# Exam 1 Review

 Lecture: Topic: 8/27 Introduction to the Class 8/27-29 First Exercise of our Modeling Tools 8/27-29 More Experience with SNAP 9/5 The Basics of Two-Phase Flow Acronyms Nomenclature Void Fraction Estimates 9/5 The Basic Model Equations 9/7 Intro. to Finite Volume and Finite Difference Methods Some Standard Difference Equations 9/7 Conservative Finite Volume Equations 9/10-12 Verification and Validation 9/12-14 Work on Richardson Error Analysis 9/17 Error Analysis for Difference Equations 9/19-21 Heated Flow Exercise 9-24 Solving the Difference Equations Simple Newton's Method Newton's Method for More than One Equation 9-26 Modeling Pumps 9-28 The TEE and its Relatives, Momentum Conservation

## Highlights

Introduction
• Why are computer simulations important to reactor safety
• What is the difference between Best Estimate and Conservative calculations.

First Exercises
• Estimation of reasonable transient time  to reach steady state.
• Use of Bernoulli Equation with loss coefficients
• Qualitative feeling for numerical diffusion

Basics of Two Phase Flow
• Three tasks of numerical simulation
• Any Acronyms that have been used regularly in class
• Key nomenclature (void fraction, mean density, 3 flavors of quality, mean velocity ...)
• Basic flow regime map and horizontally stratified flow.
• Ability to do simple geometric calculations to obtain expressions for void fraction or relate such things as droplet spacing to void fraction

The Base Model Equations
• Know the basic assumptions and  number and type of mass, momentum, and energy equations associated with each model
• Know what area averaging  is and what it can do to you.
• Be able to identify important individual terms in the equations: mass, energy, and momentum fluxes, pressure work, interfacial friction, wall friction, interfacial heat transfer, wall heat transfer

Finite Volume Equations
• Know how to apply Gausses theorem
• Know meaning of Fully Explicit (Forward), Fully Implicit (Backward), Semi-Implicit time level methods
• How do thiese methods affect numerical diffusion?
• What does numerical instability look like?
• For the above methods when (if at all) is stability a problem?
• Know upwind (or donor cell) differencing
• Understand a staggered mesh and where mass, energy, and momentum equations are evaluated on that mesh.
• Given a differential equation, be able to write a finite volume implementation of any of the above time level methods using a staggered mesh.
• What is a conservative form of a flow equation, and how does it impact the finite volume implementation?
• Mass concervation errors due to void fraction cut-off.
• When can the non-conservative form of the energy equation cause serious problems with results from simulations?

Verification and Validation

• Definitions: verification, validation, error, uncertainty, calibration
• Two major authors on V&V
• Sources of error
• Richardson error analysis
• Grid convergence index
• Key components of QA
• What is the Method of Manufactured solutions?
• What is the PIRT process?  What are its key components?
• Systematic (bias) and Stochastic error in experiments

Taylor Series Truncation Error analysis
• How do you choose the point to evaluate error?
• Given the difference form of a differential operator or differential equation, be able to evaluate truncation error.
• What is order of accuracy?
• What penalty do you pay for use of higher order methods?

Heated Flow
• Be able to apply the steady state form of a mixture energy equation to relate inlet and outlet conditions to power added.

Equation Solution
• Be able to do simple linearization (single Newton iteration). and obtain updated values for variables.
• Be able to apply the linearization process to more than one equation with more than one unknown.
• Understand basic properties of a Newton iteration (e.g. quadratic convergence, radius of convergence)
• How do you decide when to quit an iteration?

Pumps
• What are the primary types of pumps, and their characteristics?
• How many curves define the full range of pump behavior?
• What is the independent variable associated with each curve?
• What  quantity does each curve give you?

Tee Momentum sources and Momentum Conservation
• Be able to use simple conservation of momentum arguments to estimate pressure changes in simple situations.
• Have a common sense feel for impact of side flows on pressure in flows

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Authored and Maintained by John Mahaffy : jhm@psu.edu