How to calculate the thermal efficiency of an Ideal Rankine Cycle using Python and PYroMat
The content of this post is based on the video:
The problem
Consider the following question:
Determine the efficiency of a Rankine cycle using steam as the working fluid in which the condenser pressure is 10 kPa. The boiler pressure is 2 MPa. The steam leaves the boiler as saturated vapor.
Initiate PYroMat and configure its units:
The first thing that we need to do before we start with solving the problem, is to import the necessary packages and configure PYroMat to use the correct units:
import pyromat as pm pm.config["unit_pressure"] = "kPa"
pm.config["def_p"] = 100 mp_water = pm.get("mp.H2O")
To determine the cycle efficiency, we must calculate the turbine work, the pump work, and the heat transfer to the steam in the boiler. We do this by considering a control surface around each of these components in turn. In each case the thermodynamic model is the steam tables, and the process is steady state with negligible changes in kinetic and potential energies.
The pump
First, consider the pump:
#saturated liquid, thus x = 0
p1 = 10 # ←given
T1 = mp_water.Ts(p=p1)[0]
s1 = mp_water.ss(p=p1)[0]
p2 = 2000 # ←given and converted to kPa
s2= s1
v = 1/mp_water.ds(p=p1)[0]w_p = v*(p2-p1)
print(f”Work required by pump: {round(float(w_p),1)} kJ/kg”)
The work required by the pump is 2.0 kJ/kg
After the work required by the pump is calculated, the enthalpy value after the pump (point 2) can be calculated:
h1 = mp_water.hs(p=p1)[0]
h2 = h1+w_p
T2 = mp_water.T_h(p=p2,h=h2)print(f”h2 = {round(float(h2),1)} kJ/kg”)
The enthalpy at point two is 193.8 kJ/kg.
The boiler
Next, lets consider the boiler:
# steam leaves the boiler as saturated vapor, thus x = 1
p3 = p2
T3 = mp_water.Ts(p=p3)h3 = mp_water.hs(p=p3)[1]
s3dash = mp_water.ss(p=p3)[0]
T3dash = T3
s3 = mp_water.ss(p=p3)[1]
q_H = h3-h2print(f”Heat input by boiler: {round(float(q_H),1)} kJ/kg”)
The heat input by the boiler is calculated to be 2604.5 kJ/kg.
The turbine
Now, we consider the turbine:
p4 = p1
s4 = s3
T4, x = mp_water.T_s(s=s4,p=p4, quality=True)
h4 = mp_water.h(p=p4,x=x)
w_t = h3-h4
print(f”Quality of low pressure steam: {round(float(x),4)}”)
print(f”Work generated by turbine: {round(float(w_t),1)} kJ/kg”)The quality of the low pressure steam is 0.7587 and the work generated by the turbine is 791.7 kJ/kg.
The condenser
Finally, we consider the condenser:
q_L = h4-h1
print(f”Heat rejected by the condenser: {round(float(q_L),1)} kJ/kg”)The heat rejected by the condenser is calculated to be 1814.8 kJ/kg.
Thermal efficiency
We can now calculate the thermal efficiency with
eta_th = (w_t-w_p)/q_H*100
print(f”Thermal efficiency is: {round(float(eta_th),1)}%”)
The calculated thermal efficiency is 30.3%.
Plotting the Ideal Rankine Cycle
After all the points have been calculated, the Rankine Cycle can be shown visually on a graph.
import numpy as np
import matplotlib.pyplot as pltp = np.linspace(1,22063,1000)
T = mp_water.Ts(p=p)
s = mp_water.ss(p=p)font = {‘family’ : ‘Times New Roman’,
‘size’ : 22}plt.figure(figsize=(15,10))
plt.title(‘Ideal Rankine Cycle T-s Diagram’)
plt.rc(‘font’, **font)
plt.plot(s[0],T, ‘b — ‘)
plt.plot(s[1],T,’r — ‘)
plt.ylabel(‘Temperature (K)’)
plt.xlabel(‘Entropy (s)’)
plt.plot([s1, s2, s3dash, s3, s4, s1],[T1, T2, T3dash, T3, T4, T1], ‘black’)plt.text(s1-.1,T1,f’(1)\nT = {round(float(T1),2)} K\nh = {round(float(h1),1)} kJ/kg\n s = {round(float(s1),3)} kJ/kgK’,
ha=’right’,backgroundcolor=’white’)
plt.text(1.6,330,f’(2)\nT = {round(float(T2),2)} K\nh = {round(float(h2),1)} kJ/kg \ns = {round(float(s2),3)} kJ/kgK’,
ha=’left’,backgroundcolor=’white’)
plt.text(s3+.1,T3,f’(3)\nT = {round(float(T3),2)} K\nh = {round(float(h3),1)} kJ/kg \ns = {round(float(s3),3)} kJ/kgK’,
ha=’left’,backgroundcolor=’white’)
plt.text(s4+.1,T4,f’(4)\nT = {round(float(T4),2)} K\nh = {round(float(h4),1)} kJ/kg \ns = {round(float(s4),3)} kJ/kgK\nx = {round(float(x),3)}’,
ha=’left’,backgroundcolor=’white’)
Hope you have enjoyed following along with calculating the efficiency of an ideal Rankine Cycle!
