Summary
Hydrogen economy, where Hydrogen as a clean fuel replaces gasoline as the primary energy source, especially for powering cars and trucks, is a frequently discussed topic in conversations addressing a zero-emission carbon-free future. In this note, I examine the likelihood of that, using basic physical principles and deriving approximate, easy to remember numbers for the key parameters. My conclusion is that EVs operating on batteries are the likely means to move us around in the clean economy, relegating Hydrogen in transportation to niche applications.
Basic numbers
Avogadro number = 6.02E+23 molecules/gram-mole
Molar volume = 22.4 liters/mole of an ideal gas at 273.15 K temperature and 1 atm. Pressure
1 J = 6.24E+18 eV
1 kWh = 3.6E+6 J = 2.25E+25 eV
H2: 1 mole = 2 gr 1 kg = 500 moles = 3.01E+26 molecules/kg
H atom ionization energy = 13.6 eV
H-H bond energy 4.52 eV = Energy(H-H)
O=O bond energy 5.15 eV = Energy(O=O)
H2O bond energy 9.62 eV = Energy(H2O)
Energy(H2O) - Energy(H-H) -(½)Energy(O=O) = 2.5 eV
[H2O bonding: one H-HO and one H-O bond, two Hydrogen bonds per water molecule, and each H bond has an average bonding energy of 464 kJ/mole
= (2 * 464E+3(J/mole) * 6.24E+18(eV/J))/((6.02E+23(molecules/mole) = 9.62 eV/(molecule of H2O)]
To break a bond, energy needs to be supplied. Thus bond energy is negative energy. Combining an H2 molecule with one half of an O2 molecule produces water (H2O) and the H2O bond energy is lower than is the total bond energy of the combining molecules (see above) by 2.5 eV/molecule of H2O, and energy is released in the process (exothermic process).
The energy needed for electrolysis of H2O to produce H2: theoretical ideal value
= 32.9 kWh/(kg of H2) = 2.1 eV/molecule of H2,
practical (approximate actual) value = ~55 kWh/(kg of H2) = 4.1 eV/molecule of H2.
[The theoretical electricity values for electrolysis input and fuel cell output are equal, 2.1 eV/ molecule of H2 see: http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/electrol.html#:~:text=A%20fuel%20cell%20uses%20a,of%20hydrogen%20and%20oxygen%20gas.&text=Combining%20a%20mole%20of%20hydrogen,produces%20a%20mole%20of%20water]. In addition to breaking and forming molecular bonds in electrolysis and fuel cells, entropy exchange with the environment and the thermodynamics of the gases enters the energy equation.
Hydrogen cycle: water production, and electrolysis
Pure Hydrogen (H2) is most commonly produced by electrolysis of water (H2O), separating Hydrogen and Oxygen at the two electrodes, water and electricity are the inputs. Correspondingly, in the reverse process in a fuel cell, where hydrogen and oxygen combine, electricity and water are produced. Thus, in H2O electrolysis energy is consumed, and in the H2 fuel cell energy is produced. A single water molecule contains less energy, it is in a lower energy state, than the hydrogen molecule (H-H) plus one-half of an oxygen molecule ½(O=O) energy state was before recombination.
Theoretically, the electricity input to electrolyze water and produce a kg of H2 is equal to the electricity output of H2 kWh/kg from the fuel cell where oxygen is combined with Hydrogen and water is produced. In practice, it takes about 55 kWh/kg = 4.5 eV/ molecule of H2 to produce H2 and in a fuel cell, about 20 kWh/kg = 1.64 eV/molecule of H2 of electricity is recovered from H2.
Thus the Hydrogen cycle efficiency through a Hydrogen fuel cell is ~36%. Assuming a PV efficiency of 20%, the sunlight-to-electricity efficiency of the HFC (Hydrogen Fuel Cell) system is ~7.2%.
Transportation energy needs
The principal known sources of clean electricity in the 21st century are solar PV, wind, hydroelectric, and nuclear electrical-power generators. A clean economy will be an electric economy, fueled by clean electricity, where all energy needs must be supported and fulfilled by clean electricity. For stationary loads, like homes and factories, fulfilling this requirement is conceptually not too difficult. This, however, will not be that easily accomplished in the transportation sector of the economy.
In the USA 28% of the total energy usage is devoted to transportation, most of it is for ground transport. A small fraction of the ground transport is along fixed routes, like trains, trams, and trolleys. Such fixed routes can be easily supported by wired electricity delivery. But most cars and trucks require on-board transportable energy sources for their functioning.
Currently, most not-fixed-route transport is energized by oil-based products, gasoline, and other crude oil derivatives. In oil-based fuels, we have approximately 10 kilowatt-hours per kilogram or liter. (Actually, the energy content is 8.9 kWh/ litter and 12.5 kWh/kg, but 10 kWh is an easily remembered approximation in either case.)
Fuel-efficient gasoline-powered cars currently can deliver about 20 km per liter ( about 50 miles/gallon) of gasoline or about 0.5 kWh/km. Large trucks (having about 15,000 kg of gross weight) use about 4 kWh/km.
A Tesla Model3 electric car uses about 0.17 kWh/km.
The difference between the energy consumption of gasoline-powered cars and electrically driven Teslas is because the energy conversion efficiency from the chemical energy stored in the gasoline to mechanical energy moving the car is about three times lower than is the conversion from the electrical energy stored in the battery to the mechanical energy.
Most cars and trucks are used for local, in city and vicinity, as well as intercity long-distance transportation. Therefore, they must be designed for a range capability of approximately 500 kilometers. (More would be highly desirable, significantly less would limit the available markets.)
Thus the on-board transportable “starting” energy requirement (energy required at the beginning of a trip) is: for cars 85 kWh. Rounded reference figure estimates for portable and on-route replenishable (refillable) electrical energy storage requirements (kWh electricity available for use by an electric motor) for a 500 km range are shown here below.
For trucks 800 kWh. For cars 100 kWh.
There are two known options for transportable electrical energy sources: Li batteries and Hydrogen fuel cells. Li batteries for EVs (Electric Vehicles) are well developed and widely available. The cost of Li batteries is about $150/kWh in 2020, is expected to decline to $100/kWh in 2024 and $60/kWh in 2030. Thus the battery cost of a car is about $15,000 in 2020 and is expected to decline to $6,000 by 2030. A cursory review of data available on the internet suggests that car engines and electric motors of the same horsepower rating cost about the same ($2-5 thousand). Thus the cost of the battery will be additive, leading to higher prices for EVs than that of comparable performance ICE (Internal Combustion Engine) driven vehicles.
Operating costs of EVs are already lower than those of ICE vehicles. Anticipated carbon taxes are likely to further increase the operating costs of ICE engined cars and trucks, while the well-known learning curve phenomenon is likely to drive down the manufacturing cost of EVs. In any case, in a clean economy with currently known technologies, ICE vehicles are no longer an option. We need to switch to EVs, the question is how? Will batteries prevail, or is the Hydrogen fuel cell a viable option? The answer to this question will most likely depend, at least in part, on the cost and ease of the distribution and transport of Hydrogen. In principle, Hydrogen could be stored and distributed in liquid form at low temperature, or in some yet to be invented chemical storage form, or as a pressurized gas. Currently, the only known practical means for onboard storage of Hydrogen to power HFC EVs is pressurized gas.
Transporting Hydrogen
I am interested in making some quantitative estimates regarding the transportation of Hydrogen gas. Specifically, I am most interested in transporting Hydrogen to be used in fuel cells to drive passenger cars. As indicated above, for this purpose we wish to transport an amount of Hydrogen sufficient for producing 100 kWh of electrical energy. Furthermore, as estimated above, molecular Hydrogen (H2) can produce in fuel cells approximately 20 kWh/kg of H2, thus we need to transport in a car about 5 kg of H2 to have a reasonable range of 500 km.
As a useful and simple approximation, assume a spherical pressure tank for the transport of H2. The simple estimate shown below only considers tension stress in the axial direction of the spherical pressure tank. More elaborate calculations indicate the need to consider the multi-dimensional reality of the stresses, involving both tensile and shear stresses. Such more accurate calculations indicate that for thin-walled vessels, for which the radius is approximately 10 times larger than the thickness of the vessel wall, the simple calculation outlined below provides reasonable estimates for the basic properties of spherical containers
Denote: t = container wall thickness, R = radius of spherical container,
F = force pulling apart two halves of the sphere at any equatorial plane,
S = tensile stress in wall material,
Y = max allowed container wall stress for wall material used,
= ~10^9 N/m^2 AHSS steel,
D = density of wall material = ~10^4 kg/m^3 for steel,
P = transported H2 gas pressure,
Po = H2 pressure at STP ( 298 degK and 1 atm) = 1 atm,
V = volume of spherical container, Vo = volume at STP of H2 to be transported,
Vs = the skin volume of the spherical container (surface area times thickness)
D = density of wall material used,
W = weight of the spherical container
N = total number of H2 molecules in container,
T = transport temperature (same as in STP)
kT = kT (energy) = 4.11E-21 J = 45.7E-3 eV at STP (298 degK and 1 atm)
Then: F = PπR^2, S = F/(2πRt) = PR/(2t),
and using the max allowed tension the required wall thickness is t = RP/ (2Y).
At STP (Standard Temperature and Pressure), H2 is about 20 liters/(2 gr) = 10^4 liters/kg = 10 m^3/kg,
Po = 1 atm = 1E+5 N/m^2,
and then Po*Vo= 5E+6 Nm and the STP volume of the needed 5 kg of H2 is Vo = 50 m^3,
Assuming a large container with 1 m radius, then V = (4π/3)*R^3 m^3 = ~ 4 m^3,
Vo/V = 50/4 =~10 and also Vo/V = P/Po at constant temperature T,
P = Po*Vo/V = (1E+5)*10 = 1E+6 N/m^2; t = 1*1E+6/(2E+9) = 0.5E-3 meter.
The foregoing estimate indicates that it is possible to store enough H2 fuel in a spherical pressure tank of 1-meter radius to support a Hydrogen fuel-cell-powered car for a reasonable range. Larger spherical vessels would have thinner walls and lower pressures. As the following analysis shows it is not possible to reduce the weight of the spherical tank by increasing the radius and thus reducing the pressure.
The pressure in the pressurized container is P = Po*Vo/V = PoVo/((4π/3)R^3).
The skin (or wall) volume Vs of the spherical container with radius R is
Vs = t*4πR^2 = (RP/2Y)*4πR^2 = (R*(PoVo/((4π/3)*(R^3)*2Y)
= (3/2)*PoVo/Y
Note that for ideal gases PoVo = PV = NkT and then Vs = (3/2)* (NkT)/Y; thus for an ideal gas, the volume of the wall of a spherical container is equal to the translational kinetic energy of the molecules of the contained gas (Joules = Nm) divided by the allowable yield stress in the wall (N/m^2). Furthermore, note that if the container is divided into several smaller spherical parts, the total Vs is determined by the total amount of gas contained; thus making several smaller tanks does not change the total tank wall volume (or weight), if using the same material for the single large, or multiple smaller parts versions of the tank).
To store 5 kg of H2, the container skin volume is
Vs = 3/2 * 5E+6 / (1E+9) = 7.5E-4 m^3 = ~1E-2 m^3 = 10 liters
And the weight of the skin of the spherical container is
Ws = D*Vs = (1E+4 kg/ m^3) * 1E-2 = 100 kg container weight.
We need a 100 kg steel container to store 5 kg of H2 at STP temperature.
With a 1 m radius sphere, the wall thickness calculated above was 0.5 mm. If we reduced the radius fourfold to t = 0.25 m, the shell area is reduced 16 fold and the thickness will be 8 mm = ~1 cm.
In the 1 m radius sphere, the pressure was 10 atm; in the 0.25 m radius sphere, it would be 160 atm. To gain a margin of safety, the wall thickness could be doubled, reducing the tension two-fold and increasing the container weight from 100 to 200 kg for cars. For trucks, our base number for the H2 container weight is 800 kg and the double wall-thickness safer solution weighs 1,600 kg. We conclude that sufficient H2 could be a portable energy source for powering HFC-EV cars and trucks for long-distance travel. The weight of a car is about 2,000 kg; thus the above estimated H2 container weights would represent 5-10 % of the total weight; likewise, a similar fraction of the total truck weight would be the H2 storage system.
But why should we want to use EVs with energy supplied from HFC instead of Li batteries? From an electricity source-to-delivery efficiency perspective batteries are the clean winners (over 90 % versus less than 40 %). Also, HFC provides no clear cost advantage over Li batteries to power EVs. The only obvious shortcoming of battery energized EVs is the on-route charging time. Tesla states that most Superchargers deliver 75 kW charging rates; thus to fill a nearly empty battery in a car would take over an hour and several hours in a truck. For intra-city and suburban use EVs this can be easily accomplished with night-time charging, but for inter-state long-distance travel this is a major inconvenience; people are used to filling up their tanks at gas stations in minutes. Tesla announced in 2019 that new V3 Superchargers capable of 250 kW charging rates will be soon available. While this is indeed much faster than are the current 75 kW superchargers, it still implies close to half-hour charging of nearly empty-battery cars.
In principle, HFC EVs could be “gas-charged” at rates similar to those of ICE vehicle “fuel-charging”. But how to get the H2 gas to the charging station? Maybe do local electrolysis at the station with electricity supplied by the grid or by local storage supported PV? TBD. The answer is not clear and not known.
Currently, and for the foreseeable future, EVs with battery-supplied electricity are the clear winners over HFC EVs. I expect HFC EVs will serve niche markets, while battery-equipped EVs will dominate both the main car and truck markets, replacing ICE-powered vehicles by the middle of the 21st century clean economies.
Note added in proof
The Toyota Mirai serves as an available reference to check the parameters obtained in the foregoing approximations. (https://en.wikipedia.org/wiki/Toyota_Mirai accessed Dec. 23, 2020.) The 2016 model of that vehicle (First generation JPD 10; 2014), according to US EPA, has a range of 502 km on the full capacity of two tanks; each tank is “a three-layer structure made of carbon fiber reinforced plastic”; the total weight of the two tanks is 87.5 kg and their combined capacity is 5 kg of Hydrogen.
My approximation suggested 500 km on 5 kg of Hydrogen and a spherical high-tensile-strength steel tank of a weight of 100 kg. Also, my simple analysis of spherical tanks showed that for a selected container material and a given amount of gas to be stored, with properly optimized containers it does not matter whether all the gas is in one large or one small container, or in one large container or it is distributed into several smaller containers. The total weight of the required container(s) does not change. It is determined by the strength of the container and the amount of substance to be stored at a given temperature. Clearly, in a vehicle, multiple small containers are more easily accommodated than a single large one.
The Toyota Mirai is a hybrid: it derives its surge power of 113 kW (152 hp) from a 1.6 kWh battery pack rechargeable by the Hydrogen fuel cell, and the desired range is achieved by the fuel cell in combination with the on-board Hydrogen tank containing 5 kg H2. As discussed above, batteries are efficient (over 90 % of the electricity input to charge the battery is available at the output terminals of the charged battery), while HFCs are much less efficient (only 36 % of the electricity input to the water electrolysis that produces the H2 fuel is available for use at the output terminals of the HFC). The refueling of a Toyota Mirai takes 3 to 5 minutes, while a Tesla takes over an hour. If Hydrogen became widely available, and for some uses, the rate of charging would be of greater value than maximized efficiency, in an all-electric clean economy future hybrid HFC cars could be designed for the combined use of efficient urban and suburban use with night-rechargeable 100 km range (10 kWh) batteries, and also for less efficient intercity use with over 500 km range fuel cells, supported by rapid charging of 5 kg of stored H2.
