## 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.

## Coupled Natural Circulation Loops

Coupled natural circulation loops (NCLs) have not been much investigated. Single natural circulation loops, on the other hand, have been the subject of experimental, analytical and numerical research for several decades since the early 1950s. The literature is very extensive with investigations continuing to this day. Much of the research has been directed toward various systems of electric power generation by nuclear power plants.

The objectives of the present notes include; (1) development of model equations for steady-state and transient flows in coupled NCLs, (2) giving the steady state solutions for the steady state equations, (3) linearization of the transient equations for use in stability analyses, and (4) incorporation of realistic boundary-condition representations into the model equation systems for coupled NCLs.

The results are distilled to a system of equations that will be used for investigations into the stability of coupled natural circulation loops.

The design of such systems, also an interesting problem, is not addressed here.

I have uploaded a file here.

## Verify the Methods: Conservation of Water Mass

Professor Pielke Sr. has posted a comment on this paper. The complete paper is available at the URL. The citation for the paper is:

Beate G Liepert and Michael Previdi, 2012: Inter-model variability and biases of the global water cycle in CMIP3 coupled climate models, 2012: Environmental Research Letters Volume 7 Number 1 014006 doi:10.1088/1748-9326/7/1/014006

Abstract

Observed changes such as increasing global temperatures and the intensification of the global water cycle in the 20th century are robust results of coupled general circulation models (CGCMs). In spite of these successes,model-to-model variability and biases that are small in first order climate responses, however, have considerable implications for climate predictability especially when multi-model means are used. We show that most climate simulations of the 20th and 21st century A2 scenario performed with CMIP3 (Coupled Model Inter-comparison Project Phase 3) models have deficiencies in simulating the global atmospheric moisture balance. Large biases of only a few models (some biases reach the simulated global precipitation changes in the 20th and 21st centuries) affect the multi-model mean global moisture budget. An imbalanced flux of −0.14 Sv exists while the multi-model median imbalance is only −0.02 Sv. Moreover, for most models the detected imbalance changes over time. As a consequence, in 13 of the 18 CMIP3 models examined, global annual mean precipitation exceeds global evaporation, indicating that there should be a ‘leaking’ of moisture from the atmosphere whereas for the remaining five models a ‘flooding’ is implied. Nonetheless, in all models, the actual atmospheric moisture content and its variability correctly increases during the course of the 20th and 21st centuries. These discrepancies therefore imply an unphysical and hence ‘ghost’ sink/source of atmospheric moisture in the models whose atmospheres flood/leak. The ghost source/sink of moisture can also be regarded as atmospheric latent heating/cooling and hence as positive/negative perturbation of the atmospheric energy budget or non-radiative forcing in the range of −1 to +6 W m−2(median +0.1 W m−2). The inter-model variability of the global atmospheric moisture transport from oceans to land areas, which impacts the terrestrial water cycle, is also quite high and ranges from 0.26 to 1.78 Sv. In the 21st century this transport to land increases by about 5% per century with a model-to-model range from 1 to 13%. We suggest that this variability is weakly correlated to the land–sea contrast in air temperature change of these models. Spatially heterogeneous forcings such as aerosols contribute to the variability in moisture transport, at least in one model. The polewards shifts of dry zones in climate simulations of the 21st century are also assessed. It is shown that the multi-model means of the two subsets of models with negative and positive imbalances in the atmospheric moisture budget produce spatial variability in the dry zone positions similar in size to the spatial shifts expected from 21st century global warming. Thus, the selection of models also affects the multi-model mean dry zone extension. In general, we caution the use of multi-model means of E − P fields and suggest self-consistency tests for climate models.

Clearly the GCMs considered in the paper do not conserve water mass, where water means the phases of water. The ‘leaking’ and ‘flooding’ are nothing more or less than sinks and sources for water due to lack of conservation of mass for these aspects of the numerical solution methods.

Verify the methods.

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

V & V? Not so much.

For decision support all you need is numbers.

## Serial Corrections of Errors

PS. I speak as someone who has publicly got this wrong more times than most

Posted by: Gavin | September 4, 2011 1:51 PM

## 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: Single Fluid System Calculations

I have completed some additional verification of the single-node calculations, and other results, given in the previous notes here.

A short introduction to implicit function theory was given in the notes here.

In the present notes I’ve tuned up the original calculations and cleared out a few bugs as follows.

(1) I found some bugs in the summary tables of the EOS derivatives and give new Tables below in these notes. I have also posted the corrected Tables with the notes introducing implicit function theory.

(2) I have made the fluid thermodynamic state and thermophysical properties more consistent by using a single source for these numbers. The source is an old National Research Council ( Canada )/ National Bureau of Standards ( USA ) water property program. For the original calculations I had used whatever convenient values could find and these did not always correspond to the exact same fluid state. The program is based on the equations in this document.

(3) I coded the fluid properties and the primitive elements of the derivatives along with a routine to numerically evaluate the various determinants in order to compare these with my analytically derived derivatives. This process led to discovery of several coding errors that, when cleared, led to agreement with my analytical derivatives. And, yes, coding of the analytical derivatives was also verified by various means. I have abandoned the original spread sheet approach and now rely on the new coded routines.

(4) I have tuned up the wall-material properties and at the same time expanded the discussion of the interaction of the fluid and wall material relative to the base pressure-wave speed.

(5) I dug into the fluid-structure literature a little bit to see if my derivation of the effective sound speed was correct. I discovered that the formulation obtained by use of implicit function theory and the equation of state gives exactly the classical value. That equation was first obtained, as reported by Tijsseling ( 1996 ), by Korteweg in 1878. The center of the hydraulic-transient / fluid-structure-interaction universe seems to have moved from the University of Michigan and Wiggert, Wylie, and Streeter, to Tijsseling and Eindhoven University of Technology.

(6) I have investigated the numerical solution method in some more detail. The second-order explicit Euler method shows growth of the difference between the analytical and numerical solutions. The fourth-order explicit Runga-Kutta method shows much improved performance.

All these changes have been implemented and additional calculations carried out for the single-node case. The better focus on the calculations in contrast to the theory has also led to corrections in the consistency of the reported values for the calculations. The length of the node was incorrectly reported in parts of the previous notes.

The mathematical model seems to be a very good candidate for production of Method of Exact Solutions ( MES ) and Numerical Benchmark results for Verification of numerical solution methods for transient, compressible hydraulic-transient codes.

I have uploaded the file.

I see that the Table 3 did not come out too good in the file. A copy is also here. There was a bug in the first column of the third row.

## 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.