DOCSLIB.ORG

Tubular Flow with Laminar Flow (CHE 512) M.P. Dudukovic Chemical Reaction Engineering Laboratory (CREL), Washington University, St

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Tubular Flow with Laminar Flow (CHE 512) M.P. Dudukovic Chemical Reaction Engineering Laboratory (CREL), Washington University, St Tubular Flow with Laminar Flow (CHE 512) M.P. Dudukovic Chemical Reaction Engineering Laboratory (CREL), Washington University, St. Louis, MO 4. TUBULAR REACTORS WITH LAMINAR FLOW Tubular reactors in which homogeneous reactions are conducted can be empty or packed conduits of various cross-sectional shape. Pipes, i.e tubular vessels of cylindrical shape, dominate. The flow can be turbulent or laminar. The questions arise as to how to interpret the performance of tubular reactors and how to measure their departure from plug flow behavior. We will start by considering a cylindrical pipe with fully developed laminar flow. For a Newtonian fluid the velocity profile is given by r 2 u = 2u 1 (1) R u where u = max is the mean velocity. By making a balance on species A, which may be a 2 reactant or a tracer, we arrive at the following equation: 2C C D C C D A u A + r A r = A (2) z2 z r r r A t For an exercise this equation should be derived by making a balance on an annular cylindrical region of length z , inner radius r and outer radius r. We should render this equation dimensionless by defining: z r t C = ; = ; = ;c = A L R t C Ao where L is the pipe length of interest, R is the pipe radius, t = L/u is the mean residence time, C is some reference concentration. Let us assume an n-th order rate form. Ao The above equation (2) now reads: 2 D c c D L 1 c n1 c 212 + kC t c n = (2a) 2 () A0 u L u R R We define: u L L2 / D characteristic diffusion time() axial Pe = = = (3) a D L / u process time() characteristic convection time = axial Peclet number. 1 u R Pe = = radial Peclet number (4) r D u R2 R2 / D characteristic radial diffusion time = = = DL L / u characteristic convection time R Per x = ()radial Peclet number x() pipe aspect ratio L n 1 L/u process time Dan = kC A t = = (5) o 1 characteristic reaction time kCn1 Ao In terms of the above dimensionless groups which involve: characteristic reaction time, characteristic diffusion time, characteristic convection or process time and the aspect ratio, we can rewrite the above equations as: u L u dt L L (3a) Pea = = = Pe D D dt dt u R u dt R Pe Per = = = (4a) D D dt 2 1 L 1 Pe Per R 647 a4 8 64 7 4 8 2 1 d c 2 c 1 4 L 1 c n c t 21 + Da c = 2 () n Pe L Pe dt 1 L 1 Pe Per R 647 a4 8 64 7 48 2 1 d c 2 c 1 4 L 1 c n c t 21 + Da c = (2b) 2 () n Pe L Pe dt u dt where Pe = = Re Sc is the Peclet number for the flow. In terms of Pea and Per we have: D 2 1 c 2 c 1 L 1 c n c 21 Da c (2c) 2 () + n = Pea Pe r R The above equation can be simplified if we deal with a steady state reactor problem for which c = 0, or if we deal with a nonreactive tracer dynamic response, for which Da = 0. In either n 2 event we need to solve a cumbersome partial differential equation and need two boundary conditions in axial coordinate and two in radial coordinate . 4.1 Segregated Flow Model The question arises whether we really need to solve the above equation (2c) numerically at all times or whether we can find simple solutions which are valid under certain conditions. Since reactant or tracer dimensionless concentration, c, is a smooth function, based on physical arguments, the value of the function and its derivatives is of similar order of magnitude except perhaps at a finite number of points. It can be shown then that when Pea >> 1 and R Pe >> 1 the first and third term of eq (2c) can be neglected. For a steady state reactor r L problem this results in the following equation: c 21 2 Da cn = 0 (6) () n = 0 ;c=1 (7) The exit concentration c () =1, is a function of radial position, i.e of the stream line on which the reactant travels. The overall average exit concentration, or mixing cup concentration, is obtained as R R 2 r 2 ruC z = L,r dr 4 ru 1 C ()L,r dr C A () A A exit o o R c = = R = ex C R2u C Ao 2 ru C dr Ao Ao o 2 r where u = 2 u 1 . Using dimensionless variables we get: R 1 2 cex = 4 ()1 c()1, d (8) o For a first order reaction (n = 1) we readily find from eqs. (6) and (7) that Da 21() 2 c () = 1, = e (9) Using eqn (9) in eqn (8) we obtain the exit mixing cup concentration: 3 Da 1 2 21()2 cex = 4 ()1 e d (10) o 1 2 d Change variables to 2 = u; 2 = du to get 1 ()1 2 Da 2 3 Da u ()1 u e 2 Da c = 4 e 2 du = 2 du = 2 E (10a) ex 3 3 1 2 1 u 2 e xu where E ()x = du is the n-th exponential integral. n un 1 In contrast, the cross-sectional average concentration is: Da 1 1 2 ˜ 21() cex = 2 c ()1, d = 2 e d o o Da (11) u e 2 Da = du = E u2 2 2 1 Da Da Da Da c˜ = E = e 2 E (11a) ex 2 2 2 1 2 Thus, if we measure by an instrument the cross-sectional average concentration, and try to infer reactant conversion from it, our results may be in error since conversion is only obtainable from mixing cup (flow averaged) concentration and clearly there is a discrepancy between equation (11a) and eq (10a).. You should examine the deviation of eq (11a) compared to eq (10a) and plot it as a function of Da . The needed exponential integral are tabulated by Abramowitz and Stegun (Handbook of Mathematical Functions, Dover Publ. 1964). Using the following relationship among exponential integrals 1 E ()z = ez zE()z (12) n+1 n []n we get the following expression for conversion from eqn (10a) Da Da Da2 Da 1 x = c = 1 e 2 + E (10b) A ex 2 4 1 2 4 e xu e t where E ()x = du = dt 1 u t 1 x We should realize immediately, upon reflection, that by eliminating the diffusion terms in eq (2c) and in arriving at eq (6-7) we deal with the segregated flow system. Indeed, in absence of diffusional effects there is no mixing among various stream lines. The reactant that enters on a particular stream line exits on the same stream line, i.e at the same radial position , and hence is surrounded by elements of the same age as its own at all times during its journey through the reactor. Every stream line has a different residence time. The shortest residence time is experienced by the elements on the center line ()t / 2 , the mean residence time ()t is experienced by the fluid traveling on the stream line that has the mean velocity, i.e, at = 1/ 2 = 0.707, while infinite residence time is experienced by the elements on the stream line at the wall ( = 1). However, since the stream line at the wall receives infinitesimal amount of new fluid the mean residence time for the system exists and is t . We recall that for the segregated flow condition and first order reaction the exit concentration can be written as: Da (13) cex = e E () d = E ()s s = Da o This means that we have found the Laplace transform of the exit age density function for fully developed laminar flow of Newtonian fluid in a pipe s s s s2 s LE() = E ()s = 2 E = 1 e 2 + E (14) {} 3 2 2 4 1 2 However, even the extensive transform pairs of Bateman (Tables of Integral Transform Vol. 1) do not show this transform. We can, however, derive the RTD or the F function for laminar flow readily based on physical arguments. Let us imagine that we have switched from white to red fluid at the inlet at t = 0. Red fluid will appear at the outlet at z = L only starting at time t / 2 . The fraction of the outflow that is younger than t is given by the fraction of the fluid which is red. This is obtained by integrating the flow from the center stream line, where the red fluid is present at the outlet from time t /2, to the stream line at position r at which red fluid just at the outlet at time t and by dividing this flow rate by the total flow rate. 5 Recall that L L t = = u r 2 2u 1 R By definition the F curve is given by: r r 2 r' 2 r' udr' 4 r' u 1 dr' R Ft()= o = o (15) Q R2u t Ft()= 4 ' 1 '2 d' , for t (15a) () 2 o t t 1/ 2 where t = or = 1 . Hence, 21() 2 2t 2 4 2 t Ft()= 4 = 22 1 for t (16) 2 4 2 2 t 1 2 t t t t Ft()= 21 1 2t = 1 1 + = 1 2t 2 2t 2t 4t2 t 0 t < Ft t 2 2 (17) ()= 1 t 4t2 t 2 or t 2 t Ft()= 1 Ht (17a) 4t2 2 and 1 1 F() = 1 H (17b) 4 2 2 The exit age density function is: 6 dF t 2 t E(t) = = Ht (18) dt 2t3 2 The dimensionless exit age density function is: 1 1 E () = t E()t = H (18a) 2 3 2 For any reaction order we can then write for segregated flow laminar flow 1 1 c () c = c () H d = batch d (19) ex batch 23 2 2 3 o 1 2 R Previous studies have shown that this model is valid when Pe > 1000 and Pe > 85.
Load more

Recommended publications
  • Glossary Physics (I-Introduction)
    1 Glossary Physics (I-introduction) - Efficiency: The percent of the work put into a machine that is converted into useful work output; = work done / energy used [-]. = eta In machines: The work output of any machine cannot exceed the work input (<=100%); in an ideal machine, where no energy is transformed into heat: work(input) = work(output), =100%. Energy: The property of a system that enables it to do work. Conservation o. E.: Energy cannot be created or destroyed; it may be transformed from one form into another, but the total amount of energy never changes. Equilibrium: The state of an object when not acted upon by a net force or net torque; an object in equilibrium may be at rest or moving at uniform velocity - not accelerating. Mechanical E.: The state of an object or system of objects for which any impressed forces cancels to zero and no acceleration occurs. Dynamic E.: Object is moving without experiencing acceleration. Static E.: Object is at rest.F Force: The influence that can cause an object to be accelerated or retarded; is always in the direction of the net force, hence a vector quantity; the four elementary forces are: Electromagnetic F.: Is an attraction or repulsion G, gravit. const.6.672E-11[Nm2/kg2] between electric charges: d, distance [m] 2 2 2 2 F = 1/(40) (q1q2/d ) [(CC/m )(Nm /C )] = [N] m,M, mass [kg] Gravitational F.: Is a mutual attraction between all masses: q, charge [As] [C] 2 2 2 2 F = GmM/d [Nm /kg kg 1/m ] = [N] 0, dielectric constant Strong F.: (nuclear force) Acts within the nuclei of atoms: 8.854E-12 [C2/Nm2] [F/m] 2 2 2 2 2 F = 1/(40) (e /d ) [(CC/m )(Nm /C )] = [N] , 3.14 [-] Weak F.: Manifests itself in special reactions among elementary e, 1.60210 E-19 [As] [C] particles, such as the reaction that occur in radioactive decay.
    [Show full text]
  • 1 Fluid Flow Outline Fundamentals of Rheology
    Fluid Flow Outline • Fundamentals and applications of rheology • Shear stress and shear rate • Viscosity and types of viscometers • Rheological classification of fluids • Apparent viscosity • Effect of temperature on viscosity • Reynolds number and types of flow • Flow in a pipe • Volumetric and mass flow rate • Friction factor (in straight pipe), friction coefficient (for fittings, expansion, contraction), pressure drop, energy loss • Pumping requirements (overcoming friction, potential energy, kinetic energy, pressure energy differences) 2 Fundamentals of Rheology • Rheology is the science of deformation and flow – The forces involved could be tensile, compressive, shear or bulk (uniform external pressure) • Food rheology is the material science of food – This can involve fluid or semi-solid foods • A rheometer is used to determine rheological properties (how a material flows under different conditions) – Viscometers are a sub-set of rheometers 3 1 Applications of Rheology • Process engineering calculations – Pumping requirements, extrusion, mixing, heat transfer, homogenization, spray coating • Determination of ingredient functionality – Consistency, stickiness etc. • Quality control of ingredients or final product – By measurement of viscosity, compressive strength etc. • Determination of shelf life – By determining changes in texture • Correlations to sensory tests – Mouthfeel 4 Stress and Strain • Stress: Force per unit area (Units: N/m2 or Pa) • Strain: (Change in dimension)/(Original dimension) (Units: None) • Strain rate: Rate
    [Show full text]
  • Introduction to the CFD Module
    INTRODUCTION TO CFD Module Introduction to the CFD Module © 1998–2018 COMSOL Protected by patents listed on www.comsol.com/patents, and U.S. Patents 7,519,518; 7,596,474; 7,623,991; 8,457,932; 8,954,302; 9,098,106; 9,146,652; 9,323,503; 9,372,673; and 9,454,625. Patents pending. This Documentation and the Programs described herein are furnished under the COMSOL Software License Agreement (www.comsol.com/comsol-license-agreement) and may be used or copied only under the terms of the license agreement. COMSOL, the COMSOL logo, COMSOL Multiphysics, COMSOL Desktop, COMSOL Server, and LiveLink are either registered trademarks or trademarks of COMSOL AB. All other trademarks are the property of their respective owners, and COMSOL AB and its subsidiaries and products are not affiliated with, endorsed by, sponsored by, or supported by those trademark owners. For a list of such trademark owners, see www.comsol.com/trademarks. Version: COMSOL 5.4 Contact Information Visit the Contact COMSOL page at www.comsol.com/contact to submit general inquiries, contact Technical Support, or search for an address and phone number. You can also visit the Worldwide Sales Offices page at www.comsol.com/contact/offices for address and contact information. If you need to contact Support, an online request form is located at the COMSOL Access page at www.comsol.com/support/case. Other useful links include: • Support Center: www.comsol.com/support • Product Download: www.comsol.com/product-download • Product Updates: www.comsol.com/support/updates •COMSOL Blog: www.comsol.com/blogs • Discussion Forum: www.comsol.com/community •Events: www.comsol.com/events • COMSOL Video Gallery: www.comsol.com/video • Support Knowledge Base: www.comsol.com/support/knowledgebase Part number: CM021302 Contents Introduction .
    [Show full text]
  • Hydraulics Manual Glossary G - 3
    Glossary G - 1 GLOSSARY OF HIGHWAY-RELATED DRAINAGE TERMS (Reprinted from the 1999 edition of the American Association of State Highway and Transportation Officials Model Drainage Manual) G.1 Introduction This Glossary is divided into three parts: · Introduction, · Glossary, and · References. It is not intended that all the terms in this Glossary be rigorously accurate or complete. Realistically, this is impossible. Depending on the circumstance, a particular term may have several meanings; this can never change. The primary purpose of this Glossary is to define the terms found in the Highway Drainage Guidelines and Model Drainage Manual in a manner that makes them easier to interpret and understand. A lesser purpose is to provide a compendium of terms that will be useful for both the novice as well as the more experienced hydraulics engineer. This Glossary may also help those who are unfamiliar with highway drainage design to become more understanding and appreciative of this complex science as well as facilitate communication between the highway hydraulics engineer and others. Where readily available, the source of a definition has been referenced. For clarity or format purposes, cited definitions may have some additional verbiage contained in double brackets [ ]. Conversely, three “dots” (...) are used to indicate where some parts of a cited definition were eliminated. Also, as might be expected, different sources were found to use different hyphenation and terminology practices for the same words. Insignificant changes in this regard were made to some cited references and elsewhere to gain uniformity for the terms contained in this Glossary: as an example, “groundwater” vice “ground-water” or “ground water,” and “cross section area” vice “cross-sectional area.” Cited definitions were taken primarily from two sources: W.B.
    [Show full text]
  • Sediment Transport & Fluid Flow
    Sediment Transport & Fluid Flow • Downslope transport • Fluid Flow – Viscosity – Types of fluid flow – Laminar vs. Turbulent – Eddy Viscosity – Reynolds number – Boundary Layer • Grain Settling • Particle Transport Loss of intergrain cohesion caused by saturation of pores… before… …after 1 Different kinds of mass movements, variable velocity (and other factors) From weathering to deposition: sorting and modification of clastic particles 2 Effects of transport on rounding, sorting How are particles transported by fluids? What are the key parameters? 3 Flow regimes in a stream Fluid Flow • Fundamental Physical Properties of Fluids – Density (ρ) – Viscosity (µ) Control the ability of a fluid to erode & transport particles • Viscosity - resistance to flow, or deform under shear stress. – Air - low – Water - low – Ice- high ∆µ with ∆ T, or by mixing other materials 4 Shear Deformation Shear stress (τ) is the shearing force per unit area • generated at the boundary between two moving fluids • function of the extent to which the slower moving fluid retards motion of the faster moving fluid (i.e., viscosity) Fluid Viscosity and Flow • Dynamic viscosity(µ) - the resistance of a substance (water) to ∆ shape during flow (shear stress) ! µ = du/dy where, dy τ = shear stress (dynes/cm2) du/dy = velocity gradient (rate of deformation) • Kinematic viscosity(v) =µ/ρ 5 Behaviour of Fluids Viscous Increased fluidity Types of Fluids Water - flow properties function of sediment concentrations • Newtonian Fluids - no strength, no ∆ in viscosity as shear rate ∆’s (e.g.,
    [Show full text]
  • Introduction
    VERIFYING CONSISTENCY OF EFFECTIVE-VISCOSITY AND PRESSURE- LOSS DATA FOR DESIGNING FOAM PROPORTIONING SYSTEMS Bogdan Z. Dlugogorski1, Ted H. Schaefer1,2 and Eric M. Kennedy1 1Process Safety and Environment Protection Research Group School of Engineering The University of Newcastle Callaghan, NSW 2308, AUSTRALIA Phone: +61 2 4921 6176, Fax: +61 2 4921 6920 Email: [email protected] 23M Specialty Materials Laboratory Dunheved Circuit, St Marys, NSW 2760, AUSTRALIA ABSTRACT Aspirating and compressed-air foam systems often incorporate proportioners designed to draw sufficient rate of foam concentrate into the flowing stream of water. The selection of proportioner’s piping follows from the engineering correlations linking the pressure loss with the flow rate of a concentrate, for specified temperature (usually 20 oC) and pipe size. Unfortunately, reports exist that such correlations may be inaccurate, resulting in the design of underperforming suppression systems. To illustrate the problem, we introduce two correlations, sourced from industry and developed for the same foam concentrate, that describe effective viscosity (μeff) and pressure loss as functions of flow rate and pipe diameter. We then present a methodology to verify the internal consistency of each data set. This methodology includes replotting the curves of the effective viscosity vs the nominal shear rate at the wall (32Q/(πD3)) and also replotting the pressure loss in terms of the stress rate at the wall (τw) vs the nominal shear rate at the wall. In the laminar regime, for time-independent fluids with no slip at the wall (which is the behaviour displayed by foam concentrates), consistent data sets are 3 3 characterised by a single curve of μeff vs 32Q/(πD ), and a single curve of τw vs 32Q/(πD ) in the laminar regime.
    [Show full text]
  • 7. Laminar and Turbulent Flow
    CREST Foundation Studies Fundamentals of Fluid Mechanics 7. Laminar and turbulent flow [This material relates predominantly to module ELP035] 7.1 Real fluids 7.2 Examples of Laminar and turbulent flow 7.1 Real Fluids The flow of real fluids exhibits viscous effect, that is they tend to “stick” to solid surfaces and have stresses within their body. You might remember from earlier in the course Newtons law of viscosity: du τ ∝ dy This tells us that the shear stress, τ, in a fluid is proportional to the velocity gradient - the rate of change of velocity across the fluid path. For a “Newtonian” fluid we can write: du τµ= dy where the constant of proportionality, µ, is known as the coefficient of viscosity (or simply viscosity). We saw that for some fluids - sometimes known as exotic fluids - the value of µ changes with stress or velocity gradient. We shall only deal with Newtonian fluids. In this Unit we shall look at how the forces due to momentum changes on the fluid and viscous forces compare and what changes take place. 7.2 Examples of Laminar and turbulent flow If we were to take a pipe of free flowing water and inject a dye into the middle of the stream, what would we expect to happen? Would it be this.... 1 CREST Foundation Studies Fundamentals of Fluid Mechanics this.... or this? Actually all would happen - but for different flow rates. The top occurs when the fluid is flowing fast and the lower when it is flowing slowly. The top situation is known as laminar flow and the lower as turbulent flow.
    [Show full text]
  • Viscous Flow in Ducts Reynolds Number Regimes
    Viscous Flow in Ducts We want to study the viscous flow in ducts with various velocities, fluids and duct shapes. The basic problem is this: Given the pipe geometry and its added components (e.g. fittings, valves, bends, and diffusers) plus the desired flow rate and fluid properties, what pressure drop is needed to drive the flow? Reynolds number regimes The most important parameter in fluid mechanics is the Reynolds number: where is the fluid density, U is the flow velocity, L is the characteristics length scale related to the flow geometry, and is the fluid viscosity. The Reynolds number can be interpreted as the ratio of momentum (or inertia) to viscous forces. A profound change in fluid behavior occurs at moderate Reynolds number. The flow ceases being smooth and steady (laminar) and becomes fluctuating (turbulent). The changeover is called transition. Transition depends on many effects such as wall roughness or the fluctuations in the inlet stream, but the primary parameter is the Reynolds number. Turbulence can be measured by sensitive instruments such as a hot‐wire anemometer or piezoelectric pressure transducer. Fig.1: The three regimes of viscous flow: a) laminar (low Re), b) transition at intermediate Re, and c) turbulent flow in high Re. The fluctuations, typically ranging from 1 to 20% of the average velocity, are not strictly periodic but are random and encompass a continuous range of frequencies. In a typical wind tunnel flow at high Re, the turbulent frequency ranges from 1 to 10,000 Hz. The higher Re turbulent flow is unsteady and irregular but, when averaged over time, is steady and predictable.
    [Show full text]
  • Understand the Real World of Mixing
    Back to Basics Understand the Real World of Mixing Thomas Post Post Mixing Optimization and Most chemical engineering curricula do not Solutions adequately address mixing as it is commonly practiced in the chemical process industries. This article attempts to fi ll in some of the gaps by explaining fl ow patterns, mixing techniques, and the turbulent, transitional and laminar mixing regimes. or most engineers, college memories about mix- This is the extent of most engineers’ collegiate-level ing are limited to ideal reactors. Ideal reactors are preparation for real-world mixing applications. Since almost Fextreme cases, and are represented by the perfect- everything manufactured must be mixed, most new gradu- mixing model, the steady-state perfect-mixing model, or the ates are ill-prepared to optimize mixing processes. This steady-state plug-fl ow model (Figure 1). These models help article attempts to bridge the gap between the theory of the determine kinetics and reaction rates. ideal and the realities of actual practice. For an ideal batch reactor, the perfect-mixing model assumes that the composition is uniform throughout the Limitations of the perfect-mixing reactor at any instant, and gives the reaction rate as: and plug-fl ow models Classical chemical engineering focuses on commodity (–rA)V = NA0 × (dXA/dt) (1) chemicals produced in large quantities in continuous opera- tions, such as in the petrochemical, polymer, mining, fertil- The steady-state perfect-mixing model for a continuous process is similar, with the composition uniform throughout the reactor at any instant in time. However, it incorporates Feed the inlet molar fl owrate, FA0: Uniformly Mixed (–rA)V = FA0 × XA (2) The steady-state plug-fl ow model of a continuous process assumes that the composition is the same within a differential volume of the reactor in the direction of fl ow, typically within a pipe: Product (–r )V = F × dX (3) Uniformly A A0 A Mixed These equations can be integrated to determine the time required to achieve a conversion of XA.
    [Show full text]
  • The Laminar Flow Instability Criterion and Turbulence in Pipe S.L. Arsenjev, I.B. Lozovitski1, Y.P. Sirik Physical-Technical
    The laminar flow instability criterion and turbulence in pipe S.L. Arsenjev, I.B. Lozovitski1, Y.P. Sirik Physical-Technical Group Dobroljubova street 2, 29, Pavlograd, Dnepropetrovsk region, 51400 Ukraine The identification of stream in the straight pipe as a flexible rod has allowed to present the criterion expression for determination of transition of the laminar flow regime to the turbulent as a loss of stability of the rectilinear static structure of translational motion of stream in pipe and its transition to the flexural-vortical dy- namic structure of translational motion, just as a flexible rod buckling. The intro- duced criterion allows to take into account an influencing of the inlet geometry, the pipe length, the flow velocity, and also of any physical factors on stability of the rectilinear flow structure. It is ascertained, that Reynolds' number is the num- ber of local hydrodynamic similarity and it is displayed that one is constituent part of the introduced stability criterion. The developed approach to a problem of sta- bility is applicable for a problem solving on internal flow and external streamline. Pacs: 01.65.+g; 46.32.+x; 47.10.+g; 47.15Fe; 47.27Cn; 47.27Wg; 47.60.+i; 51.20.+d; 83.50.Ha Introduction The fluid has such essential property as an intrastructural mobility in contrast to the solid body. In other words the number of degrees of freedom of the fluid motion is much more, than for the solid body. In this connection the description of the fluid motion is much more complicated than description of the solid body motion.
    [Show full text]
  • Fluid Mechanics
    FLUID MECHANICS PROF. DR. METİN GÜNER COMPILER ANKARA UNIVERSITY FACULTY OF AGRICULTURE DEPARTMENT OF AGRICULTURAL MACHINERY AND TECHNOLOGIES ENGINEERING 1 5. FLOW IN PIPES 5.1.3. Pressure and Shear Stress Fully developed steady flow in a constant diameter pipe may be driven by gravity and/or pressure forces. For horizontal pipe flow, gravity has no effect except for a hydrostatic pressure variation across the pipe, 훾퐷, that is usually negligible. It is the pressure difference, ∆푃 = 푃1 − 푃2 between one section of the horizontal pipe and another which forces the fluid through the pipe. Viscous effects provide the restraining force that exactly balances the pressure force, thereby allowing the fluid to flow through the pipe with no acceleration. If viscous effects were absent in such flows, the pressure would be constant throughout the pipe, except for the hydrostatic variation. In non-fully developed flow regions, such as the entrance region of a pipe, the fluid accelerates or decelerates as it flows (the velocity profile changes from a uniform profile at the entrance of the pipe to its fully developed profile at the end of the entrance region). Thus, in the entrance region there is a balance between pressure, viscous, and inertia (acceleration) forces. The result is a pressure distribution along the horizontal pipe as shown in Fig.5.7. The magnitude of the 휕푃 pressure gradient, , is larger in the entrance region than in the fully developed 휕푥 휕푃 ∆푃 region, where it is a constant, = − < 0. 휕푥 퐿 The fact that there is a nonzero pressure gradient along the horizontal pipe is a result of viscous effects.
    [Show full text]
  • Viscous Flow in Ducts with Various Velocities, Fluids and Duct Shapes
    Flow in viscous ducts • We want to study the viscous flow in ducts with various velocities, fluids and duct shapes. • The basic problem is this: Given the pipe geometry and its added components (e.g. fittings, valves, bends, and diffusers) plus the desired flow rate and fluid properties, what pressure drop is needed to drive the flow? Integral Relations for CV M. Bahrami ENSC 283 Spring 2009 1 Reynolds number The most important parameter in fluid mechanics is the Reynolds number: The Reynolds number can be interpreted as the ratio of momentum (or inertia) to viscous forces. Laminar and turbulent regimes A profound change in fluid behavior occurs at moderate Reynolds number. The flow ceases being smooth and steady (laminar) and becomes fluctuating (turbulent). The changeover is called transition. Reynolds number regimes • 0 < Re < 1: highly viscous laminar “creeping” motion • 1 < Re < 100: laminar strong Re dependence • 100 < Re < 103: laminar, boundary layer theory useful • 103 < Re < 104: transition to turbulence • 104 < Re < 106: turbulent, moderate Re dependence • 106 < Re <∞: Turbulent, slight Re dependence Reynolds experiment • In 1883 Osborne Reynolds, a British engineering professor, observed the transition to turbulence in a pipe flow by introducing a dye streak in the flow. • Reynolds experimentally showed that the transition occurs in a pipe flow at: Internal viscous flow • Both laminar and turbulent flows can be internal and external • An internal flow is contained (or bounded) by walls and the viscous flow will grow and meet and permeate the entire flow. As a result, there is an entrance region where nearly inviscid upstream flow converges and enters the duct.
    [Show full text]