Consider an ideal gas Joule cycle, also called the Brayton cycle, shown below. Find the formula

Consider an ideal gas Joule cycle, also called the Brayton...

Key Concepts

Brayton (Joule) Cycle

The Brayton cycle, also known as the Joule cycle, is a thermodynamic cycle that underpins the operation of gas turbine engines. It consists of two adiabatic (no heat transfer) processes and two constant-pressure processes, which together model the compression, heat addition, expansion, and heat rejection stages of the engine.

Adiabatic Process

An adiabatic process is one in which there is no exchange of heat with the surroundings. In the context of an ideal gas, this results in a relationship among pressure, volume, and temperature described by PV^? = constant. Such processes are critical in the Brayton cycle for both compression and expansion phases.

Specific Heat Ratio (?)

The specific heat ratio, ?, is defined as the ratio of the specific heat at constant pressure to the specific heat at constant volume. This parameter determines how pressure and temperature change during adiabatic processes and directly influences the efficiency of the Brayton cycle.

Pressure Ratio

The pressure ratio, often defined as the ratio of the maximum pressure to the minimum pressure in the cycle, is a key factor in determining the performance of a gas turbine engine. It affects the temperature and work interactions during the cycle and is central to expressing the cycle’s efficiency in theoretical models.

Cycle Efficiency

Cycle efficiency measures the effectiveness of a thermodynamic cycle in converting heat into work. In ideal cycles like the Brayton cycle, efficiency can be explicitly expressed in terms of the pressure ratio and the specific heat ratio. This formulation highlights how changes in these parameters can maximize or limit the energy conversion performance of the cycle.

Transcript

00:02 In the question as ideal gas jule cycle, so there are path 1 to 2 as adiabetic process.

00:25 Then path 2 to 3 is as isobaric process.

00:38 Now path 3 to 4 as adiabetic expansion expansion process and path 4 to 1 is an isobaric process so so we know that efficiency of any heat engine is given by as e is equal to w upon q in and we know that the work done by a heat engine is given why w is equal to q of in minus q of out so we take this equation as first now take it as a second we will further use this equation to simplify and get the desired answer so put value of w 2 into 1 so we get simply e is equal to q input minus q output divided by q input input now e would come as 1 minus q output upon q input as equation 3 so we know that in adiabetic process in adiabetic process we know that there is no heat exchange so it means the heat added to the system q input will be added to an isoboric process in path 2 to 3 so here q of input would be equal to n cp the molar heat constant at constant pressure delta t difference from 2 to 3 now q of input will be equal to as n cp t 3 is a temperature at terminal 3 and t 2 at terminal 2 so now putting this equation as 4 the lost of system will be removed at an isobaric process at path 4 to a 4 to 1 4 to 1 so it means that q of output is equal to n cp delta t 4 to 1 now q of output will be equal to n cp t 4 minus t 1 now take it as a equation 5 so putting the value of q in and q output from 4 to 5 in equation 3 we get as e is equal to 1 minus n cp t 4 minus t 1 divided by n cp t 3 minus t 2 so e would come as 1 minus t 4 minus t1 upon t3 minus t 2 so this is the expression now further we know that for ideal gas law as pv is equal to n r t where n are the number of moles and r is constant p v and t the relationship so therefore t1 is equal to p b v1 upon nr whereas t 4 is equal to p4 v4 upon nr t 2 is equal to p3 v3 upon nr and r and 2 is equal to p2 v2 upon nr and t 3 is equal to p3 v3 upon nr now, substituting these all four equations of t1, t2, t3 and t4 into equation 1, into equation 6.

06:35 Now, e would come as 1 minus a big bracket as p4b4 upon nr minus p1 v1 upon nr minus p1 v1 upon nr...

Please give Ace some feedback