Transcript Fluids models - ROYAL MECHANICAL

Fluid Models
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Fluids
Computational Fluid Dynamics
Liquid
Gas
Ship / submarine hydrodynamics
Hydraulic systems
Tank sloshing
Blood flow
Tides
Airframe aerodynamics
Propulsion systems
Inlets / Nozzles
Turbomachinery
Combustion
Two-phase flow
• fuel droplets
• solids particles
We will concern
ourselves with
gases – mostly air.
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Fundamentals
Continuum
Consider fluids in the sense of classical thermodynamics in which a differential element of
fluid contains a very large number of molecules. Properties of the fluid (p, T, ,…) are
represented by statistical averages rather than behavior of individual molecules.
Phases
A fluid can exist in phases of liquid or gas. A two-phase flow contains both liquid and gas.
Solid particles can be suspended in a fluid with the predominant dynamics be governed by
the fluid. WIND is only capable of simulating single-phase flow.
Equilibrium
Types of equilibrium include mechanical, thermal, phase, and chemical equilibrium.
Classical thermodynamics assumes processes that keep fluids in quasi-equilibrium.
Properties
Properties define the state of a fluid and are independent of the process used to arrive at the
state. Properties are intensive (i.e. pressure) or extensive (i.e. mass).
State Principle
For a simple, compressible system consisting of a pure substance (fluid with uniform
chemical composition) two properties are sufficient for defining the state of the system.
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Basic Thermodynamic Properties
Basic fluid properties include:
p, Static Pressure ( lbf/ft2 or N/m2 )
T, Static Temperature (oR or K )
, Static Density ( slug/ft3 or kg/m3 )
 p 
2
 
c

c, Acoustic speed ( ft/sec or m/sec )
   s
e, Specific internal energy ( lbf-ft/slug or J/kg )
h, Specific enthalpy ( lbf-ft/slug or J/kg )
h  e
cp & cv, Specific heats ( lbf-ft/slug-oR or J/kg-K )
, Ratio of specific heats
WIND: 1 slug = 32.2 lbm

cp
cv

p

 e 
cv   
 T v
h
e
p   e   1
 h 
cp   
 T  p
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Perfect Gas
A perfect gas is defined as “A gas where intermolecular forces are
negligible”. A gas that is not perfect is called a real gas. Some use the
term “real gas effects” for describing high-temperature effects of
vibration, dissociation, and chemical reactions associated with high Mach
number flows. The suggested term is “high-temperature effects”.
WIND only simulates perfect gases.
Perfect Gas Equation of State:

R
M
p   RT
, Universal gas constant
M, Molecular weight
For air, R = 1716 ft lbf / ( slug oR ) = 287 J / ( kg K )
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Mixtures of Perfect Gases
A gas may be a mixture of other gases. A mixture of several species of perfect
gases is still a perfect gas. The composition of a gas mixture is defined by the
mass fraction, ci
ci 
mi  i

m 
where mi is the mass of species i and m is the total mass of all species. One
condition for a mixture with ns species is that
ns
c
i 1
i
1
The molecular weight, M, of the mixture is determined as
1
M  ns
ci

i 1 M i

R
M
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Partial Pressures
Perfect gas equation of state applies to each species to determine the partial
pressure of species i, pi
pi  i Ri T
Dalton’s Law of Partial Pressures states that
ns
p   pi
i 1
Note also that
ns
   i
i 1
ns
R   ci R i
i 1
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Specific Enthalpy & cp
The specific enthalpy for a gas mixture is determined as
ns
h   ci hi
i 1
where for each species
T
h i  h fi 
c
pi
dT
Tref
The hfi is the specific enthalpy of formation of species i at the reference
temperature Tref. The cpi is the specific heat, which is defined using a
polynomial
c pi
Ri
K
  ak i T k
k 0
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Transport Properties
The transport properties of a fluid are coefficients in physical models
that govern the “transport” of
1) molecular kinetic energy:  , molecular viscosity
2) thermal energy:  k, thermal conductivity
3) mass:  Dij, binary diffusion
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Sutherland’s Formula
Sutherland’s formula computes the coefficient of molecular viscosity  for
an perfect gas with fixed composition as a function of temperature
T 3/ 2
  C1
 T  C2 
Where C1 and C2 are constants.
WIND uses C1 = 2.269578E-08 slug / (ft-s-oR1/2) and C2 = 216.0 oR.
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Thermal Conductivity
The definition of the Prandtl number
Pr 
 cp
k
is used to compute the coefficient of thermal conductivity, k
k
 cp
Pr
where the Prandtl number is considered a constant. Thus k is only a
WIND uses Pr = 0.72.
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Wilke’s Law
Wilke’s Law computes the transport properties  and k for a mixture
ns
 
i 1

ns
k 
i 1

xi  i
ns
j 1
x j ij
xi ki
ns
j 1
x j ij
where ij is the inter-collisional parameter
1  M i 
ij 
1

8  M j 
1/ 2
  
1   i 
   j 

1/ 2
1/ 4
Mj 


 Mi 




2
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Diffusion
The coefficient of diffusion for a binary mixture of two species i and j of
dilute gases is provided by the Chapman-Enskog formula
1
1  1  T 3 / 2
Dij  0.0018583

M i M j  d ij2  p  ij
where the effective diffusion collision integral factor is
ij 
1
TD0.145

1
TD  0.52
TD 
For the mixture of gases,
Dim 
1 X i
T
T ij
d ij 
ns
Xj
j i
ij
1
di  d j 
2
D
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Air
Air is the most common gas (default gas) used in CFD simulations with
WIND. Air is a mixture of mostly nitrogen (N2) and oxygen (O2) with other
trace gases (Argon, water vapor).
Usually assume air to have a fixed composition with constant R, , cp, and cv,
which is assuming air behaving as an calorically (ideal) perfect gas.
Simulations with hypersonic Mach numbers and higher temperatures (>1440
oR) involve changes in the energy states and composition of air.
The
relations for a mixture of perfect gases need to be used and models for the
chemical reactions need to be used to determine the thermodynamic and
transport properties.
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Air
(continued)
The behavior of air with temperature can be summarized as
T < 1440 oR. Air behaves as a calorically perfect gas.
T > 1440 oR. Vibrational energy states of molecules become significant and the
specific heats become a function of temperature. Air behaves as a thermally
perfect gas.
T > 4500 oR. Air chemically reacts to dissociate O2 to form atomic oxygen O.
T > 7200 oR. Air chemically reacts to dissociate N2 to form atomic oxygen N.
T > 16200 oR. Both N and O begin to ionize to form a plasma.
The above temperatures assume the air pressure is 1 atm. If the the air pressure
is lower, the on-set temperatures (except the 1440 oR) will be lower, and vice
versa. Once air chemically reacts, the chemical composition (species) of the air
mixture will need to be tracked and the chemical reactions modeled.
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Gas Models
The following gas models provide specific relations for the thermodynamic
properties ( p, T, , h, e, … ) and the transport properties ( , k, D ):
Constant Property Fluid
Calorically Perfect (Ideal) Gas
Thermally Perfect Gas
Equilibrium Chemistry
Finite-Rate Chemistry
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Constant Property Fluid Model
• Composition remains constant.
• Specific heats cp and cv are constant, maybe the same, c.
• Molecular viscosity  and thermal conductivity k are constant.
• Density  may be constant (incompressible).
• Used for simulating liquid flow.
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Calorically Perfect Gas Model
• Ideally Perfect (Ideal) Gas
• Composition remains constant, thus R is constant.
• Assumes the specific heats cp and cv are constant
h cp
   
e cv
cp 
R
 1
cv 
R
 1
• Given  and e,
p   e   1
p
T
R
• Molecular viscosity  is computed using Sutherland’s formula
• Thermal conductivity k computed assuming a constant Prandtl number, Pr.
This model is the default model and is probably
used 95-99% of time for WIND applications.
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Thermally Perfect Gas Model
• Vibrational energy modes of gas molecules are significant.
• Composition is frozen, and so, gas constant, R, remain constant.
• Specific heats cp and cv are functions of static temperature, T.
cp
R
K
  ak T k
cv  c p  R
k 0
T
h  href   c p T  dT
Tref
• Given  and e,
h

e
p   e   1
p
T
R
• Molecular viscosity  is computed using Sutherland’s formula
• Thermal conductivity k computed assuming a constant Prandtl number, Pr.
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Equilibrium Chemistry Model
The equilibrium chemistry model considers changes in the chemical
composition; however, the chemical reactions are assumed to happen
instantaneously such that the flow remains in local thermodynamic and
chemical equilibrium.
• Chemical composition varies, but is a function of two properties (p, T)
• Curve fits (Liu & Vinokur) are available for thermodynamic and transport
properties.
• Applicable when fluid motion time scale is much greater than chemical
reaction time scale.
• For air, chemical reactions occur above 4500 oR (2500 K).
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Finite-Rate Chemistry Model
The finite-rate chemistry model considers changes in chemical composition
when the time scales of chemical reactions are comparable or greater with the
fluid motion time scale. Requires the specification of the chemical reactions
in the model
Kf n
ai X i    bi X i 

i 1
i 1
Kb
n
ai and bi are the stoichiometric mole numbers of the reactants and products of
species i, respectively.
Xi 
 ci
Mi
K f  C T exp E / K T 
n
Kb 
Kf
Ke
Each species has a continuity equation that must be solved as part of the
system of conservation equation.
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