Review: a simple Brayton cycle engine
Adiabats
pVγ = const.
http://en.wikipedia.org/wiki/Brayton_cycle
R-R Nene, Hamilton Air Museum (CEW photo)
A really simple version:
P0 ,T0
h
P0
frictionless,
massless piston
area A
mass m
air,
volume V
,, heat
P1 = P0 ; T1 = T0
Stage 1->2: weight of mass compresses air adiabatically
P2 = P1 + mg/A ; V2 = V1 (P1/P2 )1/γ ; T2 =T1 (V1/V2 )γ-1; γ ≈ 1.4 for diatomic gases like air
Stage 2->3: heat air isobarically, raise mass
V3 = V2 + Ah ; P3 = P2 ; T3 =T2 (V3/V2 )
Stage 3->4: piston shoots up (adiabatically) when mass removed, to reduce pressure in cylinder
to ambient
V4 = V3 (P3/P4 )1/γ; T4 =T3 (V3/V4 )γ-1
Stage 4->1: cool air isobarically, bring down piston
T1 =T4 (V1/V4 )
Calculate efficiency η in the ideal limit that only the enclosed air needs to be heated.
Useful work, W = mgh
Heat in Qin = ρ1V1cp(T3-T2) ;
The mass of air is ρ1V1, and the specific heat capacity of air at constant pressure (important –
see later) is cp. Note cp is sometimes defined per unit mass, and sometimes per mole.
η = W/Qin
Compare with the theoretical value: ηthy = 1 – T1/T2
A note on estimating real efficiencies
Consider my 2005 Honda Civic: on many long road trips up and down Highway 97, at an average
speed of 90 km/h, this car has consistently demonstrated a fuel economy of 5.8 L/100 km.
What is the efficiency of the engine+transmission?
The thermal power from the fuel can be found from the enthalpy of gasoline, which is typically
36 MJ/L the measured fuel economy, and the time to travel 100 km, i.e. 4000 s.
𝑃𝑡ℎ = (36 MJ/L)(5.8 L/4000 s) = 0.052 MW = 52 kW
The mechanical power can be found by estimating the total drag force and multiplying by the
speed. The drag force arises from rolling friction of the tires on the road, 𝐹𝑟 , and the
aerodynamic drag 𝐹𝑎 .
𝐹𝑟 = 𝜇𝑟 𝑚𝑔
Here 𝜇𝑟 is the rolling friction coefficient, typically 0.012 or thereabouts, and 𝑚𝑔 is the weight of
the car (𝑚 ≈ 1500 kg with two occupants and luggage).
𝐹𝑎 =
1
𝐶 𝐴𝜌𝑣 2
2 𝐷
Here 𝐶𝐷 𝐴 is the drag coefficient multiplied by the frontal area of the vehicle; these quantities
are usually quoted as a product because that is the only thing that matters. The quoted value
for a Civic (at least according to Wikipedia) is 0.654 m2. The density of air is 𝜌 ≈ 1.2 kg/m3.
Using these nominal values, 𝐹𝑟 = 176 N, and 𝐹𝑎 = 204 N, i.e. about the same. There would be
not much gain in working hard to reduce one while leaving the other untouched, and 𝐶𝐷 𝐴 is
hard to shift because of the human “form factor”.
𝑃𝑚𝑒𝑐ℎ = (𝐹𝑟 + 𝐹𝑎 )𝑣 = 9.5 kW (13 HP for the devotees of Top Gear)
The mechanical efficiency is the ratio of the two powers.
𝜂=
𝑃𝑚𝑒𝑐ℎ
≈ 0.18
𝑃𝑡ℎ
Given the uncertainties in the numbers, say 20%.
Tutorial
Write some MATLAB code (or Python, if you wish) to produce a PV diagram for the following
case:
A vertical cylinder of air has a massless, frictionless piston, area 100 cm2, sitting on 1 L of air at
20 C. The ambient pressure is 100 kPa. Place a 100 kg mass on the piston, and heat to increase
the volume to 2 L, then remove the mass. Calculate the uselful work done in raising the mass,
and thus the efficiency in terms of the work done and the required heat input, as derived from
your code output (say how you arrived at this answer) and compare to the theoretical Brayton
efficiency.
Homework (due start of class Thursday May 19th)
(a) Your solutions to questions 2 and 3 from Tuesday May 10th
(b) Properly labelled PV diagram* from the code you started writing today, with results of
the efficiency calculations, including a summary of how these numbers were derived,
with your code cut-and-pasted in at the end.
*i.e. quantities and units on axes, and the data annotated with the four stages of the cycle.
Hand in your homework on paper at the start of Thursday’s class.