(159b) Models for Multi-Phase & Single-Phase Flow in Pressure Relieving System Using Bernoulli Integration

Models for Multi-Phase & Single-Phase Flow
in Pressure Relieving System Using Bernoulli Integration

Freeman Self (feself@bechtel.com)

Sanjay Ganjam (sviswana@bechtel.com)

Bechtel Oil, Gas and Chemicals, Inc.

Gerald Jacobs (gerald@virtualmaterials.com)

Virtual Materials Group

Prepared for presentation at the

American Institute of Chemical Engineers Spring Meeting

Hilton, Austin, Texas

April 2015

Session

DIERS T1G00 Practical Methods for Two-Phase
Flow

Abstract

Introduction

The primary
fluid flow relationships are the Bernoulli (mechanical energy) equation, energy
and continuity equations, along with an equation of state.  In particular, the
Bernoulli equation is important in describing how mechanical energy is
transformed into thermal energy.  Analytical solutions of the Bernoulli
equation pose calculation complexity since the density may change considerably with
pressure.  Solving the Bernoulli equation analytically for compressible fluids requires
a set of simplifying assumptions that limit the application. The most
challenging is the assumption of an ideal gas, where gases approach ideal at
zero pressure and high temperatures.  Other assumptions are constant heat
capacity and the characterization of density changes with pressure. Further,
analytical solutions do not lend themselves conveniently to multiphase flow. 

Despite
these limitations, analytical equations are the norm and are sometimes used inappropriately. 
For example the incompressible equation is used for flashing fluids above the
cricondenbar (the pressure peak of a phase envelope typically called the
dense-phase region), although these fluids exhibit variable densities.
Additionally, the compressible gas equation is used for high-pressure gases which
do not exhibit ideal gas behavior. 

It is
becoming more prevalent to use integral methods to numerically solve the
Bernoulli equation, so that analytical equations are not necessary.   Integral
methods are of course not new; the power industry has used them for years for
single and two-phase flow of steam and water.  With increasing computational
power, excellent thermodynamic and physical property databases, and the use of
compositional models where process streams are characterized by their
individual components and not just bulk properties, integral methods provide elegant
solutions. Further integral methods easily lend themselves to use in multi-phases
models. 

This
paper will illustrate how to set up models for several common situations like
pressure relief valves, orifices, control valves, and pipe tees.  Examples and
derivations will be provided to illustrate how these models can be used for
choked / non-choked, single and multi-phase systems.

Pressure
Relief Valves             

The
integral method has proven successful for relief valves under a full range of
fluids and conditions, including flashing liquids, high-pressure gases,
dense-phase fluids and multi-phase fluids for both choking and non-choking
conditions.  Additionally the equation may be arranged to account for fluid
slip between gas and liquid phases (i.e. the velocity of the phases are not
equal) and phase non-equilibrium if appropriate data is available. 

The
mass flow relation will be written in terms of mass flux G [mass/time-area] and
area A.  To account for errors due to assumptions, a discharge coefficient is
introduced.  The area and discharge coefficient are often included in the same
parameter grouping. 

In British
Standard Units, the equation presents the mass flux G in units of lb/hr-in²,
area in inch² and pressure in psi.

                                                                 

                                          

The
mass flux at the vena contracta is evaluated from the integral as a constant
entropy process starting at the relief pressure. The mass flow is then
determined from the area at the vena contracta and mass flux.  Since the flow
from the inlet to the vena contracta is essentially frictionless, and heat
transfer is minimal, the flow may be considered constant entropy process.  The
integral method is especially beneficial since it rigorously evaluates the mass
flux for variable densities. If the fluid chokes at the vena contracta, the
choke pressure and conditions are determined.  For multi-phase flow, the mixed-phase
density at the vena contracta is expressed as a homogenous average density,
which a reasonable assumption for highly turbulent relief flow. 

                                     where X is the vapor
mass fraction

The
discharge coefficients for industrial relief valves may be obtained either from
the National Board Pressure Relief Device Certifications NB-18 (?Redbook?), or API
Standard 526 ?Flanged Steel Pressure Relief Valves?. 

· For gases, K_d is 0.975 for
many major vendors based on API Standard 526. This implies that the equation properly
predicts actual flow data.  

· For liquids, K_d varies
between vendors but is on the order of 0.75.  As opposed to gases, this implies
that the liquid equation requires adjustments to match flow data.

· For compressible gas, multi-phase
flow or choking liquids, the discharge coefficients are chosen depending on
whether the flow chokes

· If the flow chokes, only the inlet
nozzle flow of the relief valves is involved and ?gas? discharge coefficient is
used

· If the flow does not choke, both the
inlet nozzle and body downstream of the vena contracta is involved and the
?liquid? discharge coefficient is used.

Critical
Flow Orifices (square-edge) for Compressible Flow

A
critical flow orifice is an orifice which chokes in compressible flow. 
Generally for the orifice to choke, the orifice is required to have a certain thickness relative to the
orifice hole.  Thick orifices, manufactured with square edges, with thickness /
diameters greater than 1.0 will choke in compressible flow.  The same
formulation and assumptions used for relief valves may be used for orifices
with similar results. 

                                          

The
schematic illustrates the flow streamlines.

                                                                                 

                               

The
coefficient of discharge is labelled C_d.  For choking flow of compressible
gases the C_d has been experimentally determined as approximately 0.84;
values are found in handbooks. Flashing liquids and choking multi-phase flow
may also be modeled by a thick-orifice using the homogenous assumption and the
mixed density. 

 

Control
Valves

Control valve design
exhibits several unusual features. First, the coefficients of discharge for
gases and vapors are not dimensionless but have units of measure, and the value
is dependent on the units. For example, the liquid discharge coefficient Cv has
British Standard units in gallons / minutes -psi^1/2. 

Control
valve equations are generally treated as orifice equations. The pressure drop
that produces flow is that between upstream and the vena contracta, and this is
the pressure drop used in numerical integration.  However, unlike orifices, control
valve sizing is based on pressures upstream and downstream of the valve, which requires
accounting for pressure recovery in the valve outlet body. Additionally some
control valves are characterized by higher discharge friction with low pressure
recovery. 

The
challenge is to develop the control valve model with parameters utilized by the
control valve industry.  Although some models include specific terms for
friction, this paper will represent the flow similar to orifices, and the mass
flow relation will be written in terms of mass flux G and area-discharge
coefficient parameter. 

                                                                 

                                           

The
mass flux G is in units of
lb/hr-in², area is in inch² and pressure in psi. 

The
mixed density is  where X is the vapor
mass fraction.

X_T
is the choking pressure ratio. Fg
is used to adjust heat capacity ratio from air and Cg is a consolidation of terms that are
functions only of the ideal heat capacity ratio. 

The
area-discharge coefficient parameter group will be found by equating the ISA
control valve equations to traditional orifice equations.  These will be used
to define the mass flow.

 

· The group for
incompressible fluids is [ A  K ] = ( Cv / 38), which is a well know relation. 

· For compressible fluids,   [ A  K  ]
= [ ( Cv / 38)]  [(X_T Fg)
^½ /  (0.002044 Cg
]   

If the parameter C₁
is used, [ A  K ] = [ Cv / 38 ]  [ C₁ / 28.9 ]