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A thermoeconomic cost balance can conserve every dollar and still leave product costs undetermined; independent auxiliary equations and a full-rank productive structure make the solution unique.
Open the chapter contents · Thermodynamics root · Gold Standard of Education · How X Works
Chapter contents
Choose the route closest to the question, or read from Chapter 1 for the complete argument. The teaching chapters remain open; these grouped routes form the index.
Route 1 · Define fuel, product and balance
Route 2 · Work the structural ledger
Route 3 · Transfer across system structure
Route 4 · Check limits and evidence
Route 5 · Ask and continue
CHAPTER 1 OF 12 · DEFINE FUEL, PRODUCT AND BALANCE
1. A cost balance can be physically sensible and mathematically incomplete
Thermoeconomics attaches monetary or exergetic costs to streams in an energy system. For each component, the cost rate of outgoing products equals the cost rate of incoming resources plus operation, maintenance and capital charges.
When one component has several outgoing product streams, one balance supplies only one equation. The product costs remain underdetermined until independent auxiliary equations express how shared cost is assigned.
CHAPTER 2 OF 12 · DEFINE FUEL, PRODUCT AND BALANCE
2. Fuel and product definitions come first
The fuel of a component is the exergy resource it consumes to create its intended result. The product is that intended result. These are functional definitions, not simply every inlet and outlet.
SPECO records exergy additions and removals to define fuel and product, then writes the matching cost balance. If the productive purpose is vague, the cost equations can be algebraically solvable yet conceptually wrong.
CHAPTER 3 OF 12 · WORK THE STRUCTURAL LEDGER
3. Worked example: one balance and two product streams
Take a component with an incoming resource cost rate of 100 dollars per hour and a capital plus operating charge of 20 dollars per hour. It produces P1 = 60 kW and P2 = 40 kW. The balance is C1 + C2 = 120 dollars per hour.
There are two unknown outgoing cost rates and only one equation. Apply an equal-specific-cost auxiliary equation C1/60 = C2/40. The unique solution is C1 = 72 and C2 = 48 dollars per hour, so each product carries 1.20 dollars per kWh.
| Equation stage | Matrix rank | Cost solution |
|---|---|---|
| Balance C1 + C2 = 120 | 1 | Infinitely many pairs |
| Add C1/60 = C2/40 | 2 | C1 = 72, C2 = 48 |
| Add a duplicate of the balance | 1 | Still underdetermined |
CHAPTER 4 OF 12 · WORK THE STRUCTURAL LEDGER
4. Auxiliary equations encode a costing principle
Equal specific costs assign the same unit cost to both outputs. A by-product rule, extraction rule or local-average rule encodes something else. Conservation does not choose among them.
The multi-product allocation guide shows how alternative keys and credits change attributed burden while leaving the physical exergy balance intact.
CHAPTER 5 OF 12 · WORK THE STRUCTURAL LEDGER
5. Changed auxiliary rule, changed product costs
Return to the worked component. Suppose P2 is a by-product assigned 0.80 dollars per kWh by a declared replacement rule. Then C2 = 32 dollars per hour, and the balance gives C1 = 88. The specific cost of P1 becomes 1.467 dollars per kWh.
| Auxiliary rule | C1 ($/h) | C2 ($/h) | Reconciled total ($/h) |
|---|---|---|---|
| Equal unit costs | 72 | 48 | 120 |
| P2 fixed at $0.80/kWh | 88 | 32 | 120 |
| P2 receives zero assigned cost | 120 | 0 | 120 |
The calculation is not arbitrary once the rule is declared. The judgement lies in selecting and justifying that rule.
CHAPTER 6 OF 12 · TRANSFER ACROSS SYSTEM STRUCTURE
6. Structural uniqueness is a rank condition
Write the complete cost system as A c = b. A unique solution requires as many independent equations as unknown costs and a full-rank coefficient matrix. A missing equation gives a family of solutions; a contradictory equation gives no exact solution.
Numerical software may still return a least-squares vector for an ill-posed system. That output is not proof that the thermoeconomic model is structurally complete.
CHAPTER 7 OF 12 · TRANSFER ACROSS SYSTEM STRUCTURE
7. Cost formation follows the productive structure
A stream’s cost accumulates through the components that produce it. The productive structure records which fuels feed which products. Structural theory expresses this network through characteristic equations and derivatives, allowing different thermoeconomic formulations to be compared in one linear framework.
The route from resource cost to final product cost should be traceable component by component. A final number without its productive path is difficult to audit.
CHAPTER 8 OF 12 · TRANSFER ACROSS SYSTEM STRUCTURE
8. Changed context: a recirculating system
Recirculation creates coupled cost equations because a stream can be both downstream product and upstream fuel. Solving components one by one may fail; the whole linear system must be solved simultaneously.
Check for loops, zero-product components and disconnected products before inversion. Rank can fail because of structure, not because the linear solver is weak.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Changed context: dissipative components
Condensers, throttling valves and waste-treatment units may consume resources without an obvious saleable product. Forcing their cost onto an arbitrary outlet can distort the productive structure.
SPECO and structural approaches require an explicit rule for dissipative costs. The exergy-destruction guide separates physical irreversibility from the later costing decision.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Limits and misconceptions
Full matrix rank guarantees a unique algebraic solution, not a uniquely correct economic interpretation. Two full-rank models with different valid auxiliary principles can allocate cost differently.
Specific exergy cost is not market price. Capital, fuel and environmental boundaries must be declared. Negative or extreme stream costs can reveal a strong credit rule, inconsistent sign convention or structural mistake.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Evidence and verification route
Lazzaretto and Tsatsaronis set out SPECO fuel–product definitions and auxiliary equations in A Systematic and General Methodology for Calculating Efficiencies and Costs. Valero and colleagues formulate the foundations in Theory of the Exergetic Cost90006-Y).
The common linear framework is developed in Structural Theory as Standard for Thermoeconomics00057-6). Verification should publish fuel and product definitions, every balance and auxiliary equation, units, sign conventions, matrix rank, residuals and a complete reconciliation to external resource and capital costs.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Frequently asked questions
Why are auxiliary equations necessary?
One component balance cannot divide shared cost among several unknown product streams.
Does full rank prove the allocation is fair?
No. It proves structural uniqueness for the chosen equations, not the fairness of their costing principle.
Can I use a least-squares solution when equations conflict?
Only as a diagnostic. The inconsistency should be repaired before reporting stream costs.
Final transfer check and connected reading
Using the worked component, replace equal specific costs with C2 = 32 dollars per hour. Derive C1 = 88 and verify the total remains 120. Then explain why both solutions are algebraically correct but answer different allocation rules.
Next, imagine a recirculating stream and state why simultaneous solution is needed. Continue with the Thermodynamics root, multi-product allocation, exergy efficiency and fair benchmarks, the Gold Standard of Education and How X Works.
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