## Natural Circulation Loop with Heat Exchanger Boundary Conditions

The basic natural circulation loop case is modeled with heat source and sink. The high and low temperatures for the fluid are expressed in terms of the source and sink temperatures and the characteristics of the heat exchangers. The transient and steady state model equations are developed, the steady state solution presented, the model equations are given in dimensionless form, and the linearized versions of the equations developed.

Additional work is required to finish analyses of the stability of the system.

A file is here.

## Numbers? You want numbers . . . we’ve got numbers.

V & V? Not so much.

For decision support all you need is numbers.

## Two Nodes Coupled by a Link

**Method of Exact Solutions Verification Problems for Transient Compressible Flows: Two Nodes Coupled by a Link**

The analysis given in these previous notes:

Implicit Function Theory Introduction

Initial Method of Exact Solutions Calculations

Single Isolated Node Calculations

Two Fluid Systems Mechanically Coupled Through a Wall

are expanded to include the case of two fluid nodes coupled by a link. Numerical and analytical solutions results are given for an illustrative application.

The mathematical model, an exact continuous analogue of the discrete approximations used in many numerical solution methods, provides analytical and numerical-benchmark problems for verification by the Method of Exact Solutions ( MES ).

I have uploaded a file.

## Fluid Systems Mechanically Coupled Through a Wall

**Method of Exact Solutions Verification Problems for Transient Compressible Flows: Two Fluid Systems Mechanically Coupled Through a Wall**

The analysis given in these previous notes:

Implicit Function Theory Introduction

Initial Method of Exact Solutions Calculations

Single Isolated Node Calculations

are expanded to include the case of coupled fluid systems. Numerical solution results are given for an illustrative application.

The mathematical model, an exact continuous analogue of the discrete approximations used in many numerical solution methods, provides analytical and numerical-benchmark problems for verification by the Method of Exact Solutions ( MES ).

I have uploaded a file.

## Implicit Function Theory Applications; Part 1: Method of Exact Solutions

In a previous post I gave some background info about implicit function theory and how it might be useful. In these notes I have used results from applications to the equation of state to develop a few exact solutions for extremely simple transient, compressible flows that include fluid-structure interaction. These notes address the case of mechanical coupling of the fluid to a deformable / flexible wall. I have also included an introduction to the case of coupling of fluid systems through a common deformable / flexible wall. Additional notes will address the case of thermal interactions for both a single fluid system and coupled systems.

I kind of ran out of steam when I got to coupled-systems part of the present notes. There’s a lot of ground to cover for this case and I’m thinking a separate report might be the way to go. With coupled systems you get more that just twice as many things to look at compared to the single-system case.

I think these solutions might be candidates for analytical, and numerical-benchmark-grade, Method of Exact Solutions ( MES ) for verification of limited aspects of coding of transient compressible fluid flow model equation systems and solution methods.

I have uploaded a file.

Consider these notes as a rough draft of a report and let me know what you think about all aspects.

## Implicit Function Theory Applications; Part 0

I developed analytical solutions for a couple of simple transient compressible fluid flow problems that include fluid-structure interactions. I think the problems and solutions might be candidates for standard problems / benchmarks / Method of Exact Solutions.

Implicit function theory is an important aspect of the analytical solutions. Getting the required implicit function theory results was proving to take more pages and space that the analytical solutions. I decided to document those results in these separate notes. The results needed for the analytical solutions are given in the latter part of the notes, beginning with Variable Fluid Control Volume.

I have uploaded a Table that is in landscape and that I don’t now how to get into a portrait layout-document. That Table is here and should open in a separate window.

There are tons and tons of algebra associated with this work; straightforward but tedious algebra. I have checked and re-checked but maybe haven’t cleared out all the errors. If you plan to use any of this material, let me know what is of interest and I’ll work with you to ensure that the equations are correct.

The PDF file is here and should open in a separate window.

**Update February 26, 2011**

A corrected version of Table 3 is here. There was a bug in the first column of the third row.

**Update January 14, 2011**

There’s a typo in Eq. (1.8). The X_sub i in the last line ( the bottom part of the bottom ) should be Y_sub i. That’s a strange kind of bug; a typo in a nemonic device.

In the Section, The Bridgman Method, I say:

I have uploaded the table as a PDF file and provided this URL link in my post: https://edaniel.files.wordpress.com/2011/01/testbridgmantable.pdf . You’ll have to copy-n-paste the link into your browser.

That is the same Table mentioned in the Post, and you don’t have to copy-n-paste the URL; it’ll open from the PDF.

Note that Eq. (1.33) can be written in terms of the square of the sound speed.

And in a few places following Tables of derivatives, I said that the entropy derivatives had not been reduced when in fact they are shown reduced form in the tables.

Let me know if you find any problems.

## Looks like we’re getting some Traction

This is interesting; Computational science: …Error. From *Nature News*, even. Comments allowed over there.

## CRU Hack Info Dump

I’ve looked at some of the info in the CRU data dump and have concluded that the situation is much worse that I have previously envisioned..

All results, each and every number, to be used to set public policy must be Independently Reviewed and Verified. This is the world-wide operating standard for all other decisions, without exception, that have a potential to effect the health and safety of the public.

The climate change community remains the sole exception to this standard. The very organizations that continue to insist that it be exempt from this requirement have now been shown to have fatally flawed the very system that they propose be used.

Maybe we’ll see a Carbon Regulatory Agency developed and implemented by some progressive governments.

## Independent Verification and Peer Review

Steve McIntyre has discovered and presented yet another problem with a long-term temperature reconstruction: Yamal: A “Divergence” Problem. While Steve has named his blog ClimateAudit, in my opinion his work is in fact Independent Verification.

The old, and rapidly becoming tedious, “Peer Review Publication is Required” nonsense has already appeared in the comments. Ironically, it seems that no one who raises this issue is aware that Steve is in fact discovering problems in peer-reviewed publications.

A very short summary of a major problem area relative to this olde rant follows.