http://ir.mu.ac.ke:8080/jspui/handle/123456789/23 2026-04-20T18:11:32Z http://ir.mu.ac.ke:8080/jspui/handle/123456789/10001 Mathematical modeling and parameter estimation for an optimal Solar food dryer Korkoren, Kenneth Cheruiyot Food shortage in most countries is not only associated with unfavorable weather conditions, but also significantly blamed on ineffective post-harvest handling of food. This calls for an urgent need to address food insecurity in Kenya, in line with Vision 2030 and Government Big 4 Agenda. Eminent threat caused by post-harvest losses due to inadequate drying and poor storage is responsible for up to 40-60% loses of agricultural produce each season. In order to address this issue, this project seeks to model and simulate the characteristics of a solar dryer for the purpose of designing an effective and sustainable, low-cost thermal solar dryer suitable for dehydrating a variety of agricultural produce to ensure prolonged shelf life hence reduce losses. The proposed model is to be formulated using mathematical equations describing integration of four divisions, namely; solar heat collector, circulation of fluid in insulated closed loop pipe network, heat exchangers to generate heated air supplied to the drying chamber and dryer, equipped with humidity control systems, temperature, mass flow rate and energy balance. The mathematical model was formulated and simulation done in order to realize the objectives of delivering a solar drier suitable for drying a wide variety of food products. The simulation results showed that, a solar panel of it was found that a solar collector with aperture area of 𝐴𝑐 = 14.4𝑚2 and volume of 𝑉𝑐 = 500𝑙, when exposed to solar irradiation of 𝐼𝑐 = 1.367𝐾𝑊/𝑚2 at 𝜂𝑐 = 80% efficiency is able to heat water from 𝑇𝑖𝑛 = 220𝐶 to 𝑇𝑐𝑜 = 700𝐶 in 12 hours at a flow rate of 𝑣𝑐 = 1.128𝑙/𝑠, and cumulatively to1300𝐶 in 6 days. This energy if transmitted by insulated pipes to a set of 5 heat exchangers each of 𝐴 = 1𝑚2, and radiative heat transfer coefficient ℎ𝑟 = 100𝑊/𝑚2𝐾 cumulatively dissipates hot air of 2300𝐶 at 𝑣= 250𝑐𝑚3/𝑠, 1300𝐶 at � �=1000𝑐𝑚3/𝑠 and 900𝐶 at 𝑣= 2000𝑐𝑚3/𝑠 air mass flow rate. This output temperatures of dry air are regulated as desired according to the specifications of the food products to be dried. During the night or on cloudy day with minimum or no solar insolation, alternative supplementary source of heat is obtained from petroleum cooking gas, which is regulated automatically depending on the level of solar insolation. It is found that the optimal cost of the gas is 𝐾𝑠ℎ 180/𝑑𝑎𝑦 as opposed to 𝐾𝑠ℎ 560/𝑑𝑎𝑦 when used alone. This is over 67.86% reduction in cost, which makes the use of solar an ideal green energy. 2025-01-01T00:00:00Z http://ir.mu.ac.ke:8080/jspui/handle/123456789/10000 The minimal nilpotent cover of the finite symmetric, alternating and dihedral groups Kimeu, Arphaxad Ngwava Given a group G, a minimal nilpotent cover is a family of subgroups of G of minimal size subject to the property that all of the subgroups are nilpotent and their union is equal to the group G. A maximal non-nilpotent subset is a subset of G of largest size such that for any two distinct elements, the subgroup they generate is not nilpotent. There has been e ort to investigate covers of groups with particular structure, speci cally normal and abelian covers; much has not been done on nilpotent covers. The main objective of this study was to conduct an analysis of minimal nilpotent cover for each of the three families of groups; nite symmetric group Sn, alternating group An and dihedral group D2k, which are ubiquitous throughout the discipline of mathematics. The speci c objectives were to; investigate nilpotent covers, analyse their relationship with non-nilpotent sub sets, investigate the size of minimal nilpotent cover and maximal non-nilpotent subset and apply group theoretic properties in determining a general formula for the size of the minimal nilpotent cover for each family. The analysis was carried out by means of mathematical proofs based on logical approach together with results from computer algebra package GAP and properties of permutations as well as group action on a k-gon . It has been established that except for A9, the size of minimal nilpotent cover for each of Sn, An for n = 34 10, and D2k, coincide with size of maximal non-nilpotent subset. For D2k, a general formula is produced for k = 34 . The results of this study bene t scientists in identi fying redundancy in the analysis of possible organic molecular structures, solve molecular conformation problems as well as mathematicians in expanding knowl edge in group theoretic concepts. For Sn, only for n = 34 10 were analyzed since as n grows large, it becomes complex and even computer algebra package GAPcannot generate any results. We therefore recommend an investigation on a general formula for minimal nilpotent cover of the nite symmetric group Sn as well as the nite alternating group, An 2025-01-01T00:00:00Z http://ir.mu.ac.ke:8080/jspui/handle/123456789/9913 Asymptotic behaviour of solutions of discretised unbounded positive symmetric and dirac operators on Hilbert Spaces Owino, Benard Odhiambo Stability analysis has been investigated for deficiency indices and spectrum of sym- metric higher order differential operators. It has been established that deficiency indices and discrete spectrum are stable under bounded perturbations whereas sin- gular continuous spectrum is not stable even under finite rank perturbations. How- ever, little has been done to establish the stability of existence of positive self-adjoint operator extensions under unbounded perturbations. Similarly, asymptotic analysis for discretised Dirac operators with unbounded coefficients was limited. This study has added to existing knowledge on the related areas as well as analysed the stability of national income using asymptotics of the solutions. The main goal of this research was to analyse the stability of existence of positive self-adjoint operator extension of symmetric operators under unbounded perturbations and asymptotic behaviour of solutions of discrete Dirac operators. The specific objectives of this study were to: 1) Construct positive perturbed difference operators with unbounded potentials, 2) Analyse the necessary and sufficient conditions for the stability of existence of positive self-adjoint operator extension under unbounded perturbations, 3) Inves- tigate the spectral properties of self-adjoint operator extensions of the perturbed difference operators with unbounded coefficients, and 4) Examine the asymptotic behavior of solutions of the discretised Dirac system with unbounded potentials. In- ner products of Hilbert spaces was applied to obtain coefficients growth conditions for the construction of positive symmetric difference operators. Asymptotic sum- mation based on discretised Levinson theorem, and von Neumann theorem for the existence of self-adjoint operator extensions were used to determine the existence of the positive self-adjoint operator extensions of the positive symmetric operators. The M-matrix of square summable eigensolutions was constructed and its spectral measures applied to determine the spectral properties of the positive self-adjoint op- erator extensions. Finally, the matrix asymptotic summation was applied to obtain the asymptotic properties of the Dirac difference operator. Suppose L 1 and L 2 are fourth and second-order symmetric difference operators with L = L 1 –zL 2 , where z is a real constant, and if the leading coefficient of L 1 dominates the coefficients of L 2 , then L 1 and L are positive symmetric operators. Furthermore, the deficiency indices of L 1 and L are (n, n), 2 ≤ n ≤ 4, with each operator L 1 and L having positive self-adjoint operator extensions even if their coefficients are unbounded. The abso- lutely continuous spectra of H 1 and H, the positive self-adjoint operator extensions of L 1 and L respectively are subsets of (0, ∞) of spectral multiplicity two whenever the leading coefficient of L 1 is unbounded. The solutions of the Dirac system were the product of their transforming matrices and the direct product of the eigenval- ues matrix with asymptotic behavior of the solutions determined by the unbounded potentials. These results can be applied by economists to study the stability of national income via the Keynesian model. Due to the complexity in computations of roots of polynomials of higher degrees more than four, similar analysis could not be investigated for order six or more. It is thus recommended that order six or more could be analysed in future using numerical analysis techniques. 2025-01-01T00:00:00Z http://ir.mu.ac.ke:8080/jspui/handle/123456789/9858 Occurrence of selected antibiotics in sediments and surface water of river sosiani, Eldoret, Kenya and their sorptive removal using biochars Chemtai, Catherine Antibiotics as therapeutics have been widely used in both human and veterinary medicine. Their indiscriminate use and improper disposal have led to environmental pollution, exerting adverse effects on non-target organisms. Various methods have therefore been developed to remove antibiotics from environmental matrices. Adsorption using bio-based adsorbents has been a preferred method for the removal of antibiotics from water as it is eco-friendly, cost effective and relatively simple. The general objective of this study was to assess the occurrence of selected antibiotics in water sediments, and surface water of River Sosiani, Eldoret, Kenya, and their sorptive removal using modified and unmodified biochar materials derived from water hyacinth and millet husks. The specific objectives of the study were; to quantify selected antibiotics in sediments and surface water from Sosiani River in Eldoret, to prepare and characterize chemically modified and unmodified biochar materials from water hyacinth and millet husks; to determine the sorption capacity of ciprofloxacin (CIP), sulfamethoxazole (SMX) and penicillin G (PNG) antibiotics from water using the prepared biochars, and to establish sorption isotherms and kinetics for adsorption of CIP, SMX and PNG by the biochar materials. The biochars were prepared by slow pyrolysis at 350 °C and 500 °C, modified using potassium hydroxide and characterized using Fourier Transform-Infrared (FT-IR) spectroscopy and scanning electron microscopy (SEM). Batch sorption equilibrium experiments were used to determine the capacity and efficiency of adsorption and data was fitted into different isotherm and kinetic models. Quantification of the antibiotics was done using liquid chromatography tandem mass spectrometry. The results showed that 30 compounds belonging to eight antibiotics classes: fluoroquinolones, sulfonamides, macrolides, penicillins, nitroimidazoles, lincosamides, diaminopyrimidines, and salfones were present in surface water and sediments from River Sosiani. Detectable concentrations of antibiotics in water samples ranged from 0.1 - 247 ng L -1 and 0.01 - 974 μg kg -1 in the sediments. Compared to other classes of compounds, fluoroquinolones were found in higher amounts in both water (56.02 ng L -1 ) and sediment samples (16 μg kg -1 ). Sulfamethoxazole had the highest concentration in water (247 ng L -1 ), whereas penicillin G showed the highest concentrations in sediments (414 - 974 μg kg -1 ). The results of FT- IR and SEM showed that there were differences in the physical and chemical properties of the biochar pyrolyzed at 350°C and 500 °C; hence differences in observed sorption characteristics. Biochar prepared at high temperature (BC500) had high adsorption capacity (> 80%) for the three antibiotics, while activated biochar showed high removal efficiency (83-88%) compared to non-activated biochar (50-70%). Sorption kinetics was best described by pseudo-second-order kinetics (R 2 > 0.99), while the sorption process could be best described by the Freundlich isotherm (R 2 > 0.97), and intra-particle diffusion models. Therefore, multilayer sorption of tested antibiotics on heterogeneous biochar surfaces was the most plausible sorption mechanism. The study showed a high prevalence of pharmaceuticals in surface water and sediments, and are therefore potential ecological hazards. Biochar from both feedstocks showed potential as alternatives to commercial activated carbon for the removal of CIP, PNG and SMX from water. The occurrence of antibiotics in surface water and sediments of the river differed widely according to classes of the pharmaceuticals and was mainly influenced by anthropogenic activities within the sampled location. Continuous monitoring of these pollutants is imperative to inform the public on their fate in the environment and for the enactment of policies for interventions, and sustainable mitigation strategies. 2025-01-01T00:00:00Z