The General Class of Accelerating, Rotating and Charged Plebanski-Demianski Black Holes as Heat Engine
aa r X i v : . [ g r- q c ] J un The General Class of Accelerating, Rotating and ChargedPlebanski-Demianski Black Holes as Heat Engine
Ujjal Debnath ∗ Department of Mathematics,Indian Institute of Engineering Science and Technology,Shibpur, Howrah-711 103, India. (Dated: June 5, 2020)We first review the general class of accelerating, rotating and charged Plebanski-Demianski(PD) black holes in presence of cosmological constant, which includes the Kerr-Newmanrotating black hole and the Taub-NUT spacetime. We assume that the thermodynamicalpressure may be described by the negative cosmological constant and so the black holerepresents anti-de Sitter (AdS) PD black hole. The thermodynamic quantities like surfacearea, entropy, volume, temperature, Gibb’s and Helmholtz’s free energies of the AdS PDblack hole are obtained due to the thermodynamic system. Next we find the critical pointand corresponding critical pressure, critical temperature and critical volume for AdS PDblack hole. Due to the study of specific heat capacity, we obtain C V = 0 and C P ≥ µ , we determine the heating and cooling nature of PD black hole. Putting µ = 0,we find the inversion temperature. Next we study the heat engine for AdS PD black hole.In Carnot cycle, we obtain the work done and its maximum efficiency. Also we describethe work done and its efficiency for a new engine. Finally, we analyze the efficiency for theRankine cycle in PD black hole heat engine. ∗ [email protected] Contents
I. Introduction II. Thermodynamics of Plebanski-Demianski Black Hole P - V Criticality 7D. Stability 10E. Joule-Thomson Expansion 10
III. Heat Engine
IV. Discussions and Concluding Remarks References I. INTRODUCTION
The black hole thermodynamics has become an important topic of intensive research since Hawking’sradiation of black holes [1, 2] and considered as a crucial topic to gaining insight into the quantum natureof gravity. From the early discoveries that black hole area and surface gravity behave as thermodynamicentropy [3, 4] and temperature [1] respectively. Gibbons and Hawking [5] have studied the physics ofanti-de Sitter (AdS) black hole due to AdS/CFT correspondence. Hawking and Page [6] have studied thethermodynamic properties of static Schwarzschild-AdS black hole. After few years, Chamblin et al [7, 8]have investigated the physical properties of charged Reissner-Nordstrom-AdS black hole. If one considerscharge and/or rotation of the AdS black hole, the nature of the AdS black hole is similar to the Van derWaals fluid [9, 10]. In the black hole chemistry [11, 12], the negative cosmological constant (Λ <
0) isconsidered as a thermodynamic pressure P = − Λ8 π = πℓ ( ℓ is the length of AdS black hole) [13–22],has recently started to attract a growing deal of interest. In the thermodynamic system, the first law ofblack hole thermodynamics gives δM = T δS + V δP + ... with the black hole thermodynamic volume V = (cid:16) ∂M∂P (cid:17) S,... , where, M is the mass, S is the entropy and T is the temperature of the AdS black hole.The geometry of AdS black hole thermodynamics has been extensively studied by several authors [23–39].In the context of black hole chemistry, the concept of holographic heat engine has been proposed byJohnson [40] for AdS black hole, where he has considered the cosmological constant as a thermodynamicvariable. Based on the holographic heat engine proposal, Setare et al [41] have studied polytropic blackhole as a heat engine. Subsequently, Johnson [42–44] has analyzed the heat engine phenomena for theGauss-Bonnet black holes and Born-Infeld AdS black holes. Holographic heat engines for different typesof black holes have been studied in [45–53]. Zhang et al [54] have studied the f ( R ) black holes asheat engines. Heat engines of AdS black hole have been analyzed in the frameworks of massive gravity[46, 55–57], gravity’s rainbow [58] and conformal gravity [59]. Heat engine efficiency has been studiedfor Hayward [60] and Bardeen [61, 62] black holes. Heat engine in three dimensional BTZ black holehas been obtained in [63, 64]. Heat engine for dilatonic Born-Infeld black hole has been analyzed in[65]. For charge rotating dyonic black hole, the thermodynamic efficiency has been studied in [66]. Tillnow, several authors also have studied the heat engine phenomena of black holes in various occasions[67–70, 72–80].Using the standard black hole thermodynamics, it was found that the Hawking temperature of accel-erating black holes is more than Unruh temperature of the accelerated frame. Thermodynamics natureof accelerating black holes have been discussed in [81–86]. The thermodynamics properties of acceler-ating and rotating black holes have been investigated in [87, 88]. Also charged accelerating black holethermodynamics have been analyzed in [89–91]. Charged rotating and accelerating black hole thermo-dynamics have been studied in [92, 93]. The entropy bound of horizons for accelerating, rotating andcharged PlebanskiDemianski black hole have been discussed in [94]. Zhang et al [95] have studied theaccelerating AdS black holes as the holographic heat engines in a benchmarking scheme. Also Zhang etal [96] have studied the thermodynamics of charged accelerating AdS black holes and holographic heatengines. Recently Jafarzade et al [97] have investigated the thermodynamics and heat engine phenomenaof charged rotating accelerating AdS black holes. Motivated by the above works, here we’ll study thethermodynamics, P - V criticality, stability, Joule-Thomson expansion and heat engine for more generalclass of accelerating, rotating and charged Plebanski-Demianski (PD) black hole in AdS system. In sec-tion II, we write the general class of accelerating, rotating and charged AdS Plebanski-Demianski blackhole metric. Then we obtain the thermodynamic quantities, critical point, specific heat, stability andJoule-Thomson expansion. In section III, we study the phenomena of heat engine for PD black hole andstudy the Carnot cycle, Rankine cycle, work done and their efficiency. Finally, we provide the result ofthe whole work in section IV. II. THERMODYNAMICS OF PLEBANSKI-DEMIANSKI BLACK HOLEA. Plebanski-Demianski Black Hole Metric
Plebanski and Demianski [98] have presented a large class of Einstein-Maxwell electro-vacuum (alge-braic type D ) solutions, which includes Kerr-Newman black hole, Taub-NUT spacetime, (anti-)de Sitter(AdS) metric and their arbitrary combinations. The general class of accelerating, rotating and chargedPlebanski-Demianski (PD) black hole metric in AdS system is given by [94, 99–103] ds = 1Ω " − Q ρ (cid:26) dt − (cid:18) a sin θ + 4 lsin θ (cid:19) dφ (cid:27) + ρ Q dr + P ρ n adt + ( r + ( a + l ) ) dφ o + ρ P sin θdθ (1)where Ω , ρ , Q and P are given by Ω = 1 − αω ( l + a cosθ ) r , (2) ρ = r + ( l + a cosθ ) , (3) Q = ( ω k + e + g ) − M r + ǫ r − αnω r − (cid:18) α k + Λ3 (cid:19) r , (4) P = (1 − a cosθ − a cos θ ) sin θ , (5) a = 2 αaω M − α alω ( ω k + e + g ) − al Λ , (6) a = − α a ω ( ω k + e + g ) − a , (7) ǫ = ω ka − l + 4 αlω M − ( a + 3 l ) " α ω ( ω k + e + g ) + Λ3 , (8) n = ω kla − l − α ( a − l ) ω M + l ( a − l ) " α ω ( ω k + e + g ) + Λ3 , (9) k = " αlω M − α l ω ( e + g ) − l Λ ω a − l + 3 α l − (10)Here, a (= J/M ) , l, e, g, α, ω are angular momentum, NUT parameter, electric charge, magneticcharge, acceleration parameter and rotation parameter respectively. Also M is the mass of the PD blackhole and Λ is the cosmological constant. In particular, the PD black hole metric can be reduced to thefollowing black hole metrics: (i) C-metric ( a = l = 0) [104], (ii) Kerr-Newman-Taub-NUT black hole( α = g = 0) [105, 106], (ii) Kerr-Taub-NUT black hole ( α = e = g = 0) [107], (iii) Taub-NUT blackhole ( α = a = e = g = 0) [108], (iv) Kerr-Newman black hole ( α = l = g = 0) [109], (v) Kerr blackhole ( α = l = e = g = 0) [110], (vi) Riessner-Nordstrom black hole ( α = a = l = g = 0) and (vii)Schwarzschild black hole ( α = a = l = e = g = 0). B. Thermodynamic Quantities
Here, we’ll obtain the thermodynamic quantities for PD black hole in AdS system. For this purpose,the cosmological constant (Λ) can be written in terms of thermodynamic pressure ( P ) as Λ = − πP .The horizon radius r h of PD black hole can be obtained from the following equation (putting Q = 0)( ω k + e + g ) − M r h + ǫ r h − αnω r h − (cid:18) α k − πP (cid:19) r h = 0 (11)From above equation, pressure P can be expressed in terms of r h as P = 38 πω (cid:16) a r h α − ( ω − lr h α ) (cid:17) h l M r h α − l ( M + r h ) αω + 2 lr h (cid:16) e + g + M r h (cid:17) αω + (cid:16) e + g − M r h + r h (cid:17) ω − l (cid:16) e r h α + 3 g r h α + ω (cid:17) + a (cid:16) − l M r h α + 2 l ( M + r h ) αω + ω (cid:17)i × h − l r h α − l ω + 6 l r h ω + r h ω + a (cid:16) lr h α + 3 l ω + r h ω (cid:17)i − (12)Now the surface area of the PD black hole in AdS system is obtained in the form [111] A = Z Z √ g θθ g φφ dθdφ = 4 πω ( r h + ( a + l ) )( ω − lαr h ) − a α r h (13)So the Bekenstein-Hawking entropy [3, 112] on the horizon is given by S = A πω ( r h + ( a + l ) )( ω − lαr h ) − a α r h (14)From this, we can obtain the horizon radius in terms of entropy as r h = ω p f ( S ) − lαωSf ( S ) (15)where f ( S ) = α l S + (cid:16) S − ( a + l ) π (cid:17) f ( S ) (16)and f ( S ) = α ( a − l ) S + πω (17)The volume of the PD black hole is given by V = (cid:18) ∂M∂P (cid:19) S,... = 4 πω (cid:2) ω ( r h + 6 l r h − l ) − αl r h + a (2 lαr h + 6 l ω + ωr h ) (cid:3) a l α − l α + ω ) (18)which can be written in terms of entropy as in the following form V = 4 πω (cid:20) l ( a − l ) f ( S ) + ωf ( S ) f ( S ) (cid:16)p f ( S ) − lαS (cid:17) + ω (cid:16)p f ( S ) − lαS (cid:17) (cid:21) f ( S ) [3 l α ( a − l ) + ω ] (19)where f ( S ) = a (cid:20) f ( S ) + 2 lα (cid:18)q f ( S ) − lαS (cid:19)(cid:21) + 2 l (cid:20) f ( S ) − lα (cid:18)q f ( S ) − lαS (cid:19)(cid:21) (20)Now the surface gravity on the horizon of black hole is defined by κ = 12 √− h ∂∂x a (cid:18) √− h h ab ∂r∂x a (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) r = r h (21)where h ab is the second order metric constructed from the t - r components of the metric and h = det ( h ab ).The temperature on the horizon of the PD black hole is given by T = κ π = [ ω − α ( l + a ) r h ]2 πω [ r h + ( l + a ) ] (cid:20) − M + 2 ǫ r h − αnω r h − (cid:18) α k − πP (cid:19) r h (cid:21) (22)The temperature can be written in terms of entropy as in the following form T = h f ( S ) + α ( a + l ) (cid:16)p f ( S ) − lαS (cid:17)i f ( S )3 πf ( S ) f ( S ) (23)where f ( S ) = ( a + l ) f ( S ) + ω (cid:18)q f ( S ) − lαS (cid:19) (24)and f ( S ) = 3 M f ( S ) + ω (cid:18)q f ( S ) − lαS (cid:19) (cid:18) nαf ( S ) (cid:18)q f ( S ) − lαS (cid:19) +2 ω (cid:18)q f ( S ) − lαS (cid:19) (3 kα − πP ) − ǫf ( S ) ! (25)The Gibb’s free energy is given by [75] G = M − T S = ( ω + e + g ) f ( S )2 ω (cid:16)p f ( S ) − lαS (cid:17) + ǫω (cid:16)p f ( S ) − lαS (cid:17) f ( S ) − αnω (cid:16)p f ( S ) − lαS (cid:17) f ( S ) − ω f ( S ) (3 α k − πP ) (cid:18)q f ( S ) − lαS (cid:19) − S h f ( S ) + α ( a + l ) (cid:16)p f ( S ) − lαS (cid:17)i f ( S )3 πf ( S ) f ( S ) (26) r h S FIG. 1: Figure represents the plot of entropy S against PD black hole horizon radius r h . Also the Helmholtz’s free energy is obtained as [75] F = G − P V = ( ω + e + g ) f ( S )2 ω (cid:16)p f ( S ) − lαS (cid:17) + ǫω (cid:16)p f ( S ) − lαS (cid:17) f ( S ) − αnω (cid:16)p f ( S ) − lαS (cid:17) f ( S ) − ω f ( S ) (3 α k − πP ) (cid:18)q f ( S ) − lαS (cid:19) − S h f ( S ) + α ( a + l ) (cid:16)p f ( S ) − lαS (cid:17)i f ( S )3 πf ( S ) f ( S ) − πω P (cid:20) l ( a − l ) f ( S ) + ωf ( S ) (cid:16)p f ( S ) − lαS (cid:17) f ( S ) + ω (cid:16)p f ( S ) − lαS (cid:17) (cid:21) f ( S ) [3 l α ( a − l ) + ω ] (27)We have drawn the entropy S , pressure P , temperature T , volume V , Gibb’s free energy G andHelmholtz’s free energy F against PD black hole horizon radius r h in figures 1-6 respectively for theparameters a = 1 . , l = 1 . , ω = 0 . , α = 1 . , e = 1 , g = 1 , M = 10. From figure 1, we see that theentropy S first sharply decreases upto r h ≈ r h increases. We’ll choose the same values of the parameters in all the figures. From figure 2, we see thatthe pressure P increases as r h increases. From figure 3, we see that the temperature T decreases withequal slope as r h increases. From figure 4, we see that the volume V increases but maintains with nearlyequal slope as r h grows. From figure 5 and 6, we see that the Gibb’s free energy G and Helmholtz’s freeenergy F first sharply increase upto r h ≈ r h grows. C. P - V Criticality
Following [20] the idea of critical behaviour of charged AdS black holes, here we will study the criticalbehavior of PD black hole. The critical points for PD black hole can be found from the followingconditions: (cid:18) ∂P∂r h (cid:19) cr = 0 , ∂ P∂r h ! cr = 0 (28) r h P FIG. 2: Figure represents the plot of pressure P against PD black hole horizon radius r h . r h T FIG. 3: Figure represents the plot of temperature T against PD black hole horizon radius r h . r h V FIG. 4: Figure represents the plot of volume V against PD black hole horizon radius r h . From these conditions, we obtain the critical point r cr as r cr = M + 2 X − h ( a + l ) + M i + 2 − X (29)where X = M (5 a + 12 al + 4 M ) + q M (5 a + 12 al + 4 M ) −
16 (( a + l ) + M ) (30) r h G FIG. 5: Figure represents the plot of Gibb’s free energy G against PD black hole horizon radius r h . r h F FIG. 6: Figure represents the plot of Helmholtz’s free energy F against PD black hole horizon radius r h . At the critical point, the critical values S cr , P cr , T cr and V cr are obtained as S cr = πω ( r cr + ( a + l ) )( ω − lαr cr ) − a α r cr , (31) P cr = 3 (cid:2) M − r cr + π ( a + l ) T cr + πr cr T cr (cid:3) πr cr ( a + 6 l + 2 r cr ) , (32) T cr = 4 r cr − ( a + 6 l + 6 r cr ) Mπ [ a + 2 a l + 12 al ( l + r cr ) + a (7 l + 5 r cr ) + 2(3 l + r cr )] , (33) V cr = 4 πω (cid:2) ω ( r cr + 6 l r cr − l ) − αl r cr + a (2 lαr cr + 6 l ω + ωr cr ) (cid:3) a l α − l α + ω ) (34)If we choose the values of the parameters a = 1 . , l = 1 . , ω = 0 . , α = 1 . , e = 1 , g = 1 , M = 10, thenwe obtain the critical point r cr = 3 . S cr = 0 . P cr = 24 . T cr = 382 .
56 and V cr = 7 . r h C p FIG. 7: Figure represents the plot of specific heat capacity C P against PD black hole horizon radius r h . D. Stability
The specific heat capacity of the black hole thermodynamical system is the key quantity to determinethe stability of the black hole and can be written as [20] C = T (cid:18) ∂S∂T (cid:19) (35)If C ≥ C < V is constant (i.e., entropy S is constant), then the specific heat capacity C V = 0. But for constant pressure(i.e., P constant), we can obtain the specific heat capacity as in the form: C P = (cid:18) q f ( S ) f ( S ) (cid:18) − ( a + l ) α (cid:18) lαS − q f [( S ) (cid:19) + f ( S ) (cid:19) f ( S ) f ( S ) (cid:19) × (cid:2) ( a + l ) αf ( S ) f ( S ) f ( S ) f ′ ( S ) − a + l ) αf ( S ) (cid:0) f ( S ) f ( S ) f ′ ( S ) + f ( S ) (cid:0) f ( S ) f ′ ( S ) − f ( S ) f ′ ( S ) (cid:1)(cid:1) − q f ( S ) (cid:16) − l ( a + l ) Sα f ( S ) f ( S ) f ′ ( S ) + f ( S ) (cid:0) f ( S ) f ′ ( S ) − f ( S ) f ′ ( S ) (cid:1) + f ( S ) (cid:16) − l ( a + l ) Sα f ( S ) f ′ ( S ) + f ( S ) (cid:16) f ( S ) (cid:16) l ( a + l ) α + f ′ ( S ) (cid:17) + l ( a + l ) Sα f ′ ( S ) (cid:17)(cid:17)(cid:17)(cid:17) ] − (36)where dash represents derivative with respect to S . We have drawn C P against horizon radius r h infigure 7. We see that the specific heat capacity C P first sharply increases from some positive value upto r h ≈ r h grows. So our considered PD black hole is stable in nature sincefrom graph C P > E. Joule-Thomson Expansion
Joule-Thomson expansion [113, 114] is irreversible process, which explains that the temperaturechanges from high pressure region to low pressure region, while the enthalpy remains constant. Since in1the AdS space, the black hole mass may be interpreted as enthalpy [115], so the mass of the AdS blackhole remains constant during the Joule-Thomson expansion process. Joule-Thomson expansion for AdSblack hole has been studied in [116, 117]. Many authors have studied the Joule-Thomson expansion forseveral AdS black holes [57, 61, 80, 118–131]. Here, we’ll examine the Joule-Thomson expansion for AdSPD black hole. The Joule-Thomson coefficient µ is the slope of the isenthalpic curve, given by [116] µ = (cid:18) ∂T∂P (cid:19) M (37)which can be written as µ = 1 C P (cid:20) T (cid:18) ∂V∂T (cid:19) P − V (cid:21) (38)By evaluating the sign of µ , we can determine the cooling or heating nature of the black hole. Duringthe Joule-Thomson expansion process, the pressure always decreases, so the change of pressure is alwaysnegative while the change of temperature may be positive or negative. So the temperature determinesthe sign of µ . If the change of temperature is positive then µ is negative and so heating process occurs.But if the change of temperature is negative then µ is positive and therefore cooling process occurs. Nowfor AdS PD black hole, we obtain µ = 4 πω l ( a − l ) α + ω ) f ( S ) − ω (cid:18) − lαS + q f ( S ) (cid:19) + 6 l (cid:16) a − l (cid:17) f ( S )+ ω (cid:18) − lαS + q f ( S ) (cid:19) f ( S ) f ( S ) (cid:19) f ′ ( S ) + f ( S ) ω (cid:18) − lαS + q f ( S ) (cid:19) − lα + f ′ ( S )2 p f ( S ) ! + ωf ( S ) f ( S ) − lα + f ′ ( S )2 p f ( S ) ! + 18 l (cid:16) a − l (cid:17) f ( S ) f ′ ( S ) + ω (cid:18) − lαS + q f ( S ) (cid:19) f ( S ) f ′ ( S )+ ω (cid:18) − lαS + q f ( S ) (cid:19) f ( S ) f ′ ( S ) (cid:19) − ω (cid:18) − lαS + q f ( S ) (cid:19) + 6 l (cid:16) a − l (cid:17) f ( S )+ ω (cid:18) − lαS + q f ( S ) (cid:19) f ( S ) f ( S ) (cid:19) (cid:18) − (cid:18) ( a + l ) α (cid:18) − lαS + q f ( S ) (cid:19) + f ( S ) (cid:19) f ( S ) f ( S ) f ′ ( S )+ f ( S ) f ( S ) f ( S ) ( a + l ) α − lα + f ′ ( S )2 p f ( S ) ! + f ′ ( S ) ! − f ( S ) (cid:18) ( a + l ) α (cid:18) − lαS + q f ( S ) (cid:19) + f ( S )) f ( S ) f ′ ( S ) + f ( S ) (cid:18) ( a + l ) α (cid:18) − lαS + q f ( S ) (cid:19) + f ( S ) (cid:19) f ( S ) f ′ ( S ) (cid:19)(cid:19) × (cid:18)(cid:18) ( a + l ) α (cid:18) − lαS + q f ( S ) (cid:19) + f ( S ) (cid:19) f ( S ) f ( S ) (cid:19) − ! (39)We have drawn the Joule-Thomson coefficient µ against PD black hole horizon radius r h in figure 8.We observe that µ first keeps nearly parallel to r h axis upto r h ≈ r h increases. Since µ > r h , so the change of temperature isnegative and therefore cooling process occurs for PD black hole.2 r h Μ FIG. 8: Figure represents the plot of Joule-Thomson coefficient µ against PD black hole horizon radius r h . For µ = 0, we can determine the expansion process of inversion curve in a small infinite pressure. Atthe inversion temperature, put µ = 0 in (38) and inversion temperature is given by T i = V (cid:18) ∂T∂V (cid:19) P (40)So for PD black hole, we obtain the inversion temperature T i = (cid:20) lSαω f ( S ) − ω f / ( S ) − q f ( S ) (cid:16) l S α ω + ωf ( S ) f ( S ) (cid:17) + l (cid:16) l S α ω + (cid:16) − a l + 6 l (cid:17) f ( S ) + Sαωf ( S ) f ( S ) (cid:17)(cid:17) (cid:0) ( a + l ) αf ( S ) f ( S ) f ( S ) f ′ ( S ) − a + l ) αf ( S ) (cid:0) f ( S ) f ( S ) f ′ ( S ) + f ( S ) (cid:0) f ( S ) f ′ ( S ) − f ( S ) f ′ ( S ) (cid:1)(cid:1) − q f ( S ) (cid:16) − l ( a + l ) Sα f ( S ) f ( S ) f ′ ( S ) + f ( S ) (cid:0) f ( S ) f ′ ( S ) − f ( S ) f ′ ( S ) (cid:1) + f ( S ) (cid:16) − l ( a + l ) Sα f ( S ) f ′ ( S ) + f ( S ) (cid:16) f ( S ) (cid:16) l ( a + l ) α + f ′ ( S ) (cid:17) + l ( a + l ) Sα f ′ ( S ) (cid:17)(cid:17)(cid:17)i × h πωf ( S ) f ( S ) (cid:16) − f ( S ) (cid:16) l S α ω + f ( S ) f ( S ) (cid:17) f ′ ( S ) + 6 ω f ( S ) f ′ ( S )+6 lαω f / ( S ) (cid:0) f ( S ) − Sf ′ ( S ) (cid:1) + f ( S ) (cid:16) l S α ω f ′ ( S ) + f ( S ) (cid:16) − l Sα ω − ω f ′ ( S )+4 f ( S ) f ′ ( S ) (cid:1) − f ( S ) f ′ ( S ) (cid:17) + 2 lα q f ( S ) (cid:16) − l S α ω f ′ ( S ) + Sf ( S ) (cid:16) l Sα ω + 3 ω f ′ ( S ) − f ( S ) f ′ ( S ) (cid:1) + f ( S ) (cid:0) f ( S ) + Sf ′ ( S ) (cid:1)(cid:17)(cid:17)i − (41)We have drawn the inversion temperature T i against PD black hole horizon radius r h in figure 9. Weobserve that T i decreases slowly as r h increases but slopes of the curve al all points are almost samethroughout the evolution of r h . III. HEAT ENGINE
A heat engine is a thermodynamic system that converts thermal energy (or heat) and chemical energyto mechanical energy to do mechanical work. So physically, a heat engine carries heat from hot reservoir in3 r h T i FIG. 9: Figure represents the plot of inversion temperature T i against PD black hole horizon radius r h . which part of the heat converts into the physical works while the remaining part of the heat is transferredto cold reservoir. The working substance in a black hole heat engine is thought to be the black hole fluidor black hole molecules. The heat engine brings a working substance from a higher state temperatureto a lower state temperature. Then the working substance generates work while transferring heat to thecold reservoir until it reaches a low temperature state. During this process in the heat engine, someof the thermal energy is converted into work where the working substance has non-zero heat capacity.Therefore, the heat engine operates in a cyclic manner by adding energy (heat) in one part of the cycleand using that energy to do work in another part of the cycle. In the following subsections, we’ll studythe Carnot cycle and Rankine cycle of the heat engine for AdS PD black hole. A. Carnot Cycle
Carnot cycle is theoretical ideal thermodynamic cycle which was proposed by N. L. S. Carnot in1824. A Carnot heat engine is a classical thermodynamic engine that operates on the Carnot cycle whichcan be achieved during the conversion of heat into work. There are two heat reservoirs (hot and cold)forming part of the heat engine of the Carnot cycle. Here we assume, T H and T C are temperatures ofhot and cold reservoirs respectively. These are included upper and lower isothermal processes with twoadiabatic processes. The P - V diagram has been shown in Ref [40] for Carnot heat engine which forms aclosed path. From the diagram, it was shown that along the upper isotherm process, the heat flows aregenerated from 1 to 2 and which is given by [40] Q H = T H △ S → = T H ( S − S ) (42)and the exhausted heat produced from 3 to 4 along lower isothermal process is given by [40] Q C = T C △ S → = T C ( S − S ) (43)4Here PD black hole entropies S i ’s are related to volumes V i ’s satisfying as V i = 4 πω (cid:20) l ( a − l ) f ( S i ) + ωf ( S i ) f ( S i ) (cid:16)p f ( S i ) − lαS i (cid:17) + ω (cid:16)p f ( S i ) − lαS i (cid:17) (cid:21) f ( S i ) [3 l α ( a − l ) + ω ] , i = 1 , , , , (44)where f ( S i ), f ( S i ) and f ( S i ) can be calculated from the following relations f ( S i ) = α l S i + (cid:16) S i − ( a + l ) π (cid:17) f ( S i ) , i = 1 , , , , (45) f ( S i ) = α ( a − l ) S i + πω , i = 1 , , , f ( S i ) = a (cid:20) f ( S i ) + 2 lα (cid:18)q f ( S i ) − lαS i (cid:19)(cid:21) + 2 l (cid:20) f ( S i ) − lα (cid:18)q f ( S i ) − lαS i (cid:19)(cid:21) , i = 1 , , , . (47)Also the heat engine flow has been shown in Ref [40]. The total mechanical work done by the heat engineis the difference of the amount of heat energies between upper and lower isotherm processes as W = Q H − Q C (48)A central quantity, the efficiency of a Carnot heat engine is defined by the ratio of total mechanical workand the amount of heat energy along the upper isotherm process and is given by η Car = WQ H = 1 − Q C Q H (49)Since for Carnot cycle, V = V and V = V , so we have the maximum efficiency for Carnot cycle and isgiven by ( η Car ) max = 1 − T C ( S − S ) T H ( S − S ) (cid:12)(cid:12)(cid:12)(cid:12) V = V ,V = V (50)which simplifies to the following form ( η Car ) max = 1 − T C T H (51)Since T H > T C , so we have 0 < ( η Car ) max <
1. It should be noted that the ( η Car ) max is the maximumefficiency of all possible cycles between higher temperature T H and lower temperature T C . Since weknow that the Stirling cycle consists of two isothermal processes and two isochores processes, so thismaximally efficient Carnot engine is also Stirling engine. From the figure 3, if we choose T C = 250 and T H = 400 then the maximum efficiency for Carnot cycle is obtained as ( η Car ) max = 0 . W = △ P → △ V → = ( P − P )( V − V ) (52)where V and V are described in (44). The upper isobar gives the net inflow of heat which is given by Q H = Z T T C P ( P , T ) dT = Z S S h f ( S ) + α ( a + l ) (cid:16)p f ( S ) − lαS (cid:17)i f ( S )3 πf ( S ) f ( S ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P = P dS (53)The lower isobar gives the exhaust of heat which is given by Q C = Z T T C P ( P , T ) dT = Z S S h f ( S ) + α ( a + l ) (cid:16)p f ( S ) − lαS (cid:17)i f ( S )3 πf ( S ) f ( S ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P = P dS (54)The thermal efficiency for the new heat engine is described by η New = WQ H = ( P − P )( V − V ) Q H = ( P − P )( V − V ) Z S S h f ( S ) + α ( a + l ) (cid:16)p f ( S ) − lαS (cid:17)i f ( S )3 πf ( S ) f ( S ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P = P dS − (55)Since the above integration is very complicated, so we may find the value of the efficiency of the new enginefor particular values of the parameters, say a = 1 . , l = 1 . , ω = 0 . , α = 1 . , e = 1 , g = 1 , M = 10.If we choose S = 0 . , S = 0 . , P = 15 , P = 25 , V = 5 , V = 15 from the figures 1, 2 and 4, thenthe value of the thermal efficiency of the new engine is obtained as η New = 0 . B. Rankine Cycle
A Rankine cycle [71] is an idealized thermodynamic cycle of a heat engine, which converts heat intomechanical work for undergoing phase change. The Rankine cycle for black hole heat engine is shownin Ref [72]. From the Ref [72], we see that the working substance starts from A to B with increasingtemperature and adiabatic pressure. Next the working substance follows from B to E and within thesestates a phase transition occurs from C to D with constant temperature. Then the working substancereduces the temperature from E to F and returns to A by reducing its volume. For constant pressure P ,6we have dP = 0. From the first law of the black hole thermodynamics dH P = T dS for constant pressure,we have the enthalpy function H P ( S ) = R T dS . Now according to the formalism of Wei et al [71, 72],the efficiency for Rankine cycle for black hole heat engine can be expressed as η Ran = 1 − T A ( S F − S A ) H P B ( S F ) − H P B ( S A ) = 1 − T ( S − S ) H P ( S ) − H P ( S ) (56)where the subscripts A, B, F are denoted by 1 , , T and H P for PD black holeare given by T = h f ( S ) + α ( a + l ) (cid:16)p f ( S ) − lαS (cid:17)i f ( S )3 πf ( S ) f ( S ) (57)and H P ( S i ) = Z S i S h f ( S ) + α ( a + l ) (cid:16)p f ( S ) − lαS (cid:17)i f ( S )3 πf ( S ) f ( S ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P = P dS , i = 1 , η Ran = 1 − ( S − S ) h f ( S ) + α ( a + l ) (cid:16)p f ( S ) − lαS (cid:17)i f ( S )3 πf ( S ) f ( S ) × Z S S h f ( S ) + α ( a + l ) (cid:16)p f ( S ) − lαS (cid:17)i f ( S )3 πf ( S ) f ( S ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P = P dS − (59)Since the above integration is very complicated, so we may find the value of the efficiency of the Rankinecycle for particular values of the parameters, say a = 1 . , l = 1 . , ω = 0 . , α = 1 . , e = 1 , g = 1 , M =10. If we choose S = 0 . , S = 0 . , P = 15 from the figures 1 and 2, then the value of the efficiencyof the rankine cycle is obtained as η Ran = 0 . IV. DISCUSSIONS AND CONCLUDING REMARKS
We have assumed the general class of accelerating, rotating and charged Plebanski-Demianski (PD)black holes in presence of cosmological constant, which includes the Kerr-Newman rotating black holeand the Taub-NUT spacetime. We have assumed that the thermodynamical pressure ( P ) may bedescribed as the negative cosmological constant (Λ <
0) by the relation Λ = − πP and so the black holemay be formed in anti-de Sitter (AdS) PD black hole. Using the horizon radius ( r h ), the thermodynamicquantities like surface area ( κ ), entropy ( S ), volume ( V ), temperature ( T ), Gibb’s free energy ( G ) andHelmholtz’s free energy ( F ) of the AdS PD black hole have been obtained due to the thermodynamicsystem. We have drawn the entropy S , pressure P , temperature T , volume V , Gibb’s free energy G and7Helmholtz’s free energy F against PD black hole horizon radius r h in figures 1-6 respectively for theparameters a = 1 . , l = 1 . , ω = 0 . , α = 1 . , e = 1 , g = 1 , M = 10. From figure 1, we have observedthat the entropy S first sharply decreases upto r h ≈ r h increases. From figure 2, we have seen that the pressure P increases as r h increases.From figure 3, we have seen that the temperature T decreases with equal slope as r h increases. Fromfigure 4, we have observed that the volume V increases but maintains with nearly equal slope as r h grows. From figure 5 and 6, we have seen that the Gibb’s free energy G and Helmholtz’s free energy F first sharply increase upto r h ≈ r h grows.Next we found the critical point and corresponding critical entropy, critical pressure, criticaltemperature and critical volume for AdS PD black hole. In particular, for the chosen the values of theparameters a = 1 . , l = 1 . , ω = 0 . , α = 1 . , e = 1 , g = 1 , M = 10, we have obtained the criticalpoint r cr = 3 .
915 and the corresponding critical values of entropy, pressure, temperature and volume are S cr = 0 . P cr = 24 . T cr = 382 .
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