The density, or more the volumetric mass density, of a substance is its mass per unit volume. The symbol most used for density is ρ, although the Latin letter D can be used. Mathematically, density is defined as mass divided by volume: ρ = m V where ρ is the density, m is the mass, V is the volume. In some cases, density is loosely defined as its weight per unit volume, although this is scientifically inaccurate – this quantity is more called specific weight. For a pure substance the density has the same numerical value as its mass concentration. Different materials have different densities, density may be relevant to buoyancy and packaging. Osmium and iridium are the densest known elements at standard conditions for temperature and pressure but certain chemical compounds may be denser. To simplify comparisons of density across different systems of units, it is sometimes replaced by the dimensionless quantity "relative density" or "specific gravity", i.e. the ratio of the density of the material to that of a standard material water.
Thus a relative density less than one means. The density of a material varies with pressure; this variation is small for solids and liquids but much greater for gases. Increasing the pressure on an object decreases the volume of the object and thus increases its density. Increasing the temperature of a substance decreases its density by increasing its volume. In most materials, heating the bottom of a fluid results in convection of the heat from the bottom to the top, due to the decrease in the density of the heated fluid; this causes it to rise relative to more dense unheated material. The reciprocal of the density of a substance is called its specific volume, a term sometimes used in thermodynamics. Density is an intensive property in that increasing the amount of a substance does not increase its density. In a well-known but apocryphal tale, Archimedes was given the task of determining whether King Hiero's goldsmith was embezzling gold during the manufacture of a golden wreath dedicated to the gods and replacing it with another, cheaper alloy.
Archimedes knew that the irregularly shaped wreath could be crushed into a cube whose volume could be calculated and compared with the mass. Baffled, Archimedes is said to have taken an immersion bath and observed from the rise of the water upon entering that he could calculate the volume of the gold wreath through the displacement of the water. Upon this discovery, he leapt from his bath and ran naked through the streets shouting, "Eureka! Eureka!". As a result, the term "eureka" entered common parlance and is used today to indicate a moment of enlightenment; the story first appeared in written form in Vitruvius' books of architecture, two centuries after it took place. Some scholars have doubted the accuracy of this tale, saying among other things that the method would have required precise measurements that would have been difficult to make at the time. From the equation for density, mass density has units of mass divided by volume; as there are many units of mass and volume covering many different magnitudes there are a large number of units for mass density in use.
The SI unit of kilogram per cubic metre and the cgs unit of gram per cubic centimetre are the most used units for density. One g/cm3 is equal to one thousand kg/m3. One cubic centimetre is equal to one millilitre. In industry, other larger or smaller units of mass and or volume are more practical and US customary units may be used. See below for a list of some of the most common units of density. A number of techniques as well as standards exist for the measurement of density of materials; such techniques include the use of a hydrometer, Hydrostatic balance, immersed body method, air comparison pycnometer, oscillating densitometer, as well as pour and tap. However, each individual method or technique measures different types of density, therefore it is necessary to have an understanding of the type of density being measured as well as the type of material in question; the density at all points of a homogeneous object equals its total mass divided by its total volume. The mass is measured with a scale or balance.
To determine the density of a liquid or a gas, a hydrometer, a dasymeter or a Coriolis flow meter may be used, respectively. Hydrostatic weighing uses the displacement of water due to a submerged object to determine the density of the object. If the body is not homogeneous its density varies between different regions of the object. In that case the density around any given location is determined by calculating the density of a small volume around that location. In the limit of an infinitesimal volume the density of an inhomogeneous object at a point becomes: ρ = d m / d V, where d V is an elementary volume at position r; the mass of the body t
A chemical reaction is a process that leads to the chemical transformation of one set of chemical substances to another. Classically, chemical reactions encompass changes that only involve the positions of electrons in the forming and breaking of chemical bonds between atoms, with no change to the nuclei, can be described by a chemical equation. Nuclear chemistry is a sub-discipline of chemistry that involves the chemical reactions of unstable and radioactive elements where both electronic and nuclear changes can occur; the substance involved in a chemical reaction are called reactants or reagents. Chemical reactions are characterized by a chemical change, they yield one or more products, which have properties different from the reactants. Reactions consist of a sequence of individual sub-steps, the so-called elementary reactions, the information on the precise course of action is part of the reaction mechanism. Chemical reactions are described with chemical equations, which symbolically present the starting materials, end products, sometimes intermediate products and reaction conditions.
Chemical reactions happen at a characteristic reaction rate at a given temperature and chemical concentration. Reaction rates increase with increasing temperature because there is more thermal energy available to reach the activation energy necessary for breaking bonds between atoms. Reactions may proceed in the forward or reverse direction until they go to completion or reach equilibrium. Reactions that proceed in the forward direction to approach equilibrium are described as spontaneous, requiring no input of free energy to go forward. Non-spontaneous reactions require input of free energy to go forward. Different chemical reactions are used in combinations during chemical synthesis in order to obtain a desired product. In biochemistry, a consecutive series of chemical reactions form metabolic pathways; these reactions are catalyzed by protein enzymes. Enzymes increase the rates of biochemical reactions, so that metabolic syntheses and decompositions impossible under ordinary conditions can occur at the temperatures and concentrations present within a cell.
The general concept of a chemical reaction has been extended to reactions between entities smaller than atoms, including nuclear reactions, radioactive decays, reactions between elementary particles, as described by quantum field theory. Chemical reactions such as combustion in fire and the reduction of ores to metals were known since antiquity. Initial theories of transformation of materials were developed by Greek philosophers, such as the Four-Element Theory of Empedocles stating that any substance is composed of the four basic elements – fire, water and earth. In the Middle Ages, chemical transformations were studied by Alchemists, they attempted, in particular, to convert lead into gold, for which purpose they used reactions of lead and lead-copper alloys with sulfur. The production of chemical substances that do not occur in nature has long been tried, such as the synthesis of sulfuric and nitric acids attributed to the controversial alchemist Jābir ibn Hayyān; the process involved heating of sulfate and nitrate minerals such as copper sulfate and saltpeter.
In the 17th century, Johann Rudolph Glauber produced hydrochloric acid and sodium sulfate by reacting sulfuric acid and sodium chloride. With the development of the lead chamber process in 1746 and the Leblanc process, allowing large-scale production of sulfuric acid and sodium carbonate chemical reactions became implemented into the industry. Further optimization of sulfuric acid technology resulted in the contact process in the 1880s, the Haber process was developed in 1909–1910 for ammonia synthesis. From the 16th century, researchers including Jan Baptist van Helmont, Robert Boyle, Isaac Newton tried to establish theories of the experimentally observed chemical transformations; the phlogiston theory was proposed in 1667 by Johann Joachim Becher. It postulated the existence of a fire-like element called "phlogiston", contained within combustible bodies and released during combustion; this proved to be false in 1785 by Antoine Lavoisier who found the correct explanation of the combustion as reaction with oxygen from the air.
Joseph Louis Gay-Lussac recognized in 1808 that gases always react in a certain relationship with each other. Based on this idea and the atomic theory of John Dalton, Joseph Proust had developed the law of definite proportions, which resulted in the concepts of stoichiometry and chemical equations. Regarding the organic chemistry, it was long believed that compounds obtained from living organisms were too complex to be obtained synthetically. According to the concept of vitalism, organic matter was endowed with a "vital force" and distinguished from inorganic materials; this separation was ended however by the synthesis of urea from inorganic precursors by Friedrich Wöhler in 1828. Other chemists who brought major contributions to organic chemistry include Alexander William Williamson with his synthesis of ethers and Christopher Kelk Ingold, among many discoveries, established the mechanisms of substitution reactions. Chemical equations are used to graphically illustrate chemical reactions, they consist of chemical or structural formulas of the reactants on the left and those of the products on the right.
They are separated by an arrow which indicates the type of the reaction.
Stoichiometry is the calculation of reactants and products in chemical reactions. Stoichiometry is founded on the law of conservation of mass where the total mass of the reactants equals the total mass of the products, leading to the insight that the relations among quantities of reactants and products form a ratio of positive integers; this means that if the amounts of the separate reactants are known the amount of the product can be calculated. Conversely, if one reactant has a known quantity and the quantity of the products can be empirically determined the amount of the other reactants can be calculated; this is illustrated in the image here, where the balanced equation is: CH4 + 2 O2 → CO2 + 2 H2O. Here, one molecule of methane reacts with two molecules of oxygen gas to yield one molecule of carbon dioxide and two molecules of water; this particular chemical equation is an example of complete combustion. Stoichiometry measures these quantitative relationships, is used to determine the amount of products and reactants that are produced or needed in a given reaction.
Describing the quantitative relationships among substances as they participate in chemical reactions is known as reaction stoichiometry. In the example above, reaction stoichiometry measures the relationship between the methane and oxygen as they react to form carbon dioxide and water; because of the well known relationship of moles to atomic weights, the ratios that are arrived at by stoichiometry can be used to determine quantities by weight in a reaction described by a balanced equation. This is called composition stoichiometry. Gas stoichiometry deals with reactions involving gases, where the gases are at a known temperature and volume and can be assumed to be ideal gases. For gases, the volume ratio is ideally the same by the ideal gas law, but the mass ratio of a single reaction has to be calculated from the molecular masses of the reactants and products. In practice, due to the existence of isotopes, molar masses are used instead when calculating the mass ratio; the term stoichiometry was first used by Jeremias Benjamin Richter in 1792 when the first volume of Richter's Stoichiometry or the Art of Measuring the Chemical Elements was published.
The term is derived from the Ancient Greek words στοιχεῖον stoicheion "element" and μέτρον metron "measure". In patristic Greek, the word Stoichiometria was used by Nicephorus to refer to the number of line counts of the canonical New Testament and some of the Apocrypha. A stoichiometric amount or stoichiometric ratio of a reagent is the optimum amount or ratio where, assuming that the reaction proceeds to completion: All of the reagent is consumed There is no deficiency of the reagent There is no excess of the reagent. Stoichiometry rests upon the basic laws that help to understand it better, i.e. law of conservation of mass, the law of definite proportions, the law of multiple proportions and the law of reciprocal proportions. In general, chemical reactions combine in definite ratios of chemicals. Since chemical reactions can neither create nor destroy matter, nor transmute one element into another, the amount of each element must be the same throughout the overall reaction. For example, the number of atoms of a given element X on the reactant side must equal the number of atoms of that element on the product side, whether or not all of those atoms are involved in a reaction.
Chemical reactions, as macroscopic unit operations, consist of a large number of elementary reactions, where a single molecule reacts with another molecule. As the reacting molecules consist of a definite set of atoms in an integer ratio, the ratio between reactants in a complete reaction is in integer ratio. A reaction may consume more than one molecule, the stoichiometric number counts this number, defined as positive for products and negative for reactants. Different elements have a different atomic mass, as collections of single atoms, molecules have a definite molar mass, measured with the unit mole. By definition, carbon-12 has a molar mass of 12 g/mol. Thus, to calculate the stoichiometry by mass, the number of molecules required for each reactant is expressed in moles and multiplied by the molar mass of each to give the mass of each reactant per mole of reaction; the mass ratios can be calculated by dividing each by the total in the whole reaction. Elements in their natural state are mixtures of isotopes of differing mass, thus atomic masses and thus molar masses are not integers.
For instance, instead of an exact 14:3 proportion, 17.04 kg of ammonia consists of 14.01 kg of nitrogen and 3 × 1.01 kg of hydrogen, because natural nitrogen includes a small amount of nitrogen-15, natural hydrogen includes hydrogen-2. A stoichiometric reactant is a reactant, consumed in a reaction, as opposed to a catalytic reactant, not consumed in the overall reaction because it reacts in one step and is regenerated in another step. Stoichiometry is not only used to balance chemical equations but used in conversions, i.e. converting from grams to moles using molar mass as the conversion factor, or from grams to milliliters using density. For example, to find the amount of NaCl in 2.00 g, one would do the following: 2.00 g NaCl 58.44 g NaCl mol − 1 = 0.034 mol In the above example, when written out in fraction form, the units of grams form a multiplicative identity, equivalent to one, wit
In chemical processing, a packed bed is a hollow tube, pipe, or other vessel, filled with a packing material. The packing can be randomly filled with small objects like Raschig rings or else it can be a designed structured packing. Packed beds may contain catalyst particles or adsorbents such as zeolite pellets, granular activated carbon, etc; the purpose of a packed bed is to improve contact between two phases in a chemical or similar process. Packed beds can be used in a chemical reactor, a distillation process, or a scrubber, but packed beds have been used to store heat in chemical plants. In this case, hot gases are allowed to escape through a vessel, packed with a refractory material until the packing is hot. Air or other cool gas is fed back to the plant through the hot bed, thereby pre-heating the air or gas feed. In industry, a packed column is a type of packed bed used to perform separation processes, such as absorption and distillation. A packed column is a pressure vessel. Columns used in certain types of chromatography consisting of a tube filled with packing material can be called packed columns and their structure has similarities to packed beds.
The column can be filled with random dumped packing or with structured packing sections, which are arranged or stacked. In the column, liquids tend to wet the surface of the packing and the vapors pass across this wetted surface, where mass transfer takes place. Packing material can be used instead of trays to improve separation in distillation columns. Packing offers the advantage of a lower pressure drop across the column, beneficial while operating under vacuum. Differently shaped packing materials have different surface areas and void space between the packing. Both of these factors affect packing performance. Another factor in performance, in addition to the packing shape and surface area, is the liquid and vapor distribution that enters the packed bed; the number of theoretical stages required to make a given separation is calculated using a specific vapor to liquid ratio. If the liquid and vapor are not evenly distributed across the superficial tower area as it enters the packed bed, the liquid to vapor ratio will not be correct and the required separation will not be achieved.
The packing will appear to not be working properly. The height equivalent to a theoretical plate will be greater than expected; the problem is not the packing itself but the mal-distribution of the fluids entering the packed bed. These columns can contain liquid distributors and redistributors which help to distribute the liquid evenly over a section of packing, increasing the efficiency of the mass transfer; the design of the liquid distributors used to introduce the feed and reflux to a packed bed is critical to making the packing perform at maximum efficiency. Packed columns have a continuous vapor-equilibrium curve, unlike conventional tray distillation in which every tray represents a separate point of vapor-liquid equilibrium. However, when modeling packed columns it is useful to compute a number of theoretical plates to denote the separation efficiency of the packed column with respect to more traditional trays. In design, the number of necessary theoretical equilibrium stages is first determined and the packing height equivalent to a theoretical equilibrium stage, known as the height equivalent to a theoretical plate, is determined.
The total packing height required is the number theoretical stages multiplied by the HETP. Packed bed reactors can be used in chemical reactions in chemical industries; these reactors are tubular and are filled with solid catalyst particles, most used to catalyze gas reactions. The chemical reaction takes place on the surface of the catalyst; the advantage of using a packed bed reactor is the higher conversion per weight of catalyst than other catalytic reactors. The conversion is based on the amount of the solid catalyst rather than the volume of the reactor; the Ergun equation can be used to predict the pressure drop along the length of a packed bed given the fluid velocity, the packing size, the viscosity and density of the fluid. The Ergun equation, while reliable for systems on the surface of the earth, is unreliable for predicting the behavior of systems in microgravity. Experiments are underway aboard the International Space Station to collect data and develop reliable models for on orbit packed bed reactors.
The performance of a packed bed is dependent on the flow of material through it, which in turn is dependent on the packing and how the flow is managed. Electrical tomography may be used to observe the distribution of liquids at different cross sections of the vessel, or indeed the flow pattern throughout the packed column. Depending on the nature of the materials, capacitance or resistance tomography may be used. Height of a theoretical plate Continuous distillation Kozeny-Carman equation Fluidized bed Industrial Tomography Systems Dixon rings Random column packing Perry, Robert H. & Green, Don W.. Perry's Chemical Engineers' Handbook. McGraw-Hill. ISBN 0-07-049479-7
Oscillatory baffled reactor
A Continuous Oscillatory Baffled Reactor is a specially designed chemical reactor to achieve plug flow under laminar flow conditions. Achieving plug flow has been limited to either a large number of continuous stir tank reactors in series or conditions with high turbulent flow; the technology incorporates annular baffles to a tubular reactor framework to create eddies when liquid is pushed up through the tube. When liquid is on a downstroke through the tube, eddies are created on the other side of the baffles. Eddy generation on both sides of the baffles creates effective mixing while still maintaining plug flow. By using COBR higher yields of product can be made with greater control and reduced waste. A standard COBR consists of a 10-150mm ID tube with spaced baffles throughout. There are two pumps in a COBR; this design offers a control over mixing intensity that conventional tubular reactors cannot achieve. Each baffled cell acts as a CSTR and because a secondary pump is creating a net laminar flow, much longer residence times can be achieved relative to turbulent flow systems.
With conventional tubular reactors, mixing is accomplished through stirring mechanisms or turbulent flow conditions, difficult to control. By changing variable values such as baffle spacing or thickness, COBRs can operate with much better mixing control. For instance, it has been found that a spacing of 1.5 times tube diameter size is the most effective mixing condition. There are significant advantages relative to using stir tank reators in terms of size operation time, yield; the low shear rate and enhanced mass transfer provided by the COBR makes it an ideal reactor for various biological processes. For shear rate, it has been found that COBRs have an evenly distributed, five-fold reduction in shear rate relative to conventional tubular reactors. For the case of mass transfer, COBR fluid mechanics allows for an increase in oxygen gas residence time. Furthermore, the vortexes created in the COBRs causes a gas bubble break-up and thus an increase in surface area for gas transfer. For aerobic biological processes, therefore, COBRs again present an advantage.
An promising aspect of the COBR technology is its ability to scale-up processes while still retaining the advantages in shear rate and mass transfer. Though the prospect for COBR applications in fields like bioprocessing are promising, there are a number of necessary improvements to be made before more global use. There is additional complexity in the COBR design relative to other bioreactors, which can introduce complications in operation. Furthermore, for bioprocessing it is possible that fouling of baffles and internal surfaces becomes an issue; the most significant needed advancement moving forward is further comprehensive studies that COBR technology can indeed be useful in industry. There are no COBRs in use at industrial bioprocessing plants and the evidence of its effectiveness, though promising and theoretically an improvement relative to current reactors in industry, is limited to smaller laboratory-scale experiments
A heat exchanger is a device used to transfer heat between two or more fluids. Heat exchangers are used in both heating processes; the fluids may be separated by a solid wall to prevent mixing or they may be in direct contact. They are used in space heating, air conditioning, power stations, chemical plants, petrochemical plants, petroleum refineries, natural-gas processing, sewage treatment; the classic example of a heat exchanger is found in an internal combustion engine in which a circulating fluid known as engine coolant flows through radiator coils and air flows past the coils, which cools the coolant and heats the incoming air. Another example is the heat sink, a passive heat exchanger that transfers the heat generated by an electronic or a mechanical device to a fluid medium air or a liquid coolant. There are three primary classifications of heat exchangers according to their flow arrangement. In parallel-flow heat exchangers, the two fluids enter the exchanger at the same end, travel in parallel to one another to the other side.
In counter-flow heat exchangers the fluids enter the exchanger from opposite ends. The counter current design is the most efficient, in that it can transfer the most heat from the heat medium per unit mass due to the fact that the average temperature difference along any unit length is higher. See countercurrent exchange. In a cross-flow heat exchanger, the fluids travel perpendicular to one another through the exchanger. For efficiency, heat exchangers are designed to maximize the surface area of the wall between the two fluids, while minimizing resistance to fluid flow through the exchanger; the exchanger's performance can be affected by the addition of fins or corrugations in one or both directions, which increase surface area and may channel fluid flow or induce turbulence. The driving temperature across the heat transfer surface varies with position, but an appropriate mean temperature can be defined. In most simple systems this is the "log mean temperature difference". Sometimes direct knowledge of the LMTD is not available and the NTU method is used.
Double pipe heat exchangers are the simplest exchangers used in industries. On one hand, these heat exchangers are cheap for both design and maintenance, making them a good choice for small industries. On the other hand, their low efficiency coupled with the high space occupied in large scales, has led modern industries to use more efficient heat exchangers like shell and tube or plate. However, since double pipe heat exchangers are simple, they are used to teach heat exchanger design basics to students as the fundamental rules for all heat exchangers are the same. Shell and tube heat exchangers consist of a series of tubes which contain fluid that must be either heated or cooled. A second fluid runs over the tubes that are being heated or cooled so that it can either provide the heat or absorb the heat required. A set of tubes is called the tube bundle and can be made up of several types of tubes: plain, longitudinally finned, etc. Shell and tube heat exchangers are used for high-pressure applications.
This is because the tube heat exchangers are robust due to their shape. Several thermal design features must be considered when designing the tubes in the shell and tube heat exchangers: There can be many variations on the shell and tube design; the ends of each tube are connected to plenums through holes in tubesheets. The tubes may be straight or bent in the shape of a U, called U-tubes. Tube diameter: Using a small tube diameter makes the heat exchanger both economical and compact. However, it is more for the heat exchanger to foul up faster and the small size makes mechanical cleaning of the fouling difficult. To prevail over the fouling and cleaning problems, larger tube diameters can be used, thus to determine the tube diameter, the available space and fouling nature of the fluids must be considered. Tube thickness: The thickness of the wall of the tubes is determined to ensure: There is enough room for corrosion That flow-induced vibration has resistance Axial strength Availability of spare parts Hoop strength Buckling strength Tube length: heat exchangers are cheaper when they have a smaller shell diameter and a long tube length.
Thus there is an aim to make the heat exchanger as long as physically possible whilst not exceeding production capabilities. However, there are many limitations for this, including space available at the installation site and the need to ensure tubes are available in lengths that are twice the required length. Long, thin tubes are difficult to take out and replace. Tube pitch: when designing the tubes, it is practical to ensure that the tube pitch is not less than 1.25 times the tubes' outside diameter. A larger tube pitch leads to a larger overall shell diameter, which leads to a more expensive heat exchanger. Tube corrugation: this type of tubes used for the inner tubes, increases the turbulence of the fluids and the effect is important in the heat transfer giving a better performance. Tube Layout: refers to. There are four main types of tube layout, which are, rotated triangular and rotated square; the triangular patterns are employed to give greater heat transfer as they force the fluid to flow in a more turbulent fashion around the piping.
Square patterns are employed where high fouling is experienced and cleaning is more regular. B
A differential equation is a mathematical equation that relates some function with its derivatives. In applications, the functions represent physical quantities, the derivatives represent their rates of change, the equation defines a relationship between the two; because such relations are common, differential equations play a prominent role in many disciplines including engineering, physics and biology. In pure mathematics, differential equations are studied from several different perspectives concerned with their solutions—the set of functions that satisfy the equation. Only the simplest differential equations are solvable by explicit formulas. If a closed-form expression for the solution is not available, the solution may be numerically approximated using computers; the theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy. Differential equations first came into existence with the invention of calculus by Newton and Leibniz.
In Chapter 2 of his 1671 work Methodus fluxionum et Serierum Infinitarum, Isaac Newton listed three kinds of differential equations: d y d x = f d y d x = f x 1 ∂ y ∂ x 1 + x 2 ∂ y ∂ x 2 = y He solves these examples and others using infinite series and discusses the non-uniqueness of solutions. Jacob Bernoulli proposed the Bernoulli differential equation in 1695; this is an ordinary differential equation of the form y ′ + P y = Q y n for which the following year Leibniz obtained solutions by simplifying it. The problem of a vibrating string such as that of a musical instrument was studied by Jean le Rond d'Alembert, Leonhard Euler, Daniel Bernoulli, Joseph-Louis Lagrange. In 1746, d’Alembert discovered the one-dimensional wave equation, within ten years Euler discovered the three-dimensional wave equation; the Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the tautochrone problem. This is the problem of determining a curve on which a weighted particle will fall to a fixed point in a fixed amount of time, independent of the starting point.
Lagrange solved this problem in 1755 and sent the solution to Euler. Both further developed Lagrange's method and applied it to mechanics, which led to the formulation of Lagrangian mechanics. In 1822, Fourier published his work on heat flow in Théorie analytique de la chaleur, in which he based his reasoning on Newton's law of cooling, that the flow of heat between two adjacent molecules is proportional to the small difference of their temperatures. Contained in this book was Fourier's proposal of his heat equation for conductive diffusion of heat; this partial differential equation is now taught to every student of mathematical physics. For example, in classical mechanics, the motion of a body is described by its position and velocity as the time value varies. Newton's laws allow these variables to be expressed dynamically as a differential equation for the unknown position of the body as a function of time. In some cases, this differential equation may be solved explicitly. An example of modelling a real world problem using differential equations is the determination of the velocity of a ball falling through the air, considering only gravity and air resistance.
The ball's acceleration towards the ground is the acceleration due to gravity minus the acceleration due to air resistance. Gravity is considered constant, air resistance may be modeled as proportional to the ball's velocity; this means that the ball's acceleration, a derivative of its velocity, depends on the velocity. Finding the velocity as a function of time involves solving a differential equation and verifying its validity. Differential equations can be divided into several types. Apart from describing the properties of the equation itself, these classes of differential equations can help inform the choice of approach to a solution. Used distinctions include whether the equation is: Ordinary/Partial, Linear/Non-linear, Homogeneous/Inhomogeneous; this list is far from exhaustive. An ordinary differential equation is an equation containing an unknown function of one real or complex variable x, its derivatives, some