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Maximum work production from a heated gas in a cylinder with piston
Volume
72, number
MAXlMtbl
Yehuda
1
CHEhlICAL
PHYSICS
WORK PRODUCTION
FROM A HEATED
15 May 1980
LETTERS
GAS IN A CYLINDER
WITH PISTON
B. BAND
Department
of Chemlstty.
Ben-Gut-ion Unirernty
of the Neger. Beer-She&a. Israel
Oded KAFRI Nuclear Research Center Negev, Beer-Sheva. Israel
and Peter SAL_AMON * Department Recelred
of Chemists.
27 February
Tel A LIVUtrtverslty, Ramat Aviv, Israel
1980
The optimal motion of a piston IIIa cybndcr containmg a gas. pumped wrth a given rate of heatmg is determined for maxtmal work productton Reversibtbty IS not a crttenon for high effictency tn fmtte time. The efficiency reaches the clasmal thermodynarmc bound for mfinttely slow processes.
Thermodynamrcs provrdes a bound on the maximum work obtainable from a grven thermodynamrc process. However, thermodynamrc bounds are often too hrgh to be valuable from a process which operates rrreversrbly and m finite trme. In this letter we determme the maxrmum work obtainable from a prototype system operatmg at finite time dependent rates for a fmrte tune. Furthermore we determme the actual system matron yrelding the maximum work. There has been consrderable recent mterest m the optunal control of thermodynamic systems operating m finite trme [l-4]. By incorporating trme dependent heating of a thermodynamrc system into the prototype problem, we add an important new aspect to the problems in thrs field. We consrder a piston fitted m a cylinder containmg a working fluid which IS coupled with a heat bath of temperature T,, and which is pumped with a grven rate of heatmg,fir). For srmplicrty we neglect effects of the mertia of the piston and the gas, piston frictron, and we assume equilibrmm of the working fluid at each instant l
of time but we note that these effects
Present address: Department Umverslty, Tempe, Arizona
of Mathematics, 85281. USA.
Arizona
as State
well as other energy loss mechanisms can be included m the present formulation. We ask the question: what is the optimal matron of the piston III order to extract the maximum work m a given amount of time, tm, gven an mitral and final volume of the system? For our system, the first law of thermodynamics takes the form &)
= f(t)
- ti(,ct) - K [ nt)
- T=..1,
(1)
where E is the internal energy of the working fluid, FVis the work of the expansion against the piston, and K is the thermal conductivity which we take to be a constant vdependent of the fluid volume. F?(f) is given by W(f) = p(f) P(t) where p and V are the pressure and volume of the flmd. If we take the working fhnd to be an ideal gas, then p(r) = nRT(f)/V(f) and E(f) = mvT(r) where tz is the number of moles of the gas, cy :s the heat capacity, and R is the gas constant. The second !aw of thermodynamics, m the equality form [S] , is given by Sin = $, f $, f $,, where Si, is the irreversible entropy production and S,,S,, and Sp are the entropies of the gas, the bath, and the heat pump, respectively. A standard thermodynamic analysis
Volume
72, number
CHEhiICAL
1
PHYSICS
[6] shows that the work is bounded by the reversrblr work which equals the sum of the changes in the avadabtlrtres of the pump and the gas, I$‘< ru,,
= EP - Tc,SP
- (Sg
- T,,AS,),
(2a)
(5)
wlth a=-
IrIG
Ep - T,,$,
+ uRT,,
ln[ V,/V(O)].
(2b)
if we assume that the pumped beat comes from a pure work source (I e. an mfimte temperature source), S,
also vamshes and it’ < Ep + nR Te, In [ Pm / V(O)] .
@cl
Smce the operaimg condltlons are Irreversible, this bound can never be obtained (except for the degenerate case E, = 0 and t, = +m, discussed below). We now a better and e\phcitly to the to obthe The of &dt the equation can carried using Pontryagm prmcrple We that optrmal IS of (EL) of form V(t) = V(t,)[F-(t)/F(t,)]
-=vm F”*(,)dT
E(t) = E(t,) [F(t)/F(ti)]
‘*.
cy
R
where EP = lofmf(t)dt.
Drfferentratmg this e\pressron wrth respect to T(t,) shows that the maxrmum work 1s obtained when T(t,) = T,, so when T(0) = T,, this becomes
15 May 1980
LEITERS
F~2(0)Jo’mFvt(~)d~ E(0) [ V(O)] R’cr’
and b = V(0)[E(O)K/cyf(O)]cv’R
exp(Kt,/R).
(2) An EL arc from v’(0) at t = 0 to t = t,. (3) An adiabaticJump to the given fmal volume v,att=t,.
Frg. 1 shows the volume as a function of trme which optrmrzes the work for heating functronsf(t) = Ate-@ with B= 1 s, and A = 0, 103, lOa, IO6 cal/s2. Wetookt,=2s,c gas), K = 3 cal K_ 1 s_1 T(o)r.==T3R/2 (monatomrc = 300 K, V(0) = 1 Q, and Pm = 8 Q. For’zero pumzng energy, the optrmal motion is an adiabatic expansion to V(tm) = Pm_ The isothermal expansion pulls energy from the heat bath. For nonvamshing f(t), we observe that as the pumped energy, Ep, IS mcreased and becomes larger than the mrtral internal energy of the gas, the mrtral jump tncreasingly compresses the fluid. For large pumping energies the Euler-Lagrange motion IS such as to keep the gas compressed and at a Hugh temperature whale the heat IS being put m. Fig. 2 shows E(f) versus V(t) for the four examples mentioned above. In the first example the uuttal adtabat IS an expansion
1 (3W
where t, IS some uutral trme and F(t) E KT_ +f(t), and boundary arcs whrch are instantaneous adiabats gtven by E(v)
= E(V,)(V,/@‘cr’
(4
These arcs must be pieced together UI an optimal way to maximize the work. The problem of finding the optimal swrtchmg between the various stages of the solution. eqs. (3) and (4) IS called the stagmg problem m optimal control theory. The solutron of the stagmg problem for our example consists of three stages: (1) An adiabatic jump at t = 0 from V(0) to P’(O), where P’(0) 1s the solutron of the equatron 128
Fig 1. Volume versus time for the Izes the work. V(O) = 1 P, V(2 s) =
= lo3 f e-‘, (3)f(t)=
optrmal path which maxim8 P. (l)f(r) =O, (2)f(t) lo4 t e-*; (4)f(r) = lo6 t e-t
15 May 1980
CHEMICAL PHYSICS LETTERS
Volume 72, number 1
Table 1 Ep (Cal) a)
s 6)
11
G
Sh Ce9
0 594 5940 59400 5940
1 1 1 0.2
0.773 0 594 0.461 0 675 0.565
1.05 1.17 1.56 2.95 3.35
0.613 0.063 4570 554.200 3.61
a) The corresponding values of A arc. 0. 103, LO’, LO’. 1.486 X 10’ calls. respectively.
L 0
solution does not approach a reversible path as t, + =, since the motlonal arcs for t E [O, ?&I&I J are not close to thermal equilibrium with the heat bath. It IS a simple matter to determine the limit at in either case it is *m + m for this case. Furthermore, easy to show that the maximum work is a nondecreasing function of t, _ In table I we list several properties of the optimal solutions for the heat functrons given above. The efficiency I). the gam factor over other system operating conditrons, C, and the Irreversibility of the optimaI solution are hsted in table I_ Here, we choose to define the efficrency the optimal
I
I
I
0
10
“ITIters,
Fig 2. Energy versus volume along the optimal path for the four cases in fE_ 1. The ordinate scale for cases (1) and (2) LS1 urut = 1000 Cal, for case (3) the scale is 1 unit = 5000 cal, and for case (4) the scale 1s 1 umt = 10’ Cal. In each case the miual energy equals 900 Cal and the mtml volume equals 1 P.
and is followed
by an EL arc which 1s an Isothermal expansion and a final adiabatic expansion. The imtial adrabatic expansion cools the gas thereby allowmg energy to leak into the gas from the bath. In example two we already see a compromlse between the desire to extract heat from the bath and to have the gas at a Hugh temperature as the gas IS heated. Asf(t) is IIIcreased in examples three and four it IS no longer expedlent to extract energy from the bath. The mitral adlabats become compressions so as to prepare the gas to accept the heat as high-grade energy. It is interesting to note the hmlt of the optimal solutions as t, +m. Forf(t) = 0 we obtain a solution gwen by an inltlally infimtesimal adiabatic evpanslon to a temperature T= Te, 11 -(R/kt,)~[V~/V(O)I and correspondmg nential isothermal
3-
volume V’(O), followed expansion
V(t) = V’(O)exp{tln[~,/~(O)I
by an expo-
lt,3-
Thus, we see how the reversrble path of (infmlte tune) thermodynamics is reached as a limit of the present optunal solution whenf(r) = 0. For a fmtte, time dependent J(t) III the interval [O,?&, ] where and with f(t) = 0 III the interval [Tm, tm] , Fm c&l,
9 = i’lCEp
+ RTex In[~,,,lV(O)13,
thereby comparmg with the thermodynamic bound for the case where S, = 0. Table 1 indicates that the efficiency decreases, reaches a mimmum, and then hcreases as the pumping energy ISincreased_ The irreversrble entropy production behaves similarly. Note that the optlmal path is highly irreversible. For firute time operation, one does not want to operate the system close to equdlbrium. Moreover, even for f,,, + +oo, as long as finite rates of heat pumping are involved, the optimal path is highly irreversible compared with nonoptimal paths. This is in contradic!ion wth the operation of systems run for infmite times with vanishmgly small rates. Table 1 presents the gain in choosmg the optimal route of the system compared with the route given by expanding with a constant rate of change of volume equal to [V,.,, - V(O)j /r, _331~ gain increases with increasing pumping energy. Furthermore, we found the gain to increase as the width of the pumpmg pulse is decreased_ We can consider several associated problems related to the original problem but with modifications of ‘312 system constraints or the system operating conditions_ 129
Volume 71. number 1
CHEhfICAL PHYSICS LETTERS
For example, rt 1s a sample task to find the solutions wrth unconstratned rrm, wrth Em constramed but V, unconstramted. and with both V, and Em constramed. For unconstramed V,. the mavrmum work equals ‘m E(O) + $ 0
f(T)dr
+ KT,,t,
and the optimal path IS grven by an inttral adrabatrc Jump to a very large volume, v’(O) + m, which extracts all the energy of the gas, followed by an EulerLagrange arc whrch extracts all the pumped energy plus the energy KT,,t, wlwzl~leaks mto the cylmder. For both Em and Ym constramed, the optimal path is given by an adtabat, an EL arc, and an adrabat, but now the end point of the nntial adrabat IS not arbitrary and therefore not open to further optrmrzatron. For E,.,, constramed and V, unconstramed, the maxnnum work equals t7l E(0) - E, + j- f(r)dr + KT,,t,. 0 The optrmal path IS an adrabar to a very large volume, fo!lowed by an EL arc, and then a final adrabatrc compressron to energy Em _ Examples of mod&matrons of system constramts mclude constramts upon the volume or the rate of change of volume of the system. In the latter case the boundary arcs that were adrabats are now of a drfferent nature, and in both cases, there may be addrtronal stages m the solutron hloreover, we can mclude modrficatrons arising from the effects of the mertra of the piston, prston frrctron, or bounds
130
15 h¶ay 1980
upon the esternal pressure necessary to determme the matron of the piston. The problems that we have considered here are of relevance to the design of various types of engmes. Optimal operatron of external combustion engmes are duectly related to the present problem formulation. In fact, we hare applied the concepts and methods developed herem to cychcly operating external combustion engmes Furthermore, we have applied the methods presented here to Internal combustron engmes wherein the heat pumped by tire chemical reactrons occurrmg rnstde the cyhnder can depend upon the volume and temperature of the workmg fluid. It seems clear that studies of optimal operation of blologxal systems, for example optrmal muscle control or neural stimulation, can also benefit from the methods developed herein. References
111B Andresen, PI [31 rql [51 [61 171
R.S. Berry, A. Nrtzan and P. Salamon, Phys Rev A15 (1977) 2086. P Salarnon, A. N~tzan. B. Andresen and R.S Berry, Munmum Entropy Prodrlctron III Heat Engmes. Phys. Rev A, submttted for pubhcation. hl Rubin, Phys. Rev. A19 (1979) 1272,1277. D Gutko&in-Krusin. I Procaccia and J. Ross, J Chem Phys 69 (1978) 3898. R C Tclman and PC Fme, Rev. Mod. Phys 20 (1948) 51 0 Kafri and R D. Levme, Israel J. Chem 16 (1977) 342. L S. Pontryagm, V-C. Boltyanskd, R V Tamkrehdge and E F bhshchenko, The mathematmal theory of optimal processes (Wdcy, New York, 1962).
72, number
MAXlMtbl
Yehuda
1
CHEhlICAL
PHYSICS
WORK PRODUCTION
FROM A HEATED
15 May 1980
LETTERS
GAS IN A CYLINDER
WITH PISTON
B. BAND
Department
of Chemlstty.
Ben-Gut-ion Unirernty
of the Neger. Beer-She&a. Israel
Oded KAFRI Nuclear Research Center Negev, Beer-Sheva. Israel
and Peter SAL_AMON * Department Recelred
of Chemists.
27 February
Tel A LIVUtrtverslty, Ramat Aviv, Israel
1980
The optimal motion of a piston IIIa cybndcr containmg a gas. pumped wrth a given rate of heatmg is determined for maxtmal work productton Reversibtbty IS not a crttenon for high effictency tn fmtte time. The efficiency reaches the clasmal thermodynarmc bound for mfinttely slow processes.
Thermodynamrcs provrdes a bound on the maximum work obtainable from a grven thermodynamrc process. However, thermodynamrc bounds are often too hrgh to be valuable from a process which operates rrreversrbly and m finite trme. In this letter we determme the maxrmum work obtainable from a prototype system operatmg at finite time dependent rates for a fmrte tune. Furthermore we determme the actual system matron yrelding the maximum work. There has been consrderable recent mterest m the optunal control of thermodynamic systems operating m finite trme [l-4]. By incorporating trme dependent heating of a thermodynamrc system into the prototype problem, we add an important new aspect to the problems in thrs field. We consrder a piston fitted m a cylinder containmg a working fluid which IS coupled with a heat bath of temperature T,, and which is pumped with a grven rate of heatmg,fir). For srmplicrty we neglect effects of the mertia of the piston and the gas, piston frictron, and we assume equilibrmm of the working fluid at each instant l
of time but we note that these effects
Present address: Department Umverslty, Tempe, Arizona
of Mathematics, 85281. USA.
Arizona
as State
well as other energy loss mechanisms can be included m the present formulation. We ask the question: what is the optimal matron of the piston III order to extract the maximum work m a given amount of time, tm, gven an mitral and final volume of the system? For our system, the first law of thermodynamics takes the form &)
= f(t)
- ti(,ct) - K [ nt)
- T=..1,
(1)
where E is the internal energy of the working fluid, FVis the work of the expansion against the piston, and K is the thermal conductivity which we take to be a constant vdependent of the fluid volume. F?(f) is given by W(f) = p(f) P(t) where p and V are the pressure and volume of the flmd. If we take the working fhnd to be an ideal gas, then p(r) = nRT(f)/V(f) and E(f) = mvT(r) where tz is the number of moles of the gas, cy :s the heat capacity, and R is the gas constant. The second !aw of thermodynamics, m the equality form [S] , is given by Sin = $, f $, f $,, where Si, is the irreversible entropy production and S,,S,, and Sp are the entropies of the gas, the bath, and the heat pump, respectively. A standard thermodynamic analysis
Volume
72, number
CHEhiICAL
1
PHYSICS
[6] shows that the work is bounded by the reversrblr work which equals the sum of the changes in the avadabtlrtres of the pump and the gas, I$‘< ru,,
= EP - Tc,SP
- (Sg
- T,,AS,),
(2a)
(5)
wlth a=-
IrIG
Ep - T,,$,
+ uRT,,
ln[ V,/V(O)].
(2b)
if we assume that the pumped beat comes from a pure work source (I e. an mfimte temperature source), S,
also vamshes and it’ < Ep + nR Te, In [ Pm / V(O)] .
@cl
Smce the operaimg condltlons are Irreversible, this bound can never be obtained (except for the degenerate case E, = 0 and t, = +m, discussed below). We now a better and e\phcitly to the to obthe The of &dt the equation can carried using Pontryagm prmcrple We that optrmal IS of (EL) of form V(t) = V(t,)[F-(t)/F(t,)]
-=vm F”*(,)dT
E(t) = E(t,) [F(t)/F(ti)]
‘*.
cy
R
where EP = lofmf(t)dt.
Drfferentratmg this e\pressron wrth respect to T(t,) shows that the maxrmum work 1s obtained when T(t,) = T,, so when T(0) = T,, this becomes
15 May 1980
LEITERS
F~2(0)Jo’mFvt(~)d~ E(0) [ V(O)] R’cr’
and b = V(0)[E(O)K/cyf(O)]cv’R
exp(Kt,/R).
(2) An EL arc from v’(0) at t = 0 to t = t,. (3) An adiabaticJump to the given fmal volume v,att=t,.
Frg. 1 shows the volume as a function of trme which optrmrzes the work for heating functronsf(t) = Ate-@ with B= 1 s, and A = 0, 103, lOa, IO6 cal/s2. Wetookt,=2s,c gas), K = 3 cal K_ 1 s_1 T(o)r.==T3R/2 (monatomrc = 300 K, V(0) = 1 Q, and Pm = 8 Q. For’zero pumzng energy, the optrmal motion is an adiabatic expansion to V(tm) = Pm_ The isothermal expansion pulls energy from the heat bath. For nonvamshing f(t), we observe that as the pumped energy, Ep, IS mcreased and becomes larger than the mrtral internal energy of the gas, the mrtral jump tncreasingly compresses the fluid. For large pumping energies the Euler-Lagrange motion IS such as to keep the gas compressed and at a Hugh temperature whale the heat IS being put m. Fig. 2 shows E(f) versus V(t) for the four examples mentioned above. In the first example the uuttal adtabat IS an expansion
1 (3W
where t, IS some uutral trme and F(t) E KT_ +f(t), and boundary arcs whrch are instantaneous adiabats gtven by E(v)
= E(V,)(V,/@‘cr’
(4
These arcs must be pieced together UI an optimal way to maximize the work. The problem of finding the optimal swrtchmg between the various stages of the solution. eqs. (3) and (4) IS called the stagmg problem m optimal control theory. The solutron of the stagmg problem for our example consists of three stages: (1) An adiabatic jump at t = 0 from V(0) to P’(O), where P’(0) 1s the solutron of the equatron 128
Fig 1. Volume versus time for the Izes the work. V(O) = 1 P, V(2 s) =
= lo3 f e-‘, (3)f(t)=
optrmal path which maxim8 P. (l)f(r) =O, (2)f(t) lo4 t e-*; (4)f(r) = lo6 t e-t
15 May 1980
CHEMICAL PHYSICS LETTERS
Volume 72, number 1
Table 1 Ep (Cal) a)
s 6)
11
G
Sh Ce9
0 594 5940 59400 5940
1 1 1 0.2
0.773 0 594 0.461 0 675 0.565
1.05 1.17 1.56 2.95 3.35
0.613 0.063 4570 554.200 3.61
a) The corresponding values of A arc. 0. 103, LO’, LO’. 1.486 X 10’ calls. respectively.
L 0
solution does not approach a reversible path as t, + =, since the motlonal arcs for t E [O, ?&I&I J are not close to thermal equilibrium with the heat bath. It IS a simple matter to determine the limit at in either case it is *m + m for this case. Furthermore, easy to show that the maximum work is a nondecreasing function of t, _ In table I we list several properties of the optimal solutions for the heat functrons given above. The efficiency I). the gam factor over other system operating conditrons, C, and the Irreversibility of the optimaI solution are hsted in table I_ Here, we choose to define the efficrency the optimal
I
I
I
0
10
“ITIters,
Fig 2. Energy versus volume along the optimal path for the four cases in fE_ 1. The ordinate scale for cases (1) and (2) LS1 urut = 1000 Cal, for case (3) the scale is 1 unit = 5000 cal, and for case (4) the scale 1s 1 umt = 10’ Cal. In each case the miual energy equals 900 Cal and the mtml volume equals 1 P.
and is followed
by an EL arc which 1s an Isothermal expansion and a final adiabatic expansion. The imtial adrabatic expansion cools the gas thereby allowmg energy to leak into the gas from the bath. In example two we already see a compromlse between the desire to extract heat from the bath and to have the gas at a Hugh temperature as the gas IS heated. Asf(t) is IIIcreased in examples three and four it IS no longer expedlent to extract energy from the bath. The mitral adlabats become compressions so as to prepare the gas to accept the heat as high-grade energy. It is interesting to note the hmlt of the optimal solutions as t, +m. Forf(t) = 0 we obtain a solution gwen by an inltlally infimtesimal adiabatic evpanslon to a temperature T= Te, 11 -(R/kt,)~[V~/V(O)I and correspondmg nential isothermal
3-
volume V’(O), followed expansion
V(t) = V’(O)exp{tln[~,/~(O)I
by an expo-
lt,3-
Thus, we see how the reversrble path of (infmlte tune) thermodynamics is reached as a limit of the present optunal solution whenf(r) = 0. For a fmtte, time dependent J(t) III the interval [O,?&, ] where and with f(t) = 0 III the interval [Tm, tm] , Fm c&l,
9 = i’lCEp
+ RTex In[~,,,lV(O)13,
thereby comparmg with the thermodynamic bound for the case where S, = 0. Table 1 indicates that the efficiency decreases, reaches a mimmum, and then hcreases as the pumping energy ISincreased_ The irreversrble entropy production behaves similarly. Note that the optlmal path is highly irreversible. For firute time operation, one does not want to operate the system close to equdlbrium. Moreover, even for f,,, + +oo, as long as finite rates of heat pumping are involved, the optimal path is highly irreversible compared with nonoptimal paths. This is in contradic!ion wth the operation of systems run for infmite times with vanishmgly small rates. Table 1 presents the gain in choosmg the optimal route of the system compared with the route given by expanding with a constant rate of change of volume equal to [V,.,, - V(O)j /r, _331~ gain increases with increasing pumping energy. Furthermore, we found the gain to increase as the width of the pumpmg pulse is decreased_ We can consider several associated problems related to the original problem but with modifications of ‘312 system constraints or the system operating conditions_ 129
Volume 71. number 1
CHEhfICAL PHYSICS LETTERS
For example, rt 1s a sample task to find the solutions wrth unconstratned rrm, wrth Em constramed but V, unconstramted. and with both V, and Em constramed. For unconstramed V,. the mavrmum work equals ‘m E(O) + $ 0
f(T)dr
+ KT,,t,
and the optimal path IS grven by an inttral adrabatrc Jump to a very large volume, v’(O) + m, which extracts all the energy of the gas, followed by an EulerLagrange arc whrch extracts all the pumped energy plus the energy KT,,t, wlwzl~leaks mto the cylmder. For both Em and Ym constramed, the optimal path is given by an adtabat, an EL arc, and an adrabat, but now the end point of the nntial adrabat IS not arbitrary and therefore not open to further optrmrzatron. For E,.,, constramed and V, unconstramed, the maxnnum work equals t7l E(0) - E, + j- f(r)dr + KT,,t,. 0 The optrmal path IS an adrabar to a very large volume, fo!lowed by an EL arc, and then a final adrabatrc compressron to energy Em _ Examples of mod&matrons of system constramts mclude constramts upon the volume or the rate of change of volume of the system. In the latter case the boundary arcs that were adrabats are now of a drfferent nature, and in both cases, there may be addrtronal stages m the solutron hloreover, we can mclude modrficatrons arising from the effects of the mertra of the piston, prston frrctron, or bounds
130
15 h¶ay 1980
upon the esternal pressure necessary to determme the matron of the piston. The problems that we have considered here are of relevance to the design of various types of engmes. Optimal operatron of external combustion engmes are duectly related to the present problem formulation. In fact, we hare applied the concepts and methods developed herem to cychcly operating external combustion engmes Furthermore, we have applied the methods presented here to Internal combustron engmes wherein the heat pumped by tire chemical reactrons occurrmg rnstde the cyhnder can depend upon the volume and temperature of the workmg fluid. It seems clear that studies of optimal operation of blologxal systems, for example optrmal muscle control or neural stimulation, can also benefit from the methods developed herein. References
111B Andresen, PI [31 rql [51 [61 171
R.S. Berry, A. Nrtzan and P. Salamon, Phys Rev A15 (1977) 2086. P Salarnon, A. N~tzan. B. Andresen and R.S Berry, Munmum Entropy Prodrlctron III Heat Engmes. Phys. Rev A, submttted for pubhcation. hl Rubin, Phys. Rev. A19 (1979) 1272,1277. D Gutko&in-Krusin. I Procaccia and J. Ross, J Chem Phys 69 (1978) 3898. R C Tclman and PC Fme, Rev. Mod. Phys 20 (1948) 51 0 Kafri and R D. Levme, Israel J. Chem 16 (1977) 342. L S. Pontryagm, V-C. Boltyanskd, R V Tamkrehdge and E F bhshchenko, The mathematmal theory of optimal processes (Wdcy, New York, 1962).