A double bounded key identity for Goellnitz's (big) partition theorem

Given integers i,j,k,L,M, we establish a new double bounded q-series identity from which the three parameter (i,j,k) key identity of Alladi-Andrews-Gordon for Goellnitz's (big) theorem follows if L, M tend to infinity. When L = M, the identity yields a strong refinement of Goellnitz's theorem with a bound on the parts given by L. This is the first time a bounded version of Goellnitz's (big) theorem has been proved. This leads to new bounded versions of Jacobi's triple product identity for theta functions and other fundamental identities.Comment: 17 pages, to appear in Proceedings of Gainesville 1999 Conference on Symbolic Computation

The Optimum Position of Water Heat Transfer Coils Downstream of a Radial Swirler in a 20kW Heater

A 76mm outlet diameter radial swirler with 8mm vane depth was investigated in a 140mm combustor diameter condensing 20 kW ultra-low NOx boiler. The aim was to show that small turbulent flames could achieve compact ultra-low NOx water heating. Low NOx was achieved using lean well mixed low flame temperature combustion with a 0.7 equivalence ratio (Ø). Thermal NOx formation was also minimised by cooling the flame downstream of the swirler outlet. A water cooled heat transfer coil was traversed into the flame to determine how close to the swirler exit the heat transfer could occur, without a major increase in the combustion inefficiency. This was shown to be 70mm from the radial swirler throat outlet. Rapid fuel and air mixing was achieved using fuel injection through the wall of the 76mm swirler outlet throat, assisted by a 41mm diameter outlet orifice at the exit of the 76mm internal diameter wall fuel injector. This created swirling flow with higher axial velocities and a more concentrated high turbulence region downstream of the orifice outlet. A 4 mb burner pressure loss was used, which is typical of domestic forced draught combustion systems. The air inlet temperature was 400K, which is typical of reverse air flow cooled combustion chambers at domestic water heater conditions. The strong swirling flow interaction with the heat exchanger coil give an 89% thermal efficiency with the front of the coil 70mm from the swirler outlet. The emissions measurements showed that the combustion inefficiency was below 0.1%, the CO/CO2 ratio <0.001 and the NOx emissions were 5ppm at 0% oxygen with the heat exchanger at 70mm from the radial swirler outlet. This design easily met the 2018 EU legislation for eco-design of domestic water heaters

Influence of Fuel Injection Location in a Small Radial Swirler Low NOₓ Combustor for Micro Gas Turbine Applications

The influence of fuel injection location in a low NOₓ (1) micro-gas turbine [MGT] in the ∼50kWe (kW electric) size range was investigated, for NG and propane, to extend the power turn down using a pilot fuel injector. The low NOx main combustor (1) was a radial swirler with vane passage fuel injection and had ultra-low NOₓ emissions of 3ppm at 15% O2 at 1800K with natural gas, NG at a combustion intensity of 11.2 MW/m2bara (MW thermal). This was a 40mm diameter outlet eight bladed radial swirler in a 76mm diameter combustor, investigated at 740K air temperature at atmospheric pressure. However, power turn down was poor and the present work was undertaken to determine the optimum position of pilot fuel injection that would enable leaner mixtures to be burned at low powers. Central injection of pilot fuel was investigated using 8 radial outward holes. This was compared with pilot fuel injected at the 76mm wall just downstream of the 40mm swirler outlet. It was show that the central injection pilot was poor with a worse weak extinction than for radial passage fuel injection. The 76mm outlet wall injection was much more successful as a pilot fuel location and had a weak extinction of 0.18Ø compared with 0.34Ø for vane passage fuel injection. NOₓ emissions were higher for wall fuel injection, but were still relatively low at 16ppm at 15% oxygen for natural gas. This indicates that wall fuel injection could be combined with vane passage fuel injection to improve the micro-gas turbine low NOₓ performance across the power range

A generalization of the q-Saalschutz sum and the Burge transform

A generalization of the q-(Pfaff)-Saalschutz summation formula is proved. This implies a generalization of the Burge transform, resulting in an additional dimension of the ``Burge tree''. Limiting cases of our summation formula imply the (higher-level) Bailey lemma, provide a new decomposition of the q-multinomial coefficients, and can be used to prove the Lepowsky and Primc formula for the A_1^{(1)} string functions.Comment: 18 pages, AMSLaTe