Mathematics puzzles in modelling technology for energy

Suppose you need to calculate the amount of carbon in coal for someone who was pondering creating hydrogen from it. 

One way is as follows: 

1. Find out how much CO₂ a coal-fired power station emits for each megawatt-hour of electricity it sends out. 

2. Find out how many tonnes of coal the same coal-fired power station burns for each megawatt-hour of electricity it sends out. 

3. From the above pieces of information, calculate how much CO₂ is produced from each tonne of coal the power station burns. 

4. Calculate the amount of carbon in the CO₂ that is produced from each tonne of coal that is burned. 

The Australian Government's National Greenhouse and Energy Reporting site provides a sample of some of the information -

Australian Government's National Greenhouse and Energy Reporting
Yallourn Power Station

Energy Australia provided the corresponding information for coal consumption Yallourn Power Station on its web site a few years ago. This is a sample retrieved from the WayBack Machine - 

Coal consumption at Yallourn coal-fired power plant

 

The above information is sufficient for the calculations suggested above - 

1. CO₂ per MWh : 1.34 tonnes.

2. Coal per MWh : (1,480 MWh from burning 2,400 tonnes of coal) => 1.62 tonnes per MWh. 

3. The amount of CO₂ for each tonne of coal? 

1.34 tonnes of CO₂ are produced from burning 1.62 tonnes of brown coal. 

So burning 1 tonne of coal produces 0.83 tonnes of carbon dioxide.

4. The formula weight of carbon dioxide shows that 44 grams of carbon dioxide is made of 12 grams of carbon and 32 grams of oxygen. In other words the amount of carbon in carbon dioxide is (12/44) times the mass of the carbon dioxide.

So the carbon in 0.83 tonnes of carbon dioxide is 0.225 tonnes. (= 0.83 tonnes x (12 / 44).)

0.225 tonnes is 225 kgs of carbon in each tonne of brown coal burned at Yallourn Power Station. 

This should not be a controversial or surprising answer. 

The calculations to estimate that there are 225 tonnes of carbon using publicly available information about Victorian brown coal are not terribly complex. 

These following calculations are surprising:

1. In a 2019 report "Evaluation of options for production of low-cost CO₂ - free hydrogen from Victorian brown coal" the details for option 4: "Brown coal gasification plant using oxygen blown entrained flow gasifier followed by shift reactor for H2 production " given on page 35 are listed in the table below. 

The two lines of special interest are the ones showing -

• Wet coal of 893 tonnes per hour and 
• Total CO₂ generation of 518 tonnes per hour. 

A few calculations show that each tonne of coal in this modelling exercise is assumed to produce 0.58 tonnes of carbon dioxide, and

This means that the coal for the purpose of this model have only 160 kgs of carbon in each tonne. 

Target production of H₂

Wet coal requirement

Dry coal

Excess char to refinery

Tar production

Total CO₂ generation

CO₂ capture efficiency

Steam requirement 

32.1 tons/hour

893 tons/hour

332 tons/hour

Nil

Nil

518 tons/hour

88%

344 tons/hour

If the calculation of carbon coal burned by the Yallourn power station - 225 kgs per tonne - is correct, then the carbon in 427 tonnes of wet brown coal is sufficient to produce 32 tonnes of hydrogen by reaction with steam. 

This is less than half of the 893 tonnes of wet brown coal the 2019 modelling exercise found to be needed. 

2. The web site for the Hydrogen Energy Supply Chain pilot project says that it is to use 150 tonnes of brown coal to produce 3 tonnes of hydrogen. 

This is an even greater amount of coal per tonne of hydrogen than the 2019 modelling reported. 

The Hydrogen Energy Supply Chain project also estimated that it would produce 100 tonnes of carbon dioxide in producing the hydrogen. This is about double the rate of carbon dioxide produced per tonne of hydrogen that the 2019 modelling reported. 

That estimate has since been greatly increased to around 140 tonnes.

The assumptions and calculations in these two examples - the 2019 report, and the Hydrogen Energy Supply Chain project - cannot be reconciled with the information available about the use of coal by the Yallourn Power Station.

Hydrogen from Renewable Energy

 It may be surprisingly simple to make hydrogen commercially viable with renewable energy. 

In a renewable energy powered electrolyzer 91 tonnes of water can be decomposed into 80 tonnes of oxygen and 10 tonnes of hydrogen leaving a residue of 1 tonne of water. 

See for example "Alkaline Water Electrolysis Powered by Renewable Energy: A Review" by Jörn Brauns and Thomas Turek, Institute of Chemical and Electrochemical Process Engineering, Clausthal University of Technology, Leibnizstr. 17, 38678 Clausthal-Zellerfeld, Germany. 

At the target price of $2 per kilogram, the 10 tonnes of hydrogen has a sale value of $20,000.

The residue of 1 tonne of water contains a little over 7 kilograms of deuterium oxide. 

At $1,500 per kilogram, the 7 kilograms of deuterium oxide has a sale value of $10,500. 

The 80 tonnes of oxygen has a number of potential uses. 

One use is to generate electricity in a gas turbine while partially oxidising 80 tonnes of biomethane into 20 tonnes of hydrogen and 140 tonnes of carbon monoxide. 

See for example "Integrated Coproduction of Power and Syngas from Natural Gas to Abate Greenhouse Gas Emissions without Economic Penalties" by Mikhail Granovskiy, Southern Research, Laboratory of Sustainable Chemistry and Catalysis, Birmingham, Alabama, USA. 

At the top-left of the above schematic diagram, the "Air Separation Unit" is replaced with an alkaline water electrolyzer that produces oxygen, doing away with the need for an "Air Separation Unit".

At the target price of $2 per kilogram, this 20 tonnes of hydrogen has a sale value of $40,000.  

The power generated may be sold or used in powering the electrolyzer. 

The 140 tonnes of carbon monoxide can be combined with 90 tonnes of water to produce 230 tonnes of formic acid. Formic acid can be used in various industrial processes. 

At a price of $500 a tonne, the formic acid has a sale value of $115,000. 

Note that there are no carbon dioxide emissions. 

There is no carbon capture and storage required.