Thermal Engineering

Example: Isentropic Expansion in Gas Turbine

Assume an isentropic expansion of helium (3 → 4) in a gas turbine. Since helium behaves almost as an ideal gas, use the ideal gas law to calculate outlet temperature of the gas (T_4,is). In this turbines the high-pressure stage receives gas (point 3 at the figure; p₃ = 6.7 MPa; T₃ = 1190 K (917°C)) from a heat exchanger and exhaust it to another heat exchanger, where the outlet pressure is p₄ = 2.78 MPa (point 4).

Solution:

The outlet temperature of the gas, T_4,is, can be calculated using p, V, T Relation for isentropic process (reversible adiabatic process):

In this equation the factor for helium is equal to κ=c_p/c_v=1.66. From the previous equation follows that the outlet temperature of the gas, T_4,is, is:

See also: First Law of Thermodynamics

See also: Ideal Gas Law

See also: What is Enthalpy

Example: Isentropic Expansion in Gas Turbine

Let assume the ideal Brayton cycle that describes the workings of a constant pressure heat engine. Modern gas turbine engines and airbreathing jet engines also follow the Brayton cycle.

Ideal Brayton cycle consist of four thermodynamic processes. Two isentropic processes and two isobaric processes.

1. isentropic compression – ambient air is drawn into the compressor, where it is pressurized (1 → 2). The work required for the compressor is given by W_C = H₂ – H₁.
2. isobaric heat addition – the compressed air then runs through a combustion chamber, where fuel is burned and air or another medium is heated (2 → 3). It is a constant-pressure process, since the chamber is open to flow in and out. The net heat added is given by Q_add = H₃ – H₂
3. isentropic expansion – the heated, pressurized air then expands on turbine, gives up its energy. The work done by turbine is given by W_T = H₄ – H₃
4. isobaric heat rejection – the residual heat must be rejected in order to close the cycle. The net heat rejected is given by Q_re = H₄ – H₁

As can be seen, we can describe and calculate (e.g. thermal efficiency) such cycles (similarly for Rankine cycle) using enthalpies.

See also: Thermal Efficiency of Brayton Cycle

See also:

Isentropic Process

Isentropic Efficiency – Turbine, Compressor, Nozzle

In previous chapters we assumed that the gas expansion is isentropic and therefore we used T_4,is  as the outlet temperature of the gas. These assumptions are only applicable with ideal cycles.

Most steady-flow devices (turbines, compressors, nozzles) operate under adiabatic conditions, but they are not truly isentropic but are rather idealized as isentropic for calculation purposes. We define parameters η_T,  η_C, η_N, as a ratio of real work done by device to work by device when operated under isentropic conditions (in case of turbine). This ratio is known as the Isentropic Turbine/Compressor/Nozzle Efficiency.

These parameters describe how efficiently a turbine, compressor or nozzle approximates a corresponding isentropic device. This parameter reduces the overall efficiency and work output. For turbines, the value of η_T is typically 0.7 to 0.9 (70–90%).

Example: Isentropic Turbine Efficiency

Assume an isentropic expansion of helium (3 → 4) in a gas turbine. In this turbines the high-pressure stage receives gas (point 3 at the figure; p₃ = 6.7 MPa; T₃ = 1190 K (917°C)) from a heat exchanger and exhaust it to another heat exchanger, where the outlet pressure is p₄ = 2.78 MPa (point 4). The temperature (for isentropic process) of the gas at the exit of the turbine is T_4s = 839 K (566°C).

Calculate the work done by this turbine and calculate the real temperature at the exit of the turbine, when the isentropic turbine efficiency is η_T = 0.91 (91%).

Solution:

From the first law of thermodynamics, the work done by turbine in an isentropic process can be calculated from:

W_T = h₃ – h_4s     →     W_Ts = c_p (T₃ – T_4s)

From Ideal Gas Law we know, that the molar specific heat of a monatomic ideal gas is:

C_v = 3/2R = 12.5 J/mol K and C_p = C_v + R = 5/2R = 20.8 J/mol K

We transfer the specific heat capacities into units of J/kg K via:

c_p = C_p . 1/M (molar weight of helium) = 20.8 x 4.10^-3 = 5200 J/kg K

The work done by gas turbine in isentropic process is then:

W_T,s = c_p (T₃ – T_4s) = 5200 x (1190 – 839) = 1.825 MJ/kg

The real work done by gas turbine in adiabatic process is then:
W_T,real = c_p (T₃ – T_4s) . η_T = 5200 x (1190 – 839) x 0.91 = 1.661 MJ/kg

See also:

Isentropic Process

Example: Isentropic Turbine Efficiency

Assume an isentropic expansion of helium (3 → 4) in a gas turbine. In this turbines the high-pressure stage receives gas (point 3 at the figure; p₃ = 6.7 MPa; T₃ = 1190 K (917°C)) from a heat exchanger and exhaust it to another heat exchanger, where the outlet pressure is p₄ = 2.78 MPa (point 4). The temperature (for isentropic process) of the gas at the exit of the turbine is T_4s = 839 K (566°C).

Calculate the work done by this turbine and calculate the real temperature at the exit of the turbine, when the isentropic turbine efficiency is η_T = 0.91 (91%).

Solution:

From the first law of thermodynamics, the work done by turbine in an isentropic process can be calculated from:

W_T = h₃ – h_4s     →     W_Ts = c_p (T₃ – T_4s)

From Ideal Gas Law we know, that the molar specific heat of a monatomic ideal gas is:

C_v = 3/2R = 12.5 J/mol K and C_p = C_v + R = 5/2R = 20.8 J/mol K

We transfer the specific heat capacities into units of J/kg K via:

c_p = C_p . 1/M (molar weight of helium) = 20.8 x 4.10^-3 = 5200 J/kg K

The work done by gas turbine in isentropic process is then:

W_T,s = c_p (T₃ – T_4s) = 5200 x (1190 – 839) = 1.825 MJ/kg

The real work done by gas turbine in adiabatic process is then:
W_T,real = c_p (T₃ – T_4s) . η_T = 5200 x (1190 – 839) x 0.91 = 1.661 MJ/kg

See also:

Isentropic Process

Adiabitic vs. Isentropic Process

In comparison to the isentropic process in which the entropy of the fluid or gas remains constant, in the adiabatic process the entropy changes. Therefore the adiabatic process is considered to be irreversible process. The isentropic process is a special case of an adiabatic process. The isentropic process is a reversible adiabatic process. An isentropic process can also be called a constant entropy process. In engineering such an idealized process is very useful for comparison with real processes.

One way to make real processes approximate reversible process is to carry out the process in a series of small or infinitesimal steps or infinitely slowly, so that the process can be considered as a series of equilibrium states. For example, heat transfer may be considered reversible if it occurs due to a small temperature difference between the system and its surroundings. But real processes are not done infinitely slowly. Reversible processes are a useful and convenient theoretical fiction, but do not occur in nature. For example, there could be turbulence in the gas. Therefore, heat engines must have lower efficiencies than limits on their efficiency due to the inherent irreversibility of the heat engine cycle they use.

See also: First Law of Thermodynamics

See also: Ideal Gas Law

See also: What is Enthalpy

Isentropic Efficiency – Turbine, Compressor, Nozzle

In previous chapters we assumed that the gas expansion is isentropic and therefore we used T_4,is  as the outlet temperature of the gas. These assumptions are only applicable with ideal cycles.

Most steady-flow devices (turbines, compressors, nozzles) operate under adiabatic conditions, but they are not truly isentropic but are rather idealized as isentropic for calculation purposes. We define parameters η_T,  η_C, η_N, as a ratio of real work done by device to work by device when operated under isentropic conditions (in case of turbine). This ratio is known as the Isentropic Turbine/Compressor/Nozzle Efficiency.

These parameters describe how efficiently a turbine, compressor or nozzle approximates a corresponding isentropic device. This parameter reduces the overall efficiency and work output. For turbines, the value of η_T is typically 0.7 to 0.9 (70–90%).

See also:

Adiabatic Process

Adiabatic Process

An adiabatic process is a thermodynamic process, in which there is no heat transfer into or out of the system (Q = 0). The system can be considered to be perfectly insulated. In an adiabatic process, energy is transferred only as work. The assumption of no heat transfer is very important, since we can use the adiabatic approximation only in very rapid processes. In these rapid processes, there is not enough time for the transfer of energy as heat to take place to or from the system.

In real devices (such as turbines, pumps, and compressors) heat losses and losses in the combustion process occur, but these losses are usually low in comparison to overall energy flow and we can approximate some thermodynamic processes by the adiabatic process.

See also: First Law of Thermodynamics

See also: Ideal Gas Law

See also: What is Enthalpy

Adiabatic Expansion – Adiabatic Compression

See also: What is an Ideal Gas

In an ideal gas, molecules have no volume and do not interact. According to the ideal gas law, pressure varies linearly with temperature and quantity, and inversely with volume.

pV = nRT

where:

• p is the absolute pressure of the gas
• n is the amount of substance
• T is the absolute temperature
• V is the volume
• R  is the ideal, or universal, gas constant, equal to the product of the Boltzmann constant and the Avogadro constant,

In this equation the symbol R is a constant called the universal gas constant that has the same value for all gases—namely, R =  8.31 J/mol K.

The adiabatic process can be expressed with the ideal gas law as:

pV^κ = constant

or

p₁V₁^κ = p₂V₂^κ

in which κ = c_p/c_v is the ratio of the specific heats (or heat capacities) for the gas. One for constant pressure (c_p) and one for constant volume (c_v). Note that, this ratio κ  = c_p/c_v is a factor in determining the speed of sound in a gas and other adiabatic processes.

Other p, V, T Relation

On a p-V diagram, the process occurs along a line (called an adiabat) that has the equation p = constant / V^κ. For an ideal gas and a polytropic process, the case n = κ  corresponds to an adiabatic process.

Example of Adiabatic Expansion

Assume an adiabatic expansion of helium (3 → 4) in a gas turbine. Since helium behaves almost as an ideal gas, use the ideal gas law to calculate outlet temperature of the gas (T_4,real). In this turbines the high-pressure stage receives gas (point 3 at the figure; p₃ = 6.7 MPa; T₃ = 1190 K (917°C)) from a heat exchanger and exhaust it to another heat exchanger, where the outlet pressure is p₄ = 2.78 MPa (point 4).

Solution:

The outlet temperature of the gas, T_4,real, can be calculated using p, V, T Relation for adiabatic process. Note that, it is the same relation as for the isentropic process, therefore results must be identical. It this case, we calculate the expansion for different gas turbine (less efficient) as in case of Isentropic Expansion in Gas Turbine.

In this equation the factor for helium is equal to κ=c_p/c_v=1.66. From the previous equation follows that the outlet temperature of the gas, T_4,real, is:

See also: Mayer’s relation

See also:

Adiabatic Process

Example of Adiabatic Expansion

Assume an adiabatic expansion of helium (3 → 4) in a gas turbine. Since helium behaves almost as an ideal gas, use the ideal gas law to calculate outlet temperature of the gas (T_4,real). In this turbines the high-pressure stage receives gas (point 3 at the figure; p₃ = 6.7 MPa; T₃ = 1190 K (917°C)) from a heat exchanger and exhaust it to another heat exchanger, where the outlet pressure is p₄ = 2.78 MPa (point 4).

Solution:

The outlet temperature of the gas, T_4,real, can be calculated using p, V, T Relation for adiabatic process. Note that, it is the same relation as for the isentropic process, therefore results must be identical. It this case, we calculate the expansion for different gas turbine (less efficient) as in case of Isentropic Expansion in Gas Turbine.

In this equation the factor for helium is equal to κ=c_p/c_v=1.66. From the previous equation follows that the outlet temperature of the gas, T_4,real, is:

See also: Mayer’s relation

See also: First Law of Thermodynamics

See also: Ideal Gas Law

See also: What is Enthalpy

Adiabatic Process in Gas Turbine

Let assume the Brayton cycle that describes the workings of a constant pressure heat engine. Modern gas turbine engines and airbreathing jet engines also follow the Brayton cycle.

The Brayton cycle consist of four thermodynamic processes. Two adiabatic processes and two isobaric processes.

1. adiabatic compression – ambient air is drawn into the compressor, where it is pressurized (1 → 2). The work required for the compressor is given by W_C = H₂ – H₁.
2. isobaric heat addition – the compressed air then runs through a combustion chamber, where fuel is burned and air or another medium is heated (2 → 3). It is a constant-pressure process, since the chamber is open to flow in and out. The net heat added is given by Q_add = H₃ – H₂
3. adiabatic expansion – the heated, pressurized air then expands on turbine, gives up its energy. The work done by turbine is given by W_T = H₄ – H₃
4. isobaric heat rejection – the residual heat must be rejected in order to close the cycle. The net heat rejected is given by Q_re = H₄ – H₁

As can be seen, we can describe and calculate (e.g. thermal efficiency) such cycles (similarly for Rankine cycle) using enthalpies.

See also: Thermal Efficiency of Brayton Cycle

See also:

Adiabatic Process

Adiabatic Process

An adiabatic process is a thermodynamic process, in which there is no heat transfer into or out of the system (Q = 0). The system can be considered to be perfectly insulated. In an adiabatic process, energy is transferred only as work. The assumption of no heat transfer is very important, since we can use the adiabatic approximation only in very rapid processes. In these rapid processes, there is not enough time for the transfer of energy as heat to take place to or from the system.

In real devices (such as turbines, pumps, and compressors) heat losses and losses in the combustion process occur, but these losses are usually low in comparison to overall energy flow and we can approximate some thermodynamic processes by the adiabatic process.

Free Expansion – Joule Expansion

These are adiabatic processes in which no transfer of heat occurs between the system and its environment and no work is done on or by the system. These types of adiabatic processes are called free expansion. It is an irreversible process in which a gas expands into an insulated evacuated chamber. It is also called Joule expansion. For an ideal gas, the temperature doesn’t change (see: Joule’s Second Law), however, real gases experience a temperature change during free expansion. In free expansion Q = W = 0, and the first law requires that:

dE_int = 0

A free expansion can not be plotted on a P-V diagram, because the process is rapid, not quasistatic. The intermediate states are not equilibrium states, and hence the pressure is not clearly defined.

See also:

Adiabatic Process

Adiabatic Process

An adiabatic process is a thermodynamic process, in which there is no heat transfer into or out of the system (Q = 0). The system can be considered to be perfectly insulated. In an adiabatic process, energy is transferred only as work. The assumption of no heat transfer is very important, since we can use the adiabatic approximation only in very rapid processes. In these rapid processes, there is not enough time for the transfer of energy as heat to take place to or from the system.

In real devices (such as turbines, pumps, and compressors) heat losses and losses in the combustion process occur, but these losses are usually low in comparison to overall energy flow and we can approximate some thermodynamic processes by the adiabatic process.

Adiabatic Curve – Adiabat

See also: What is an Ideal Gas

In an ideal gas, molecules have no volume and do not interact. According to the ideal gas law, pressure varies linearly with temperature and quantity, and inversely with volume.

pV = nRT

where:

• p is the absolute pressure of the gas
• n is the amount of substance
• T is the absolute temperature
• V is the volume
• R  is the ideal, or universal, gas constant, equal to the product of the Boltzmann constant and the Avogadro constant,

In this equation the symbol R is a constant called the universal gas constant that has the same value for all gases—namely, R =  8.31 J/mol K.

The adiabatic process can be expressed with the ideal gas law as:

pV^κ = constant

or

p₁V₁^κ = p₂V₂^κ

in which κ = c_p/c_v is the ratio of the specific heats (or heat capacities) for the gas. One for constant pressure (c_p) and one for constant volume (c_v). Note that, this ratio κ  = c_p/c_v is a factor in determining the speed of sound in a gas and other adiabatic processes.

Other p, V, T Relation

On a p-V diagram, the process occurs along a line (called an adiabat) that has the equation p = constant / V^κ. For an ideal gas and a polytropic process, the case n = κ  corresponds to an adiabatic process.

See also:

Thermodynamic Processes

Isothermal Process

An isothermal process is a thermodynamic process, in which the temperature of the system remains constant (T = const). The heat transfer into or out of the system typically must happen at such a slow rate in order to continually adjust to the temperature of the reservoir through heat exchange. In each of these states the thermal equilibrium is maintained.

For an ideal gas and a polytropic process, the case n = 1 corresponds to an isothermal (constant-temperature) process. In contrast to adiabatic process , in which n = κ  and a system exchanges no heat with its surroundings (Q = 0; ∆T≠0), in an isothermal process there is no change in the internal energy (due to ∆T=0) and therefore ΔU = 0 (for ideal gases) and Q ≠ 0. An adiabatic process is not necessarily an isothermal process, nor is an isothermal process necessarily adiabatic.

In engineering, phase changes, such as evaporation or melting, are isothermal processes when, as is usually the case, they occur at constant pressure and temperature.

Isothermal Process and the First Law

The classical form of the first law of thermodynamics is the following equation:

dU = dQ – dW

In this equation dW is equal to dW = pdV and is known as the boundary work.

In isothermal process and the ideal gas, all heat added to the system will be used to do work:

Isothermal process (dU = 0):

dU = 0 = Q – W    →     W = Q      (for ideal gas)

Isothermal Expansion – Isothermal Compression

See also: What is an Ideal Gas

In an ideal gas, molecules have no volume and do not interact. According to the ideal gas law, pressure varies linearly with temperature and quantity, and inversely with volume.

pV = nRT

where:

• p is the absolute pressure of the gas
• n is the amount of substance
• T is the absolute temperature
• V is the volume
• R  is the ideal, or universal, gas constant, equal to the product of the Boltzmann constant and the Avogadro constant,

In this equation the symbol R is a constant called the universal gas constant that has the same value for all gases—namely, R =  8.31 J/mol K.

The isothermal process can be expressed with the ideal gas law as:

pV = constant

or

p₁V₁ = p₂V₂

On a p-V diagram, the process occurs along a line (called an isotherm) that has the equation p = constant / V.

Boyle–Mariotte Law

Boyle-Mariotte Law is one of the gas laws. At the end of the 17th century, Robert William Boyle and Edme Mariotte independently studied the relationship between the volume and pressure of a gas at constant temperature. The results of certain experiments with gases at relatively low pressure led Robert Boyle to formulate a well-known law. It states that:

For a fixed mass of gas at constant temperature, the volume is inversely proportional to the pressure.

That means that, for example, if you increase the volume 10 times, the pressure will decrease 10 times. If you halve the volume, you will double the pressure.

You can express this mathematically as:

pV = constant

or

p₁V₁ = p₂V₂

Yes, it seems to be identical as isothermal process of ideal gas. In fact, during their experiments the temperature remain constant as was assumed by Mariotte. These results are fully consistent with ideal gas law, which determinates, that the constant is equal to nRT.

pV = nRT

where:

• p is the absolute pressure of the gas
• n is the amount of substance
• T is the absolute temperature
• V is the volume
• R  is the ideal, or universal, gas constant, equal to the product of the Boltzmann constant and the Avogadro constant,

In this equation the symbol R is a constant called the universal gas constant that has the same value for all gases—namely, R =  8.31 J/mol K.

Example of Isothermal Process

Assume an isothermal expansion of helium (i → f) in a frictionless piston (closed system). The gas expansion is propelled by absorption of heat energy Q_add. The gas expands from initial volume of 0.001 m³ and simultaneously the external load of the piston slowly and continuously decreases from 1 MPa to 0.5 MPa. Since helium behaves almost as an ideal gas, use the ideal gas law to calculate final volume of the chamber and then calculate the work done by the system, when the temperature of the gas is equal to 400 K.

Solution:

The final volume of the gas, V_f, can be calculated using p, V, T Relation for isothermal process:

p_iV_i = p_fV_f ⇒ V_f = p_iV_i / p_f = 2 x 0.001 m³ = 0.002 m³

To calculate such processes, we would need to know how pressure varies with volume for the actual process by which the system changes from state i to state f. Since during this process the internal pressure was not constant, the p∆V work done by the piston must be calculated using the following integral:

By convention, a positive value for work indicates that work is done by the system on its surroundings. A negative value indicates that work is done on the system by its surroundings. The pΔV work is equal to the area under the process curve plotted on the pressure-volume diagram.

Free Expansion – Joule Expansion

These are adiabatic processes in which no transfer of heat occurs between the system and its environment and no work is done on or by the system. These types of adiabatic processes are called free expansion. It is an irreversible process in which a gas expands into an insulated evacuated chamber. It is also called Joule expansion. For an ideal gas, the temperature doesn’t change (this means that the process is also isothermal), however, real gases experience a temperature change during free expansion. In free expansion Q = W = 0, and the first law requires that:

dE_int = 0

A free expansion can not be plotted on a P-V diagram, because the process is rapid, not quasistatic. The intermediate states are not equilibrium states, and hence the pressure is not clearly defined.

See also:

Thermodynamic Processes

Isothermal Process

An isothermal process is a thermodynamic process, in which the temperature of the system remains constant (T = const). The heat transfer into or out of the system typically must happen at such a slow rate in order to continually adjust to the temperature of the reservoir through heat exchange. In each of these states the thermal equilibrium is maintained.

For an ideal gas and a polytropic process, the case n = 1 corresponds to an isothermal (constant-temperature) process. In contrast to adiabatic process , in which n = κ  and a system exchanges no heat with its surroundings (Q = 0; ∆T≠0), in an isothermal process there is no change in the internal energy (due to ∆T=0) and therefore ΔU = 0 (for ideal gases) and Q ≠ 0. An adiabatic process is not necessarily an isothermal process, nor is an isothermal process necessarily adiabatic.

In engineering, phase changes, such as evaporation or melting, are isothermal processes when, as is usually the case, they occur at constant pressure and temperature.

Isothermal Expansion – Isothermal Compression

See also: What is an Ideal Gas

In an ideal gas, molecules have no volume and do not interact. According to the ideal gas law, pressure varies linearly with temperature and quantity, and inversely with volume.

pV = nRT

where:

• p is the absolute pressure of the gas
• n is the amount of substance
• T is the absolute temperature
• V is the volume
• R  is the ideal, or universal, gas constant, equal to the product of the Boltzmann constant and the Avogadro constant,

In this equation the symbol R is a constant called the universal gas constant that has the same value for all gases—namely, R =  8.31 J/mol K.

The isothermal process can be expressed with the ideal gas law as:

pV = constant

or

p₁V₁ = p₂V₂

On a p-V diagram, the process occurs along a line (called an isotherm) that has the equation p = constant / V.

Example of Isothermal Process

Assume an isothermal expansion of helium (i → f) in a frictionless piston (closed system). The gas expansion is propelled by absorption of heat energy Q_add. The gas expands from initial volume of 0.001 m³ and simultaneously the external load of the piston slowly and continuously decreases from 1 MPa to 0.5 MPa. Since helium behaves almost as an ideal gas, use the ideal gas law to calculate final volume of the chamber and then calculate the work done by the system, when the temperature of the gas is equal to 400 K.

Solution:

The final volume of the gas, V_f, can be calculated using p, V, T Relation for isothermal process:

p_iV_i = p_fV_f ⇒ V_f = p_iV_i / p_f = 2 x 0.001 m³ = 0.002 m³

To calculate such processes, we would need to know how pressure varies with volume for the actual process by which the system changes from state i to state f. Since during this process the internal pressure was not constant, the p∆V work done by the piston must be calculated using the following integral:

By convention, a positive value for work indicates that work is done by the system on its surroundings. A negative value indicates that work is done on the system by its surroundings. The pΔV work is equal to the area under the process curve plotted on the pressure-volume diagram.

See also:

Isothermal Process