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reduction in the quasi-potential. The restorative influences of non-equilibrium continually try to reduce
the quasi-potential, so the rock travels downhill until no further Yq reduction can occur, stalling the
trajectory motion, trapping the rock at rest, within reversible fluctuations and uncertainty tolerances.
The unresolved trajectory route(s) and the un-relinquished constraint (curve) represent different
notions, except when the rock is trapped. Without those constrained equilibrium circumstances there is
no differential restriction on quasi-potential Yq, and therefore insufficient information for the work-
principle technique to render the thermodynamic state of the working fluid. The restoration influences
perform work to force the return of equilibrium even under imposed constraints, but when those
restoration forces are not otherwise tracked, the related trajectory motion remains unresolved. For the
work-principle formulation, the trajectory motion remains unresolved except for the trajectory terminal
states, which is why it is referred to as an unresolved trajectory or non-trajectory algorithm.

TDRC fume results resolved for other thermodynamic states, which were not trajectory terminal (key)
states, would probably not be relevant, and therefore rather worthless. Root-finding techniques are
quite useful for determining the restriction on the thermodynamic state represented by the minimum
trap and the imposed constraint, though that won’t reveal the fume spectrum. Numerical results
utilizing TDRC reveal the fume spectrum for the restricted key state, though they are not rigged
internally for the requisite root finding. Therefore key-state results were reckoned by trial-and-error,
taking tentative trial choices compatible with constraint, computing the working-fluid state functions
with TDRC, hoping for further reductions in the quasi-potential and discarding the numerous wrong
results on the way.

REDUCTIONS FOR GIBBS AND HELMHOLTZ

Within this section, the quasi-potential Yq is restricted in form, so that the work Principle can render
the traditional inequality requirements for the Helmholtz energy F or G Gibbs energy.*'? The reservoir
conditions must reflect the wanted working fluid conditions rather than the fume reservoir conditions.
When the working fluid density is rigidly restrained (constant p) and there is thermal equilibrium
between the working fluid and the reservoir T = Tp, then Yg= U - Tr‘S + Pr/p = U - T-S + Constant =
F + Constant. Under transitional change the residual constant disappears and is therefore irrelevant. The
requirement AY < 0 reduces to AF < 0, where the reduction in F registers the restoration work. For the
other requirement, take thermal equilibrium (Tp=T) and mechanical equilibrium (PR=P) between the
working fluid and the reservoir working fluid, then Yg= U - Tp-‘S + Pe/p =U-T-S+P/p =G. The
requirement AYQ < 0 reduces to AG < 0, where the reduction in G registers the restoration work. Under
the two restricted transformations, the reservoir characteristics were replaced with working fluid
characteristics, yielding two state functions F and G respectively. The restrictions illustrated underlying
the minimization routines utilized within TDRC, depending on the circumstance wanted.'?'*

REPRESENTATION OF WANTED PROCESS

Reaction fume process(s) fluid are represented on a trajectory graph using two state functions for the
working fluid, like the energy U and 1/Rho (or I/p) the reciprocal density. The representative
thermodynamic diagram and the work-principle components are shown in Figure 1. For rigid
confinement, the density would remain constant and the working fluid would transition from the
reactant ingredient (initial) state or I-point to the J-point. For rock blasting, some notable reduction in U
would occur in conjunction with the reduction in p, yielding the transition to the R-point. When the
reduction in the resultant (or net) interaction energy is neglected, the work-principle technique yields


of-eb-a} I>

The bracket terms were rewritten so the working fluid to reservoir pressure ratio P/Pp could be
compared with the reciprocal of the brace term, including the constant T/T. term. Recall that W was
required positive, so if the brace term within the bracket term were negative or zero the density would
rise. For the remaining conditions the brace term is regarded positive, and the reciprocal is taken. When
the ratio P/Pp is higher (lower) than the reciprocal, the bracket term would be positive, so the working
fluid density would fall (rise). The gamma factors represent dimensionless forms of higher-order
thermodynamic derivatives, and therefore are not necessarily constant. The [}= W/CyT remains
positive, where 1/Cy = 6T/GUly = GLn(T)/ASly is the reciprocal of the specific heat at constant density.
The Gruneisen gamma is normally positive and has two useful representations ig = 6W/6Uy =
@Ln(T)/Gy|s. For a working fluid on the reservoir isotherm, the bracket term reduces to [1-Pp/P].
regardless of the state equation EOS. For thermodynamic domains where the working fluid reduces to
the ideal-gas form, Ig = Ig, the bracket term reduces to the bracket term resolved for the zero-net
interaction. Whence the internal energy U reduces to just a function of the temperature T, and therefore
it is logical that the work-principle yield identical indications.

Fume RESULTS WITHOUT REACTANT AIR

Reaction fume results were computed for the Z-point to check the worthiness of the work-principle
technique at this stage of model development. Three types of ANFO were resolved at the Z-point,
depending on their hydrocarbon fuel, referred to as rocket-fuel kerosene RFK, jet-fuel kerosene JFK,
and number-two heating fuel oil HFO. The representative formula for RFK or dodecane was C)2H2.,
for JFK or Jet-A it was C12H23, and for HFO was C)4Hy, (rather than C7H)2). The heat of formation for
RFK was -350.9 kJ/mol as referenced and for HFO was -247 KJ/mol as estimated from the heavier
hydrocarbon for that H/C ratio.'® The heat of formation for JFK was -303.4 kJ/mol, from NASA
chemical equilibrium reaction code utilized for rendering the TDRC results.°

The underground fumes chamber contains 274 m? or 324 kg of air, and the explosive mass is 4.54 kg or
roughly 1.4% proportional weight. The toxic fumes molar concentration would normally be under
0.25%, when transformed to carbon monoxide CO equivalent toxicity.’ The Z-point fume temperature,
the CO and the nitrogen-oxides NOX results were graphed against the hydrocarbon fuel [%], shown in
Figures 3, 4, and 5 respectively. The results were not sensitive to the choice of the hydrocarbon fuel,
within the range of 1% to 10%. The CO rises monotonically under fuel rich conditions while the NOX
tends to rise under opposite conditions with a maximum near 4%, Theoretical CIs |-state fume-spectrum
and temperature results for prill ANFO with different fuel [%] were reported.'* Results were unit-
transformed, the NOX=NO+N0Q) was computed, and the CJ-state results were plotted for comparison
on our relevant Z-point graphs. The Z-point temperature is only ~20% smaller compared to the CJ-
state, which is over thousand times denser. The two CO trends were comparable throughout, and the
two NOX trends were comparable at very rich and stoichiometric conditions. Otherwise there are
noteworthy differences, which might be understood with further theoretical refinements.

Toxic fumes concentrations [cc/g] taken during underground hard-rock metal mine operations, yield
results somewhat different than typical gallery cannon fixtures. Their range of tabulated mean
concentration results for six different explosives including two semi-gel dynamites, three slurry
explosives, and mostly ANFO (*some dynamite) were transformed to metric units and displayed in the


from the thermal-mechanical state functions like W, y, T, S, (or U), rather than from reaction variables
like r-bar, Q, or the concentrations (Ny). The unrestrained y taken while holding « constant would
remain otherwise unrestricted were it not for reductions in the quasi-potential. The change of the quasi-
potential Yq or negative work function Wr is resolved according to a Taylor’s expansion in
unrestrained y, while « remains constant. The capital-delta (small-case) notation represents finite
(infinitesimal) differences, with normal (not-reverse) chronology understood.

Technically the subscripts should include Q to bear witness to the restriction of chemical equilibrium.
That requirement is universally implied for terminal state resolution, so the reduced notation just
displaying « thermal-mechanical constraint is tractable. For transformations that are infinitesimal, there
is retention of only the lowest-order-J non-zero terms. Recognition of the reduction in form is noted
with the grave mark on the summation symbol, with later illustrations showing relevant results.

For resolving the minimum trapping state, the partial derivative JY q/6x|k would reduce to zero for the
terminal state, so the next non-trivial term determines the stable nature of the stationary condition. For
the trajectory trapping minimum the second-order derivative PY oak is positive. The relational
inequality suggests that unrestrained variation squared multiplying the second-order derivative would
then have the wrong sign, for compliance. The only workable solution would require that the otherwise
unrestrained y vanish, which represents stalled trajectory motion, which is the interpretation rendered
for the rock rolling down the hill. The constraint curve obtained by graphing AYq with constant K over
a wider range with respect to unrestrained x, like in Figure 2, reveals the minimum trapping condition,
with downward motion regardless of the pathway, since the route is unresolved.

Worthwhile conditions occur when the quasi-potential trapping minimum is reached, stalling trajectory
motion (no choice - downward), until the constraint is replaced with a different restriction or left
unrestrained. For the work-principle formulation, other types of stationary results are unregulated or
worthless, since they refer to unstable or meta-stable conditions. The thermodynamic quasi-potential
‘rock’ rolls downhill, not uphill into a Yo maximum, unless something is ill resolved. For the non-
traditional maximum starting condition, the trajectory is initially unregulated, with multiple branches
downhill. Inflection point circumstances would be resolved in a similar fashion, using higher-order
terms to understand the nature of the stability. Un-trapped trajectory motion would render the
constraint worthless (one choice - downward) or unregulated (multiple choices - downward).

NON TRAJECTORY AND THERMODYNAMIC UNCERTAINTY

Irreversible and reversible fluctuations render the chronological occurrences that force the reduction of
the quasi-potential, though there is other rationale for the retention of pathway uncertainty. There is a
quantum mechanics uncertainty principle and there is a thermodynamic uncertainty principle, though
the latter is not as widely known.'* Uncertainty principles tend to render strict trajectory notions
obsolete and force worst-case minimum tolerances on canonically conjugate variables. When


the Z-point, taken for predicting toxic fumes in underground chamber test. Future renditions could
utilize a formulated constraint to replace the neglected interaction with the requisite energy reduction,
yielding the trajectory motion to the N-point. Whether those refinements are warranted for fume
spectrum results, or just for occasionally confirming the reliability of the Z-point results, remains
unclear. The N-point would probably represent the last remnant of worthwhile equilibrium for the
working fluid, which then drifts downward through the cooler out-of-equilibrium conditions ultimately
reaching the reservoir conditions. The reservoir conditions are taken as the, Q*-point, or the
measurement state conditions, when they are readjusted to the reference standards [25 °C and 1
atmosphere].

' Mechanical

Agent

Reservoir.

1/Rho

Figure 1. Work Principle Component Model, &

Thermodynamic Diagram with Wanted Processes.
The named points (except T) represent trajectory terminal (or key) states, and only the trajectory route
through the Z-point displays the multiple segments needed for reaching the reservoir conditions. The
TDRC with wanted reactant ingredients is resolved for some trial state, the T-point, with the constraint
of zero-net interaction, which requires the retention or return of the working fluid energy U to the
original U; upon reaching the trajectory key state. The working-fluid state functions for the tentative
T-point yields the corresponding Yq quasi-potential. The trial point T is readjusted as noted previously
until there is no further reduction in the Yq quasi-potential. The tentative T-point results shown in
Figure 2 show the downward trend(s) in the quasi-potential which was utilized to find the trapping
minimum, Z-point, the only worthwhile point. The Z-point name is remindful of the zero-net-
interaction energy, with zero tolerance for readjustment, and a logical zenith state since reduction of
energy U would occur with non-zero-net interaction. Refinements in work-principle technique can only
be rendered by readjustments in the constraints representing the wanted thermodynamic process, so
while the work principle overcomes some hard issues, it has its own unique form of intractability.

RENDERING OF IMPOSED CONSTRAINTS

The rendering of imposed constraints refers to the utilization of relative minimum trapping conditions
to stall the trajectory motion. The imposed constraints are represented as YQ partial derivatives with
respect to the unrestrained x, while holding x constant. The variable functions x or x are constructed


domain under investigation. When the zero-net interaction Z-point constraint is removed, non-
equilibrium prevails with unresolved circumstances, though the restorative influences must ultimately
tecover the requisite Kr circumstances at the reservoir conditions or Q*-point measurement state. The
work principle renders a trapping minimum for the Z-point where K = Kg and forecasts that it would
replicate K = Kz when the working fluid reaches the reservoir conditions. Real-gas formulations, like
@=W-Mw/R*T, evaluated at the K-function Z-point condition reduce to the ideal form (=1), wholly
compatible with the rarified-hot state resulting from no energy reduction and small-rho minimum. For
such circumstances, the Z-point restriction K = Kr further requires that the Z-point molar density p/Mw
= Kp = 40.874 mol/m*. Under the circumstances, the work principle renders the Z-point molar density
of the reaction fumes, regardless of the charge formulation, and forecasts it would ultimately replicate
that result for the reservoir conditions. The working fluid for the Q*-point, taken at reservoir
conditions, normally includes notable quantities of condensed phases, like water, which ruin the ideal-
gas presumption for the reservoir conditions.

ILLUSTRATION OF OTHER CONSTRAINTS

With the work principle, the only way to stipulate or refine the representation of some wanted natural
process is to compose and readjust the constraints. Other renditions of trapping conditions, though
rudimentary, are illustrated to represent the way mode! development transpires. With the work
principle, the constraint must render the minimum trap for the downhill drift of the quasi-potential,
represented theoretically by bracket terms that can reduce to zero. Though the bracket terms arise
regardless of the choice of the unrestrained variable, the related multiplier could include zeros or poles
that relate to the stability of the working fluid, which remains outside the scope of this report. For the
results illustrated the unrestrained variable was chosen so that the multiplier would have a tractable
form with obvious fixed sign. For isochoric process or constant density (or logarithm thereof) take y =
x and x = U, then compute @Y9/dUly. The subscript y is capitalized as are others for readability.

a, T
-W, ]|, = 8Yo] ou Fol -su[1-B]so

For natural processes this relationship dictates that the energy U of the working fluid would fall (rise)
when it has a temperature T higher (lower) than Tr the reservoir temperature. Until this constraint is
relinquished, there is no requirement that the working fluid pressure match that of the reservoir. For
isentropic process take S = x and y = U, then compute @Y9/AU|s.

su. =6U- -B]o
aU |, P

For natural processes this relationship dictates the energy U of the working fluid would fall (rise) when
the working fluid has a pressure P higher (lower) than Pg the reservoir pressure. Until this constraint is
relinquished there is no requirement that the working fluid temperature match that of the reservoir.

3{-w, ], = 8Y,

For an isothermal process take T = x, and x = y, then compute 6Y0/@y|r. Notice that for temperature
conditions, some reductions in form could occur when the constraint constant x refers to the Tr
reservoir isotherm.


Work Principle For Predicting Explosive
Toxic Fumes

Michael S. Wieland

ABSTRACT

The work-principle method was developed for theoretically resolving toxic fumes from non-ideal
charge formulations. The whole description of a wanted process refers to macroscopic changes
resulting from removing and imposing constraints that work one at a time to chronologically render
trajectory terminal (key) states. The trajectory resolution utilizes a thermodynamic quasi-potential or
negative work function, and numeric results require some appropriate standard thermodynamic
(equilibrium) reaction code TDRC. The underlying method is unable to restrict reactions, tailor the
work output or resolve states within the trajectory transitions, so the trajectory terminology is used
somewhat figuratively. The work-principle model has the working fluid (reaction material) interacting
with a tremendous reservoir (chamber air and walls), while forecasting the work performed upon a
mechanical agent.

Within the underground test chamber, the widening fume zone undergoes a decline in temperature, a
reduction in reaction rates, resulting in out-of- equilibrium circumstances at the Q*-point, representing
the measurement state. Rather than coping with the hard issues of non-equilibrium, the toxic fumes are
resolved numerically for the Z-point, rendered by a zero-net- interaction energy constraint. Though
other techniques require complex equation-of-state EOS for the whole trajectory, the work-principle
formulation eludes the issue, requiring an EOS that works just for the terminal-state. Under the
circumstances, the Z-state is reconcilable with the ideal-gas domain, and relevant TDRC results yield
the restricted state in a numerical way for different charge formulations. The work principle makes it
hard to remedy or to tamper with the resulting Z-point fumes: non-macroscopic trajectory readjustment
is forsaken, terminal states are results rather than stipulations, and resetting the EOS formula constants
would be ineffectual. Though the non-equilibrium portion remains unresolved, it is noteworthy that the
work principle rigidly fixes the Z-state, thereby reducing the uncertainty for rendering reaction-rate
models characterizing the non-equilibrium Z-Q* transition. Though more refinements are needed, some
Z-point fume results are included to illustrate the threshold (minimum) resolving capacity of the work-
principle formulation.

INTRODUCTION

The NIOSH underground fumes facility for toxic fume measurements has a 274-m? chamber,
Tepresenting a thermodynamic reservoir of walls and 324-kg air, and therefore quite massive compared
to the 4.54 kg of mining explosives under test.' The coal-mine walls are a rather indefinite reservoir
component, which poses no difficulty, and they confine the fumes and help regulate their temperature.
The mining explosive is confined within a cylindrical steel pipe with a 10.2-cm diameter for proper
detonation. Except for the rupture and fragmentation of the cylinder, the explosive works upon and
transfers heat to the reservoir. The work-principle technique rendered within this report is tailored to


regarded as rather remarkable, since the TDRC was not tuned for the trapping minimization technique
that is utilized in conjunction with the work principle.

With such rarefied Z-point indications, it is worthwhile working with a TDRC developed specifically
for the ideal gas form.° Though it removes the nuisance of readjustable constants and the complex
formula with worthless terms, it raises the likelihood of reliable results. The related reference material
and tables can range over more types of molecular species, which noticeably raises the types of
reactions considered in TDRC. The more unrestricted the reaction possibilities, the more likely the
work-principle results reflect reality, and for toxic fumes this reliability is wanted.

CONCLUSIONS AND RECOMMENDATIONS

Whether or not the work-principle with zero-net-interaction energy technique is used to render toxic
fumes or recognition of more refined interaction is required remains unclear. The zero-net interaction
constraint was noted to resemble test conditions for the underground fumes chamber rather than those
for rock blasting. Foremost this restriction renders fume spectrum resolution without readjustable
constants, optional choices, or selection of EOS constants, when the reservoir and original charge state
are taken at reference standard conditions, [25 °C and 1 atmosphere]. The remarkable characteristics of
the zero-net interaction Z-point are noteworthy and were rendered in theoretical form for the working
fluid of any charge formulation, regardless of its equation of state EOS. The K function at the Z-point
reduces to the K-function result for a reservoir Kg = Pp/R*Tp = 40.874 ymol/cc or the reciprocal of
24.470 cc/mol. It was recognized that the K-function, though unresolved in trajectory transition, would
ultimately replicate the result Kp for the cool temperature Q*-point reservoir or measurement state. It
was forecast that this result would occur regardless of the EOS taken for the working fluid, or the cool
temperature.

For the working fluid at the Z-point, the temperature is over two thousand degrees, the trial density is
roughly 1 mg/cc, and the fraction of condensed phase material is rather small or negligible. The
reciprocal of K or R*I/P = 24,470 cc/mol at the Z-point represents a thermodynamic regime where the
real-gas EOS forms reduce to the ideal-gas form. Reckoned as an ideal gas, the working fluid for the
reaction products would have a molar density of 40.874 mol/cc at the Z-point, regardless of the
charge formulation. Furthermore, the working fluid molar density would return to that numeric value
upon ultimately reaching the reservoir conditions, were the ideal form retained. Unlike the former
forecast, the latter one would falter, since the working fluid is not a simple fluid that can always be
reckoned as an ideal gas. The working fluid representing the fumes has a rather complex nature, and a
noticeable number of the mixture components would transform into condensed products on the way to
the reservoir conditions, characterizing the Q*-state.

For mining explosives, which have non-ideal performance characteristics, there remain notable
uncertainty in the rationale and the underlying presumptions regarding their detonation reaction or
related fume results. The work principle requires different suppositions, and thereby overcomes some
of the hard issues, though it has its own unique intractability and philosophical difficulty. It remains
unclear what discrepancies will remain when the work principle is refined with non-zero interaction
terms and out-of-equilibrium reaction models are utilized to render cool-temperature fume results, for it
is an unfinished work in progress.


next table,[added NOX=NO+NO)].’° The hard rock mine shots and closed-gallery results correspond
to Q*-point results rather than Z-point results. The Z-Q* numerical difference for some fume species is
rather minor, suggesting that further model refinements would be worthwhile.

“Toxic Fumes Concentrations [¢¢/g] Underground: Hard: -Rock! Mine
= Range CO; - CO - $022
Overall _|°47-106 | 3.9-18.5 | 0. 1-2.9
~ANFO 58. | 46. or

One major theoretical discrepancy that was noted (not graphed) was that the Z-point NO2/NOX ratio is
negligible, though it is a noticeable fraction for most Q*-point observations. Resolving the
transformation of NOX components requires non-equilibrium reaction code models formulated to yield
the cool temperature results that occur within the hour of measurement. The work principle rigidly
requires that K is numerically fixed (=K) at the Z-point regardless of the working fluid’s EOS or the
charge formulation, which ought to facilitate the formulation of the non-equilibrium reaction-rate
kinetic model for rendering Q*-point results from Z-point results.

FUME RESULTS WITH REACTANT AIR

The work principle can incorporate reactant-air as a working fluid ingredient, while retaining the rest as
non-reactive reservoir material. The ratios of reactant-air mass to explosive mass RAX were taken as
0%, 25%, 50%, 75%, and 100%. For the highest RAX of 100%, the total working-fluid mass doubles
to roughly 9 kg, while the original 324-kg of chamber air drops to 320 kg non-reactant-air mass, hardly
readjusting the tremendous reservoir (walls and air) required for the work-principle model. For the jet-
fuel kerosene with 4%, 5%, 6%, and 7% fuel, the trend parameters were resolved with respect to RAX
[%]. The Z-point graphs of temperature, density, CO and NOX are shown in Figures 6, 7, 8, and 9
respectively. The temperature which were roughly 2500 °K for the 4% to 7% fuel range for zero RAX,
diminishes rapidly with respect to rising RAX [%]. The NOX roughly doubled and the CO rapidly
disappeared with rising RAX [%], so fume sensitivity with respect to RAX is rather high. Normally
when resolving toxic fumes with some TDRC, reactant-air is neglected, so RAX = 0% represents
traditional results. The trends from the work-principle warn of difficulty with the traditional
presumption, if the Q*-point measurement results were noticeably influenced by reactions with the
surrounding air.

COMPARISON TO THEORETICAL & RELIABILITY

For wide variations in trial density the trapping profile of the quasi-potential Yq were obvious, but near
the terminal state it was harder to resolve because of slight irregularities in the working fluid’s state
functions that were transmitted into the quasi-potential. Trials were normally run until the Z-point until
deviations were under 5% and often under 2%. The restriction on K resolved theoretically for the zero-
net-interaction energy constraint could not be utilized directly, since the K and U would not yield a
workable pair of input conditions for the TDRC. Utilizing the graphic and tabular minimization of the
quasi-potential Yq, the working fluid state functions were obtained for Z-point. The theoretical
restriction K = Kp was checked for different charge formulation including those with inert additives.
The K reciprocal for the Z-point fumes was compared with the theoretical Kr reciprocal or R*Tp/Pr =
24,470 cc/mol. This comparison was confirmed numerically, with the worst-error under 5%. This was


multiplying the uncertainty tolerances for the canonical coordinates, the multiplication result in
quantum mechanics has units of Heisenberg’s constant (or action), while the multiplication result in
thermodynamics has units of Boltzmann’s constant (or entropy). While our restrained and unrestrained
variable-functions do not necessarily form a canonically conjugate pair, there is still underlying
uncertainty in the working fluid state functions from which they were formed. Resolution of trajectory
motion remains the relevant issue, not the fact that the TRDC results are fraught with much greater
numerical errors than the theoretical uncertainty. Although there are profound philosophical
differences. it is worth noting that the work principle tends to work like the non-trajectory requirements
of quantum mechanics, with terminal (key) states rendered as the only recognized or observable
operations, and with intervening transitional states remaining unresolved or unobserved.

ZERO-NET-INTERACTION ENERGY

For zero-net-interaction energy, the terminal trajectory state is resolved by requiring that the working
fluid retain or recover the working fluid energy U; of the I-point trajectory state. Non-trivial influence
is possible, since the work-heat transfer with the reservoir is unrestricted except for registering the
zero-net energy-change for the macroscopic process. There is no restriction on the working fluid’s
entropy change, thanks to the reservoir, though there must be a reduction in the quasi-potential. The
restriction on the terminal trajectory is written U = « for theoretical resolution, with x = U; for the
numerical resolution using the relevant TDRC. Utilizing hindsight, the forthcoming results were
reduced in form by defining the K-function, K = P/R*T, which remains useful regardless of the
working fluid’s EOS. R* is the Universal Gas Constant. Upon reaching the reservoir state, the K-
function would equal the constant Kp = Pp/R*Tp = 40.874 mol/cc (or mol/m?). The reciprocal of Kg
would equal 24,470 cc/mole for reservoir conditions taken at the reference standard. The unrestrained
variation could be x = y or x = S yielding,

T, #&P, TW K,
= W-dy.)-8--8 |= [8 |.gy.]1-“2} <0
bs a B/-(AS) | z|

a, P,_T, K
: =T-8S-| 2-2 |=(-T._).88-J1-22 |<
a , [? z| ( »)35 z| °

The working fluid ratio W/T remains positive for a thermal mechanical stable fluid. The reservoir-fluid
state functions Tr and Pg are likewise restricted. Zero temperature is unattainable according to the third
law of thermodynamics. The bracket term arises, regardless of the way x was chosen, provided that it
was unrestrained for x constant. When the Z-point K-function is larger (smaller) than the reservoir
constant Kg, the brackets are positive (negative). Under those circumstances the entropy must rise (fall)
and the density must fall (rise). The terms without parenthesis refer to the explosion-type processes,
while the reverse cases refer to implosion-type processes. Transformations that obey the inequality
occur, while reverse inequality conditions that violate the reduction of the quasi-potential Yq are
forbidden.

With the work principle, the theoretical restrictions or bracket terms, deduced from the quasi-potential
minimum-trap requirement, hold for any working fluid regardless of its equation of state EOS. The
tenability of the restrictions remains unaffected when studied numerically, though such results are
worthwhile only if the TDRC and related EOS characterize the working fluid in the thermodynamic

Q



WorKING FLUID STATE FUNCTION NOTATION

Just like the generalized coordinates of analytical mechanics ultimately prove their worth, the
circumstances that logically render the model development of the work principle can benefit from some
non-traditional and generalized notation. The logarithmic density y = Ln[p/RHO] is utilized in
preference to the density Rho (or p). The density and its reciprocal are required non-negative and non-
zero for logical and theoretical reasons. The higher-order derivatives of the working fluid state
functions with respect to y tend to remain constant over a wider thermodynamic range. The constant
RHO is removable from any intermediate expressions and therefore is not stipulated. The reaction
concentrations are not utilized for stipulating the thermal-mechanical type constraint-relations, so the
working fluid has a differential form that can be rewritten in a useful way.

au=T-as+?. yyy an,
PP

W=P/p, y=Ln[p/RHO]

Qe the Ye az = Nx
My N-vy

dU =T-dS+W-dy+Q-d¥

The traditional first relation with terms per unit working-fluid mass was readjusted by expanding the
reciprocal density term. The multiplier that works with the y differential, W, is referred to as the flow
work per unit mass. The reaction terms in contracted form utilize the reaction coordinate r-bar with Q
as the multiplier. The reaction sum Q of stoichiometric coefficients vx’s times their respective chemical
potentials jix’s is divided by the formula molecular weight Mw for the working fluid mixture. W and Q
are recognized as working-fluid state functions, Q is not restricted in sign, and W and T remain positive
for stable fluids.'' There are no constancy requirements imposed on the working-fluid state functions
for unresolved trajectory conditions.

The two choices for reaction conditions that are quite tractable for trajectory terminal points are those
that would nullify the chemical-work: the requirement of r-bar constant represents non-reactive or
frozen flow and the requirement of zero Q represents chemical equilibrium. The latter would occur
when the reverse reactions match their respective forward reactions, which is normally presumed for
the TDRC. The requirement for zero Q arises strictly for the terminal state, not within the unresolved
transitional trajectory, where the restorative influences for non-zero Q could readjust the reaction
chemistry for the transitory conditions.

ROCK ON A HILL - THE QUASI-POTENTIAL

The reduction of the quasi-potential Yq or the rise in the working function Wy registers the restoration
work necessary to undo the deterioration from the original thermodynamic state. The quasi-potential
YQ is analogous to a rock that would roll downhill for irreversible processes, remain at rest for
reversible processes, and would not roll uphill for forbidden processes. For natural processes with their
recognizable finite irreversible transition(s), the rock’s downhill motion would register notable


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requirement utilizes a quasi-potential Yq or negative work function [-Wr]. The work-principle relation
was written per unit working-fluid mass, since the TDRC render their numeric results that way. The
work function Wr terminology tends to be redundant, though it has some relevance and is therefore
retained. The defining relation for the work function Wr or the quasi-potential Yq is

-W, =¥, =[u-1,.5+%/]

The quasi-potential Yq (or Wr) depends upon the working fluid’s energy per unit mass U, entropy per
unit mass S and density p (or Rho). Neither the quasi-potential nor the work function qualifies as a
state function, because of the reservoir parameters, temperature Tp and pressure Pa, despite their
constancy. The resulting construction of the quasi-potential, represented as a superposition of
working-fluid state functions with constant reservoir-multipliers, is trajectory pathway independent,
regardless of the irreversibility, and therefore has formidable theoretical capacity. The work-principle
inequality relation taken from the noted references ”* is then

Work < AW, = A[-Yo]

The Work term represents the work done by the working fluid upon the mechanical agent, regardless
of the type of process, and would change depending upon whether the process was reversible (the
equality) or irreversible (the inequality). Reversible paths taken with infinitesimal quasi-static steps
would take forever to render a finite thermodynamic transition. For a reversible path, the work done
upon the mechanical agent would equal the work function rise AWs or A[-Yq]. Restoration work refers
to work rendered by (not upon) the mechanical agent when returning the working fluid to the original
state along the reversible path in the opposite direction. The work function or quasi-potential register
transitions like state functions, and taken over a closed cycle must return to their original numerical
value. When closed-cycle integration is taken over the relationship, the Wr or Yq terms yield zero
results. For the cyclic integration then, the resulting work done upon the mechanical agent would be
zero or negative, Work < 0. Therefore, it is the mechanical agent that would have to render the
restoration work. The restoration work reveals the system’s propensity for chemical reaction work.'®

It is worthwhile to invoke the terminology that natural processes refer to finite transitions with some
irreversibility, however slight. Reversible infinitesimal transformations would have non-infinite
response time and therefore are not excluded, though they could not wholly account for macroscopic
transitions. For natural-process work, with fumes, the mechanical agent is not removed from the work-
principle technique, though it is rendered inoperative or dysfunctional, so Work = 0. Since the
mechanical agent is rendered moot, no device in the rampant reaction combustion process need be
identified with it. With the dysfunction restriction, the work-principle relation reduces to

AW,20 or AY, <0

Without the mechanical agent to render restoration work, returning to the starting state upon
undergoing a natural process (inequality) is impossible. Restoration work remains an illusion that
requires a reversible process taken in reverse time travel (or reverse chronology). With normal
chronology, the reduction in the quasi-potential A[-Yo] or the rise in the work function AW, registers
the restoration work that would be needed to un-do the deterioration.


model the test circumstances, and not rock blasting. The reaction of mining explosives is quite non-
ideal, and not reconcilable with the traditional requirements of the Zeldovich-von Neumann-Doering
ZND theory of detonation.>* For the traditional ZND theory, the reaction thermicity term in the
referenced relation must vanish at the CJ condition. Non-ideal explosives undergo reaction through the
CJ, so the zero-thermicity requirement is a recognized conflict. This is rendered moot by the
introduction of some representative loss term in the theoretical formulation, so the relevant
relationships remain numerically bounded.’ The resolution raises questions about the proper choice of
process description, and forms a little (philosophical) room for the unorthodox techniques undertaken
with the work-principle.

For rock mining or fume testing, the restoring influences that try to maintain mechanical and thermal
equilibrium rapidly widen the fume zone within the reservoir, via Le Chaterlier’s principle.” Under the
transitory circumstances with declining fume zone temperatures, the restorative influences that try to
maintain chemical equilibrium falter. For the post-detonation combustion process, total equilibrium
remains plausible only while the fume zone stays hot, keeping chemical relaxation times small. With
cooler temperatures, the relaxation times grow drastically, precipitating out-of-equilibrium
circumstances within the working fluid. Wall or dust interactions and the residual non-equilibrium
yield reactions that persist noticeably at the cooler temperatures within the underground chamber at the
hour of measurement, ruin the presumption of non-reacting flow. TDRC yield results that require total
equilibrium conditions for the working fluid and they would be quite inappropriate for resolving the
non-equilibrium Q*-point measurement state.

This raises the question of the former (not Q*-point) thermodynamic state that was the last respectable
remnant of total equilibrium, and therefore worthy of fume spectrum resolution with TDRC. Non-
equilibrium conditions tend to remain rather intractable, so that non-work-principle techniques reckon
reaction components as frozen for different temperatures on the late-stage (post CJ) trajectory.
Numerical fume-spectrum results tracked downward from the high-density region could be wrong or
unreliable, if the EOS formula constants are reckoned badly, which is undesirable for ranking fume
toxicity. The work principle can not be tampered with or remedied in this fashion, which could be
regarded as a disadvantage or a useful circumstance. Under the circumstances noted later the Z-points
were thermally hot and rarefied and therefore the TDRC chosen for resolving the fumes was the NASA
Lewis Complex Chemical Equilibrium Code.

THEORY UNDERLYING WORK PRINCIPLE

The work principle characterizes a thermodynamic system of three interacting components that
otherwise remains isolated from the rest of the universe, and therefore is governed by the laws of
thermodynamics for isolated systems. The technique utilizes constraints that chronologically work one
at a time for representation of process, but they are not required until the work function or quasi-
potential is resolved. The working fluid, the reservoir, and the mechanical agent constitute the
interacting components, though the latter two are unable to interact directly. The reservoir must retain
constant thermal-mechanical characteristics, and therefore have tremendous (mass) proportions
compared to the other components. The working fluid may have a reactive composition, transfer heat
and work upon the reservoir or work upon the mechanical agent in a reversible (non-expansion) way.
None of the components interchange particles, so the working fluid mass remains constant.

The work-principle formulation is one of those ubiquitous inequality relationships of traditional
thermodynamics that remains unchanged regardless of the process undertaken.”* The work-principle


REFERENCES

' Mainero, R. J. leasurin is Produced by Blasting Agents. Proc. 23rd Ann. Conf.
Expl. Blast. Tchnq., Soc. Expl. Eng., Cleveland, OH, P.595-604, Feb. 1997.

? Cook, M. A. The Science of High Explosives, Reinhold Publishing Corporation, P. 48-59, 123-133, 166-170,

238-244, 1958.

* Persson P-A., R. Holmberg, and J. Lee. Rock Blasting and Explosives Engineering. CRC Press, Boca Raton,
P. 100-114, 1994.

* Dussel E. The Non-Ideal Detonation. Seventh High-Tech Seminar, Blasting Analysis International, Inc.

Allentown, Pennsylvania, P.403-418, Revised copy, July 1997.
5 Lavenda, B. H. Probabilistic roach. John Wiley & Sons, Inc., NY, P. 153, 1991.

° McBride, B. J. and S. Soa Computer Program for Calculation of Complex Chemical Equilibrium
Compositions and Applications, Il. Users Manual and Program Description. NASA Reference Publication

1311, NASA Lewis Research Center, Cleveland, Ohio, June 1996.
7 Landau and Lifshitz. Statistical Physics. 3" Edition, Part 1, Course of Theoretical Physics, Volume 5,
Pergamon, P. 57-65, 1980.

® Reference 8, P. 46-49, 59, 1991.

* Chang, R. General Chemistry. Random House, NY, P.683, 1986.

© Langford, C. H. and R. A. Beeby. The Development of Chemical Principles. Dover Publications, Inc. NY, 1969.
[Unabridged and corrected replication, p 166, 1995]

1 Reference 7, P. 42-44, 63-65, 1980.

"2 Whalen, J. W. Molecular Thermodynamics: A Statistical Approach. John Wiley & Sons, Inc., Chptr. 2, 4,
1991.

13 Reference 12, Chptr. 9, 10, 1991.

4 Gordon, S and B. J. McBride. Computer Pr: mplex Chemical E

Compositions and Applications, I. Analysis. NASAL iene Publication 1311, NASA Lewis Research
Center, Cleveland, Ohio, June 1996.

'5 Reference 5, P. 150-160, 191-199, 1991.

* Lide, D. R. and H. P. R. Frederikse. CRC Handbook of Chemistry and Physics. CRC Press, Boca Raton, P.
§:5-60, 1996.

‘' Wieland, M. S. Understanding Toxic Fumes from Mining Explosives. Seventh High-Tech Seminar, Blasting
Analysis International, Inc. Allentown, Pennsylvania, P.93-108, July 1997.

‘8 Chaiken, R. F., E. B. Cook and T. C. Ruhe. Toxic Fumes from Explosives: Ammonium Nitrate-Fuel Oil
Mixtures. Report of Investigation 7867, P.17-22, 1974.

'* Abata, D. L., J. H. Johnson, B. G. Bunting, and J. Robb. Monitoring of Mine Air Blast Pollutants from Six
Inder .. Michigan Technological Univ., Houghton, MI, Bureau of Mines,

Washington DC, Grant Agreement No. *GOnTT ISD, Table 19, Oct 1, 1978.


Work Principle For Predicting
Explosive Toxic Fumes

-- Erratum

Michael S. Wieland

Two relationships that represent Taylor expansions were written incorrectly,
in the report. The terms within the summations are missing J—factorials [J!] in
the denominator. The relationships should be replaced by the following two
formulas, respectively:

J or

A[-W, ], = 4%], = oe <0
(8x) HY,

a{-w,], =8Y,|, == Gr. a <0

The J—factorial errors were not propagated, and therefore modifications to the
other relationships are not required.