Electric equivalent circuit (EEC) models have been widely used to interpret the innerdynamics of all type of batteries. Added to this, they also have been used to estimate state of charge(SOC) and state of health (SOH) values in combination with different methods. Four EEC models areconsidered for enhanced flooded lead acid batteries (EFB) which are widely used in micro hybridvehicles. In this study, impedance and phase prediction capabilities of models throughout a frequencyspectrum from 1 mHz to 10 kHz are compared with those of experimental results to investigate theirconsistency with the data. The battery is charged, discharged, and aged according to appropriatestandards which imitates a lifetime of a micro hybrid vehicle battery under high current partialcycling. Impedance tests are repeated between different charge and health states until the end of thebattery’s lifetime. It is seen that adding transmission line elements to mimic the high porous electrodeelectrolyte interface to a double parallel constant phase element resistance model (ZARC) can increasethe model data representing capability by 100%. The mean average percentage error (MAPE) ofthe conventional model with respect to data is 3.2% while the same value of the transmission lineadded model found as 1.6%. The results can be helpful to represent an EFB in complex simulationenvironments, which are used in automobile industry.
A thermoelectric voltage is induced in a junction, constituted of two dissimilar materials under a temperature gradient. Similarly, a thermosize voltage is expected to be induced in a junction made by the same material but having differentsizes, so-called thermosize junction. This is a consequence of dissimilarity in Seebeck coefficients due to differencesin classical and/or quantum size effects in the same materials with different sizes. The studies on thermosize effectsin literature are mainly based on semi-classical models under relaxation time approximation or even simpler localequilibrium ones where only very general ideas and results have been discussed without considering quantum transport approaches and specific materials. To make more realistic predictions for a possible experimental verification, here,we consider ballistic thermosize junctions made by narrow and wide (n-w) pristine graphene nanoribbons with perfectarmchair edges and calculate the electronic contribution to the thermosize voltage, at room temperature, by using the Landauer formalism. The results show that the maximum thermosize voltage can be achieved for semiconducting nanoribbons and it is about an order of magnitude larger than that of metallic nanoribbons. In the semiconducting case, the thermosize voltage forms a characteristic plateau for a finite range of gating conditions. We demonstrate, throughnumerical calculations, that the induced thermosize voltage per temperature difference can be in the scale of mV/K,which is high enough for experimental measurements. Owing to their high and persistent thermosize voltage values,graphene nanoribbons are expected to be good candidate for device applications of thermosize effects.
The current density of states (DOS) calculations do not take into account the essential discretenessof the state space, since they rely on the unbounded continuum approximation. Recently, discrete DOS based on the quantum-mechanically allowable minimum energy interval has been introducedfor quadratic dispersion relation. In this work, we consider systems exhibiting photonic (photon-like) dispersion relation and calculate the related density and number of states (NOS). Also, a Weyl's conjecture-based DOS function is calculated for photons and acoustic phonons at low frequency limit,by considering the bounded continuum approach. We show that discrete DOS function reducesto expressions of bounded and unbounded continua in the appropriate limits. The uctuationsin discrete DOS completely disappear under accumulation operators. It's interesting that relativeerrors of NOS and DOS functions with respect to discrete ones have exactly the same character withthe ones of quadratic dispersion relation. Furthermore, the application of discrete and Weyl DOS for the calculation of internal energy of a photon gas is presented and importance of discrete DOSis discussed. It's shown that discrete DOS function given in this work needs to be used wheneverthe low energy levels of a physical system are heavily occupied.
An ideal Maxwell–Boltzmann gas confined in various rectangular nanodomains is considered underquantum size effects. Thermodynamic quantities are calculated from their relations with the partitionfunction, which consists of triple infinite summations over momentum states in each direction. Toobtain analytical expressions, summations are converted to integrals for macrosystems by acontinuum approximation, which fails at the nanoscale. To avoid both the numerical calculation ofsummations and the failure of their integral approximations at the nanoscale, a method which gives ananalytical expression for a single particle partition function (SPPF) is proposed. It is shown that adimensional transition in momentum space occurs at a certain magnitude of confinement. Therefore,to represent the SPPF by lower-dimensional analytical expressions becomes possible, rather thannumerical calculation of summations. Considering rectangular domains with different aspect ratios, acomparison of the results of derived expressions with those of summation forms of the SPPF is made.It is shown that analytical expressions for the SPPF give very precise results with maximum relativeerrors of around 1%, 2% and 3% at exactly the transition point for single, double and triple transitions,respectively. Based on dimensional transitions, expressions for free energy, entropy, internal energy,chemical potential, heat capacity and pressure are given analytically valid for any scale.
By considering the quantum-mechanically minimum allowable energy interval, we exactly count number of states (NOS) and introduce discrete density of states (DOS) concept for a particle in a box for various dimensions. Expressions for bounded and unbounded continua are analytically recovered from discrete ones. Even though substantial fluctuations prevail in discrete DOS, they’re almost completely flattenedout after summation or integration operation. It’s seen that relative errors of analytical expressions of bounded/unbounded continua rapidly decrease for high NOS values (weak confinement or high energy conditions), while the proposed analytical expressions based on Weyl’s conjecture always preserve their lower error characteristic.
Intrinsic discrete nature in thermodynamic properties of Fermi gas appears in strongly confined and degenerate conditions. For a rectangularconfinement domain, thermodynamic properties of an ideal Fermi gas are expressed in their exact summation forms. For 1D, 2D and 3D nanodomains; variations of both number of particles and internal energy per particle with chemical potential are examined. It is shown that theirrelation with chemical potential exhibits a discrete nature which allows them to take only some definite values. Furthermore, quasi-irregularoscillatory-like sharp peaks are observed in heat capacity. New nano devices can be developed based on these behaviors.
We propose an analytical model for the prediction and accurate calculation of size and density dependent quantum oscillations in thermodynamic and transport properties of confined and degenerate Fermi gases. Our model considers only half-vicinity states of Fermi level. We show that the half-vicinity model quite accurately estimates quantum oscillations depending on confinement and degeneracy. Periods of quantum oscillations are even analytically expressed for one-dimensional case. Furthermore, similarities between functional behaviors of total occupancy variance and conventional density of states functions at Fermi level are discussed.
For part I see DOI: 10 .1016 /j.physleta.2018.02.006. Size and density dependent quantum oscillations appear in Fermi gases under strong confinement and degeneracy conditions. We provide a universal recipe that explicitly separates oscillatory regime from non-oscillatory (stationary) one. A phase diagram representing stationary and oscillatory regimes on degeneracy-confinement space is proposed. Analytical expressions of phase transition interfaces are derived. The critical point, which separates entirely stationary and oscillatory regions, is determined and its dependencies on aspect ratios are examined for anisometric domains. Accuracy of the half-vicinity model and the phase diagram are verified through the quantum oscillations in electronic heat capacity and its ratio to entropy.
Thermodynamic properties of confined systems depend on sizes of the confinement domain due to quantum nature of particles. Here we show that shape also enters as a control parameter on thermodynamic state functions. By considering specially designed confinement domains, we demonstrate how shape effects alone modify Helmholtz free energy, entropy and internal energy of a confined system. We propose an overlapped quantum boundary layer method to analytically predict quantum shape effects without even solving Schrödinger equation or invoking any other mathematical tools. Thereby we reduce a thermodynamic problem into a simple geometric one and reveal the profound link between geometry and thermodynamics. We report also a torque due to quantum shape effects. Furthermore, we introduce isoformal, shape preserving, process which opens the possibility of a new generation of thermodynamic cycles operating at nanoscale with unique features.
In this study, a new analysis method is proposed for estimating thermal conductivity of a ground by using the constant-temperature thermal response test data. The new method is based on an analytical solution of heat transfer rate per unit borehole length by using the Laplace transformation for constant-temperature thermal response tests. Its advantage is that it allows one to estimate thermal conductivity directly from the slope of the logarithmic time dependency of inverse unit-heat-transfer rate value without making an estimation of volumetric heat capacity. The method has been verified by using a numerical model and applied to different experimental data based on different test temperatures and compared with the classical thermal response test method. The results show that the proposed method reliably and effectively estimates thermal conductivity of ground.
In ground source heat pump (GSHP) applications, borehole drilling cost constitutes an important part ofthe investment cost and it can be reduced by improving borehole performance. In vertical GSHP applications,usually double-U tube configurations are used to improve the heat transfer rate per unit length of aborehole, (unit HTR value). To determine the optimal number of U-tubes which maximizes the commercialand engineering benefits of multi U-tube applications, cost and performance analyses of multi U-tubeboreholes are crucial. In this study, a triple U-tube is used in a borehole of 50 m depth. Time variation ofunit HTR value of the borehole is experimentally measured when single, double and triple U-tubes are inoperation separately. Furthermore a computational model is calibrated by fitting the computationalresults to the experimental ones, and effects of using four and five U-tubes in a borehole are computationallyinvestigated. The relations between number of U tubes and time variation of unit HTR value ofa borehole as well as investment cost are analyzed. Long term borehole performance predictions aremade and compared for multi U-tube applications. Both experimental and computational results showedthat performance improvements are remarkable for 2U-tube and 3U-tube configurations while it isnearly insignificant for 4U and 5U ones. If the investment cost per thermal power is considered, 2U-tubeconfiguration is the optimal one if the prices of polyethylene pipes are relatively high, like in Turkey.When the cost of pipes decreases, then 3U-tube or even 4U–tube configuration can be the cheapestsolution.
Classical thermosize effects arise due to different transport characteristics of macro and nano partsof a macro/nano combined system. In this paper, classical thermosize effects are considered forFermi and Bose gases. The Knudsen law is generalized for ideal gases including the quantum onesand some analytical expressions for thermosize effects are obtained for degenerate Fermi and Bosegases. The influence of the quantum degeneracy on the classical thermosize effects are analyzedand the comparison with the results of the Maxwellian case is given. It is seen that degenerationimproves the classical thermosize effects for a Bose gas while it deteriorates the effects for a Fermi gas.
Classical thermosize effects arise due to different transport regimes of gases confined in macroand nano structures. Pressure is the constant quantity of a gas confined in a closed macro domainunder a temperature gradient while the pressure over square root of temperature is the constantquantity (Knudsen law) in case of nano domain. These two different behaviors give rise classicalthermosize effects which are similar to thermoelectric effects. Thermodynamic power and refrigerationcycles based on the thermosize effects are proposed. Cycles consist of two isothermal, oneisobaric and one Knudsen processes. These thermosize cycles are thermodynamically analyzedand expressions for work and heat exchange, efficiency and COP values are derived. The resultscan be useful to design some new nano devices.
Liquid hydrogen (LH2) storage has the advantage of high volumetric energy density, whileboil-off losses constitute a major disadvantage. To minimize the losses, complicatedinsulation techniques are necessary. In general, Multi Layer Insulation (MLI) and a Vapor-Cooled Shield (VCS) are used together in LH2 tanks. In the design of an LH2 tank with VCS,the main goal is to find the optimum location for the VCS in order to minimize heatleakage. In this study, a 2D thermal model is developed by considering the temperaturedependencies of the thermal conductivity and heat capacity of hydrogen gas. The developedmodel is used to analyze the effects of model considerations on heat leakagepredictions. Furthermore, heat leakage in insulation of LH2 tanks with single and doubleVCS is analyzed for an automobile application, and the optimum locations of the VCS forminimization of heat leakage are determined for both cases.
The installation cost and performance of ground source heat pump (GSHP) systems can greatly be affectedby application parameters of ground heat exchangers (GHEs). These parameters affecting the helical GHE’sperformance are their spacing, major diameter and pitch length. In large scale GSHP applications, morethan one GHE is needed. Therefore determining the distance between GHEs (spacing) becomes as animportant issue. In this work, the effect of distance between vertical helical GHEs on the heat transferrate (HTR) is studied. Performance of helical GHE is determined for different spacing and the properdistance is examined. Furthermore, the influences of the pitch length (Lp) as well as major diameter (D)of GHE on HTR value are numerically studied in COMSOL environment. The available experimental dataare used to validate the numerical results. Computational results show that they are in good agreementwith the experimental one. The proper distance between GHEs is suggested as 7 m and more. Furthermorethe results of the simulations prove that 100% changes in Lpand D can affect the performance of GHE onlyin order of 10% although the excavation cost increase more than 10%. The results provide useful guidancefor optimum design of vertical helical GHE for GSHP systems.
In nano domains, thermodynamic properties of gases considerably differ from those in macrodomains. One of the reasons for this difference is the quantum size effects, which becomeimportant when the thermal de Broglie wavelength of particles is not negligible in comparisonwith domain size. In this study, it is shown that quantum forces may appear in gases confinedin nano structures due to the quantum boundary layer caused by quantum size effects. In thecase of experimental verification of these quantum forces, a macroscopic manifestation of theeffect of the quantum boundary layer on the thermodynamic behavior of gases can beconfirmed.
In nanoscale, gas density is not really homogenous even in thermodynamic equilibrium especially ina region near to the domain boundaries due to the wave character of gas particles. This inhomogeneous region is called quantum boundary layer (QBL) since its thickness goes to zero when the Planck’s constant goes to zero. QBL can be neglected and density is assumed to be homogenous as long as thermalde Broglie wavelength (lT) of particles is negligible in comparison with the domain sizes. In nanoscale, however, this condition breaks down and QBL changes the thermodynamic behaviour of gases considerably. In literature, density distribution of a Maxwellian gas has been examined for only a rectangular domain to obtain the analytical results. In this study, density distribution is examined for some regularand irregular domain geometries for which the analytical solution is not possible. It is shown that QBL covers the whole surface of the domain and both thickness and density profile of QBL are independent ofthe domain geometry. It is concluded that QBL has a universal thickness and density profile for a Maxwellian gas. Furthermore, an effective quantum potential is introduced to explain the inhomogeneous density distribution in thermodynamic equilibrium.
For an ideal gas confined in a rectangular domain, it has been shown that the density is not homogenous even in thermodynamic equilibrium and it goes to zero within a layernear to the boundaries due to the wave character of particles. This layer has been called the quantum boundary layer (QBL). In literature, an analytical expression for the thickness of QBL has been given for only a rectangular domain since both energy eigenvalues andeigenfunctions of the Schrödinger equation can analytically be obtained for only a rectangulardomain. In this study, ideal Maxwellian gases confined in spherical and cylindrical domainsare considered to investigate whether the thickness of QBL is independent of the domain shape. Although the energy eigenvalues are the roots of Bessel functions and there is noanalytical expression giving the roots, the thickness of QBL is expressed analytically by considering the density distributions and using some simplifications based on the numerical calculations. It is found that QBL has the same thickness for the domains of different shapes.Therefore, QBL seems to have a universal thickness independent of the domain shape for anideal Maxwellian gas.
Friedel oscillations appear in density of Fermi gases due to Pauli exclusion principle and translational symmetry breaking nearby a defect or impurity. In confined Fermi gases, this symmetry breaking occurs also near to boundaries. Here, density oscillations of a degenerate and confined Fermi gas are considered and characterized. True nature of density oscillations are represented by analytical formulas for degenerate conditions. Analytical characterization is first done for completely degenerate case, then temperature effects are also incorporated with a finer approximation. Envelope functions defining the upper and lower bounds of these oscillations are determined. It is shown that the errors of obtained expressions are negligible as long as the system is degenerate. Numbers, amplitudes, averages and spatial coordinates of oscillations are also given by analytical expressions. The results may be helpful to efficiently predict and easily calculate the oscillations in density and density-dependent properties of confined electrons at nanoscale.
The thermodynamic behavior of gases confined in nano structures is considerably different than those in macro onesdue to the effects of both particle-wall interactions and the wave character of particles. The homogeneous density distribution of a gas at thermodynamic equilibrium is disturbed by these effects. Because of particle-wallinteractions, the local density of a gas changes drastically near the domain boundaries. Also, the wave character ofthe particles causes an inhomogeneous density distribution, especially near the boundaries. Consequently, the apparent density (number of particles over the domain volume) is different than the real one. All the density dependent thermodynamic properties are affected by the inhomogeneity in the density distribution. Therefore, it is important to consider these effects on local density to analyze the thermodynamic behaviors of gases confined innano structures. The detailed analysis of these effects on local density also gives a base of knowledge for the experimental verification of quantum size effects on local density due to the wave character of particles. In thisstudy, the density distributions of classical (Maxwellian) and quantum (both Fermi and Bose) gases are calculated and investigated by considering both particle-wall interactions and quantum size effects. The results can be used for experimental verification of quantum size effects on gas density as well as the modeling of nano heat engines.
In large-scale ground-source heat pump applications, a large number of borehole heat exchangers are used and performance losses become an important issue due to thermal interactions. Dependency of total performance losses on borehole spacing can analytically be expressed by using thermal interaction coefficient. For a given application field, interaction coefficient depends on number of boreholes (N), aspect ratio of borehole's arrangement geometry and operation time. In this study, functional dependencies of interaction coefficient on N and aspect ratio are investigated by considering different rectangular borehole arrangements. Dependencies of both thermal interaction coefficient and total heat transfer rate on aspect ratio are computationally examined. Also, the effects of number of boreholes and operation time on interaction coefficient are studied. The results showed that the values of both interaction coefficient and performance losses decrease with the decrease of aspect ratio of a borehole field. Aspect ratio dependency of total unit heat transfer rate becomes more evident in case of shorter borehole spacing. Furthermore, a strong dependency of interaction coefficient on N is observed when N is much smaller than a critical value, N_{c}, although an asymptotic behavior appears and dependency on N becomes negligible for N > N_{c}. Some empiric expressions are proposed for aspect ratio and N dependency of interaction coefficient as well as N_{c}. The results and the proposed expressions can be used to make an energy efficient and optimal design of a BHE field by maximizing the total performance while minimizing the field allocation and the thermal losses.
Effect of borehole spacing on Heat Transfer Rate per unit borehole length (unit HTR) is computationallyinvestigated for multiple Borehole Heat Exchangers (BHE). Experimentally verified computational modelis used to analyze different configurations consisting of various number of BHE. To determine the performanceloss due to mutual thermal interactions of BHE, the averaged unit HTR value of the most criticalborehole in each configuration is compared with that of a single borehole alone for various borehole spacingand operation durations of 1800 and 2400 h. Effect of thermal conductivity of ground on the relationbetween performance loss and borehole spacing is also examined. Furthermore, variations of total HTRvalues of a borehole field with borehole spacing are compared with each other for different configurations.It is seen that 4.5 m spacing is enough to keep the total performance losses less than 10% evenfor 2400 h non-stop operation. An analytical formulation is proposed for total HTR value which dependson thermal interaction coefficient (d) as well as spacing and number of BHE (N). Dependence of d on N isalso examined for different operation durations. Variations of dimensionless HTR value with boreholespacing are analyzed for different values of N. The results can be used during the engineering designstages of a BHE field to predict the variation of total HTR value of the field with borehole spacing andnumber of boreholes.
There are numerous experimental and numerical studies about quantum size effects on Seebeckcoefficient. In contrast, in this study, we obtain analytical expressions for Seebeck coefficientunder quantum size effects. Seebeck coefficient of a Fermi gas confined in a rectangular domainis considered. Analytical expressions, which represent the size dependency of Seebeckcoefficient explicitly, are derived in terms of confinement parameters. A fundamental form ofSeebeck coefficient based on infinite summations is used under relaxation time approximation.To obtain analytical results, summations are calculated using the first two terms of Poissonsummation formula. It is shown that they are in good agreement with the exact results based ondirect calculation of summations as long as confinement parameters are less than unity. Theanalytical results are also in good agreement with experimental and numerical ones in literature.Maximum relative errors of analytical expressions are less than 3% and 4% for 2D and 1D cases,respectively. Dimensional transitions of Seebeck coefficient are also examined. Furthermore, adetailed physical explanation for the oscillations in Seebeck coefficient is proposed byconsidering the relative standard deviation of total variance of particle number in Fermi shell.
For the low temperature applications of Bi2Te3 based thermoelectric power generators (TEGs), determinationof low temperature material behaviors is important. In the temperature range of 100–375 K, temperaturedependency of Seebeck coefficient and electrical conductivity of a Bi2Te3 based TEG areexperimentally determined. Furthermore, for a constant temperature difference, the variation of maximumpower output with the mean temperature is analytically examined based on the experimentallymeasured material properties. It is observed that 250 K seems a critical mean operating temperaturefor the considered Bi2Te3 module. Correlations for temperature dependency of material quantities arealso given for the temperature region of 100–375 K. The results can be used for the low temperatureapplications of Bi2Te3 based TEGs.
A thermosize junction consists of two different sized structures made using the same material. Classical and quantum thermosize effects (CTSEs and QTSEs), which are opposite to each other, induce a thermosize potential in a thermosize junction. A semi-analytical method is proposed to calculate thermosize potentials in wide ranges of degeneracy and confinement by considering both CTSEs and QTSEs in thermosize junctions made using semiconductors. Dependencies of thermosize potential on temperature, size, and degeneracy are examined. It is shown that a potential difference in millivolt scale can be induced as a combined effect of CTS and QTS. The highest potential is obtained in nondegenerate limit where the full analytical solution is obtained. The model can be used to design semiconductor thermosize devices for a possible experimental verification of CTSEs and QTSEs, which may lead to new nano energy conversion devices.
Thermal and potential conductivities of ideal Maxwellian, Fermi and Bose gases are derived by considering the small corrections due to the wave character of gas particles. Potential conductivity is regarded as conductivity due to any potential gradient like electrical, gravitational or chemical ones. A long rectangular channel is considered as a transport domain. The size of the domain in the transport direction is much longer than the mean free path of particles l while the sizes in transverse directions are shorter than l. On the other hand, all sizes of the domain are assumed to be larger than the thermal de Broglie wavelength ofparticles. Therefore, quantum size effects (QSE) are weak enough to be considered as smallcorrections on conventional terms. Corrections on thermal and potential conductivities areexamined. It is seen that the size and shape of the transport domain become additional control parameters on both conductivities. Since the size dependencies of thermal and electrical conductivities are different, the Lorenz number becomes size and shape dependent and deviations from the Wiedemann–Franz law may be expected in nanoscale due to QSE. Variations of the corrections with chemical potential are analysed.
The thermodynamic and transport properties of gases confined at the nano scale are considerably different than those at the macro scale. At the nano scale, quantum size effects (QSE) become important and changes the behavior of gases. In this study, the diffusion coefficients of monatomic Fermi and Bose gases are analytically derived by considering QSE. The influences of QSE and quantum degeneracy on the diffusion coefficients are examined separately to analyze these effects individually. The variations of the ratio of diffusion coefficients of He3 and He4 gases with the concentration of He3 are analyzed for both low and high density conditions.
The effect of quantum degeneracy on the work output from a Stirling cycle working at quantum degeneracy conditions (QDCs) is analyzed. Expressions for net work outputs of Stirling power cycles working with monatomic ideal Bose and Fermi gases are derived by using the quantum ideal gas equation of state. Ratios of net work outputs of Stirling cycles working with Bose and Fermi gases to the net work output of a classical Stirling cycle (RBW and RWF, respectively) are obtained. Variations of RBW and RFW with TH are examined for a given temperature ratio (τ=TL/TH) and a specific volume ratio (rν=νH/νL). At QDC, it is seen that RBW has a maximum value, which is greater than unity. On the other hand, there is no maximum or minimum point for RFW and RFW⩽1 for any values of TH. Consequently, the use of Bose gas as a working fluid in a Stirling cycle provides an advantage since it causes the net work output per cycle to increase by consuming more heat energy. This fact is seen to be in the opposite direction for a Stirling cycle working with Fermi gas.
Reactivity worth of the fuel elements of I.T.U. TRIGA MARK-IIreactor are calculated by using both one-group perturbation theory and a one dimensional, two-group diffusion computer code, TRIGAP. Results of both methods are compared with those measured experimentally. Although the perturbation theory is obtained from the diffusion theory by making some assumptions, it is seen that one-group perturbation theory gives the results with better accuracy in comparison to the results of TRIGAP. This situation shows that TRIGAP cannot reflect the effect of local material changes on the multiplication factor well enough. In order to calculate fuel elements reactivity worth of TRIGA type reactors, one-group perturbation theory can be preferred to TRIGAP due to its simplicity and accuracy, when the flux and group constants are known.
The contribution from thermally generated electron–positron pairs to the radiativelosses from a black body is considered near the temperature corresponding to the electron’srest mass energy Tc= mec2/k. The correction factors are defined as the ratio of correctedexpressions (which also include the contribution from thermal pairs) to the classical expressions(which include merely photons). The correction factors for energy, free energy, and entropyfluxes have different values about 0.5Tc, while they have the same value at T << Tc and T >> Tc. The Stephan–Boltzmann’s constant becomes temperature dependent due to thecontribution of thermal pairs. According to the classical expressions of radiative losses, the ratioof energy flux to the absolute value of free energy flux is a constant and it is equal to 3. On thecontrary, it is shown that it is a function of temperature about Tc and it has a maximum about0.38Tc. The correction factor for mean energy per emitted thermal particle has also a maximumabout T = 0.42Tc.
An upper limit for surface temperature of a static and spherical body in steady state is determined by considering the gravitational temperature drop (GTD). For this aim, abody consisting of black body radiation (BBR) only is considered. Thus, it is assumedthat body has minimum mass and minimum GTD. By solving the Oppenheimer-Volkoff equation, density distribution of self-gravitating thermal photon sphere with infinite radius is obtained. Surface temperature is defined as the temperature at distance of R from centre of this photon sphere. By means of the density-temperature relation of BBR, surface temperature is expressed as a function of central temperature and radius R. Variation of surface temperature with central temperature is examined. It is shown that surface temperature has a maximum for a finite value of central temperature. For this maximum, an analytical expression depending on only the radius is obtained. Since a real static and stable body with finite radius has much more mass and much more GTD than their values considered here, obtained maximum constitutes an upper limit for surface temperature of a real body. This limitation on surface temperature also limits the radiative energy lose from a body. It is shown that this limit for radiative energy lose is a constant independently from body radius and central temperature. Variation of the minimum mass with central temperature is also examined. It is seen that the surface temperature and minimum mass approach some limit values, which are less than their maximums, by making damping oscillations when central temperature goes to infinity.
The Casimir-like size effect rises in ideal gases confined in a finite domain due to the wave character of atoms. By considering this effect, thermodynamic properties of an ideal gas confined in spherical and cylindrical geometries are derived and compared with those in rectangular geometry. It is seen that anideal gas exhibits an unavoidable quantum surface free energy and surface over volume ratio becomes a control variable on thermodynamic state functions in microscale. Thermodynamics turns into non-extensive thermodynamics and geometry difference becomes a driving force since the surface over volumeratio depends on the geometry.
Thermosize effects have been proposed in literature by considering the quantum size effects (QSE)induced by the wave character of particles. These effects appear only if nano and macro domains are connectedto each other when they are under a temperature gradient. QSE are noticeable in nano domain while they arealmost negligible in macro domain. This difference causes thermosize effects, which may be called quantumthermosize effects (QTSE) because of their pure quantum origin. On the other hand, also classical thermosizeeffects (CTSE) appear as a result of different transport regimes in nano and macro domains, and they can benoticeable even if QTSE are negligible. As long as the mean free path (l) is much greater than the mean deBroglie wave length of particles (λ), which is almost the case in practice λ/l < 1, the principal effects areCTSE. QSE cause only small corrections on CTSE when the scale is down to nanoscale. On the other hand, in literature, QTSE and CTSE have been examined individually although it is not possible to observe QTSE alone in practice except for the extreme case of λ/l >> 1. Furthermore, the constant pressure assumption andthe Knudsen law have been used during the derivations of QTSE and CTSE, respectively, although the properassumption at nanoscale is the modified Knudsen law, which considers QSE. In this study, QSE on CTSE areconsidered and the modified Knudsen law is derived and used to obtain the more realistic results for thermosizecoefficients. The results can be used for a possible experimental verification of thermosize effects as well asto design some new devices based on these effects.
The wave character of atoms can produce Casimir-like size effects in gases confined in a narrow box. Thus the pressure tensoris not isotropic anymore and size difference becomes a driving force for isothermal diffusion by a permeable wall. Such sizeeffects give rise to “thermosize effects” similar to thermoelectric effects.
Density distribution of an ideal Maxwellian gas confined in a finite domain is not uniform even in thermodynamic equilibrium. Near to theboundaries, there is a layer in which the density goes to zero. Existence of this boundary layer explains the shape and size dependence of thethermodynamic quantities in nano scale.
The Ericsson power cycles working with ideal Bose and Fermi monoatomic gasesare examined. They are conveniently called the Bose and Fermi cycles. Efficiencies of Bose and Fermi cycles are derived (etaB and etaF respectively). Variations of them with the temperature ratio (tau) and pressure ratio of the cycle are examined. A comparison of theefficiencies with each other and that of the classical Ericsson cycle (etaCl) is made. In thedegenerate gas state it is seen that etaB < etaF < etaCl, although etaB = etaF = etaCl in the classical gas state. In a Bose cycle, it is shown that there is an optimum value for the lowest temperature (TL) at which the efficiency reaches its maximum value for a given pressureratio. Furthermore, Bose–Einstein condensation restricts the value of TL of a Bose cycle for agiven value of PH . In a Fermi cycle, there is no an optimum value for TL. However, etaF goesto a finite value of less than unity when tau goes to zero.
The contribution of thermal electmn-posibon pairs to the thermodynamio propertiesof blackbody radiation (BBR) is considered. This contribution is examined in the vicinity of the temperature corresponding to the electron rest mass energy Te = mec2/k, in which it becomes appreciable. The corection factors are defined as the ratio of extended themdynamic expressions of BBR (which also include the contributions from thermal pairs) to their familiar expressions. Then the variation of these factors with temperame is given. It is shown that while they have the same value for each property in both high and low temperatme regimes (T >> Tc and T << Tc), they have different values about 0.5 Tc. It is found that for BBR, the ratio of the energy density to the pressure has a maximum abaut 0.33 Tc and also the ratio of the specific heats has a minimum about 0.4 Tc.