Frequently Asked Questions 
10. Thermodynamics – Advanced Topics
Conventional thermal power plants use the temperature at the bottom of the atmosphere, typically 30°C, as heat sink. The AVE uses the temperature at the top of the troposphere, typically -60°C, as the heat sink.
The lowest available heat sink temperature for conventional thermal power plants is the temperature of surrounding air or water. The efficiency of thermal power plant is proportional to the difference in temperature between the hot source and the cold source (another name for the cold sink); it is therefore advantageous to operate with as hot a heat source as possible. The heat to work conversion efficiency is limited by the second law of thermodynamics. The maximum heat to work conversion efficiency of modern gas fired power around 55%. The efficiency of coal fired power plant is typically between 35% and 45%. The efficiency of nuclear power plant is typically around 35% to keep the fuel well below its melting point. The efficiency of small engines such as automobile gasoline engines is typically around 20%.
Conventional thermal engines need a difference in temperature of at least 50°C between the cold and hot source. The temperature difference required across heat exchangers at both the hot and cold source and other losses can easily consume a 50°C temperature difference leaving no temperature difference to run the engine. Solar thermal power plants require concentrators in order to produce temperatures sufficient to power an engine. The concentrators in order of increasing hot source temperatures can be parabolic trough, parabolic dish or mirrors aimed a central tower. Glass covered collectors without concentrators are adequate for heating water or buildings but do not produce temperatures high enough to operate an engine.
The solar chimney is an exception to the above 50°C rule. In the Manzanares solar chimney the temperature of the air was increased by 20°C in a circular green house. Convective air currents above warm surfaces and rising smoke from cigarettes demonstrate that a difference of air temperature of a few degrees is sufficient to produce motion. In the solar chimney, the cold source temperature is the temperature at the top of the chimney. The difference in ambient temperature between the bottom and top of the 200 m chimney was 2°C therefore the difference in temperature between the hot and cold source temperatures was only 2°C.
The fundamental difference between conventional thermal engines and the AVE is that the cold source temperature is the temperature at the bottom of the atmosphere, typically 30°C, in the first case and the temperature at the top of the troposphere, typically -60°C in the second case. For a conventional thermal engine, the temperature of the hot source must be 50°C or more higher than the temperature at the bottom of the atmosphere. For the AVE, the hot source temperature can be the temperature at the bottom of the atmosphere and even slightly less when the air is very humid.
The high hot source temperatures required by conventional engines usually require fuel as the heat source. In practice the heat is supplied at as high a temperature as practical in order to maximize efficiency. The low hot source temperatures required by the AVE are low enough to be produced without using fuel or even a solar collector. The heat content of the air can be increased sufficiently by contacting it with a dry surface at temperatures of 30°C to 40°C or by spraying water at temperatures of 28°C to 32°C into the air. Heat sources at these temperatures are widely available in nature there is no need to burn fuel to warm the air.
The atmosphere is usually close to equilibrium. Surface air often has sufficient heat content to become buoyant when raised. Heating the air with a 35°C dry heat or with a 28°C wet heat source is usually more than sufficient to initiate upward heat convection. Once a threshold heat content is passed, there is a potential of converting approximately 30% of the heat supplied to work. Once surface air becomes buoyant it usually has the potential of remaining buoyant right up to the tropopause. The heat to work efficiency is fairly constant at around 30% because efficiency depends essentially on the temperature of the hot source and the temperature of the cold source.
A conventional thermal engine is an engine that humans work on; an atmospheric vortex engine is the engine humans live in.
10.2 What is the thermodynamic cycle of the AVE process ?
Higher resolution version of the above figure - PNG (565 KB)
The thermodynamic cycle for the AVE process can also be illustrated using P-v (Pressure-Volume) and T-s (Temperature-Entropy) diagrams. P-v and T-s diagrams are often used to describe thermodynamic cycles of heat engines. Refer to the following figure for P-v and T-s diagrams for the idealized thermodynamic cycle of the AVE process. The range of operating conditions of the AVE process is most notable. In the example below, operating pressures range from 100 kPa to 20 kPa and termperature from +20C to -80C. The cold source temperature available in the upper atmosphere is what makes the AVE process possible.
Higher resolution version of the above figure - PNG (142 KB)
The thermodynamic cycle of the AVE is very similar to the Brayton gas turbine cycle. The figures below show a gas-turbine ideal cycle and an AVE or gravity ideal cycle. The working fluid in both cases is dry air assumed to behave as an ideal gas.
Link to presentation containing the figures for question 10.2 of the FAQ in PDF format
In both ideal cycles, the air is heated from 290 K to 300 K at a pressure of 100 kPa and cooled from 189.4 K to 183.1 K at the 20 kPa level. The gas is heated a constant pressure, expanded in the turbine, cooled at constant pressure, and compressed back to its original pressure. The work produced by the expansion of the warm gas is more than required to compress the cooled gas and the excess is available to drive a load.
The thermodynamic calculations were carried out using standard thermodynamic techniques. The efficiency of an ideal gas-turbine is solely a function of the pressure ratio. For the pressure ratio of 5 used in the figures the ideal cycle efficiency is 36.9%.
The temperatures and temperature differences are much lower than in typical gas-turbine cycle. The turbines and compressors are reversible and adiabatic and therefore isentropic. The low efficiency of real turbines and compressor would make it impossible for a real gas-turbine to operate with the low temperature differences used in the gas-turbine figure.
In the gas-turbine cycle, the turbine is divided in two parts: a part driving the compressor, and a part supplying the net cycle work. Note that the compressor work is much larger than the net cycle work. Losses in compressors and turbines with 85% efficiencies would easily exceed net cycle work.
In the gravity or AVE cycle, the compressor and the turbine driving it are replaced with tall columns of air. Otherwise the operating conditions are the same in both cycles. Both cycles are reversible. No entropy is produced; the entropy received from the external system (Sei) is equal to the entropy given up (Seo). Heat is assumed to be received and given up at the temperature of the working fluid.
The fact that the atmospheric heat to work conversion cycle is similar to other thermodynamic cycle means that all the techniques and shortcuts used by engineers can be used to analyze how wind energy is produced and dissipated. It is standard practice in analyzing thermodynamic engines to initially assume that velocities are negligible, that frictional losses are negligible, that the kinetic energy of the working fluid is negligible, and that the cross sectional area and length of the conduit do not matter. The cross sectional area of the downward conduit can be much larger than the cross sectional area of the upward conduit. Once the ideal cycle has been understood it is easy to introduce the effect of irreversibility, friction losses and velocities. Thermodynamic calculations are normally done on a per unit mass basis.
Compressing a gas by slowly lowering it in a large diameter conduit is far more efficient than in conventional mechanical compressor. The compression efficiency can approach 100% while the efficiency of conventional compressors is usually around 85%. Compressing a gas from the pressure at the top of the troposphere to the pressure at the bottom of the troposphere with a conventional compressor would require an enormous machine.
In the gas-turbine cycle, both turbines are isentropic therefore it does not matter whether the two turbines are combined or have separate shafts. In the case of the AVE cycle, expansion in the turbine and expansion in the upward conduit are both isentropic processes; therefore the turbine could be at the bottom or the top of the conduit or anywhere in the conduit. The turbine could be divided into 100 smaller turbines spread over the upward flow conduit. The turbine could even be in the downward conduit.
The efficiency of an ideal cycle is equal to the efficiency of a Carnot engine with hot and cold source temperatures equal to the log average temperatures at which the heat is received and given up. All the processes in the cycle must be reversible – which means that the expansion process must be constrained. The efficiency of reversible cycle depends only on the average temperature at which heat is received and given up.
The key to understanding energy conversion processes is simplification. It is difficult to understand engines without the concept of isentropic compressors and expanders. Sadi Carnot used polytropic reversible process to understand how heat can be converted to work. While the isentropic process concept is natural for engineers who work with engines, the concept is foreign to atmospheric scientists. An engineer knows that an isentropic expander must have a shaft to take work out of the system. Atmospheric scientists sometimes assume that expansion can be isentropic without providing means of removing work from the system.
The figure below by Kerry Emanuel of MIT shows the thermodynamic cycle of the hurricane process. The air is heated by the sea surface at a temperature of 300 K and cooled at high elevation at lower temperatures. Colors represent entropy. Entropy increases as the air converges towards the base of the hurricane while receiving heat from the surface; entropy remains constant during the upflow process; and the entropy decreases as the air subsides. For an average hot source temperature of 300 K and an average source temperature of 200 K, the efficiency is 33%. There is no expander in the process making it difficult to figure out what happens to the work. The work is assumed to eventually dissipate through friction. Since no net work taken out of the system, the ultimate efficiency of the process must be zero.
The next figure replaces the turbine in the AVE cycle with a restriction or nozzle to show how easy it is to dissipate work. The kinetic energy of the fluid coming out of a restriction dissipates quickly unless there is a turbine blade immediately downstream of the nozzle. An engineer assumes that an expansion process is isenthalpic unless there is an expander with a rotating shaft to capture the work. When there is an expander, isentropic expansion is assumed as a first approximation. In the irreversible restriction case below, entropy is produced internally; the entropy leaving the system is greater than the entropy entering the system.
Like in the turbine case, the single restriction could be replaced with 100 restrictions with larger openings. The kinetic energy of the air coming out of each restriction or nozzle would only be one hundredth of the kinetic energy in the single restriction case. The multiple restrictions concept shows that the dissipation can be distributed over the complete circuit. Mechanical energy dissipation occurs mainly when air rises and expands unconstrained. The kinetic energy of horizontal wind may be less than one hundredth of the work that would be produced in a reversible process.
For more information on the atmospheric cycle, see:
Michaud
2000: Thermodynamic cycle of the atmospheric upward heat convection
process.
In 1905, Austrian thermodynamist Max Margules used isolated, piston covered, closed thermodynamic systems to analyze how mechanical energy is produced when air masses of uniform entropy exchange positions isentropically.
Margules calculated the work produced when air masses of uniform entropy initially on top of one another exchange position. The figure below shows an example of a rearrangement process. Potential temperature (θ) is equivalent to entropy and is the temperature that air would attain if brought isentropically to a pressure of 100 kPa. The potential temperature of each mass remain constant during the rearrangement process. The mechanical energy is equal to the reduction in the enthalpy of the system. In the example, the enthalpy reduction is 872 J/kg, enough to produce a velocity of 42 m/s in all of the air in the system.
Link to presentation containing the figures for question 10.3 of the FAQ in PDF format
Margules also calculated the work produced when air masses of uniform entropy initially side by side were re-arranged so that they finished on top of one another. The figure below shows an example of a rearrangement where the air masses are initially side by side. Work in the air masses initially side-by-side case is half the work of the air masses initially on top of one another because the side-by-side case is simply an intermediate step. Half as much work is produced because half as much high potential temperature air is raised.
The relationship between heat and work is not readily apparent from Margules’ large air masses examples. In order to see the relationship between heat and work it is necessary to use a small air mass as shown in the following figure. With a small air mass it is readily apparent that the work is equal to the heat received multiplied by the Carnot efficiency calculated using the temperature at which heat is received and given up as the hot and cold source temperatures.
Once the work to heat relationship has been realized, it is easy to show that the work produced in the large air cases is heat that must be supplied to restore the initial condition multiplied by the Carnot efficiency calculated using the average (log average) of the temperature at which heat must be supplied and given up to restore the initial condition.
Note that in the three states unit mass process the work is produced when the heat is carried upward by convection and not during the heating or cooling processes.
Unfortunately Margules side-by-side case was used to defend the argument that work production is the result of North-South re-arrangements. The next figure shows that as far as energy conservation is concerned, it makes no difference whether the high potential temperature air mass is raised directly overhead or raised at a long distance away.
The next figure gives equations used to calculate the enthalpy of large air masses. The Margules method is silent about what happens to the mechanical energy. The mechanical is presumed to somehow increase the kinetic energy of the air. What happens to the mechanical energy can be explained by examining specific reversible processes.
For more information on closed
thermodynamic system, see:
Michaud
1995:
Heat to work conversion during upward heat convection.Part I: Carnot
engine method.
Open systems such as the one shown in the figure below are commonly used to calculate the maximum quantity of work that can be produced when a fluid is moved adiabatically from one reservoir to another; or the minimum quantity of work required to move a fluid adiabatically from one reservoir to another. The equation is a reduced form of the total energy equation which is universally used to calculate work produced or consumed in moving fluid from one location to another.
The maximum work is simply equal to the reduction in the enthalpy minus the increase in potential energy. In case of a gas-turbine the change in potential energy is zero and the maximum work is simply the reduction in enthalpy. In the case of hydraulic turbines the change in enthalpy is zero and the maximum work is simply the reduction in potential energy. In the case of the AVE or of atmospheric convection both terms are important, the maximum work is the small difference between two large numbers.
Note that the maximum work is same as in the Margules thin layer case. The net work is the reduction in the total enthalpy of the air in the system - the reduction in the enthalpy of the small raised air mass minus the increase in the enthalpy of the large air mass. The increase in the enthalpy of the large air mass is equal to the potential energy of the raised unit mass in its final state!
As a first approximation or limiting case, it is standard practice to assume that velocities and kinetic energy are negligible, that there are no frictional loss and that there is no heat transfer. The results can later be modified to account for these additional factors.
The use of open flow system and of the total energy equation greatly simplifies energy calculations. Calculating heights is much easier than calculating the total enthalpy of the large air mass in both its initial and final state. Calculating work from the total energy is also much simpler than calculating CAPE. In addition the total energy equation has the advantage of applying whether the air is pure or whether the air contains water irrespective of the phase of the water.
In the figure below, the pressure difference between ambient pressure and the pressure at the bottom of the tube is used to accelerate the gas in a nozzle; a turbine blade located immediately downstream of the nozzle then captures the kinetic energy of the jet. An expander can be a turbine or a cylinder with a piston. When the expander is a cylinder with a piston, the process can take place slowly without producing any appreciable velocity or kinetic energy.
An expander captures work produced by the expansion of a gas.
An expanders can can be either: a cylinder with a piston, or
a turbine with nozzles and blades. The figure below uses a cylinder and
piston system to show that work is a maximum when the expansion is
reversible and that the work can be zero when the expansion is
irreversible.
Link to presentation containing the figures for question 10.5 of the FAQ in PDF format
Hendrick C. Van Ness, distinguished professor of chemical
engineering thermodynamics at Renselaer Polytechnid Institute,
considered the concept of mechanical constraint to be so
important that he dedicated a chapter ot it in his book: Understanding
Thermodynamics. For links to Van Ness publications see:
Understanding
Thermodynamics
Introduction
to Chemical Engineering Thermodynamics
Reversibility requires that the process be at constraint (also known as being in mechanical equilibrium) which means that the force on the two sides of the piston must be equal at all times. Early thermodynamists used the concept of an automat to explain mechanical equilibrium. An automat is a reversible mechanism for capturing and storing work, for using some of this work as work of compression, and for using the remainder of the work to drive a load. The work is maximized by having the automat hold back the piston and only letting the piston move slowly. The work is zero when the piston is unrestrained. Dissipating the work is as simple as letting go of the piston.
The above process was assumed to be adiabatic. A reversible process does not have to be adiabatic. A Carnot cycle uses two reversible expansion processes: an adiabatic constant entropy process, and a polytropic constant temperature process. What is important is that the process is at mechanical equilibrium. Polytropic expansion processes are reversible; polytropic work is the maximum work that can be produced in view of the fact that heat exchange occurs during the expansion.
The above figure applies to any process for transferring air from a pressure of 100 kPa to 95 kPa. For the purpose of illustrating how work is produced in the AVE, the 100 kPa source could represent the pressure at the bottom of a column of ambient air and the 95 kPa could represent the pressure at the bottom of the column of rising air.
The next figure shows what happens if the air in a cylinder piston system initially at 100 kPa and 30°C is allowed to expand against a pressure of 95 kPa.
In the reversible case when the piston is restrained by the automat; the expansion is isentropic. The total work done by the gas is 4461 J/kg of which 3085 J/kg is used to push the 95 kPa surrounding air out of the way and of which 1346 J/kg is captured by the automat. In the irreversible case when the piston is not held back by the automat and when the latch is simply let go, the total work done by the gas is 3384 J/kg. All the work is used to push the 95 kPa surrounding air out of the way; none of the work is captured by the automat. The temperature of the air in the cylinder after expansion is 24.6°C in the reversible case and 25.6°C in the irreversible case. The decrease in the enthalpy of the air in the cylinder is equal to the total work and is less in the irreversible case because the total work is less. In the irreversible case 1047 J/kg, about 22% of the potential work, remains in the cylinder as heat (enthalpy) and the opportunity to produce mechanical energy is lost.
Work of expansion can not exceed the work required to move the load; energy which does not leave the system as work remains in the system. The lesson is that no work is done unless there is an opportunity of doing work. If there is no possibility of doing external work the energy remains in the system as heat.
Approximately 20% of the work that would produced with isentropic expansion is lost irrespective of the change in pressure. In the above example the air was expanded from 100 to 95 kPa. If the pressure change had been from 100 to 99.9 kPa, the work would still be only 80% of the reversible work.
The above figure is valid for either a stationary cylinder or for a rising cylinder. Consider a bubble of buoyant air rising in a loose weightless plastic bag. There is only so much work required to push away the surrounding air, energy beyond what is required to push the surrounding air away remains within the bag as heat. Dissipating work is easy. Since the work of expansion produced by the rising warm air is much greater than the net cycle work dissipating less than 10% of this work of expansion is sufficient to dissipate all of the net work of convection.
The calculation basis for the above figure is described in the figure below.
A reversible adiabatic expansion process is a constant entropy process. An irreversible adiabatic expansion is a constant enthalpy process. Process simulators use a single "Unit of Operation" which can represent either a piston and cylinder or a turbine. A reversible adiabatic expander is an expander with an efficiency of 100%. An irreversible adiabatic expander is an expander with an efficiency of 0%. A nozzle can be represented as either an expander with zero efficiency or as a restriction. The figure below shows that a turbine is simply a nozzle with a turbine blade downstream of the nozzle to capture the kinetic energy of the fluid exiting the nozzle.
The figure below shows the equations applicable to a reversible turbine.
The figure below shows the equations applicable to an irreversible nozzle.
The figure below shows that the lifting process can be isentropic or isenthalpic. In order for a lifting process to be reversible it requires an expander.
After the reversible process is understood, it is easy to conceive a multitude of processes that produce less work that the reversible process. A reversible process could have two turbines sized so that one produces 90% of the work and the other 10% of the work. Similarly an irreversible process could have two restrictions sized so that 90% of the work is dissipated by one restriction and 10% of the work is dissipated by the other. Fuel burned in an engine produces mechanical energy; fuel burned in the open produces little mechanical energy. Simulating open fires does not lead to understanding how to improve the performance of engines. Simulations of the atmospheres do not lead to understanding how mechanical energy can be produced during upward heat convection. The kinetic energy of the wind is usually only a small fraction of the energy that could be produced in an ideal convection process. The quantity of mechanical energy produced can vary widely. Like the understanding of other thermodynamic engines, the understanding of the atmospheric process should start with ideal processes. Understanding engines is a prerequisite for understanding the atmosphere.
Fig. 10.6 shows the ideal AVE process consisting of: constant entropy expansion processes 1‑2, constant pressure heating process 2‑3, and constant entropy lifting process 3‑4. The water spray represents the wet cooling tower where enthalpy is transferred from water to air. The total energy equation:
is
used to calculate the energy received
and produced in each of the three processes, where w
is work, q
is heat, h
is the enthalpy of the air including the enthalpy of its water content,
g is the acceleration of gravity, z is height, and v is velocity.
The net work is transferred to the point where the flow is restricted namely expansion process 1-2. The work produced during process 3‑4 is used to lift the air. The key to solving the problem is realizing that shaft work in process 3‑4 is zero. The pressure at the base of the vertical tube is calculated by assuming an approach to equilibrium at state 3, calculating the work during process 3-4 for two P3 guesses, and then interpolating to determine the value of P3 required the make the work w34 zero. A second iteration can be used to find the value of P4 that maximizes w12; work is maximized when state 4 corresponds to the level of neutral buoyancy which is usually close to the tropopause.
Table
10.6 - Vortex engine sample process calculations
Air
Properties (Unit) / Case
Case 1
Case 2
Case 3
Case 4
P1
(kPa)
101.1
101.1
101.1
101.1
T1
(°C)
25.80
25.80
25.80
33.66
r1
= r2
(g/kg)
16.87
16.87
16.87
16.87
U1
(%)
80.0
80.0
80.0
50.1
s1
= s2
J/(K·kg)
241.0
241.0
241.0
267.7
h1
(J/kg)
68910
68910
68910
76990
P2
= P3
(kPa)
101.1
97.72
97.70
97.73
DPi
= P1 - P2
(kPa)
0.00
3.38
3.40
3.37
T2
(°C)
25.80
22.92
22.91
30.60
U2
(%)
80.0
92.3
92.3
57.6
h2
(J/kg)
68910
65930
65920
73940
T3
(°C)
25.80
24.50
30.70
30.60
U3
(%)
80.0
97.0
57.4
57.6
r3
= r4
(g/kg)
16.87
19.57
16.87
16.87
h3
(J/kg)
68910
74430
73990
73940
s3
= s4
J/(K·kg)
241.0
269.7
268.0
267.7
P4
(kPa)
10.0
10.0
10.0
10.0
T3
(°C)
-87.1
-80.92
-82.23
-82.30
z4
(m)
16570
16570
16570
16570
h4
(J/kg)
-96210
-91130
-91150
-91150
Heat
In: q
= h3
- h2
0
8500
8070
8080
Work:
wi
= h1
- h2
0
2980
3000
3050
Velocity:
vi
= sqrt (2 W)
(m/s)
0
77.2
77.4
78.1
Efficiency:
n = w12
/ q23
n/a
35.1
37.1
37.8
Efficiency:
n = 1
- T4
/ T3
n/a
35.4
37.2
37.2
Heat
source
None
26 °C
water at P2
36 °C
dry heat at P2
40 °C dry heat at P1
Excel
spreadsheet with more
case
Thermodynamic calculations can be carried out using a wide variety of computer and calculator programs. The HP48SX calculator program described in the following link is a very effective tool for understanding atmopheric thermodynamics: http://vortexengine.ca/misc/AT%20User.pdf
Computer
mathematic programs such as MathCad give identical results and
are
easier to document. See: http://vortexengine.ca/Isabel/TEE_MPI%20ISABEL275d.pdf
10.8 Can process calculations be carried out
with engineering process simulatators?
AVE process calculations can readily be carried out with standard chemical engineering process simulators. Simulartors can be used to analyze a wide variety of units of operation. Modern simulators can be drag and drop programs wherein units of operation can be selected and linked. Simulators ensure that mass and energy balances are respected. Chemical process simulators include programs for calculating thermodynamic properties of pure substance and of mixtures. Chemical simulators are well suited for AVE calculations because their ability to handle mixtures of air and water. Chemical simulators permit the use of a variety of thermodynamic property systems. The thermodynamic system used in these PROII simulation is SIMSCI SRK; the datum for entropy and enthalpy is not the same as the one used in the previous section. Simulator results are slightly different than those of the thermodynamic program of the previous section because the thermodynamic system is different and because the heat produced by the freezing of water is not considered in the simulator program. The effect of heat addition on energy production are similar for both approaches.
The figure below shows a simulation done on Simulation Science PROII. Expansion takes place in isentropic expanders EX1 and EX2. EX1 work can be used to drive a generator. Expander EX2 represents the upward flow process. EX2 work is used to lift the air and is not available to do other work. Air and water are brought to equilibrium at reduced pressure in flash drum F1. Calculator CA calculates the net work during the upward flow process as expander EX2 work minus the lifting work (Mgz). Controller CN finds the pressure P2 required to make the net work during the upward flow process zero.
The
table shows stream
properties; the items below the table show work and heat transfer duty
both of
which are calculated from the change in total enthalpy. The simulation is
based on a
nominal flow of 1000 kg/s
(1 t s-1) of air with a mixing ration of 16.87 g/kg. In
flash
drum F1, 1 t s-1 of air at 25.8 °C is mixed isenthalpically with
1 t s-1 of water at 27.5 °C. Expander #1 work is
3 MW. Work and duty are proportional to airflow; increasing
the airflow
to 10 t s-1 would increase the
power
output to 30 MW.
Link to pdf file with 7 Pro2 cases
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