text:
Open access peer-reviewed chapter

# Inside the Phenomenological Aspects of Wet Granulation: Role of Process Parameters

By Veronica De Simone, Diego Caccavo, Annalisa Dalmoro, Gaetano Lamberti, Matteo d’Amore and Anna Angela Barba

Submitted: February 15th 2018Reviewed: June 29th 2018Published: October 24th 2018

DOI: 10.5772/intechopen.79840

## Abstract

Granulation is a size-enlargement process by which small particles are bonded, by means of various techniques, in coherent and stable masses (granules), in which the original particles are still identifiable. In wet granulation processes, the powder particles are aggregated through the use of a liquid phase called binder. The main purposes of size-enlargement process of a powder or mixture of powders are to improve technological properties and/or to realize suitable forms of commercial products. A modern and rational approach in the production of granular structures with tailored features (in terms of size and size distribution, flowability, mechanical and release properties, etc.) requires a deep understanding of phenomena involved during granules formation. By this knowledge, suitable predictive tools can be developed with the aim to choose right process conditions to be used in developing new formulations by avoiding or reducing costs for new tests. In this chapter, after introductive notes on granulation process, the phenomenological aspects involved in the formation of the granules with respect to the main process parameters are presented by experimental demonstration. Possible mathematical approaches in the granulation process description are also presented and the one involving the population mass balances equations is detailed.

### Keywords

• granular materials
• wet granulation process parameters
• granule growing
• granule breakage
• granulation mathematical description

## 1. Introduction

Granulation, also known as agglomeration, pelletization, or balling, is a “size-enlargement process” of small particles into larger coherent and stable masses (granules), in which the original particles are still identifiable [1]. The aim of the granulation process is to improve the properties of the final product compared to the powder form, such as giving better flow properties for safer and cheaper transport and storage, lowering of caking and lump formation (especially for hygroscopic materials), improving heat transfer features, obtaining a more uniform distribution of the active molecule, lowering powder dispersion in the environment, linked to a reduced inhalation, handling and explosion risks hazard [1, 2]. Granules have, therefore, received a great interest in many industrial fields, from mineral processing to agricultural products, detergents, pharmaceuticals, foodstuffs, nutraceuticals, cosmetics and zootechnical products [3].

In pharmaceutical field, solid dosage forms remain an important part of the overall drug market, despite the success and the development of new pharmaceutical forms. The oral solid dosage forms market was of $571 million in 2011 and projected to reach$870 million at the end of 2018 [4]. In particular, among novel drugs approved by FDA, 46% in 2014 and 32% in 2016 were solid dosage products [5], most of them made of granules. The most important pharmaceutical industries, such as Patheon, Aesica, Rottendorf Pharma GmbH, Catalent Pharma, continually do investments in oral solid manufacturing solutions, including the development of granulation processing methods [6]. Granulated products are also highly used in the fertilizers field: about 90% of fertilizers are applied as solids, less as powder, more in granular form. The global demand for fertilizer nutrients was estimated to be 184.02 million tons in 2015, and it is forecast to reach 201.66 million tons by the end of 2020 [7]. The animal feed additives global market was estimated at 256.8 kilo tons in 2015, and in particular, industries aim to develop new technologies [8], very often based on granulation principles, to provide stabilization and effective protection of the active components in the finished products [9].

In spite of its great importance and over 40 years of research, granulation process is still based on practical experience, i.e., there is a qualitative understanding of both the granule growth mechanisms and the effects of different variables on agglomeration phenomena. This constitutes a great problem for industries exploiting many and frequently changing formulations with widely varying properties (e.g., food, pharmaceuticals and agricultural chemicals). Thus, new formulations always need expensive and lengthy laboratory and pilot-scale testing. Moreover, even when pilot-scale testing is ok, there is still a significant failure rate during scale up to the industrial production [10]. Over the past decade, however, design, scale-up, and operation of granulation processes have been considered as quantitative engineering and significant advances have been made to quantify the granulation processes [3]. Firstly, granulation must be recognized as an example of “powder particle design”: granule final features are controlled by a perfect combination of formulation design, i.e., choice of the feed-material properties, and process design, i.e., choice of equipment type and operating conditions [11, 12]. The granule final properties are also determined by the interaction of phenomena coexisting in the granulation process simultaneously [1]: physical transformations of the powder particles with relevant kinetic mechanisms (wetting and nucleation, consolidation and growth, and breakage and attrition) and aggregation rate are controlled by operating conditions and feeding material properties [3]. Therefore, for the quantitative analysis of the granulation process, both a careful characterization of the feed-material properties and knowledge of operating parameters and phenomenological aspects are needed [3, 13]. Basically, the granulation methods are divided into dry and wet ones. Dry granulation is based on the use of mechanical compression (slugs) or compaction (roller compaction), while the wet granulation exploits a granulation liquid phase called “binder” to agglomerate powder particles by formation of a wet mass by adhesion. This second process involves a final step of granules stabilization by removal of the wetting phase to make the relevant bonds permanent. Among these two techniques, wet granulation is the most widely used [2, 10, 14]. To produce granular structures with tailored features (in terms of size and size distribution, flowability, mechanical and release properties, etc…), a deep understanding of phenomena involved during granule formation is required. This can also lead to optimized processes in terms of production costs and other types of resources involved.

The purpose of this chapter is to emphasize the phenomenological aspects involved in the formation of the granules with respect to the main parameters of the wet granulation process. To this aim, experimental demonstrative campaigns of wet granulation runs were performed by changing several formulations and process parameters to detail the effect of each single parameter both on the granulation stages and on the obtained granular materials features. Mathematical modeling approaches’ description of granulation process was then presented. Usually, in literature, the approach to the granulation process is based on either experimental tests (to study granules properties) or on modeling studies. The novelty of this work is the integrated approach exploiting both experimental studies to understand the involved basic phenomena in powders aggregation, with the aim to optimize the process, and modeling aspects to verify and predict the experimental results.

## 2. Wet granulation process: apparatus and process parameters

### 2.1. Apparatus features

As briefly introduced, the wet granulation technique allows the formation of granules through the addition of a binding phase to a powder bed. It is usually performed in four steps: (1) homogenization of dry powders, (2) wetting by binder addition, (3) wet massing when liquid feeding system switches off, and (4) drying of the finished product [15]. The most common apparatuses used for granules production are: tumbling granulators, both batch and continuous low and high shear mixers, fluid bed granulators [16]. In tumbling granulators (including discs, drums, pans, and similar equipment), the particles motion is assured by the tumbling action caused by the balance between gravity and centrifugal forces. In particular, the powder feed is fed to the disc, typically at the edge of the rotating granular bed, and the binder is added through a series of nozzles distributed across the face of the bed. Discs and drums generally operate continuously and have large throughputs; thus, they are extensively used in mineral processing and fertilizer granulation [17]. Low and high shear mixers are mechanically agitated containers that promote an efficient mixing, especially of cohesive materials. Such mixers exert intense local shear force actions on the powder, which break the small cohesive aggregates, promoting good dispersion of the liquid and effective consolidation of the product [15]. In fluid bed granulators, the powder bed is first fluidized by a flow of air injected upward through a distributor plate at the base of the granulator, and then the liquid binder is sprayed through a nozzle onto the fluidized bed to agglomerate powder in granules. When binder spraying is stopped, the granules continue to dry in the fluidizing airstream, avoiding the use of a following drying step [18]. This type of granulator is flexible, relatively easy to scale, difficult for cohesive powders, and good for coating applications [16]. Wet granulation has also witnessed various technical and technological innovations such as steam granulation, moist granulation, thermal adhesion granulation, melt granulation, freeze granulation, foamed binder granulation, and reverse wet granulation. For example, steam granulation exploits water steam as binder, providing a more rapid diffusion into the powder and a more favorable thermal balance during the drying step [14].

In general, three fundamental sets of rate processes determine wet granulation behavior: (1) wetting and nucleation, (2) consolidation and growth, (3) breakage and attrition (see schematization in Figure 1). Wetting/nucleation is the initial step where the liquid binder comes in contact with the dry powder bed causing the adhesion among particles to obtain a distribution of small aggregates (nuclei). During the consolidation and growth phase, the particles collide in the granulator and the nuclei begin to grow (particles increase in size and volume) for the deposition of additional material on the nuclei surface. Finally, attrition and breakage phase is characterized by the rupture of granules with relevant formation of small particles, due to both the impacts in the granulator and product handling. These mechanisms coexist in all the wet granulation processes, even if their importance is related to the process type. For example, in the fluid-bed granulation, the wetting phase prevails while in the high-shear granulation, the consolidation step is predominant. To estimate what will be the characteristics of the granules, it is necessary to know that each of the phases presented has a fundamental role that must be predictable. In fact, once these processes have been analyzed, it is possible to predict, at least theoretically, the type of equipment and operating conditions to be used to obtain a good granulation [2].

### 2.2. Effect of process parameters

Several parameters can play a fundamental role on the basic mechanisms of wet granulation and therefore on the product final properties [19]. In particular, granule features depend on ingredients formulation (binder and powder properties and their interaction and proportion), process, and equipment parameters, these last two depending on the kind of the used apparatus. Thus, both material and operating variables together define the kinetic mechanisms and rate constants of wetting, growth, consolidation, and attrition [16].

#### 2.2.1.1. Effect of binder addition rate

Due to the use of a liquid phase as binder, the ratio between the liquid and solid phases would affect the granule final properties [20]. If the liquid/solid ratio increases (due to a high amount of used liquid), nucleation is favored, but at the same time, it is possible that overwetting phenomena may occur, with consequent formation of a mixture and not granules. Moreover, as the quantity of the added liquid increases, the granule saturation, i.e., the ratio between the liquid volume and the interstitial granule volume, changes. However, the addition of too much liquid implies a larger granule size because of a high saturation; in the same way, a low saturation does not allow the granules to growth. The needed amount of liquid must be increased as the size of the powder particles decreases [19, 21, 22, 23].

#### 2.2.1.2. Effect of binder delivery method

The addition of the wetting phase can take place in three different ways, i.e., by pouring, by spraying, or by making it melt, and it is closely linked to the nucleation regime, which has in turn a substantial effect on the product final features [24, 25]. A uniform liquid spray with small droplets size will have the greatest coverage throughout the powder bed and will prevent localized overwetting of the granules, which can result in oversized particles [16]. Moreover, both when pouring and when spraying, the particles size distribution (PSD) is initially bimodal and it tends to be unimodal for high granulation times [26]. In melting technique, however, the obtained granules will be less coarse and only for high granulation times a bimodal distribution will develop [19, 27].

#### 2.2.1.3. Effect of binder properties

Binder viscosity and surface tension are the properties, which more influence the granulation process because the collision energy necessary to agglomerate particles depends on them [28, 29, 30, 31]. In particular, a higher binder solution viscosity could lead to larger granule size and less needed binder amount to start the granule growth in both high-shear [32, 33] and fluid-bed [34] granulation processes. This is due to the fact that a high liquid viscosity requires more energy to break up the liquid droplets (less binder spreadability); hence, larger droplets are formed, which consequently give larger granules. However, at too high viscosity, droplets will be unable to spread throughout the bed causing the reduction in collisions and relevant growth [35]. Surface tension and capillary forces always act to pull particles together, and their magnitudes depend on the liquid bridge formed between the particles [36]. Reducing binder surface tension causes the decrease in the capillary suction pressure and friction resistance, leading therefore to an improved wettability and spreading efficiency [10]. Moreover, also the solvent used for the formulation of the binder solution (only water, alcoholic, or hydroalcoholic solutions are usually used) could significantly change granule properties for its impact on binder wettability and spreadability [28].

#### 2.2.1.4. Effect of powder particles’ size and solubility

Powder particles’ size influences the amount of binder to be used: a larger liquid amount is required to establish liquid bridges between the powders of lower size, thus with a high surface area [37, 38]. Moreover, the high surface area allows the availability of more contact points between colliding particles bringing as final result to stronger granules, which, however, have a more porous structure [10, 29, 30]. Perhaps the larger surface area allows also a higher growth tendency of the smaller particle fraction probably due to a more efficient nucleation and coalescence [39]. Moreover, a larger solubility of the solid excipient in the granulating solvent is able to decrease the solvent amount needed for granule formation, and granules with uniform particle size distribution and a reduced friability will be formed [28].

#### 2.2.2. Equipment and process parameters

In general, equipment and process variables impact on mixing, agglomerating, and drying operations. For example, equipment variables in fluid bed granulators are related to the apparatus design with the aim to fluidize, thus granulate and dry the product. Therefore, air distribution plate must be appropriately designed depending on the powder properties. For example, a product with low bulk density will require a low fluidizing velocity, thus a distributor plate having a small open area. Moreover, a blower with appropriate pressure drop will fluidize the process material adequately. A proper use and cleaning of filters to retain entrained particles also must be carried out in fluidized bed [16]. However, for both mixers and fluid beds, the bowl geometry is also considered a factor with large impact on the agglomeration process. Moreover, for low and high shear mixers, the impeller and chopper design affects the flow patterns and powder flow dynamics in the bowl, by varying the volume of powder mixture swept out by the impeller itself: a high-swept volume provokes high densification of the agglomerates and narrow granule size distributions [40].

Process variables in fluid bed granulators agglomeration is highly dependent on: process inlet air temperature, atomization air pressure, fluidization air velocity and volume, liquid spray rate, nozzle position and number of spray heads, and product and exhaust air temperature [41]. Inlet-process air temperature depends on both the binder type and the heat sensitivity of powder bed. For example, higher temperatures will cause binder faster evaporation with the relevant production of smaller and friable granules [42]. Process variables in high/low-shear mixers are essentially related to the impeller and chopper relative speed, granulating solution addition rate, both global granulation and wet-massing time [43]. In general, we can conclude that for all types of apparatuses, both equipment and process parameters define the two most important operating conditions in wet granulation process, i.e., mixing performance and residence time.

## 3. Inside wet granulation runs: experimental evidences

The impact that the aforementioned process parameters have on final granule properties is continuously studied to predict the final product quality. A possible approach to study the wet granulation process was that of a recent study conducted by [44], devoted to first planning experiments by the design of experiments (DoEs) and then to use the results to give correlations between product properties and process parameters. In particular, the central composite design (CCD) statistical protocol was applied for planning the experimental campaign about the production of hydroxypropyl methylcellulose (HPMC 20, Pentachem Srl, San Clemente RN-Italy) granules with distilled water as the binder phase and using a bench scale low-shear granulator apparatus. In brief, a given amount of HPMC 20 powder was placed in the low-shear granulator, and then the addition of the binder phase was carried out by spraying it by an ultrasonic device. The produced granulates were stabilized by dynamic drying, using hot air (65°C) for 1 h, collected, and then separated by a manual sieving with cut-off sizes as follows: 2 mm, 0.45 mm, and a metal collection pan. Three particles fractions were obtained: a fraction of “big scrap,” i.e., particles with size larger than 2 mm, a fraction of “small scrap,” i.e., particles with size smaller than 0.45 mm, and a fraction of “useful,” i.e., particles with size between 0.45–2 mm. The range size 0.45–2 mm was considered as the fraction of interest being a size typical range of commercial granulated food, pharmaceutical, and zootechnical products. Finally, only the fraction of useful was subjected to characterization protocols carried out by adopting the American Society for Testing and Materials (ASTM) standards.

Firstly, screening experiments were performed in order to determine the independent variables and their interactions playing a significant role on the dependent variables, i.e., on the final features of granulated product, such as granulation yield (defined as mass percentage of dry granules with size between 0.45 and 2 mm) and flowability properties (Carr Index, Hausner Ratio and Angle of Repose). The screening work showed that some parameters can be fixed, such as the powder mass (50 g) and the process time of 20 min (by observing that longer times caused granule-breaking phenomena). Instead, the impeller rotation speed, the binder volume at constant mass, and the binder flow rate were the parameters (factors) with larger influence on the final granule properties. The factors have a values limited range: high binder volume (greater than 100 ml) or high binder flow rate (greater than 58 ml/min) involved overwetting phenomena, low amounts of liquid (lesser than 50 ml) did not form granules, high impeller rotation speed (greater than 112 rpm) generated solid particulate breaking. At this point, for each factor, three intensities (levels) were used, i.e., the minimum, medium, and maximum values, and combined by the CCD method. The performed runs have underlined that there are process operating conditions, which combined together can produce granules with size smaller than the requested one, i.e., failure of the aggregation phenomena (Figure 2A).

Others, instead, can achieve clusters of powder and binder, i.e., overwetting phenomena, that is a condition to avoid (Figure 2C). The best conditions of granulation, able to produce granules with a defined size (0.45–2 mm) and good flowability together with a high granulation process yield, were obtained by working with a high impeller rotation speed, i.e., 112 rpm, a high binder volume, i.e., 100 ml, and a low binder flow rate, i.e., 17 ml/min (Figure 2B). Then, semiempirical correlations between granule properties and process parameters were developed by describing the experimental data with several model equations. Akaike information criterion and R-square calculations showed that the best comparison between experimental data and model predicted values was attained by using the second-order polynomial equation. The proposed correlations were then validated by new granulation tests, not included in the work plan, underlining their ability to predict the granule final properties in terms of flowability and granulation yield. It is important to note that several studies in the literature describe the correlations between process parameters and granule properties; however, they use different apparatuses and final products (tablets). It was the first time that, for such similar granulation systems, semiempirical correlations were able to give the combined effect of impeller rotation speed, binder volume, and binder flow rate on granulation yield and flowability [44].

Moreover, the effect of the incorporation of a lipophilic vitamin was also tested by using the loading method 1. Due to its lipophilic properties, a binder composed of a solution made by distilled water and ethanol was used. Results showed that the use of ethanol gave a reduced granulation yield and granules with less defined shape, smaller dimensions, more friable structure, worse flowability, and slightly faster polymer erosion. It was demonstrated that the molecule solubility did not affect either granules’ physical or mechanical properties, but it had effect on the molecule release mechanism (diffusion for the hydrophilic molecule and erosion for the lipophilic one) [46].

A deep understanding of the phenomena involved during the wet granulation process can lead to optimized processes that obtain the maximum benefits without increasing the production costs. In light of that, studies on the evolution of particle size distribution (PSD) during the granulation process were performed by using an ad hoc dynamic image analysis (DIA) device, based on the free falling particle scheme of particles per unit of volume. It resulted evident that nucleation, agglomeration, and breakage phenomena occur simultaneously during all the process time [47], phenomena that therefore must be taken into account in the modeling approach.

## 4. Mathematical approach to describe the phenomena involved in granulation process

Mathematical modeling of the granulation process can play a dual role: can help to understand and to underline the observed physical phenomena and to predict properties (size and size distribution) of granular materials. It is important to note that the predictive ability is a powerful way to define suitable process conditions, which can minimize costs and optimize process yields, avoiding onerous experimental tests.

Modeling approaches mainly consist in three types:

• Empirical models: they are obtained from the regression of large set of experimental data, most of the time derived from the design of experiments statistic technique [44, 48, 49]. Empirical models have the advantages of being very simple (often polynomial equations), easy to obtain (regression of data), quite reliable within the investigated range. The drawback of this approach is that it is a black box approach: it provides very little (or none) information on the underlying mechanisms.

• Discrete element method (DEM) models: this is the most detailed type of mathematical model for particulate systems [50]. Thanks to mass and momentum balances on each particle of the powder bulk, DEM models aim to describe the evolution of the analyzed system. The advantage is that the description is very detailed (at a single particle level), but it is so computational power demanding that rarely it is applied to systems of more than hundreds of thousand particles. They are often used to obtain parameters useful for higher scale models [51].

• Population balance equation (PBE)-based models: the powder bulk is described as a population with a distribution of certain characteristics (internal coordinates i.e., size, binder content) that can vary in space (external coordinates) and time [52]. Therefore, the PBE-based models describe the evolution of the number of particles of a given characteristic in time and space [53]. The advantages of such an approach, with respect to DEM models, are enormous in terms of computational power requirement, despite the complexity of the involved equations (integro-differential). Nowadays, this is still the most used approach in modeling granulation processes and several numerical methods have been developed to “easily” solve the PBEs. Among them, the discretization of the PBEs (DPBEs) in classes (multiclass method) is the most used to describe the evolution of the distribution of the internal variables (i.e., size: particle size distribution) [54, 55, 56, 57, 58]. Less computational demanding methods, like the method of moment (which focus only on the moments of distributions), are preferred when the PBE has to be combined with computational fluid dynamics (CFD) models to describe the multiphase flow (i.e., the movements inside the granulator) [59].

### 4.1. Discretized population balance equations modeling

#### 4.1.1. From the continuous to the discretized form of PBEs

In the following, a modeling approach based on the one-dimensional (one internal coordinate: particle size) discretized PBEs (DPBEs) will be shown. Such kind of models turns to be the best choice when dealing with granulation processes, being very descriptive, since the evolution of the entire size distribution is considered, including all the phenomena that cause variation of the particle size distribution, and computational efficient (with the modern computational power), which allows to integrate them in flow sheet models of entire processes.

Considering the entire granulator as control volume, the PBEs do not depend on spatial variables (external coordinates), and it can be written as [60]:

nvtt=Q̇inVninvQ̇outVnoutvGAnvtv+BNucvt+BAggvtDAggvt+BBrvtDBrvtI.C.nv0=n0vB.C.n0t=0E1

where nvtis the probability density function [b−1 x−1], with “b“ basis of calculation that can be the total mass of powder [kg] or the total volume [m3] and x the particle volume [m3] or particle diameter [m], depending on the internal coordinate chosen. Indeed, this last v[x] can be the particles volume v[m3] or the particles diameter l[m]. Q̇inand Q̇outare the flow rate of the inlet and outlet currents [b t−1], Vis the volume or mass in the granulator [b], and Gand Aare the growth as layering and the attrition [x t−1], respectively. BNucis the birth by nucleation [b−1 x−1 t−1], which creates particles within the PSD of interest. BAggand DAggare the birth and death by agglomeration of particle of size v, and analogously BBrand DBrare the birth and death by breakage phenomena of particle of size v. Disregarding layering and the attrition phenomena, for a batch granulator:

nvtt=BNucvt+BAggvtDAggvt+BBrvtDBrvtI.C.nv0=n0vE2

The nucleation, agglomeration, and the breakage phenomena are described by the equations:

BNucvt=vE3
BAggvt=120vβuvunutnvutduE4
DAggvt=0βuvnutnvtduE5
BBrvt=vbuvSunutduE6
DBrvt=SvnvtE7

where Kis a constant that multiply a function of the internal coordinate v(i.e., Dirac delta function). βis the coalescence kernel [b t−1], which describes the frequency of collision between particles of internal coordinate uand vand the influence of the internal coordinates on the efficiency of agglomeration. On the other hand, bis the breakage function [x−1], which describes the probability of formation of particles of internal coordinates vfrom the collision and breakage of particles of internal coordinates u(u>v). Sis a selection function [t−1], which describes the frequency at which particles of a given internal coordinates are broken.

To solve Eq. (2), a numerical method has to be used: the class methods of zero order [61],

Ni=vivi+1nvdv=nivi+1viE8

where Niis the number of particles per unit base [b−1] within a class with size range vivi+1. The characteristic size of each class is vi: vivivi+1. With this approach, instead of solving for a continuous (particle size) distribution, a system of ordinary differential equations (ODEs) is generated to describe the evolution of the number of particles within the classes. Increasing the number of classes, the number of ODEs increases as well as the computational power requirements: the most used discretization uses a geometric progression with common ratio vi+1/vi=21/q(or equivalently li+1/li=23q), where qis an integer ≥ 1.

#### 4.1.2. Modeling the agglomeration phenomena

The most used discretized form of the agglomeration phenomena is the one proposed by Hounslow et al. [55] (vi+1/vi=2), updated by Litster et al. [62] to consider geometric progression with q1(vi+1/vi=21/q). The described agglomeration phenomenon is binary (interaction between two particles), and the birth terms are due to the collisions and coalescence between particles of lower dimensions with respect to the considered class. The death terms account for the interactions and coalescence of particles belonging to the considered class, producing their disappearance (and their birth in upper classes).

BAggitDAggit=j=1iSq12ji+1q21q1βi1jNi1Nj++k=2qj=iSqk+2k+1iSqk+1k2ji+1q1+2k+1q21q1βikjNi1Nj++0.5βiqiqNiq2++k=2qj=iSqk+2k+2iSqk+1k+12jiq2k+1q+21q21q1βik+1jNik+1Nj+j=1iSq2jiq21q1βijNiNjj=1Sq+1hβijNiNjE9

where Sq=p=1qp. For q=1, the Hounslow discretization can be obtained.

#### 4.1.2.1. The kernel of coalescence

The coalescence kernel is the most important parameter when describing granulation processes. A body of literature is present on this kernel, proposing several expressions ranging from purely empirical, semiempirical, and model-based kernels. In general, the coalescence kernel is split into two parts:

βuvt=β0tβuvE10

where the first is the “aggregation rate” term and the latter describes the dependence of the coalescence kernel on the dimensions of the granules. In the first term, various system parameters are incorporated (i.e., granulator geometry, operating conditions, formulation properties, etc.).

The nature ofβuv, most of the time an homogeneous function, establishes how the agglomeration modify the internal coordinate, in particular whether or not the transformation is self-preserving in PSD and whether or not a gelling behavior should be expected. Analyzing the degree of homogeneity λ(βcucv=cλβuv), which expresses the strength of the dependence of βuvon its argument, it is possible to distinguish between nongelling and leading to a self-preserving size distribution kernels (λ1) and gelling (and non-self-preserving PSD) kernels (λ>1) [46]. The most used kernel in literature is the equikinetic energy (EKE) kernel, which is a nongellingand leading to a self-preserving size distribution kernels, as it can be seen from Table 1. In Figure 3, the (normalized) EKE kernel is reported: as it can be seen the maximum probability of coalescence in a binary process is between a big particle (high classes) and a small particle (low classes).

KernelEquationλ
Constant kernelβuv=10
Sum kernelβuv=u+v1
Product kernelβuv=uv2
Coagulation kernelβuv=u23+v232/3
Equikinetic energy kernelβuv=u13+v1321u+1v1/6

### Table 1.

Examples of coalescence kernels and degrees of homogeneity .

#### 4.1.3. Modeling the breakage phenomena

The discretized form of the breakage phenomena can be obtained by substituting continuous with discrete functions and by using Eq. (8):

BBritDBrit=j=i+1bvivjSvjNjΔvjSiNiE11

However, as suggested in Vanni [63], to satisfy the mass conservation (valid for both agglomeration an breakage), for all the possible discretizations (i.e., for a geometric progression of the type vi+1/vi=21/q), the equation has to be corrected:

BBritDBrit=j=i+1Γi,jSjNjCj1Ci2SiNiE12

where

Ci1=vij=1i1vjΓjiCi2E13
Ci2=11vi+1vivivi+1vvi+1bvqdqdvE14
Γij=vivi+1bvivjdvE15

with C12=0and vithe characteristic size x of the class in volume [m3]. In case the distribution density function has been obtained (nlt) and discretized (Ni=nili+1li) with the diameter lias caracteristic size, the relation vili3can be used to adapt Eqs. (13), (14), and (15).

#### 4.1.3.1. The selection and breakage function

Several functions can be chosen for the selection function and breakage function (continuous or discrete), usually of semiempirical nature: the breakage theory is not well developed as the agglomeration theory [63].

An example of selection function can be the power law form: Si=kviγ, where kand γ(1/3γ2) are adjustable parameters.

A flexible form of the breakage function is the parabolic form:

bvivj=Cvj1+1C283vi23vi+1vj13122vi1vj12+6vj1E16

that, depending on C, can simulate different behaviors: concave parabola (0C<2), it is more likely the formation of unequal fragments, convex parabola (2<C<3), it is more likely the formation of equal fragments, and uniform distribution (C=2), in which it is equally likely to form a child particle of any size.

#### 4.1.4. Modeling the nucleation phenomenon

Nucleation phenomenon occurs because small particles lower than the considered minimum size class can suddenly form granules (i.e., due to the action of binder droplets) within the considered size range. In light of this, it is clear that nucleation is not a mass conservative mechanism. It can be modeled as:

BNucit=ktfiE17

where ktis a function of time, describing for how long this phenomenon is present. fiis a function of the size class and it individuates the class interested by nucleation.

#### 4.1.5. Model results

The resulting DPBE is a system of ordinary differential equations:

dNidt=BAggitDAggit+BBritDBrit+BNucitI.C.Ni0=Ni0,E18

which can be solved numerically with the well-known discretization techniques (i.e., explicit Runge-Kutta). The results give the evolution of Niduring the granulation process, allowing to follow the (discretized) particles size distribution (Figure 4).

The agglomeration process leads to the lowering and translation toward higher dimension of the PSD, because the particles diminish in number increasing their size. On the contrary, breakage phenomena lead to an increase of the number of particles per unit volume, with a translation toward lower dimensions of the PSD. Finally, nucleation locally increases the number of particles per unit volume.

## 5. Conclusions

Granular materials represent a relevant form of commercial products on the worldwide market. A rational organization of manufacturing, based on phenomenological knowledge rather than practical trials, can allow high granulation performance in terms of yields, product features, and manufacturing costs.

In this chapter, wet granulation process is described presenting the role of several factors, such as apparatus features, formulation, and operative parameters on granules final features.

Experimental campaigns of wet granulation runs, performed by using a low-shear granulator, changing several formulation and process parameters, have been performed to detail each single parameter effect both on the granulation stages and on the obtained granular materials features.

Mathematical modeling approaches of the granulation process have been thus introduced with the aim both to understand and underline the observed physical phenomena, and to propose predictive tools able to forecast granulation results in terms of size and size distribution of obtained granular materials. The predictive ability is a powerful way to define suitable process conditions, which can minimize costs and optimize process yields, avoiding onerous experimental tests. In this work, particular attention was given to models based on population balance equations (PBEs), for which appropriate mathematical descriptive functions have been presented.

## Conflict of interest

The authors declare that they have no conflict of interests.

## Abbreviations

 ASTM American society for testing and materials CCD central composite design CFD computational fluid dynamics DEM discrete element method DIA dynamic image analysis DoE design of experiments DPBE discretized population balance equation EKE equikinetic energy HPMC hydroxypropyl methylcellulose ODE ordinary differential equation PBE population balance equation PSD particle size distribution Symbols β coalescence kernel A attrition b breakage function BAgg birth by agglomeration phenomena of particle of size v BNuc birth by nucleation phenomena BBr birth by breakage phenomena of particle of size v DBr death by breakage phenomena of particle of size v DAgg death by agglomeration phenomena of particle of size v G growth as layering K constant that multiply a function of the internal coordinate v n probability density function Ni number of particles for class “i” per unit of volume of solid q integer ≥ 1 Qiṅ flow rate of the inlet currents Qouṫ flow rate of the outlet currents S selection function v internal coordinate (particle volume or particle diameter) V volume or mass in the granulator

chapter PDF
Citations in RIS format
Citations in bibtex format

## More

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## How to cite and reference

### Cite this chapter Copy to clipboard

Veronica De Simone, Diego Caccavo, Annalisa Dalmoro, Gaetano Lamberti, Matteo d’Amore and Anna Angela Barba (October 24th 2018). Inside the Phenomenological Aspects of Wet Granulation: Role of Process Parameters, Granularity in Materials Science, George Kyzas and Athanasios C. Mitropoulos, IntechOpen, DOI: 10.5772/intechopen.79840. Available from:

### chapter statistics

1Crossref citations

### Related Content

#### Granularity in Materials Science

Edited by George Kyzas

Next chapter

#### Introductory Chapter: Granularity in Adsorption

By George Z. Kyzas and Athanasios C. Mitropoulos

First chapter

#### Introductory Chapter: Nanomaterials in the 2020s

By George Z. Kyzas and Athanasios C. Mitropoulos

We are IntechOpen, the world's leading publisher of Open Access books. Built by scientists, for scientists. Our readership spans scientists, professors, researchers, librarians, and students, as well as business professionals. We share our knowledge and peer-reveiwed research papers with libraries, scientific and engineering societies, and also work with corporate R&D departments and government entities.


text: Image text transcribed for accessibility: The node-voltage equation for this circuit may be written in the following matrix form: Determine the elements of the 2 times 2 matrix A and the elements of the right-hand side vector b. The mesh-current equations for this circuit may be written in the following matrix form: Determine the elements of the 2 times 2 matrix A and the elements of the right-hand side vector b. Solve the circuit and determine all of the node voltages: VA, vB, Vc. and Vp.

format: plain


text:
## Studying ‘VBA’

Summary VBA is an abbreviation of Visual Basic for Application. Abbreviated as VBA, Visual Basic for Applications is a programming language that is similar to Visual Basic, only it is embedded in an individual Microsoft application such as Excel or Access. Using VBA you can create macros or small programs that perform tasks within the […]

## Implementing the File System | Assignment 3 | Operating System

Brief Information Title: Implementing the File System Hand-in date: 2016-05-17 Tue. Due date (1st): 2016-05-29 Sun. Due date (2nd): 2016-05-31 Tue. Due date (3rd): 2016-06-26 Sun. How to do the assignment: (pdf files) Homework 3 Homework 3: appendix 과제3 변경 및 중요사항(Homework 3: changes and important notices) Trace 2016-05-21 Sat. 23:13: Read the instructions. Try to implement […]

## Black-Scholes Model

The Black-Scholes Equation $latex \frac{\partial V}{\partial t}+\frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2}+rS\frac{\partial V}{\partial S}-rV=0 &s=3$   $latex S$: the price of the stock $latex V=V(S,t)$: the price of a derivative(e.g., option) $latex K$: the strike price of the option $latex r$: the annualized risk-free interest rate $latex \sigma$: the standard deviation of the stock’s return \$latex […]

## Deriving the General Form of the Tower of Hanoi Sequence

Deriving the General Form of the Tower of Hanoi Sequence.pdf


text:
# cirq.DepolarizingChannel¶

class cirq.DepolarizingChannel(p: float)[source]

A channel that depolarizes a qubit.

__init__(p: float) → None[source]

The symmetric depolarizing channel.

This channel applies one of four disjoint possibilities: nothing (the
identity channel) or one of the three pauli gates. The disjoint
probabilities of the three gates are all the same, p / 3, and the
identity is done with probability 1 - p. The supplied probability must
be a valid probability or else this constructor will raise a ValueError.

This channel evolves a density matrix via

$$\rho \rightarrow (1 - p) \rho + (p / 3) X \rho X + (p / 3) Y \rho Y + (p / 3) Z \rho Z$$

Parameters

p – The probability that one of the Pauli gates is applied. Each of the Pauli gates is applied independently with probability p / 3.

Raises

ValueError – if p is not a valid probability.


text:
On the Hardness of Average-case k-SUM

Noah Stephens-Davidowitz
Cited by: 0|Bibtex|Views5

Abstract:

In this work, we show the first worst-case to average-case reduction for the classical \$k\$-SUM problem. A \$k\$-SUM instance is a collection of \$m\$ integers, and the goal of the \$k\$-SUM problem is to find a subset of \$k\$ elements that sums to \$0\$. In the average-case version, the \$m\$ elements are chosen uniformly at random from some inter...More

Code:

Data:

Full Text


text:
# Opposing electrical magnets

Discussion in 'General Electronics Chat' started by Gdrumm, Aug 2, 2014.

1. ### Gdrumm Thread Starter Distinguished Member

Aug 29, 2008
684
36
Can electrical magnets be made to oppose each other the way earth magnets do?

Thanks

Gary

2. ### tom_s Member

Jun 27, 2014
301
381
i would assume the answer is yes, my understanding - reversing the current flow would change the polarity.

3. ### Alberto Active Member

Nov 7, 2008
169
36
When your coil is powered it will produce a magnetic field with north in front and south on the back ( depending on polarity if DC if AC you will have N and S changing at the AC frequency)

Alberto

4. ### studiot AAC Fanatic!

Nov 9, 2007
5,005
519
An electromagnet has north and south poles in just the same way as a permanent magnet.

A linear solenoid generates the same field configuration as a bar magnet.

The direction is given by the right hand screw rule, with the North at the point of the screw as shown in the second part of the sketch on the right.

• ###### coil1.jpg
File size:
28.7 KB
Views:
15
Gdrumm likes this.
5. ### ErnieM AAC Fanatic!

Apr 24, 2011
7,907
1,789
Of course they can. A magnet is a magnet is a magnet.

6. ### shortbus AAC Fanatic!

Sep 30, 2009
4,549
2,588
That's how a "wound field" electric motor works.

7. ### wayneh Expert

Sep 9, 2010
13,623
4,416
An electromagnet in a black box can not be differentiated from a permanent magnet in a black box. As Ernie noted, a magnet is a magnet.

8. ### Gdrumm Thread Starter Distinguished Member

Aug 29, 2008
684
36
What would be the determining factors regarding how much of a gap I could achieve between the magnets, when they are opposing; more voltage, more core wire?

I'm thinking of an application where a machine, when in set up mode, would have two electrical magnets opposing each other, creating a gap (of about 1/8") through which I could thread a substrate.

Then, when switched to the run mode, the magnets would come together, grabbing the substrate, then by mechanical assist, feed the substrate into a die, to be cut.

The unit already exists, and has an electromaget that feeds the material into the die.
The bottom portion is the electrical magnet, the top portion is simply a steel plate.

But I'm looking for a way to thread the material into the magnet area during set up, that does not involve touching the magnet carriage mechanism with human hands.

If I can aviod touching the carriage mechanism during set-up, I can avoid having to lock out the machine, for OSHA concerns.

I've tried a simple spring attached to the top portion, and it does create some seperation, when the magnet is off, but not enough to do what I need to do.

If I increase the spring pressure, then when the unit is turned back on, it is not powerful enough to overcome the spring, and re-engage the substrate.

Thanks,
Gary

9. ### wayneh Expert

Sep 9, 2010
13,623
4,416
I don't see why that would not work.

Is it conceivable to replace the steel plate with a permanent magnet? Then all you'd have to do is reverse the current in the existing electromagnet and it would go from attraction to repulsion. A negative, from a safety point of view, is that there would still be attraction with all power off.

Another approach would be to simply replicate the existing electromagnet and replace the steel plate, but you probably don't need as much field strength as this would enable. The 2nd magnet can probably be a much lower field strength.

Field strength is proportional to amp-turns. The usual challenge is to choose which wire gauge to wind so that, in the volume you have to place the coil, you get a reasonable compromise of DC resistance and turns; both go up with wire gauge and length. Of course current - and thus field strength - goes down with DC resistance. Heat dissipation is another concern.

10. ### studiot AAC Fanatic!

Nov 9, 2007
5,005
519
To a first approximation you can adapt the formula, which is used to calculate the lifting capacity of lifting electromagnets.

$P = \frac{{{B^2}A}}{2{\mu}}$

Where B is the field induced by a single electromagnet by current I.
A is the cross sectional area of the facing steel areas.
P is the force developed between the faces.

Last edited: Aug 5, 2014
11. ### Gdrumm Thread Starter Distinguished Member

Aug 29, 2008
684
36
Replacing the steel plate with a permenant magnet in this case would create a problem. There must be a release between the two during the return stroke, otherwise it would pull the paper (substrate) back out of the die.

I will sleep on it.

It might be best to try to make one, and see how much seperation I get, using parts I have in stock.

It might be a Rube Goldburg too!

Thanks,
Gary


text:
# Interface between c# GUI and FPGA

#### Qamar Shafi

##### Newbie level 5
I'm new to FPGA i'm a undergraduate student i'm into my final year project where i have to interface my GUI with the FPGA or please just tell me how to i use my Gui which is made in c sharp using visual studio how i can program GUI that i can transmit anything and should able to recieve it on FPGA

#### KlausST

##### Super Moderator
Staff member
Hi,

You need to provide useful informations.
Not clear:
* what your GUI is (OS, functions...)
* what your FPGA is (type, function...)
* what you want to control
* what interface is between GUI and FPFA (data rate, protocol ...)

In any case: draw a sketch with good description. Show examples of what you want to achieve...
The more detailed the better

Klaus

#### FlyingDutch

##### Full Member level 4
Hello,

if you are using "Windows Forms" application (C# .NET on Windows OS) - then you can use class "SerialPort" from "System.IO.Ports". You can program the virtual serial port using this class (on PC computer). On the FPGA board you have to use such konverter (USB-UART):

Then you also have to implement UART on your FPGA project. Then one side of converter you are attaching to UART pins (implemented on FPGA chip) and the second side -> usb socket you are connecting to your PC computer. This is likely that you have to also use logic level converters (low voltage on FPGA side). Then you be able to transmit data between PC box and FPGA board. Some FPGA boards has hardware UART implemented - then you can just use it.

https://www.codeproject.com/Articles/678025/Serial-Comms-in-Csharp-for-Beginners

it should be helpful for C# serial port programming (PC box).

Regards

#### Qamar Shafi

##### Newbie level 5
yes i have coded in c# but how to ensure coding on FPGA how FPGA will understand what is coming from pc

#### TrickyDicky

You need to write the logic that interprets the incoming commands + data and does something with them.

#### Qamar Shafi

##### Newbie level 5
where i have to write logic in verilog??

#### FlyingDutch

##### Full Member level 4
where i have to write logic in verilog??
Hello,

you didn't write what is purpose of your desing in FPGA. You also didn't attached any code in HDL of your project - even skeleton. We are not fortune-tellers. Maybe describe what tasks your project have to pursue, then someone will be able to give you hints.

Regards

#### Qamar Shafi

##### Newbie level 5
how to make LED's of the FPGA blink by using a GUI built in C sharp windows forms
yes i have to add serial port in GUI
but what i write in that GUI and then how to code in verilog for FPGA so that lights should blink
if u want me to read something to do this plzz tell me the website where i can study this and able to interface between FPGA and PC

#### KlausST

##### Super Moderator
Staff member
Hi,

You don't show a drawing.
You don't answer our questions. (It seems you did not even read them)
We see no effort on your own.
Thus we don't know what you need.

We want to help, but you make it impossible.

*******
I can only recommend:
There are many circuits, reference circuits, developmemt boards with schematics in the internet.
Every FPGA development board offer an interface to a PC....
Look through some of them, find out which one suits you most.
Give us the link to it.
Tell us what you want to modify and why.
Search the internet for solutions.

A "half line post" surely is not enough information. And not a 1 line, and not a 4 line....

Klaus

#### Qamar Shafi

##### Newbie level 5
which type of drawing you want ???

#### KlausST

##### Super Moderator
Staff member
Hi,

On a sheet of paper...with or without colors...computer drawn ...it does not matter.

****
Honestly: We write lenghty posts, try to explain, ask for informations, try to help. But we see no effort from your side.
If you are not willing to provide useful informations or a drawing, no internet link.. show what you have done so far ...
... then don't expect that I (we?) spend more time on this thread.

Klaus

#### Qamar Shafi

##### Newbie level 5
Let me tell you actually my project is to Design and Implementation of Test Controller (ASIC) for X Band Quad Transmit Receive Module
but for that i have to make a gui so that i can send commands via FPGA but first thing is to interface the computer and FPGA for that my advisor gave me a task to learn how to interface both and just blink Artix-7 LED's from GUI.My GUI is looks like

#### Qamar Shafi

##### Newbie level 5
now tell me how to send commands to FpGA and how that will respond by using it
FPGA is Artix-7
communication protocol is via UART

#### TrickyDicky

You need to write an FPGA design that receives commands (defined by you), decodes the commands and turns on/off the LEDs. You will also need code for a UART controller.
There are many UART examples out there....

#### Qamar Shafi

##### Newbie level 5
that i know but how to do it i'm asking that

#### KlausST

##### Super Moderator
Staff member
Hi,

UART is no protocol. UART is a PART of an interface.
--> So I must assume you even didn´t do an internet search "What is an UART".
--> you have to do so simple internet searches on your own!

A UART can be implemented in an FPGA. Over one million hits for an internet search "How to implement UART on a FPGA".
Documents, libraries, descriptions, examples, even videos. What more do you need?
Are you too lazy to go through ten (or so) of the first hits? - At least you didn´t tell, although I asked for it..
Why don´t you tell us which of them you´ve read, what is clear and what is not clear...

It seems we need to ask for each item several times .. you just look for someone to do your job.
I like to support people who try to find a solution on their own, who show their effort, show how hard they work.

I don´t see this here, thus I leave the thread.
The same time I honestly hope that you can convince other people that you work hard and thus motivate them to help you. Good luck

#### dpaul

that i know but how to do it i'm asking that
Ok, given that you are new in this field, your best approach would be to split up the "todo"s.
As others have pointed out, there is a design implementation part on FPGA (Verilog/VHDL) and command interfacing part using GUI. In my option it is better to start with the design impl part. How to pass the commands from GUI can be the next task. So plan accordingly.

GUI --> Serial ctrl i/f (TeraTerm/Putty) --> UART data tx to FPGA --> Save data to registers --> Read register values to blink LEDs

This is what I can imagine as a flow. Now it is your task to come up with ideas as to how to implement the above stages.

Last edited:

##### Super Moderator
Staff member
Now it is your task to come up with ideas as to how to implement the above stages.
What if they don't know anything about logic design and hardware?

From what I've read in this thread, I don't think they have any background in digital design, they appear to only know software. I think they first need to go through all the simple tutorials on using FPGAs (and do some studying on logic design) before tackling this assignment.

If you type "FPGA tutorial" in a google search and you will find there are many resources on the subject.

#### MarkPh

##### Newbie level 6
I'm new to FPGA i'm a undergraduate student i'm into my final year project where i have to interface my GUI with the FPGA or please just tell me how to i use my Gui which is made in c sharp using visual studio how i can program GUI that i can transmit anything and should able to recieve it on FPGA
If you want to do it very fast you can use Instant SoC.
You simply write a few lines of C++ code and the tool generates the VHDL code.
There are ready UART classes etc.
Here you can see an example using UART.

#### FlyingDutch

##### Full Member level 4
If you want to do it very fast you can use Instant SoC.
You simply write a few lines of C++ code and the tool generates the VHDL code.
There are ready UART classes etc.
Here you can see an example using UART.
Hello,
I think that such solution might be not accepted by the teacher (I think it should be project in pure HDL language).

I suggest Qamar start from implementing simple UART on FPGA and test it on FPGA board - using for example similiar USB-Serial converter I mentioned in my first post. For first tests 'Putty" application can be used. We should know on which FPGA chip this project will be implemented because the choice off software tool for synthessis dependes. There are many open-source implementation of UART on Opencores.org website:
https://opencores.org/projects?expanded=Coprocessor,Communication controller

Unfortunately most of them are implemented in VHDL (Qmar said that it should be Verilog).

There is one very simple implementation in Verilog (without hardware FIFO):

https://opencores.org/projects/mmuart

This is simplest implementation it can be - only two files in Verilog (uart.v and uart_transceiver.v):

Code:
/*
* Milkymist VJ SoC
* Copyright (C) 2007, 2008, 2009, 2010 Sebastien Bourdeauducq
*
* This program is free software: you can redistribute it and/or modify
* the Free Software Foundation, version 3 of the License.
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this program.  If not, see <http://www.gnu.org/licenses/>.
*/

module uart #(
parameter clk_freq = 100000000,
parameter baud = 115200
) (
input sys_clk,
input sys_rst,

input [13:0] csr_a,
input csr_we,
input [31:0] csr_di,
output reg [31:0] csr_do,

output rx_irq,
output tx_irq,

input uart_rx,
output uart_tx
);

reg [15:0] divisor;
wire [7:0] rx_data;
wire [7:0] tx_data;
wire tx_wr;

reg thru;
wire uart_tx_transceiver;

uart_transceiver transceiver(
.sys_clk(sys_clk),
.sys_rst(sys_rst),

.uart_rx(uart_rx),
.uart_tx(uart_tx_transceiver),

.divisor(divisor),

.rx_data(rx_data),
.rx_done(rx_irq),

.tx_data(tx_data),
.tx_wr(tx_wr),
.tx_done(tx_irq)
);

assign uart_tx = thru ? uart_rx : uart_tx_transceiver;

/* CSR interface */
wire csr_selected = csr_a[13:10] == csr_addr;

assign tx_data = csr_di[7:0];
assign tx_wr = csr_selected & csr_we & (csr_a[1:0] == 2'b00);

parameter default_divisor = clk_freq/baud/16;

always @(posedge sys_clk) begin
if(sys_rst) begin
divisor <= default_divisor;
csr_do <= 32'd0;
end else begin
csr_do <= 32'd0;
if(csr_selected) begin
case(csr_a[1:0])
2'b00: csr_do <= rx_data;
2'b01: csr_do <= divisor;
2'b10: csr_do <= thru;
endcase
if(csr_we) begin
case(csr_a[1:0])
2'b00:; /* handled by transceiver */
2'b01: divisor <= csr_di[15:0];
2'b10: thru <= csr_di[0];
endcase
end
end
end
end

endmodule
Code:
/*
* Milkymist VJ SoC
* Copyright (C) 2007, 2008, 2009, 2010 Sebastien Bourdeauducq
* Copyright (C) 2007 Das Labor
*
* This program is free software: you can redistribute it and/or modify
* the Free Software Foundation, version 3 of the License.
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this program.  If not, see <http://www.gnu.org/licenses/>.
*/

module uart_transceiver(
input sys_rst,
input sys_clk,

input uart_rx,
output reg uart_tx,

input [15:0] divisor,

output reg [7:0] rx_data,
output reg rx_done,

input [7:0] tx_data,
input tx_wr,
output reg tx_done
);

//-----------------------------------------------------------------
// enable16 generator
//-----------------------------------------------------------------
reg [15:0] enable16_counter;

wire enable16;
assign enable16 = (enable16_counter == 16'd0);

always @(posedge sys_clk) begin
if(sys_rst)
enable16_counter <= divisor - 16'b1;
else begin
enable16_counter <= enable16_counter - 16'd1;
if(enable16)
enable16_counter <= divisor - 16'b1;
end
end

//-----------------------------------------------------------------
// Synchronize uart_rx
//-----------------------------------------------------------------
reg uart_rx1;
reg uart_rx2;

always @(posedge sys_clk) begin
uart_rx1 <= uart_rx;
uart_rx2 <= uart_rx1;
end

//-----------------------------------------------------------------
// UART RX Logic
//-----------------------------------------------------------------
reg rx_busy;
reg [3:0] rx_count16;
reg [3:0] rx_bitcount;
reg [7:0] rx_reg;

always @(posedge sys_clk) begin
if(sys_rst) begin
rx_done <= 1'b0;
rx_busy <= 1'b0;
rx_count16  <= 4'd0;
rx_bitcount <= 4'd0;
end else begin
rx_done <= 1'b0;

if(enable16) begin
if(~rx_busy) begin // look for start bit
if(~uart_rx2) begin // start bit found
rx_busy <= 1'b1;
rx_count16 <= 4'd7;
rx_bitcount <= 4'd0;
end
end else begin
rx_count16 <= rx_count16 + 4'd1;

if(rx_count16 == 4'd0) begin // sample
rx_bitcount <= rx_bitcount + 4'd1;

if(rx_bitcount == 4'd0) begin // verify startbit
if(uart_rx2)
rx_busy <= 1'b0;
end else if(rx_bitcount == 4'd9) begin
rx_busy <= 1'b0;
if(uart_rx2) begin // stop bit ok
rx_data <= rx_reg;
rx_done <= 1'b1;
end // ignore RX error
end else
rx_reg <= {uart_rx2, rx_reg[7:1]};
end
end
end
end
end

//-----------------------------------------------------------------
// UART TX Logic
//-----------------------------------------------------------------
reg tx_busy;
reg [3:0] tx_bitcount;
reg [3:0] tx_count16;
reg [7:0] tx_reg;

always @(posedge sys_clk) begin
if(sys_rst) begin
tx_done <= 1'b0;
tx_busy <= 1'b0;
uart_tx <= 1'b1;
end else begin
tx_done <= 1'b0;
if(tx_wr) begin
tx_reg <= tx_data;
tx_bitcount <= 4'd0;
tx_count16 <= 4'd1;
tx_busy <= 1'b1;
uart_tx <= 1'b0;
ifdef SIMULATION
\$display("UART: %c", tx_data);
endif
end else if(enable16 && tx_busy) begin
tx_count16  <= tx_count16 + 4'd1;

if(tx_count16 == 4'd0) begin
tx_bitcount <= tx_bitcount + 4'd1;

if(tx_bitcount == 4'd8) begin
uart_tx <= 1'b1;
end else if(tx_bitcount == 4'd9) begin
uart_tx <= 1'b1;
tx_busy <= 1'b0;
tx_done <= 1'b1;
end else begin
uart_tx <= tx_reg[0];
tx_reg <= {1'b0, tx_reg[7:1]};
end
end
end
end
end

endmodule
You said something about Artix-7 FPGA - in such case the software tool for synthessis it will be Xilinx Vivado. Just try to install free Vivado Webpack on your computer and try to implement this UART project. If you have it working (UART) we can talk about next steps to finish your project.

Regards

Last edited:
FvM


text:
# 5. Гігієна, ооз 1 a number of viable fetuses per 1000 women at the age between 15 and 44 is determined by: A

 Назва 5. Гігієна, ооз 1 a number of viable fetuses per 1000 women at the age between 15 and 44 is determined by: A Сторінка 1/4 Дата 25.09.2012 Розмір 0.5 Mb. Тип Документи
1   2   3   4
 Крок 2 Medicine5. Гігієна, ООЗ 1 A number of viable fetuses per 1000 women at the age between 15 and 44 is determined by: A Genital index B Reproductive level C Birth rate D Perinatal rate E Obstetric rate 2 Point out the unit for statistical observation for the determination of blood sugar level influence on a wound surface healing during postoperative period. A The patient in a postoperative period B Blood sugar level C Blood test D The patient who has a wound surface E The patient who was discharged to outpatient treatment 3 What methods of the collecting of the information is preferable for study of housing conditions of students of medical HIGH SCHOOL for a training period? A Questioning B Interviewing C Selecting of materials D A method of the directed selection E Statistical 4 What information gathering method is preferable to study housing conditions of medical students during training period? A Questionaire B Interviewing C Materials selection D Directed selection method E Statistical 5 Choose a method of a graphic representation of monthly information about number of the registered cases of acute intestinal infection and their comparisons to the average monthly values, obtained for 5 previous years: A The linear diagram B The radial diagram C The sector diagram D The figured diagram E The curvilinear disgram 6 The parameter of infantile mortality for the last year was - 16,3, in present year - 15,7. Name a kind of the diagram that can be used for a graphic representation of it: A Stylar B Linear C Intrastylar D Sector E Radial 7 The average body lenth of newborn boys is 50,9 cm at a sigma 1,66; and average mass - 3432 at a sigma 5,00. What criterion is necessary in order to compare degree of variability of these signs? A Coefficient of variation B Sigma C Limit D Amplitude E Coefficient of association 8 What method is applied to establish rate of correlation between age of men and their mortality due to myocardial infarction? A Method of grade correlation (Spirman) B Correlation ratio C The quadrate method (Pirson) D The Indirect method (Student) E Method of graduated correlation (Armler) 9 Indicate the registration medical document for the patient, who 21.02. was addressed to the doctor with diagnosis ARVD for the first time in this year: A The statistical coupon is to be filled in and it is necessary to deliver on a sign (+) B The statistical coupon for registration of final diagnosis is not necessary C The statistical coupon is to be filled in, but a sign (+) is not necessary to be put in D It is necessary to fill in the emergency notice on a case of a contagion E The necessary registration form is not indicated 10 Define the basic registration document at the profound study of a case rate with temporary lost labor ability at the industrial enterprise: A A card of the personal account of a case rate B "The Report on reasons of a temporary lost labor ability" C The sick-leave certificate D A ambulatory medical card E The inpatient medical record 11 Head of a department and a trade-union group have appealed to the head of a hospital about dismissal of the senior nurse who has 17 year record of service. The facts of charge were confirmed and recognized by the nurse herself. This nurse lives with a daughter (who is divorced and unemployed) and a 9-month-old grandson. Make an administrative decision: A To continue the worker in office with a warning of dismissal in case of repeated violation of labor discipline B To discharge the worker, i.e. to satisfy demands of the collective C To issue the sick list D To shift the solution of this problem on other officials or public organizations E - 12 25 unorganized children in the age 2-3 year will be observed on a pediatric district it in the current year. What scheduled number of initial visitations will make to this group of children? A 50 B 20 C 40 D 100 E 200 13 A child is 6 years old. Within one year of observation he had URI that lasted 8 days. Physical state is satisfactory. Specify hi health group: A I B II C III (a) D III (b) E III (c) 14 A 38 year old man was admitted to a hospital from his working place on July 19 because of hip fracture. He was invalid till November 19. Requires prolongation of treatment. Who decides on the issue of further temporary invalidity? A Specialized (traumatologic) MSEC B DCC C The head physician of a polyclinic D Interregional general MSEC E Regional MSEC 15 What is the maximum duration of medical certificate in case of tuberculosis? A 2 months B Week C 2 weeks D Month E 10 months 16 An engineer-chemist at the age of 47 often fells ill with an occupational skin disease. Who makes a decision to transfer him to other job accepts? A DCC B A head physician C The attending physician D The chief of shop E MSEC 17 A patient with high temperature came to a first-aid post in the evening. The fact of temporary disability was established. Indicate the order of examination in this case: A The night duty doctor should issue a medical certificate, which will be subsequently used for issuing a sick list from the date of the previous day B The sick list for 1 day should be issued C The sick list for up to 3 days should be issued D The sick list for 3 days should be issued E Any document shouldn't be issued 18 A sample of milk was taken for testing from a 5 ton milk batch. Lab analysis showed the following: fat content 2\%, specific density- 1,04 $g/cm^3$, acidity $21^0C$, reductase probe – weak positive. What way the product is to be used in? What would you advise? A Sell but inform customers about milk quality B Write the product off for animal feeding C Utilize technically D Sell without limitations E Annihilate the product 19 A 5 tons milk batch was sampled. The lab analysis revealed: fat content 2\%, specific density - 1,04 g/$cm^3$, acidity - $21^0$Т, reductase probe - weak-positive. What way is the product to be used in? A Sell but inform customers about milk quality B Discard for animal feeding C Technical utilization D Sell without limitations E Do the product away 20 The student has the following devices: Geiger counter, Ebert counter, Krotov's apparatus, Mischuk device, Ebert device. What device can he use to assess air germ pollution? A Krotov's apparatus B Ebert's counter C Geiger's counter D Mischuk's device E Ebert's device 21 Thyreotoxicosis patient is in the two-place hospital ward of therapeutic department. The area of the ward is $18 m^2$, height is 3 m, ventilation rate is 2,5/h. Air temperature is $20^0С$, relative humidity is 45\%, air movement velocity is 0,3 m/sec, light coefficient is 1/5, noise level constitutes 30 dB. Make a hygienic assessment of these conditions. A Discomfortable microclimate B Non-effective ventilation C Poor lighting D High level of noise E All conditions are OK 22 A 9 y.o. girl has an average height and harmonic growth development. She was ill with acute respiratory infection for five times. Define the group of her health. A 2nd group B 1st group C 3rd group D 4th group E 5th group 23 A student lives in the modern house in the flat with a complete set of sanitary equipment (WC, bath, shower, local water heater). How much water consumption has he got? A 160-200 L/day B 10-15 L/day C 50-100 L/day D 300-400 L/day E 500 -600 L/day 24 A 30-year-old patient with complaints of occipital headache, disturbed sleep with nightmares came to a policlinic. BP was 150/95 mm Hg. He was diagnosed with hypertensic crisis. The patient should be registered in the following dispensary group for arterial hypertension surveillance: A In the second B In the first C In the fourth D In the third E In the fifth 25 A young patient who came to a policlinic was diagnosed with the 1 stage of hypertension. How often should he undergo the medical check-up? A Twice a year B Once a year C 3 times a year D 4 times a year E 5 times a year 26 25 children at the age of 2-3 years who don't attend any child welfare institutions should be observed by a district pediatrician within the current year. How many initial visits of this group of children should be planned? A 50 B 20 C 40 D 100 E 200 27 Studying of pulmonary tuberculosis incidence provided data about patients' socioeconomic living conditions and bad habits. What method allows to estimate the impact of these factors on tuberculosis incidence? A Calculation of correlation coefficient B Calculation of correspondence index C Calculation of regression coefficient D Standardized index calculation E Calculation of reliability coefficient 28 A teacher of a secondary school was diagnosed with pulmonary tuberculosis. What is the maximum duration of his medical certificate? A Ten months B Five months C Four months D Two months E A month 29 An employee of a private company was ill with acute respiratory viral infection. Consulted a district doctor, who determined the fact of temporary loss of working ability, but refused to issue a sick-list, arguing that the patient worked in the private and not state-owned company. Should the sick-list be issued to the employees of private companies? A Issued regardless of company's ownership B Issued only to empties of state-owned companies C Issued only on condition of payment guarantee by the company's proprietor D Issued a medical certificate of a set form E Issued a medical certificate of a free form 30 Clinic of a research instutute for occupational diseases examined a worker who works at a concentration plant and diagnosed him with chronic dust bronchitis. The case is investigated by a commission including the representatives of: the plant, clinic, territorial SES, department of Social Insurance Fund, trade union. According to the "regulation on investigation of…", the commission should be headed by the representative of the following authority: A Territorial SES B Plant C Social Insurance Fund D Trade union E Clinic 31 Basing upon the data of laboratory assessment of sanitary state of soil in a certain territory, the soil was found to be low-contaminated according to the sanitary indicative value; contaminated according to the coli titer; low-contaminated according to the anaerobe titer (\emph{Cl. Perfringens}). This is indicative of: A Fresh fecal contamination B Insufficient intensity of soil humification C Old fecal contamination D Constant entry of organic protein contaminations E Insufficient insolation and aeration of soil 32 A maternity hospital registered 616 live births, 1 stillbirth, 1 death on the 5th day of life over a 1 year period. What index allows the most precise estimation of this situation? A Perinatal mortality B Crude mortality rate C Natality D Neonatal mortality E Natural increase 33 A patient complained about problems with pain and tactile sensitivity, pain in the nail bones at the end of the working day. He works at a plant with mechanical devices. What pathology can be suspected? A Vibration disease B Caisson disease C Noise disease D Overwork symptoms E Hypovitaminosis of $B_1$ 34 Bacterial analysis of air in a living space in winter period by means of Krotov's apparatus revealed that total number of microorganisms in 1$m^3$ of air was 7200. What is the permissible number of microorganisms for the air to be characterized as "pure"? A Up to 4500 B Up to 2500 C Up to 3500 D Up to 5500 E Up to 7500 35 Maximum permissible concentration of carbon dioxide in the air is considered to be a sanitary index of air purity in a classroom. What concentration of carbon dioxide in the air is accepted as maximum permissible? A 0,1\% B 0,05\% C 0,15\% D 0,2\% E 0,3\% 36 Study of actual diet of an adult revealed the following: proteins make up 16\% of energy value of daily ration, fats - 25\%, carbohydrates - 59\%. Evaluate compliance of protein, fat and carbohydrate share in the energy value of daily ration with the recommended shares of these nutrients? A Carbohydrate share is insufficient, there is excess of proteins B Fat share is insufficient C Carbohydrate share is insufficicent D Carbohydrate share is excessive E Nutrient content complies with the recommended shares of energy value 37 A patient who has been consuming refined foodstuffs for a long time complains about headache, fatiguability, depression, insomnia, irritability. Objectively: muscle asthenia, pain and cramps in the gastrocnemius muscles, during walking the patient lands onto his heel first, then on the external edge of foot. Cardiovascular system exhibits tachycardia, hypoxia, dystrophic changes of myocardium. There are also gastrointestinal disorders. What is the most likely diagnosis? A Hypovitaminosis $B_1$ B Hypovitaminosis $B_2$ C Hypovitaminosis $B_{12}$ D Hypovitaminosis $B_6$ E Hypovitaminosis $B_{15}$ 38 A patient who had eaten mushrooms in the morning was delivered to the infectious diseases hospital at night. The disease development was rapid. The patient presented with stomach pain, frequent diarrhea, intractable vomiting, burning thirst, headache and dizziness. He died on the third day. What mushrooms are most likely to have caused mycetismus? A Deadly amanita B Morels C Fly agarics D Sulfur-tufts E Russules 39 Periodical survey of a worker of a chemicals plant revealed a malignant neoplasm on the urinary bladder. This occupational disease was the most probably caused by contact with the following industrial poison: A Benzidine B Vinyl chloride C Nickel carbonyl D Asbestos E Arsenic 40 A 37 year old patient applied to a local therapeutist. As a result of exacerbation of chronic obstructive bronchitis the patient had been temporarily disabled for 117 days within 1 year. What tactics will be legally correct? A The patient should be referred to the medicosocial expertise B The therapeutist should extend a medical certificate C The patient should be referred to the medical consultation comission for extension of medical certificate D The therapeutist should issue a new medical certificate E The patient shoul be referred to the sanatorium-and-spa treatment 41 A 46-year-old patient once took part in elimination of breakdown at an atomic power plant. Currently he is being treated at an in-patient hospital. He was diagnosed with progressing vegetative insufficiency. This disease relates to the following group of ionizing radiation effects:
1   2   3   4

## Схожі:

 5. Гігієна, ооз 1 5. Гігієна, ооз 1Комплексна оцінка здоров’я дітей, які проживають в Закарпатському регіоні України 5. Гігієна, ооз 1Комплексна оцінка здоров’я дітей, які проживають в Закарпатському регіоні України Модуль 2 з гігієни та екології для студентів 3 курсу медичного факультетуГігієна праці“, “Гігієна дітей і підлітків”, “Екологія людини”, “Здоровий спосіб життя, особиста гігієна та психогігієна”, “Гігієна... Графік консультацій та відпрацювань “нб” практичних занять з соціальної медицини та ооз студентами ІV курсу, медичних факультетів №1, №2 та №3 у осінньому семестріОоз студентами ІV курсу, медичних факультетів №1, №2 та №3 у осінньому семестрі Category " women " in publychnoe and chastnoe right drevneho rima Language discrimination against women as reflection of sociocultural patterns Cod number 2013 Normal random number generator Reference Number Country Area
Додайте кнопку на своєму сайті:
Документи

База даних захищена авторським правом ©zavantag.com 2000-2013
При копіюванні матеріалу обов'язкове зазначення активного посилання відкритою для індексації.
звернутися до адміністрації


text:
# 7.3 Unit circle  (Page 7/11)

 Page 7 / 11

Access these online resources for additional instruction and practice with sine and cosine functions.

## Key equations

 Cosine $\mathrm{cos}\text{\hspace{0.17em}}t=x$ Sine $\mathrm{sin}\text{\hspace{0.17em}}t=y$ Pythagorean Identity ${\mathrm{cos}}^{2}t+{\mathrm{sin}}^{2}t=1$

## Key concepts

• Finding the function values for the sine and cosine begins with drawing a unit circle, which is centered at the origin and has a radius of 1 unit.
• Using the unit circle, the sine of an angle $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ equals the y -value of the endpoint on the unit circle of an arc of length $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ whereas the cosine of an angle $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ equals the x -value of the endpoint. See [link] .
• The sine and cosine values are most directly determined when the corresponding point on the unit circle falls on an axis. See [link] .
• When the sine or cosine is known, we can use the Pythagorean Identity to find the other. The Pythagorean Identity is also useful for determining the sines and cosines of special angles. See [link] .
• Calculators and graphing software are helpful for finding sines and cosines if the proper procedure for entering information is known. See [link] .
• The domain of the sine and cosine functions is all real numbers.
• The range of both the sine and cosine functions is $\text{\hspace{0.17em}}\left[-1,1\right].$
• The sine and cosine of an angle have the same absolute value as the sine and cosine of its reference angle.
• The signs of the sine and cosine are determined from the x - and y -values in the quadrant of the original angle.
• An angle’s reference angle is the size angle, $\text{\hspace{0.17em}}t,$ formed by the terminal side of the angle $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ and the horizontal axis. See [link] .
• Reference angles can be used to find the sine and cosine of the original angle. See [link] .
• Reference angles can also be used to find the coordinates of a point on a circle. See [link] .

## Verbal

Describe the unit circle.

The unit circle is a circle of radius 1 centered at the origin.

What do the x- and y- coordinates of the points on the unit circle represent?

Discuss the difference between a coterminal angle and a reference angle.

Coterminal angles are angles that share the same terminal side. A reference angle is the size of the smallest acute angle, $\text{\hspace{0.17em}}t,$ formed by the terminal side of the angle $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ and the horizontal axis.

Explain how the cosine of an angle in the second quadrant differs from the cosine of its reference angle in the unit circle.

Explain how the sine of an angle in the second quadrant differs from the sine of its reference angle in the unit circle.

The sine values are equal.

## Algebraic

For the following exercises, use the given sign of the sine and cosine functions to find the quadrant in which the terminal point determined by $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ lies.

$\text{sin}\left(t\right)<0\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\text{cos}\left(t\right)<0$

$\text{sin}\left(t\right)>0\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\mathrm{cos}\left(t\right)>0$

I

$\text{sin}\left(t\right)>0\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\mathrm{cos}\left(t\right)<0$

$\text{sin}\left(t\right)>0\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\mathrm{cos}\left(t\right)>0$

IV

For the following exercises, find the exact value of each trigonometric function.

$\mathrm{sin}\text{\hspace{0.17em}}\frac{\pi }{2}$

$\mathrm{sin}\text{\hspace{0.17em}}\frac{\pi }{3}$

$\frac{\sqrt{3}}{2}$

$\mathrm{cos}\text{\hspace{0.17em}}\frac{\pi }{2}$

$\mathrm{cos}\text{\hspace{0.17em}}\frac{\pi }{3}$

$\frac{1}{2}$

$\mathrm{sin}\text{\hspace{0.17em}}\frac{\pi }{4}$

$\mathrm{cos}\text{\hspace{0.17em}}\frac{\pi }{4}$

$\frac{\sqrt{2}}{2}$

$\mathrm{sin}\text{\hspace{0.17em}}\frac{\pi }{6}$

$\mathrm{sin}\text{\hspace{0.17em}}\pi$

0

$\mathrm{sin}\text{\hspace{0.17em}}\frac{3\pi }{2}$

$\mathrm{cos}\text{\hspace{0.17em}}\pi$

-1

$\mathrm{cos}\text{\hspace{0.17em}}0$

$\mathrm{cos}\text{\hspace{0.17em}}\frac{\pi }{6}$

$\frac{\sqrt{3}}{2}$

By the definition, is such that 0!=1.why?
(1+cosA+IsinA)(1+cosB+isinB)/(cos@+isin@)(cos$+isin$)
hatdog
Mark
how we can draw three triangles of distinctly different shapes. All the angles will be cutt off each triangle and placed side by side with vertices touching
bsc F. y algebra and trigonometry pepper 2
given that x= 3/5 find sin 3x
4
DB
remove any signs and collect terms of -2(8a-3b-c)
-16a+6b+2c
Will
Joeval
(x2-2x+8)-4(x2-3x+5)
sorry
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
(X2-2X+8)-4(X2-3X+5)=0 ?
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
Y
master
master
Soo sorry (5±Root11* i)/3
master
Mukhtar
2x²-6x+1=0
Ife
explain and give four example of hyperbolic function
What is the correct rational algebraic expression of the given "a fraction whose denominator is 10 more than the numerator y?
y/y+10
Mr
Find nth derivative of eax sin (bx + c).
Find area common to the parabola y2 = 4ax and x2 = 4ay.
Anurag
y2=4ax= y=4ax/2. y=2ax
akash
A rectangular garden is 25ft wide. if its area is 1125ft, what is the length of the garden
to find the length I divide the area by the wide wich means 1125ft/25ft=45
Miranda
thanks
Jhovie
What do you call a relation where each element in the domain is related to only one value in the range by some rules?
A banana.
Yaona
a function
Daniel
a function
emmanuel
given 4cot thither +3=0and 0°<thither <180° use a sketch to determine the value of the following a)cos thither
what are you up to?
nothing up todat yet
Miranda
hi
jai
hello
jai
Miranda Drice
jai
aap konsi country se ho
jai
which language is that
Miranda
I am living in india
jai
good
Miranda
what is the formula for calculating algebraic
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda


text:
# Something to think upon?

√2 + √3 = 3.14 (approximately). Does this have any significance or is it just a coincidence?

Note by Harshit Joshi
4 years ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$

Sort by:

well even i was thinking of the same thing today after solving a problem which involved √3 &√2.. can there be any value of π which is not just some unknown term but a specific value which involved combination of √ , * , + , / , other operations..!

- 3 years, 12 months ago