A gas turbine operates under the Brayton cycle, which is...

A gas turbine operates under the Brayton cycle, which is depicted in the $P V$ diagram of Fig. $28 .$ In process ab the air-fuel mixture undergoes an adiabatic compression. This is followed, in process bc, with an isobaric (constant pressure) heating, by combustion. Process cd is an adiabatic expansion with expulsion of the products to the atmosphere. The return step, da, takes place at constant pressure. If the working gas behaves like an ideal gas, show that the efficiency of the Brayton cycle is $$e=1-\left(\frac{P_{\mathrm{b}}}{P_{\mathrm{a}}}\right)^{\frac{1-\gamma}{\gamma}}$$

A gas turbine operates under the Brayton cycle, which is depicted in the $P V$ diagram of Fig. $28 .$ In process ab the air-fuel mixture undergoes an adiabatic compression. This is followed, in process bc, with an isobaric (constant pressure) heating, by combustion. Process cd is an adiabatic expansion with expulsion of the products to the atmosphere. The return step, da, takes place at constant pressure. If the working gas behaves like an ideal gas, show that the efficiency of the Brayton cycle is
$$e=1-\left(\frac{P_{\mathrm{b}}}{P_{\mathrm{a}}}\right)^{\frac{1-\gamma}{\gamma}}$$

Key Concepts

Ideal Gas Behavior and Specific Heat Ratio (?)

Ideal gas behavior assumes no intermolecular forces and that the gas obeys the ideal gas law (PV = nRT). The specific heat ratio, ? (gamma), defined as the ratio of specific heat at constant pressure to that at constant volume, is critical in determining the temperature changes during adiabatic processes. This parameter directly influences the performance and efficiency calculations of thermodynamic cycles.

Brayton Cycle

The Brayton cycle is a thermodynamic cycle that models the operation of gas turbines and jet engines. It consists of four processes: adiabatic compression, isobaric heat addition, adiabatic expansion, and isobaric heat rejection. Understanding the cycle is key to analyzing systems that operate under constant pressure conditions during heat exchange and to evaluating the overall performance of gas turbines.

Adiabatic Process

An adiabatic process is one in which no heat is transferred to or from the working fluid. In this process, changes in pressure and volume lead directly to changes in temperature. The adiabatic relations, such as PV^? = constant for an ideal gas, are central in analyzing the compression and expansion parts of the Brayton cycle.

Isobaric Process

An isobaric process occurs at constant pressure, meaning any heat addition or rejection directly changes the temperature and volume of the working fluid. In the Brayton cycle, the heat addition during combustion and the subsequent heat rejection occur under constant pressure, which simplifies the analysis of energy transfer and work.

Thermodynamic Efficiency of Cycles

The efficiency of a thermodynamic cycle represents the proportion of heat input that is converted into useful work. For the Brayton cycle, efficiency is expressed in terms of temperature or pressure ratios derived from the adiabatic relations of the working fluid. This relationship shows how improvements in component design and process control can optimize the performance of gas turbine systems.

Transcript

00:01 We need to find the efficiency of the brayton cycle.

00:03 So we know that heat transfer occurs only along the isobaric processes.

00:24 So in this case, the isobaric processes would be from b to c and from a to d, or rather from d to a.

00:30 And we know that the heat transfer doesn't occur from a to b or from c to d because again, these are adiabatic expansions.

00:39 So given this, we can say that, that q sub h is going to be equal to, rather the heat input would be equal to q sub b, c, which would be equal to n, the number of moles times the specific heat, rather the molar heat capacity at a constant pressure times the change in temperature.

01:01 So t sub c minus the temperature at b.

01:05 The heat exhaust is going to be equal to the heat transfer associated with process d .a.

01:12 And this is going to to be equal to the number of moles times the molar heat capacity at a constant pressure, times the difference between the temperature at d minus the temperature at a.

01:24 And so finally, we also know that the efficiency is going to be equal to 1 minus q sub l divided by q sub h.

01:33 So at this point, we can say that e is going to be equal to 1 minus n sub c p times t sub d, minus t sub a divided by n sub c p t sub c minus t sub b these are going to cancel out and we finally have this expression now we're going to use the ideal gas law in order to substitute for these and say that e is now going to be equal to one minus p sub d v sub d minus p sub a v sub a divided by p sub c v sub c minus p sub b b b this is going to equal furthermore one minus p sub a times v sub d minus v sub a divided by p sub b v v sub b v v c minus v sub b this is we can cancel out we other we can factor out a p a pressure sub a and a pressure sub b because these are again i so barric processes, which means that these have a constant pressure.

02:48 Now, we are going to say that process a, b is, of course, adiabatic as it's labeled.

03:00 And we can say that p sub a, v .a to the gamma power would be equal to p sub b, v sub b to the gamma power.

03:08 So we can say that v.

03:09 A is going to equal v.

03:11 Sub b times p sub b, divided by p sub a, to the one over gamma power.

03:19 The same exact thing for process cd.

03:25 Process cd is also adiabatic...