design and control of thermal fluid systems: … · design and control of thermal fluid systems:...

Design and Control of Thermal Fluid Systems:

Applications and Opportunities

John A. Burns

Interdisciplinary Center for Applied Mathematics Virginia Polytechnic Institute and State University

Blacksburg, Virginia 24061-0531

Computational Methods for Control of Infinite-dimensional Systems Institute for Mathematics and its Applications

University of Minnesota Minneapolis, MN

IMA March 17, 2016

People and Goals People who do all the real work

Virginia Tech - ICAM – Jeff Borggaard, Gene Cliff, Terry Herdman, Lizette Zietsman – Weiwei Hu (IMA), Boris Kramer (MIT), C. Rautenberg

(Humboldt U.) Texas Tech

– Eugenio Aulisa, David Gilliam Carnegie Mellon - Chemical Engineering

– Larry Biegler United Technologies (UT Aerospace, Carrier, UTRC)

– Rui Huang, Clas Jacobson, Satish Narayanan, Slaven Peles (LLNL), Draguna Vrabie

Outline 1) Some history of lessons learned (and free advice) 2) My take on some “important” problems (predictions 3) Progress and challenges 4) Words of wisdom (clearly not mine

IMA March 17, 2016

Overview Talk Requirements

IMA March 17, 2016

The talk should be understandable by a general math / engineering / science audience

– Don’t need a Ph.D. in math to understand 1st sllide

Goal: Highlight potential areas for future research – What areas? – Which problems and Why?

Goal: Motivate young people to work in these areas and on important problems (prediction of “important problems” ?

The talk is NOT about “me” …

“You and Your Research”

It is about

Well … actually it is MY opinion & “predictions” about

“You and Your Research”

Free Advice

Richard Hamming, You and Your Research, New School Economic Review, Vol. 31, (2008), 5--26.

On March 7, 1986, Dr. Richard Hamming gave the talk “You and Your Research”

to Bellcore staff members and visitors at the Morris Research and Engineering Center. His talk centered on Hamming’s observations and research on the question:

IMA March 17, 2016

“Why do so few scientists make significant contributions and so many are forgotten in the long run?”

“If you do not work on important problems, it’s unlikely that you’ll do important work. It’s perfectly obvious.”

Hamming commented that:

This may sound arrogant, but Hamming went on to say that “important problem” must be phrased carefully”.

What is an Important Problem?

Hamming was talking about “… guaranteed a Nobel Prize and any sum of money you want to mention”.


IMA March 17, 2016

Hamming did not consider impact (or “consequences”) to be important.

“It’s not the consequence that makes a problem important; it is that you have a reasonable attack. That is what makes a problem important.

Using Hamming’s definition … in my early career I worked on important problems because I had a reasonable attack on very general optimization problems defined in topological vector spaces … AND

Given that mathematicians do not win Nobel Prizes (nor receive any sum of money we want) maybe we need to rethink the definition of “important problems” ?

NONE of this work had any consequences or impact on real problems

“Problems of Engineers” (Some History) “The control of distributed parameter (DP) systems represents a real challenge, both from a theoretical and a practical point of view, to the system engineer. Distributed parameter systems arise in various applications areas, such as chemical process systems, aerospace systems, magneto-hydrodynamic systems, and communication systems, just to mention a few. Thus, there is sufficient motivation for research directed toward the analysis, synthesis, and design techniques for DP systems”.

“MISSED” MEMS, Nano-Technology, Medical Sciences, Computing, Biological Sciences & Buildings

“Toward a Practical Theory for Distributed Parameter Systems”, IEEE Transactions on Automatic Control,

Michael Athans

April, 1970.

BUT

Smart Materials & Large Space Structures Fluid Flow Control & Combustion

IMA March 17, 2016

“Problems of Engineers” (Some History) “All physical systems, however, are intrinsically distributive in nature, and we should anticipate that a closer inspection of the physical laws governing system behavior would result in a mathematical model involving families of partial differential and/or difference equations. Moreover, in such diverse physical plants as distillation columns, chemical and nuclear reactors, large-scale air-conditioning systems, continuous furnaces, and compressible or elastic control actuators the spatial energy distributions preclude approximation by lumped parameter models. Thus it is not surprising that distributive problems have occupied an increasing share of the literature”.

Buildings and HVAC

Modern Foundations of Systems Engineering, W. A. Porter

Macmillan Company, 1968.

IMA March 17, 2016

I suggest that “important problems” in applied mathematics should have impact beyond mathematics … to engineering, science and / or society.

What Fields & Which Problems ?

In general I agree with Hamming … except for the narrow definition of “important problems”.

← are you serious ?

Examples Cure cancer (at least reduce the risk Save the planet Contribute to the national defense Build a better air conditioner

IMA March 17, 2016


Look for problems in … Engineering Physics & Chemistry Biology and Life Sciences Medicine

Look at industry Look for big and important problems with impact

? Modern air conditioning / refrigeration ?

In 1902, the first modern electrical air conditioning unit was invented. This invention was incredibly significant. It has and continues to save lives … more than the CT scanner Dramatically improved work effort/output in the past 100 years Rebecca Rosen, Associate Editor at The Atlantic: July 18, 2011 …

“Air conditioning hasn't just cooled our rooms -- it's changed where we live, what our houses look like, and what we do on a hot summer night”. “Many of the central changes in our society since World War II would not have been possible were air conditioning not keeping our homes and workplaces cool.

The ability to preserve food and medicine changed the world AC is needed for many medical, scientific laboratories, industrial processes and for keeping computers cool. AC is required for modern more-electric airplanes (787, F35, B-21)

Often listed as one of the top 10 inventions that changed the world forever. IMA March 17, 2016

Vapor Compression Systems

“The vapor compression systems for the Boeing 787 environmental control system (ECS) produces enough cooling to cool more than 25 typical New England homes". - Tom Pelland, VP, UTC AM Systems.

Heating, Ventilation and Air Conditioning (HVAC) & aircraft Environmental Control Systems (ECS) for More Electric Aircraft (commercial & DOD)

IMA March 17, 2016

http://abcnews.go.com/International/pilot-diverts-international-flight-save-dog-board/story?id=33807759

A Big and Important Problem with Impact How about a problem that helps

Cure cancer (at least reduce the risk Save the planet Contribute to the national defense Build a better air conditioner

MODELING, CONTROL, OPTIMIZATION & DESIGN

of THERMAL-FLUID SYSTEMS

Clearly this problem must be important and have impact ! What are the corresponding “important” technical research problems & areas Applied and & computational mathematics

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BUILD A BETTER VCS

HELPS ALL FOUR ? HOW ?

More Electric Aircraft

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Thermal Management Challenges

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Thermal Management Systems (TMS)

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Two Phase Heat Transfer

IMA March 17, 2016

“Next generation aircraft have a significant increase in thermal loads due to a transition to more electric aircraft, more powerful electronics, and the use of more composite structures. Two phase heat transfer is being used to provide thermal management for these aircraft. This can provide energy transfer that is orders of magnitude more efficient than single phase cooling thus allowing increased heat fluxes with almost isothermal surfaces at 1% of the mass flow rate. Vapor compression cooling systems utilizing two phase heat transfer are seen as essential components for the INVENT program to reach its goals.” https://applyafrlsffp.sysplus.com/SFFP/contact/sublab.aspx?sublabid=13

“INVENT was established to address these thermal management challenges in modern survivable military aircraft, from a vehicle energy perspective, through new system integration and optimization approaches. These new aircraft have three to five times the heat load of legacy platforms while being limited in their ability to reject heat to the environment.”

Building System

IMA March 17, 2016

Energy Efficiency & Buildings Buildings are responsible for more than one third of global greenhouse gas emissions, both in developed and developing countries

Buildings produce 48% of U.S. carbon emissions Buildings are responsible for more than 40 percent of global energy use

Buildings consume - 39% of total U.S. energy - 71% of U.S. electricity - 54% of U.S. natural gas

The only energy end-use sector showing growth in energy intensity

Sources: Ryan and Nicholls 2004, USGBC, USDOE 2004

Energy Intensity by Year Constructed Energy Breakdown by Sector

IMA March 17, 2016

Impact of Energy Efficient Buildings

A 10 percent reduction in buildings’ energy usage is equivalent to all renewable energy generated in the U.S. each year. A 50 percent reduction in buildings’ energy usage would be equivalent to taking every passenger vehicle and small truck in the United States off the road. A 70 percent reduction in buildings’ energy usage is equivalent to eliminating the entire energy consumption of the U.S. transportation sector.

HUGE

Need control to manage the uncertainty

Performance less important than efficiency

Control, Optimization & Design of

Whole Building Systems is key to

Achieving & Maintaining Efficiency

IMA March 17, 2016

Save the Planet & Reduce Health Risks

Exposure to UV rays has also increased the cases of cataracts which in turn affects people’s vision and could also cause an increase in people becoming blind. Depletion of ozone layer and increase in UV rays also causes DNA damage which can be catastrophic. Freon (also known as R-22) has been the refrigerant of choice for residential heat pump and air-conditioning systems for more than four decades. It has been described as one of the most significant environmental pollutants and causes of ozone depletion.

R-22 is being phased out and 2015 regulating standards will apply to all manufacturers, distributors, and consumers. No exemptions will exist.

These regulations & standards are forcing us to …

DESIGN & BUILD A BETTER AIR CONDITIONER

Thinning of ozone layer means getting direct in touch with ultra violet rays which can cause skin cancer or skin irritation which can lead to death. A decrease in 1% of ozone layer can cause 5% increase in cases of skin cancer.

IMA March 17, 2016

Save the Planet: Climate Goals

IMA March 17, 2016

To Get Just 1 of the 40 Gigaton Reduction

Install 1,000 sequestration sites like Norway’s Sleipner project (1 MtCO2/year) Only 3 sequestration projects of this scale exist today

Geologic Sequestration

Build 273 “zero-emission” 500 MW coal-fired power plants

Equivalent to about 7% of current global coal-fired capacity of 2 million MW Coal-Fired

Power Plants

Convert a barren area about 2 times the size of the UK (over 480,000 km2), using existing production technologies

Biofuels

Install about 750 GW of solar PV Roughly 125 times current global installed capacity of 6 GW*

Solar Photovoltaics

Actions that Provide One Gigaton CO2/ Year of Mitigation or Offsets Technology

Convert a barren area greater than the combined size of Germany and France (over 900,000 km2)

CO2 Storage in New Forest

Install about 270,000 1 MW wind turbines Roughly 3 times the global total installed wind capacity at end of 2007.

Wind Energy

Deploy 273 million new cars at 40 miles per gallon (mpg) instead of 20 mpg New “CAFÉ” rules would accomplish about half that

Efficiency

Build 136 new nuclear 1 GW power plants instead of coal-fired without CCS Equivalent to about one third of existing worldwide nuclear capacity of 375 GW

Nuclear

Source: Climate Change Technology Program Strategic Plan, September 2006. IMA March 17, 2016

To Get Just 1 of the 40 Gigaton Reduction

Install 1,000 sequestration sites like Norway’s Sleipner project (1 MtCO2/year) Only 3 sequestration projects of this scale exist today

Geologic Sequestration

Build 273 “zero-emission” 500 MW coal-fired power plants

Equivalent to about 7% of current global coal-fired capacity of 2 million MW Coal-Fired

Power Plants

Convert a barren area about 2 times the size of the UK (over 480,000 km2), using existing production technologies

Biofuels

Install about 750 GW of solar PV Roughly 125 times current global installed capacity of 6 GW*

Solar Photovoltaics

Actions that Provide One Gigaton CO2/ Year of Mitigation or Offsets Technology

Convert a barren area greater than the combined size of Germany and France (over 900,000 km2)

CO2 Storage in New Forest

Install about 270,000 1 MW wind turbines Roughly 3 times the global total installed wind capacity at end of 2007.

Wind Energy

Reduce GHG emissions due to buildings by 7% The building sector has the largest potential for significantly reducing greenhouse gas emissions

Efficiency

Build 136 new nuclear 1 GW power plants instead of coal-fired without CCS Equivalent to about one third of existing worldwide nuclear capacity of 375 GW

Nuclear

Source: Buildings and Climate Change - United Nations Environment Programme, 2009 IMA March 17, 2016

BIG Science with Hugh Impact MATHEMATICS OF ENERGY

EFFICIENT BUILDINGS

BUILDING SYSTEMS ARE COMPLEX, MULTI-SCALE, NON-LINEAR, UNCERTAIN, INFINITE DIMENSIONAL DYNAMICAL SYSTEMS WITH 100’S OF COMPONENTS

REQUIRES COMBINING MODELING, COMPLEX MULTI-SCALE DYNAMICS, CONTROL, OPTIMIZATION, DESIGN, SENSITIVITY AND UNCERTAINTY ANALYSIS, HPC …

THINGS THAT APPLIED & COMPUTATIONAL MATHEMATICIANS DO

” “I don’t need a mathematical model or an expensive supercomputer to tell me where to put a thermostat”.

A CHALLENGE … INTRODUCING “HIGH TECH METHODS” INTO A HISTORICALLY “LOW TECH INDUSTRY”

IMA March 17, 2016

Some “Math” - Sensor Placement

Courtesy of Bryan Eisenhower

Where is the thermostat ?

Y” “I don’t need a mathematical model or an expensive supercomputer to tell me where to put a thermostat”.

REALLY ???

IMA March 17, 2016

DO WE NEED PDE MODELS - WILL SIMPLE MODELS BE ENOUGH ?

“Simplifying Assumptions”

IMA March 17, 2016

NONLINEAR SPRING

2 2

2 ( , ) ( , ) ( , )w t s w t s w t ss s t st

ρ τ γ ∂ ∂ ∂ ∂

= + ∂ ∂ ∂ ∂∂

2 23

1 22 ( , ) ( , ) ( , ) ( , ) [ ( , )] ( ) ( )m w t l w t l w t l w t l w t l u t ts t st

τ γ α α η ∂ ∂ ∂

= − + − − + + ∂ ∂ ∂∂

Assumption (A): Consider only the linearized system

Assumption (B): No disturbance

Assumption (C): Point displacement sensor ˆ( ) ( , )y t w t s=

Beautiful mathematics (unbounded output operators)

RATINGS Math Geek Design Engr.

√ √ √

wonderful

“OK”

√ MAYBE X

who cares? I care X

even more wonderful

XXX goodbye

We just lost credibility with the design engineer

Not realistic (no physical device) Produces “silly results” like … “a fly lands on the cable and observability changes”

BUT … “engineers” are also guilty

Basic VCS

IMA March 17, 2016

FV and Moving Boundary Modeling

IMA March 17, 2016

Rasmussen and Shenoy, (2012) “Review Article: Dynamic modeling for vapor compression systems--Part II: Simulation tutorial”, HVAC&R Research, 18:5, 956-973..

Simple Reduced VCS Modeling

IMA March 17, 2016

Schurta, Hermes, Neto, “A model-driven multivariable controller for vapor compression refrigeration systems”, Journal of Refrigeration, Vol. 32 (2009) 1672-1682.

Simple Reduced VCS Modeling

IMA March 17, 2016

Schurta, Hermes, Neto, “A model-driven multivariable controller for vapor compression refrigeration systems”, Journal of Refrigeration, Vol. 32 (2009) 1672-1682.

( ) ( ) ( ), ( ) ( ) ( ) ( )x t x t u t t y t x t u tη= + + = +A B W , C D

, ,( ) [ ]Te e e o c c c ox t p L h p L h=

SIX STATES WITH TWO CONTROLS

NOT ORDE A R REDUCED MODEL!

A REDUCED MODEL

? “Simple Models” ?

IMA March 17, 2016

Qi Qi, Shiming Deng, “Multivariable control of indoor air temperature and humidity in a direct expansion (DX) air conditioning (A/C) system”, Building and Environment 44 (2009) 1659–1667.

? “Simple Models” ?

IMA March 17, 2016

Qi Qi, Shiming Deng, “Multivariable control of indoor air temperature and humidity in a direct expansion (DX) air conditioning (A/C) system”, Building and Environment 44 (2009) 1659–1667.

2 1 2 1 3 1 2 1 2 2 1 3[ ] ( ) [ ] ( ) [ ( ) ( )] ( ) [ ( ) ( )] ( ) [ ( ) / 2]a h h fg a fg wf fVC V T t V h W t C T t T t t h W t W t t A T T TVρ ρ ρ ρ α+ = − + − + − +

1 3 2 3 1 1 2 3[ ] ( ) [ ( ) ( )] ( ) [ ( ) / 2]a h a f wC V T t C T t T t t A T T TVρ ρ α= − + − +

2 11 1 2 3 2 2 1 3[ ] ( ) [ (t) ( ( ) (t)) / 2] [ ( ) ( ( ) ( )) / 2] [ ( ) ( )] ( )a w w w w r r csC V T t A T T t T A T t T t T Vt b h t h t tρ α α= − − + − − + − −

1 1( ) [2*(0.0198) /1000] ( ) 0W t T t− =

2 1 2[ ] ( ) [W ( ) ( )] ( )fV W t t W t t MVρ ρ= − +

1/( / )(1 0.015[( / ) 1])com sb V v Pc Pe β= − −

2 1 2[ ] ( ) [ ( ) ( )] ( ) ( )fa a spl f loadC V T t C T t T t t k t QV Vρ ρ= − + +

( ) ( ) ( )ri a i fg ih t C T t h W t= +

Heat Pipes

IMA March 17, 2016

Flow in a Pipe The Darcy–Weisbach equation is a phenomenological equation, which relates pressure loss (due to friction along a given length of pipe) to the average velocity of the fluid flow. The Fanning friction factor is an empirical function of the Reynolds number that allows one to compute the appropriate pressure loss at different flow conditions.

Some widely recognized issues: 1) The models are often constructed from very simplified physics & ignore axial conduction (convection dominates so why not?) 2) Current models have problems for zero or low flows 3) Many of the empirical functions are discontinuous.

v

RefF =

=

friction factorReynolds number

7Re 10 0.012fF−< =,

Smooth PipeTurbulent Flow

Re 2, 000<Laminar Flow

7Re 10 0.04fF−< =,

Rough Pipe

Basic Heat Pipe: Vapor Model

IMA March 17, 2016

Typical Counter Flow Heat Exchanger

[ ( , )][( ( , )]( , ) ( ( , ) ( , )) 0w t x V t xt x t x v t xt x R

ρρ ρ∂ ∂+ + =

∂ ∂

continuity equation

2 43[( ( , )( ( , )) ( , ) ( ( , ) / )] 2 ( , )( ( , ) ( , )) 0wt x v t x p t x v t x x t xt x v t x

t x Rρ µ τρ ∂ + − ∂ ∂∂

+ + =∂ ∂

momentum equation

W. Jerry Bowman, “Numerical Modeling of Heat-Pipe Transients”, Journal of J. Thermophysics, Vol. 5 (1991) 374-379.


IMA March 17, 2016

( ( , ) ( , )) [( ( , )( ( , )( ( , ) ( , ) / ( , )) ( ( , ) / )]

( , ) ( , )[ ( , ) ( , ) / ( , )]0w w

t x E t x t x v t x E t x p t x t x T t x xt x

t x V t x E t x p t x t xR

ρ ρ ρ κ

ρ ρ

∂ ∂ + − ∂ ∂+

∂ ∂+

+ =

energy equation

( )2( , ) ( , ) ( ( , )) / 2E t x e t x v t x= −

Re Re( , ) 2 ( , )( ( , )) /t x t x v t x Rρ µ= =

( , ) ( . x)ve t x c T t=

( , ) ( , ) ( . )p t x R t x T t xρ=

( )2( , ) (Re) ( , )( ( , )) / 2w t x fF t x v t xτ ρ= shear stress :

friction model


IMA March 17, 2016

( ( , ) ( , )) [( ( , )( ( , )( ( , ) ( , ) / ( , )) ( ( , ) / )]

( , ) ( , )[ ( , ) ( , ) / ( , )]0w w

t x E t x t x v t x E t x p t x t x T t x xt x

t x V t x E t x p t x t xR

ρ ρ ρ κ

ρ ρ

∂ ∂ + − ∂ ∂+

∂ ∂+

+ =

[ ( , )][( ( , )]( , ) ( ( , ) ( , )) 0w t x V t xt x t x v t xt x R

ρρ ρ∂ ∂+ + =

∂ ∂

( )

2

2

43[( ( , )( ( , )) ( , ) ( ( , ) / )]( ( , ) ( , ))

2 (Re) ( , )( ( , )) / 20

t x v t x p t x v t x xt x v t xt x

fF t x v t x

R

ρ µρ

ρ

∂ + − ∂ ∂∂+

∂ ∂

+ =

(Re), Re 3, 000(Re)

(Re), 3, 000 ReL

T

fFfF

fF<

= <

16(Re)ReLfF = 10

1 1.2564.0 log3.7(Re) Re (Re)

d

T TfF fF

= − +

δ

Even “Simple Math” Can Help

IMA March 17, 2016

(Re)LfF (Re)TfF

16

Re(Re)LfF =

101 1.2564.0 log

3.7(Re) Re (Re)d

T TfF fF

= − +

δ


IMA March 17, 2016

(Re)LfF (Re)TfF

IGNORANCE

Irving H. Shames, Mechanics of Fluids, McGraw-Hill, New York, 1992

101 1.2564.0 log

3.7(Re) Re (Re)d

T TfF fF

= − +

δ

NOT CORRECT IN

THIS REGION

16

Re(Re)LfF =


IMA March 17, 2016

3,000300

==

αβ

(Re) (1 (Re)) (Re) (Re) (Re)L TfF fF fF∞ = − +σ σ

FICTION … BUT SMOOTH … AND AGREES WITH KNOWN PHYSICS

NOT CORRECT IN

THIS REGION

( )/1( )

1 xxe α βσ − −=

+

Use sigmoid to smooth the fF

Heat Pipes

IMA March 17, 2016

Flow in a Pipe The Darcy–Weisbach equation is a phenomenological equation, which relates pressure loss (due to friction along a given length of pipe) to the average velocity of the fluid flow. The Fanning friction factor is an empirical function of the Reynolds number that allows one to compute the appropriate pressure loss at different flow conditions.

Some widely recognized issues: 1) The models are often constructed from very simplified physics & ignore axial conduction (convection dominates so why not?) 2) Current models have problems for zero or low flows 3) Many of the empirical functions are discontinuous.

v

RefF =

=

friction factorReynolds number

7Re 10 0.012fF−< =,

Smooth PipeTurbulent Flow

Re 2, 000<Laminar Flow

7Re 10 0.04fF−< =,

Rough Pipe

Counter Flow Heat Exchangers

IMA March 17, 2016

( )1 11 1 2 1 1 1

( , ) ( , ) ( , ) ( , ) , ( ,0) ( ) T t x T t xv T t x T t x T t g tt x

κ∂ ∂= − + − =

∂ ∂

( )2 12 2 1 2 2 2

( , ) ( , ) ( , ) ( , ) , ( , ) ( ) T t x T t xv T t x T t x T t L g tt x

κ∂ ∂= + + − =

∂ ∂

( )2

1 1 11 1 1 2 1 1 12

( , ) ( , ) ( , )( , ) ( , ) , ( , 0) ( ),

T t x T t x T t xv T t x T t x T t g t

t xxε κ

∂ ∂ ∂= − + − =

∂ ∂∂

( )2

2 1 12 2 2 1 2 2 22

( , ) ( , ) ( , )( , ) ( , ) , ( , ) ( ),

T t x T t x T t xv T t x T t x T t L g t

t xxε κ

∂ ∂ ∂= + + − =

∂ ∂∂

11

( , )0

T t Lx

ε∂

=∂

22

( , 0)0

T tx

ε∂

− =∂

1) Current models have problems for zero or low flows

Axial conduction may be “small”, but sometimes small things matter!

Axial conduction is “small” …

Full Flux Model

IMA March 17, 2016

( )2

1 1 11 1 1 2 1 1 12

( , ) ( , ) ( , )( , ) ( , ) , ( , 0) ( ),

T t x T t x T t xv T t x T t x T t g t

t xxε κ

∂ ∂ ∂= − + − =

∂ ∂∂

( )2

2 1 12 2 2 1 2 2 22

( , ) ( , ) ( , )( , ) ( , ) , ( , ) ( ),

T t x T t x T t xv T t x T t x T t L g t

t xxε κ

∂ ∂ ∂= + + − =

∂ ∂∂

11

( , )0

T t Lx

ε∂

=∂

22

( , 0)0

T tx

ε∂

− =∂

1 2ˆ0 , 1ε ε ε< < <<

1 1 2 2( ) ( ) ( ), ( ) w ( ), ( ) w ( ),a a a a aw t x t u t g t t g t t= + = =A B H H

,( ) ( ) [ ( ) ( )] ( ) ( ) ( )v z t v z t z t u tε εΣ = + + +

( )( )v

ε

Basic Idea : Approximate the selfadjoint operator using finite elements andthe non - selfadjoint operator using a upwind finite volume scheme.

P. Deuring, R. Eymard, and M. Mildner, “L2-stability independent of diffusion for a finite element - finite volume discretization of a linear convection-diffusion equation”, SIAM Journal on Numerical Analysis, 53 (2015), 508--526

,( ) ( ) [ ( ) ( ) ] ( ) ( )N N N N N Nv FE FV A Az t v z t u tε εΣ = + + +

Benefits

IMA March 17, 2016

,( )NvεΣ What we have now is a full flux model that is (i) valid for low

or zero flow and (ii) satisfies a dual convergence condition neededfor optimization & control.

.

For details see " ", J. A. Burns and B. Kramer, 2015 American Control Conference, Chicago, IL, 2015, 577 - 582

Full Flux Models For Optimization and Control of Heat Exchangers

CLOSING COMMENTS (WISDOM)

Some Wise People & Predictions

“… In brief, the flight into abstract generality must start from and return again to the concrete and specific.” Richard Courant

“A theory has only the alternative of being right or wrong. A model has a third possibility: it may be right, but irrelevant.” Manfred Eigen

“Without education, we are in a horrible and deadly danger of taking educated people seriously.” G. K. Chesterton

“Results! Why, man, I have gotten a lot of results. I know several thousand things that won't work.” Thomas Edison

“It isn't that they can't see the solution. It is that they can't see the problem.” G. K. Chesterton

Same for math results !! IMA March 17, 2016

Some Wisdom about Predictions Clarke's Three Laws are three “laws” of prediction formulated by the British science fiction writer Arthur C. Clarke. They are:

1. When a distinguished but elderly scientist states that something is possible, he is almost certainly right. When he states that something is impossible, he is very probably wrong.

2. The only way of discovering the limits of the possible is to venture a little way past them into the impossible.

3. Any sufficiently advanced technology is indistinguishable from magic.

“It's tough to make predictions, especially about the future.” Yogi Berra

“I think there is a world market for maybe five computers.” Thomas Watson, President of IBM (1943)

“Two years from now, spam will be solved.” Bill Gates, founder of Microsoft (2004 )

“Radio has no future. Heavier-than-air flying machines are impossible. X-rays will prove to be a hoax.” William Thomson, Lord Kelvin, British scientist (1899)

IMA March 17, 2016

Some Wisdom about Predictions

“It's tough to make predictions, especially about the future.” Yogi Berra

“I think there is a world market for maybe five computers.” Thomas Watson, President of IBM (1943)

“Two years from now, spam will be solved.” Bill Gates, founder of Microsoft (2004 )

“Radio has no future. Heavier-than-air flying machines are impossible. X-rays will prove to be a hoax.” William Thomson, Lord Kelvin, British scientist (1899)

IMA March 17, 2016

Clarke's Three Laws are three “laws” of prediction formulated by the British science fiction writer Arthur C. Clarke. They are:

1. When an elderly scientist states that something is possible, he is almost certainly right. When he states that something is impossible, he is very probably wrong.

2. The only way of discovering the limits of the possible is to venture a little way past them into the impossible.

3. Any sufficiently advanced technology is indistinguishable from magic.

THANK YOU

CONTACT INFORMATION

John A. Burns Interdisciplinary Center for Applied Mathematics

West Campus Drive Virginia Tech

Blacksburg, VA 24061

540 – 231 – 7667 [email protected]

IMA March 17, 2016

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