Latent Power Turbine Theory
1 The Rankine Cycle
This provides a thermodynamic summary of how existing steam turbine units
work.
Figure 1. Conventional steam turbine
power systems
such as that illustrated in Figure 1 on the main Latent Power Turbine page
operate on the Rankine cycle.
This graph shows the key features of a Rankine Cycle displayed on a P-V diagram.
One kilogram of water is considered as it moves round the cycle
Figure 2. In order to harness the latent heat released when saturated vapour condenses conventional thinking on steam turbine design needs to be reversed. Instead of the volume inside the turbine increasing as the steam passes through successive sets of turbine blades, the volume needs to decrease.
As we will explain below, simply tapering the
turbine profile is not sufficient because the system would be very unstable,
encouraging the vapour to drift into an unproductive unsaturated condition.
In
order to understand the cause of this instability we need to look at the
differences between saturated and unsaturated vapours passing through nozzles or
constrictions.
2 Key differences when
saturated and unsaturated vapours flow through nozzles or
constrictions
With the exception of the heat pump shown in Figure 4 below, we will assume that all conduit sections are well lagged, so that processes are adiabatic.
2.1 Negligible viscous friction (drag)
When an unsaturated vapour or gas passes through a constriction in a length of conduit the fluid has to move faster through the constriction in order to maintain a steady rate of mass flow.
Figure 3, the passage of gas or unsaturated vapour through a constriction. The pressure and temperature drop inside the tapered section, then increase again as the cross sectional area of the flared section increases.
Many textbooks explain how the pressure and velocity changes can be related using Bernoulli's equation, but references to the temperature change are less common. However the existence of the Bernoulli heat pump provides practical evidence that a significant temperature change takes place inside a constriction.
Figure 4. The Bernoulli heat pump principle.
Its mode of operation relies on the temperature of a gas or unsaturated vapour
falling as it passes through a constriction.
What happens if a
saturated vapour enters a Bernoulli heat pump?
When saturated vapour passes through any form of constriction the temperature
changes are minimal because as soon as the vapour starts to cool condensation is
triggered. This releases latent heat, re-warming the vapour.
Figure 5, Passage of a saturated vapour through a constriction. The temperature and pressure drops are minimal. "Dry" saturated vapour refers to one that is just saturated, with no water droplets present. The vapour becomes "wet" when some vapour partially condenses out to produce a fine mist of water droplets.
Figure 6. Temperature changes on passing though a constriction. The temperature change for the saturated vapour is very small-just sufficient to trigger condensation.
If we consider unit mass (1 kg) of dry saturated steam entering the constriction and plot the volume changes due to condensation, then evaporation on a P-V graph, the condensation phase corresponds to the plateau in Figure 2 above,
Figure 7. The pressure remains approximately constant when a saturated steam + water droplet mixture passes through flares and constrictions. As we will see below, the important feature in terms of the Latent Power Turbine design is the constriction where latent heat is released, minimising the drop in steam temperature.
2.2 Significant viscous friction (drag)
Viscous drag converts some of the forward kinetic energy of the gas/saturated or unsaturated vapour into heat. This is an irreversible process that takes place at all cross sections along the constriction.
At a molecular level heating is explained as an increase in the random movement of the molecules. At a macroscopic level this is manifested as a slight increase in pressure.
Figure 8. Passage of initially dry saturated vapour/gas through a real constriction. where friction increases the fluid temperature.
In order to maintain forward motion with v2 = v1, a counterbalancing external pressure drop has to be applied.
In contrast, if wet
saturated steam enters the constriction then drag causes evaporation and an
increase in volume, rather than an increase in pressure.
In this case, the
condition v2 = v1 can only be met if the
conduit broadens out.
Figure 9. Passage of wet saturated vapour through a real constriction. In order to maintain forward motion with v2 = v1, the cross sectional area of the wide part of the conduit has to increase. Additional work has to be done by the notional pump in order to cope with the increased volume.
What happens if the cross sectional area of the conduit does not increase?
Figure 10. The rate of mass flow must remain constant. This can only be achieved if the notional pump accelerates the wet saturated steam to a velocity v2 that is higher than v1.
In the following sections an ideal zero drag turbine design will be developed, then modified to allow for drag effects.
3 Single Turbine analysis, ideal zero drag conditions
3.1 Gas or permanently or unsaturated steam flow
This is the starting point for our turbine design, with no drag problems and no refinements to harness the release of latent heat.
Figure 11. The rate of mass flow at B and C must be identical. But momentum has been transferred from the gas to the turbine blades, so the forward velocity of the gas is reduced. To accommodate this, the exit port has to have a larger cross section than the nozzle(s).
Can we extract all of the
kinetic energy from the moving gas?
No, the best we can do is reduce the velocity by 50% in order to extract 75%
of the kinetic energy.
External work has been done by the gas passing through the turbine, so if we compare two points on either side of the turbine where the velocities are the same, the Law of Conservation of Energy requires the temperature of the gas to fall at the exit.
3.2 Saturated
vapour.
If the gas is replaced by a saturated vapour, then condensation occurs and the temperature
drop is minimal, thanks to the release of latent
heat.
Figure 12.
The saturated vapour does external work as it passes through
the turbine. Any tendency of the vapour to cool and lose pressure is offset by
the release of latent heat.
Water is far denser than steam, so the rate of volume flow exiting the turbine
is less than on entering.
If we want to maintain the same flow velocity before and after the turbine, then
the cross sectional area at D needs to be less than at A.
4 A saturated vapour turbine chain: (i) No friction damping
Figure
13. The turbines and related apertures decrease in
capacity in a calculated way so that the pressures inside the first and final
turbines are the same.
The system will need to be primed in order to create the initial vapour flow and
at the far end of the chain a small fraction of the vapour will need to be
pumped out in order to maintain a steady state of flow.
A sample calculation on turbine chain length is provided as an Appendix below.
5 A turbine chain that copes with viscous drag
5.1 The basic design modification required
Viscous drag produces a heating effect, causing a small fraction of the water droplets to evaporate. We have argued above with reference to Figure 9 that as a consequence the connecting conduits need to be slightly enlarged in order to maintain a steady rate of vapour flow.
A similar argument applies to the turbine units. In reality, the tapering off of turbine size along the chain will need to be slightly less than for a drag free system, to allow for friction induced evaporation.
Figure 14.
Successive turbines must taper off in size in order to ensure that the
pressure remains constant as external work is done and vapour condenses out.
But, in order to allow for partial re-evaporation caused by drag, the
taper will be slightly less than the ideal model predicts,.
It is important to note that evaporation caused by friction does not add energy
to the system, it is a nuisance that has to be dealt with.
5.2 Coping with a variable power output
Our patent literature describes a number of ways for coping with varying saturated vapour inputs. One simple technique is to short circuit the early stages of the turbine chain when the input drops.
Figure 15. When the saturated vapour input drops, the early turbine stages are short-circuited. The vapour passes through a conduit adjacent to the chain of turbines and only enters the chain at a point where its flow rate matches that at the same point when the chain is operating under maximum floe rate conditions.
6 Guaranteeing a continuous flow of steam through the system
Once primed, the steam should continue to flow under its own inertia, provided that fresh steam is supplied at a steady rate and that the residual steam at the end of the chain is condensed or pumped out.
However, even in the worst case scenario, when unforeseen forces bring the steam to a halt after passing through a turbine, the design is flexible enough to cope.
Figure 16. In this worst case scenario, impeller pumps have been added after each turbine. We will assume for the sake of argument that the pumps operate in chambers that have a cross sectional area ten times the cross sectional area of the following set of nozzles. Kinetic energy is proportional to the square of the velocity, so the work done by each pump is only 1% of the kinetic energy at the next set of nozzles. As in the previous examples described above, latent heat has to be liberated to offset the increase in kinetic energy.
7 P-V diagram for a system that incorporates a superheating boiler and a chain of Latent Power Turbines.
The sections of the path in blue are analogous to the Rankine cycle. Red lines relate to the new features.
Figure 17.
The Latent Power Turbine (LPT)
Cycle.
At the point on the isotherm plateau where the volume of steam becomes too low
to process economically the residual steam can be condensed out. This is a
simple but inefficient way of dealing with the residual steam. It is more energy
efficient to compress the steam back up to boiler pressure. Work has to be done
in this process, but the latent heat is preserved.
The following diagram indicates how energy calculations can be done for a Latent Power Turbine Cycle.
Figure 18. Calculation diagram. When phase changes occur, the latent heat of vaporisation has to be taken into account. Latent heat and steam density values vary with temperature.
8 T-s diagram for the above system
The general shape of the temperature (T)-entropy (s) diagram for a Latent Power Turbine system incorporating a superheating boiler is the same as for a Rankine cycle system.
Figure 19. The T-s diagram shown assumes that the residual steam at the end of the Latent Power Turbine chain condenses out, as for a conventional Rankine cycle.
APPENDIX
Estimating the number of turbine required in a Latent Power Turbine Chain.
Figure 20. The rate of saturated vapour mass flow drops off along the chain. If viscous drag is negligible, there is a corresponding drop in turbine chamber volume along the chain.
This graph assumes the following conditions
(i) The turbine converts 75%
of kinetic energy of steam into work.
(ii) Temperature of saturated steam = 100oC
(iii) Mass of saturated steam entering first turbine = 1 kg
(iv) Nozzle velocity, v = 450 m/s
(The velocity must remain below the speed of sound to avoid Laval nozzle type
expansion. For steam at 100oC, this is 473 m/s.)
Values used
Specific latent heat = 2.26 x 106
J/kg
Density of steam = 0.6 kg/m3
Sample calculation
Work done on turbine = ¾ x ½ x mass x
(velocity)2
= ¾ x ½ x 1x 450 x 450
= 76 x 103 J
Mass of steam condensing = 76 x 103 J / Specific latent heat of steam
= 76 x 103 J /2.26 x 106 J/kg
= 33.6 x 10-3 kg
Mass of saturated steam entering second turbine = (1 - 33.6 x 10-3) kg
= 0.966 kg
Volume of steam/second entering first turbine
= volume of 1 m3/450 m/s
= 1/(0.6 x 450)
= 3.7 x 10-3 m3