The Basic Model Equations for Flow

Assignment :

None

I'm going to introduce a large number of very complicated equations (see the PDF file). You are not expected to memorize all of the equations, but should know the basic contributing variables (e.g. mean density, mean velocity, vapor density, etc.) and recognize the equations and the physical meaning of the terms when you see them. You should become very familiar with the mass conservation equation.

• Homogeneous Equilibrium model (HEM)
• A number of options exist for choosing the primary set of unknowns to characterize the fluid state. I focus on pressure, temperature, and velocity for the liquid and gas, plus the void fraction, giving a total of seven unknown state variables at each point in space and time. I then obtain the other state variables such as density and internal energy from an equation of state. My choice of primary state variables is governed primarily by the needs for numerical solution of the flow equations, rather than by specifics of the differential equations.
• HEM uses 3 flow equations, one each for conservation of mass, energy, and momentum. It needs 4 more equations at each point in space to cover the number of unknown state variables.
• Equal pressure assumption p_l=p_g
• Equal velocity assumption v_l=v_g
• This has trouble in large pressure gradients.
• Bad assumption in vertical flows such as bubble rise in pools or droplets falling in slowly moving gas.
• Equal temperature assumption T_l=T_g=T_sat
• The equal temperature assumption is a particular problem when modeling accidents that result in voiding large regions of the reactor.
• Cold water in saturated or superheated steam is common during ECCS injection.
• Often see liquid drops at saturation temperature rising in superheated steam in the upper core during reflood.
• The Mean equations are always valid (except perhaps the friction model in the mean momentum equation)
• The equations can be derived from complex mathematics, starting with the Boltzmann equation and moving through time or volume averages (Ishii)
• Simple Thermodynamics and Mass and Energy conservation arguments get you to the same point
• Consider a volume of space V bounded by a surface S
• How do we get change of mass with time in this volume?
• The Finite Volume method makes simplifying approximations to relate the density at the surface to the volume mean density and does a finite difference approximation for the time derivative.
• Average 1-D flow equations
• Start with a general 3-D equation
• Integrate across the pipe
• Discuss laminar and turbulent flow (see the PDF file of MathCad calculations)
• The largest problem is in the momentum flux term.
• Much worse in Laminar than Turbulent flow because the turbulent velocity profile is relatively flat.
• Problems can obviously arise in an annular-drop regime where the wall liquid film may be falling while the drops are being carried up by a strong gas flow.
• The Drift flux model
• 2 mass equations, 2 energy equations, and one momentum equation
• The first assumption is that liquid and vapor velocity differ by an amount that is obtained from a correlation dependent only on local conditions.
• Liquid and gas temperatures are allowed to differ requiring separate energy equations.
• Note that equations are correct but we will later see that they can wreak havoc on numerical methods.
• The Two-Fluid Model
• Note that the equations are nearly equivalent to previous ones.
• The difference is an implied Time evolution equation for relative velocity
• This has an unfortunate side effect of creating a set of equations that are formally ill-posed
• New jargon to learn from Mathematics
• A Small (bounded) change in initial conditions can produce large (unbounded) change in answers
• In practice the finite-volume solution techniques effectively introduce terms to make the system Well-posed.
• The problem can also be fixed by various modifications, including the unequal pressure model, but no such change has been well accepted.
• Important individual terms
• interfacial drag
• friction
• interfacial heat transfer
• wall heat transfer terms
• pressure work term

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