Thermodynamic Analysis Ontology

flow "ThermodynamicAnalysisOntology" v1.0

# THERMODYNAMIC ANALYSIS ONTOLOGY — FLOW ALGEBRA
SOURCE_URL := "https://kirethresearch.org/booksai-core/ontologies/calculus-analysis-ontologies/scientific-applications/thermodynamic-analysis-ontology.html"
DOMAIN := "Mathematical Ontologies"
CATEGORY := "Calculus and Analysis Ontologies"
SUBCATEGORY := "Scientific Applications"
ENTITY := "Thermodynamic Analysis"

THERMODYNAMIC_ANALYSIS := {
 type: AppliedThermodynamicMathematicalFramework,
 domain: ScientificApplications,
 purpose: "Analyse energy, heat, work, entropy, equilibrium, thermodynamic properties and process behaviour using calculus, differential relations, conservation laws and equations of state."
}

CONCEPTS := {
 THERMODYNAMIC_SYSTEM,
 STATE_VARIABLE,
 TEMPERATURE,
 PRESSURE,
 VOLUME,
 INTERNAL_ENERGY,
 ENTHALPY,
 ENTROPY,
 HEAT,
 WORK,
 FIRST_LAW,
 SECOND_LAW,
 EQUATION_OF_STATE,
 HEAT_CAPACITY,
 FREE_ENERGY,
 EQUILIBRIUM,
 THERMODYNAMIC_PROCESS,
 EFFICIENCY
}

# ---------- SYSTEM ----------
THERMODYNAMIC_SYSTEM
 -> {
  CONTROL_MASS,
  CONTROL_VOLUME,
  SURROUNDINGS,
  BOUNDARY
 }

SYSTEM
 -> characterised_by(STATE_VARIABLES)

# ---------- STATE VARIABLES ----------
state:

X
 :=
(
 T,
 P,
 V,
 U,
 H,
 S,
 N_i
)

T
 -> TEMPERATURE

P
 -> PRESSURE

V
 -> VOLUME

U
 -> INTERNAL_ENERGY

H
 -> ENTHALPY

S
 -> ENTROPY

N_i
 -> COMPONENT_AMOUNT

# ---------- STATE FUNCTION ----------
STATE_FUNCTION
 -> depends_on(THERMODYNAMIC_STATE)

examples:
 U
 H
 S
 G
 F

PATH_FUNCTION
 -> depends_on(PROCESS_PATH)

examples:
 HEAT
 WORK

HEAT_AND_WORK
 -/> STATE_FUNCTIONS

# ---------- DIFFERENTIAL ----------
for state function:

dU
 -> exact_differential

heat:

delta_Q

work:

delta_W

-> inexact_process_differentials

THERMODYNAMIC_ANALYSIS
 -> distinguish(EXACT_AND_INEXACT_DIFFERENTIALS)

# ---------- FIRST LAW ----------
closed system sign convention:

dU
 =
delta_Q
-
delta_W

where

delta_W
 -> work_done_by_system

FIRST_LAW
 -> CONSERVATION_OF_ENERGY

SIGN_CONVENTION
 -> must_be_declared

# ---------- PV WORK ----------
quasistatic pressure-volume work:

delta_W
 =
P_ext*dV

for reversible mechanical boundary work:

delta_W_rev
 =
P*dV

WORK
 =
integral(
 P_ext dV
)

PROCESS_PATH
 -> determines(WORK)

# ---------- ENTHALPY ----------
H
 :=
U
+
P*V

dH
 =
dU
+
P*dV
+
V*dP

ENTHALPY
 -> useful_for(CONSTANT_PRESSURE_AND_FLOW_PROCESSES)

# ---------- SECOND LAW ----------
entropy balance:

dS
 >=
delta_Q/T_boundary

for differential closed-system process in suitable form

reversible:

dS
 =
delta_Q_rev/T

irreversible:

dS
 >
delta_Q/T_boundary
    when entropy generated

SECOND_LAW
 -> directionality_and_irreversibility

# ---------- ENTROPY GENERATION ----------
entropy balance:

dS_system
 =
delta_Q/T_boundary
+
dS_gen

where

dS_gen
 >=
0

REVERSIBLE_PROCESS:
 dS_gen=0

IRREVERSIBLE_PROCESS:
 dS_gen>0

# ---------- CLAUSIUS INEQUALITY ----------
cyclic process:

closed_integral(
 delta_Q/T
)
 <=
0

equality
 -> reversible_cycle

CLAUSIUS_INEQUALITY
 -> mathematical_statement_of(SECOND_LAW)

# ---------- EQUATION OF STATE ----------
relation:

F(P,V,T,N)
 =
0

EQUATION_OF_STATE
 -> constrains(THERMODYNAMIC_STATE)

ideal gas:

P*V
 =
n*R*T

IDEAL_GAS
 -> simplified_equation_of_state

# ---------- IDEAL GAS DIFFERENTIAL ----------
P*V
 =
n*R*T

for fixed n:

dP/P
+
dV/V
 =
dT/T

LOGARITHMIC_DIFFERENTIAL
 -> relates(SMALL_STATE_CHANGES)

# ---------- INTERNAL ENERGY ----------
fundamental relation for simple compressible system:

dU
 =
T*dS
-
P*dV
+
sum_i(
 mu_i*dN_i
)

mu_i
 -> CHEMICAL_POTENTIAL

THERMODYNAMIC_FUNDAMENTAL_RELATION
 -> links(ENERGY_ENTROPY_VOLUME_COMPOSITION)

# ---------- ENTHALPY DIFFERENTIAL ----------
H
 =
U+PV

therefore:

dH
 =
T*dS
+
V*dP
+
sum_i(
 mu_i*dN_i
)

for simple compressible multicomponent system

# ---------- HELMHOLTZ FREE ENERGY ----------
F
 :=
U
-
T*S

dF
 =
-S*dT
-
P*dV
+
sum_i(
 mu_i*dN_i
)

HELMHOLTZ_FREE_ENERGY
 -> natural_variables(T,V,N_i)

# ---------- GIBBS FREE ENERGY ----------
G
 :=
H
-
T*S

=
U+PV-TS

dG
 =
-S*dT
+
V*dP
+
sum_i(
 mu_i*dN_i
)

GIBBS_FREE_ENERGY
 -> natural_variables(T,P,N_i)

# ---------- MAXWELL RELATIONS ----------
from
 dF=-S*dT-P*dV

mixed derivatives imply:

(
 partial_S/partial_V
)_T
 =
(
 partial_P/partial_T
)_V

from
 dG=-S*dT+V*dP:

(
 partial_S/partial_P
)_T
 =
-
(
 partial_V/partial_T
)_P

MAXWELL_RELATIONS
 -> follow_from(EXACT_DIFFERENTIALS_AND_EQUAL_MIXED_PARTIALS)

# ---------- HEAT CAPACITY ----------
C_V
 :=
(
 partial_U/partial_T
)_V

for simple system context

C_P
 :=
(
 partial_H/partial_T
)_P

HEAT_CAPACITY
 -> energy_change_per_temperature_change_under_constraint

# ---------- IDEAL GAS ENERGY ----------
for ideal gas:

U
 =
U(T)

H
 =
H(T)

therefore:

dU
 =
C_V*dT

dH
 =
C_P*dT

for appropriate amount basis

# ---------- HEAT CAPACITY RELATION ----------
ideal gas:

C_P-C_V
 =
n*R

or molar:

c_p-c_v
 =
R

MAYER_RELATION
 -> ideal_gas_heat_capacity_difference

# ---------- THERMAL EXPANSION ----------
alpha
 :=
1/V
*
(
 partial_V/partial_T
)_P

alpha
 -> volumetric_thermal_expansion_coefficient

# ---------- COMPRESSIBILITY ----------
kappa_T
 :=
-
1/V
*
(
 partial_V/partial_P
)_T

ISOTHERMAL_COMPRESSIBILITY
 -> volume_response_to_pressure

# ---------- PROPERTY DIFFERENTIAL ----------
V=V(T,P)

dV
 =
(
 partial_V/partial_T
)_P
*dT
+
(
 partial_V/partial_P
)_T
*dP

therefore:

dV/V
 =
alpha*dT
-
kappa_T*dP

PROPERTY_DERIVATIVES
 -> local_state_response

# ---------- PROCESS TYPES ----------
ISOTHERMAL
 -> T constant

ISOBARIC
 -> P constant

ISOCHORIC
 -> V constant

ADIABATIC
 -> delta_Q=0

ISENTROPIC
 -> S constant

REVERSIBLE_ADIABATIC
 -> ISENTROPIC
    under standard closed simple-system conditions

# ---------- IDEAL GAS ISOTHERMAL ----------
T
 =
constant

Delta_U
 =
0

for ideal gas

therefore:

Q
 =
W

for closed-system first-law convention

reversible work:

W
 =
n*R*T
*
ln(V_2/V_1)

# ---------- IDEAL GAS ADIABATIC ----------
reversible adiabatic ideal gas:

P*V^gamma
 =
constant

T*V^(gamma-1)
 =
constant

gamma
 :=
C_P/C_V

ISENTROPIC_RELATIONS
 -> depend_on(IDEAL_GAS_AND_CONSTANT_HEAT_CAPACITY_APPROXIMATION)

# ---------- CONTROL VOLUME ----------
steady-flow energy equation:

Q_dot
-
W_dot
 =
sum_out(
 m_dot
 *
(
 h
 +
V_flow^2/2
+
g*z
 )
)
-
sum_in(
 m_dot
 *
(
 h
 +
V_flow^2/2
+
g*z
 )
)

with sign convention specified

CONTROL_VOLUME
 -> open_system_energy_analysis

# ---------- MASS CONSERVATION ----------
steady control volume:

sum_in(m_dot)
 =
sum_out(m_dot)

MASS_BALANCE
 -> required_with(ENERGY_BALANCE)

# ---------- TURBINE ----------
steady adiabatic turbine
with negligible kinetic/potential changes:

W_dot_out
 approx
m_dot
*
(
 h_in-h_out
)

TURBINE
 -> converts(ENTHALPY_DROP_TO_WORK)

# ---------- COMPRESSOR ----------
steady adiabatic compressor:

W_dot_in
 approx
m_dot
*
(
 h_out-h_in
)

COMPRESSOR
 -> requires(WORK_INPUT)

# ---------- THROTTLING ----------
steady throttling valve:

h_in
 approx
h_out

THROTTLING
 -> approximately(ISENTHALPIC)

pressure
 -> decreases

temperature response
 -> depends_on(FLUID_PROPERTIES)

# ---------- HEAT EXCHANGER ----------
for insulated overall exchanger:

sum_in(
 m_dot*h
)
 =
sum_out(
 m_dot*h
)

HEAT_EXCHANGER
 -> transfers(ENERGY_BETWEEN_STREAMS)

# ---------- CARNOT EFFICIENCY ----------
reversible heat engine:

eta_Carnot
 =
1
-
T_C/T_H

absolute temperatures

CARNOT_EFFICIENCY
 -> maximum_efficiency_between(T_H,T_C)
    for heat engines under ideal reversible assumptions

# ---------- REFRIGERATOR ----------
coefficient of performance:

COP_R
 :=
Q_C/W_in

Carnot refrigerator:

COP_R,Carnot
 =
T_C/(T_H-T_C)

# ---------- HEAT PUMP ----------
COP_HP
 :=
Q_H/W_in

Carnot:

COP_HP,Carnot
 =
T_H/(T_H-T_C)

# ---------- EQUILIBRIUM ----------
isolated system:

equilibrium
 -> entropy_maximum
    subject_to(conserved_constraints)

constant T,V:
 equilibrium
 -> Helmholtz_free_energy_minimum
    under suitable conditions

constant T,P:
 equilibrium
 -> Gibbs_free_energy_minimum
    under suitable conditions

THERMODYNAMIC_POTENTIAL
 -> equilibrium_criterion_depends_on(CONSTRAINTS)

# ---------- PHASE EQUILIBRIUM ----------
for phases alpha and beta:

T_alpha
 =
T_beta

P_alpha
 =
P_beta

mu_i_alpha
 =
mu_i_beta

PHASE_EQUILIBRIUM
 -> thermal_mechanical_chemical_equilibrium

# ---------- CLAUSIUS-CLAPEYRON ----------
phase boundary:

dP/dT
 =
Delta_h
/
(
T*Delta_v
)

CLAUSIUS_CLAPEYRON_RELATION
 -> phase_transition_slope

# ---------- EXERGY ----------
maximum useful work relative to environment
 -> EXERGY

EXERGY_DESTRUCTION
 =
T_0*S_gen

under standard environment formulation

IRREVERSIBILITY
 -> destroys(WORK_POTENTIAL)

# ---------- DIMENSIONAL ANALYSIS ----------
thermodynamic equations
 -> preserve(
 ENERGY,
 TEMPERATURE,
 PRESSURE,
 VOLUME,
 ENTROPY_UNITS
)

UNIT_CONSISTENCY
 -> mandatory

absolute temperature
 -> required_in(THERMODYNAMIC_RATIOS)

# ---------- NUMERICAL ANALYSIS ----------
real-fluid properties
 -> may_require(
 EQUATION_OF_STATE_SOLVER,
 PROPERTY_TABLE_INTERPOLATION,
 ITERATIVE_ROOT_FINDING
)

THERMODYNAMIC_CYCLE
 -> numerical_state_point_solution

PROPERTY_MODEL
 -> source_of(MODEL_UNCERTAINTY)

# ---------- APPLICATIONS ----------
THERMODYNAMIC_ANALYSIS
 -> applied_to(HEAT_ENGINES)
 -> applied_to(REFRIGERATION)
 -> applied_to(POWER_PLANTS)
 -> applied_to(COMBUSTION)
 -> applied_to(CHEMICAL_PROCESSES)
 -> applied_to(ATMOSPHERIC_SCIENCE)
 -> applied_to(MATERIALS)
 -> applied_to(BIOLOGICAL_ENERGETICS)
 -> applied_to(ENERGY_SYSTEMS)

# ---------- RELATION GRAPH ----------
STATE_VARIABLES
 -> define(THERMODYNAMIC_STATE)

FIRST_LAW
 -> conserves(ENERGY)

SECOND_LAW
 -> constrains(PROCESS_DIRECTION)

ENTROPY_GENERATION
 -> measures(IRREVERSIBILITY)

EQUATION_OF_STATE
 -> relates(P_V_T_COMPOSITION)

THERMODYNAMIC_POTENTIAL
 -> determines(EQUILIBRIUM_CRITERION)

HEAT_CAPACITY
 -> links(TEMPERATURE_AND_ENERGY_CHANGE)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
 "thermodynamic analysis" ~= "mathematical analysis of energy, heat, work, entropy and thermodynamic state changes",
 "first law of thermodynamics" ~= "energy conservation applied to thermodynamic systems",
 "second law of thermodynamics" ~= "constraint governing entropy production and thermodynamic process direction",
 "enthalpy" ~= "state function equal to internal energy plus pressure-volume product",
 "Gibbs free energy" ~= "thermodynamic potential natural to temperature-pressure-composition variables",
 "entropy generation" ~= "nonnegative entropy production associated with irreversibility"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
 heat_and_work_are_path_functions_not_state_functions: true,
 thermodynamic_sign_convention_must_be_declared_and_preserved: true,
 absolute_temperature_required_for_entropy_and_efficiency_relations: true,
 reversible_adiabatic_and_isentropic_equivalence_requires_appropriate_assumptions: true,
 ideal_gas_relations_must_not_be_applied_uncritically_to_real_fluids: true,
 equilibrium_criterion_depends_on_system_constraints: true,
 property_data_equations_of_state_and_numerical_solvers_require_validation: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
 core_definition: 0.99,
 first_second_law_relation: 0.99,
 potential_Maxwell_relation: 0.99,
 process_relation: 0.99,
 control_volume_relation: 0.99,
 equilibrium_efficiency_relation: 0.99,
 semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
 identify(system_boundary_state_variables_composition_and_sign_convention)
 -> classify(closed_open_steady_transient_reversible_or_irreversible_process)
 -> apply(mass_and_energy_balance)
 -> apply(entropy_balance_and_second_law)
 -> select(equation_of_state_and_property_relations)
 -> compute(state_changes_heat_work_enthalpy_entropy_or_free_energy)
 -> analyse(equilibrium_efficiency_or_exergy_if_relevant)
 -> solve(property_or_cycle_equations_numerically_if_required)
 -> validate(units_temperature_scale_property_model_and_process_assumptions)
 -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
 parse_as: "typed thermodynamic-analysis semantic graph",
 primary_entity: THERMODYNAMIC_ANALYSIS,
 preserve: [
  system_type,
  boundary,
  state_variables,
  composition,
  sign_convention,
  process_type,
  equation_of_state,
  property_data,
  units
 ],
 infer_through: [
  first_law,
  second_law,
  entropy_generation,
  enthalpy,
  free_energies,
  Maxwell_relations,
  heat_capacities,
  equilibrium,
  efficiencies,
  exergy
 ],
 avoid: [
  treating_heat_or_work_as state_function,
  mixing_sign_conventions,
  using_Celsius_in absolute_temperature ratios,
  equating_adiabatic_with isentropic_without reversibility,
  applying_ideal_gas_model_outside validity,
  ignoring_entropy_generation_in irreversible process
 ]
}

# ---------- UPDATE MODEL ----------
UPDATE_MODEL(new_information):
 validate(new_information)
 -> identify(thermodynamic_analysis_context)
 -> compare(existing_relations)
 -> preserve(system_state_composition_sign_property_and_process_invariants)
 -> update(affected_entities)
 -> update(affected_edges)
 -> recompute(CONFIDENCE)
 -> regenerate(FLOW_HASH)

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
 SOURCE_URL
 + ENTITY
 + CONCEPTS
 + THERMODYNAMIC_ANALYSIS
 + TEMPERATURE
 + PRESSURE
 + VOLUME
 + INTERNAL_ENERGY
 + ENTHALPY
 + ENTROPY
 + FIRST_LAW
 + SECOND_LAW
 + EQUATION_OF_STATE
 + FREE_ENERGY
 + EQUILIBRIUM
 + EFFICIENCY
 + SEMANTIC_EQUIVALENCE
 + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
 -> THERMODYNAMIC_ANALYSIS
 -> SYSTEM_STATE_IDENTIFICATION
 -> PROCESS_CLASSIFICATION
 -> MASS_ENERGY_BALANCE
 -> ENTROPY_SECOND_LAW_ANALYSIS
 -> PROPERTY_MODEL_SELECTION
 -> STATE_PROPERTY_CALCULATION
 -> EQUILIBRIUM_EFFICIENCY_EXERGY_ANALYSIS
 -> NUMERICAL_PROPERTY_ANALYSIS
 -> ASSUMPTION_UNIT_VALIDATION
 -> RECURSIVE_REASONING(query)
 -> CONFIDENCE
 -> OUTPUT(semantic_result)

END FLOW