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Compact sizing of fin-and-tube heat exchangers is quite possible by integrating longitudinal vortex generators with the baseline fins. As geometry of the vortex generators is an important design parameter, the thrust of this three-dimensional numerical investigation is to identify the most favorable geometry of winglet type generators, if it exists. For that purpose, a thermo-hydraulic assessment of all the geometric designs, without any bias in the selection of potential contenders, is obligatory. Although past studies have reported the effect of winglet geometry on the performance augmentation, the selected designs were either biased or too less to draw conclusions. Each winglet design may be identified based on the geometric aspect ratio of the generators, which is varied over a wide range (0.5–1.5) in this study. Multiple favorable positions are considered, for installing the winglets around the tubes, before arriving at the favorable geometric design(s). Although highest thermal augmentation is delivered by the generators bearing smallest aspect ratio, linearized highest thermo-hydraulic augmentation is achieved when the ratio equals 0.85 irrespective of the generators’ position. After ascertaining the promising designs, their applicability is examined over a range of Reynolds number which spans from 2122 to 6367. Later part of the study discusses the change in the thermal and flow characteristics of the heat exchanger, due to variations in the winglet geometry. A study of the flow structures makes it evident that the spread of the longitudinal vortices widens, with eventual dissipation, in the flow direction, which has a discernible effect on the heat flux distribution over the fin surface. The study pertaining to the local changes suggests that the surfaces of both fins as well as the tubes undergo heat transfer augmentations, and the degree of augmentation grows with the decreasing aspect ratio of the vortex generators.

A microchannel with a 90° bend is one of the essential components of a complex microchannel in microdevices. Therefore, a fundamental understanding of flow through a microchannel with bends is required. This paper investigates gaseous flow experimentally and numerically through a microchannel with bends in the slip regime. A microdevice with 20 microchannels of dimensions 393.96 × 1.03 × 25 200 µm3 with double 90° bends has been fabricated using standard silicon microfabrication processes. These channels have a common inlet and outlet plenum, and nitrogen gas is used as the working fluid. The mass flow rate through the microchannel is measured for various pressure ratios ranging between 1.30 and 1.55 at different outlet Knudsen numbers in the slip regime. Three-dimensional simulations are performed for rarefied isothermal flow using Ansys fluent 18.1 with slip boundary conditions on the top and bottom walls. The slip effect is quantified by comparing the measured data of the mass flow rates with the corresponding no-slip boundary conditions. To the authors’ best knowledge, experimental data in the slip regime covering the Knudsen number range 0.0662–0.0805 have not been reported previously. The gas flow was modelled numerically using the first-order model with an appropriate slip coefficient (C 1 = 1.15), and the equivalent straight length per bend in the slip regime was also calculated.

Reactivity controlled compression ignition (RCCI) is an emerging premix low temperature combustion philosophy. Contemporary understanding suggests that RCCI concept with a premix high octane (low reactivity) fuel and direct injected high cetane (high reactivity) fuel ensures in-cylinder stratification of equivalence ratio as well as fuel reactivity. This stratification of reactivity coupled with partial premixing ensures simultaneous reduction of oxides of nitrogen and smoke. Furthermore, the induced delay in combustion phasing and high compression ratio ensures diffusive flames away from piston surface resulting in higher thermal efficiency. In the present work, an experimental investigation was carried out using port injected gasoline and anhydrous ethanol as low reactivity pilot fuels and direct injected diesel as high reactivity main fuel under various energy share. A comparative study of both the pilot fuels were carried out in terms of engine performance, emission and in-cylinder behavior in both representative and statistical perspective.

In this paper, we have investigated spatially homogeneous and anisotropic Kantowski–Sachs cosmological model in Lyra (Math. Z. 54:52, 1951) manifold in the presence of bulk viscous fluid containing one dimensional Cosmic strings. To obtain a determinate solution of the field equations, we have used some plausible physical conditions. The solution obtained represents a bulk viscous string cosmological model in Lyra geometry. Some physical and geometrical properties of the model are also discussed.

Modern coal fired power plants are required to handle a variety of coal types and accommodate large load changes. Coal mills grind the coal to required fineness and primary air dries and supplies the pulverized fuel to the burners. The dynamic response of coal mills is poor due to simple controls and various faults occurring inside the milling system. In this paper, an approach for time series prediction of n-step ahead values of important variables associated with the milling system is provided. A simple, data driven, non parametric technique i.e. k-NN regression is used for the prediction. The prediction of mill variables is helpful for improving controls and optimizing the mill operation. The proposed approach is applied for 5 minute ahead prediction and validated using the actual data obtained from a coal fired power plant in Gujarat, India.

Steam boiler provides appropriate quality and quantity of steam to the turbine for power generation. In drum type boiler, water and steam flow in the drum is controlled in such a way that the water level in the drum is maintained fairly at a preset value. While very low water level in the drum may lead to starvation and explosion of boiler, too high level may cause carryover of water particle and damages the super-heaters. Therefore Drum level is an important parameter for safe operation of a boiler. Drum level in a drum type boiler is influenced by number of factors, such as sudden change in load demand and changed in heat input to the boiler etc. During Boiler Feed Pump (BFP) runback, the boiler steam generation is reduced through fuel cut off while steam out flow to the turbine is controlled by pressure controller. Under such circumstance, due to mismatch between steam out and feed water in, it becomes difficult in maintaining water level in the drum to its pre set value. This paper has suggested an innovative drum level control by adopting the concept of related rate variation. The related rate module uniquely co-relates two independent time varying parameters namely steam flow and feed water flow in real time and accordingly apply correction of steam flow to match steam and water flow during BFP runback. Simulation result shows real time correction of steam flow maintains drum level within its safe operating limit during BFP runback. The innovative related rate module can be used as a control block in the future boiler controls for coordination between many parameters varying independently with respect to time.

Given an integer n ≥ 2, let λ(n) := (log n)/(log γ(n)), where γ(n) = Πp|n p, stand for the index of composition of n, with λ(1) = 1. We study the distribution function of (λ(n) – 1) log n as n runs through particular sets of integers, such as the shifted primes, the values of a given irreducible cubic polynomial and the shifted powerful numbers.

A CHARACTERIZATION OF UNITARY CONVOLUTION THROUGH ITS NON-TRIVIAL IDENTICAL EQUATION

SOME PROPERTIES OF PERFECT PARTITIONS

ON A CLASS OF ψ-PRODUCTS PRESERVING MULTIPLICATIVITY — II

The asymptotic expansion of ]]> ]]> ]]> \sum\limits_{n\le x}e^*(n)$ is given, where $e^*(n)$ is defined by $\sum{\frac{{e^*(n)}}{{n^s}}}\eg \prod\limits_{n=2}^\infty (1-1/n^s)$.

We give the asymptotics of the sum \(\sum\nolimits_{x \leqslant n \leqslant x + h} {f(n)}\), where \(h \gg x^{7/12 + \varepsilon }\), for the multiplicative functions \(f(n) = z^{\omega (n)} , z^{\omega (n)} |\mu (n)|, 1/d_k (n)\), and \(nd(n)\).

The amount of pine needles (pinus roxburgii ) potentially available for use as energy feedstock in the Central Himalayan state of Uttarakhand in India has been estimated. It involves estimating the gross annual amount of pine needle yield followed by a comprehensive identification and quantification of the factors that affect the net annual pine needle yield available as energy feedstock. These factors include considerations such as accessibility, alternative uses, forest fires, other losses, etc., that are influenced by aspects ranging from physical constraints to traditional societal traits. Tree canopy cover method has been used for estimating the gross annual pine needle yield. The information on canopy density is obtained from remote sensing data, that forms the basis for forest classification. The annual gross pine needle yield has been estimated at 1.9 million tonnes while the annual net pine needle yield at 1.33 million tonnes. The annual primary energy potential of pine needles available as energy feedstock has also been estimated. For annual net energy potential estimation, thermal and electrical routes are considered. Electrical energy generation from pine needles using thermochemical conversion has been examined and the corresponding potential for electricity generation been estimated. An installed capacity of 789 MW can be supported with pine needles feedstock for supplying electricity in rural areas for five hours a day. For round the clock generation, an installed capacity of 165 MW can be supported by the pine needle energy feedstock.

We have investigated the influence of the incorporation of Ca and Nd ions on the structural and superconducting properties of the system by Rietveld refinement of x-ray diffraction patterns of seven samples, i.e. , 0.10, 0.20, 0.30, 0.40, 0.50 and 0.60 together with a.c. susceptibility and oxygen content measurements. X-ray data confirm the occurrence of a single-phase tetragonal structure with a space group of for all the seven samples and all Nd ions exclusively occupy La sites. Ca substitutes onto the La sites with a concomitant displacement of La onto the Ba sites for and for ; most of the Ca, %, enters La sites with some fraction occupying Ba sites. Samples with are superconducting and is between 42 and 80 K. With increasing x = y, both the hole concentration in the -O layers and increase to a maximum value of 0.309 and 80 K respectively at . An approximate correlation between and the hole concentration is suggested.

the following assertion is proved. If f is a completely multiplicative function taking values of modulus 1, such that the set e of limit points of \(f(n + 1)\bar f(n)\) has k ≨ 3 distinct values, then f(n) = nirF(n) with some r e R, and Fk(n) = 1 for every n.

Let l ( n ) l(n) denote, as usual, the length of a shortest addition chain for a given positive integer n. The most famous unsolved problem in addition chains is Scholz’s 1937 conjecture that for all natural numbers n , l ( 2 n − 1 ) ≤ l ( n ) + n − 1 n,l({2^n} - 1) \leq l(n) + n - 1 . While this conjecture has been proved for certain classes of values of n, its validity for all n is yet an open problem. In this paper, we put forth a new conjecture, namely, that for each integer n ≥ 1 n \geq 1 there exists an addition chain for 2 n − 1 {2^n} - 1 whose length equals l ( n ) + n − 1 l(n) + n - 1 . Obviously, our conjecture implies (and is stronger than) Scholtz’s conjecture. However, it is not as bold as conjecturing that l ( 2 n − 1 ) = l ( n ) + n − 1 l({2^n} - 1) = l(n) + n - 1 , which is known to hold, so far, for only the twenty-one values of n which were obtained by Knuth and Thurber after extensive computations. By utilizing a series of algorithms we establish our conjecture for all n ≤ 128 n \leq 128 by actually computing the desired addition chains. We also show that our conjecture holds for infinitely many n, for example, for all n which are powers of 2.

Let p*(n) denote the number of product partitions, that is, the number of ways of expressing a natural number n > 1 as the product of positive integers ≥ 2, the order of the factors in the product being irrelevant, with p*(1) = 1. For any integer if d is an ith power, and = 1, otherwise, and let . Using a suitable generating function for p*(n) we prove that

On the set F of complex-valued arithmetic functions we construct an infinite family of convolutions, that is, binary operations ψ of the formso that (F, +, ψ) is a commutative ring, for which the unity is unbounded. Here + denotes pointwise addition.

Steam turbines in coal fired power plants are designed for some fixed operating conditions. When operated at those fixed conditions, the turbines are supposed to give maximum cycle efficiency. But off-design conditions occur several times due to condenser variation, environmental change, plant aging, etc. Performance at those off-design conditions affects economy of operation. Hence prediction of performance at off-design conditions is necessary. In this work a 210 MW steam turbine is designed using available design data. Calculated data matched available data quite satisfactorily. Design condition heat cycle data and other available technical data were used for design. Three off-design conditions, varying steam flow rate, varying condenser pressure, and varying inlet steam parameters were considered for study of off-design performance. These effects were suitably incorporated to recalculate performance of the same turbine at these conditions. Performance characteristics are obtained by varying one off-design condition at a time. Specific examples also are solved by simultaneously considering all three mentioned off-design variables using actual field data. Off-design performance predictions were found to be satisfactory.Copyright © 2007 by ASME

If n is a natural number, then a partition of n is a representation of n as the sum of one or more natural numbers. For example, the equation: $$ 10 = 5 + 3 + 2 $$ (1) defines a partition of 10. It is customary to use the following notation: a partition of n will be given by $$ n = {\lambda _1} + {\lambda _2} + \cdots + {\lambda _r} $$ (2) where the λ i are natrual numbers such that $$ {\lambda _1} \ge {\lambda _2} \ge \cdots \ge {\lambda _r} $$ (3) . The summands λ i are called the parts of the partition, and the integer r is the number of parts. A good reference on partitions is [1].

The celebrated seventy year old, innocent looking problem of D.H. Lehmer [5] asking for composite numbers, if any, satisfying the relation φ(n)|(n − 1), where φ(n) is the Euler totient, is still unsolved. This is easily seen to be equivalent to asking the

The product expansion given by \(\prod\limits_{n = 2}^\infty {\left( {1 - {n^{ - 3}}} \right)} = 1 + \sum\limits_{n = 2}^\infty {e*} \left( n \right){n^{ - 3}},\operatorname{Re} \;s > 1,\)defines e*(n) as a product partition function. Namely, e*(n) represents the excess of the number of representations of n > 1 as a product of an even number of distinct factors (also called parts in the sequel) > 1 over those into an odd number of such parts, the order of the parts in the product being ignored. To our knowledge, this function has not been considered in the literature so far. In this preliminary study, we obtain in Sections 1 and 2 a basic property of this function. Let f(k) denote the value of e*(p 1…p k ), where p 1,…, P k are distinct arbitrarily chosen primes. We show that as \(k \to \infty ,\;\log \mid f\left( k \right)\mid /k\)is unbounded, and in fact \(\mathop {\lim \;\sup }\limits_{k \to \infty } \;\frac{{\log \mid f\left( k \right)\mid }}{{k\;\log \;k}} = 1.\)More generally, a similar result holds for a certain weighted partition function associated with e*(n). In Section 3 we make several remarks on the expansion of infinite products associated with certain additive and multiplicative partition functions. Some open problems are mentioned at the end of this note.

Let K be a totally real agebraic number field of degree n. Let K (i) (1 ≤ i ≤ n) be the conjugate fields of K.