Example calculations

  1080 5600 6170 8450 14130 kW
Dimensions:
Cylinder arrangement linear ("Boxer")  
Number of cylinder pairs 4 8 8 8 8  
Inside cylinder diameter 0.25 0.35 0.35 0.35 0.40 m
Stroke length 0.31 0.44 0.44 0.44 0.56 m
Piston rod diameter 0.15 0.21 0.21 0.21 0.24 m
Displacement 0.04 0.22 0.22 0.22 0.36 m3
Length of the crankshaft apr. 3.0 apr. 9.45 apr. 9.45 apr. 9.45 apr. 10.8 m
Width machine apr. 2.65 apr. 3.7 apr. 3.7 apr. 3.7 apr. 4.7 m
Medium data
Medium (working gas) Helium  
Density at 1 bar 0.179 kg/m3
Molar volume 22.43 m3/kmol
Isochore spec. heat capacity cv 3.13 kJ/(kg·K)
Individual gas constant Rs 2.08 kJ/(kg·K)
Molecular weightMx 4.003 kg/kmol
Operating data:
Number of revolutions 400 400 400 600 600 rpm
Medium piston speed 4.17 5.83 5.83 8.75 11.20 m/s
Absolute pressure at bottom dead center 3 bar
Absolute pressure at bottom top center 75 bar
Compression ratio 25  
Temperatures:
tBurner 800 °C
tmax 780 °C
tCooling water temperature 15 °C
tmin 100 100 30 50 50 °C
ηCarnot 0.65 0.65 0.72 70 70 %
Volumes and masses:
Mass flow / revolution 0.042 0.231 0.231 0.231 0.386 kg
Mass flow 0.28 1.54 1.54 2.31 3.86 kg/s
Volume flow rate at pmin 0.524 2.874 2.874 4.31 7.206 m3/s
Volume flow rate at pmax 0.021 0.115 0.115 0.172 0.288 m3/s
Amount of substance nx = mx/Mx 0.005 0.029 0.029 0,029 0,048 mol
Power:
Gained work:
W=n·R·ln(Vmax/Vmin)·(Tmax-Tmin) 		1800 5600 6170  8450 14130 kW
Piping:
Flow velocity max. 18 18 18 18 18 m/s
DMR Collecting pipe at pmin 192 451 451 552 714 mm
DMR Collecting pipe at pmax 38 90 90 110 143 mm
DMR Single pipe at pmin 48 56 56 69 89 mm
DMR Single pipe at pmax 15 32 32 39 50 mm
Regenerator:
Temperatures:
Inlet temperature hot shell side
at pmin: TH 		780 780 780 780 780 °C
Outlet temperature cold shell side
at pmin: TC Reg 		250 250 50 150 150 °C
Inlet temperature cold pipe side
at pmax: TC 		100 100 30 50 50 °C
Outlet temperature hot pipe side
at pmax: TH Reg 		630 630 760 680 680 °C
Geometry:
Heat transfer surface A 47.0 162.3 103.1 441 704 m2
Inner diameter shelli 344 444 444 697 797 mm
Total length of inner tubes 10850
(=3x apr.3617) 		13690
(=2x6845) 		10850
(=2x5425) 		16880
(=2x8440) 		20510
(=2x10255) 		mm
Distance between baffles 2350 1660 2420 mm
Inner tube dimensions 25 x 2 25 x 2 25 x 2 25 x 2 25 x 2 mm
Pitch angle 60 60 45 45 45 °
Number of pipes inside 69 121 151 333 437  
Heat transfer and flow:
Heat transfer coefficient (inside) αi 592.8 669.7   537.4 W/(m2·K)
Heat transfer coefficient (outside) αa 308.7 302.4   356.1 W/(m2·K)
Heat transfer coefficient k 109.7 180.3 191.2   193.5 W/(m2·K)
Dynamic viscosity shell side ηa 0.03992 0.03915 0.03915   0.0374 mPa·s
Dynamic viscosity inner pipe side ηi 0.0338 0.03377 0.03114   0.03377 mPa·s
Thermal conductivity inside li 0.2461 0.2642 0.2433   0.2642 W/(m·K)
Thermal conductivity outside la 0.3057 0.2918 0.3057   0.2918 W/(m·K)
Reynolds number shell side Rea   3828 4022   4037  
Reynolds number pipeside Rei   18308 24784   15857  
Flow velocity outside (shell side) 29.95 29.91   30 m/s
Flow velocity inside (pipe side) max 15 14.75 20.2 max 15 14.76 m/s
Performance:
Regenerator efficiency
ηReg = (TH - TC Reg) / (TH - TC) 		78 78 97 86 86 %
Heat output Q 4240 4240 7560 12633 kW
Regenerator loss
LReg = 1-ηReg 		22.1 21.4 2.6 14.2 14.2 %
Cycle:
Expansion cylinder:
Outlet temperature heater = Inlet temperature T1b 1053 1053 1053 1053 1053 K
Outlet pressure heater = Inlet pressure p1b 75 75 75 75 75 bar
Regenerator (Shell side):
Outlet temperature cylinder = Inlet temperature Regenerator T2 1053 1053 1053 1053 1053 K
Outlet pressure cylinder = Inlet pressure Regenerator p2 3 3 3 3 3 bar
Cooler:
Outlet temperature regenerator = Inlet temperature cooler T3a 523 523 323 423 423 K
Outlet pressure Regenerator = Inlet pressure cooler p3a 1.409 1.409 0.920 1.205 1.205 bar
DTCooler 150 150 30 100 100 K
Heat to be removed DQ/Dt 1218.41 1198.62 159.8 1198.62 2003.90 kW
Compression cylinder:
Outlet temperature cooler = inlet temperature compression cylinder T3b 373 373 303 323 323 K
Outlet pressure cooler = inlet pressure compression cylinder p3b 1.063 1.063 0.863 0.92 0.92 bar
Regenerator (Pipe side):
Outlet temperature compression cylinder = inlet temp. Regenerator pipe side T4 373 373 303 323 323 K
Outlet pressure compression cylinder = inlet pressure regenerator pipe side p4     26.57 21.58 23.01 23.01 bar
Heater:
Outlet temperatureRegenerator = Inlet temperature heater T1a 903 903 1033 953 953 K
DTHeater 150 150 20 100 100 K
Outlet temperature heater T1b 1053 1053 1053 1053 1053 K
Outlet pressure regenerator = pressure heater p1a 64 64 68 68 68 bar
Heater outlet pressure p1b 75 75 75 75 75 bar


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Conclusion:

  • In all of the examples listed, there is still room for improvement: the leeway for speeds, pressures or compression is far from being exhausted.
  • The increase in efficiency is not primarily hampered by physical limits, as is the case with internal combustion engines, but is mainly determined by a suitable design of the heat exchangers. In particular, there are no limits to the optimization of the regenerator: For example, by increasing the shell diameter, the number of inner tubes and the heat transfer surface can be increased, which at the same time reduces the flow velocity and pressure loss.