Long Paper

Thermodynamic Analysis of Steam Expansion in a Cylinder using MATLAB

Name: Prithvi Suresh

Matriculation No.: 10061919

Technical Publications and Presentations

MSc. Systems Engineering and Engineering Management

Contents

I. Introduction……………………………………………………………………………………………………………. 1

II. Theory……………………………………………………………………………………………………………………. 1

A. Ideal Gas Law………………………………………………………………………………………………… 1

B. Adiabatic Expansion……………………………………………………………………………………….. 2

C. Solving the Differential Equation……………………………………………………………………… 3

III. Simulation Model……………………………………………………………………………………………………. 3

A. Variable Description……………………………………………………………………………………….. 3

B. Computational Module……………………………………………………………………………………. 4

1. F_Start……………………………………………………………………………………………………………. 5

2. F_Volume………………………………………………………………………………………………………… 5

3. F_HeatEnergy…………………………………………………………………………………………………… 5

4. F_dQCalculation……………………………………………………………………………………………….. 5

5. F_Sum…………………………………………………………………………………………………………….. 5

6. XSteam……………………………………………………………………………………………………………. 6

7. F_Equation………………………………………………………………………………………………………. 6

8. F_NewValues……………………………………………………………………………………………………. 6

IV. Results……………………………………………………………………………………………………………………. 7

V. Conclusion……………………………………………………………………………………………………………… 8

VI. Bibliography…………………………………………………………………………………………………………… 9

VII. Appendix………………………………………………………………………………………………………………. 10

I. Abstract

The primary advantages of steam engine use come from applications where current technologies are either not appropriate or cannot be scaled down in size. This research presents the simulation procedure developed for the expansion process in a steam engine. The mathematical model uses a double-acting piston steam engine which was manufactured in the early 1900’s. All the data required for the research have been obtained by conducting experiments on this steam engine. The mathematical model is implemented in the MATLAB software and the results obtained from the simulated model have been compared with the theory presented in literature. Result shows that simulation model exhibits similar expansion curve when compared to the adiabatic expansion curve for ideal gas. These results have been compared and discussed in the final part of the paper.

II. Introduction

The forerunner of the steam engine was the piston steam pump. This pump was invented around the year 1700 by Thomas Newcomen. It had a minimal power output and was used for pumping out water. In the year 1865, James Watt invented a new generation of steam engine, in which he interconnected all the previous discoveries and inventions and was able to improve the efficiency of the steam engine crucially 6.

The development of the steam engine led to the 1st Industrial Revolution, and since then there have been innumerable inventions which influenced our world into this era, where producing energy is one of the necessities without which nothing seems to be possible. Though we have the most advanced fuel engines over the years, the fundamental principle of the steam engines remains the same.

Hence, this research develops a simulation model of a steam expansion process in a cylinder which could then be incorporated into various other processes. The simulation is carried out using the thermodynamic model explained in the dissertation of Michael Grill 1. This is explained in the first section of the paper which is then followed by the implementation of the mathematical model for the simulation process. Finally, the are presented along with some discussions and comparisons.

This paper uses the MATLAB/Simulink software to create the simulation model of the steam engine. The software contains numerous tools and applications to develop complex numerical systems which are mainly used in the field of research and development for faster analysis of a dynamic model 2.

III. Theory

This section explains some of the basic principles of thermodynamics relating to the steam engines and then followed by the explanation of the main differential equations used in the programming of the simulation model.

A. Ideal Gas Law

A physical law which describes the thermodynamic properties for a constant molar amount of substance is given by the relation (1). This is known as the “Ideal Gas Law”.

(1)

The Ideal Gas Law results from the combination of two experimentally determined gas laws from Boyle-Mariotte and Gay-Lussac 3.

(2)

For a 1 kg mass of the ideal gas, it results:

(3)

In the case of real gas, the equation (3) is not equal to one. Hence, it is replaced by the real gas factor Z 3:

(4)

B. Adiabatic Expansion

The adiabatic expansion is a thermodynamic process that changes the condition of the system without any heat exchange, and due to the reversibility of the process, the entropy is regarded as constant. The process is explained by the following equation 4:

(5)

For an ideal gas, is equal to:

(6)

The adiabatic expansion process of an ideal gas (1-2) and real gas (1-2´) is shown in fig. 1.

Fig. 1 Adiabatic expansion process in p-V and h-s Diagram 4

C. Solving the Differential Equation

To determine the unknown variables, change in pressure change in volume and change in temperature of each system, a set of equations is set up. This is carried out by constructing the total differentiation for a 1-Zone system according to Grill 1:23.

(7)

The second equation is obtained by constructing the total differentiation for the gas constant R.

(8)

It is important to note that R and u are the functions of temperature, pressure and gas mixture. In the equations (7) and (8), are unknown. Hence, the computation requires a third equation to solve the three unknowns. Since the volume of the system can be calculated using the position of the piston, it is used to determine the change in volume of the whole system which is given by (9) 1.

(9)

IV. Simulation Model

This section describes the variables and the important functions used in the Simulink model.

A. Variable Description

1. System Parameter

The structure array struct KDM stores the system parameters of the steam engine which facilitates the use of variables of different data types in a single array. An example of this structure array is shown in table 1. Such variables are saved in the structure array and is possible to change these variables manually without affecting any other data. After carrying out the initialization, the struct KDM loads automatically with all the datatypes available to be used by the computational modules in the workspace.

Table 1. Example of a structure-array *

Structure Array

Name of the variable

Description

KDM.s0

Hub of the cylinder

KDM.D_z

Diameter of the cylinders

KDM.phi_k

Piston angle

KDM.p_start

Initial value for pressure

…

…

2. Computational Variables

All the computing parameters such as volume, pressure, and temperature, are stored as a double type variable. A group of instructions with a definite number of iterations is implemented in a for-loop, whereby successively PS1 PS2 the thermodynamic quantities are calculated according to the computing parameters.

B. Computational Module

For the clarity of the computations within the EMF (Embedded Function), the computation steps are stored in separate functions, which makes the editing of the modules easier and autonomous. Fig. 3 shows the sequence of the computational modules in a flow chart. The structure array KDM containing the system parameters of the steam engine is needed as inputs to the EMF (as shown in Table 2).

Table 2. State variables as input parameters *

State variables and their descriptions

p:

Pressure

h:

Specific enthalpy

T:

Temperature

u:

Specific internal Energy

R:

Specific gas constant

m:

Mass

V:

Volume

From the thermodynamic properties, the differentiation equations from the section II.C are computed, which further changes the state variables to the previous time step, which in turn gives out the parameters for the next computation step. In the first computation step, an inquiry takes place, whether the loop variable COUNT equals zero. This condition is specified in the MEMORY blocks and checks if any computation has taken place yet. If not, then the loop goes into the function F_Start to obtain the initial parameters.

3. F_Start

This function returns the initial values for the pressure p and the temperature T as specified in the structure array. Furthermore, once the function is implemented by the steam-tables, which gives the caloric values R, u, h and cp and cv. After this function is called, the process of computation continues according to the flowchart in fig. 3.

4. F_Volume

Based on the piston’s position obtained from the p-V-diagram of the experiments, the change in system volume dV is calculated using the increment of the piston position dphi and the new system volume V.

5. F_HeatEnergy

The movement of the piston generates heat in the system volume due to the friction between the piston rings and the cylinder wall. The heat from the heating jacket is computed using the formulas of thermal conduction and heat transfer.

6. F_dQCalculation

In this function, the change in heat flow is computed which occurs during the expansion process. The change in heat flow takes place due to the difference between the current system volume and the previous system volume.

7. F_Sum

In this function, the thermal energy is summed up, and the specific change in enthalpy and specific change in mass flow are computed. The value of COUNT is also changed for the following iteration.

8. XSteam

The steam table is taken up from the MATLAB community, which is programmed according to the IAPWS IF-97 standard, to obtain the caloric values of water vapor 2.

9. F_Equation

To solve the differential equations as described in section II.C, the computed masses, the heat flow, enthalpy, caloric values, variables of state, and the change in volume are needed as inputs. Since is already known, the two unknown quantities and is computed using the other two sets of equations.

10. F_NewValues

This function saves the resultant differentials of the state variables in the memory blocks and automatically updates them to the current thermodynamic properties.

Fig.3 Flow-chart of the Simulink-Modell *

V. Results

The Simulink model has been programmed for the pure expansion process occurring inside the steam engine, and the model is run using the initial conditions which were obtained from the experiments on the steam engine. The pure expansion process starts at a point where the intake valve closes. The endpoint of the pure expansion process ends at a point where the change in pressure is negligible. The resultant p-V diagram from the Simulink model is shown in Fig. 4. For a better understanding of the results, a p-V graph is additionally obtained from the Model.

Fig.4 p-V Scope from the Simulink Model *

A comparison of the adiabatic curve and the p-V scope of the Simulink model is made, and it can be observed clearly that both the curves almost coincide with each other as shown in Fig. 5. Hence, it can be concluded that the Simulink model recreates the expansion process of the steam engine, and can be used for further analysis of the complete steam cycle.

VI. Conclusion

Fig. 5 p-V Diagram from the Model *

VII. Bibliography

1 Grill, Michael: “Objektorientierte Prozessrechnung von Verbrennungsmotoren”, Dissertation, Universität Stuttgart, 2006.

2 Pietruszka, W.S.: “MATLAB und Simulink in der Ingenieurpraxis: Modellbildung, Berechnung und Simulation”, Springer Fachmedien Wiesbaden, 2014.

3 Doering, Ernst., Schedwill, Herbert., Dehli, Martin: “Grundlagen der Technischen Thermodynamik, Lehrbuch für Studierende der Ingenieurwissenschaften”, 8. Ed. Esslingen, 2016, pp 30-150.

4 Zich, Alexej: “Steigerung der Effizienz der Expansionsprozesse im Rahmen der Gasverteilung”, Dissertation, TUB Freiburg, 2014.

5 Holmgren, Magnus: “MathWorks – Makers of MATLAB and Simulink, Community – File Exchange”, 01 August 2007, 10 November 2017

6 Matschoss, Conrad: “Geschichte der Dampfmaschine, Ihre kulturelle Bedeutung, technische Entwickelung und ihre großen Männer”, Ed. 4. Hildesheim, Gerstenberg, 1987, pp. 1-30.

* Own illustrations.

VIII. Appendix

Symbols

Symbols

Units

Descriptions

A

mm2

Area

b

mm

Width

cp

kJ/kgK

Specific heat capacity (constant Pressure)

cv

kJ/kgK

Specific heat capacity (constant Volume)

D, d

mm

Diameter

H

kJ

Enthalpy

h

kJ/kg

Specific. Enthalpy

k

–

Isentropic exponent

la

mm

Indicator diagram hub

M, m

kg

Mass

n

min-1

Cycles per minute

P

N/m2

Pressure

Q

kJ

Heat supply

R

kJ/kgK

Specific Gas constant

T

K

Temperature

U

kJ

Internal Energy

u

kJ/kg

Specific Internal Energy

V

m3

Volume

Z

–

Real Gas factor

?

W/m2K

Heat convection coefficient

?

W/mK

Heat conduction coefficient

?

degrees

Angle

?

–

Adiabatic index

PS1Check

PS2