Avaliação da potencialidade dos resíduos de gesso de revestimento incorporados em formulações de massas cerâmicas

In building, during the implementation process of major or even minor works, there is a considerable waste of plaster in the steps of coating, making it is a negative factor because of the loss of these processes constructive remains incorporated into buildings, as component, whose final dimensions are higher than those projected. Another negative factor is the disposal of waste gypsum in inappropriate places, thus contributing to the degradation of environmental quality, due to the leaching of this waste and may trigger the formation of sulfuric acid. Therefore, based on this picture, processing and reuse of waste coating, combined with the ceramics industry, which is a strong potential in the reuse of certain types of waste, promote mutual benefits. Thus the overall objective of this work is to conduct a search with scientific and technological aspects, to determine the effect of the incorporation of the residue of plaster for coating, from the building, the formulation of bodies for red ceramic. The residue of plaster coating was collected and characterized. They were also selected raw materials of two ceramic poles of the state of Rio Grande do Norte and formulations have been made with the intention of obtaining those with the best physical and mechanical properties, the residue was added the percentage of 5%, 10%, 15%, 20%, 25% and 30%, in the best formulation of ceramic industry 1 and, according the properties analyses, 5%, 10% and 15% as the best results of ceramic industry 2. The samples were sintered at temperatures of 850 ºC, 950 °C and 1050 °C, the heating rate of 5 ºC / min with isotherm of two hours. They were submitted to testing technology, such as lineal shrinkage, water absorption, apparent porosity, apparent density and bending resistence. The residue incorporation best results in the formulations of mass in red ceramic, were observed between the temperatures of 850 ºC and 950 ºC, in those formulations that have illite clays and medium plastic in their composition, in the range of 0% to 15% residue incorporated

Avaliação de concretos reforçados por confinamento com manta de fibra de vidro estrutural e pré-impreganada

Due to the occurrence of diseases in the use of structural reinforcements in composites, with presentation of concrete blanket detachment, has been identified the need to evaluate the performance of concrete reinforced with glass fiber. This study aims to evaluate these concretes by means of testing methodologies, using concrete with low resistance with structural reinforcement for confinement by preimpregnated glass fiber and traditional fiberglass blanket. The first stage of work was the development of methodologies for analysis, opting for four types, such as the acoustic survey, strength to compressive, the pull-off and ultrasound. Next, tests were carried out using the four selected methodologies in 30 of proof-of-specimens by 5x10 cm, 15 were reinforced with the traditional fiberglass blanket with 5specimens exposed to test a marine environment of marine coastline of Natal-RN and 15 were reinforced with a pre-impregnated glass fiber blanket, as well as 5specimens exposed to a test environment of the marine coastline of Natal-RN. After conducting the acoustic survey, it has been verified a lack of delaminating and air bubbles in the samples, confirming the absence of gross shortcomings in the implementation of the ribs both the traditional fiberglass blanket and in the preimpregnated fiber glass blanket. After carrying out methods of pull-off and compressive strengthening test it was observed that the reinforced proof-bodies with pre-impregnated glass blanket showed maximum stresses higher than the traditional fiberglass blanket; consequently a greater grip with the formation of a smaller area of . fracture, unlike traditional glass mat, which showed lower maximum stresses, with a greater area of fracture. It was also found that the traditional fiberglass blanket presented detachment of blanket-concrete interface, unlike the pre-impregnated fiberglass blanket, which showed a better grip on the blanket-concrete interface. In the trial of ultrasound there was no presence of cracks in the blanket-concrete interface, yielding to both blankets good compactness of the concrete. At the end of this work, they were developed and proposed two methods of testing for evaluation of reinforced concrete structures with composites, for standardization, the acoustic survey and pull-off

Estudo da adição do resíduo proveniente da extração de minério de ferro em argilas do Rio Grande do Norte

The mining industry is responsible for the generation of waste from their natural process of extraction. The mining impacts in urban areas are of special importance due to the high urban occupation, which are exacerbated due to the proximity of the mined areas and populated areas. Some solutions to wastedisposal have the potential to significantly reduce the environmental risks and liabilities, but represent higher costs in the stages of deployment and operation. The addition of mining waste as raw material in the development of commercial products reduces the environmental impacts, transforming the waste into a positive element in the generation of employment and income. This thesis studies the incorporation of waste iron ore in two clays, one from the ceramic industry of the City of Natal and the other from the ceramic industry of the Seridó Region, both in the State of Rio Grande do Norte, Brazil. Percentages of iron ore waste of 5%, 10% , 15%, 20%, 25% and 30% were used in the tested ceramic matrix. The two clays and the iron ore waste used as part of this investigation were characterized by X-ray diffraction tests, X-ray fluorescence tests, differential thermal analysis, thermogravimetric analysis and dilatometric analysis. The samples were sintered under temperatures of 850 °C, 950 °C and 1050°C at a heating rate of 5 °C/min with isotherms of two hours. The following tests were performed with the samples: linear shrinkage, water absorption, apparent porosity, apparent density, mass loss in fire and bending resistance in order to obtain their physical and mechanical properties. An amount of 5% of waste iron ore in the matrix clay at a temperature of 850 0C resulted in na increase of about 65% in the tensile strength of the clay samples from the Natal ceramic industry. A linear shrinkage of only 0.12% was observed for the samples, which indicates that the physical properties of the final product were not influenced by the addition of the waste

Obtenção de uma porcelana dielétrica a partir de matérias-primas do Rio Grande do Norte

The dielectric porcelain is usually obtained by mixing various raw materials proportions and is used in the production of electronic equipment for various applications, from capacitors of high and low Power to insulators for low, medium, high and extra high voltage, which are used in distribution lines and transmission of electricity.This work was directed to the s tudy of technological properties of technic porcelain, made from raw materials extracted from pegmatites found in the regions of Seridó and the Alto Oeste of Rio Grande do Norte, which are made of kaolin, quartz and feldspar, abundant and high quality in these regions. The technic ceramics were obtained by mixing in appropriate levels, kaolin, feldspar, quartz and clay, the last item from a pottery in the city of Sao Gonçalo do Amarante, Rio Grande do Norte. During the development the following characterizations correlated to raw materials were made: laser particle sizing, x-ray diffraction, DTA and TG. The compositions studied were formed by uniaxial pressing at a pressure of 50 MPa and sintered at temperatures ranging from 1150 to 1350ºC and levels (times) of sintering between 30, 60, 90 and 120 minutes. The characterization of the samples were taken from the analysis of weight loss, linear shrinkage, porosity, stoneware curve, bulk density, flexural strength of three points, SEM and X-ray diffraction, TMA, Dielectric and cross Resistivity. The studied materials can be employed in producing the objects used in electrical engineering such as: insulators for low, medium and high-voltage electrical systems, command devices, bushing insulation for transformers, power capacitors, spark plugs, receptacles for fluorescent and incandescent light bulbs and others

Estudo do efeito da adição da escória de aciaria no processamento e propriedades finais de telhas cerâmicas

This research presents an overview of the addition steelwork dust of ceramic shingles in order to contribute to the utilization use of such residue. The ceramic industry perspective in the Brazilian State of Piauí is quite promising. Unlike other productive sectors, the ceramic industry uses basically natural raw materials. Its final products are, in short, the result of transforming clay compounds. These raw materials are composed primarily of aluminum oxide, silicon, iron, sodium, magnesium, end calcium, among others. It was verified that steelwork dust is composed primarily of these same oxides, so that its incorporation in to structural ceramics is a very reasonable idea. Both clay and steelwork powder were characterized by AG, XRF, XRD, TGA and DTA. In addition, steelwork dust samples containing (0%, 5%, 10%, 15%, 20% and 25%) were extruded and burned at 800°C, 850°C, 900°C and 950°C. Then t echnological tests of linear shrinkage, water uptake, apparent porosity, apparent density and flexural strengthwere carried at. The results showed the possibility of using steelwork powder in ceramic shingles until 15% significant improvement in physical and mechanical properties. This behavior shows the possibility of burning at temperatures lower than 850ºC, thus promoting a product final cost reduction

Síntese de zeólita A a partir de diatomita como fonte de aluminossilicato

The synthesis of zeolites from natural sources of silicon and aluminum are promising alternative routes to obtain porous or zeolite MCM family. Such materials are typically used in catalytic processes and / or adsorption is to obtain new products or for separation and purification processes thereof. Environmental legislation is becoming stricter and requires the use of materials more efficient, aiming to achieve pollution prevention, by gas or liquid contaminants in the environment. In order to obtain a material with environmentally friendly features, this study aimed at the synthesis of zeolite A, from an amorphous sediment, diatomite, which is found in abundance in the northeast region of Brazil, may be substituted for conventional products the production of zeolite, involving higher costs. The methodology for obtaining the "Zeolite A" using as a source of silica and alumina diatomite is simple, since this is a source of silicon, not requiring therefore a structural driver, but also by heat treatment, only drying conventional to remove water. The "zeolite A" was obtained from diatomite, but as an intermediate step we obtained the sodalite. The characterization was made by the following techniques: EDX, XRD, FT-IR, SEM and determining a specific area by the BET method and the BJH method for checking the diameter of pores. By characterization of the obtained material was first demonstrated the achievement of sodalite and after modification of the same, there was obtained zeolite A

Estudo de massa cerâmica triaxial visando à otimização do ciclo de queima da indústria de cerâmica estrutural

Over recent years the structural ceramics industry in Brazil has found a very favorable market for growth. However, difficulties related to productivity and product quality are partially inhibiting this possible growth. An alternative for trying to solve these problems and, thus, provide the pottery industry the feasibility of full development, is the substitution of firewood used in the burning process by natural gas. In order to contribute to this process of technological innovation, this paper studies the effect of co-use of ceramic phyllite and kaolin waste on the properties of a clay matrix, verifying the possible benefits that these raw materials can give to the final product, as well as the possibility of such materials to reduce the heat load necessary to obtain products with equal or superior quality. The study was divided into two steps: characterization of materials and study of formulations. Two clays, a phyllite and a residue of kaolin were characterized by the following techniques: laser granulometry, plasticity index by Atterberg limits, X-ray fluorescence, X-ray diffraction, mineralogical composition by Rietveld, thermogravimetric and differential thermal analysis. To study the formulations, specifically for evaluation of technological properties of the parts, was performed an experimental model that combined planning involving a mixture of three components (standard mass x phyllite x kaolin waste) and a 23 factorial design with central point associated with thermal processing parameters. The experiment was performed with restricted strip-plot randomization. In total, 13 compositional points were investigated within the following constraints: phyllite ≤ 20% by weight, kaolin waste ≤ 40% by weight, and standard mass ≥ 60% by weight. The thermal parameters were used at the following levels: 750 and 950 °C to the firing temperature, 5 and 15 °C/min at the heating rate, 15 and 45min to the baseline. The results showed that the introduction of phyllite and/or kaolin waste in ceramic body produced a number of benefits in properties of the final product, such as: decreased absorption of water, apparent porosity and linear retraction at burn; besides the increase in apparent specific mass and mechanical properties of parts. The best results were obtained in the compositional points where the sum of the levels of kaolin waste and phyllite was maximal (40% by weight), as well as conditions which were used in firing temperatures of 950 °C. Regarding the prospect of savings in heat energy required to form the desired microstructure, the phyllite and the residue of kaolin, for having small particle sizes and constitutions mineralogical phases with the presence of fluxes, contributed to the optimization of the firing cycle.

Preparação e caracterização de filmes cerâmicos para cátodos de células a combustível de óxido sólido

Alternative and clean energy generation research has been intensified in last decades. Among the alternatives, fuel cells are one of the most important. There are different types of fuel cells, among which stands out intermediate temperature solid oxide fuel cell (IT-SOFC) matter of the present work. For application as cathode on this type of devices, the ceramic Ba0.5Sr0.5C0.8Fe0.2O3-δ doped with rare earth ions (Nd, Sm) have been quite promising because they show good ionic conductivity and operate at relatively low temperatures (500 - 800°C). In this work, Ba0.5Sr0.5Co0.8Fe0.2O3-δ, (BaSr)0.5Sm0.5Co0.8Fe0.2O3-δ and (BaSr)0.5Nd0.5C0.8Fe0.2O3-δ were obtained by modified Pechini method, making use of gelatin as polymerizing agent. The powders were characterized by X-Ray Diffraction (XRD), Temperature Programmed Reduction (TPR) and Scanning Electron Microscopy (SEM). The perovskite phase was observed in all X-ray patterns for the materials Ba0.5Sr0.5C0.8Fe0.2O3-δ doped with rare earth ions (Nd, Sm). The SEM images showed that the materials have a characteristics porous, with very uniform pore distribution, which are favorable for application as cathodes. Subsequently, screen-printed assymmetrical cells were studied by impedance spectroscopy, to assess the kinetics of the cathode for the reduction reaction of oxygen. The best resistance to the specific area was found for the cathode BSSCF sintered at 1050 °C for 4 hours with around 0.15 Ω.cm2 at 750 °C as well as cathodes BSNCF and BSCF obtained resistances specific area of 0.2 and 0.73 Ω.cm2, respectively, for the same conditions. The polarization curves showed similar behavior to the best cathodes BSSCF and BSNCF, such combination of properties indicates that the film potentially depict good performance as IT-SOFC cathodes

Preparação e caracterização de catalisadores à base de titânio suportado em MCM-41 para produção de compostos oxigenados através da pirólise catalítica do capim elefante

In recent years, the area of advanced materials has been considerably, especially when it comes to materials for industrial use, such as is the case with structured porosity of catalysts suitable for catalytic processes. The use of catalysts combined with the fast pyrolysis process is an alternative to the oxygenate production of high added value, because, in addition to increasing the yield and quality of products, allows you to manipulate the selectivity to a product of interest, and therefore allows greater control over the characteristics of the final product. Based on these arguments, in this work were prepared titanium catalysts supported on MCM-41 for use in catalytic pyrolysis of biomass, called elephant grass. The reactions of pyrolysis of biomass were performed in a micro pyrolyzer, Py-5200, coupled to GC / MS, the company CDS Corporation, headquartered in the United States. The catalysts Ti-MCM-41 in different molar ratios were characterized by XRD, TG / DTG, FT-IR, SEM, XRF, UV-visible adsorption of nitrogen and the distribution of particle diameter and specific surface area measurement by the BET method. From the catalytic tests it was observed that the catalysts synthesized showed good results for the pyrolysis reaction.The main products were obtained a higher yield of aldehydes, ketones and furan. It was observed that the best reactivity is a direct function of the ratio Si/Ti, nature and concentration of the active species on mesoporous supports. Among the catalysts Ti-MCM-41 (molar ratio Si / Ti = 25 and 50), the ratio Si / Ti = 25 (400 ° C and 600 ° C) favored the cracking of oxygenates such as acids , aldehydes, ketones, furans and esters. Already the sample ratio Si / Ti = 50 had the highest yield of aromatic oxygenates

Compósitos de matriz metálica à base níquel com adição de TaC e NbC produzidos via metalurgia do pó

Carbide reinforced metallic alloys potentially improve some important mechanical properties required for the overall use of important engineering materials such as steel and nickel. Nevertheless, improved performance is achieved not only by composition enhancement but also by adequate processing techniques, such as novel sintering methods in the case of powder metallurgy. The method minimizes energy losses in addition to providing uniform heating during sintering. Thus, the general objective of this study was to evaluate the density, hardness, flexural strength, dilatometric behavior and to analyze the microstructure of metal matrix composites based nickel with addition of carbides of tantalum and / or niobium when sintered in a conventional furnace and Plasma assisted debinding and sintering (PADS). Initially, were defineds best parameters of granulation, screening and mixing procedure. After, mixtures of carbonyl Ni and 5%, 10% and 15 wt.% NbC and TaC were prepared in a Y-type mixer under wet conditions during 60 minutes. The mixtures were then dried and granulated using 1.5 wt. % paraffin diluted in hexane. Granulates were cold pressed under 600 MPa. Paraffin was then removed from the pressed pellets during a pre-sintering process carried out in a tubular furnace at 500 °C during 30 min. The heating rate was 3 ºC/min. The pellets were then sintered using either a plasma assisted reactor or a conventional resistive tubular furnace. For both methods, the heating rate was set to 8 ºC/min up to 1150 °C. The holding time was 60 minutes. The microstructure of the sintered samples was evaluated by SEM. Brinell hardness tests were also carried out. The results revealed that higher density and higher hardness values were observed in the plasma-assisted sintered samples. Hardness increased with the concentration of carbides in the Ni-matrix. The flexural strength also increased by adding the carbides. The decline was larger for the sample with addition of 5% 5% TaC and NbC. In general, compositions containing added carbide 10% showed less porous and more uniform distribution of carbides in the nickel matrix microstructural appearance. Thus, both added carbide and plasma sintering improved density, hardness, flexural strength and microstructural appearance of the composites

Avaliação da incorporação combinada dos resíduos de cascalho de perfuração de poço de petróleo e de polimento do porcelanato em cerâmica vermelha

Among the industries, those that produce ceramic porcelain for use in construction industry and oil, during the exploration and production period, play an important role in the production of waste. Much research has been carried out both by academia and the productive sector, sometimes reintroducing them in the same production line that generated them, sometimes in areas unrelated to their generation, as in the production of concrete and mortar for the construction, for example, but each one in an isolated way. In this research, the aim is to study the combined incorporation of the waste drill cuttings of oil well and the residue of the polishing of porcelain, generated in the final stage of finishing of this product in a clay matrix, for the production of red pottery, specifically bricks, ceramic blocks and tiles. The clay comes from the municipality of São Gonçalo, RN, the drilling waste is from the Natal basin, in Rio Grande do Norte, and the residue of the polishing proceeds from a ceramic porcelain of the State of Paraíba. For this purpose, we used a mixture of a plastic clay with a non-plastic, in a ratio of 50% each, settling formulations with the addition of these two residues in this clay matrix. In the formulations, both residues were incorporated with a minimum percentage of 2.5% and maximum of 12.5%, varying from 2.5% each, in each formulation, which the sum of the waste be no more than 15%. It should be noted that the residue of the polishing of ceramic porcelain is a IIa class (not inert). The materials were characterized by XRF, XRD, TG, DTA, laser granulometry and the plasticity index. The technological properties of water absorption, apparent porosity, linear shrinkage of burning, flexural tensile strength and bulk density were evaluated after the sintering of the pieces to 850 °C, 950 °C and 1050 °C, with a burning time of 3 hr, 3 hr and 30 minutes, and 3 hr and 50 minutes, respectively, with a heating rate of 10 °C/minute, for all formulations and landing of 30 minutes. To better understand the influence of each residue and temperature on the evaluated properties, we used the factorial planning and its surfaces of response for the interpretation of the results. It was found that the temperature has no statistical significance at a 95% of reliability level in flexural tensile strength and that it decreases the water absorption and the porosity, but increases the shrinkage and the bulk density. The results showed the feasibility of the desired incorporation, but adjusting the temperature to each product and formulation, and that the temperatures of 850 °C and 950 °C were the one that responded to the largest number of formulations

Estudo da viabilidade do processo de reciclagem do material descartado do sistema Zirkonzanh® de produção de próteses dentárias

At present, the material of choice for performing aesthetic dental prosthetic work is in the ceramic. Among them, the ceramic base of stabilized zirconia with 3% yttria (3Y - TZP) stand out for having excellent physical and mechanical properties. During the machining of blocks of zirconia in the laboratory to prepare the various types of prostheses, much of the material is given off in the form of powder, which is subsequently discarded. The waste of this material results in financial loss, reflecting higher final cost treatment for patients, as well as damage to the environment, thanks to the processes involved in the manufacture and disposal of the ceramic. This research, pioneered the recycling of zirconium oxide powder obtained during milling of dental crowns and bridges, we highlight the social and environmental aspects and aims to establish a protocol for the reuse of waste (powder of zirconia Zirkonzahn® system) discarded to obtain a new block of compacted zirconia to maintain the same mechanical and microstructural properties of commercial high-cost imported material. To compare with the commercial material, samples were uniaxially (20 MPa) and isostatically (100 MPa), and its mechanical and microstructural characterization was performed through tests of density, porosity, dilatometry, X-ray diffraction (XRD), hardness, fracture toughness, resistance to fracture electron microscopy (SEM) and analysis of grain size. The results observed in the samples were isostatically pressed similiares those obtained with samples from the commercial material demonstrating the viability of the process

Avaliação do efeito da adição de resíduo de borracha de pneu e brita calcária na formulação de compósitos cimentíceos

Researches have shown that the introduction of rubber in concrete improves the features of its deformability, as well as contributes to environmental disposal of waste generated in the tire retreading process. Furthermore, there is a high availability of limestone within RN and CE country. Ignorance about this stone, does not allow its wide use as aggregate, leaving, this abundant supply idle. A composite of limestone gravel, with proportions of tire rubber waste which could be used as concrete would be an alternative to concrete for low applications. Therefore, this research aims to evaluate the characteristics of concrete containing limestone gravel and proportions of little aggregate replacement (sand) by tire rubber waste. To this goal, the material components of the concrete were characterized, concrete specimens with limestone gravel were made, from the dash 1.0: 2.5: 3.5, varying the water/cement ratio, and inserting a commercial plasticizer, without a proportion of residue, known as reference. From this, concrete with and without the presence of the additive in the same proportions were chosen, as well as these with the use of granite gravel, for being the most used. Selected the references, to these, replacements of little aggregate (sand) were added replaced by rubber waste from the tire retreading process, treated with 1M NaOH in proportions from 5.0 to 20.0 % by mass, cured and exposed to the semiarid environment. The results indicate the possibility of using limestone gravel in the concrete composition with workability correction using plasticizer. There was a decrease in the mechanical properties of the concrete with increments of waste rubber, but there is an improvement in toughness and deformability of the composite, which makes it interesting for the construction of non-structural concrete floors, as well as, the rubber waste delayed the hardening process, continuing to gain resistance after 28 days

Efeito da velocidade de aquecimento nas propriedades de produtos da cerâmica estrutural

Heating rate is one of the main variables that determine a fire cycle. In industrial processes that use high temperatures, greater fire great can reduce the cost of production and increase productivity. The use of faster and more efficient fire cycles has been little investigated by the structural ceramic industry in Brazil. However, one of the possibilities that aims at modernizing the sector is the use of roller kilns and the inclusion of natural gas as fuel. Thus, the purpose of this study is to investigate the effect of heating rate on the technological properties of structural ceramic products. Clay raw materials from the main ceramic industries in the state of Rio Grande do Norte were characterized. Some of the raw materials characterized were formulated to obtain the best physical and mechanical properties. Next, raw materials and formulations were selected to study the influence of heating rate on the final properties of the ceramic materials. The samples were shaped by pressing and extrusion and submitted to rates of 1 °C/min, 10 °C/min and 20 °C/min, with final temperatures of 850 °C, 950 °C and 1050 °C. Discontinuous cycles with rates of 10 °C/min or 15 °C/min up to 600 °C and a rate of 20 °C/min up to final temperature were also investigated. Technological properties were determined for all the samples and microstructural analysis was carried out under a number of fire conditions. Results indicate that faster and more efficient fire cycles than those currently in practice could be used, limiting only some clay doughs to certain fire conditions. The best results were obtained for the samples submitted to slow cycles up to 600 °C and fast fire sinterization up to 950 °C. This paper presents for the first time the use of a fast fire rate for raw materials and clay formulations and seeks to determine ideal dough and processing conditions for using shorter fire times, thus enabling the use of roller kilns and natural gas in structural ceramic industries

Influência da substituição de areia natural por pó de pedra no comportamento mecânico, microestrutural e eletroquímico de concretos

The construction industry is one of the largest consumers of natural raw materials, and concrete is considered today the most used material wide. This accentuated consumption of natural resources has generated concern with the preservation of the environment, and has motivated various studies related to the use of resid ues, which can partially or entirely substitute, with satisfactory performance, some materials such as the aggregate, and in so doing, decrease the impact on the environment caused by the produced residues. Research has been done to better understand and improve the microstructure of concrete, as well as to understand the mechanism of corrosion in reinforced steel. In this context, this work was developed aiming at discovering the influence of the substitution of natural sand by artificial sand, with rega rd to mechanical resistance, microstructure, and durability. To obtain the electrochemical parameters, an adaptation was made to the galvanostatic electrochemical method to study the corrosion in reinforced steel. Concretes of categories 20 MPa and 40 MPa were produced, containing natural sand, and concretes of the same categories were produced with artificial sand substituting the natural sand, and with the addition of sodium nitrate and sodium chloride. Due to the use of rock dust reject (artificial sand), an evaluation was made of its environmental risk. The results indicate that the concretes of category 20 MPa present a better performance than the concrete made with natural sand, thus making it a viable substitute. For the category 40 MPa, the better performance is from the concrete containing natural sand. The adaptation of the galvanostatic electrochemical technique to the study of the corrosion of reinforced steel within concrete proved to be valid for obtaining electrochemical parameters with a high degree of reliability, considering the number of degrees of freedom