Thermodynamic cycles: calculations on the Internet

Proceedings of the 7-th Conference «Power System Engineering, Thermodynamics & Fluid Flow», June 26-27, 2008, Pilsen, Czech Republic, pp. 165-173

, A.A. Alexandrov, K.A. Orlov, A.V. Ochkov - Moscow Power Engineering Institute (TU) — (www.mpei.ru)

New approaches of thermodynamic cycle calculation are described using modern network information technologies that can change calculation methods and instruction principles in heat engineering

Today mathematical packages are used widely in education and engineering practice that demand revision of calculation techniques and new instruction methods.

The main base of calculations in various scientific disciplines (in particular, thermodynamics) is a set of formulae and algorithms, that was formed for hand computation in pre-computer (and even in pre-calculator) era. This puts a kind of seal on formulae and methods of their application that impedes their use in computer calculations and to understanding of a scientific discipline by students. The formulae were often excessively simplified. Moreover, some approaches were omitted in lots of formulae related with so named “technology” reasons, i.e., with methods and technical means of calculations used in times of formula deducing that don’t connect with its physical essence.

There is a classical example from thermodynamics. Fig. 1 shows an Internet site where with calculation of the thermal efficiency of a thermodynamic Otto cycle formed by two isochoric and two isentropic processes.  A few people remember how this formula was developed and that it is correct if cp and cv (specific isobaric and isochoric heat capacities of working medium) are related with temperature. It is an ideal case which does not correspond even to a gas with constant composition. In reality, working substances for power plants are mixtures with changing composition – fuel-air mixtures, combustion products, etc.

Figure 1. Calculation of thermal efficiency of thermodynamic Otto cycle by a simplified formula

Returning to the thesis noted in the beginning of the article, and connecting it to thermodynamic cycle calculations, it is possible to regenerate the initial formula (more precisely, a set of formulae and algorithms). Modern mathematical systems (Mathcad, in particular) can give us not only Otto cycle efficiency but all intermediate values –parameters of working medium at various points of the cycle. Fig. 2 shows this calculation.

Figure 2. Calculation of Otto cycle thermal efficiency by initial formulae

Fig. 2 shows that for an Otto cycle calculation in Mathcad, a built-in operator was used.  This was a definite integral type and the block (Given-Find) for system solving, in this case, for a system of integral-algebraic equations. These tools allow us to compute the main relations of thermodynamics: Mendeleev-Clapeyron equation PV =RT, integral over temperature and pressure for calculations of entropy S, enthalpy H, and integral over temperature for calculations of internal energy U. These provide the possibility to find the values of internal energy U at all four points of an Otto cycle and to calculate the  cycle efficiency using these values. Fig. 2 shows that the constants cp and cv are not taken out of integrals. For more complicated (real) calculations we can substitute appropriate functions of one (temperature), two (temperature and pressure) or more (temperature, pressure, composition of gas mixture) variables for these constants. Fig. 3 shows the calculation where a user can choose the working medium, its parameters (pressure p1, temperature T1, the volumetric compression ratio r, maximal temperature T3), and specify the axes of the diagram visualizing the cycle, or a point in the cycle of internal-combustion power plant.

Figure 3. Calculation of Otto cycle where specific heat of working medium depends on temperature

Fig. 3 shows two diagrams in the end of an Otto cycle calculation. One of them is the usual p,V-diagram (this and other diagrams are possible to plot by choosing proper buttons on the axes).  Another is a cp,p-diagram which represents the dependence of specific isobaric heat capacity of the working medium (air, on Fig. 3) on temperature within limits of its changing in the Otto cycle. In this calculation a user can choose a point of the cycle: first point is the beginning of compression process, point 2 is the end of compression, point 3 is the end of isochoric combustion of fuel-air mixture (here the heat capacity values depend on temperature and on composition of the working substance), and the last point 4 is the end of piston stroke. The chosen phase determines the current shot of animation, the outlined point on the diagram, and represented working substance parameters.

In the network document shown on Fig. 3 the calculations of the working substance parameters are carried out not only for four points as it shown on Fig. 2 but also for intermediate points with a given step. These allow us to create vectors which elements are parameters of the working substance (p, T, v, s, h, U, cp, and cv) and also to plot volumetric graphs in various systems of coordinates (for example, see Fig. 6 below).

There is a second difference between the calculations presented on Fig. 2 and Fig. 3. Fig. 3 shows calculations based on functions provided by the WaterSteamPro  package (www.wsp.ru) without using relations for ideal gas.  For example, cp =aR where a is a constant related with gas composition.

The WaterSteamPro™ package has been created as a tool for users to calculate heat-transfer properties of water and steam in program languages C, Fortran, BASIC, Pascal, tabular processor Excel, and mathematical packs such as Mathcad, MatLab, et cetera. Additionally, the list of functions with prefix wsp was increased and the functions with prefix wspg were added for calculations of thermodynamic properties of individual gases and gas mixtures.These enable us to apply the WaterSteamPro™ package for calculating steam power, gas turbine and also combined-cycle plants. The functions of the WaterSteamPro package are accessible for use on the Internet. Fig. 4 shows the Internet site in which a user can introduce values in text boxes, press the  Recalculate button and obtain numerical results –the values of entropy, enthalpy and other properties of the working medium.

Figure 4. Calculation of parameters of the working media in power engineering

Mathcad Calculation Server technology, supplemented with the author’s technology of access to a broadened list of built-in functions of numerical and symbolic mathematics of Mathcad and Maple packages, allows us to change educational sites concerned with the theoretical principles of heat engineering.

Let us return to Figures.1 –3 and try to change the problem statement solved on these sites. Figures. 1 –3 show that a user can change input data, send it to the calculation server (Mathcad Calculation Server), press the Recalculate button and obtain new results as numerical data, graphs, and pictures. Mathcad Calculation Server technology also gives us the possibility to create sites for distributing knowledge such as for distance education. Fig. 5 shows the site (upper part) where a visitor (for example, a student solving thermodynamic problem in time of examination) must input the formulae for the  correct calculation of the thermal efficiency of an Otto cycle and other parameters. If the formulae were input correctly (int is integral, diff is differential and other – here it is used as notations and built-in functions of mathematical packs Mathcad and Maple which have now become a standard “man-computer” interface) then value in fields “Your answer” and “Correct answer” (see the bottom part of Fig. 5) must coincide.

Figure 5. Network template and correctly filled network template of a thermodynamics problem

The main feature of the text task presented on Fig. 5 and Fig. 6 is that a student should work with formulae but not with numbers. A lot of test systems demand some formula evaluations without entering the formula itself. As a result, it is often impossible to determine where the mistakes were done: in calculations or in selection of formulae.

The author’s method of network testing including analytic transformation and numerical calculations enables a student to give various answers. For example, the variants may be R*int(1/p, p=p0…p1) or R*(p/p0) or R*ln(p) –R*ln(p0). By the way, the last two expressions are not adequate if variables p and p0 have dimension of pressure. This is a result of the reasons mentioned in the beginning of this paper: an attempt to simplify calculations and substitute subtraction for division.

There is another form of Internet education for heat-and-power engineers, as mentioned before: demonstrated through three-dimensional graphics.

Figure 6. Three-dimensional graphics of Rankine cycle

For example, Fig. 6 shows diagrams built on the base of WaterSteamPro™ functions for a Rankine cycle.

We can make network calculations both for scalar values and for vectors i.e. with a number of elements of initial data using “vectorization” of calculations. Fig. 7 shows such a calculation and the optimization of a steam plant with two reheats.

Figure 7. Calculation and optimization of steam unit cycle with two reheats

Using the Internet sites we can carry out the calculations of cycle efficiency quickly and clearly. Different gases can be introduced as the working medium. For example, we can determine that for all the gasses (with the exception of monatomic), thermal efficiency of the Otto cycle depends on the volumetric compression ratio and also on the maximum temperature of gas. Thus, the difference between the cycle thermal efficiency and those values calculated by the simplified formula may amount to 5% or more. Obviously, that inaccuracy is intolerable in engineering calculations and students must be trained to avoid the application of simplified formula. It is especially important to take into account the dependence of cycle efficiency on parameters of the working medium in optimized calculations.

Changing simplified formulae to base algorithms of calculations (it may be called calculation Renaissance) enables us to return from ideal processes to real.

Other calculations of thermodynamic cycles accessible on the Internet (http://twt.mpei.ac.ru/TTHB/2/ThermCycleMCS.html):

discussed above for their improvement.

References:

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