2018 Chemical PE Study Notes

I am happy to note I passed the PE exam, which I took on Pi day (3/14) this year (2018). I took some notes on the 2018 PE Chemical Practice Exam, the NCEES PE Chemical Reference Handbook (v 1.2) and on general ChE items I had forgotten, thought fundamental, or found interesting. I pasted them below, relatively unadulterated from notes I had maintained while studying, with the hope that they might be useful. Please comment or contact me (danielgputt@gmail.com) with anything I may have missed or gotten wrong. Note that I have posted no information with respect to the actual exam to avoid legal liability.


PE Study



  1. The ideal gas approximation is best at low pressures and high temperatures. Intermolecular potentials have less of an effect, and space between molecules is great.
  2. Fourier transform decomposes an arbitrary wave into a series of sines and cosines.
  3. Equations in the reference manual are often presented in rigorous and/or differential form, followed later in the text by integrated and/or special case forms. The latter are generally easier to use and should be sought.

Material and Energy Balances

  1. Full mass balance: Input + Gen. – Consumption – Output = Accumulation
  2. Getting net heating value from combustion requires using combustion heat, and then a steam heater to condense.
  3. Positive heat of reaction: products have greater enthalpy. Reaction is endothermic.
  4. Breakdown of problems in practice test (number in the practice test is noted in parenthesis):
    1. 8 total
    2. 2 combustion (1, 6)
    3. 1 solid/liquid phase diagram & Clausius/Clapeyron (2)
      1. Really the Clapeyron equation. Rearrange the equation into form y = mx to make the slope more quickly obvious. Phase change will have a positive enthalpy if solid to liquid. Delta V will determine slope. If delta V is negative, specific volume of the liquid is lower than the solid. Therefore, density of the liquid is higher, and density of the solid is lower. Last mile: careful in not fat fingering density and specific volume.
    4. 1 conversion of mole percent to mass percent (3)
      1. There is a quick formula for this on page 102 of the reference manual. Search Mole fraction.
    5. 1 basic process with recycle (4)
    6. 1 simple mass balance from a distillation tower (5)
    7. 2 reactions (know selectivity vs. yield) (7,8)


  1. Reference manual considers work positive if it is outward. Makes sense, in a sense, as systems are generally created to perform work rather than to be worked upon.
  2. Q – W = dH
  3. Work/Heat are proportional to the natural log of pressure/volume in an isothermal process.
  4. Polytropic process equations can be directly derived from the ideal gas law. See the Wikipedia article.
  5. Entropy decreases with increasing pressure and increases with increasing temperature. Since temperature increases with increasing pressure and irreversibility, entropy will tend to increase.
  6. Compressibility factor is generally less than one. This means at a given pressure, a real gas generally takes up less volume than would an ideal gas. The real gas is thus naturally “compressed” by that factor.
  7. There is a formula in the reference manual to convert from wet basis to dry basis (page 102.)
  8. Dalton’s law was empirical and published in 1801. Pressure was probably the only thing accurately measurable. Dalton’s law does not hold for some real gases.
  9. Fugacity
    1. Units of pressure
    2. When liquid and vapor fugacities are equal, equilibrium is reached
    3. The fudge factor for a pure component is based on the equation of state (phi)
    4. The rigorous version is based on Gibbs free energy
    5. The fudge factor for mixtures (in addition) is based on an activity coefficient (gamma)
    6. The vapor phase does not have a gamma correction.
    7. “fudge-gacity” may be an appropriate moniker
  10. A minimum-boiling azeotrope boils at the lowest temperature of any composition. Like a eutectic point.
  11. Problems not present on practice exam:
    1. Detailed equilibrium calculations using non-ideal fluids (e.g. cubic EOS)
    2. Thermo problems using ideal gases (e.g. isothermal expansion)
  12. Breakdown of problems in practice test:
    1. 10 total (9-18)
    2. 2 reading phase diagrams (9, 11)
    3. 3 enthalpy balances (mCp * deltaT); one with reaction (10, 12, 14)
      1. In heat of reaction problems, watch if basis is based on reaction mixture, or on one component.
    4. 1 Clausius/Clapeyron (13)
    5. One heat of reaction via heat of formation (15)
      1. This was at standard conditions, may have to calculate at non-standard. Reference book has formula.
    6. One misc. phase equilibrium (conceptual, no calcs) (16)
    7. 2 steam turbines’ energy production (17, 18)


Heat Transfer

  1. Typical approach temp, countercurrent exchanger: T_hot_out – T_cold_in. As approach temperature approaches 0, exchanger area approaches infinity.
  2. Maximum heat transfer is the hot stream mass * Cp multiplied by the delta T between the hot inlet stream (max temp of system) by the cold inlet stream (min temp of the system.) If the hot outlet stream reached the temperature of the cold inlet stream, all heat of that stream would have been transferred, leaving no dT driving force, achieved by an infinitely large exchanger.
  3. Per manual, heat capacities vary over a factor of about 40 from 0.03 to 1.2 BTU/(lb*F), centered around 0.5, by mass.
  4. Thermal conductivity varies over 5 orders of magnitude from metals to gases.
  5. Overall heat transfer coefficient for exchangers varies over 3 orders of magnitude from 2 BTU/ft^2*F to 1300 BTU/ft^2*F. Condensing oil is 40-100 BTU/ft^2*F
  6. Prandtl number is diffusion of momentum/diffusion of heat.
  7. Oxidized steel has an emissivity of 0.85! The laser on the IR gun does not measure the temperature. It is just a guide, like a scope.
  8. When faced with an integral, solve for the end bound of integral, visualize at that constant value, then backtrack. Solve the simple problem first.
  9. For simple overall heat transfer (e.g. through a plate window or cylinder) use Q = UA*deltaT. For heat exchangers, use Q = UA* deltaT_log_mean. In special cases, when log-mean leads to 0/0, use the arithmetic mean.
  10. Problems not present on practice test but possible:
    1. LMTD calculation with heat exchanger (F) correction factor.
    2. More Nusselt/h correlations.
    3. Radiation problem using T^4 relation.
    4. Thermal resistance in parallel
    5. Fin heat transfer
    6. Heat exchanger NTUs
  11. Breakdown of problems in practice test:
    1. 13 total (19-31)
    2. 5 on Overall resistance through both cylinders and walls (combined heat transfer through different materials via conduction/convection) (19, 20, 22, 27, 30)
      1. In calculating area of pipe in heat transfer, watch that the surface area of a cylinder is used (pi*d*L) not the volume (pi*r^2*L)
      2. Watch for non-traditional sources of resistance (defined fouling factors, etc.)
    3. 2 on heat exchange equipment Q = UA * LMDT (26, 28)
      1. (26) stated an azeotrope temperature at a different pressure than the problem conditions. That temperature had to be used for the problem. Azeotrope temperatures are sensitive to temperature, so this may be an error. LMTD was fairly straightforward.
      2. (28) if using the LMTD, 0/0 is obtained. Thus the equation Q = UA*DeltaT must be used. This is a common problem. Careful to use heat of vaporization when calculating steam use, not enthalpy of the vapor alone.
    4. 1 on radiation (23)
    5. 1 on correlations for convective heat transfer coefficient (h) (21)
    6. 2 conceptual (no calcs) (boiling and MW change) (24, 31)
    7. 2 Q = mCp * deltaT (25, 29)


  1. Conversion is a measure of reaction completeness. A conversion of 1 (100%) signifies a remaining concentration of the initial reactant of 0.
  2. At equilibrium, forward and reverse rates of reaction are equal
    1. Rate is a reaction constant * concentration of reactants
    2. Therefore equilibrium constant is reaction constant of forward rate / reaction constant of reverse rate.
    3. Also equal to products over reactants, but the former makes more sense.
  3. A PFR is infinitely many differential CSTRs in series.
  4. Breakdown of problems in practice test:
    1. 9 total (32-40)
    2. 3 conceptual (no calcs) (32, 36, 40)
      1. Rate law given, state adsorption pattern
      2. Minimization of Gibbs Free Energy, what product obtained
      3. Relationship between rate/activation energy/temperature
  1. 1 rate law from elementary mechanisms (33)
  2. 1 conversion of rate constant to different units (34)
    1. May have gotten somewhat lucky here. Just used unit conversions, but solution used ideal gas laws.
  3. 1 Physical Chemistry equilibrium problem, using activity coefficients (35)
  4. 2 Spacetime calculations (PFR & CSTR) (37, 39)
    1. (37) is a reaction of shifting order. The spacetime calculations are in the reference manual for these already integrated, a potential time savings.
    2. (39) is straightforward
  5. 1 CSTR rate (38)



  1. Shear stress is proportional to the velocity difference between two layers of fluid. Like dragging a board over rough concrete. A better analogy would be two alike surfaces. Viscosity is the measure of roughness.
  2. Mass balance -> continuity equation. A1 v1 = A2 v2 is the simplified form. Rigorous is rho * A * v = rho * A * v
  3. When thinking of dividing or multiplying by S.G. in conversion h = 2.31 * psi/S.G., think of how much height liquid neutron star could be pumped to. Or think (h * S.G.) / 2.31 = psi.
  4. Reynold’s number is the ratio of inertial forces to viscous forces.
  5. Viscosity of motor oil (the most dramatic of the fluids listed) varies over 2 orders of magnitude for ~300 degrees F.
  6. Roughness factors vary over 3 orders of magnitude from PVC pipe to wood.
  7. Moody friction factor varies over about 1 order of magnitude.
  8. Fitting losses vary from L/D of 10 for a wide open gate valve to 830 for a “lift” check valve.
  9. Boats float because their average density is less than that of water.
  10. Terminal velocity is the settling velocity. If upward velocity is less than the settling velocity, the particles will settle.
  11. Head loss for a porous bed goes as (1 – porosity) / porosity^3. Curve knuckles upward around 0.4. Can be used to model pressure drop in plugging fixed bed reactor.
  12. In slug flow, slugs move faster than the bulk fluid. Presumably, this could impinge upon and shock equipment.
  13. Y – expansion factor – is charted on page 218.
  14. There is a simple equation for temperature rise of liquid across a pump on page 225.
  15. An equal percentage valve is called such as it is exponential – the flow increases by an equal percentage for each incremental valve position movement. E.G. from 10% to 20% valve position, flow increases by 50%. From 20% to 30% valve position, flow also increases by 50%. When combined with a centrifugal pump, whose pressure output decreases with increasing flow, a EP valve provides a more linear response to opening. Need to consider pump curve and Cv curve together in hydraulic calculations.
  16. Orifice plate flow equations on page 257. Fairly straightforward.
  17. Good graph of permanent pressure loss across an orifice on page 259.
  18. Problem types not present but possible:
    1. Conceptual questions on types of viscous fluids (Newtonian, shear thinning/thickening, etc.)
    2. Capillary rise
    3. Conservation of momentum
    4. Pressure drop through fittings
    5. Stokes flow
  19. Breakdown of problems in practice test:
    1. 13 total (41-53)
    2. 3 conceptual (no calcs)
      1. 2 Laminar vs turbulent flow profiles, friction factor (41, 43)
      2. 1 flowmeter dP (52)
    3. 1 dP vs. flow amenable to industrial calculations (42)
    4. 2 miscellaneous, odd geometry calculation heavy (packed bed, bob viscometer) (44, 53)
    5. 1 v * A = Volumetric flow rate (45)
    6. 2 Bernoulli equations (46, 49) (watch out for velocity head)
      1. (46) is strange. Do not understand why the higher density at the outlet leads to lower hp in total.
      2. (49) Page 229 lays out pump head calculations in common units. Use this to take velocity head into account. Read the question carefully. Looks for head imparted into fluid. Efficiency of the pump does not have to be considered.
    7. 2 Darcy pressure drop (47, 50)
      1. In (50) exchanger passes are doubled (from 2 to 4.) Velocity doubles, but so does length. Doubling of length doubles the pressure drop, doubling of velocity quadruples the pressure drop. Total increase is 8x.
    8. 1 pump curve (48)
    9. 1 manometer/orifice plate (51)


Mass Transfer

  1. Sum of liquid fractions (xis) have to equal one when calculating dew pressure or dew temperature. The vapor fractions are already known. Creating a drop of liquid will not change the vapor fractions significantly. Pressure and temperature will affect the proportions of liquid, which will have to sum to equal one, by varying P and T.
  2. Conversely, in bubble point calculations, liquid fractions are known, and the composition of the first bubble has to be calculated with compositions which to sum to one.
  3. Diffusion is four orders of magnitude faster in gases than in liquids. On the order of the density difference between water (1000 kg/m^3) and atmospheric steam (0.6 kg/m^3)
  4. Diffusion is proportional to a constant and the concentration gradient.
  5. Colburn analogy: Eddies transfer heat, momentum and mass in a similar manner.
  6. Reactions can be controlled by kinetics, diffusion, or both.
  7. Start at the top right of a tower.
    1. V1 and Lo are related by mass balance. (Must be a line.)
    2. Lo and V1 mix intimately
    3. Lo becomes L1 by mass transfer. Moves to equilibrium line, with composition of V1.
    4. L1 returns to the mass balance line with crossing stream V2.
    5. Repeat as needed. Mass balance, equilibrium, mass balance, equilibrium, etc…
  8. Refluxing more for a given number of trays (operating scenario) must increase purity. Think of total reflux when the operating line is 45 degrees.
  9. Page 355 has an equation to estimate drop diameter in a dispersed phase of a liquid-liquid separator. Might be useful in Stokes’ Law calculations.
  10. Drying is faster with airflow perpendicular to the surface than with airflow parallel to the surface.
  11. Problem 62 uses a log-mean equation on page 299. The (Yin) and (Xin) values should be subscripts. As written, it appears these should be multiplied this is not correct.
  12. Breakdown of problems in practice test:
    1. 11 total (54-64)
    2. 1 Henry’s law (similar problem: Raoult’s law?) (54)
    3. 1 Saturation pressure and activity coefficient (55)
    4. 1 ternary phase diagram (56)
    5. 2 conceptual
      1. 1 dew point/composition (57)
      2. 1 tray hardware (60)
    6. 1 column bottom mass balance (58)
    7. 1 absorber traditional (59)
      1. Able to solve quickly with one key assumption: the carrier gas is not absorbed. After that, simple mass balance.
    8. 1 McCabe Thiele (61)
    9. 2 packed towers (62, 63)
    10. 1 adsorption (64)
      1. Key: adsorption is increased by increasing concentration. Le Chatlier’s principle.

Plant Design and Operation

  1. Breakdown of problems in practice test:
    1. 16 total (65-80)
    2. 1 financial (NPV) (65)
    3. 1 P&ID (66)
    4. 2 distillation high level (could be classified as mass transfer) (67, 68)
    5. 3 safety (RV, confined space, reactor) (69, 70)
    6. 1 instrumentation (bubble tube, ideal gas law) (71)
    7. 1 Process control (what variable to manipulate) (72)
    8. 2 inspection/corrosion/metallurgy (73,74)
    9. 1 pump curve (could be in fluids) (75)
    10. 1 misc. re: optimal changeout frequency of filters (required integrating exp. Fcn.) (76)
      1. Calculator can perform integrals. Integral of exp(ax) is number 22 on page 57 of the manual.
    11. 1 tank dike sizing (77)
    12. 1 volitalization of substance into a room (78)
    13. 1 Environmental regulation (80)



Mom, The Molecules She Set in Motion

This is the first Mother’s day I’ll spend without my mom. I’m comforted by the fact that the molecules she set in motion—nitrogen, oxygen, water vapor and argon—tumble into one another as she left them.

There are enough of them that by chance they move in metaphorical ways: physics recapitulating biography.

One of them traces the foot path she took as a toddler chasing a cat.
One holds still, the way she sat with her coffee and the morning sun.
One moves clumsily, the way she bumped into corners with her walker.

I dream that one of those molecules catches the wind, down to a cool night, where I watch a steam plume from a Texas refinery. It floats up, folding over on itself, visible until it flits from its companions. It may give life or stay inert. It may shine or stay in darkness. It will return to the randomness from which it came, where I will someday go as well.



Mountain Streams and Calculation Daydreams

Who among us hasn’t watched a flowing mountain stream, and wondered at its flow rate? And who hasn’t wondered how that flow rate compares to the national consumption of say, milk, or gasoline? Indeed mankind has asked these questions since the time of the ancients.

Here is a picture of a stream I was fortunate enough to visit, not far from Yosemite National Park:photo 5How can we measure this stream’s flow rate? There are many ways. There are a number of devices used to measure flow. Some meters work on Bernoulli’s principle, working on pressure drops across orifices or venturis. Some use the number of rotations of a set of paddles, like a wind meter. Of course, we could lace the stream with radioactive isotopes and buy a suitable detector, or just divert the whole lot into a large tank for some time interval.

Isotopes are increasingly hard to find for the everyday consumer, as are giant tanks and earth-moving equipment. Fortunately another option exists: a volumetric flow rate can be calculated as the product of a flowing stream’s cross-sectional area and its velocity.

Q = Av

This makes sense from an intuitive point of view. Think about a large stream, like the Mississippi. It has a large cross-sectional area. Imagine pinching it down, until it became whitewater rapids. The volumetric flow rate would stay the same, with the loss in area being made up for by the increase in velocity.  And the units check out (ft² * ft/s = ft³/s).

I set out to measure the stream’s cross-sectional area first. If the stream happened to have a simple geometry, like that of a rectangular drainage ditch, this would be a simple matter. We could measure width and depth, and multiply. However, most natural streams do not have simple geometries. We can use a bit of numerical integration to estimate area, however. That’s a fancy term for chunking our stream into a series of rectangles (or other, more intricate shapes, if we want greater accuracy) and adding the area of each individually.

To do this, we start by measuring depth at regular intervals. I did, and here’s what this stream looks like:

Water Depth

I was lucky in that I had a little foot bridge over the stream, and could use the boards to get a regular width at which to measure depth. What’s going on at measurement point 4, you ask? That’s a rock.

This is the measuring instrument used. The units are in feet.

measuring stick

photo 1 (1)

Now I had to measure the width. Fortunately the boards of the bridge looked to be spaced fairly regularly:

Displaying photo 4.JPG

Of course it was good practice to make sure. I measured four boards to get a representative sample, and they were all within 3/8 of an inch of one another. The average came out to be 0.47 ft (.14 m).

Multiplying this width by each of the individual depth measurements, and adding the results together, we get the area:

A = 5.31 ft² (0.493 m²)

Now to measure velocity. This is a bit more involved. I found a method for doing so on a US Environmental Protection Agency site. To use this method, you measure out a length of stream and throw in a floating object (not a human or animal that can’t swim. That’d be breaking the law). Then, use a stopwatch to record the time it takes to get from from beginning to end. I was lucky enough to have a load of pine cones lying around, and grabbed 15 to get a good estimate.

Displaying photo 4.JPGA stream will generally flow more quickly at its center than at its edges. I tried to introduce the pine cones at different points in the width of the stream to get a representative sample. Here are the data I got:

Pine Cone Transit

Note the missing points at runs four and fifteen. The pine cones got caught in eddies in those cases, greatly extending transit times. I decided to throw these data out, though I’m not sure how intellectually honest this was. Surely if the pine cone is held up, some of the water is held up as well. And if the water is held up, it’s not flowing forward, affecting flow rate. I’d be interested to hear how this is dealt with in practice, but for the time being, I punted.

Taking an average of the transit times, 4.54 seconds, and dividing it into the run length, 11.5 feet, I got the average velocity:

v = 2.54 ft/s (0.77 m/s)

We can now solve using the formula first outlined:

Q = Av
= 5.31 ft² * 2.54 ft/s
= 13.5 ft³/s
Q = 108 gallons per minute (409 l/min)

This is what the 108 gpm stream looks like:

So, answering the questions of our ancestors:

Americans consume 368.51 million gallons of gasoline per day (EIA).  That’s 2370 of these streams.

How about milk? American production was 201 billion pounds in 2013. That’s 425 of these streams.

Lunch to whoever finds the product produced at a rate closest to this stream! Cheers!


The irony of a fouled catalyst

Merriam Webster’s defines irony as an ‘incongruity between the actual result of a sequence of events and the normal or expected result.’ In their excellent book, A Working Guide To Process Equipment, Norman and Liz Lieberman write about the end of run condition of a chemical reactor, giving an example of an ironic process.

The reaction in Lieberman’s example mixes diesel fuel and hydrogen together as reactants. This occurs at a relatively high temperature over a catalyst. The products yielded are low-sulfur diesel and hydrogen sulfide. The catalyst increases the rate of reaction, and the reaction rate increases with increasing temperature. The reaction takes place on the surface of the catalyst, so it is advantageous for the catalyst to have a large surface area.

After the reaction is run for some time, the catalyst begins to foul. That is, coke, a hard black substance that is mostly carbon, begins to deposit on the catalyst. This reduces the available surface area of the catalyst, and the reaction proceeds at a slower rate.

When the operator of the process becomes aware of the slower reaction rate, she increases the reaction temperature. When the temperature is increased, the reaction rate increases, and all is well. However, with this increase in temperature, the fouling of the catalyst by deposited coke also increases. When the catalyst is further fouled, the reaction rate slows. It becomes necessary to increase the temperature further to raise the reaction rate to set point. This fouls the catalyst…and on and on. Finally the catalyst becomes completely fouled, and the end of run is reached.

In the case of the fouled catalyst, the irony is that increasing the temperature of the reaction is needed to increase reaction rate, but it is in fact that higher temperature that fouls the catalyst, hastening the end of the process’s effectiveness. The cure to the disease hastens the disease itself. 

There are certainly processes outside of chemical engineering that show a similar tendency toward irony:

  • Antibiotic use: Antibiotics admirably combat infection in individuals on short time scales, while creating resistant bacteria in the long time scales. Ironically, by curing individuals now, we risk being unable to cure many more later. 

  • Struggling cities: As budgets in a struggling city are squeezed, taxes are raised to increase revenue. Rates of revenue collection show an initial increase. However higher taxes drive residents out, ultimately decreasing revenue on large time scales. Ironically, the steps taken to balance the budget in the short-term end up causing further shortfalls later. 

  • Addiction: An addict takes an amount of drug that interacts with neuroreceptors, leading to a release of psychological and physical unease short-term. The body reacts by creating more receptors, creating more unease in the long-term. Ironically, the more drug the addict takes, the more he or she desires. 

  • Arms races: One short-term, rational decision for any actor faced with an enemy with greater arms than itself is to become more armed in turn. The opposite actor then will become more armed in response. In the long-term, more destruction is often the result. Ironically, the rational choice for individual actors is often detrimental to the whole.

Each of these ironic processes rests on what is termed a positive-feedback loop. As the variable in question – be it temperature, antibiotics, revenues, drugs or arms – is increased, the process responds by demanding a further increase in turn. You can see why these types of processes are often called ‘death spirals,’ with death being, in each case respectively, a slow reaction, incurable infection, an insolvent city, an insatiable craving, and destruction of a population.

Crucially, ironic processes are ironic in that the increase of the process variable,while hastening the approach to ‘death’ in the long-term like any positive-feedback process, gives a temporary reprieve from the ‘death’ in the short-term. The short-term ‘cure’ accelerates the long-term ‘disease.’ From the definition of irony, the long term decay is contrary to the expectations of relief.


Process Gain and Conspiracy Theories



Recently I’ve been studying process control, one of the last classes in the chemical engineering curriculum. In process control there is a useful and simple concept called process gain. Process gain is defined as the ratio of a process output to a process input, or mathematically:

Process Gain = Output / Input

It’s easy to imagine an example of process gain around the house. Increasing a knob associated with a burner on the stove top from 0 to 10 could increase the temperature of water in a pot on that burner from 70 degrees to 212 degrees F. The process gain would then be:

Gain = (212 – 70) degrees F / (10 – 0) knob units

= 15.2 degrees / knob unit.

A more interesting example applies the idea of process gain outside of the engineering realm. Think about the famous rhyme:

For Want of a Nail

For want of a nail the shoe was lost.
For want of a shoe the horse was lost.
For want of a horse the rider was lost.
For want of a rider the message was lost.
For want of a message the battle was lost.
For want of a battle the kingdom was lost.
And all for the want of a horseshoe nail.

For the lowly input of a nail, the output was the loss of a kingdom, or:

Gain = – kingdom / – nail. 

I’ll leave it to the diligent reader to do the unit conversion.

Now to conspiracy theories. It has been said that many people are apt to believe conspiracy theories because it is difficult for them to accept the idea that events so consequential can be perpetrated by people whom are not so. Using the idea of process gain, we can communicate this in a different manner. People expect outputs to be commensurate with inputs, or that process gain should be somehow reasonable. 

In other words, if the numerator of the gain equation is the kingdom, the denominator should be something near in value to a kingdom. A numerator as world-altering as 9-11 should have been perpetrated by a denominator with similar weight, like a superpower government. An assassination of someone as powerful as a president should have been committed by a group with similar power, like the mob. Process gain should be on the order of 1, not on the order of 1 billion.

On WDET Detroit today, Craig Fahle featured author Jeff Greenfield talking about his book If Kennedy LivedThe author speculated on that counterfactual: perhaps the conflict in Vietnam would have been wound down, perhaps Lyndon Johnson would not have been able to pass through great-society reforms. He goes on: just before Kennedy was to drive that fateful route in Dallas, it had been raining hard enough that the top needed to be down on his convertible. It of course cleared up slightly thereafter. Who knows how large of a process gain has been obtained by a flicker of a rain cloud with respect to  the output of history?