(\kappa +1)}{2 \zeta_1 \kappa}}. (128)

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}}. (129)

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 } (130)

and

\Delta_1^2 F = \zeta \frac{\dot{m}^2}{2 g \rho^2 A_2^2 }, (131)

respectively.

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

Figure: Geometry of a long orifice restrictor
\begin{figure}\epsfig{file=Long_orifice.eps,width=11cm}\end{figure}

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 [45]. 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.