Air-Standard Brayton Cycle Example

Imports

[1]:
from thermostate import State, Q_, units
from numpy import arange
%matplotlib inline
import matplotlib.pyplot as plt

Definitions

[2]:
substance = 'air'
p_1 = Q_(1.0, 'bar')
T_1 = Q_(300.0, 'K')
T_3 = Q_(1700.0, 'K')
p2_p1 = Q_(8.0, 'dimensionless')
p_low = Q_(2.0, 'dimensionless')
p_high = Q_(50.0, 'dimensionless')

Problem Statement

An ideal air-standard Brayton cycle operates at steady state with compressor inlet conditions of 300.0 K and 1.0 bar and a fixed turbine inlet temperature of 1700.0 K and a compressor pressure ratio of 8.0. For the cycle,

  1. determine the net work developed per unit mass flowing, in kJ/kg
  2. determine the thermal efficiency
  3. plot the net work developed per unit mass flowing, in kJ/kg, as a function of the compressor pressure ratio from 2.0 to 50.0
  4. plot the thermal efficiency as a function of the compressor pressure ratio from 2.0 to 50.0
  5. Discuss any trends you find in parts 3 and 4

Solution

1. the net work developed per unit mass flowing

In the ideal Brayton cycle, work occurs in the isentropic compression and expansion. Therefore, the works are

\[\begin{aligned} \frac{\dot{W}_c}{\dot{m}} &= h_1 - h_2 & \frac{\dot{W}_t}{\dot{m}} &= h_3 - h_4 \end{aligned}\]

First, fixing the four states

[3]:
st_1 = State(substance, T=T_1, p=p_1)
h_1 = st_1.h.to('kJ/kg')
s_1 = st_1.s.to('kJ/(kg*K)')

s_2 = s_1
p_2 = p_1*p2_p1
st_2 = State(substance, p=p_2, s=s_2)
h_2 = st_2.h.to('kJ/kg')
T_2 = st_2.T

p_3 = p_2
st_3 = State(substance, p=p_3, T=T_3)
h_3 = st_3.h.to('kJ/kg')
s_3 = st_3.s.to('kJ/(kg*K)')

s_4 = s_3
p_4 = p_1
st_4 = State(substance, p=p_4, s=s_4)
h_4 = st_4.h.to('kJ/kg')
T_4 = st_4.T

Summarizing the states,

State T p h s
1 300.00 K 1.00 bar 426.30 kJ/kg 3.89 kJ/(K kg)
2 540.13 K 8.00 bar 670.65 kJ/kg 3.89 kJ/(K kg)
3 1700.00 K 8.00 bar 2007.09 kJ/kg 5.19 kJ/(K kg)
4 1029.42 K 1.00 bar 1206.17 kJ/kg 5.19 kJ/(K kg)

Then, the net work is calculated by

[4]:
W_c = h_1 - h_2
W_t = h_3 - h_4
W_net = W_c + W_t

Answer: The works are \(\dot{W}_c/\dot{m} =\) -244.35 kJ/kg, \(\dot{W}_t/\dot{m} =\) 800.92 kJ/kg, and \(\dot{W}_{net}/\dot{m} =\) 556.57 kJ/kg

2. the thermal efficiency

[5]:
Q_23 = h_3 - h_2
eta = W_net/Q_23

Answer: The thermal efficiency is \(\eta =\) 0.42 = 41.65%

3. and 4. plot the net work per unit mass flowing and thermal efficiency

[6]:
eta_l = []
W_net_l = []
p_range = arange(p_low, p_high, 1)
for p_ratio in p_range:
    s_2 = s_1
    p_2 = p_1*p_ratio
    st_2 = State(substance, p=p_2, s=s_2)
    h_2 = st_2.h.to('kJ/kg')
    T_2 = st_2.T

    p_3 = p_2
    st_3 = State(substance, p=p_3, T=T_3)
    h_3 = st_3.h.to('kJ/kg')
    s_3 = st_3.s.to('kJ/(kg*K)')

    s_4 = s_3
    p_4 = p_1
    st_4 = State(substance, p=p_4, s=s_4)
    h_4 = st_4.h.to('kJ/kg')
    T_4 = st_4.T

    W_c = h_1 - h_2
    W_t = h_3 - h_4
    W_net = W_c + W_t
    W_net_l.append(W_net.magnitude)

    Q_23 = h_3 - h_2
    eta = W_net/Q_23
    eta_l.append(eta.magnitude)
/home/travis/miniconda/envs/py37/lib/python3.7/site-packages/pint/quantity.py:1377: UnitStrippedWarning: The unit of the quantity is stripped.
  warnings.warn("The unit of the quantity is stripped.", UnitStrippedWarning)
[7]:
fig, work_ax = plt.subplots()
work_ax.plot(p_range, W_net_l, label='Net work per unit mass flowing', color='C0')
eta_ax = work_ax.twinx()
eta_ax.plot(p_range, eta_l, label='Thermal efficiency', color='C1')
work_ax.set_xlabel('Pressure ratio $p_2/p_1$')
work_ax.set_ylabel('Net work per unit mass flowing (kJ/kg)')
eta_ax.set_ylabel('Thermal efficiency')
lines, labels = work_ax.get_legend_handles_labels()
lines2, labels2 = eta_ax.get_legend_handles_labels()
work_ax.legend(lines + lines2, labels + labels2, loc='best');
_images/air-standard-brayton-cycle-example_18_0.png

We note from this graph that the thermal efficiency of the cycle increases continuously as the pressure ratio increases. However, because there is a fixed turbine inlet temperature, the work per unit mass flowing has a maximum around \(p_2/p_1\) = 20.