Work-Principle Model for Predicting Toxic Fumes of 

Nonideal Explosives 

Michael S. W1eland6 

National Institute of Occupational safety and Health, Pittsburgh Research Center, P.O. Box 18070, Pittsburgh, PA 

15236-0070 (USA) 

Abstract 

The work-principle from thermodynamics was used to formulate a mode1 for predicting toxic fumes from mining explosives in 

underground chamber tests, where rapid turbulent combustion 

within the surrounding air noticeably changes the resulting 

concentrations. Two model constants were required to help 

characterize the reaction zone undergoing rapid chemical transformations in conjunction with heat transfer and work output: a 

stoichiometry mixing fraction and a reaction-quenching temperature. Rudimentary theory with an unsteady uniform concentration gradient was taken to characterize the combustion zone, 

yielding 75% for the mixing fraction. Four quenching temperature 

trends were resolved and compared to test results of ammonium 

nitrate compositions with different fuel-oil percentages (ANFO). 

The quenching temperature 2345 K was the optimum choice for 

fitting the two major components of fume toxicity: carbon 

monoxide (CO) and total nitrogen oxides (NOx). The resulting 

two-constant model was used to generate comparisons for test 

results of ANFO compositions with additives. Though respectable 

fits were usually found, charge formulations which reacted weakly 

couid not be resolved numerically. The work-principle model 

yields toxic concentrations for a range of charge formulations, 

making it a useful tool for investigating the potential hazard of 

released fumes and reducing the risk of unwanted incidents. 

Ke-words:Toxicity; Fumes; Explosives; Models; Mining-industry; Health-hazards; Toxic-gases; Toxic-vapors Toxicity; Fumes; Explosives; Models; 

1 Introduction 

The mining explosives tested in the underground chamber 

generate reaction zones that undergo turbulent mixing and 

rapid combustion with the chamber air, further transforming the fume concentrations, unlike rock blasting where the 

late-stage burning is inhibited. The NIOSH underground 

test chamber is 274 m3, holding 324 kg air at standard 

conditions. Galvanized tubing or steel pipe is used to retain 

the 4540 g of .mining explosive under test. The total charge' 

mass, nominally 4712 g, includes a 172 g booster with 50- 50 

weight percent TNTand PE'I??. The resulting fume toxicity 

is normally dominated by carbon monoxide CO and total 

nitrogen oxides NOx, taken as the sum of nitric oxide (NO) 

and nitrogen dioxide (NOz). The traditional Russian 

formula for the relative fume toxicity (RFT) is their 

weighted sum: RFI' = CO + 6.5 NOx [I, 21. 

Traditional detonation codes are unable to account for the 

unfinished or transformed reactions' with air ingestion, 

without notable and questionable modifications. Test results 

for CO and NOx could not be reconciled with the theoretical 

predictions by detonation codes that rest just upon the 

traditional requirements of the Zeldovich-von NeumannDoering (ZND) theory [3,4]. For ZND theory, the reaction 

therrnicity disappears at the Chapman Jouguet (U) condition to remove a numerical singularity in the related flow 

equation. The requirement of zero-themicity is a Tecognized conflict for nonideal explosives that undergo uninterrupted transitions from detonation to deflagration [S]. 

The conflict can be rendered moot by postulating some type 

of loss term (degradation) that effectively removes the CJ 

singularity [6]. ZND theory requires a real-gas equation-ofstate (EOS) [7] for the isentropic trajectory from the CJ 

regime to some quenching temperature(s), whereupon the 

relevant fume concentrations are regarded as frozen. For the 

underground tests, the turbulent mixing and rapid combustion are not isentropic processes. Rather than postulating a 

questionable rendition of ZND theory with a non-isentropic 

trajectory and a requisite loss term, a different theoretical 

platform was chosen. 

The work-principle from traditional thermodynamics 

with imposed constraints was utilized to render a computational algorithm for predicting toxic fumes for a wide range 

of explosive formulations. The work-principle capitalizes on 

the restorative influences that constantly try to render total 

thermodynamic equilibrium, diminishing the role of process 

history. Though the model has theoretical disadvantages 

when compared to the rigorous structure of detonation 

theory, there are issues worth noting: First, the whole 

thermodynamic trajectory for the nonideal process is not 

required, just that for the region near the quenching 

transition. Utilizing a real-gas EOS, which is itself a 

questionable issue, is unnecessary, as the renowned ideal 

form suffices. Second, the work-principle model incorporates a loss term, which is ultimately determined rather than 

stipulated. Third, the thermodynamic trajectory found for 

the quenching state is normally nonisentropic. revealing the 

irreversible consequences of turbulent mixing, rapid combustion and other reaction-zone losses. 
This report discusses the work-principle technique, the Zstate constraint for the reaction-fume zone without interaction and the N-state constraint for reckoning loss. The 

rudimentary mixing rule with its 75% fraction and the trials 

that yield the optimum quenching temperature are reviewed. The graph of the loss rate versus fuel percent, which 

resembles a ranking trend for the charge performance, is 

worth noting. Lastly, the two-constant model is used to 

generate comparisons for test results of ANFO compositions with additives. 

2 Reaction Process in Work-Principle Terms 

The work-principle technique recasts the second law of 

thermodynamics for a composite system, which is isolated 

from the rest of the universe and therefore governed by the 

four laws of thermodynamics [8, 91. The work-principle 

construction retains thermodynamic interaction in restricted 

internal ways for its three components: the working fluid, the 

reservoir and the mechanical agent. The working fluid 

transfers heat or works upon the reservoir in reversible or 

irreversible ways It works reversibly upon the mechanical 

agent in other ways (non-expansion). No mass transfer 

occurs, so the working fluid retains constant mass while 

undergoing reaction or transformation. The working-fluid is 

regarded as restored or wound-up thermodynamically to the 

wanted (or initial) state within the isolated composite system 

via reversible work, rendered by the mechanical agent. 

The work-principle model for toxic fumes takes the zone 

. of transforming reactants as the working-fluid, their surroundings as the reservoir and the mechanical agent 

ultimately remains nondescript. The working-fluid is originally composed of the explosive ingredients with a tiny (or 

zero) quantity of chamber air, called reactant-air to 

distinguish it from the remaining portion taken as a 

reservoir component. Reactant-air is not a nebulous quantity, but rather the proportion tabulated in the thermodynamic reaction code (TDRC). The rupture and fragmentation of the cylinder pipe or tube is not wholly ignored, 

though it is reckoned with rather indirectly through the loss 

term in the work-principle model. 

Restoring influences continually try to maintain mechanical, thermal, and chemical equilibrium within the reaction 

zone. For an underground test, the zone expands rapidly, the 

temperature drops quickly and the reaction rates fall 

precipitously, so the chemical equilibrium falters, resulting 

in non-equilibrium concentrations. Wall or dust interactions 

or transformations those remain noticeable over the hour of 

measurement ruin the notion of wholly £rozen concentrations The rusty-tinted oxidation that rapidly transforms NO 

into NO, at cooler temperatures conserves the NOx, which 

is lost more slowly otherwise. Concentrations that remain 

useful for toxic fume comparisons stay relatively unchanged 

from the quenched state, like CO and NO,, rather than NO 

or NO2. 

The reservoir conditions are taken equal to the standards 

utilized for reducing the underground test data: temperature TR = 298.15 K (25 OC) and pressure PR = 101325 Pa 

(1 atmosphere). The work-principle is more utilitarian when 

the reservoir retaibs those constant characteristics, implying 

that it has tremendous mass compared to the other two 

components. This restriction is compatible with the underground test circumstances, where the working fluid consists 

of the total charge, -4.7 kg, plus the reactant-air, with 

typically less mass. The resultant mass is'tiny compared to 

the reservoir mass which includes the remaining chamber 

air, roughly 320 kg, without worrying about the waIis or 

other inert objects that help sustain the reservoir corzdltlons. 

With the restriction of constant reservoir conditions, the 

differential work principle can be rendered in integrated 

form with a useful quasi-potential. 

3 Working-Fluid State ~uncti'on Notation 

Throughout the report, the zone-summing (or explicit) 

variables are taken per unit constant mass of the working 

fluid, which is the type of data used with the TDRC. 

Reduction in the notation is convenient for rendering 

imposed constraints. The resultant sum of chemical potentials pK's times their respective stoichiometry coefficients 

vK9s yields Q, which works with the reaction coordinate 2. 

The reciprocal of the molecular weight A, the sum of NK, 

resets the units, while the negation (minus) means that Q 

normally has the sign of Z (i.e., spontaneous reaction in an 

insulated rigid box). The flow work W, or ratio of pressure P 

to densityp, works with the natural logarithmic density Y= 

In [p/RWO], where RHO is just a units constant. The 

traditional differential relation for the internal energy U is 

rewritten in contracted form, relation (1). The notation (S or 

d means infinitesimal variation or differential respectively, 

while A implies non-infinitesimal change. 

RHO = Const 

Real-gas EOS usually reduce to the Gr-iineisen form, W= 

J (U - I), where W is proportional to U with coefficient 

functions J and I that remain roughly constant over a modest 
M. S. Wieland 

thermodynamic domain. Working-fluid state functions W 

and T remain positive for stable fluids [8], while Q can 

undergo sign reversal. The third law of thermodynamics 

stipulates that zero temperature is unattainable, so Tand TR 

are never regarded as zero [lo]. There are two ways to null@ 

the chemical-work [Q, Z] term: the restriction to constant Z. 

which yields frozen concentrations or the restriction to 

(constant) zero Q, which implies chemical equilibrium. 

Y = U - TR . S + P,/p, TR, PR remain constant (2) 

The quasi-potential Y in definition (2) incorporates the 

working-fluid properties U, p, and entropy S per unit 

working-fluid mass in conjunction with two constant 

reservoir properties TR and PR. Rigorously speaking, the 

retention of the reservoir influence means that Y is not a 

thermodynamic state function [ll], though it retains the 

zone-summing character of the working-fluid state functions. This renders Y with a formidable nature that is 

quantifiable throughout the whole of thermodynamic space, 

regardless of the process undertaken or imagined. 

4 Theory Underlying the Work-Principle 

The work-principle theorem with the restriction to 

constant reservoii: conditions is succinctly stated in relation 

(3). using the Y-definition (2). The rule requires that the 

resultant for the work done upon the mechanical agent, . 

WORK, plus the quasi-potential change, AY, remain zero 

for reversible processes or negative for irreversible processes [8,9]. For reversible processes, the WORK done upon 

the mechanical agent would equal ( = ) the reduction in the 

quasi-potential, [ - (AW)]. Natural processes refer to recognizable (non-infinitesimal) changes or jumps in thennodynamic variables over (finite) time, so they are irreversible, 

however slight, and the inequality (<) holds. Reversible 

transformations with their innumerable quasi-static steps 

would take forever to render such a transition, though they 

represent the unreachable limit. 

the working-fluid ultimately reaches the reservoir circumstance, yielding no information regarding the route taken. 

The notion of removing and imposing hypothetical constraints, to tentatively stall the trajectory descent at intermediate circumstances (key-states), circumvents the noinformation trap and represents a way to characterize the 

. wanted process. 

5 Resolution of the 2-State and N-State Constraint 

The work-principle model' for resolving toxic fumes 

imposes just two constraints upon the reaction zone: the 

zero-loss constraint for the Z-state and the nonzero-loss 

constraint for the rate-quenching N-state. The notation 

X = constant represents a generic thermodynamic constraint upon some function of working-fluid variables. The 

technique requires an unrestrained variable, taken as the 

logarithmic density Y, since the rigid-box condition, 

Y= constant, is not required. The TDRC yields circumstances for chemical equilibrium rather than frozen concentrations, so Q = zero = constant rather than Z = constant. For manipulating constraint relations, the working-fluid 

state variables are regarded as functions of [X, Y; Q] rather 

than [S, Ir: Z]. Relations (1) are worthwhile for rearranging 

terms. For the quasi-potential minimum, the thermodynamic trajectory is tangent to the constraint surface, so GX, Q, 

6Q terms disappear, while the unrestrained GY terms 

remain, yielding the Y-minimum for the generic-X constraint, relation (4). 

For natural closed-cycle processes, the working-fluid 

returns to its original state, restoring the quasi-potential, ' AY = 0, so reversible work must be rendered by (not upon) 

the mechanical agent, WORK < 0. The WORK inequality 

would occur on every cycle, without the reverse possibility, 

removing any prospect of perpetual motion (2nd kind), 

regardless of the thermodynamic process undertaken. 

In the non-cyclic rendition for modeling fumes, the 

working-fluid is taken to the wanted (initial) reactant state, 

where the mechanical agent is rendered moot, so it is 

unnecessary to identify it with some underground object. 

With WORK = 0, the work-principle formula (3) reduces to 

A$! ,< 0. Without restoration work, Y is unable to increase, 

though it decreases from natural fluctuations until it reaches 

a trapping minimum condition. Without some constraint, 

The K-function definition, where '3 is the Universal 

Gas Constant, was utilized for restating the terms in Eq. 

(4). For reservoir conditions, K= KR=40.874 molin?. For 

zero interaction, the working fluid retains its original internal 

energy, X= U = constant = Uo, so the loss tern disappears, 

a,=a, =0, via its definition. With W.6Y nonzero, the 

residual bracket term reduces to zero upon reaching the 

minimum, yielding KZ = KR, where the Z subscript refers to 

the Zstate. The triplet [Uz, Kz, Q,] = [Uo KR, 01 characterizes the 2-state for any thermodynamic working-fluid, 

regardless of its EOS representation or structure or composition [12]. 
The N-state was resolved for nonzero reduction ( - dU), 

taken as work output upon or heat transfer to the reservoir, 

during the expansion ( - dY), yielding a non-unique though 

positive loss rate ax. The N-state constraint was reckoned in 

log-log [T, K] form, relation (9, by requiring that [J- a,] 

remain constant [JeaL] and integrating the differential 

equation noted in Appendix A. This restriction was 

rendered plausible by recognizing that J reduces to the 

Grilneisen gamma rG function, which remains relatively 

constant over a modest thennodynamic domain. The Nstate constraint relation was taken back though the 2-state 

to remove the unwanted constant of integration. For the 

rough approximations utilized in the work-principle technique, the Zstate rG was taken, rather than recursively 

solving for the N-rG. though that is a tractable refinement. 

where K, = KR, TN = TQ, J, a~ = Const. 

With T,, Kz and J ( = T,,) known, the Y-minimurn,[T, 

bracket and logarithmic [T, I(1 constraint still contain three 

unknowns [Khh aL, TN], where the N subscript refers to the 

N-state. Hypothetically, the N-state represents the last 

remnant of total thermodynamic equilibrium, where the 

relevant rates drop precipitously with zone temperature, 

quenching further transitions and regulating the outcome. 

The two relations yield solutions with the requirement that 

the N-state temperature TN refer to a unique quenching 

temperature Tp, which was thought worth trying. Within the 

work-principle model therefore, the Z and N states were 

resolved without taking into account the whole thermodynamic trajectory or process history. 

6 Thermodynamic Reaction Code, TDRC 

Though historically TDRC have utilized rate-kinetic 

formulations, more recently they use numerical minimization, so two theorems derived from the traditional workprinciple are quite renowned, namely the 'minimizations of 

Gibbs energy and Helmholtz energy [9]. Restrictions within 

the TDRC that correspond to the working fluid under 

thermal and mechanical equilibrium invoke Gibbs minimization. Replacing the mechanical equilibrium with the 

restriction of constant density therein would invoke Hehholtz minimization. The type of minimization utilized within 

the TDRC routines is regulated internally by the choices of 

the TDRC input [13], which were readjusted in a compatible 

way to resolve the trapping Y-minimum under an imposed 

constraint. The TDRC output for the Y-minimum then 

yields the remaining thermodynamic state functions and 

concentrations for chemical equilibrium. The Z-state results 

are utilized for resolving the rate-quenching N-state, where 

Conc.CO vrs. Fuel[%] 

0.0 2.5 5.0 7.5 10.0 

Fuel Oil [%] - No Additive 

Figure l. Types of theoretical CO concentration versus fuel 

percent, with test results. 

Conc.NQx vrs. Fue I[%] 

10.0 ! 1 

Fuel Oil I%] - No Additive 

Figure 2. Types of theoretical NOx concentration versus fuel 

percent, with test results. 

relevant frozen concentrations are comparable to underground fume measurements. 

For ZND results without ingested air, the reference code 

was old [14], so the charge formulations were rerun with 

updated code and real-gas EOS, without trying to implement 

non-standard modifications [IS]. Work-principle Z or N-state 

results for typical reactants were rarefied, KN < KZ = KR, with 

high temperature, TZ > TN= Tp and minor proportions of 

condensed phases Under such conditions any real-gas EOS 

rendered in the virial K-series from the virial P-series [7], 

would reduce to the universal ideal form, because the higherorder terms remain negligible. Logically the thermodynamic 

circumstances warrant using the NASA Lewis Complex 

Chemical Equilibrium Code for ideal gases with condensed 

phases taking up zero volume [16]. Registered ingredients in 

the TDRC data files were used, when possible. The notable 

exception was the fuel oil, which had the molecular formula 

C14H24 and heat of formation - 247 kJ/mol, taken from a 

regression fit for related hydrocarbons 

The NASA-TDRC has the wonderful installed capacity to 

resolve the ratio of fuel-to-oxidizer (ROF) mass fractions 

for stipulated r-equivalence (P), a type of common valance. 

The TDRC yields the ROF for stoichiometry with r* restricted 

to unity, which would require 5.68% fuel oil in our ANFO 

composition, when reactant-air and the booster are ignored 
240 M. S. Wieland 

When the TDRC ingredienfs without reactant-air yield r* > 1, 

the TDRC can resolve the requisite proportion for the 

reduction to F =l, referred to as F-air. For this trick, the 

ingredients were tagged 'fuels' regardless of their nature, with 

reactant-air the only 'oxidizer'. The trick won't work when noair results yield rt 5 1, since the requisite quantity of ingested 

air would be negative or zero, whereupon none is taken. 

The CO and NOx concentration versus fuel percent are 

shown in Figures 1 and 2, respectively. Theoretical trends 

are reported for the detonation code with standard runs, 

work-principle results for no-air, r*-air, their mixture and 

the test results (averages) [17]. Concentrations represent the 

specie volume per unit total charge mass [(Lkg) or (cm3/g)] 

taken at the stipulated reservoir conditions. To reduce the 

graphical clutter, only the work-principle results for the 

optimum (in retrospect) quenching temperature are displayed. The graphs reinforce the modeling problems noted 

previously: the unmodified detonation code and no-air 

trends would not fit the test results. Fuel-rich test data were 

trapped between the r*-air and no-air trends, indicating that 

a weighted sum (mixture) should work. 

7 Rudimentary Mixing Theory 

For the rapid turbulent air combustion, the mixture 

concentration CJ of the J-th species was taken as a weighted 

superposition of the TDRC results for cJ(No) without air 

and cJ(r*) for r*-air stoichiometry. The relative proportions 

depend upon a 'mixing' fraction, which was resolved with 

the following rudimentary hypothesis. The reaction-fume 

zone was regarding as spherical, with radius R = R(t), at time 

t. The centre concentration cJ(0) was taken as cj(N0). The 

surface concentration cj(R) was taken as cJ(r*). The 

unsteady concentration gradient k = k(r) was regarded as 

uniform and therefore equal to the concentration difference 

[cj(R) - cj(0)] divided by the radius R. Integration with 

respect to r over the range, 0 5 r 5 R, yields the mixing rule, 

relation (6): the zone-average concentration < cJ> or CJ 

equals c,(No) times 25% plus cJ(r*) times 75%. The 75% 

is referred to as the mixing fraction, while the other 

multiplier, 25%, is just the difference from unity (100%). 

The mixing rule is rendered trivial when r*-equivalence 

implies a negative or zero quantity of reactant-air, since 

cJ(r*) reduce to cJ(No) results. Replicated tests had sufficient concentration dispersion that refinements in the 

mixture-rule theory were not warranted [17]. 

where cJ(r) = cJ(0) + k . r 

Conc.CO vrs. Fuel[%] 

(5.3 2.5 5.0 7.5 18.8 

FUU I Oil [%] - NO Add itive 

Figure 3. Quenching-temperature CO concentration versus fuel 

percent, with test results. 

Conc.NOx vrs. Fue I[%] 

0.0 2.5 5.0 ' 7.5 10.0 

Fire I Oil [%I - No Add itive 

Figure 4. Quenching-temperature NOx concentration versus 

fuel percent, with test results. 

8 Quenching Temperature Optimum 

The mixture-rule CO and NOx concentrations versus the 

ANFO fuel percent are shown in Figures 3 and 4, respectively. Theoretical trends were generated for four quenching-temperatures: 2145 K, 2274 K, 2345 IS and 2400 K. 

Normally, comparisons refer to the rich (over 7%) or mid 

(4% to 7%) fuel range rather than the lean (under 4%) fuel 

range, where the CO was negligible and the NOx was hard to 

fit. For the CO gaph, the richhid results were in the 2274 - 

2400 K range. For the NOx graph, the rich results were in the 

2345-2400 K range. With the underlying test-data uncertainty and wayward (or no) fits into the lean range, 

temperature interpolation was regarded as unwarranted, 

so the quenching optimum was taken as 2345 K. 

Unfortunately, the work- principle was unable to predict 

N-state results for explosives that react weakly (coolly), 

which is why there are graphical regions without theoretical 

results. There is no numerical resolution possible when the 

Z-state temperature TZ is less than the quenching temperature Tp It would be wrong to infer that the work-principle, 

which rests upon the four laws of thermodynamics, or the 

restrictions of constant reservoir or NO WORK were faulty, 

rather than the underlying constraints or stipulations used to 

resolve them. For the unsolved cool circumstances, it 
remains unclear whether total thermodynamic equilibrium 

was unattainable or the last remnant was reached prior to 

the Z-state, which thereupon would be relegated to the non- 

.equilibrium regime. Refinements that might circumvent or 

ultimately overcome the tractability issue are noted later. 

9 Role of Ingredient Composition for the Loss Rate 

For the underground tests, the reaction-zone energy 

transferred to the retaining fixture, the reservoir air and 

chamber walls is wasteful, while for rock blasting, it would 

fragment and heave rocks. The working-fluid's capacity to 

transfer energy to its surroundings is reckoned from the loss 

rate a,, shown plotted against fuel percent in Figure 5. The 

graph resembles a ranking trend for the charge performance 

(aL) versus ingredient composition (fuel-oil), where the 

result near 6% represents the charge formulation that 

should render the most work (maximum-a,). 

GeneraLly, the N-state solutions yield positive a,, under or 

over unity, thereby revealing that typical trajectories 

through the quenching temperature TQ are not isentropic. 

Relation (7) where X=S= constant for an isentropic 

trajectory yields the requirement that ax = as = 1, which 

would occur only rarely (coincidentally). Therefore, the 

work-principle technique, with its propensity for a nonLass Rate a~ vrs. Fuel[%] 

Figure 5. Quenching-temperature loss-rate constant, versus fuel 

percent. 

Conc .CO vrs. Add itivc 1% ] 

60 1 -'@ - &rcr lnss 

+T. 2400 K 

-T= 2345 K / 

-T=2274K d* 1, +T-214SK 

@ Tcst ksult 

\5 4' $ 4% 4' *\Q $ $0 +,\ eO 

Add itive [%] With 6% Fuc 1 Oil 

Figure 6. Quenching-temperature CO concentration versus additive percent, with test results. 

Conc.NOx trrs. Add itivc[% 1 

12.5 ' -.-firobss 

n T . 24110 K 

a 10.0 * -T= 235 ); ,1 - J = 2274 ! '\ 

2 7.5 . +T- 2145 K 

- * Tcst ksult * *\ , /' 

,\s 4' &' 4% 4' &\Q $' $? .+oK 

Add itive 1% J With 6% Fuc 1 Oil 

Figure 7. Quenching-temperature NOx concentration versus 

additive percent, with test results. 

isentropic trajectory, differs notably from the traditional 

unmodified ZND theory. 

10 Charge Formulation with Additives 

The notation used for graphing the results of ANFO with 

6% fuel oil and an additive was: RD15 for 15% rock dust, 

- Not for none, W for water and A for aluminum, where the 

number represents the percentage. The mixture-rule concentrations for CO and NOx versus the type of additive are 

shown in Figures 6 and 7, respectively. The CO fits were 

respectable on the whole, except for A10. The NOx fits for 

- Not, A5 and A10 were decent, those for RD15, W1, W3 and 

W5 were marginal, while those for W7 and W10 were poor. 

Compositions with higher additive percentage might not 

have reacted completely in the underground tests, contrary 

to TDRC presumptions, making those comparisons questionable: RD15, W7, W10 and A10. mere were no 

ingredient circumstances for which the TDRC numerical 

convergence was ruined by too many condensed phases, 

though it would not be unreasonable to omit condensed 

phases that are unlikely to form in a rapid explosion. 

11 Unresolved Quenching N-States 

For rough approximations, universally fixing the loss rate 

rather than the quenching temperature resolves the tract- . 

ability issue with cool explosives, though that technique 

tends to render worse fits to the test results. The resolved 

N-state trends and NOx test results in Figure 4 have roughly 

positive concavity, whereas the wholly tractable zero-loss 

trend with restricted (zero) constant at has a reversing 

profile with a negative concavity, illustrating wrongful fit. 

The resolved loss rates for rock-dustlwater (aluminum) 

results in Figure 7 were under (over) 0.5. Were a, uniformly 

taken as 0.5 instead, the unrestricted N-state temperature 
and related NOx would decrease (increase), universally 

worsening their fit-discrepancy in Figure 7. The wrong-way 

changes can be inferred from the correlated infonnation in 

the multiple-temperature trends of Figures 5 and 7 (or 4). 

When the resolved loss rate is near 0.5, like the Not result, 

the choice of restriction is hardly relevant. Though the workprinciple technique using the quenching-temperature stipulation remains more tenuous, with residual tractability 

issues, the resolved trends yield more consistent fits than 

those from trying to uniquely stipulate the loss rate. The 

logical recorrrse far cool explosives would be to improvise 

constraints that work in the thermodynamic regime denser 

than the Z-state and study their ramifications. 

32 Conclusions and Recommendations 

The work-principle model as formulated rests upon 

relatively few postulates and restrictions with constraints 

that were tailored for underground chamber tests, not rock 

blasting. Two requisite model constants were resolved: the 

75% mixing fraction from rudimentary theory and the 

trial-optimum quenching temperature TQ = 2345 K. The 

quenching optimization refers to fits for the two major 

components of fume toxicity, CO and NOx, not other types 

of reaction products. With the unique quenching temperature, the unstipulated reaction-zone loss rate was resolved, though it was found to depend upon the ingredient 

composition of the charge formulation. The graphical 

trend for the loss rate aL versus some ingredient proportion 

(fuel oil) was recognized as useful for ranking the work 

output of the charge formulations, revealing an optimum 

choice (-6%). The work-principle model would not yield 

N-state results when the Z-state temperatures were less 

than the quenching temperature, which occurred for 

weakly reacting (cool) explosives. Though the tractability 

issue disappears when the loss rate was uniquely fixed 

instead of the quenching temperature, the trends have the 

wrong profile and render inferior fits overall. The twoconstant model, originally recommended, has respectable 

resolving power for predicting toxic release hazards or 

investigating ingredient changes that could reduce the risks 

of unwanted fume incidents. 

Recommended extensions for the work-principle technique logically include trying to impose constraints that 

characterize different types of rock blasting or resolve cool 

explosives in the underground tests. Hypothetically, this 

would require using a TDRC with a real-gas EOS for the 

higher density regime, where the internal energy is not 

wholly regulated by temperature. Though the ingestion of 

air could remain relevant in the underground case, the rock 

or stratum would normally inhibit air from reaching the 

zone of transforming fumes, reducing the modeling difficulty. Non-ideal reaction, requisite loss-rate, non-isentropic 

trajectory and the retention of total thermodynamic equilibrium will remain issues worth investigating. 

13 Appendix A. Rendering Form to the hposed 

Constraint , 

With the work-principle, there is no requirement for the 

thermodynamic trajectory to 'ride' the imposed constraint 

representing the wanted process, except in proximity to the 

trapping minimum of the quasi-potential. The constraint 

resolution presumes that the real-gas EOS reduces to 

Griineisen form in relations (a), with coefficient functions 

J and I that remain relatively constant over a modest 

thermodynamic domain, near the Z-stat,., or N-state. The 

K-function was rewritten in terms of the imperfect function 

0, which represents the .real-gas departure from ideality. 

When @ reduces to unity and the molecular weight is typical, 

-24.47 g/mol, the requirement KZ = KR = 40.874 mollm3 

yields the rarefied Z-state density, -1 kg/rn3. 

The N-state constraint relation was formulated with the 

logarithmic derivative of temperature Twith respect to K at 

constant X and Q. Reduction in form occurs for the 

reasonable presumptions that the thermod ynamic derivatives of A, @, J and I remain negligible in comparison, 

yielding the J. ax result in relation (9). Restricting J. ax to 

constant J . aL yields the logarithmic T-K constraint relation 

(5) by integration. The unwanted X-Q integration constant 

(function) was removed by tracking the N-state constraint 

through the Z-state result. 

=awl l.El =J.ax 

au x,, ay ,Q 

Where aW( X.Q 

Recognizing that the thermodynamic derivatives for J 

reduce to the Griineisen gamma function, requires taking 

the differentiation of the reduced EOS in two ways, noted by 

the subscript shift. The requirement that the Jand I deviations 

remain negligible was invoked previously. The resolved Zstate r, is utilized for J, rather than the unresolved N-state r, 

which requires recursive solutions, though that represents a 

tractable and worthwhile refinement. 
The Gruneisen gamma TG in relation (9) is usually not 

reported in the numerical TDRC output, though the related 

thennodynamic derivative called GAMMAS (Ts) is reported. Utilizing the rule (10) requires that the @ and A 

deviations remain negligible, as noted previously, yielding 

the wanted relationship, TG zz =:Ts - 1, in transposed form. 

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