Apparently, there is consternation amongst some in the climate skeptic crowd over a basic equation known as the Kaya Identity. First the equation, then some explanation, then some hilarity.

credit: Wiki commons |

**The Equation**

Kaya Identity

C = P x G/P x E/G x
C/E

Where

C = global CO2 emissions, tons/year

P = world population, billions

G = world GDP, trillions of dollars per year

E = global energy consumption, Trillion Btu/year

**Using The Equation**

The Kaya equation is used, apparently, by those who are overly-concerned that the world will soon overheat due to man's emission of carbon dioxide, CO2, into the atmosphere from burning fossil fuels.

The idea is that for a given population and GDP, energy use is required to maintain the average standard of living. Where energy use produces CO2, this results in man-made CO2 increase in the atmosphere.

The equation can be used to determine "what if" scenarios, in which reductions in CO2 can be accomplished by changing any of the variables. Reducing P, the population, would work but that is inhumane. Reducing GDP/P would work, but that requires reducing the standard of living. Reducing Energy use per unit of GDP would work, but that also reduces the standard of living. Therefore, the policy-makers focus on reducing the CO2 produced per each unit of Energy consumed. This is the basis for the drive to eliminate coal and oil in favor of nuclear power, wind, solar, and various bio-fuels such as ethanol, methane capture from landfills, and a few others.

The idea is that population will grow in the next century, perhaps to 14 or 15 billion. Even if all other factors remain unchanged, doubling the population would double the amount of man-made CO2 placed into the atmosphere. The climate change alarmists cannot have that, so they focus on the carbon intensity of energy use, the C/E term in the Kaya equation.

**The Hilarity**

A character who regularly posts essays and comments on WattsUpWithThat, Willis Eschenbach, again (how many times is it now?) shows his massive ignorance and lack of formal education with this statement on a recent post on the Kaya Identity on that blog:

*"Here’s why I laughed. Lets apply the usual rules of math to that equation. We know that if a variable occurs both on the top and bottom of a fraction, we can cancel it out. Starting from the left, Population on the top cancels Population on the bottom. Then GDP on the top cancels GDP on the bottom. Then Energy on the top cancels Energy on the bottom … and we’re left with …*

*CO2_{emissions} = CO2_{emissions}*

*Pretty profound, huh? CO2 emissions are equal to CO2 emissions. Who knew?"*

Eschenbach then goes on to dismiss the equation as useless, saying that it cannot be used to prove anything.

If the Eschenbach "discovery" were true, then engineers could not design and construct the infrastructure, process plants, power plants, and the rest of everything we build.

Hold the presses. Somebody call Fluor, and Bechtel, and KBR, and the other huge design firms and let them know that their basic engineering equations are useless. After all, the inestimable Eschenbach says so.

So, why do I get such a laugh at Eschenbach's expense?

An example of two of the most fundamental of all heat transfer equations used by both chemical engineers and mechanical engineers will illustrate. These may be found in any first-year heat transfer college textbook.

Equation 1: Q = m x Cp x DT, with units of Btu/h = lb/hr x Btu/lb/F x F

This is the fundamental equation for heat transferred when a mass flowing at m pounds per hour, having a heat capacity of Cp, changes temperature by an amount F in degrees F.

Using the Eschenbach "discovery," this equation reduces to "pretty profound, huh?"

The units on the right-hand side do indeed cancel, with lb in the numerator and lb in the denominator, also F in the numerator and F in the denominator. So, per Eschenbach's "discovery," indeed, Btu/h is equal to Btu/h. Per Eschenbach The Brilliant, this makes the equation useless. Absolutely stunning in its brilliance.

The second equation is much like the first,

Equation 2: Q = U x A x DT, where Q again is Btu/h, U is Btu/h/sq-ft/F, A is square feet, and DT is temperature change in degrees F. A similar cancellation of units can be performed, leaving Btu/h = Btu/h.

For Eschenbach's information, these equations are indeed the fundamentals of convective heat transfer. Upon them, literally millions of heat exchangers have been designed, built, placed into operation, and work quite well around the world. They have done so for many, many decades.

These are but two of the hundreds, indeed thousands, of proven engineering equations that are routinely used by competent engineers world-wide.

Thanks for the laugh, Eschenbach.

A word of advice: talk to a competent, educated engineer sometime.

Roger E. Sowell, Esq.

Marina del Rey, California

## 2 comments:

Roger, as you say, of course the UNITS on each side of the equation must be the same, and your example uses different VARIABLES, which is how useful equations are indeed written. But the Kaya Identity has the SAME variable on each side of the equation ("C"). Algebraically, the other terms just cancel out and you just get C=C.

Dr. Spencer, the Kaya Identity, and similar equations, are indeed useful even with the same variable on each side, and the other terms do cancel out. I will shortly post an article illustrating.

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