Combustion Physics (Day 1 Lecture) Chung K. Law Robert H. Goddard Professor Princeton University
Princeton-CEFRC-Combustion Institute Summer School on Combustion June 20-24, 2016
1 Day 1: Chemical Thermodynamics and Kinetics
1. Chemical Thermodynamics • Chemical equilibrium
• Energy conservation & adiabatic flame temp., Tad 2. Chemical Kinetics • Reaction rates and approximations • Theories of reaction rates • Straight and branched chain reactions 3. Oxidation Mechanisms of Fuels • Hydrogen, CO, hydrocarbons
2 1. Chemical Thermodynamics
3 Chemical Equilibrium (1/2)
• First and Second Laws
d E T d S - p d V (1.2.4)*
• Thermodynamic function: 퐸 = 퐸(푆, 푉, 푁푖)
N (1.2.6) d E = T d S - p d V + d N ii i 1 • Criterion for chemical equilibrium N (1.2.11) dN 0 ii i 1
* Equation numbers refer to those in Combustion Physics 4 Chemical Equilibrium (2/2)
• From element conservation for a general chemical reaction NN vvMM e.g.: H + O ↔ OH + O i ii i 2 ii11 dN dN i j (1.2.14) we have v v v v i i j j • Consequently, chemical equilibrium: N N dN 0 (vv ) 0 ii i i i (1.2.17) i 1 i 1
5 Equilibrium Constant for Reaction
(,)T po () T R o T In(/), p p o • Chemical potential: i i ii (1.2.29) N : ( vv ) 0 • Apply equilibrium criterion i i i N ()vv i 1 pKii T (), ip i 1 (1.2.30) N K( T ) e x p ( v v ) o ( T ) ()RTo pi i i (1.2.31) i 1 o (LHS; RHS): Function of (concentrations; temperature)
o 퐾푝(푇): Tabulated for a given reaction Equilibrium composition of a (fuel, oxidizer, inert) mixture at T and p can be calculated using the 퐾푝(푇)’s for all reactions • Comment: Procedure is cumbersome: needs tabulation for each reaction; there could be 100’s to 1000’s of reactions for oxidation of a fuel 6
Equilibrium Constant for Formation
• Simplification: Relate 퐾푝(푇) to formation reaction of
each species L v MMo i ,, j i ji (1.2.32) j 1
o M ij , : element in standard state o o o (1.2.33) KTTRTp, i() exp[ i ()/( )] ()vv N ii KTKT ()() o (1.2.34) p p, i i 1 • Comment: Number of species ~10’s to 100’s => Much reduced tabulation
7
Energy Conservation in Adiabatic System (State 1 State 2 )
N N N h T ; T o N h T ; T o • i ,1 i 1 i , 2 i 2 (1.4.17) i 1 i 1
h T;; To h o T o h s T T o • i i i (1.4.16) totalenthalpy heatofformation sensibleheat 0 o Heat of Formation, h i ( T ) : heat required to form one mole of reactants from its elements in their standard state, at a L reference temperature T o , v M o M . j i i , j i , j i Sensible Heat: energy required to raise the temperature of T o substance from T to T , hso(;)(). T T c T d T (1.4.14) io p, i T Specific Heat: function of degree of excitation,
c T h s T . (1.4.13) p ,i i p
NNNN • NhT o()()(;)(;). o NhT o o NhTT s o NhTT s o (1.4.18) i,1 i i , 2 i i , 2 i 2 i ,1 i 1 i1 i 1 i 1 i 1 8
Adiabatic Flame Temperature (1/2) • For a mixture (fuel, oxidizer, inert) of given composition and
temperature T1 , determine the final, equilibrium temperature, Tad , and product composition; the process is conducted adiabatically and at given pressure p.
o Tad calculated using 퐾푝(푇), energy conservation, and element conservation o This is the single, most important parameter for a reacting mixture
• Tad is most sensitive to mixture composition ( F / O ) o Fuel/oxidizer equivalence ratio, f . ( F / O ) st o Tad peaks slightly on the rich side of f = 1
o Excess reactants (CO, H2 for rich mixtures) serve as inert.
o Amount and cp of inert serving as a heat sink. 9 Adiabatic Flame Temperature (2/2)
• Slope asymmetry of Tad (f) largely due to asymmetrical definition of f ;
lean ; rich 0 f 1; 1 f
• Near-symmetry attained by using normalized equivalence f ratio, ; 1 f
lean ; rich 0 0 .5; 0 .5 1
• Tad slightly increases with increasing pressure due to reduced product dissociation
10
2. Chemical Kinetics
11 Introduction
• Chemical thermodynamics: relates the initial to the final equilibrium states of a reactive mixture; does not distinguish the path and time in the process (e.g.: acceptable cycle
analysis for i.c. engines, but not NOx formation)
• Chemical kinetics describes the path and rates of individual reactions and reactants; can be extremely complex – 103 intermediates and 104 elementary reactions.
NN k vv M f • For a single-step forward reaction: i ii i , ii11 ˆ o Molar rate of change: i dc i / dt ˆ ˆ j o i and j related by: i , (2.1.3) v v v v • Law of mass action: i i j j
o Reaction rate proportional to product of concentrations (ci) N o Scaled reaction rate is given by: (2.1.4) k T c i f i i 1
o Proportionality constant kf(T) : reaction rate constant; only function of temperature
• Example: H + HO 2→OH + OH
dd[H ]d [HO]2 1 [O H ] = k [H][HO ]. = = = , f 2 d t d t2 d t
13 Reverse Reaction
• Every forward reaction has a backward reaction: NN k vvMM b i i i i ii11 o Net reaction rate: iif ,, ib ()()()v iif v b v ii v
NN vv k cii k c f i b i (2.1.6) ii11 N k f ()vv = cKii o At equilibrium: 0 ic(2.1.7,8) k b i 1
NN vv 1 k cii K c o Implying: f i c i (2.1.9) ii11 • Preferred to (2.1.6) because Kc can be determined more accurately than kb N
v i • Irreversible reaction approximation: kcf i i
14
Multiple Reactions • Practical reactions involving Reactants → Products
e.g.: 2H2+O2→2H2O Rarely (never!) occurs in one step between reactants (e.g.: two H2 directly reacting with one O2)
• Reality: For H2-O2: (at least) 19 reversible reactions and 8 species (H2, O2, H, O, OH, H2O, HO2, H2O2)
• Generalized expression:
NN k kf, v M v M, k 1, 2 , , K , i,, k i i k i k kb, ii11
NN vv k ci,, k k c i k , k k,, fii k b ii11
K ().vv i i,, k i k k (2.1.14) 15 k 1 Rational Approximations
• Approximations based on comparison of rates of certain reaction entities
o Quasi-steady-state (QSS) species approximation
o Partial equilibrium (PE) reaction approximation
16 QSS Species Approximation
• Some chain carriers are generated and consumed at rapid rates such that their concentrations remain at low values and their net change rates are very small.
dci i = =, ii o For dt
dci o If (,),ii dt . o Then ii • Consequence: (implicit) algebraic instead of differential solution
• Note: dci/dt may not be negligible compared to other rates
17 Partial Equilibrium Approximation
• If both the forward and backward rates of a reaction k is much larger than its net reaction
rate, NN vv k ci,, k k c i k 0 o then set: k k,, f i k b i ii11
NN vv o such that k ci,, k k c i k k,, f i k b i (2.1.17) ii11
o which yields an algebraic relation between ci’s.
• Note: k not necessarily small compared to i. • Example: formulation of the transition state theory of reaction rates, to be shown later.
18 Approximation by Global and Semi- global Reactions • Successive application of QSS species and PE reactions will eventually lead to a one-step global reaction (at least theoretically!). o The process is tedious; results still depend on the individual reaction rate parameters most of which are not known. o Solution also requires iterations. • May as well just start with a one-step reaction Fuel + Oxidizer k Products N n kci , described by i i 1 where ni is the reaction order, and is empirical
19
Reaction Order and Molecularity
• Molecularity, 휈푖 : number of colliding molecules in an elementary reaction
o 휈푖 is a fundamental parameter; 휈푖 = 1, 2, 3. o 휈푖 = 3 is important for recombination reactions (negative influence on progression of reaction
e.g. H+O2+M → HO2+M. • Reaction order, 푛푖 : influence of concentration of i on the reaction rate
o 푛푖 is an empirical parameter
o 푛푖 mostly < 2
o ni can be negative! o 푛푖=푛푖(푝).
20 The Arrhenius Law
• Prescribes the dependence of the reaction rate constant on temperature:
d ln k ( T ) E a , d T Ro T 2
o ERTa / o For constant 퐸푎: k( T ) = A e
o Modified form: AA(T)BT = =
21 The Activation Energy
• 퐸푎,푓: minimum energy needed to initiate a reaction
o Large 퐸푎 leads to temperature sensitivity o Some reactions (e.g. 3-body termination) have 퐸푎 ≡0.
22 The Arrhenius Number, Ar
ET Ar aa o RTTm a x m a x
e x p (Ta / TT ) m a x = e x p Ar 1 e x p ( Ta / TTm a x ) • Reaction is temperature sensitive for Ar≫1 T/Tmax • Combustion systems: Ar≫1 • Ar≫1 => localizes reaction regions (spatial or temporal)
o dc ERT/ B c e a dt
23 Collision Theory of Reaction Rate (1/3)
• Assumptions: o Equilibrium Maxwell velocity distribution o Two-body hard-sphere collision o Reaction occurs if collision (translational) energy exceeds activation energy
24 Collision Theory of Reaction Rate (2/3)
m m m/() m m • Reduced mass: i, j i j i j
= ( + )/ 2 • Collision diameter: i, j i j 1 / 2 8 kTo • Collision velocity: V ij, , m 1 /2 i o 2 8 kT Z = n n . • Collision frequency per volume: i,, j i j i j m ij, * n o • Boltzmann velocity distribution: = e ERT*/ . n • Collision frequency with energy in excess of
(Ei+Ej=Ea) 1 /2 o o dn * 2 * * 8 kT ERT/ dni j a Z = i,, j n n Z i j e = i, j i j m d t d t ij, 25 Collision Theory of Reaction Rate (3/3)
0 • Relating cii n /A
1 /2 o oo dci o 2 8 kT ERTERT// = = A c c eaa A ( T ) c c e . i, j i j i j d t m ij, • Comparing: 1 / 2 o o 2 8 kT ATA( ) ij, , 1 / 2 m ij, • Deviation from theory accounted by steric factor
AZ
26 Transition State Theory of Reaction Rate (1/3) • Overall reaction consists of two steps:
N k1, f vR R ‡ o Activation: ii k i 1 1,b N ‡ k RP , 2 v o Product formation: ii i 1
‡ • Reaction rates for Ri and R
N dcR v i j v k c v k c ‡ i1, f j i 1, b R dt j 1
N dc ‡ R v j k c k c‡‡ k c . 1,f j 1, b RR 2 dt j 1
27 Transition State Theory of Reaction Rate (2/3)
• Assumptions: o Partial equilibrium for activation step N ‡ v j c‡ k c d c/0 d t R cj R i j 1 ‡ o Steady-state for activated complex, R
dc dc ‡ R i R v k c ‡ . i 2, R d t d t
dc ‡ dcR R i yields d t d t
N dc ‡ R v i j v k c‡ v k K c , i2R i 2 , c j dt j 1
N dcR v ‡ i v k c j • Compared with ij yields k k2 K c dt j 1
28 Transition State Theory of Reaction Rate (3/3)
• To estimate k2: o Kinetic energy = Vibrational energy 1 2 (ko T ) h 0 2 o Assume products form during one vibration k 2 • Therefore:
kTo kK ‡ . o c h
29 Theory of Unimolecular Reactions
A unimolecular reaction
RP , k is really the high pressure limit of a second order reaction R + M P + M where M is a collision partner
k k constant as p dc R First-order reaction kcR dt k k00 p~ k cR a s p 0 Second-order reaction
30 Straight Chain Reactions: Halogen-Hydrogen System (1/3)
• Straight chain: The consumption of one radical produces another radical
• Hydrogen-halogen system (X2: I2, Br2, Cl2, F2)
k1, f X 2 + M X + X + M Chain initiation (X1f )
k 2,f X + H2 HX + H Chain carrying (X2f )
k 3,f H + X2 HX + X Chain carrying (X3f )
k X + X + M 1,b X2 + M Chain termination (X1f )
k H + HX 2,b X+ H2 Chain carrying (X2 f )
31 Straight Chain Reactions: Halogen-Hydrogen System (2/3) • Reaction rates: • Apply steady- state assumption d [H]2 = -kk2 , f [X][H2 ] + 2 ,b [H][HX] dt for H and X: d [X] 2 -k [X ][M] - k [H][X ] + k [X]2 [M] 1 , f22 3 , f 1 ,b dt d [H] d [H] k2 , f [X][H22 ] - k 3 , f [H][X ] - k 2 ,b [H][HX] dt 0 dt d [X] d [X] = 2k1 , f [X2 ][M]- k 2 , f [X][H 2 ] + k 3 , f [H][X 2 ] dt 0 dt 2 + kk2 ,b [H][HX] - 2 1 ,b [X] [M] d [HX] k2 , f[X][H22 ] + k 3 , f [H][X ] - k 2 ,b [H][HX] dt
32 Straight Chain Reactions: Halogen-Hydrogen System (3/3)
• Detailed analysis yields: 1 /2 1 /2 d [HX] 2k2 ,f ( k 1, f / k 1, b ) [H 2 ][X 2 ] = . (2.4.8) d t1+( k / k )[HX]/[X ] 2 ,b 3, f 2 • One-step reaction yields: k H + X 0 2HX (X0) 22
d [HX] = 2k 0 [H 2 ][X 2 ], dt (2.4.1)
• Comparing, detailed analysis shows o Complex instead of linear dependence on [X2] o Inhibiting effect of [HX] 33
Branched Chain Reactions:
H2-O2 System • The consumption of one radical generates more than one radical
H + O2 → OH + O Chain branching (H1)
O + H2 → OH + H Chain branching (H2)
OH + H2 → H2O + H Chain carrying (H3) • The net of (H1) to (H3) yields 2H per cycle:
3H2 + O2 → 2H2O +2H • Different radicals have different reactivities => Chain carrying steps can be weakening
H + O2 +M→ HO2 +M (HO2 less reactive than H)
CH4 + H → CH3+H2 (CH3 less reactive than H) 34 Branched Chain Reactions: Pressure Effect (1/2)
k n R 1 C In itia tio n
k R C 2 a C P Chain branching cycle
k CRRP g Gas termination
k CP w Wall termination
d [C ] n 2 = k12 [R] + ( a 1) k [R][C] kgw [R][C] k [C] dt
n k1 [R] + k 2 [R]( a a c )[C]
2 kkgw[R ] + a c = 1 + . k 2 [R]
35 Branched Chain Reactions: Pressure Effect (2/2)
d [C ] n • = k1 [R] + k 2 [R]( a a c )[C], (2.4.15) dt
o blows up for a > ac
o delays for a < ac
2 kkgw[R ] + • a c = 1 + k 2 [R]
k 1 + w a s p 0
k 2 [R]
k g [R ] 1 + a s p
k 2 36 3. Oxidation Mechanisms of Fuels
37 Oxidation of Hydrogen (1/2)
38 Oxidation of Hydrogen (2/2) • Initiation: q 104 kcal/mole H2 + M → H+ H+ M p
O2 + M → O + O + M 118kcal/mole
H2 + O2 → HO2 + H 55kcal/mole; preferred • First-limit chemistry: o Wall termination of H as p • Second-limit chemistry o Branching & carrying:
H + O2 → O + OH
O + H2 → H + OH
OH + H2 → H + H2O o Termination:
H + O2 + M→ HO2 + M (3-body reaction promoted as p↑)
HO2 → wall termination 39
Second-limit Chemistry
d [H] = -k1 [H][O2 ] + k 2 [O][H 2 ] + k 3 [OH][H 2 ] - k 9 [H][O 2 ][M] dt d [O] = kk12 [H][O22 ] - [O][H ] dt d [OH] = k1 [H][O2 ] + k 2 [O][H 2 ] - k 3 [OH][H 2 ]. dt • Assume steady state for O and OH:
d [H] (2kk19 - [M])[H][O2 ] = 2 1 - 9 dt (3.2.4,5)
(2kk - [M])>0 o System explodes if 19 2 k p1 R T . o Second limit: 2kk19 = [M ] k 9
40 Third-Limit Chemistry
• [HO2] increases with increasing pressure, leading to:
HO2 + H2 → H2O2 + H
H2O2 + M → OH + OH + M • At high temperatures and pressures, more radicals are produced, leading to radical-radical reactions
HO2 + HO2 → H2O2 + O2
HO2 + H → OH + OH
HO2+ O → OH + O2
41
Role of Initiation Reaction
Homogeneous System Diffusive System with Flame o Shock tube, flow o Radical produced in high- reactor,… temperature flame o No diffusive transport o Radical back diffuses and reacts with incoming o Radicals generated from reactant original reactants o e.g.: H + O2 → OH + O o e.g.: H2 + O2 → HO2 + H o Different (lower) global activation energy
Flame H + O H ,O H H ,O 22 HO, H 22 H + O 2 2 2 2
42 Oxidation of Carbon Monoxide
• Direct oxidation rarely relevant:
CO + O2 → CO2 + O o High activation energy (48 kcal/mole) o No branching o Hence no dry CO oxidation • Dominant oxidation route:
CO + OH → CO2 + H (CO3)
o Integrated to the H2-O2 chain
o H2, H2O are catalysts for CO oxidation o Extremely sensitive to moisture content
43 General Considerations of Hydrocarbon Oxidation • Most important reactions in HC oxidation:
o Chain initiation: (H, HO2) o Chain branching: H + O2 O H + O o Heat release: C O + O H C O2 + H • HC oxidation is hierarchical:
o Large HC molecule breaks down into smaller C1,C2,C3 fragments in the low-temperature region/regime, with small heat release, which subsequently undergo massive oxidation with large heat release
• Dominant low-temperature chemistry: HO2 chemistry
• Dominant high-temperature chemistry: H2-O2 chain
44 Methane Oxidation
45 Methane Initiation Reactions
• High-temperature route:
CH4 + M → CH3 + H + M
H + O2 → OH + O
CH4 + (H, O, OH) → CH3 + (H2, OH, H2O)
o CH4+H →CH3+H2 is retarding (exchange H by CH3)
Ignition delay time increases with [CH4] ! • Low-temperature route:
CH4 + O2 → CH3 + HO2
CH4 + HO2 → CH3 + H2O2
H2O2 + M → OH + OH + M
46 Methyl (CH3) Reactions
• Oxidation path:
o Continuous stripping of H eventually leads to CO2
o Oxidation of H leads to H2O • Growth path:
o CH3 + CH3 + M→C2H6 + M latched onto the ethane oxidation path
o CH3 + CH3 →C2H5 + H latched onto the ethyl oxidation path
47 Closing Remarks of Day 1 Lecture (1/2)
• Adiabatic flame temperature, Tad , and equilibrium composition, ci o Consequence of chemical equilibrium and energy conservation o Single most important property of a combustible mixture; peaks around f = 1
o Is there a Tad for nonpremixed systems?
48 Closing Remarks of Day 1 Lecture (2/2) • Oxidation mechanisms of fuels are described by a plethora of coupled, nonlinearly interacting elementary reactions o Track radicals in branching, carrying & termination reactions
o (H, HO2) are key radicals in (high, low) temperature regimes
o Increasing T favors 2-body,large Ea reactions; increasing pressure favors 3-body, termination reactions, with Ea=0
o Combustion is characterized by large values of overall Ea; i.e. the Arrhenius number, Ar, when referenced to Tad o Large Ar confines reactions to narrow spatial regions or
time intervals 49
! Daily Specials !
50 Day 1 Specials
1. Significance of normalized parametric representation (e.g.: scatters in laminar flame speed determination) 2. Growth & reduction of complexity in combustion CFD
3. Cubic description of the H2-O2 Z-curve: towards “analytical” chemistry
51 1. Significance of Normalized Parametric Representation: The Equivalence Ratio
• Assessment of premixed systems frequently are based on equivalence ratio: (퐹/푂) 휙 = ; Lean (0 < 휙 < 1); Rich (1 < 휙 < ∞) (퐹/푂)푆 • A normalized, symmetrical definition: 휙 훷 = ; Lean (0 < Φ < 0.5); Rich (0.5 < Φ < 1) 1+휙
Example: Hydrogen/Air Flame Speeds
• Hydrogen/oxygen kinetics forms the backbone of hydrocarbon chemistry • Laminar flame speed is a key component in validating the mechanism • When plotted in f, tight correlation on lean side versus wider scatter on rich side has led to concern on adequacy of rich chemistry Normalized (x, y) Representations
• Use of moderates difference in scatters • Furthermore, since regime of lean flames consists of more low flame speeds, normalizing the flame speeds (y-axis) shows even larger scatter on the lean side! • As such, lean/rich chemistry are equally accurate/inaccurate! Further Thoughts
• Could the use of f instead of have caused misinterpretation in other combustion phenomena (e.g. sensitivity of NOx formation)? • In scientific investigations, parameters representing the ratio of two processes/factors are frequently used to simplify/focus analysis & understanding; but it could skew interpretation & hinder unification • Consider the following parameters: Re, Gr, Pe, Da, Ka,… • Try a formulation of the N-S equations based on a
normalized Re, as ReN = Re/(1+Re) 2. The Unrelenting Growth of Size of Reaction Mechanism!
• Exponential growth of mechanism Lu-Law Diagram size with increasing fuel size o N ~ 103; K ~ 104 o Empirically: K ≈ 5N • Size prohibitively large: o For insight even with sensitivity analysis o For CFD: Not even possible for simple 1D flames; compete with fluid- scale resolution for turbulent flames • Is this growth forever enslaving us to the super-computer? • Need systematic reduction in all aspects of computation • Hope & goal: Chemistry-limited asymptotic size 56 Additional Concerns: Inadequate State of Mechanism Development
100
10 n-Heptane-Air
p = 1 atm 1 f = 1
0.1
0.01
Ignition Delay (sec) , DelayIgnition (sec) without low-T chemistry 0.001 with low-T chemistry
0.0001 0.5 1 1.5 2 1000/T (1/K)
Example: Various versions of • Adoption of mechanisms GRI-Mech yield qualitatively beyond applicability range different results • One generally cannot get the right answer with the wrong reactions! 57 Further Challenges in Computation
~K2 ~K 10-6 4 Shortest flow time 10 PRF Diffusion is o -o c ta n e n -h e p ta n e Chemistry -9 Ethylene, 10 iso-octane, skeletal p = 1 atm
T0 = 1000K n-heptane, skeletal
3 10 C 1 -C 3 1,3-Butadiene -12 n-Heptane, 10 DME p = 50 atm G R I3 .0
T0 = 800K Number of exp functions G R I1 .2 C h e m is try D iffu s io n 10-15 2 Shortest SpeciesShortest Scale,Sec Time 1000 2000 3000 10 1 2 3 10 10 10 Temperature, K Number of species , K Stiffness in reaction rates Crossing point: K~20 Evaluation of diffusion coefficients can overwhelm that of chemistry
58 Strategy for Facilitated Computation
Detailed mechanisms Skeletal mechanisms Reduced mechanisms
CH4: 30 species CH4: 13 species CH4: 9 species
C2H4: 70 species C2H4: 30 species C2H4: 20 species nC7H16: 500 species nC7H16: 80 species nC7H16: 60 species
DRG, DRGASA Isomer Lumping CSP QSSDG
Skeletal reduction Time scale reduction Analytical QSS solution Minimal diffusive species C2H4: 9 groups Reduced mechanism nC7H16: 20 groups • Fewer species/reactions CCR DCL • Less stiff • Faster simulation Computation cost reduction Diffusion reduction
WP Air Force Base Georgia Tech SANDIA
C2H4, VULCAN C2H4, 3D, LES CH4, 3D, DNS
CFD simulations
Time savings: factors of 10 ~ 100 3. Structure & Cubic Description of the
H2-O2 Explosion Z-Curve
• Explicit expression was derived for the 2nd limit. How about the entire Z-curve? • Shape of Z-curve suggests 3rd degree polynomial description • Explicit expression can be used to rule out unphysical situations
60 Structure of H2-O2 Z-Curve
1st and 3rd limits due to The backbone 2nd limit surface deactivation
61 Cubic Description of H2-O2 Z-Curve
• Assume steady state for O, OH and H2O2 • Detailed analysis yields:
d H H 퐼 = 퐴H,HO2 + d푡 HO2 HO2 퐼
2푘 O − 푘 O M − 푘 3푘 H 퐴 = 1 2 9 2 H 17푏 2 H,HO2 푘 O M −푘 H − 푘 9 2 17푏 2 HO2 • Neutral condition yields cubic equation in [M] ~ p: a(T)[M]3+b(T)[M]2+c(T)[M]+d(T) = 0 • Cubic equation yields explicit expressions for three limits, high & low pressure parabola, & loss of non-monotonicity 62