Efficiency of simple quantum engines: The Joule-Brayton and Otto cycles L. Guzman-Vargas*1, V. Granadosf and R. D. Mota* * Unidad Profesional en Ingenieria y Tecnologias Avanzadas, Institute Politecnico National, Av. IPNNo. 2580, Col. Laguna Ticomdn, 07340, Mexico D.E T Departamento de Fisica, Escuela Superior de Fisica y Matemdticas, Institute Politecnico National, Edif. No. 9, U.P. Zacatenco, 07738, Mexico D.E Abstract. Following the formalism of a quantum engine proposed by Bender et al.1 for a single particle of mass m confined to an infinite one-dimensional potential well of width L, we construct the isobaric and isochoric quantum analogous processes in order to analyze the efficiency of two classical thermodynamic engines: The Joule-Brayton and Otto cycles. We find that the efficiency are analogous to those obtained for classical engines.

INTRODUCTION Reversible thermodynamics applied to the analysis of heat engines gives results based on equilibrium processes. Real heat engines have efficiencies less than the ideal Carnot value for the same heat reservoirs. Quantum engine models have been studied during the last two decades [1, 2, 3, 4]. This new field takes into account the analogies between quantum systems and macroscopic engines. Some of these models consider dynamic equations of working fluid and finite time of operation [3, 4, 5]. Recently, Bender et al. [6] proposed a model of a cyclic engine based on a single quantum mechanical particle confined an infinite one-dimensional potential well of width L. This model uses the fact that the walls of the containing potential play the role of the piston in a cylinder containing an ideal gas and then they construct quantum mechanical equivalents of isothermal and adiabatic processes. Bender et al. defined the force F exerted on the walls as, (1) F = _« and based on this force, it is possible to define several quantum processes analogous to those used in reversible thermodynamics. The role of the temperature here is replaced by the energy given by the expectation value of the Hamiltonian. Classically, an isobaric process is one in which the system evolves remaining the pressure constant, that is, the average forces against the walls of the container are constant, even when the system is compressed or expanded. In the quantum mechanical case, if we assume that the system at the initial state \i/(x) of volume L is a linear combination of eigenstates n(x), the expectation value of the Hamiltonian must change as the walls of the well moves, Bender C. M., Brody C. B. and Meister K. B., J. Phys. A 33, 4427 (2000)

CP643, Quantum Limits to the Second Law: First International Conference, edited by D. P. Sheehan © 2002 American Institute of Physics 0-7354-0098-9/02/$ 19.00 291

Downloaded 09 Jul 2003 to 129.105.205.149. Redistribution subject to AIP license or copyright, see http://proceedings.aip.org/proceedings/cpcr.jsp

fej

.1%

FIGURE 1. The Joule-Brayton cycle

then the instantaneous force exerted on the walls can be obtained using the relation (1). During the isobaric process this force remains constant, that is, the rate of change of the energy with respect to changes in size of the well is constant. The energy value as a function of L can be written as \an\2En

E(L) =

(2)

n=l

where the energy spectrum is En = ^j^r and the coefficients \an\ are constrained to £~=1 \an 2 = L. On the other hand, an isochoric process is one in which the volume of the potential well is constant. During this process the system can increase or diminish its energy and the force exerted on the walls also changes according with the energy pumping by an external source.

THE JOULE-BRAYTON AND OTTO CYCLES The classical Joule-Brayton cycle is composed by two isobaric and two adiabatic processes (see Fig. 1). From 1 —>• 2 we have an isobaric process. If the system is in the state one at point 1 and expands isobarically, the system is excited to the second level. Thus, the state of the system at any time is given by a linear combination of the two energy eigenstates, where (/^ and 0 2 are the wavefunctions of state one and two, respectively. The coefficients satisfy the condition a\ |2 +|#2 2 = 1 • The expectation value of the Hamiltonian is calculated by means of E — (\\f\H\ iff), which result is, (4)

292

Downloaded 09 Jul 2003 to 129.105.205.149. Redistribution subject to AIP license or copyright, see http://proceedings.aip.org/proceedings/cpcr.jsp

where we have used the relation \a\ \2 + a^ — 1. The force during this process remains constant and its value is given in terms of its definition (eq. 1), -3ai2).

(5)

The force at point 1 as a function of the length LI is

FB = ^J

raL-j

(6)

Then, equalling eq. (5) and (6) we can conclude that

L 3 = L?(4-3|ai |2)

(7)

Now, it is easy to show that the maximum value of L = £2, is LI — 41/3Li , which happens when a\ — 0 (point 2). By combining eqs. (4) and (7), we find that the energy as a function of L can be written as 2mL\

(8,

V ;

or

f ~ 2~^rJ = const-> mat i§» me isobaric analogous equation. During the process 2 —>• 3 we have an adiabatic expansion. That is, no external energy enter to the system, so that the particle remains in the second state [6] and the change in the internal energy equals the work performed against the walls of the well. The expectation value of the Hamiltonian is E = 2^ 2 , and the force is given by (9)

In eq. (9) we can observe that the the product L3F(L) is a constant and is considered the quantum analogue of the classical adiabatic processes [6]. From 3 -> 4 we have again an isobaric process. At the starting point (3) the system is in the second state. Thus as the compression is taking place the system return to the base state. The ground state is reached at point £4 = -j^- . The force during this isobaric compression is given by A^n-2^2

(10)

and the energy as a function of L is E = 2nmLf 3 L . 3 During the last step from 4 -> 1 an adiabatic compression is performed. The system remains in the base state and the expectation of the Hamiltonian is given by ^|j and the force exerted on the walls is ^-2^2

(ID

293

Downloaded 09 Jul 2003 to 129.105.205.149. Redistribution subject to AIP license or copyright, see http://proceedings.aip.org/proceedings/cpcr.jsp

Now, we calculate the work performed during one closed cycle, along the four steps described above. By using the eqs. (6), (9), (10) and (11), we have

4!/3Li

W

= fFBdL+

L3

^173

LI

/ F 2 (L)dL+/F A dL+ / F4(L)dL _*2_ 4 l/3

+F A L 3 (^)+4jF A L3-iFBL 1 (12) where we have used eqs. (6) and (10) to rewrite some terms. The heat input QH along the first isobaric process (1 —> 2) can be calculated as QH — W\2 + A£"i2, where W\2 and AE\2 are the work performed and the change in the internal energy along the isobaric branch. This change in the internal energy can be calculated using the eq. (8),

(13) where we used eq. (6), and the work W\2 is given by the first term of the right of eq. (12). Thus, the heat input \Qn\ can be expressed as

QH = \FaLi (41/3 - l)

(14)

Finally, the efficiency of the closed cycle is defined as

W

where W is the work and Q# is the heat input given in the eqs. (12) and (14), respectively. After some elementary steps we find that,

If we take the quotient between the forces given by the eqs. (6) and (10), we obtain,

£3 _ 41/3 / F& \ Using this relation into eq. (16) and defining the ratio rp — |£, the efficiency can be written as V

=

l

--j/3

(17)

We can observe that this efficiency is analogous to the efficiency of a classical JouleBrayton cycle [7]. As is shown in Fig (2), the Otto cycle is composed by two isochoric and two adiabatic processes. Starting at point 1, from 1 —> 2 we have an isochoric process. Again, if the system is in the state one at point 1 and is compressed isochorically (length constant),

294

Downloaded 09 Jul 2003 to 129.105.205.149. Redistribution subject to AIP license or copyright, see http://proceedings.aip.org/proceedings/cpcr.jsp

F *

FIGURE 2. The Otto cycle

then the system is excited to the second level. Thus, as we did in the Joule-Brayton cycle, the expectation value of the Hamiltonian and the expression of the force are equal to eqs.(4) and (5), respectively. From eq. (5) we can express the length in terms of the force as

mF

-3 h i 2 ) -

V

(18)

Thus, the length at point 1 is 7t2H2

(19)

by equalling eqs. (19) and (20), we can conclude that F = Fi(4-3|a!| 2 )

(20)

Now, it is easy to show that the maximum value of FI, is that FI = 4Fi, it occurs when ai — 0 (point 2). By combining eq.(4) and eq.(21), we find that the energy can be written as

this expression can be written as — ~^ — const., that is the isochoric analogous equation. During the process 2 —>> 3 we have an adiabatic expansion and is valid equation (9). From 3 —> 4 we have again an isochoric process. At the starting point 3, the system is in the second state. Thus as the system is yielding heat the system return to the base level. The ground state is reached at point in which F^ = ^. The length of the well is given by (22)

and the energy as a function of L is

E=

2n2H2F

(23)

295

Downloaded 09 Jul 2003 to 129.105.205.149. Redistribution subject to AIP license or copyright, see http://proceedings.aip.org/proceedings/cpcr.jsp

During the last step from 4 —>• 1 an adiabatic compression is performed and is valid equation (11). Since for one closed cycle there is no change in the internal energy, the work performed along the one closed cycle, can be related to the heat exchanged as W= \QH\-\Qc\

(24)

where \Qn and \Qc\ are the heat input and output during the process 1—>• 2 and 3 —)> 4, respectively. This quantities can be calculated by using eqs. (22) and (24), 4F

i

\QH\ = AEU= f§dF = %£r F\

(25)

l

and |2c| = A£34= / §<^=S

(26)

^3

Finally, using the definition of efficiency (eq. (16)) and eqs. (26) and (27), we can conclude that Qc _ i _ i

QH

(27)

where we have introduced the compression ratio, ri — ^-. This is the analogous efficiency to the classical Otto cycle [7].

CONCLUSIONS The conversion of heat into work, is the main objective of a thermal engine. In this work, we showed that if we use a quantum mechanical particle confined to a potential well, as a working fluid, we can construct analogous equations to classical isobaric and isochoric processes. Besides, following the analogous processes that compose the Joule-Bray ton and Otto cycle, we found that the efficiencies of these two quantum cycles, are analogous to the well known efficiencies from classical thermodynamics.

ACKNOWLEDGMENTS We thank to F. Angulo-Brown for comments. This work was supported by COFAA-IPN.

REFERENCES 1. 2. 3. 4. 5. 6. 7.

Kosloff R., J. Chem. Phys. 80(4), 1625, (1984) Feldmann T, Geva E., Kosloff R. and Salamon P., Am. J. Phys., 64(4), 485, (1996). Geva E., Kosloff R., /. Chem. Phys., 96(4), 3054, 1992 Sisman A. and Saygin H., J. Phys. D: Appl. Phys. 32, 664-670, (1999). Saygin H., Sisman A., J. Appl. Phys. 90,6,(2001) 3086-3089. Bender C. M., Brody C. B. and Meister K. B., /. Phys. A 33, 4427 (2000). Zemansky M.W., Heat and Thermodynamics, McGraw-Hill, New York, 1968.

296

Downloaded 09 Jul 2003 to 129.105.205.149. Redistribution subject to AIP license or copyright, see http://proceedings.aip.org/proceedings/cpcr.jsp