AIM

To write code in MATLAB that can solve a Rankine cycle and make plots for it.

INTRODUCTION

1.       THEORY OF RANKINE CYCLE: [1]

The Rankine cycle is a model used to predict the performance of steam turbine systems. It was also used to study the performance of reciprocating steam engines. The Rankine cycle is an idealized thermodynamic cycle of a heat engine that converts heat into mechanical work while undergoing phase change. It is an idealized cycle in which friction losses in each of the four components are neglected. The heat is supplied externally to a closed loop, which usually uses water as the working fluid.

The Rankine cycle closely describes the process by which steam-operated heat engines commonly found in thermal power generation plants generate power. Power depends on the temperature difference between a heat source and a cold source. The higher the difference, the more mechanical power can be efficiently extracted out of heat energy, as per Carnot's theorem. The heat sources used in these power plants are usually nuclear fission or the combustion of fossil fuels such as coal, natural gas, and oil, or concentrated solar power. The higher the temperature, the better.

Figure 1. Classical Carnot Heat Engine

The efficiency of the Rankine cycle is limited by the high heat of vaporization of the working fluid. Also, unless the pressure and temperature reach super critical levels in the steam boiler, the temperature range the cycle can operate over is quite small: steam turbine entry temperatures are typically around 565 °C and steam condenser temperatures are around 30 °C. This gives a theoretical maximum Carnot efficiency for the steam turbine alone of about 63.8% compared with an actual overall thermal efficiency of up to 42% for a modern coal-fired power station. This low steam turbine entry temperature (compared to a gas turbine) is why the Rankine (steam) cycle is often used as a bottoming cycle to recover otherwise rejected heat in combined-cycle gas turbine power stations.

Figure 2. Rankine Cycle Layout [1 – Pump, 2 – Boiler, 3 – Turbine, 4 – Compressor]

The cold source (the colder the better) used in these power plants are usually cooling towers and a large water body (river or sea). The efficiency of the Rankine cycle is limited on the cold side by the lower practical temperature of the working fluid. The working fluid in a Rankine cycle follows a closed loop and is reused constantly. The water vapor with condensed droplets often seen billowing from power stations is created by the cooling systems (not directly from the closed-loop Rankine power cycle). This 'exhaust' heat is represented by the "Q_out" flowing out of the lower side of the cycle shown in the T–s diagram below. Cooling towers operate as large heat exchangers by absorbing the latent heat of vaporization of the working fluid and simultaneously evaporating cooling water to the atmosphere.

While many substances could be used as the working fluid in the Rankine cycle, water is usually the fluid of choice due to its favorable properties, such as its non-toxic and unreactive chemistry, abundance, and low cost, as well as its thermodynamic properties. By condensing the working steam vapor to a liquid the pressure at the turbine outlet is lowered and the energy required by the feed pump consumes only 1% to 3% of the turbine output power and these factors contribute to a higher efficiency for the cycle. The benefit of this is offset by the low temperatures of steam admitted to the turbine(s). Gas turbines, for instance, have turbine entry temperatures approaching 1500 °C. However, the thermal efficiency of actual large steam power stations and large modern gas turbine stations are similar.

2.       PROCESSES: [1]

• PROCESS 1-2:

The working fluid is pumped from low to high pressure. As the fluid is a liquid at this stage, the pump requires little input energy. In other words Process 1-2 is called an Isentropic compression.

Figure 3. T–s diagram of a typical Rankine cycle operating between pressures of 0.06 bar and 50 bar. Left from the bell-shaped curve is liquid, right from it is gas, and under it is saturated liquid–vapor equilibrium.

• PROCESS 2-3:

The high-pressure liquid enters a boiler, where it is heated at constant pressure by an external heat source to become a dry saturated vapor. The input energy required can be easily calculated graphically, using an enthalpy–entropy chart (h–s chart, or Mollier diagram), or numerically, using steam tables. In other words, Process 2-3 is called Constant pressure heat addition in boiler.

• PROCESS 3-4:

The dry saturated vapor expands through a turbine, generating power. This decreases the temperature and pressure of the vapor, and some condensation may occur. The output in this process can be easily calculated using the chart or tables noted above. In other words, Process 3-4 is called an Isentropic expansion.

• PROCESS 4-1:

The wet vapour then enters a condenser, where it is condensed at a constant pressure to become a saturated liquid. In other words, Process 4-1 is called Constant pressure heat rejection in condenser.

In an ideal Rankine cycle the pump and turbine would be isentropic, i.e., the pump and turbine would generate no entropy and hence maximize the net-work output. Processes 1–2 and 3–4 would be represented by vertical lines on the T–s diagram and more closely resemble that of the Carnot cycle. The Rankine cycle shown here prevents the state of the working fluid from ending up in the superheated vapor region after the expansion in the turbine, which reduces the energy removed by the condensers.

The actual vapor power cycle differs from the ideal Rankine cycle because of irreversibilities in the inherent components caused by fluid friction and heat loss to the surroundings; fluid friction causes pressure drops in the boiler, the condenser, and the piping between the components, and as a result the steam leaves the boiler at a lower pressure; heat loss reduces the net-work output, thus heat addition to the steam in the boiler is required to maintain the same level of net-work output.

3.       GOVERNING EQUATIONS: [1]

• VARIABLE DETAILS:

The list of variables is used to define the thermodynamic equations of the Rankine Cycle are mentioned below:

• THERMODYNAMIC EQUATIONS:

Application of the First law of thermodynamics to the control volume (pump, steam generator, turbine and condenser), gives: [2]

Work done on pump, per kg of water:

Heat energy added in steam generator:

Work delivered by turbine:

Heat energy rejected in the condenser:

Net-Work done is given by:

The thermal efficiency of the Rankine cycle is given by:

Also, the thermal efficiency can also be identified as:

Back Work Ratio is given by:

Specific Steam Consumption:

OBJECTIVES

• To write a code in MATLAB that creates a T-S and H-S diagrams for Rankine Cycle.
• To calculate the Net-work and back work ratio of the Rankine Cycle.

PROGRAM – CODE WRITTEN IN MATLAB

PROBLEM STATEMENT:

It is required to create a Rankine cycle by writing the necessary code in MATLAB. The cycle under consideration takes the following inputs from the user:

• Pressure at the inlet of the turbine (bar)
• Temperature at the inlet of the turbine (Celsius)
• Pressure at the condenser (bar).

The net-work, back work ratio, efficiency, specific steam consumption, heat input must be calculated, and TS and HS plots are to be created with saturation curves supporting the plots.

CODE SIMULATION:

%RANKINE CYCLE SIMULATOR
%CALCULATING THE NETWORK OF THE RANKINE CYCLE
%CALCULATING THE BACK WORK RATIO OF THE RANKINE CYCLE
%TAKING VALUES FROM THE USER
 
clear all
close all
clc
 
%DISPLAYING THE CONTENT 
fprintf("\n                                RANKINE CYCLE SIMULATOR            \n");
fprintf("\n Process 1-2 : Isentropic expansion in the Turbine \n");
fprintf("\n Process 2-3 : Isobaric heat rejection by the Condenser \n");
fprintf("\n Process 3-4 : Isentropic compression in the Pump \n");
fprintf("\n Process 4-1 : Isobaric heat addition by the Boiler \n");
 
%REQUESTING THE USER TO GIVE THE INPUTS
p1 = input('\n Enter the pressure at the Turbine inlet (in bar) = ');
t1 = input('\n Enter the temperature at the Turbine inlet (in Celsius) = ');
p2 = input('\n Enter the pressure at the Condenser (in bar) = ');
 
%TURBINE INLET PARAMETERS
tsat_1 = XSteam('Tsat_p',p1);
if t1 > tsat_1
    h1 = XSteam('h_pt',p1,t1);
    s1 = XSteam('s_pt',p1,t1);
elseif t1 == tsat_1
        h1 = XSteam('hV_p',p1);
        s1 = XSteam('sV_p',p1);
    else 
        fprintf("\n Enter higher value of Turbine inlet temperature (in Celsius)!!! \n\n");
end
 
 
%CONDITIONS AT TURBINE OUTLET
%BASED ON THE LOCATION OF POINT 2 THE FOLLOWING VALUES/PROPERTIES ARE
%EVALUATED
s2 = s1; %isentropic expansion in the turbine
ssat_2 = XSteam('sV_p',p2);
if s2 < ssat_2
    sf_2 = XSteam('sL_p',p2);
    sg_2 = XSteam('sV_p',p2);
    %let x denote the dryness fraction of the steam in the system
    x = (s2 - sf_2)/(sg_2 - sf_2);
    hf_2 = XSteam('hL_p',p2);
    hg_2 = XSteam('hV_p',p2);
    h2  = hf_2 + x*(hg_2 - hf_2);
    t2 = XSteam('Tsat_p',p2);
else
    x = 1;
    h2 = XSteam('hV_p',p2);
    t2 = XSteam('Tsat_p',p2);
end
 
 
%CONDITIONS AT PUMP INLET
p3 = p2;
t3 = XSteam('Tsat_p',p3);
h3 = XSteam('hL_p',p3);
s3 = XSteam('sL_p',p3);
v3 = XSteam('vL_p',p3);
 
 
%CONDITIONS AT BOILER INLET
p4 = p1;
s4 = s3; 
v4 = XSteam('v_ps',p4,s4);
cp4 = XSteam('Cp_ps',p4,s4);
cv4 = XSteam('Cv_ps',p4,s4);
gamma = cp4/cv4;
G = (gamma - 1)/gamma;
t4 = t3*(p2/p1)^G;
 
%WORK REQUIRED BY THE PUMP
W_p = v3*(p4 - p3)*100;
%W_p = h4 - h3
h4 = W_p + h3;
 
 
 
%CONDITION OF INTERMEDIATE STATES: STATE 5 AND STATE 6 IN (4-1) STAGE
t5 = XSteam('Tsat_p',p4);
s5 = XSteam('sL_p',p4);
t6 = t5;
s6 = XSteam('sV_p',p4);
 
 
%WORK DONE BY THE TURBINE
W_t = h1 - h2;
 
%HEAT INPUT
Q_in = h1 - h4;
 
%HEAT OUTPUT
Q_out = h2 - h3;
 
%NET-WORK OBTAINED
W_net = W_t - W_p;
 
%Efficiency of the Rankine Cycle
efficiency = (W_net*100)/Q_in;
 
%BACK WORK RATIO
BWR = W_p/W_t;
 
%SPECIFIC STEAM COSUMPTION
SSC = 3600/W_net;
 
 
%DISPLAYING THE OBTAINED INFORMATION
fprintf("\n                                RESULTS            \n");
fprintf("At State Point 1:                              At State Point 4: \n P1 = %.2f bar                                 P4 = %.2f bar \n T1 = %.2f C                                  T4 = %.2f C  \n H1 = %.2f kJ/kg                             H4 = %.2f kJ/kg  \n S1 = %.2f kJ/kg-C                              S4 = %.2f kJ/kg-C \n",p1,p4,t1,t4,h1,h4,s1,s4);
fprintf("\nAt State Point 2:                              At State Point 3: \n P2 = %.2f bar                                  P3 = %.2f bar \n T2 = %.2f C                                   T3 = %.2f C  \n H2 = %.2f kJ/kg                             H3 = %.2f kJ/kg  \n S2 = %.2f kJ/kg-C                              S3 = %.2f kJ/kg-C \n",p2,p3,t2,t3,h2,h3,s2,s3);
 
fprintf("\n Work required by the pump in the Rankine Cycle = %.2f kJ/kg\n",W_p);
fprintf("\n Work produced by the turbine in the Rankine Cycle = %.2f kJ/kg\n",W_t);
fprintf("\n Net Work obtained in the Rankine Cycle = %.2f kJ/kg\n",W_net);
fprintf("\n Heat input to the Rankine Cycle = %.2f kJ/kg\n",Q_in);
fprintf("\n The efficiency of the Rankine Cycle = %.2f \n",efficiency);
fprintf("\n Back Work Ratio in the Rankine Cycle = %.9f \n",BWR);
fprintf("\n Specific Steam Consumption in the Rankine Cycle = %.2f kg/kWh \n",SSC);
 
 
%PLOTTING THE CURVES
 
%Plotting Rankine Cycle TS DIAGRAM
%Plotting Saturation Curve
T = linspace(0,373.946,100); % critical point of water occurs at 373.946 degreeC
figure(1);
hold on
for i = 1 : length(T)
    s_l(i) = XSteam('sL_T',T(i));
    s_g(i) = XSteam('sV_T',T(i));
    h_l(i) = XSteam('hL_T',T(i));
    h_g(i) = XSteam('hV_T',T(i));
end
 
plot(s_l,T,'linewidth',3,'color','g');
plot(s_g,T,'linewidth',3,'color','g');
plot([s1 s2],[t1 t2],'linewidth',1,'color','k');
text(s1,t1,'1'); 
plot([s2 s3],[t2 t3],'linewidth',1,'color','r');
text(s2,t2,'2');
plot([s3 s4],[t3 t4],'linewidth',1,'color','k');
text(s3,t3,'3');
plot([s4 s5],[t4 t5],'linewidth',1,'color','r');
text(s4,t4,'4');
plot([s5 s6],[t5 t6],'linewidth',1,'color','r');
text(s5,t5,'5');
plot([s6 s1],[t6 t1],'linewidth',1,'color','r');
text(s6,t6,'6');
xlabel('Entropy (kJ/kg-C)');
ylabel('Temperature (C)');
title('Entropy versus Temperature Curve - Rankine Cycle');
legend('Entropy (s)kJ/kg-C','Temperature (t)C');
grid on
hold off
 
 
%Plotting Rankine Cycle HS DIAGRAM
%Plotting Saturation Curve
figure(2);
hold on
plot(s_l,h_l,'linewidth',3,'color','c');
plot(s_g,h_g,'linewidth',3,'color','c');
plot([s1 s2],[h1 h2],'linewidth',1,'color','k');
text(s1,h1,'1');
plot([s2 s3],[h2 h3],'linewidth',1,'color','r');
text(s2,h2,'2');
plot([s3 s4],[h3 h4],'linewidth',1,'color','k');
text(s3,h3,'3');
plot([s4 s1],[h4 h1],'linewidth',1,'color','r');
text(s4,h4,'4');
xlabel('Entropy (kJ/kg-C)');
ylabel('Enthalpy (kJ/kg)');
title('Enthalpy versus Entropy Curve - Rankine Cycle');
legend('Entropy (s)kJ/kg-C','Enthalpy (h)kJ/kg');
grid on
hold off

EXPLANATION:

• The code begins with the introduction of the Rankine Cycle to the user and mentioning the process and their nature.
• The user is then posed with remarks to feed the data. Which takes in the values of p1,t1 and p2.
• The turbine inlet parameters are defined. Initially, the saturation temperature is found out at pressure p1, by invoking a MATLAb defined function i.e, XSteam.
• If the turbine inlet temperature is higher than the saturation temperature at pressure p1, then, the values of h1 and s1 are invoked using XSteam at the pressure p1 and temperature t1.
• And if the temperature t1 is equal to the saturation temperature then the enthalpy and entropy occur in the vapour region and the values can be invoked by XSteam at the pressure p1.
• These values are computed using the if and else-if conditions.
• If the temperature does not satisfy either of these conditions, then another value of turbine inlet temperature is required to run the Rankine Cycle.
• It is evident from the process that stage 1 to 2 comprises Isentropic Expansion i.e., s2 = s1, in the Turbine where work is produced.
• On the parallel lines as the temperature if the entropy at state 2 is less than the saturated entropy value at the pressure p2 i.e., s2 < ssat2 , then dryness fraction must be considered.
• Entropy of saturated liquid and saturated vapour are calculated using XSteam at the pressure p2.
• The value of dryness fraction is calculated as follows:

• The value of the enthalpy at state 2 is calculated using the dryness fraction and the values of enthalpy hf and hg which are obtained using XSteam function at the pressure p2.
• The temperature t2 corresponds to the saturation temperature at the pressure of p2.
• Otherwise, the dryness fraction is 1. Then, the values of enthalpy and temperature i.e., h2 corresponds to the enthalpy of the saturated liquid at pressure p2 and t2 is the saturated temperature at pressure p2.
• The above values are computed using if and else conditions.
• From the Rankine cycle theory, stage 2 to 3 is isobaric heat rejection and therefore, p3 = p2.
• The pump inlet conditions are calculated in a very straightforward manner of directly invoking XSteam function at pressure p3. The values of temperature(saturation) , specific volume, enthalpy, entropy is calculated.
• The process 3-4 is Isentropic compression which occurs in the pump. In this, the entropy of state 3 and 4 are equal i.e., s4 = s3.
• The process 4-1 is an isobar heat addition process that occurs in the steam generator and therefore, the pressure at state 1 and 4 are the same i.e., p4 = p1.
• The specific volume at the stage 4 is calculated by using XSteam function with value to be calculated at pressure p4 and entropy s4. The same values of pressure and entropy are used to evaluate the values of specific heat at constant pressure and volume at state 4 using XSteam function.
• The adiabatic index is calculated by taking the ration of specific heat at constant pressure to that at constant volume.
• By the means of adiabatic gas law, the temperature of the state 4 is calculated.
• The work done by the pump is calculated using the values v3, p4 and p3.
• Using the work done by the pump, and the known value of h3, the value of enthalpy at state 4 is calculated as mentioned in the Thermodynamic relations section.
• The temperature of the process 4-1 intersects the saturated liquid line on TS curve at 5, where the temperature corresponds to the saturated temperature at p4 and the entropy at state 5 is the entropy of the saturated liquid at p4.
• The same process 4-1 meets the saturated vapour line at point 6 where the temperature remains the same as that at point 5 i.e., t6 = t5. And the entropy at the point 6 corresponds to the entropy of the saturated vapour at p4.
• The values of Work done by turbine, heat input, heat rejected, net-work, efficiency, back work ratio and specific steam consumption are calculated as mentioned in the earlier section.
• To plot the saturation curve in TS diagram, a numerical array T is assigned and using linspace the values range from 0^oC to 373.946^oC (critical point of the water) where in the range is divided into 100 equal parts.
• The saturated values to be plotted on figure(1) origin from a for loop, where a for loop is ran along the length of the numerical temperature array T on the values of entropy and enthalpy at saturated liquid and saturated vapour phases.
• While plotting the saturation curve on TS curve, the values of entropy from the for loop are quoted along with the temperature array separately for saturated liquid and saturated vapour. The linewidth is maintained at 3 and is coloured in green.
• The other points are plotted as mentioned below:

• A legend is written along with labelling the axes. The plot is titled as “Temperature versus Entropy Curve - Rankine Cycle.”
• Similarly, the HS curve is plotted. While plotting the saturation curve on HS curve, the values of enthalpy from the for loop are quoted along with the temperature array separately for saturated liquid and saturated vapour. The linewidth is maintained at 3 and is coloured in cyan.
• The other points are plotted as mentioned below:

• A legend is written along with labelling the axes. The plot is titled as “Enthalpy versus Entropy Curve - Rankine Cycle.”
• The values of the pressure, temperature, enthalpy and entropy are calculated for each state and printed in the command window along with the thermodynamic properties.

 OBSERVATION AND CONCLUSION:

• The code is pretty straight forward and totally in line with the theoretical ideal Rankine cycle.
• It is particularly important to use the XSteam function with appropriate input parameters like pressure, temperature. This will be achieved by thorough understanding of the Rankine Cycle theory.
• The output of the data in the command window is as follows:

RESULT

The TS and HS plots of the Rankine cycle according to the problem statement have been plotted by from an appropriate code written in MATLAB as shown in the figures below. The thermal efficiency of the Rankine cycle is evaluated and was found to be 36.17 % and the back work ratio was found to be 0.00268.

TS plot of Rankine Cycle – MATLAB

HS plot of Rankine Cycle - MATLAB

CONCLUSION

By modifying the above code, reheat and regenerative cycles of Rankine cycle can be simulated.

BIBLIOGRAPHY

1. https://en.wikipedia.org/wiki/Rankine_cycle#:~:text=The%20Rankine%20cycle%20is%20a,work%20while%20undergoing%20phase%20change.
2. http://home.iitk.ac.in/~suller/lectures/lec29.htm