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49-453: PFR may refer to: Science and technology [ edit ] Plug flow reactor model , a reactor simulation model Prototype Fast Reactor , a nuclear reactor at Dounreay PFR, Phosphate Flame Retardant, a type of Flame Retardant pFR, a form of the light-sensing pigment phytochrome found in plants pFR, polymeric Flame Retardant, a type of Flame Retardant Polynomial Freiman-Ruzsa ,

98-408: A Chinese dish Portable Font Resource Power finesse ratio , a statistic for baseball pitchers Prism fusion range , a clinical eye test to assess motor fusion Pro Football Reference , a statistics database for professional American football Promoted from Reserves (badminton) Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with

147-466: A PFR may be modeled as flowing through the reactor as a series of infinitely thin coherent "plugs", each with a uniform composition, traveling in the axial direction of the reactor, with each plug having a different composition from the ones before and after it. The key assumption is that as a plug flows through a PFR, the fluid is perfectly mixed in the radial direction but not in the axial direction (forwards or backwards). Each plug of differential volume

196-710: A conjecture in Mathematics Other [ edit ] Partito Fascista Repubblicano , a former political party in Italy Persons and Family Relations , one of the subjects covered in Civil Law on the Philippine Bar Examinations PFFR , an alternative rock group PFR (band) (Pray for Rain), a Christian music group PFR, product feature request Pontefract Baghill railway station , England; National Rail station code PFR Pork fried rice ,

245-494: A fixed residence time: Any fluid (plug) that enters the reactor at time t {\displaystyle t} will exit the reactor at time t + τ {\displaystyle t+\tau } , where τ {\displaystyle \tau } is the residence time of the reactor. The residence time distribution function is therefore a Dirac delta function at τ {\displaystyle \tau } . A real plug flow reactor has

294-399: A residence time distribution that is a narrow pulse around the mean residence time distribution. A typical plug flow reactor could be a tube packed with some solid material (frequently a catalyst ). Typically these types of reactors are called packed bed reactors or PBR's. Sometimes the tube will be a tube in a shell and tube heat exchanger . When a plug flow model can not be applied,

343-581: A result, the latter notation is a convenient abuse of notation , and not a standard ( Riemann or Lebesgue ) integral. As a probability measure on R , the delta measure is characterized by its cumulative distribution function , which is the unit step function . H ( x ) = { 1 if  x ≥ 0 0 if  x < 0. {\displaystyle H(x)={\begin{cases}1&{\text{if }}x\geq 0\\0&{\text{if }}x<0.\end{cases}}} This means that H ( x )

392-531: A single point, where it is infinite. To make proper sense of the Dirac delta, we should instead insist that the property ∫ − ∞ ∞ F Δ t ( t ) d t = P , {\displaystyle \int _{-\infty }^{\infty }F_{\Delta t}(t)\,dt=P,} which holds for all Δ t > 0 {\displaystyle \Delta t>0} , should continue to hold in

441-431: A value equal to τ {\displaystyle \tau } . The stationary PFR is governed by ordinary differential equations , the solution for which can be calculated providing that appropriate boundary conditions are known. The PFR model works well for many fluids: liquids, gases, and slurries. Although turbulent flow and axial diffusion cause a degree of mixing in the axial direction in real reactors,

490-453: Is a continuous analogue of the Kronecker delta function, which is usually defined on a discrete domain and takes values 0 and 1. The mathematical rigor of the delta function was disputed until Laurent Schwartz developed the theory of distributions, where it is defined as a linear form acting on functions. The graph of the Dirac delta is usually thought of as following the whole x -axis and

539-451: Is able to calculate the motion of the ball, by only considering the total impulse of the collision, without a detailed model of all of the elastic energy transfer at subatomic levels (for instance). To be specific, suppose that a billiard ball is at rest. At time t = 0 {\displaystyle t=0} it is struck by another ball, imparting it with a momentum P , with units kg⋅m⋅s . The exchange of momentum

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588-401: Is considered as a separate entity, effectively an infinitesimally small continuous stirred tank reactor , limiting to zero volume. As it flows down the tubular PFR, the residence time ( τ {\displaystyle \tau } ) of the plug is a function of its position in the reactor. In the ideal PFR, the residence time distribution is therefore a Dirac delta function with

637-543: Is given by: Thus (σ θ ) can be evaluated from the experimental data on C vs. t and for known values of ( σ θ ) 2 {\displaystyle (\sigma _{\theta })^{2}} , the dispersion number ( 1 / P e ) {\displaystyle (1/P_{e})} can be obtained from eq. (3) as: Thus axial dispersion coefficient D L can be estimated (L = packed height) As mentioned before, there are also other boundary conditions that can be applied to

686-551: Is infinite, δ ( x ) ≃ { + ∞ , x = 0 0 , x ≠ 0 {\displaystyle \delta (x)\simeq {\begin{cases}+\infty ,&x=0\\0,&x\neq 0\end{cases}}} and which is also constrained to satisfy the identity ∫ − ∞ ∞ δ ( x ) d x = 1. {\displaystyle \int _{-\infty }^{\infty }\delta (x)\,dx=1.} This

735-999: Is integrable if and only if g is integrable and the integrals of f and g are identical. A rigorous approach to regarding the Dirac delta function as a mathematical object in its own right requires measure theory or the theory of distributions . Joseph Fourier presented what is now called the Fourier integral theorem in his treatise Théorie analytique de la chaleur in the form: f ( x ) = 1 2 π ∫ − ∞ ∞     d α f ( α )   ∫ − ∞ ∞ d p   cos ⁡ ( p x − p α )   , {\displaystyle f(x)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }\ \ d\alpha \,f(\alpha )\ \int _{-\infty }^{\infty }dp\ \cos(px-p\alpha )\ ,} which

784-459: Is merely a heuristic characterization. The Dirac delta is not a function in the traditional sense as no extended real number valued function defined on the real numbers has these properties. One way to rigorously capture the notion of the Dirac delta function is to define a measure , called Dirac measure , which accepts a subset A of the real line R as an argument, and returns δ ( A ) = 1 if 0 ∈ A , and δ ( A ) = 0 otherwise. If

833-413: Is no function having this property, modelling the delta "function" rigorously involves the use of limits or, as is common in mathematics, measure theory and the theory of distributions . The delta function was introduced by physicist Paul Dirac , and has since been applied routinely in physics and engineering to model point masses and instantaneous impulses. It is called the delta function because it

882-554: Is not absolutely continuous with respect to the Lebesgue measure —in fact, it is a singular measure . Consequently, the delta measure has no Radon–Nikodym derivative (with respect to Lebesgue measure)—no true function for which the property ∫ − ∞ ∞ f ( x ) δ ( x ) d x = f ( 0 ) {\displaystyle \int _{-\infty }^{\infty }f(x)\,\delta (x)\,dx=f(0)} holds. As

931-866: Is not actually instantaneous, being mediated by elastic processes at the molecular and subatomic level, but for practical purposes it is convenient to consider that energy transfer as effectively instantaneous. The force therefore is P δ ( t ) ; the units of δ ( t ) are s . To model this situation more rigorously, suppose that the force instead is uniformly distributed over a small time interval Δ t = [ 0 , T ] {\displaystyle \Delta t=[0,T]} . That is, F Δ t ( t ) = { P / Δ t 0 < t ≤ T , 0 otherwise . {\displaystyle F_{\Delta t}(t)={\begin{cases}P/\Delta t&0<t\leq T,\\0&{\text{otherwise}}.\end{cases}}} Then

980-546: Is tantamount to the introduction of the δ -function in the form: δ ( x − α ) = 1 2 π ∫ − ∞ ∞ d p   cos ⁡ ( p x − p α )   . {\displaystyle \delta (x-\alpha )={\frac {1}{2\pi }}\int _{-\infty }^{\infty }dp\ \cos(px-p\alpha )\ .} Later, Augustin Cauchy expressed

1029-437: Is the concentration of species A at the inlet to the reactor, appearing from the integration boundary condition. PFRs are used to model the chemical transformation of compounds as they are transported in systems resembling "pipes". The "pipe" can represent a variety of engineered or natural conduits through which liquids or gases flow. (e.g. rivers, pipelines, regions between two mountains, etc.) An ideal plug flow reactor has

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1078-509: Is the integral of the cumulative indicator function 1 (−∞, x ] with respect to the measure δ ; to wit, H ( x ) = ∫ R 1 ( − ∞ , x ] ( t ) δ ( d t ) = δ ( ( − ∞ , x ] ) , {\displaystyle H(x)=\int _{\mathbf {R} }\mathbf {1} _{(-\infty ,x]}(t)\,\delta (dt)=\delta \!\left((-\infty ,x]\right),}

1127-475: Is usually used in the form of: The RTD is determined experimentally by injecting an inert chemical, molecule, or atom, called a tracer, into the reactor at some time t = 0 and then measuring the tracer concentration, C, in the effluent stream as a function of time. The RTD curve of fluid leaving a vessel is called the E-Curve. This curve is normalized in such a way that the area under it is unity: The mean age of

1176-1558: The order of integration is significant in this result (contrast Fubini's theorem ). As justified using the theory of distributions , the Cauchy equation can be rearranged to resemble Fourier's original formulation and expose the δ -function as f ( x ) = 1 2 π ∫ − ∞ ∞ e i p x ( ∫ − ∞ ∞ e − i p α f ( α ) d α ) d p = 1 2 π ∫ − ∞ ∞ ( ∫ − ∞ ∞ e i p x e − i p α d p ) f ( α ) d α = ∫ − ∞ ∞ δ ( x − α ) f ( α ) d α , {\displaystyle {\begin{aligned}f(x)&={\frac {1}{2\pi }}\int _{-\infty }^{\infty }e^{ipx}\left(\int _{-\infty }^{\infty }e^{-ip\alpha }f(\alpha )\,d\alpha \right)\,dp\\[4pt]&={\frac {1}{2\pi }}\int _{-\infty }^{\infty }\left(\int _{-\infty }^{\infty }e^{ipx}e^{-ip\alpha }\,dp\right)f(\alpha )\,d\alpha =\int _{-\infty }^{\infty }\delta (x-\alpha )f(\alpha )\,d\alpha ,\end{aligned}}} where

1225-712: The unit impulse , is a generalized function on the real numbers , whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one. Thus it can be represented heuristically as δ ( x ) = { 0 , x ≠ 0 ∞ , x = 0 {\displaystyle \delta (x)={\begin{cases}0,&x\neq 0\\{\infty },&x=0\end{cases}}} such that ∫ − ∞ ∞ δ ( x ) d x = 1. {\displaystyle \int _{-\infty }^{\infty }\delta (x)dx=1.} Since there

1274-441: The δ -function is expressed as δ ( x − α ) = 1 2 π ∫ − ∞ ∞ e i p ( x − α ) d p   . {\displaystyle \delta (x-\alpha )={\frac {1}{2\pi }}\int _{-\infty }^{\infty }e^{ip(x-\alpha )}\,dp\ .} A rigorous interpretation of

1323-561: The PFR model is appropriate when these effects are sufficiently small that they can be ignored. In the simplest case of a PFR model, several key assumptions must be made in order to simplify the problem, some of which are outlined below. Note that not all of these assumptions are necessary, however the removal of these assumptions does increase the complexity of the problem. The PFR model can be used to model multiple reactions as well as reactions involving changing temperatures, pressures and densities of

1372-478: The Quantum Dynamics and used in his textbook The Principles of Quantum Mechanics . He called it the "delta function" since he used it as a continuous analogue of the discrete Kronecker delta . The Dirac delta function δ ( x ) {\displaystyle \delta (x)} can be loosely thought of as a function on the real line which is zero everywhere except at the origin, where it

1421-407: The amalgamation into L. Schwartz's theory of distributions (1945) ...", and leading to the formal development of the Dirac delta function. An infinitesimal formula for an infinitely tall, unit impulse delta function (infinitesimal version of Cauchy distribution ) explicitly appears in an 1827 text of Augustin-Louis Cauchy . Siméon Denis Poisson considered the issue in connection with

1470-601: The concentration of species i , C i (mol/m ) can be introduced as: where v ˙ {\displaystyle {\dot {v}}} is the volumetric flow rate. On application of the above to Equation 1, the mass balance on i becomes: 2. A t u [ C i ( x ) − C i ( x + d x ) ] + A t d x ν i r = 0 {\displaystyle A_{t}u[C_{i}(x)-C_{i}(x+dx)]+A_{t}dx\nu _{i}r=0\,} . When like terms are cancelled and

1519-692: The delta function is conceptualized as modeling an idealized point mass at 0, then δ ( A ) represents the mass contained in the set A . One may then define the integral against δ as the integral of a function against this mass distribution. Formally, the Lebesgue integral provides the necessary analytic device. The Lebesgue integral with respect to the measure δ satisfies ∫ − ∞ ∞ f ( x ) δ ( d x ) = f ( 0 ) {\displaystyle \int _{-\infty }^{\infty }f(x)\,\delta (dx)=f(0)} for all continuous compactly supported functions f . The measure δ

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1568-646: The delta function is often manipulated as a kind of limit (a weak limit ) of a sequence of functions, each member of which has a tall spike at the origin: for example, a sequence of Gaussian distributions centered at the origin with variance tending to zero. The Dirac delta is not truly a function, at least not a usual one with domain and range in real numbers . For example, the objects f ( x ) = δ ( x ) and g ( x ) = 0 are equal everywhere except at x = 0 yet have integrals that are different. According to Lebesgue integration theory , if f and g are functions such that f = g almost everywhere , then f

1617-500: The dispersion model giving different relationships for the dispersion number. From the safety technical point of view the PFR has the advantages that The main problems lies in difficult and sometimes critical start-up and shut down operations. Plug flow reactors are used for some of the following applications: Dirac delta function In mathematical analysis , the Dirac delta function (or δ distribution ), also known as

1666-524: The dispersion model is usually employed. The residence-time distribution (RTD) of a reactor is a characteristic of the mixing that occurs in the chemical reactor. There is no axial mixing in a plug-flow reactor, and this omission is reflected in the RTD which is exhibited by this class of reactors. Real plug flow reactors do not satisfy the idealized flow patterns, back mix flow or plug flow deviation from ideal behavior can be due to channeling of fluid through

1715-399: The exit stream or mean residence time is: When a tracer is injected into a reactor at a location more than two or three particle diameters downstream from the entrance and measured some distance upstream from the exit, the system can be described by the dispersion model with combinations of open or close boundary conditions. For such a system where there is no discontinuity in type of flow at

1764-475: The exponential form and the various limitations upon the function f necessary for its application extended over several centuries. The problems with a classical interpretation are explained as follows: Further developments included generalization of the Fourier integral, "beginning with Plancherel's pathbreaking L -theory (1910), continuing with Wiener's and Bochner's works (around 1930) and culminating with

1813-687: The flow. Although these complications are ignored in what follows, they are often relevant to industrial processes. Assumptions: A material balance on the differential volume of a fluid element, or plug, on species i of axial length dx between x and x + dx gives: Accumulation is 0 under steady state; therefore, the above mass balance can be re-written as follows: 1. F i ( x ) − F i ( x + d x ) + A t d x ν i r = 0 {\displaystyle F_{i}(x)-F_{i}(x+dx)+A_{t}dx\nu _{i}r=0} . where: The flow linear velocity, u (m/s) and

1862-418: The idea of instantaneous transfer of momentum. The delta function allows us to construct an idealized limit of these approximations. Unfortunately, the actual limit of the functions (in the sense of pointwise convergence ) lim Δ t → 0 + F Δ t {\textstyle \lim _{\Delta t\to 0^{+}}F_{\Delta t}} is zero everywhere but

1911-456: The limit dx → 0 is applied to Equation 2 the mass balance on species i becomes 3. u d C i d x = ν i r {\displaystyle u{\frac {dC_{i}}{dx}}=\nu _{i}r} , The temperature dependence of the reaction rate, r , can be estimated using the Arrhenius equation . Generally, as the temperature increases so does

1960-408: The limit. So, in the equation F ( t ) = P δ ( t ) = lim Δ t → 0 F Δ t ( t ) {\textstyle F(t)=P\,\delta (t)=\lim _{\Delta t\to 0}F_{\Delta t}(t)} , it is understood that the limit is always taken outside the integral . In applied mathematics, as we have done here,

2009-518: The model situation of an instantaneous transfer of momentum requires taking the limit as Δ t → 0 , giving a result everywhere except at 0 : p ( t ) = { P t > 0 0 t < 0. {\displaystyle p(t)={\begin{cases}P&t>0\\0&t<0.\end{cases}}} Here the functions F Δ t {\displaystyle F_{\Delta t}} are thought of as useful approximations to

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2058-574: The momentum at any time t is found by integration: p ( t ) = ∫ 0 t F Δ t ( τ ) d τ = { P t ≥ T P t / Δ t 0 ≤ t ≤ T 0 otherwise. {\displaystyle p(t)=\int _{0}^{t}F_{\Delta t}(\tau )\,d\tau ={\begin{cases}P&t\geq T\\P\,t/\Delta t&0\leq t\leq T\\0&{\text{otherwise.}}\end{cases}}} Now,

2107-414: The point of tracer injection or at the point of tracer measurement, the variance for open-open system is: Where, which represents the ratio of rate of transport by convection to rate of transport by diffusion or dispersion. Vessel dispersion number is defined as: The variance of a continuous distribution measured at a finite number of equidistant locations is given by: Where mean residence time τ

2156-401: The positive y -axis. The Dirac delta is used to model a tall narrow spike function (an impulse ), and other similar abstractions such as a point charge , point mass or electron point. For example, to calculate the dynamics of a billiard ball being struck, one can approximate the force of the impact by a Dirac delta. In doing so, one not only simplifies the equations, but one also

2205-584: The rate at which the reaction occurs. Residence time, τ {\displaystyle \tau } , is the average amount of time a discrete quantity of reagent spends inside the tank. Assume: After integration of Equation 3 using the above assumptions, solving for C A (x) we get an explicit equation for the concentration of species A as a function of position: 4. C A ( x ) = C A 0 e − k τ {\displaystyle C_{A}(x)=C_{A0}e^{-k\tau }\,} , where C A0

2254-471: The study of wave propagation as did Gustav Kirchhoff somewhat later. Kirchhoff and Hermann von Helmholtz also introduced the unit impulse as a limit of Gaussians , which also corresponded to Lord Kelvin 's notion of a point heat source. At the end of the 19th century, Oliver Heaviside used formal Fourier series to manipulate the unit impulse. The Dirac delta function as such was introduced by Paul Dirac in his 1927 paper The Physical Interpretation of

2303-617: The theorem using exponentials: f ( x ) = 1 2 π ∫ − ∞ ∞   e i p x ( ∫ − ∞ ∞ e − i p α f ( α ) d α ) d p . {\displaystyle f(x)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }\ e^{ipx}\left(\int _{-\infty }^{\infty }e^{-ip\alpha }f(\alpha )\,d\alpha \right)\,dp.} Cauchy pointed out that in some circumstances

2352-468: The title PFR . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=PFR&oldid=1255643631 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Plug flow reactor model Fluid going through

2401-400: The vessel, recycling of fluid within the vessel or due to the presence of stagnant region or dead zone of fluid in the vessel. Real plug flow reactors with non-ideal behavior have also been modelled. To predict the exact behavior of a vessel as a chemical reactor , RTD or stimulus response technique is used. The tracer technique , the most widely used method for the study of axial dispersion,

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