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ChemEngineering 2019, 3(2), 36; https://doi.org/10.3390/chemengineering3020036
- Find the dynamic laws of heat transfer with sources for which the efficiency of the machine in the maximum power mode does not depend on the dynamic coefficients
- Determine the laws of dynamics in which the cycle corresponding to the maximum efficiency at a given heat consumption consists of two and three isotherms.
- If the optimal cycle consists of three isotherms, then sources two of them are realized?
2. Averaged Optimization and Convex Hulls of a Function
2.1. Convex Hulls of a Function
2.2. Averaged Problem of Conditional Optimization
3. “Natural” Dynamic Functions and Optimal Cycles Structure
- are continuous and differentiable with respect to their arguments everywhere except for . and hold true (these are the monotonicity conditions). The same applies to the signs of the derivatives on .
- The sign of the heat flux coincides with the sign of the temperature difference (the second law of thermodynamics). at holds true.
- The dynamic dependence can be written in the form , where the heat transfer coefficient is proportional to the contact surface of the working fluid with the source. For example, the dependency of type does not match this requirement. This type of dynamic was investigated in . The optimal cycle consisted of three isotherms and three adiabats in this work.
Statement of the Problem and Solutions Structure Corresponding to the Boundary of Realizability Region
- The optimal solution of this problem can take no more than three (base) values of the vector ; i.e., no more than three isothermal sections with adiabatic transitions from one isotherm to another, where is the base solution index.
- The duration of each isothermal section is a certain part of the cycle time, and the adiabatic transitions occurs instantaneously.
4. Graphic Interpretation of the Solution
5. The Optimal Solution
The Relation between the Power and the Entropy Production in the System
6. The Realizability Region Boundary
6.1. Characteristic Points of the Boundary
- The origin of coordinates where and p tend to zero. The energy and conversion efficiency for any heat exchange law at this point is .
- The point at which the power of the machine is maximal. This point is denoted by . The corresponding energy conversion efficiency and heat flux will be denoted by and .
- The point at which the irreversibility of the processes is so great that the power of the machine turns out to be zero. The value of the heat flux corresponding to this point is denoted by .
6.2. Boundary Construction
6.3. Conditions of Independence Efficiency of the Machine in the Mode of Maximum Power from Coefficients of Heat Exchange with Sources
6.4. Novikov Case
7. The Boundary of the Realizability Region for Two Types of Heat Transfer Dynamics
7.1. Newtonian Heat Transfer Laws
7.2. The Linear Phenomenological Law
Conflicts of Interest
Appendix A. The Problem on Conditional Maximum of Function and the Averaged Statement of This Problem
- Write the function in the next form (Lagrange function)
- Search for its unconditional maximum on the set allocated only by the restrictions on the components of the vector x. This maximum will depend on the Lagrange indefinite multiplier.
- Solve the equation for
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