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Defining the heat capacity at constant pressure

You must define the heat capacity at constant pressure for the ideal gas. It can be defined either in polynomial or tabular form. The polynomial form is based on the Shomate equation according to the National Institute of Standards and Technology. The constant pressure molar heat capacity can be expressed as

\tilde {c} _ {p} = \tilde {a} + \tilde {b} (\theta - \theta^ {Z}) + \tilde {c} (\theta - \theta^ {Z}) ^ {2} + \tilde {d} (\theta - \theta^ {Z}) ^ {3} + \frac {\tilde {e}}{(\theta - \theta^ {Z}) ^ {2}},

where the coefficients , , , \tilde { d } , and are gas constants. These gas constants together with molecular weight are listed in Table 11.5.2–1 for some gases that are often used in airbag simulations. The constant pressure heat capacity can then be obtained by

c _ {p} = \frac {\tilde {c} _ {p}}{M W}.

The constant volume heat capacity, c _ { v } . , can be determined by

c _ {v} = c _ {p} - R.

Table 11.5.2–1 Properties of some commonly used gases (SI units).

Gas	MW	$\tilde{a}$	$\tilde{b}$ $(\times 10^{-3})$	$\tilde{c}$ $(\times 10^{-6})$	$\tilde{d}$ $(\times 10^{-9})$	$\tilde{e}$ $(\times 10^{6})$	$\theta$ (kelvin)
Air	0.0289	28.110	1.967	4.802	-1.966	0.0	273–1800
Nitrogen	0.028	26.092	8.218	-1.976	0.1592	0.0444	298–6000
Oxygen	0.032	29.659	6.137	-1.186	0.0957	-0.219	298–6000
Hydrogen	0.00202	33.066	-11.36	11.432	-2.772	-0.158	273–1000
Carbon monoxide	0.028	25.567	6.096	4.054	-2.671	0.131	298–1300
Carbon dioxide	0.044	24.997	55.186	-33.691	7.948	-0.136	298–1200
Water vapor	0.0180	32.240	1.923	0.105	-3.595	0.0	273–1800

You can use the polynomial form for specifying the heat capacity at constant pressure, in which case you enter the coefficients \tilde { a } , \tilde { b } , \tilde { c } , \tilde { d } , and . Alternatively, you can define a table of constant pressure heat capacity versus temperature and any predefined field variables.

Input File Usage:

Use the following option to specify the heat capacity in polynomial form:*CAPACITY, TYPE=POLYNOMIAL $\tilde{a}, \tilde{b}, \tilde{c}, \tilde{d}, \tilde{e}$ Use the following option to specify the heat capacity in tabular form:*CAPACITY, TYPE=TABULAR, DEPENDENCIES=n $\tilde{c}_{p}, \text{temperature, field\_variable\_1, etc...}$ ...

Abaqus/CAE Usage:

Use the following option to specify the heat capacity in polynomial form:Interaction module: Create Interaction Property: Fluid cavity:Definition: Pneumatic, toggle on Specify molar heat capacity:Polynomial, Polynomial Coefficients: $\tilde{a}, \tilde{b}, \tilde{c}, \tilde{d}, \tilde{e}$ Use the following option to specify the heat capacity in tabular form:Interaction module: Create Interaction Property: Fluid cavity:Definition: Pneumatic: toggle on Specify molar heat capacity:Tabular: enter the molar heat capacityUse the following options to include temperature and field variable dependence in the table:Toggle on Use temperature-dependent data, Number of field variables: n

A mixture of ideal gases

Abaqus/Explicit can model a mixture of ideal gases in the fluid cavity. For ideal gas mixtures the Amagat-Leduc rule of partial volumes is used to obtain an estimate of the molar-averaged thermal properties (Van Wylen and Sonntag, 1985). Let each species have constant pressure and volume heat capacities, c _ { p _ { i } } and c _ { v _ { i } } ; molecular weight, M W _ { i } ; and mass fraction, f _ { i } . The constant pressure and volume heat capacities for the mixed gas are then given by

c _ {p} = \sum_ {i} f _ {i} c _ {p _ {i}},

c _ {v} = \sum_ {i} f _ {i} c _ {v _ {i}},

and the molecular weight is given by

M W = 1 / \sum_ {i} \frac {f _ {i}}{M W _ {i}}.

The specific energy and enthalpy for the mixed gas are then given by

E = \sum_ {i} f _ {i} E _ {i},

H = \sum_ {i} f _ {i} H _ {i}.

The energy flow entering the fluid cavity is given by

\dot {m} _ {i n} H _ {i n} = \sum_ {i} \dot {m} _ {i n _ {i}} H _ {i n _ {i}},

and the energy flow out of the fluid cavity is given by

\dot {m} _ {o u t} H _ {o u t} = \sum_ {i} \dot {m} _ {o u t _ {i}} H _ {o u t _ {i}}.

Using the properties of a mixture of ideal gases as given above, the pressure and temperature can be obtained from the ideal gas law and the energy equation.

Averaged properties for multiple fluid cavities

If the output of the state of the fluid inside the cavity is requested for a node set that contains more than one fluid cavity, the averaged properties of the multiple fluid cavities will also be output automatically. The average pressure is calculated by volume weighting cavity pressure contributions. The average temperature for an adiabatic ideal gas is obtained by mass weighting cavity temperature contributions. Let each fluid cavity have pressure p _ { k } , temperature \theta _ { k } , volume V _ { k } , gas constant R _ { k } , and mass m _ { k } . The average pressure of the fluid cavity cluster is then defined as

p _ {a v g} = \sum_ {k} p _ {k} V _ {k} / \sum_ {k} V _ {k},

and the average temperature is

\theta_ {a v g} = \sum_ {k} R _ {k} \theta_ {k} m _ {k} / \sum_ {k} R _ {k} m _ {k}.

Additional reference

• Van Wylen, G. J., and R. E. Sonntag, Fundamentals of Classical Thermodynamics, Wiley, New York, 1985.

11.5.3 FLUID EXCHANGE DEFINITION

Products: Abaqus/Standard Abaqus/Explicit Abaqus/CAE

References

• “Surface-based fluid cavities: overview,” Section 11.5.1
• “Fluid cavity definition,” Section 11.5.2
• *FLUID EXCHANGE
• *FLUID EXCHANGE PROPERTY
• *FLUID EXCHANGE ACTIVATION
• “VUFLUIDEXCH,” Section 1.2.17 of the Abaqus User Subroutines Reference Guide
• “VUFLUIDEXCHEFFAREA,” Section 1.2.18 of the Abaqus User Subroutines Reference Guide
• “Defining a fluid exchange interaction,” Section 15.13.12 of the Abaqus/CAE User’s Guide, in the HTML version of this guide
• “Defining a fluid exchange interaction property,” Section 15.14.5 of the Abaqus/CAE User’s Guide, in the HTML version of this guide

Overview

A fluid exchange definition:

• can be used to model flow between a single fluid cavity and its environment or flow between two fluid cavities;
• can be used to prescribe mass- or volume-based flux into or out of a cavity;
• can model the venting of a cavity through an exhaust orifice;
• can model flow through cavity walls such as leakage through a porous fabric;
• can be used to prescribe heat loss through a cavity surface due to heat transfer;
• can take the local material state into account;
• can account for blockage due to contacting boundary surfaces; and
• has a name that can be used to identify history output of mass flow rates out of a cavity.

Defining fluid exchange

The fluid exchange capability is very general and can be used to define flow in and out of a cavity either as a prescribed function or based on the pressure difference arising from analysis conditions. The flow behavior in Abaqus/Standard is based on mass fluid flow, and the behavior in Abaqus/Explicit can be based on mass fluid flow or heat energy flow. You must associate the fluid exchange definition with a name.

Input File Usage: *FLUID EXCHANGE, NAME=name

Abaqus/CAE Usage: Interaction module: Create Interaction: Fluid exchange, Name: name

Flow between a single cavity and its environment

To define flow between a fluid cavity and its environment in Abaqus/Explicit, specify the single reference node associated with the fluid cavity. In the discussion that follows this fluid cavity is referred to as the primary cavity. When the flow is defined as a prescribed function, the flow can either be into or out of the primary cavity. If the flow is into the cavity, the properties of the material flowing in are assumed to be the instantaneous properties of the material in the cavity itself. When the flow behavior is based on analysis conditions, the mass flow can occur only out of the primary cavity but the heat energy flow can be either into or out of the primary cavity. For the case of mass flow Abaqus will use the fluid cavity pressure and the specified constant ambient pressure to calculate the pressure difference used to determine the mass flow rate. For the case of heat energy flow Abaqus/Explicit will use the fluid cavity temperature and the specified constant ambient temperature to calculate the temperature difference used to determine the heat energy flow rate.

Input File Usage: Use the following options:

*FLUID CAVITY, NAME=primary_cavity_name,
REF NODE=primary_cavity_reference_node
*FLUID EXCHANGE, NAME=fluid_exchange_name
primary_cavity_reference_node 

Abaqus/CAE Usage: Interaction module: Create Interaction: Fluid exchange:

Definition: To environment, Fluid cavity interaction: name, Fluid exchange property: name 

Flow between two fluid cavities

To define flow between two fluid cavities, specify the reference nodes associated with the primary and secondary fluid cavities. When the flow is based on analysis conditions, the fluid will flow from the high pressure or upstream cavity to the low pressure or downstream cavity and the heat energy will flow from the high temperature to the low temperature.

Input File Usage: Use the following options:

*FLUID CAVITY, NAME=primary_cavity_name,
REF NODE=primary_cavity_reference_node
*FLUID CAVITY, NAME=secondary_cavity_name,
REF NODE=secondary_cavity_reference_node
*FLUID EXCHANGE, NAME=fluid_exchange_name
primary_cavity_reference_node, secondary_cavity_reference_node 

Abaqus/CAE Usage: Interaction module: Create Interaction: Fluid exchange: Definition:

Between cavities, Fluid cavity interaction 1: name, Fluid cavity interaction 2: name, Fluid exchange property: name 

Specifying the effective area in an Abaqus/Explicit analysis

The flow rate from the primary cavity for any fluid exchange property is proportional to the effective leakage area. The leakage area may represent the size of an exhaust orifice, the area of a porous fabric enclosing the cavity, or the size of a pipe between cavities.

In an Abaqus/Explicit analysis you can specify the value of the effective leakage area directly. Alternatively, you can define a surface that represents the leakage area by specifying the name of the surface on the boundary enclosing the primary fluid cavity. The effective area for fluid exchange is based on the area of the surface unless you specify the area directly or define the effective area with user subroutine VUFLUIDEXCHEFFAREA. If both the effective area and a surface are specified, the area of the surface is used only to determine blockage; see “Accounting for blockage due to contacting boundary surfaces,” below. If neither area is specified, the effective area defaults to 1.0.

You can also define the effective leakage area with user subroutine VUFLUIDEXCHEFFAREA (see “VUFLUIDEXCHEFFAREA,” Section 1.2.18 of the Abaqus User Subroutines Reference Guide) if leakage needs to be modeled as a function of the material state in the underlying elements of the specified surface. For example, this subroutine can be used to define the leakage area at an element level for modeling fabric permeability in uncoated airbags where the leakage can vary locally depending on the strains in the yarn directions and the angle between the fabric yarns. Only membrane elements are supported for use with VUFLUIDEXCHEFFAREA.

Input File Usage: Use the following option to specify the effective leakage area directly and to specify a surface that represents the leakage area:

*FLUID EXCHANGE, EFFECTIVE AREA=effective_area, SURFACE=surface_name

Use the following option to define the effective leakage area with a user subroutine:

*FLUID EXCHANGE, EFFECTIVE AREA=USER,SURFACE=surface_name

Abaqus/CAE Usage: Interaction module: Create Interaction: Fluid exchange:

Effective exchange area: effective_area

User subroutine VUFLUIDEXCHEFFAREA is not supported in Abaqus/CAE.

Application of fluid cavity pressure on a fluid exchange surface

You can control how the effect of the cavity pressure on a fluid exchange surface is accounted for in Abaqus/Explicit. By default, the cavity pressure generates forces at all of the fluid exchange surface nodes, using the same method as for other portions of the fluid cavity. Optionally, the resultant force of the cavity pressure on the fluid exchange surface can be distributed among only the nodes that lie on the perimeter of the fluid exchange surface (for example, of the nodes shown on the fluid exchange surface in Figure 11.5.3–1, only the nodes at locations A and B lie on the perimeter). This option can be used to avoid local bulging of a vent surface that will cause inaccurate computation of the leakage area. Figure 11.5.3–2 shows an example of bulging when cavity pressure forces are distributed among all nodes of a vent surface.

text_image

Initial configuration Fluid cavity A Fluid exchange surface (vent) B

Figure 11.5.3–1 Initial configuration of a fluid exchange surface.

Deformed configuration, with bulging at vent, when cavity pressure forces are distributed among all nodes of fluid exchange surface

text_image

Fluid cavity A B

Figure 11.5.3–2 Deformed configuration of a fluid exchange surface.

Input File Usage: Use the following option (default) to indicate that the fluid pressure should generate forces on all nodes of a fluid exchange surface:

*FLUID EXCHANGE, CAVITY PRESSURE=SURFACE,SURFACE=surface_name

Use the following option to indicate that the fluid pressure should generate force only on perimeter nodes of a fluid exchange:

*FLUID EXCHANGE, CAVITY PRESSURE=PERIMETER,SURFACE=surface_name

Abaqus/CAE Usage: You cannot change the default pressure application in Abaqus/CAE. The pressure is always applied to all of the fluid exchange surface nodes.

Defining the fluid exchange property

There are several different types of fluid exchange properties available in Abaqus to define the rate flow from a fluid cavity to the environment or between two cavities. The fluid exchange property can be as simple as prescribing the mass or volume flow rate directly. More complex leakage mechanisms such as those found on automotive airbags can be modeled by defining the mass or volume leakage rate as a function of the pressure difference, \Delta p ; the absolute pressure, \tilde { p } ; and the temperature, . The heat loss due to heat transfer through the surface of the cavity can be modeled in Abaqus/Explicit by prescribing the heat energy flow rate directly or by defining the heat energy flow rate as a function of the temperature difference, \Delta \theta ; the absolute pressure, \tilde { p } ; and the temperature, . Alternatively, in Abaqus/Explicit the mass flow rate and/or heat energy flow rate can be specified in user subroutine VUFLUIDEXCH.

For the purposes of evaluating the mass flow rate between two cavities, the absolute pressure and temperature are taken from the high pressure or upstream cavity. The mass flow is always in the direction from the high pressure cavity to the low pressure or downstream cavity, and the heat energy flow is always in the direction from the high temperature cavity to the low temperature cavity. The cavity absolute pressure and temperature are always used to calculate the flow between a cavity and the environment.

You must associate the fluid exchange property with a name. This name can then be used to associate a certain property with a fluid exchange definition.

Input File Usage: Use the following options:

*FLUID EXCHANGE, NAME=fluid_exchange_name, PROPERTY=property_name *FLUID EXCHANGE PROPERTY, NAME=property_name

Abaqus/CAE Usage: Interaction module: Create Interaction Property: Fluid exchange, Name: property_name

Specifying a mass or volume flux

Fluid flux into or out of the primary fluid cavity can be defined directly by prescribing the mass flow rate per unit area, \dot { m } . The mass flow rate is

\dot {m} = \dot {\bar {m}} A,

where A is the effective area.

Fluid flux can also be defined by prescribing a volumetric flow rate per unit area, \dot { \bar { V } } . . The mass flow rate is

\dot {m} = \rho \dot {\bar {V}} A,

where is the density.

A negative value for or \dot { \bar { V } } will generate flux into the primary fluid cavity. When a second fluid cavity is not defined, the state of the fluid flowing into the primary cavity is assumed to be that of the fluid already present in the primary cavity.

Input File Usage: To prescribe a flux based on mass flow rate:

*FLUID EXCHANGE PROPERTY, TYPE=MASS FLUX

To prescribe a flux based on volumetric flow rate:

*FLUID EXCHANGE PROPERTY, TYPE=VOLUME FLUX

Abaqus/CAE Usage: Interaction module: Create Interaction Property: Fluid exchange:

Definition: Mass flux or Volume flux

Specifying the flow rate using the viscous and hydrodynamic resistance coefficients

The mass flow rate, , can be related to pressure difference by both viscous and hydrodynamic resistance coefficients such as

\Delta p A = C _ {V} \dot {m} + C _ {H} \dot {m} | \dot {m} |,

where \Delta p is the pressure difference, A is the effective area, C _ { V } is the viscous resistance coefficient, and C _ { H } is the hydrodynamic resistance coefficient. The resistance coefficients can be functions of the average absolute pressure, average temperature, and average of any user-defined field variables. A positive value of corresponds to flow out of the first cavity.

Input File Usage: *FLUID EXCHANGE PROPERTY, TYPE=BULK VISCOSITY, DEPENDENCIES=n

viscous resistance coefficient ( C _ { V } ) , , hydrodynamic resistance coefficient ( C _ { H } )

Abaqus/CAE Usage: Interaction module: Create Interaction Property: Fluid exchange: Definition: Bulk viscosity: Viscous coefficient:

C _ { V } \mathbf { \Psi } . : Hydrodynamic coefficient: C _ { H }

Use the following options to include pressure, temperature, and field variable dependence:

Toggle on Use pressure-dependent data, toggle on Use temperature-dependent data, Number of field variables: n

Specifying the flow rate through a vent or exhaust orifice

The mass flow rate through a vent or exhaust orifice that can be approximated by one-dimensional, quasisteady, and isentropic flow is given (Bird, Stewart and Lightfoot, 2002) by