A Generalization of Partition Identities for First Differences of Partitions of $n$ Into at Most $m$ Parts

We show for a prime power number of parts $m$ that the first differences of partitions into at most $m$ parts can be expressed as a non-negative linear combination of partitions into at most $m-1$ parts. To show this relationship, we combine a quasipolynomial construction of $p(n,m)$ with a new partition identity for a finite number of parts. We prove these results by providing combinatorial interpretations of the quasipolynomial of $p(n,m)$ and the new partition identity. We extend these results by establishing conditions for when partitions of $n$ with parts coming from a finite set $A$ can be expressed as a non-negative linear combination of partitions with parts coming from a finite set $B$.

CFD Analysis of Modified Rushton Turbine Impeller

Mixing units is one of the prime components in various chemical processing industries. The mixing equipment produces a product by combing different liquids at various properties. Generally, the mixing units are axial and radial flow types. The rushton turbine is one of the radial flow types and had higher power number due to fluid resistance as disadvantage. This works aims to investigate the four bladed rushton turbine impellers at eight angles for finding optimum power without compromising radial flow patterns. The shaft eccentricity hydrodynamics effect is also analysed on the unbaffled stirred vessels. The variance of eccentric and coaxial agitation is clearly evaluated on the several power number and impeller blade angle by experimental and numerical analysis. The CFD analysis is made on the mixing chamber to find the optimum inputs of the mixing chamber numerically by varying the blade angles.

Validation of Novel Lattice Boltzmann Large Eddy Simulations (LB LES) for Equipment Characterization in Biopharma

Detailed process and equipment knowledge is crucial for the successful production of biopharmaceuticals. An essential part is the characterization of equipment for which Computational Fluid Dynamics (CFD) is an important tool. While the steady, Reynolds-averaged Navier–Stokes (RANS) k − ε approach has been extensively reviewed in the literature and may be used for fast equipment characterization in terms of power number determination, transient schemes have to be further investigated and validated to gain more detailed insights into flow patterns because they are the method of choice for mixing time simulations. Due to the availability of commercial solvers, such as M-Star CFD, Lattice Boltzmann simulations have recently become popular in the industry, as they are easy to set up and require relatively low computing power. However, extensive validation studies for transient Lattice Boltzmann Large Eddy Simulations (LB LES) are still missing. In this study, transient LB LES were applied to simulate a 3 L bioreactor system. The results were compared to novel 4D particle tracking (4D PTV) experiments, which resolve the motion of thousands of passive tracer particles on their journey through the bioreactor. Steady simulations for the determination of the power number followed a structured workflow, including grid studies and rotating reference frame volume studies, resulting in high prediction accuracy with less than 11% deviation, compared to experimental data. Likewise, deviations for the transient simulations were less than 10% after computational demand was reduced as a result of prior grid studies. The time averaged flow fields from LB LES were in good accordance with the novel 4D PTV data. Moreover, 4D PTV data enabled the validation of transient flow structures by analyzing Lagrangian particle trajectories. This enables a more detailed determination of mixing times and mass transfer as well as local exposure times of local velocity and shear stress peaks. For the purpose of standardization of common industry CFD models, steady RANS simulations for the 3 L vessel were included in this study as well.

Effect of Impeller Design on Power Characteristics and Newtonian Fluids Mixing Efficiency in a Mechanically Agitated Vessel at Low Reynolds Numbers

The mixing process is a widespread phenomenon, which plays an essential role among a large number of industrial processes. The effectiveness of mixing depends on the state of mixed phases, temperature, viscosity and density of liquids, mutual solubility of mixed fluids, type of stirrer, and, what is the most critical property, the shape of the impeller. In the present research, the objective was to investigate the Newtonian fluids flow motion as well as all essential parameters for the mechanically agitated vessel with a new impeller type. The velocity field, the power number, and the pumping capacity values were determined using computer simulation and experimental measurements. The basis for the assessment of the intensity degree and the efficiency of mixing had to do with the analysis of the distribution of velocity vectors and the power number. An experimental and numerical study was carried out for various stirred process parameters and for fluids whose viscosity ranged from low to very high in order to determine optimal conditions for the mixing process.

Performance Analysis of Conventional, Hybrid and Optimal PV Array Configurations of Partially Shaded Modules

In this paper, modeling and performance analysis of conventional configurations are Series-Parallel (SP), BridgeLinked (BL), Honey-Comb (HC), Total-Cross-Tied(TCT) and proposed hybrid configurations are SP-TCT, BL-TCT, HC-TCT, BL-HC and modified BL(MBL), modified HC(MHC), proposed optimal interconnection type configurations of a 5x5 size solar PV array under ten different partial shading cases it causes shading losses and compare the best configuration with respect to array power, number of interconnections or ties required between shaded modules in the array. The proposed optimal interconnection method reduces the number of ties required between modules and these ties are based on the position of number of shaded modules in the entire solar PV array. For the performance analysis of above 11 configurations, total ten shading cases are considered and compare the result with one un-shaded case-U of an irradiance 1000 W/m2 . The PV module parameters of Vikram Solar ELDORA 270 are used for modeling of above 11 conventional and proposed PV array configurations and simulate the models in MATLAB/ Simulink software.

CFD Study on the Flow Field and Power Characteristics in a Rushton Turbine Stirred Tank in Laminar Regime

A computational fluid dynamics (CFD) simulation was performed to study the hydrodynamics characteristics in a Rushton turbine stirred tank in laminar regime. The effects of operating condition, working medium and geometrical parameter on the flow field and power number characteristics were investigated. It is found that the two-loop flow pattern is formed in laminar regime when the impeller is not very close to tank bottom, while its shape and size vary with Reynolds number and impeller diameter. For a given geometrical configuration, the flow pattern, power number and dimensionless velocity profile are mainly depended on Reynolds number, and do not change with working medium and scale-up for a constant Reynolds number. When impeller off-bottom clearance is too low and Reynolds number is relatively high, the fluid flow would transit from two-loop flow pattern to sing-loop flow pattern as that occurs in turbulent regime. Power number falls for larger impeller in laminar regime. Surprisingly, in laminar regime, power number in the baffled tank with small impeller is almost identical to that in the unbaffled tank.

A comparative study to ascertain the efficacy of intravenous dexmedetomidine versus intravenous tramadol to prevent post-operative shivering after laparoscopic surgery

Background: Shivering is commonly encountered both after regional and general anaesthesia (GA) with a little higher incidence in patients receiving GA. The aim of study was to compare the effectiveness of dexmedetomidine and tramadol in decreasing postoperative shivering in patients undergoing laparoscopic surgery.Methods: Total 120 patients were included in this study. In order to get a 5% level of significance and 80% power number of patients required in each group was 40, with a total of 120 patients. Randomization of groups was done based on closed envelope method. Patients were allocated into three groups group I, II and III of 40 patients each. Patients in group I and group II were administered 0.75 μg/kg of dexmedetomidine and 1.5 mg /kg of tramadol in 100 ml NS respectively half a before extubation, while patients in group III did not receive any pharmacological intervention.Results: All three groups were comparable regarding distribution of age, gender, ASA grade and temperature at beginning and end of surgery and were non-significant.Conclusions: Dexmedetomidine seems to possess anti-shivering properties and was found to reduce the occurrence of shivering in patients undergoing general anaesthesia with minimal side effects although its anti-shivering effect was not superior to tramadol.