In this study, we define the unrestricted Pell and Pell-Lucas quaternions. We give generating functions, Binet formulas and some generalizations of well-known identities such as Vajda’s, Catalan’s, Cassini’s d’Ocagne’s identities.

Application of nanoparticles have been proven to aid heat transfer in engineering systems. This work experimentally investigated the performance of a domestic refrigerator under the influence of Al₂O₃ nanoparticles dispersed in mineral oil based lubricant at different charges (40, 60 and 80 g) of LPG refrigerant. The performance of the system was then investigated using test parameters including: power consumption, evaporator air temperature (pull-down time), to attain the specified International Standard Organisation (ISO) requirement for standard evaporator air temperature with small refrigerator size. Results showed improved pull down time and steady state evaporator air temperatures for the nano-lubricant based LPG. Improvement of about 11.79% in coefficient of performance (COP) was obtained with Al₂O₃-lubricant based LPG at 40g charge on the refrigerator system, while reduction of about 2.08% and 4.41% in COP were observed at 60 and 80 g charge of LPG based on Al₂O₃-lubricant respectively. Furthermore, reduction of about 13.4% and 19.53% in the power consumption of the system were observed at 40 and 60g charges of Al₂O₃-lubricant based LPG, whereas at 80 g, an increase of about 1.28% was recorded. Using Al₂O₃-LPG nano-refrigerant in domestic refrigerators is economical  and also a better alternative to pure LPG.

This research study explores the addition of chromium (Cr⁶⁺) ions as a nucleating agent in the alumino-silicate-glass (ASG) system (i.e., Al₂O₃-SiO₂-MgO-B₂O₃-K₂O-F). The important feature of this study is the induction of nucleation/crystallization in the base glass matrix on addition of Cr⁶⁺ content under annealing heat treatment (600 ± 10 °C) only. The melt-quenched glass is found to be amorphous, which in the presence of Cr⁶⁺ ions became crystalline with a predominant crystalline phase, Spinel (MgCr₂O₄). Microstructural experiment revealed the development of 200–500 nm crystallite particles in Cr⁶⁺-doped glass-ceramic matrix, and such type microstructure governed the mechanical properties. The machinability of the Cr-doped glass-ceramic was thereby higher compared to base alumino-silicate glass (ASG). From the nano-indentation experiment, the Young’s modulus was estimated 25(±10) GPa for base glass and increased to 894(±21) GPa for Cr-doped glass ceramics. Similarly, the microhardness for the base glass was 0.6(±0.5) GPa (nano-indentation measurements) and 3.63(±0.18) GPa (micro-indentation measurements). And that found increased to 8.4(±2.3) (nano-indentation measurements) and 3.94(±0.20) GPa (micro-indentation measurements) for Cr-containing glass ceramic.