# Simplified multicomponent phase transition model

01 Jan 1979-

Abstract: A method is presented that will allow one to generate approximate solutions to N- component phase transition problems. There are N-1 species equations and one energy equation, coupled at the phase transition interface by a thermodynamic relationship. The equivalent Couette flow boundary layer equations are utilized to relate the mass and energy fluxes to the overall driving forces. Various simplifying assumptions are made; however, the method is sufficiently general that no simplifying assumptions need be made. A completely rigorous treatment would only introduce additional nonlinearity into the system, rather than change the method.

Topics: Phase transition (53%), Nonlinear system (53%), Couette flow (51%)

## Summary (2 min read)

Jump to: [INTRODUCTION] – [II. THEORY A.] – [TYPICAL TEMPERATURE OR MASS FRACTION PROFILE] – [The subscripts g and o refer to vapor phase and condensed phase, respectively. The condensed phase heat transfer coefficient, H , will include the condensed film, if it exists, and any other layers of material which may exist between the liquid vapor interface (i) and the condensed phase reservoir (o). If a situation involving time or position dependence or both exists, then the heat transfer coefficients H and H Q and the temperatures T^, T^ and T Q will be functions of time or position or both. B. Thermodynamic Equilibrium of the Interface] – [Hence the corrected mass transfer coefficient is] and [VV]

### INTRODUCTION

- Current practice in nuclear reactor safety analysis is to make use of large computer codes that predict the behavior of a reactor under accident conditions.
- The solutions will,in general, have two uses.
- These methods give quite good results but are computationally time consuming and require a detailed specification of the flow field and geometry.
- First, the boundary layer equations are solved by the use of heat and mass transfer coefficients which have been corrected to account for material flowing through the boundary layer.
- Second, the species and energy equations are all coupled at the phase transition interface due to the thermodynamic behavior of the materials involved.

### II. THEORY A.

- The mass flux boundary condition is found simply by evaluating Eq. (1) at the interface i.
- A direct linear mode of coupling exists through the convective term pv.
- The convective term is the net mass flux through the boundary layer, which in turn is the sum of all the individual mass fluxes.

### TYPICAL TEMPERATURE OR MASS FRACTION PROFILE

- First, the net flow through the boundary layer will have an influence upon the mass fraction gradient.
- This form of coupling is nonlinear in nature and will be discussed further in the action entitled "The Correction Factor".
- The last mode of coupling is rough the thermodynamic behavior of the materials in the condensed phase, vapor mass fraction of species K at the vapor-liquid interface is governed (5).

### The subscripts g and o refer to vapor phase and condensed phase, respectively. The condensed phase heat transfer coefficient, H , will include the condensed film, if it exists, and any other layers of material which may exist between the liquid vapor interface (i) and the condensed phase reservoir (o). If a situation involving time or position dependence or both exists, then the heat transfer coefficients H and H Q and the temperatures T^, T^ and T Q will be functions of time or position or both. B. Thermodynamic Equilibrium of the Interface

- Relationships giving the mass fraction of species K as a function of the interface temperature and component miscibility will be derived.
- For a completely miscible system each component will be assumed to exert a vapor pressure equal to its saturation pressure multiplied by its mole fraction in the condensed phase, this is sometimes called Raoult's Law.
- Evaluating the mass fractions of the various species at tine surface relies upon the assumption of constant pressure through the boundary layer normal to the interface.
- Explosive vapor formation and condensation of alkali metals at low pressure may be possible exceptions to this assumption.
- If the multicomponent mixture is completely immiscible then the boiling point is called the eutectic temperature.

### Hence the corrected mass transfer coefficient is

- Note that the mass transfer coefficient is species dependent and can be used in this manner provided that the appropriate species mass diffusivity V is defined for the multicomponent mixture (Eq. ( 18)).
- Alternatively one could use the unit Lewis number assumption, in which case the mass transfer coefficient is not species dependent and G reduces to G =.

### VV

- The energy equation also has a correction factor that can be derived in the same manner.
- Calculate all of the variables which do not change during the iteration such as the free stream mass fractions m and the eutectic temperature of the multicomponent mixture.
- A simple method of solving for the surface temperature is to start at just under the upper limit, (that is, the eutectic temperature) and increment downwards until a functional, defined in Step 6, changes sign.
- The interface temperature can be updated by defining e variable Film physics can be very important whenever condensation, or absorbtion, is occurring Films can exert control over a condensation process through either a heat or mass transfer mechanism.
- Under these conditions, the vapor mass fractions at the interface are not a simple function of temperature as they are with an immiscible system of components.

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LA-7557-MS

Informal Report

Simplified Multicomponent Phase Transition Model

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CO

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MS®

LOS ALAMOS SCIENTIFIC LABORATORY

Post Office

Box 1663 Los

Alamos.

New

Mexico 87545

DISTRIBUTION

OF

TTHS tlOCTTMTSNT

19

UNI-lMITEIi

LA-7557-MS

Informal Report

UC-79p (Base Technology)

Issued:

January

1979

Simplified Multicomponent Phase Transition Model

AhtiJ.Suo-Anttila

- NOTICE-

This repoil

was

prepired

ai an

account

of

work

sponsored

by the

Uniied States Government. Neither

the

United Slates

nor the

Unitec Stiles Department

of

Energy,

noi any nf

their employe

«, nor any of

their

'

zn

'actors, subect-'racton.

or

t.ieir employees, makes

any warranty, express

or

implied,

or

assumes

any

legal

Liability

or

responsibility

for the

accuracy, completeness

or usefulness

of any

information, apparatus, product

or

process disclosed,

or

represents that

its UK

would

no:

infringe privately owned rights.

^Lit*-??

CONTENTS

Page

ABSTRACT 1

I. INTRODUCTION 1

II.

THEORY 3

A. The Flux Boundary Conditions 3

B.

Thermodynamic Equilibrium of the Interface 6

C. The Mass Transfer Coefficient 9

D.

The Correction Factor 12

E.

Solution Techniques for Simultaneous Heat and Mass Transfer 15

III.

MODELING CONCERNS 20

IV. SUMMARY AND CONCLUSIONS 23

V. REFERENCES 23

iv

SIMPLIFIED MULTICOMPONENT PHASE TRANSITION MODEL

by

Ahti J. Suo-Anttila

ABSTRACT

A metnod is presented that will allow one to

generate approximate solutions to N- component

phase transition problems. There are N-l species

equations and one energy equation, coupled at the

phase transition interface by a thermodynamic

relationship. The equivalent "Couette flow"

boundary layer equations are utilized to relate

the mass and energy fluxes to the overall driving

forces. Various simplifying assumptions are

made; however, the method is sufficiently general

that no simplifying assumptions need be made. A

completely rigorous treatment would only intro-

duce additional nonlinearity into the system,

rather than change the method.

I. INTRODUCTION

Current practice in nuclear reactor safety analysis is to make use of

large computer codes that predict the behavior of a reactor under accident

conditions. These codes model a number of physical phenomena among which

evaporation and condensation, or simultaneous heat and mass transfer, play an

important role. The complexity and variability of accident conditions force

one to make use of a general but simple predictive mass transfer theory capa-

ble of yielding approximate solutions. The solutions will,in general, have two

uses.

First, the solutions will usually be of sufficient accuracy for engi-

neering or safety analysis purposes. Second, such solutions can serve as an

initial step in deciding whether a more refined analysis to the problem is

necessary.

The analysis of convective heat and mass transfer is most often confined

to the boundary layer. The prediction of transfer rates through t^o boundary

layer normally proceeds in two steps. First, the boundary conditions are

specified, and, second, the boundary layer equations are solved subject to

some tolerable degree of approximation.

1

The boundary conditions are relatively easy to specify when heat

transfer is the only mechanism playing a role. However, when mass transfer is

important, the boundary conditions can be quite difficult to specify,especial-

ly when multicomponent systems with simultaneous heiat transfer and possible

chemical reactions are occurring. In this case a simultaneous solution in-

volving heat transfer, thermodynamics and chemical kinetics is involved. For

this reason a major portion of the theoretical development in this report will

involve setting up the proper boundary conditions for

t*ie

simultaneous heat

and mass transfer problem associated with multicomponent systems.

The solutions to the boundary layer equations can take various levels of

approximation. First, there are various exact analytical solutions for

certain specified flow fields and geometries. The flow fields and geometries

are usually fairly specific and are not applicable to the general case. The

next degree of approximation involves the numerical methods. These methods

give quite good results but are computationally time consuming and require a

detailed specification of the flow field and geometry. In addition, when the

flow field is turbulent, useful solutions are very difficult to generate. The

last and simplest level of approximation is the development of experimental

correlations of the flow field and geometry under investigation. The corre-

lations will, when properly evaluated, yield heat and mass transfer con-

ductances (coefficients) which, when combined with the boundary conditions

(driving

force),

yield the rates of transfer. This last method is applicable

to both laminar and turbulent flow fields and gives answers usually accurate

to within 20-30%, provided the limits of the correlation have not been

exceeded.

Most experimental correlations have been developed for heat transfer,

and due to the similarity of the boundary layer equations, mass transfer coef-

ficients can be derived from the correlations. This analogy between heat and

mass transfer is not completely straightforward because in the mass transfer

case there is a net flow of material through the boundary layer. This flow of

material can be accounted for by a correction factor that forces the mass

transfer coefficient to satisfy a simplified form of the boundary layer

equations.

The model which is presented herein is a simplified phase transition

model for the following reasons. First, the boundary layer equations are

solved by the use of heat and mass transfer coefficients which have been

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