Boosting thermodynamic performance by bending space-time Emily E. Ferketic andSebastian Deffnera Department of Physics University of Maryland Baltimore County Baltimore MD 21250 USA

2025-04-30 0 0 476.4KB 6 页 10玖币

侵权投诉

Boosting thermodynamic performance by bending space-time

Emily E. Ferketic and Sebastian Deffner (a)

Department of Physics, University of Maryland, Baltimore County, Baltimore, MD 21250, USA

Abstract –Black holes are arguably the most extreme regions of the universe. Yet, they are

also utterly inaccessible to experimentation, and even just indirect observation poses signiﬁcant

technical challenges. The phenomenological approach of thermodynamics is uniquely suited to

explore at least some of the physical properties of such scenarios, and this has motivated the

study of so-called holographic engines. We show that the eﬃciency of an endoreversible Brayton

cycle is given by the Curzon-Ahlborn eﬃciency if the engine is fueled by a 2-dimensional ideal

gas; and that the eﬃciency is higher, if the working medium is a (2+1)-dimensional BTZ black

hole. These ﬁndings may be relevant not only in the quest to unlock the mysteries of black holes,

but also for potential technological applications of graphene.

As pretentious as it might sound, every single time we

navigate our cars towards a gas station we are forced

to make a decision of non-trivial thermodynamic conse-

quences – which fuel should we buy? Typically, gas sta-

tions oﬀer the mundane garden variety of rather similar

chemical compositions, which oﬀer only small diﬀerences

in maximal power and engine eﬃciency.

The situation is a lot more interesting at the forefront

of technological development, where more and more quan-

tum fuels are being investigated [1]. In fact, it has been

recognized early on that engines operating with quantum

fuels can do things that classical engines cannot [2,3], and

that this is perfectly legal under the laws of thermodynam-

ics [4]. As a phenomenological theory, thermodynamics

only has very limited knowledge of the underlying micro-

scopic structure of the considered systems [5]. Any dif-

ferences in engine performance are thus fully derived from

the fundamental relation, or a complete set of equations of

state. This fundamental relation can be obtained empiri-

cally as, for instance, for the ideal gas, or it can be derived

from microscopic theories such as statistical mechanics or

quantum ﬁeld theory [6].

Owing to the universality of the thermodynamic laws,

deriving statements about the thermodynamic character-

istics becomes then almost only an exercise. However,

such an approach does permit to unveil further statements

about, e.g., the question which fuel has the best perfor-

mance in what scenario. In particular, we have recently

shown that endoreversible Otto engines can outperform

classical engines, if they operate with quantum fuels [7,8],

(a)E-mail: deffner@umbc.edu

whereas corrections from special relativity do not support

an additional boost of thermodynamic performance [9].

Thus, the rather natural question arises whether in even

more extreme scenarios thermodynamic performance is

governed by the properties of space-time. Such a question

is not quite as esoteric as it might appear, since black hole

thermodynamics [10,11] has been a particular fruitful ap-

proach in elucidating the properties of such mysterious re-

gions of space in the universe. The central quantity is the

Bekenstein-Hawking entropy [12–14], which is the amount

of entropy that must be assigned to a black hole in order

for it to comply with the laws of thermodynamics as they

are interpreted by external observers. To further investi-

gate the thermodynamic properties of such black holes, it

has now become an almost common practice to analyze so-

called holographic engines [15]. These are engine cycles, in

which the cosmological constant, i.e, the energy density of

the considered region of space [16], is treated as thermo-

dynamic variable akin to pressure. Such research is often

motivated by attempts to develop a more complete under-

standing of the universe [15]. Yet, it is somewhat unlikely

that our civilization will ever reach a state in which one

would use cosmological black holes as fuel in technological

applications [17]. However, special and general relativity

have found close analogies in condensed matter [18] and

optical systems [19,20], which are clearly going to ﬁnd

near-term applications in nanotechnology.

In the present analysis, we thus investigate whether sin-

gularities in space-time can be exploited as a thermody-

namic resource. More speciﬁcally, we show that the ef-

ﬁciency at maximum power of an endoreversible Brayton

cycle is higher, when operated with a Ba˜nados-Teitelboim-

p-1

arXiv:2210.03652v1 [cond-mat.stat-mech] 6 Oct 2022

Emily E. Ferketic and Sebastian Deﬀner

Zanelli (BTZ) black hole [6] than when fueled by an ideal

gas. The Brayton cycle consists of two isentropic and two

isobaric strokes, and hence it is most conveniently treated

in the entalphy representation [5]. Since also the funda-

mental relation of black holes is particularly accessible in

enthalpy form [21,22], the Brayton cycle appears as the

somewhat natural choice for holographic engines. As main

results, we ﬁnd that for the 2-dimensional ideal gas the

eﬃciency at maximum power is identical to the Curzon-

Ahlborn eﬃciency [23], and that for the BTZ black hole

the eﬃciency is larger. All results are derived analytically

and in closed form.

Endoreversible Brayton cycle. – We start by es-

tablishing notions and notations. The ideal Brayton or

Joule cycle is consists of two isentropic and two isobaric

strokes [5]. It is closely related to the Otto cycle, in

that the two isochoric strokes are replaced by two isobaric

strokes, which is more apt as a design principle for, e.g.,

gas turbines. The endoreversible Brayton cycle can then

be constructed in full analogy to the endoreversible Otto

cycle [7]. In particular, we have:

Isobaric heating (A→B). For the isobaric strokes it is

most convenient to work with the enthalpy representation

of the fundamental equation, H=H(S, P ). Recall that

in diﬀerential form we have [5],

dH =T dS +V dP and thus QA→B=HB−HA.(1)

Correspondingly, we can express the work done during

isobaric heating as

WA→B= ∆EA→B−QA→B=PA(VA−VB) (2)

which is nothing but the ﬁrst law of thermodynamics, and

where we used the thermodynamic identity H=E+P V ,

and that PA=PB.

In complete analogy to the endoreversible Carnot [23]

and Otto [7–9] cycles, we now assume that the working

substance is in a state of local equilibrium at a tempera-

ture below the temperature of the hot reservoir. Thus we

write,

T(0) = TAand T(τ) = TBwith TA< TB≤Thot ,

(3)

where τis the ﬁnite duration of the stroke. In linear ap-

proximation, the time-dependence of the temperature is

then given by Fourier’s law [7],

dt =−α(T(t)−Thot) (4)

where αis a constant depending on the heat conductivity

and heat capacity of the working substance.

Equation (4) can be solved exactly, and we obtain

TB−Thot = (TA−Thot) exp (−α τ).(5)

Isentropic expansion (B →C).During the isentropic

strokes the working substance is decoupled from the en-

vironment. Therefore, the endoreversible description is

identical to the equilibrium cycle. Again from the ﬁrst

law of thermodynamics, ∆E=Q+W, we simply have,

QB→C= 0 and WB→C=EC−EB(6)

where QB→Cis the heat exchanged, and WB→Cis the

work performed during the expansion.

Isobaric cooling (C →D). Heat and work during the

isobaric cooling read,

QC→D=HD−HCand WC→D=PC(VC−VD),(7)

and we now have

T(0) = TCand T(τ) = TDwith TC> TD≥Tcold .

(8)

For the sake of simplicity, we assume that the stroke time

of the isobaric cooling is identical to the stroke time of

the isobaric heating, τ. The generalization to diﬀerent

strokes times is straight forward, but does signiﬁcantly

increase the clutter in the formulas, cf Ref. [7,8] for the

corresponding Otto cycle.

Similarly to above (4) the heat transfer is described by

Fourier’s law

dt =−α(T(t)−Tcold),(9)

where αis a constant characteristic for the cold stroke.,

which again we choose to be identical to the heating stroke.

From the solution of Eq. (9) we now obtain

TD−Tcold = (TC−Tcold) exp (−α τ),(10)

which describes the cooling from TCto TD.

Isentropic compression (D →A). The cycle is com-

pleted by isentropic expansion, for which we have

QD→A= 0 and WD→A=ED−EA.(11)

Brayton eﬃciency. As always, the eﬃciency of a

thermodynamic cycle is deﬁned as ratio of work output,

Wout = (QA→B−QC→D), and (hot) heat input, QA→B.

For the present Brayton cycle we immediately ﬁnd

η= 1 + QC→D

QA→B

= 1 −HC−HD

HB−HA

,(12)

which cannot be further simpliﬁed without knowledge of

the fundamental equation, H=H(S, P ).

The ideal gas. – As a point of reference and to build

intuition we begin with an endoreversible Brayton cycle

that operates with an ideal gas as working medium. The

well-known equation of state, aka the ideal gas law reads

P V =N kBT , (13)

p-2

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

Boostingthermodynamicperformancebybendingspace-timeEmilyE.FerketicandSebastianDeffner(a)DepartmentofPhysics,UniversityofMaryland,BaltimoreCounty,Baltimore,MD21250,USAAbstract{Blackholesarearguablythemostextremeregionsoftheuniverse.Yet,theyarealsoutterlyinaccessibletoexperimentation,andevenjustindire...

展开>> 收起<<

Boosting thermodynamic performance by bending space-time Emily E. Ferketic andSebastian Deffnera Department of Physics University of Maryland Baltimore County Baltimore MD 21250 USA.pdf

共6页,预览2页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Boosting thermodynamic performance by bending space-time Emily E. Ferketic andSebastian Deffnera Department of Physics University of Maryland Baltimore County Baltimore MD 21250 USA

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: