Ideal Rankine Cycle
===================

.. post:: Apr 21, 2020
   :exclude:
   :tags: tutorial, thermodynamics, rankine-cycle, PYroMat
   :author: adriaan

How to calculate the thernal efficiency of an Ideal Rankine Cycle using Python and PYroMat

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:

.. code-block:: python
   
   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:

.. code-block:: python

   #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")

.. code-block:: bash

   Work required by pump: 2.0 kJ/kg
  

After the work required by the pump is calculated, the enthalpy value after the pump (point 2) can be calculated:

.. code-block:: python

   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")

.. code-block:: bash

   h2 = 193.8 kJ/kg
   

The Boiler
----------

Next, lets consider the boiler:


.. code-block:: python

   # 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-h2

   print(f"Heat input by boiler: {round(float(q_H),1)} kJ/kg")


.. code-block:: bash

   Heat input by boiler: 2604.5 kJ/kg


The Turbine
-----------

Now, we consider the turbine:


.. code-block:: python

   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")


.. code-block:: bash

   Quality of low pressure steam: 0.7587
   Work generated by turbine: 791.7 kJ/kg
    

The Condenser
-------------

Finally, we consider the condenser:


.. code-block:: python

   q_L = h4-h1
   print(f"Heat rejected by the condenser: {round(float(q_L),1)} kJ/kg")


.. code-block:: bash

   Heat rejected by the condenser: 1814.8 kJ/kg
  

Thermal Efficiency
------------------

We can now calculate the thermal efficiency with :math:`\eta_{th}=\frac{w_{net}}{q_H}`


.. code-block:: python

   eta_th = (w_t-w_p)/q_H*100
   print(f"Thermal efficiency is: {round(float(eta_th),1)}%")


.. code-block:: bash

   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. 

.. code-block:: python

   import numpy as np
   import matplotlib.pyplot as plt

   p = 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')


.. figure:: /images/ideal-rankine-cycle-output.png
   :width: 80%