Thermodynamic cycles: calculations on the Internet

A.A. Alexandrov, K.A. Orlov, V.F. Ochkov (http://twt.mpei.ac.ru/ochkov)

Moscow Power Engineering Institute (TU) — (www.mpei.ru)

The article was published in Jourmal “Vestnik MEI” No 1, 2007

English editing by Richard Jueschke

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^{[1]} are used widely in education and engineering practice
[1, 2] 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 [3]. 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 [4], 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^{ }^{[2]} 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 c_{p} and c_{v} (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*

(http://twt.mpei.ru/MCS/Worksheets/Thermal/Otto-cycle-1e.xmcd)

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**(http://twt.mpei.ru/MCS/Worksheets/Thermal/Otto-cycle-2e.xmcd)

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 c_{p}
and c_{v} 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 p_{1}, temperature T_{1}, the volumetric
compression ratio r, maximal temperature T_{3}), 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 c_{p},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^{ }^{[3]}, 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, c_{p}, and c_{v}) 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™^{ }^{[4]} package (www.wsp.ru) without using
relations for ideal gas. For example, c_{p} =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
[5].These enable us to apply the WaterSteamPro™ package for calculating steam
power, gas turbine and also combined-cycle plants [6]. The functions of the WaterSteamPro
package are accessible for use on the Internet. Fig. 4
shows the Internet^{ }^{[5]} 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*

(http://twt.mpei.ac.ru/MAS/Worksheets/wsp_TextBox.mcd)

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 [7].

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^{ }^{[6]}.

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

(http://twt.mpei.ac.ru/MAS/Worksheets/Therm/otto_ideal_cycle_test.mcd)

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=p_{0}…p_{1})
or R*(p/p_{0}) or R*ln(p) –R*ln(p_{0}). By the way, the last two expressions are
not adequate if variables p and p_{0} 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^{ }^{[7]}.

**
Figure 6. Three-dimensional
graphics of Rankine cycle**(http://twt.mpei.ac.ru/MAS/Worksheets/Rankine3D.mcd)

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*(http://twt.mpei.ac.ru/MAS/Worksheets/PTU-2RegP_Plot.mcd)

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:

- Carnot cycle http://twt.mpei.ac.ru/MAS/Worksheets/Term/carno_Cp_T.mcd
- Diesel cycle http://twt.mpei.ac.ru/MAS/Worksheets/Term/diesel.mcd
- Trinkler cycle http://twt.mpei.ac.ru/MAS/Worksheets/Term/trinkler_Cp_T.mcd
- Brayton cycle http://twt.mpei.ac.ru/MAS/Worksheets/Term/GTU_Calc.mcd
- Steam reheat cycle http://twt.mpei.ac.ru/MAS/Worksheets/RankinePP.mcd
- Steam regenerative cycle http://twt.mpei.ac.ru/MAS/Worksheets/RankineRP.mcd
- Steam-gas cycle http://twt.mpei.ac.ru/MAS/Worksheets/PGU.mcd
- Real steam cycle of power plant http://twt.mpei.ac.ru/MAS/Worksheets/Exp_Steam_K_300_240_LMZ.mcd
- Real steam cycle of nuclear power plant http://twt.mpei.ac.ru/MAS/Worksheets/Term/RBMK_1000.mcd

A lecturer can show efficiency dependences of various thermodynamic cycles during lectures and seminars using the network calculations. The corresponding Mathcad files can be downloaded from the sites discussed above for their improvement.

*References:*

1. V.F. Ochkov. Mathcad Application Server: The experience of exploitation in Russia // SoftLine Direct.2006, No 11,P.96 –98

2. V.F. Ochkov. Mathcad 14 for students and engineers // SPb: BHV-Petersburg, 2007

3. A.A. Alexandrov. Thermodynamic principles of power plant cycles // M.: Publishing House of MEI, 2004, 158 pp.

4. A.A. Alexandrov, K.A. Orlov, V.F. Ochkov. Mathematical packages – new approach of calculations in thermodynamics and other scientific disciplines // Izv. vys. ucheb. zav. Problems of energetics. 2005, No 11-12, P.80-86

5. A.A. Alexandrov, V.F. Ochkov, K.A. Orlov. Equations and programs for calculations of properties of gases and combustion products //Thermal Engineering, 2005, Vol. 52, No 3, P. 28-37

6. A.A. Alexandrov, K.A. Orlov, V.F. Ochkov. Investigation of schemes of steam-gas plants with injection of steam in gas flow on base of elaborated applied programs for properties of working substances of PGU //New in Russian electroenergetics, 2004, No 4, P.28-37

7. V.F. Ochkov. Mathematical packages and network interactive reference book: problems and solutions // Thermal engineering, 2006, Vol. 53, No 6, P.71-77