Question

Two different fuels are being considered for a 2.5 MW (net output) heat engine which can operate between the highest temperature produced during the burning of the fuels and an atmospheric temperature of 300 K. Fuel A burns at 2,500 K, delivering 50 MJ/kg (heating value) and costs $2.00/kg. Fuel B burns at 1,500 K, delivering 40 MJ/kg and costs $1.50/kg.

Compare the fuel costs per hour of fuel A and fuel B, assuming that the heat engine operates
(a) at Carnot efficiency
(b) at 40% of Carnot efficiency

Answer 1

Answer:

If the heat engine operates for one hour:

a) the fuel cost at Carnot efficiency for fuel 1 is $409.09 while fuel 2 is $421.88.

b) the fuel cost at 40% of Carnot efficiency for fuel 1 is $1022.73 while fuel 2 is $1054.68.

In both cases the total cost of using fuel 1 is minor, therefore it is recommended to use this fuel over fuel 2. The final observation is that fuel 1 is cheaper.

Explanation:

The Carnot efficiency is obtained as:

$\epsilon_{car}=1-\frac{T_c}{T_H}$

Where $T_c$ is the atmospheric temperature and $T_H$ is the maximum burn temperature.

For the case (B), the efficiency we will use is:

$\epsilon_{b}=0.4\epsilon_{car}$

The work done by the engine can be calculated as:

$W=\epsilon Q=\epsilon H_v\cdot m_{fuel}$ where Hv is the heat value.

If the average net power of the engine is work over time, considering a net power of 2.5MW for 1 hour (3600s), we can calculate the mass of fuel used in each case.

$m=\frac{P\cdot t}{\epsilon H_v}$

If we want to calculate the total fuel cost, we only have to multiply the fuel mass with the cost per kilogram.

$TC=m\cdot c$