Math Encounters Blog

Quote of the Day

Keep away from people who try to belittle your ambitions. Small people always do that, but the really great people make you feel that you too, can become great.

— Mark Twain. As always, he said it just right.

Introduction

Figure 1: Relative Humidity Vs Temperature For
a Fix Specific Humidity (24 grams of Water
Vapor/kg of Air).

I often have to interpret odd test requirements. In a test specification based on GR-487, a humidity test is called out where we need to have a fixed specific humidity (i.e. 24 grams of water vapor per per kilogram of air). For a given specific humidity, the relative humidity will vary with temperature. Since my test gear can control relative humidity, I need to derive a relationship between relative humidity and the specific humidity, which I show in Figure 1.

This was a quick Friday afternoon calculation. I was a bit surprised that I could not find a table that contained this information – I pulled out Mathcad and solved the problem on my own.

Here is my Mathcad work if you are interested.

Background

GR-487 Test Requirement

Here is an excerpt from GR-487 that calls out the specific humidity requirement.

At temperatures above 32° F (90°F), the relative humidity may be limited to that corresponding to a specific humidity of 0.024 kg of water per kg of dry air. During the period of descending temperatures – i.e. 32° C (90°F) to 4.4° C (40° F) – the relative humidity shall be 80-95%.

There is a small error in this statement. Strictly speaking, the requirement is specified in terms of kg of water vapor versus kg of dry air. Technically, this term is known as the mixing fraction, but it is very close in value to the specific humidity. Both terms are described below.

Definitions

mixing fraction (symbol w) AKA Humidity Ratio (HR)

Ratio of the mass of water vapor to the mass of dry air. It is often expressed in terms of gram of water vapor per kg of dry air. Symbolically it is defined as , where m_v is the mass of water vapor, and m_d is the mass of dry air (Source).

Dew Temperature (symbol tdew)

The dew temperature is the temperature at which dew forms and is a measure of atmospheric moisture. It is the temperature to which air must be cooled at constant pressure and water content to reach saturation (Source).

Relative Humidity (symbol RH)

Relative humidity is the ratio of the partial pressure of water vapor to the equilibrium vapor pressure of water at the same temperature. Relative humidity depends on temperature and the pressure of the system of interest (Source).

Absolute Humidity (symbol AH)

Absolute humidity is the total mass of water vapor present in a given volume of air. It does not take temperature into consideration. Absolute humidity in the atmosphere ranges from near zero to roughly 30 grams per cubic meter when the air is saturated at 30 °C (Source).

Specific Humidity (symbol SH)

Specific humidity is the ratio of water vapor mass (m_v) to the air parcel's total (i.e., including dry) mass (m_a) and is sometimes referred to as the humidity ratio. Specific humidity is approximately equal to the "mixing ratio", which is defined as the ratio of the mass of water vapor in an air parcel to the mass of dry air for the same parcel. We can express the specific humidity as .

Analysis

Relative Humidity Given Dew Point and Temperature

Figure 2 shows how I computed the relative humidity given the air temperature and dew point. For references, see the documents in Appendix A or click on the links in Figure 2 (brown color).

Figure 2: Relative Humidity Versus Temperature and Dew Point.

I perform a verification of this routine in Appendix B.

Calculation of Specific Humidity (SH) Given RH and Temperature (T) and Pressure (P)

Figure 3 shows how I calculated the specific humidity given the air temperature and relative humidity given the model in this paper. Here is a numerical example that I used to check my result. Further checks are done in Appendix C.

Figure 3: Specific Humidity Function.

In Appendix D, I compare the the formula from Figure 3 to a standard psychrometric curve that relates the humidity ratio (HR) = SH/(1-SH) to relative humidity and temperature. The agreement is good.

Calculation of RH Given T and SH

In Figure 3, I have a relationship between SH vs RH, T, and P. In Figure 1, I need to invert this relationship to obtain RH vs SH, T, and P. I perform this inversion numerically (i.e. root function) in Figure 4 to generated Figure 1.

Figure 4: Mathcad Code to Plot Figure 1.

Conclusion

Our testing of an assembly was held up while we discussed the required RH required. I view this as another example of the endless number of unit conversions that I end up doing.

Appendix A: Key Reference Material

I am going to use Appendix A to store some useful reference material.

• The Use of Dew-Point Temperature in Humidity Calculations
• Humidity Conversion Calculations
• Good Lecture on the Topic
• NASA Technical Note on Humidity from Dew Point

Appendix B: Reference Material For Checking Results

Figure 5 shows a table of dew points that I found on the web. I will duplicate that table using my routine for computing relative humidity given temperature and dew point.

Figure 5: Reference of Dew points Values for Different Temperatures and Relative Humidities.

Figure 6 shows the same type of table generated by my Mathcad routine. The agreement at high temperatures (>20 °C) is excellent – less so at low temperatures. Since my GR-487 work is at high temperature, my results will be accurate.

Figure 6: My Dew Point Calculation Results.

Figure 7 shows the Mathcad code that generated Figure 6.

Figure 7: Mathcad code for Generating Reference Table.

Appendix C: More Reference Material For Checking Results

Figure 8 shows a simple comparison of a table of specific humidity values generated using my SH formula for 100% humidity at various temperatures. A comparable formula from the web is also shown.

Figure 8: Second Table Used for Checking My Model.

Figure 9 shows the Mathcad code that generated the results of Figure 8.

Figure 9: Mathcad Code for Generating Specific Humidity Example.

Appendix D: Psychrometric Curve vs Fig 3 Formula.

Figure 10 shows a comparison between the formula shown in Figure 3 and a standard psychrometric chart.

Figure 10: Psychrometric Chart vs Fig. 3 Formula.

Quote of the Day

The cave you fear to enter holds the treasure you seek.
— Joseph Campbell

Introduction

Figure 1: Dry Storage of Spent Fuel (Wikipedia).

A coworker was telling me about a relative of his who is an engineer at a nuclear power plant. One of his relative's many jobs is to babysit nuclear waste casks (Figure 1) – a task which includes monitoring their temperature. These casks are warmed from the inside by the radioactive decay of the waste they hold. As I understand it, this job has good long-term security because these casks are going to be a safety hazard for tens of thousands of years.

I thought it would be an interesting exercise to estimate the amount of spent fuel generated by nuclear plants in the US each year. My analysis is rough, but I will be able to compare my estimate with the reported waste generation rates as a check on my math.

Background

Definitions

Gigawatt (GW)

10⁹ W or 1000 Mega-Watts (MW).

tonne

I use this term for a metric ton or 1000 kg. A US ton is 2000 pounds or 907 kg. Both of these units are built-in Mathcad.

Nuclear Plant Sites in the US

The US has 99 active power reactors (Figure 2), with five units under construction and eighteen more planned.

Figure 2: Map of Nuclear Power Plants in the USA (Source).

The Energy Information Administration (EIA) publishes information on all these plants. This information, coupled with some basic physics, will allow us to estimate the amount of spent fuel generated per year in the US.

Analysis

Amount of Waste Generated Per GW Per Year

Figure 3 shows my estimate for the amount of nuclear waste generate per GW per year. This estimate of 27.9 tonne/GW-year is rough because it assumes that 100% of the fuel is burned, which is very unlikely. However, we are just looking to generate an estimate.

Figure 3: Uranium Fuel Burn Rate Per Year Per GW.

Estimate Comparison with Reality

The EIA reports that 39.7 GW-days of energy are generated per metric tonne of fuel, which is the reciprocal of what I estimated. Let's compare this value to my calculation (Figure 4).

Figure 4: Comparing My Estimate to EIA Estimate.

Annual US Spent Fuel Generation Rate

Figure 5 shows the amount of MW-hours per month of energy generated in the US. We can use this chart to estimate the amount of spent fuel generated each year.

Figure 5: US Electrical Energy Generation Per Month (Source).

Figure 6 shows my estimate for the amount of spent fuel generated annual in the US.

Figure 6: My Estimate for the US Annual Spent Fuel Generation Rate.

The EIA published annual spent fuel data, which I show in Figure 7. Note how our rate of accumulated spent fuel has linear growth – you would expect that because we have not been building new plants.

Figure 7: EIA Data on Annual Spent Fuel Generation Rate (Source).

While Figure 7 is a bit hard to read exactly, it appears to show ~2450 tonnes of spent fuel are generated annually, which agrees very well with my estimate.

Conclusion

This was a quick calculation that had surprisingly good agreement with reality. I am floored by the magnitude of the spent fuel storage problem – we are talking about  thousands of tonnes of dangerous stuff that will be around for thousands of years.

Quote of the Day

All of the great leaders have had one characteristic in common: it was the willingness to confront unequivocally the major anxiety of their people in their time. This, and not much else, is the essence of leadership.

— John Kenneth Galbraith

Introduction

Figure 1: Block Diagram of a
Combined Cycle Power Plant
(Source).

While crawling around the Energy Information Administration (EIA) web page, I found some data on the energy conversion efficiencies of power plants based on the fuel that they use. I thought the data was interesting and worth going through here.

From my standpoint, the most interesting part about this information was the efficiency of electricity production using the combined cycle power generation approach (Figure 1). I had no idea that power generation using natural gas could be ~10% more efficient than traditional coal, petroleum, and nuclear generation approaches. It looks like the higher temperature exhaust available in a natural gas system can be used to generate steam to drive a secondary generator and recover some of the energy normally lost in the exhaust. In some respects, it reminds me of a turbocharger because of the focus on using the energy present in the exhaust.

Background

Definitions

Conversion Efficiency

Energy conversion efficiency (η) is the ratio between the useful output of an energy conversion machine and the input, in energy terms.

Combined Cycle Power Generation

A combined-cycle power plant uses both a gas and a steam turbine together to produce up to 50 percent more electricity from the same fuel than a traditional simple-cycle plant. The waste heat from the gas turbine is routed to the nearby steam turbine, which generates extra power (Source).

Theoretical Conversion Efficiency

The maximum possible efficiency from a heat engine, which is given by the formula

where T_h is the absolute temperature of the hot source and T_c that of the cold sink, usually measured in Kelvin (Source).

This formula tells us that having a very hot source and a very cold sink will make a heat engine more efficient. The impact on efficiency of a high-temperature source and a low-temperature sink likely drives the efficiency improvement seen with natural gas combined cycle plants.

EIA Data

The EIA actually reports on the nominal thermal energy required (in BTU)  for every kilo-Watt (in kW) of electrical power generation by fuel type – a metric known as the operating heat rate. Observe how the natural gas operating heat rate is reducing every year. This reflects the introduction of combined cycle technology, which is more efficient.

Figure 2: EIA Efficiency Data By Year and Fuel.

I realize that "BTU/kW-hr" is a strange unit. In the US, we usually use BTU when we are talking about thermal energy and kW-hr for electrical power. This allows us to know what type of energy is being discussed just by the units being used. Unfortunately, it also means that we have a bit of unit conversion to go through in order to get the efficiency number we want.

Analysis

Unit Analysis

Figure 3 shows how to convert the EIA data into conversion efficiency.

Figure 3: Unit Conversions on EIA Data

Efficiency Calculation

Figure 4 shows how the EIA data was converted to efficiencies.

Figure 4: Efficiency Calculations.

Conclusion

I find it interesting that coal, petroleum, and nuclear have similar conversion efficiencies of ~33%. Natural gas is the outlier with 10% more efficiency than the others.

The following quote describes the rise of efficiency in electricity production using natural gas.

An aggressive buildup of high-efficiency “combined-cycle” natural gas power plants in the early 2000s began changing the game. Over the past 10 years, the average efficiency of natural gas plants has been improving continuously as more of these technologically superior systems were built and called upon to deliver power.

Although there is chatter in monthly numbers, the U.S. fleet of natural gas power plants is now achieving about 44 per cent efficiency. Rising from 32 per cent to 44 per cent is highly significant; compared to 10 years ago, an average utility today needs 27 per cent less natural gas to generate the same amount of electricity.

I can confirm the 27% reduction in fuel use for the same amount of electrical output power with the following calculation (Figure 5).

Figure 5: Reduction in Natural Gas Usage for a Given Electrical Energy Output.

Quote of the Day

Our Richie? The world’s smartest man? God help us!

— Lucille Feynman, mother of RP Feynman. This was her reaction to the statement in Omni Magazine (1979) that her son was "Smartest Man in the World." No man has a brain cell in the eyes of his wife, mother, or sister. Some things never change.

Introduction

Figure 1: Richard P. Feynman,
1965 Nobel Prize Winner in
Physics (Source).

There is almost a cottage industry in RP Feynman quotes and stories.  For an engineer, there is much gold to mined in his approach to problem-solving. Long ago, I read his book Surely You're Joking Mr. Feynman, and I have treated him as source of inspiration ever since.   I was drawn to the joy he expressed about solving problems, and his skill in sharing that joy.

As you would expect for a person of this intellect, his career is surrounded by some controversy. Here are three articles worth reading to get a feel for his legacy.

• The Jaguar and the Fox
• The Feynman Behind the Myth
• 9 Lesson Richard Feynman Taught Us About Creativity

As a mere mortal, I really cannot duplicate his problem-solving approach, but I can learn from it. I wanted to share the parts of his problem-solving approach that I work to emulate – admittedly poorly.

His Problem Solving Lessons

You need to work problems

Feynman said:

You do not know anything unless you have practiced.

I often hear people say that "I read the book, but I cannot solve any of the problem." Reading is different than understanding – understanding requires much more engagement.

You need to work lots of problems

Feynman's last writing on a blackboard said:

Know how to solve every problem that has been solved. What I cannot create, I do not understand.

This is a tough one for a mere mortal, but I work on problems all the time to help retain my proficiency.

Focus

Feynman said:

To do high, real good physics work you do need absolutely solid lengths of time, so that when you’re putting ideas together which are vague and hard to remember, it’s very much like building a house of cards and each of the cards is shaky, and if you forget one of them the whole thing collapses again. You don’t know how you got there and you have to build them up again, and if you’re interrupted and kind of forget half the idea of how the cards went together—your cards being different-type parts of the ideas, ideas of different kinds that have to go together to build up the idea—the main point is, you put the stuff together, it’s quite a tower and it’s easy [for it] to slip, it needs a lot of concentration—that is, solid time to think—and if you’ve got a job in administrating anything like that, then you don’t have the solid time.

I believe that multi-tasking is a myth. I work hard to focus my mental energy, but it is hard when you are in a management position. Sometimes it is hard to get thirty seconds of uninterrupted thought, but you have to find a way to focus.

Work to develop you own point of view

Feynman said:

Science alone of all the subjects contains within itself the lesson of the danger of belief in the infallibility of the greatest teachers in the preceding generation. Learn from science that you must doubt the experts. As a matter of fact, I can also define science another way: Science is the belief in the ignorance of experts.

and

Do not read so much, look about you and think of what you see there.

I regularly read statements by experts that later prove to be wrong. I have made some doozies. In fact, I just found an error in a published derivation performed by one of today's best electrical engineers. I contacted him, he immediately saw the error, and thanked me for helping him out. We all make mistakes.

Here is a good essay on the topic.

Always look for simpler viewpoints

Here is a quote from Feynman on his inability to explain why spin one-half particles obey Fermi Dirac statistics:

You know, I couldn't do it. I couldn't reduce it to the freshman level. That means we really don't understand it.

It is always important to simplify things as much as possible. When I worked at HP, they used to provide a bounty for bugs found just before code release. The champion bug finder used to read the source code and look for complex code sections. He said that complexity usually meant that the software engineer did not really understand what he was doing and that bugs were likely in that section of code.

I always review my solutions to see if there is an easier way to get the same or approximately same result. There is a lot of insight to be gained from reviewing your results.

I should also mention that I spend quite a bit of time on understanding the problem statement. I constantly restate the problem in alternative ways to try to gain insight into what the critical factors are.

Keep a list of problems you have not been able to solve

Feynman said:

You have to keep a dozen of your favorite problems constantly present in your mind, although by and large they will lay in a dormant state. Every time you hear or read a new trick or a new result, test it against each of your twelve problems to see whether it helps. Every once in a while there will be a hit, and people will say, 'How did he do it? He must be a genius!'

I keep of list of problems that have bothered me for a long time – decades in some case. I regularly review the list and a few drop off every year. Unfortunately, a few get added every year as well. The list never shrinks.

Give yourself a break

Feynman said:

I know how hard it is to get to really know something; how careful you have to be about checking the experiments; how easy it is to make mistakes and fool yourself. I know what it means to know something.

Getting to really know something is not easy. For example, it took me a long time to become comfortable as a circuit designer, and I continue to learn a scary amount on a regular basis.