Influence of boundary conditions on the optimization of a geothermal heat pump studied using a thermodynamic model

This paper is the logical follow-up to a work [1] whose results were presented at the 28th French Thermal Congress which was to be held in Belfort in 2020. The model developed at that time is completed in this proposal to consider the specificity of the geothermal heat pump. This is a machine operating upon a mechanical vapor compression cycle, the limit of which is an inverse Carnot cycle. Its specificity consists of a cold loop at the source with the geothermal exchanger and the evaporator, then a hot loop at the sink with the condenser and a floor heat exchanger in the application considered here. We are particularly concerned with the optimal sizing of these heat exchangers through their effectiveness. The parametric sensitivity of this distribution to various boundary conditions is studied, especially by focusing on different conditions at the source: (1) imposed soil temperature, corresponding to a Dirichlet condition, (2) imposed heat flux (including adiabatic case), corresponding to a Neumann condition, (3) imposed mechanical power consumed by the heat pump, and (4) imposed coefficient of performance COP, to all cases being associated a finite thermal capacity in thermal contact with the geothermal exchanger operating in steady-state conditions.

Calculations of The Change of State of a Gas By Integrating The Partial Derivatives

It will not be denied that the calculations of the change of state for a gas is highly important in most engineering applications. For determining the gas’s properties such as the pressure (P), the volume (V) and the temperature (T), engineers and scientists uses the Boyle’s, Charles’s and Gay-Lussac’s (B-C-G) law of P1V1/T1=P2V2/T2. Although the B-C-G law provides the accurate property values of a gas, it give no detailed information embedded in the process when a gas changes its state. In this study, the author theoretically carried out the integrations of the partial differentials when differentiating the B-C-G law, which has not been tried by anyone up to now. The integration results of this study were thoroughly compared with the experimentally measured data and it was confirmed that the integration methods suggested in this study accurately provides the differential properties on ΔP, ΔV and ΔT. In addition to it, through the stepwise analysis of the integration of the partial differentials, it revealed that the efficiency in the change of state of a gas inherently exists higher than the Carnot cycle, which is operating between the same conditions. Therefore, the results of this study can be lead to the conclusion that all changes of state of all materials inevitably accompanies an energy loss and it is a natural phenomenon.

Development of the Concept of Entropy: A Simpler Alternative to Carnot Cycle

The relationship between entropy and reversible heat and temperature is developed using a simple cycle, in which an ideal gas is subjected to isothermal expansion and compression and heated and cooled between states. The procedure is easily understood by students if they have knowledge of calculations involving internal energy, reversible work, and heat capacity for an ideal gas. This approach avoids the more time-consuming Carnot cycle. The treatment described here illustrates how the total entropy change resulting from an irreversible process is always positive.

The Connection between Carnot and CAPE Formulations of TC Potential Intensity

Tropical cyclone (TC) potential intensity (PI) theory has a well known form, consistent with a Carnot cycle interpretation of TC energetics, which relates PI to mean environmental conditions: the difference between surface and TC outflow temperatures and the air–sea enthalpy disequilibrium. PI has also been defined as a difference in convective available potential energy (CAPE) between two parcels, and quantitative assessments of future changes make use of a numerical algorithm based on this definition. Here, an analysis shows the conditions under which these Carnot and CAPE-based PI definitions are equivalent. There are multiple conditions, not previously enumerated, which in particular reveal a role for irreversible entropy production from surface evaporation. This mathematical analysis is verified by numerical calculations of PI’s sensitivity to large changes in surface-air relative humidity. To gain physical insight into the connection between the CAPE and Carnot formulations of PI, we use a recently developed analytic theory for CAPE to derive, starting from the CAPE-based definition, a new approximate formula for PI which nearly recovers the previous Carnot PI formula. The derivation shows that the difference in undilute buoyancies of saturated and environmental parcels which determines CAPE PI can in fact be expressed as a difference in the parcels’ surface moist static energy, providing a physical link between the Carnot and CAPE formulations of PI. This combination of analysis and physical interpretation builds confidence in previous numerical CAPE-based PI calculations that use climate model projections of the future tropical environment.

A Review of the System-Intrinsic Nonequilibrium Thermodynamics in Extended Space (MNEQT) with Applications

The review deals with a novel approach (MNEQT) to nonequilibrium thermodynamics (NEQT) that is based on the concept of internal equilibrium (IEQ) in an enlarged state space SZ involving internal variables as additional state variables. The IEQ macrostates are unique in SZ and have no memory just as EQ macrostates are in the EQ state space SX⊂SZ. The approach provides a clear strategy to identify the internal variables for any model through several examples. The MNEQT deals directly with system-intrinsic quantities, which are very useful as they fully describe irreversibility. Because of this, MNEQT solves a long-standing problem in NEQT of identifying a unique global temperature T of a system, thus fulfilling Planck’s dream of a global temperature for any system, even if it is not uniform such as when it is driven between two heat baths; T has the conventional interpretation of satisfying the Clausius statement that the exchange macroheatdeQflows from hot to cold, and other sensible criteria expected of a temperature. The concept of the generalized macroheat dQ=deQ+diQ converts the Clausius inequality dS≥deQ/T0 for a system in a medium at temperature T0 into the Clausius equalitydS≡dQ/T, which also covers macrostates with memory, and follows from the extensivity property. The equality also holds for a NEQ isolated system. The novel approach is extremely useful as it also works when no internal state variables are used to study nonunique macrostates in the EQ state space SX at the expense of explicit time dependence in the entropy that gives rise to memory effects. To show the usefulness of the novel approach, we give several examples such as irreversible Carnot cycle, friction and Brownian motion, the free expansion, etc.

The investigation of Carnot engine in the presence of deformed formalism

In this paper, after introducing a kind of [Formula: see text]-deformation in quantum mechanics, first [Formula: see text]-deformed form of Schrödinger equation for a single particle in a box is derived. Then, the energy eigenvalues and wave function in Schrödinger equation are studied. Also, we discuss the Carnot cycle by using of the energy eigenvalues. We obtain the thermodynamic properties such as force, heat transferred, work done and efficiency in the cycle. Finally, all results have satisfied what we had expected before.

Economic Cycles of Carnot Type

Originally, the Carnot cycle was a theoretical thermodynamic cycle that provided an upper limit on the efficiency that any classical thermodynamic engine can achieve during the conversion of heat into work, or conversely, the efficiency of a refrigeration system in creating a temperature difference by the application of work to the system. The first aim of this paper is to introduce and study the economic Carnot cycles concerning Roegenian economics, using our thermodynamic–economic dictionary. These cycles are described in both a Q−P diagram and a E−I diagram. An economic Carnot cycle has a maximum efficiency for a reversible economic “engine”. Three problems together with their solutions clarify the meaning of the economic Carnot cycle, in our context. Then we transform the ideal gas theory into the ideal income theory. The second aim is to analyze the economic Van der Waals equation, showing that the diffeomorphic-invariant information about the Van der Waals surface can be obtained by examining a cuspidal potential.

Optimal Heat Exchanger Area Distribution and Low-Temperature Heat Sink Temperature for Power Optimization of an Endoreversible Space Carnot Cycle

Using finite-time thermodynamics, a model of an endoreversible Carnot cycle for a space power plant is established in this paper. The expressions of the cycle power output and thermal efficiency are derived. Using numerical calculations and taking the cycle power output as the optimization objective, the surface area distributions of three heat exchangers are optimized, and the maximum power output is obtained when the total heat transfer area of the three heat exchangers of the whole plant is fixed. Furthermore, the double-maximum power output is obtained by optimizing the temperature of a low-temperature heat sink. Finally, the influences of fixed plant parameters on the maximum power output performance are analyzed. The results show that there is an optimal temperature of the low-temperature heat sink and a couple of optimal area distributions that allow one to obtain the double-maximum power output. The results obtained have some guidelines for the design and optimization of actual space power plants.

Action and Entropy in Heat Engines: An Action Revision of the Carnot Cycle

Despite the remarkable success of Carnot’s heat engine cycle in founding the discipline of thermodynamics two centuries ago, false viewpoints of his use of the caloric theory in the cycle linger, limiting his legacy. An action revision of the Carnot cycle can correct this, showing that the heat flow powering external mechanical work is compensated internally with configurational changes in the thermodynamic or Gibbs potential of the working fluid, differing in each stage of the cycle quantified by Carnot as caloric. Action (@) is a property of state having the same physical dimensions as angular momentum (mrv = mr2ω). However, this property is scalar rather than vectorial, including a dimensionless phase angle (@ = mr2ωδφ). We have recently confirmed with atmospheric gases that their entropy is a logarithmic function of the relative vibrational, rotational, and translational action ratios with Planck’s quantum of action ħ. The Carnot principle shows that the maximum rate of work (puissance motrice) possible from the reversible cycle is controlled by the difference in temperature of the hot source and the cold sink: the colder the better. This temperature difference between the source and the sink also controls the isothermal variations of the Gibbs potential of the working fluid, which Carnot identified as reversible temperature-dependent but unequal caloric exchanges. Importantly, the engine’s inertia ensures that heat from work performed adiabatically in the expansion phase is all restored to the working fluid during the adiabatic recompression, less the net work performed. This allows both the energy and the thermodynamic potential to return to the same values at the beginning of each cycle, which is a point strongly emphasized by Carnot. Our action revision equates Carnot’s calorique, or the non-sensible heat later described by Clausius as ‘work-heat’, exclusively to negative Gibbs energy (−G) or quantum field energy. This action field complements the sensible energy or vis-viva heat as molecular kinetic motion, and its recognition should have significance for designing more efficient heat engines or better understanding of the heat engine powering the Earth’s climates.