1. 1 D r a f t Life Cycle Assessment A product-oriented method for sustainability analysis UNEP LCA Training Kit Module h – The mathematics of LCI

2. Representation of processes Representation of goal definition Balancing Environmental interventions Overall LCI structure Advantages New concepts Contents D r a f t

3. Representation of processes (1) Production of electricity: –in flow diagram terms: –in mathematical terms: 10 kWh of electricity 1 kg of CO 2 0.1 kg of SO 2 2 liter of fuel D r a f t

4. Representation of processes (2) Production of fuel: –in flow diagram terms: –in mathematical terms: 100 liter of fuel 10 kg of CO 2 2 kg of SO 2 50 liter of crude oil D r a f t

5. Representation of goal definition Functional unit/reference flow: –in flow diagram terms: –in mathematical terms: 1000 kWh of electricity ? kg of CO 2 ? kg of SO 2 ? liter of crude oil (? liter of fuel) D r a f t

6. 6 6 Balancing (1) Need to match supply and demand: 100 liter2 liter10 kWh 1000 kWh  100  2 1000 kWh 200 liter 10 kg CO 2 1 kg CO 2 100 kg CO 2 20 kg CO 2 ? kg CO 2 100 kg CO 2 120 kg CO 2

7. 7 7 D r a f t Balancing (1) Need to scale processes: –production of electricity by a factor of 100 –production of fuel by a factor of 2 More formally: balance equations: Or in matrix terms:

8. 8 8 D r a f t Balancing (2) From concrete equations with matrix coefficients … to abstract equations with symbols … that can be solved by standard techniques … and becoming concrete again

9. 9 9 D r a f t Environmental interventions (1) Apply same scaling factors to environmental flows: In matrix terms

10. 10 D r a f t Environmental interventions (2) From concrete equations with matrix coefficients … to abstract equations with symbols … in which we can insert the previous result … and becoming concrete again

11. 11 D r a f t Overall LCI structure Combining balancing and interventions: with the intensity matrix

12. 12 D r a f t Advantages of mathematical treatment Explicit treatment, transparent Guidance for software implementation New insight in LCA –e.g., allocation Advanced theoretical concepts: –perturbation theory –covariance structure –stochastic theory

13. 13 D r a f t New concepts Terms: –technology matrix, scaling vector, intervention matrix, final demand vector, intensity matrix, … Techniques: –matrix conditioning, principal components analysis, …