Researching a Shock-wave Tesla
Steam Turbine Locomotive
The concept of a boundary-layer turbine originated about a century ago, in the research of Nikola Tesla. Tesla's version of a boundary-layer turbine consists of a stack of closely spaced discs. A high-velocity of fluid is injected tangentially into the spaces between these discs, flowing inwardly in a spiral toward a centrally located exhaust. The drag between the surface of the discs and the fast moving fluid results in the conversion of fluid flow to mechanical power.
Most tests involving Tesla turbines have used subsonic flows of liquids or gases, with less than spectacular results. Unlike conventional bladed turbines that are subject to blade erosion, boundary layer turbines can operate when a partially saturated fluid is injected at supersonic speeds. Superheated steam is one fluid that can be injected into Tesla turbines under high-pressure and at supersonic speeds. When a fluid is accelerated to supersonic speeds, it undergoes a drop in both temperature and pressure. To generate the supersonic speed from a high-pressure region, the fluid first flows through a converging duct into a throat (minimum area cross-section) where sonic speed will be reached, then into a diverging duct where supersonic speeds will occur if the downstream pressure is low enough.
Supersonic flow develops when there is a large enough difference between the upstream and downstream pressures. Pressure reductions will occur in both the throat area as well as in the diverging duct. To ensure a high enough upstream pressure, a water tube boiler would need to generate at least 1,000-psia at 1,000-degrees F. Power output may be varied by adjusting steam mass-flowrates by using a series of valves. By varying mass-flowrates at constant pressure and temperature, steam flow speeds will remain essentially constant. The constant high flow speed over a wide range of mass-flowrates is essential to maximizing fluid friction within a boundary layer turbine. The dominant term in the mathematical equation of boundary layer drag friction, is the velocity-squared factor.
The power output and efficiency of a boundary layer Tesla Turbine increases with both fluid flow velocities as well as with physical size. The length, over which the fluid flows, as well as the surface area with which it interacts, is also dominant factors. One of the terms that needs to be calculated, is the Reynold's number (a dimensional number that contains factors such as density, gravity, flow velocity, viscosity and a length factor). The higher the Reynold's number, the lower the friction or coefficient of friction. However, the product of velocity-squared and friction (Cf x V-squared) increases even as the friction factor decreases at higher flow speeds. One of the idiosyncrasies of Tesla turbines is that if they are to develop higher levels of efficiency, they have to be of very large diameter (to maximize disc surface area). To reduce parasitic losses against the inside of the casing, they need to be of low height at the casing's inside circumferential surface, with at least 1-inch clearance between the casing and the upper and lower disc surfaces.
As flow speed exceeds the speed of sound, shock waves begin to appear. At sonic speed, the shock wave will appear in the throat section at lower upstream steam pressures. As steam pressure increases, the shock wave moves down toward the exit of the diffuser section, where it is called a "normal" shock wave. As the difference between upstream and downstream pressures increase, the shock wave becomes oblique at the diffuser (nozzle) exit (an over-expanded nozzle). Larger upstream and downstream pressure differences result in expansion waves forming at the exit (an under-expanded nozzle). In order to maintain a high relative speed between the rotating boundary-layer turbine and the incoming jet of fluid, very high subsonic and supersonic fluid flow speeds would be essential. Whereas shock waves (pre-ignition "pinging") can shatter pistons in internal combustion engines, Tesla turbines are immune to shock wave damage.
If a Tesla multi-disc turbine of 6'4" outer disc diameter (20-ft circumference) rotates at 4500-RPM or 1500-ft/sec at the disc edge, steam injected tangentially at a speed of 3500-ft/sec (Mach 2.0) into the discs would result in a relative speed of 2000-ft/sec. Steam leaving the water-tube boiler at 1,000-psia at 1000-deg F would drop to 540.5-psia and 793.7-deg F in the throat section and 420.2-deg F at the nozzle exit. The critical exit pressures are 129.7-psia and 573.7-psia. If the backpressure in the turbine is between these 2-values, oblique shock waves will emanate from the nozzle exit into the turbine. If the backpressure is above the higher value, a normal shock wave will appear at the exit. If the backpressure is below the lower value (e.g.; 50-psia downstream "resistance" pressure), expansion shock waves will blast forth from the nozzle exit and into the turbine at 3500-ft/sec. This will happen if the nozzle exit area is 1.746-times the throat area. Supersonic speed steam flowrates entering the Tesla turbine discs could be partially saturated, however, boundary-layer turbines can operate under such conditions (shock waves plus saturated steam) without damage.
The efficiency of turbines is a function of size, that is, large and more powerful turbines are more efficient. In a boundary layer turbine, the amount of surface contact area the working fluid is in contact with maximizes performance. If 30-discs were spaced 1/4-inch (0.25"), 29 x 2-sides would be presented to the vortex. If the main working area of the disc is 3'-outer diameter and 2-ft inner diameter, there would be 15.7-sq.ft area per side or a total of 900-sq.ft of interactive surface area. The effective working radius would be 2.5-ft. The density of the steam at 400-deg F is 0.25-lb/cu.ft and its kinematic viscosity is 3.56E-7 lbf-sec/sq.ft. With the discs rotating at 4800-RPM, the relative maximum speed between the steam vortex and the discs at a radius of 2.5’ is 1500-ft/sec.
The power output of the turbine would be a function of the square of the relative velocity (between vortex and discs) x (the interactive surface area) x (the drag coefficient) x (the steam density) divided by 2 x gravity. The drag coefficient is a function of the Reynolds Number, which for a relative speed of 1500-ft/sec would be Re = (1500 x 2.5 x.2 x pi)/3.56E-7 = 6.6152E10. To calculate friction coefficient, part of the Prandtl-Schlichting equation will be used, 0.455/ (log Re) exp2.58 = 0.000976. For a relative velocity of 1500-ft/sec, drag would be (1500 x 1500 x 900 x 0.000976 * 0.25 / 2*32.2) = 7676-lbf at 2.5-ft radius, or 19,190-lbf.ft torque at 4800-RPM (17,445-Horsepower). At a relative fluid/disc speed of 500-ft/sec, the drag coefficient would increase to 0.0011 and deliver 958-lbf at 2.5-ft for 2395-lbf-ft torque at 4800-RPM (2177-Horsepower). At a relative speed of 1000-ft/sec, torque would be 8897-lbf-ft at 4800-RPM (8131-Horsepower). With an electrical conversion efficiency of 89%, 7237-Hp would reach the rim of the locomotive drive wheels, with 6,400-Hp being available at the drawbar to pull a train at 60-mi/hr on level track.
A variable geometry nozzle of rectangular cross-section (narrow and high) would be used to inject the steam (tangentially) into the turbine. Variable geometry rectangular intakes are used on supersonic aircraft. Adjustable cross-section areas for both the throat as well as the nozzle exit could maintain a constant area-ratio (1.746 for Mach 2.0) between the two, by using a curved plate on a pivot and connected to a mechanical linkage. The inlet would admit supersonic steam flow into the vortex at 3500-ft/sec over a wide range of mass flowrates, resulting from varying the cross section of the throat to a maximum of 1.5-square inches and allowing 16.56-lbs/sec of steam (1000-psia at 1000-degrees F at the boiler has 1505.4-Btu's/lb) with 24,930-Btu/sec to leave the boiler (35,260-horsepower).
If this flowrates corresponds to 17,445-Hp, efficiency between boiler and turbine shaft would be 49.47%. If this corresponds to the lower level of 8131-Hp, the turbine would operate at 23%-efficiency, between boiler and turbine shaft. At 6400-Hp at the drawbar, overall locomotive efficiency would be 18%. It is quite possible for a single-pass turbine to operate as high as 32%-efficiency. The water pump will require less than 1.5% of total output energy for a 1,000-psia water tube boiler. If the pump can deliver 17-lb/sec of water (62.4-lb/cu.ft), or 0.272-cu.ft/sec at 1000-psi (144000-lb/sq.ft) = 39,230-lbf-ft/sec = 71.32-Hp (100% -eff) or at 85%-efficiency = 84-Hp water pump demand to deliver 6,400-Hp at 60-mi/hr.
The centrifugal forces generated within a Tesla turbine tend to push higher density compressible fluid toward the outer edges of the discs. This increased density increases the skin friction between the fluid and the discs. The injection of shock waves into the high-density fluid sustains a rapidly swirling vortex, with the less dense fluid being pushed toward the central exhaust. When the injected steam passes through the shock waves, almost instantaneous pressure and temperature rises occur. The high density swirling mass presents a high backpressure, which needs to be overcome, despite pressure drops in the nozzle. The high-pressure water-tube boiler compensates for this loss in pressure.
The supersonic injection speed of the steam would produce shock waves upon initial entry into the turbine. As the vortex begins to swirl and the turbine accelerates, sub-sonic relative speed could develop between the vortex and the outer disc areas. Due to the parallel flow between the vortex and the circumference of the discs, any shock waves that result between the vortex and the discs would have very small deflection angles and result oblique shock waves. These weak shock wave regions will allow supersonic vortex velocities to continue downstream of the shock waves. This supersonic vortex speed allows for higher disc (turbine) rotational speeds and much higher turbine output power for its size. The subsonic relative speeds between turbine and outer vortex allows the fast moving steam flow to enter the spaces between the discs, with either no shock wave or a weak shock wave would developing, the latter having minimal efficiency losses.
At the low relative vortex/disc speeds, the flow may transform from turbulent to laminar as the vortex moves inward toward the central exit. A high-energy exhaust stream could be reheated and re-expanded in a second, lower-pressure turbine. The exhaust from the second turbine could drive a third turbine, which would drive a compressor, re-compressing the exhaust steam leaving the higher-pressure power turbine and entering the lower-pressure power turbine. Such an arrangement would raise overall thermal energy efficiency of a Tesla turbine system. If a single large Tesla turbine can exceed 30% efficiency using supersonic speed steam flow in its vortex, a single-pass steam expansion may be cost-effective when low-cost renewable energy is converted.
To counter the gyroscopic effects of a Tesla turbine, a vertical shaft will need to be used (the unit may need to be mounted in gimbals, inside a 10-ft wide locomotive car body). An electric alternator may be mounted above the turbine. Flexible couplings may need to used in the steam lines leading to the Tesla turbine, to compensate changes caused by locomotive pitching (gradient changes) and roll (tilting on curves). Boundary layer turbines are among the many options available to expand steam for traction generation in a renewable-fuel modern steam locomotive. There is a great deal of controversy involving the overall thermal efficiency of such turbines. The example described in this article is by no means an optimized system. There is scope to optimize its performance and refine its operation.
This fantastic article was written by Mr. Harry Valentine, harrycv@hotmail.com
