The International Steam Pages
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-flowrate by using a series of valves. By varying mass-flowrate at constant pressure and temperature, steam flow speeds will remain essentially constant. The constant high flow speed over a wide range of mass-flowrate 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 turbine increase with both fluid flow velocity as well as with physical size. The length over which the fluid flows, as well as the surface area with which it interacts, are also dominant factors. One of the terms that needs to be calculated, is the Reynold's number (a dimensional number which 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 idiosyncracies 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 bounday-layer turbine and the incoming jet of fluid, very high subsonic and supersonic fluid flow speeds would be essentail. 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 back-pressure in the turbine is between these 2-values, oblique shock waves will eminate from the nozzle exit into the turbine. If the back-pressure is above the higher value, a normal shock wave will appear at the exit. If the back-pressure is below the lower value (eg; 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.
If a locomotive Tesla turbine has 8-discs with 7-spaces between them, the nozzles could be arranged in a 1:2:4 mass-flowrate ratio, which would correspond to the cross-section area ratios of the three throats. The largest nozzle would fire expansion shock waves into 4-spaces, the mid-sized nozzle into 2-spaces and the small one into one inter-disc space. This arrangement allows for 7-combinations, corresponding to 7-equally spaced steam mass-flowrates and in turn 7-equally spaced power levels, all at maximum pressure and temperature. Alternatively, a stack of 16-discs may be used with 15-spaces between them. The nozzles may then be arranged in a 1:2:4:8-mass-flowrate ratio, giving 15-combinations which will correspond to 15-equally spaced power levels. Nozzles will be either "on" or "off", depending on locomotive power requirements.
The number of disc spaces per nozzle may be doubled or tripled, increasing the interactive surface area (using the same disc radius) to increase fluid friction and overall engine efficiency. In view of the temperature drop incurred in accelerating steam to supersonic speed, it is possible to heat the nozzle's diffuser downstream of the throat, by using steam at 1000-deg F from the boiler. Two sides of the (high and narrow) fixed diffusers may be heated and insulated, raising exit steam temperature by an extra 100-deg F and exit speed to 3700-ft/sec (Mach 2 at the higher temperature). The steam used to heat the diffuser may used for feed heating and also for pre-heating water.
A variable geometry nozzle of rectangular cross-section is also possible. 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 (except only one side of the diffuser could then be heated). Variable steam mass-flowrate would result from varying the cross section of the throat to a maximum of 1.5-square inches, which would allow 16.56-lbs/sec of steam (1000-psia at 1000-degrees F at the boiler has 1505.4-Btu's/lb) to flow through, allowing 24,930-Btu/sec to leave the boiler (35,260-horsepower). At an estimated conversion efficiency of 21% between boiler and turbine, a maximum of 7400-Hp would be available to drive an electrical generator (91%-efficiency), developing an estimated 6700-Hp at the rim of the locomotive drive wheels. Higher overall thermal efficiencies and power output levels are possible using a single-pass system, though a higher-efficiency compound-reheat variant is quite possible (some Tesla turbine researchers claim conversion efficiencies exceeding 30% from boiler to turbine output, for a single pass layout).
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 back-pressure which needs to be overcome, despite pressure drops in the nozzle. The high-pressure water-tube boiler compensates for this loss in pressure. The water pump will require less than 1.5% of total output energy for a 1,000-psia water tube boiler.
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 carbody). 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). Variations of boundary layer turbines other than the inward radial flow Tesla turbine are possible, including a variant that uses a small axial component (a tornado swirling in an annulus). In this variant (a tubular boundary-layer turbine), a very large surface area interacts with the vortex which swirls in between the spaces (annuli) of several concentric tubes. The outer (largest diameter) annulus may taper to a small diameter at the votex entry point, where the injection nozzle would be located (steam would be injected tangentially, with a small axial component).
The vortex would swirl inside the outer annulus (level 1) from the entry point (region 1) to the opposite end (region 2), where it would progress into a smaller diameter annulus (level 2) and spiral back toward region 1. When it reaches the end of the smaller annulus (level 2), the vortex would spiral into an even smaller diameter annulus (level 3) and spiral back toward region 2, where outward radial flow curved diffusers would reverse the flow direction to a larger diameter. The exhaust steam would exit the tubular turbine in a direction opposite to the steam injection entry direction. Spiral blading similar to the spiral blade on agricultural augers (used to drill large holes into the ground) may be included inside the annuli to increase interactive surface and raise the fluid drag friction, in turn increasing overall turbine efficiency (into the 26% to 32% range).
The use of the spiral restricts the fluid flow within a measurable cross section, changing the way the Reynold's number is calculated, from flow length to the hydraulic radius. The hydraulic radius of a rectangular cross section equals 4-times cross-section area divided by the "wetted perimeter". Steam at 800-deg. F has a kinematic viscosity of 5.056 x 10E-7 (0.0000005056) lbf-sec/sq.ft . The Reynolds number can be calculated by multiplying the steam flow speed by the hydraulic radius, then dividing by the kinematic viscosity. The friction coefficient derives from the Reynold's number and would typically be in the range of 0.0015 to 0.002 in this situation. The total drag would equal (density x speed squared x total "wetted" surface area x drag coefficient) divided by 2 x gravity. If the spiral has a cross section of 2" x 2" (8" perimeter or 2/3-ft) and a total internal length of 240-ft, total area would be 160-sq.ft. For a relative speed of 1,500-ft/sec, steam density of 0.25-lb/cu-ft. and a drag coefficient of 0.0015, total drag would be 3144.4-lbf. At a radius of 2.5-ft and a rotational speed of 4800-RPM, this would yield 7180-Hp. Like a Tesla multi-disc turbine, the tubular/annular turbine would also have to be mounted using a vertical shaft to counter gyroscopic effects.
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 Both the proponents and sceptics may still have to prove their case. The boundary layer turbine is one of the few engine concepts that can utilize the energy from shock waves.
Harry Valentine, Transportation Researcher, firstname.lastname@example.org
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