Read the 2 pages below that explain how the intermediary flows (in the A technology matrix)...

  

  

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Read the 2 pages below that explain how the intermediary flows (in the A technology matrix) and elementary flows (in the B matrix of direct emissions and extractions) can be combined and structure using matrix inversion. 1. Theory: Generalization and process matrix approach The system boundary must be carefully delimited and include all relevant background processes necessary for the direct processes considered. For example, the total emissions and extractions associated with the production of one kg of aluminum must include those associated with its extraction, fabrication, disposal, and any other important stages. But should we also account for the creation of the machines that built the infrastructure necessary to extract the raw materials? And should we account for the energy needed to create these machines? And the aluminum needed to extract the energy to create the machines? If so, the chain of processes to consider for the provision of 1 kg of aluminum becomes infinitely long with closed loops (Fig.1). Spaget Figure 1. Flow chart of background processes. Boxes represent processes which are connected by elementary flows represented by horizontal arrows, with elementary flows of extractions indicated by lower arrows and of emissions indicated by upper arrows. Accounting for this infinitely long process chain can be addressed with the approach taken by the ecoinvent inventory database, which uses a model constituted of the technosphere (economic system) and the ecosphere (environmental system). Each unit process can exchange intermediary flows with any number of the m total unit processes of the technosphere and can be associated with any of the elementary flows extracted from or emitted to the environment. The technology matrix A (m x m) is a square matrix consisting of a row and column entry for every unit process in the economy, and the environmental matrix B (n x m) has a column for every unit process and a row for every elementary flow from or to the environment. Use and a row for every elementary flow from or to the environment. A B 1 4 (b E bai a 1 G b (eg. 4. (eg. 4.2) The element a, (row i, columnj) of the A matrix represents the amount of technological process i used by process j, and the element by, of the matrix B is the elementary flow of substance k extracted from the environment or emitted in the environment through processj. In other words, column j of the A matrix contains the amount of all processes used by the process. Similarly, column / of the B matrix contains all extractions and emissions directly associated with process. The matrix E of aggregated emis on and extraction factors for each unit process (equivalent to Table 4.1 or Table 4.5) is a sum of the following infinite chain: • the direct elementary flows for each first tier unit process (B) • the elementary flows associated with the processes needed for each unit process (BA) (second tier e.g., the flows associated with the machine needed) . the elementary flows associated with the processes that are needed for the processes needed for each unit process (BA) (e.g., the flows associated with the machine that made the machine needed) and so on. This can be expressed as follows: E B (1+A+A²+A+A...) B (1-A) (eq.4.4) Where I is the identity matrix with entries of 1 along the diagonal (and zeros everywhere else); just as 1 + x²+x-(1-x) for x < 1, (I + A+ A²+ A²+A¹...)-(1-) for a 1 for all This matrix inversion allows the inclusion of an infinite process chain in theory, but practically. the system is still truncated because certain unit processes are just not taken into account in the process LCA. The emissions and extractions inventory vector u is calculated by multiplying matrix E by the demand vector y that quantifies first tier intermediary flows or inputs per functional unit. (eq. 4.5) u-Ey-B(I-A)¹ y 2. Simplified example of process matrix For aluminum manufacturing, you are given the set of simplified matrices below, describing the technology matrix (A), the demand vector for 1 kg aluminum (d), the matrix of direct emissions and extractions (B), and the aggregated matrix of emissions and extraction factors (E). Electricity Oil A Alu kg kWh kg 0 0 Electricity kWh 15 0 0.3 0.25 SEB kg 0.05 0.04 0 0 0 Oil Gas (I-A)¹ Alu kg Electricity kWh Oil Gas B Energy CO₂ 1 E-B*(1-A)* 1¹ CO₂ MJ kg Energy MJ Alu kg Alu kg 1 15.2 0.66 0 kg 1.66 kWh 1E-19 162 1.01 9.5 0.04 0 Electricity Oil Gas 1² kWh 2.57 0.30 8.22 Alu Electricity Oil kg kWh kg 1 0 10.5 0 0.45 0 0.30 Gas 1.01 0 0 0 0 0 0.25 0.01 1 Gas kg 1 53.8 40.6 3.54 2.69 d 1 0 0 0 (I-A) 'd 1 15.2 0.66 0 d 1 0 Alu Electricity Oil Gas kg === 56.9 43.2 3.67 2.8 0 x=(1-A)'d 0 1 15.2 0.66 0 b-B*(1-A) 'd 162 9.5 Alu kg Electricity kWh 162 9.5 Oil Gas Energy CO₂ b-Ed-B*(1-A)'d Energy CO₂ kg 1 MJ kg MJ kg 3. Homework questions 1. Answer the following questions to see if you understand the formulation a) How many kWh electricity are directly consumed (by tier 1 processes) in the Aluminum manufacturing process per kg Aluminum? b) How many kWh electricity are consumed per kg Aluminum, including upstream processes? c) What are the direct CO2 emissions associated with 1 kWh electricity? d) What are the aggregated CO2 emissions (including upstream processes) associated with 1 kWh electricity? e) What are the aggregated CO2 emissions (including upstream processes) associated with 0.5 kg Aluminium? f) What are the respective contributions of each the Aluminum, Electricity and Oil sector to the aggregated CO2 emissions per kg aluminum? Read the 2 pages below that explain how the intermediary flows (in the A technology matrix) and elementary flows (in the B matrix of direct emissions and extractions) can be combined and structure using matrix inversion. 1. Theory: Generalization and process matrix approach The system boundary must be carefully delimited and include all relevant background processes necessary for the direct processes considered. For example, the total emissions and extractions associated with the production of one kg of aluminum must include those associated with its extraction, fabrication, disposal, and any other important stages. But should we also account for the creation of the machines that built the infrastructure necessary to extract the raw materials? And should we account for the energy needed to create these machines? And the aluminum needed to extract the energy to create the machines? If so, the chain of processes to consider for the provision of 1 kg of aluminum becomes infinitely long with closed loops (Fig.1). Spaget Figure 1. Flow chart of background processes. Boxes represent processes which are connected by elementary flows represented by horizontal arrows, with elementary flows of extractions indicated by lower arrows and of emissions indicated by upper arrows. Accounting for this infinitely long process chain can be addressed with the approach taken by the ecoinvent inventory database, which uses a model constituted of the technosphere (economic system) and the ecosphere (environmental system). Each unit process can exchange intermediary flows with any number of the m total unit processes of the technosphere and can be associated with any of the elementary flows extracted from or emitted to the environment. The technology matrix A (m x m) is a square matrix consisting of a row and column entry for every unit process in the economy, and the environmental matrix B (n x m) has a column for every unit process and a row for every elementary flow from or to the environment. Use and a row for every elementary flow from or to the environment. A B 1 4 (b E bai a 1 G b (eg. 4. (eg. 4.2) The element a, (row i, columnj) of the A matrix represents the amount of technological process i used by process j, and the element by, of the matrix B is the elementary flow of substance k extracted from the environment or emitted in the environment through processj. In other words, column j of the A matrix contains the amount of all processes used by the process. Similarly, column / of the B matrix contains all extractions and emissions directly associated with process. The matrix E of aggregated emis on and extraction factors for each unit process (equivalent to Table 4.1 or Table 4.5) is a sum of the following infinite chain: • the direct elementary flows for each first tier unit process (B) • the elementary flows associated with the processes needed for each unit process (BA) (second tier e.g., the flows associated with the machine needed) . the elementary flows associated with the processes that are needed for the processes needed for each unit process (BA) (e.g., the flows associated with the machine that made the machine needed) and so on. This can be expressed as follows: E B (1+A+A²+A+A...) B (1-A) (eq.4.4) Where I is the identity matrix with entries of 1 along the diagonal (and zeros everywhere else); just as 1 + x²+x-(1-x) for x < 1, (I + A+ A²+ A²+A¹...)-(1-) for a 1 for all This matrix inversion allows the inclusion of an infinite process chain in theory, but practically. the system is still truncated because certain unit processes are just not taken into account in the process LCA. The emissions and extractions inventory vector u is calculated by multiplying matrix E by the demand vector y that quantifies first tier intermediary flows or inputs per functional unit. (eq. 4.5) u-Ey-B(I-A)¹ y 2. Simplified example of process matrix For aluminum manufacturing, you are given the set of simplified matrices below, describing the technology matrix (A), the demand vector for 1 kg aluminum (d), the matrix of direct emissions and extractions (B), and the aggregated matrix of emissions and extraction factors (E). Electricity Oil A Alu kg kWh kg 0 0 Electricity kWh 15 0 0.3 0.25 SEB kg 0.05 0.04 0 0 0 Oil Gas (I-A)¹ Alu kg Electricity kWh Oil Gas B Energy CO₂ 1 E-B*(1-A)* 1¹ CO₂ MJ kg Energy MJ Alu kg Alu kg 1 15.2 0.66 0 kg 1.66 kWh 1E-19 162 1.01 9.5 0.04 0 Electricity Oil Gas 1² kWh 2.57 0.30 8.22 Alu Electricity Oil kg kWh kg 1 0 10.5 0 0.45 0 0.30 Gas 1.01 0 0 0 0 0 0.25 0.01 1 Gas kg 1 53.8 40.6 3.54 2.69 d 1 0 0 0 (I-A) 'd 1 15.2 0.66 0 d 1 0 Alu Electricity Oil Gas kg === 56.9 43.2 3.67 2.8 0 x=(1-A)'d 0 1 15.2 0.66 0 b-B*(1-A) 'd 162 9.5 Alu kg Electricity kWh 162 9.5 Oil Gas Energy CO₂ b-Ed-B*(1-A)'d Energy CO₂ kg 1 MJ kg MJ kg 3. Homework questions 1. Answer the following questions to see if you understand the formulation a) How many kWh electricity are directly consumed (by tier 1 processes) in the Aluminum manufacturing process per kg Aluminum? b) How many kWh electricity are consumed per kg Aluminum, including upstream processes? c) What are the direct CO2 emissions associated with 1 kWh electricity? d) What are the aggregated CO2 emissions (including upstream processes) associated with 1 kWh electricity? e) What are the aggregated CO2 emissions (including upstream processes) associated with 0.5 kg Aluminium? f) What are the respective contributions of each the Aluminum, Electricity and Oil sector to the aggregated CO2 emissions per kg aluminum?

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To answer the homework questions we must refer to the matrices provided in the simplified example of the process matrix and apply the logic from the t... View the full answer