Flow of Fluids in Pipes

1. Flow of Fluids in Pipes Dr. Lajos Gulyás, Ph.D. college professor ”The life is too short to drink bad wine.”

2. Friction LossesFlow through Conduits Incompressible Flow

3. Goals • Calculate frictional losses for laminar and turbulent flow through circular and non-circular pipes • Define the friction factor in terms of flow properties • Calculate the friction factor for laminar and turbulent flow • Define and calculate the Reynolds number for different flow situations • Derive the Hagen-Poiseuille equation

4. Introduction • Average velocity in a pipe • Recall - because of the no-slip condition, the velocity at the walls of a pipe or duct flow is zero • We are often interested only in Vavg, which we usually call just V (drop the subscript for convenience) • Keep in mind that the no-slip condition causes shear stress and friction along the pipe walls Friction force of wall on fluid

5. Vavg Vavg same same same Introduction • For pipes of constant diameter and incompressible flow • Vavg stays the same down the pipe, even if the velocity profile changes • Why? Conservation of Mass

6. For pipes with variable diameter, m is still the same due to conservation of mass, but V1 ≠ V2 D1 D2 m V1 V2 m 2 1 Introduction

7. LAMINAR AND TURBULENT FLOWS • Laminar flow: characterized by smooth streamlines and highly ordered motion. • Turbulentflow: characterized by velocity fluctuations and highly disordered motion. • The transition from laminar to turbulent flow does not occur suddenly; rather, it occurs over some region in which the flow fluctuates between laminar and turbulent flows before it becomes fully turbulent.

8. Reynolds Number • The transition from laminar to turbulent flow depends on the geometry, surface roughness, flow velocity, surface temperature, and type of fluid, among other things. • British engineer Osborne Reynolds (1842–1912) discovered that the flow regime depends mainly on the ratio of inertial forces to viscous forces in the fluid. • The ratio is called the Reynolds number and is expressed for internal flow in a circular pipe as

9. Reynolds Number • At large Reynolds numbers, the inertial forces are large relative to the viscous forces  Turbulent Flow • At small or moderate Reynolds numbers, the viscous forces are large enough to suppress these fluctuations  Laminar Flow • The Reynolds number at which the flow becomes turbulent is called the critical Reynolds number, Recr. • The value of the critical Reynolds number is different for different geometries and flow conditions. For example, Recr = 2300 for internal flow in a circular pipe.

10. Reynolds Number • For flow through noncircular pipes, the Reynolds number is based on the hydraulic diameter Dhdefined as Ac= cross-section area P = wetted perimeter • The transition from laminar to turbulent flow also depends on the degree of disturbance of the flow by surface roughness, pipe vibrations, and fluctuations in the flow.

11. Reynolds Number • Under most practical conditions, the flow in a circular pipe is • In transitional flow, the flow switches between laminar and turbulent randomly.

12. LAMINAR FLOW IN PIPES • In this section we consider the steady laminar flow of an incompressible fluid with constant properties in the fully developed region of a straight circular pipe. • In fully developed laminar flow, each fluid particle moves at a constant axial velocity along a streamline and no motion in the radial direction such that no acceleration (since flow is steady and fully-developed).

13. LAMINAR FLOW IN PIPES • Now consider a ring-shaped differential volume element of radius r, thickness dr, and length dx oriented coaxially with the pipe. A force balance on the volume element in the flow direction gives • Dividing by 2pdrdx and rearranging,

14. LAMINAR FLOW IN PIPES • Taking the limit as dr, dx → 0 gives • Substituting t = -m(du/dr) gives the desired equation, • The left side of the equation is a function of r, and the right side is a function of x. The equality must hold for any value of r and x; therefore, f (r) = g(x) = constant.

15. LAMINAR FLOW IN PIPES • Thus we conclude that dP/dx = constant and we can verify that • Here twis constant since the viscosity and the velocity profile are constants in the fully developed region. Then we solve the u(r) eq. by rearranging and integrating it twice to give r2

16. LAMINAR FLOW IN PIPES • Since u/r = 0 at r = 0 (because of symmetry about the centerline) and u = 0 at r =R, then we can get u(r) • Therefore, the velocity profile in fully developed laminar flow in a pipe is parabolic. Since u is positive for any r, and thus the dP/dx must be negative (i.e., pressure must decrease in the flow direction because of viscous effects). • The average velocity is determined from

17. LAMINAR FLOW IN PIPES • The velocity profile is rewritten as • Thus we can get • Therefore, the average velocity in fully developed laminar pipe flow is one half of the maximum velocity.

19. Pressure Drop and Head Loss • The pressure drop ∆P of pipe flow isrelated to the power requirements of the fan or pump to maintain flow. Since dP/dx = constant, and integrating from x =x1 where the pressure is P1 to x =x1 + L where the pressure is P2 gives • The pressure drop for laminar flow can be expressed as • ∆P due to viscous effects represents an irreversible pressure loss, and it is called pressure loss ∆PLto emphasize that it is a loss.

20. Pressure Drop and Head Loss • In the analysis of piping systems, pressure losses are commonly expressed in terms of the equivalent fluid column height, called the head loss hL. (Frictional losses due to viscosity)

21. Friction Losses The resulting pressure (energy and head) losses are usually computed through the use of modified Fanning’s friction factors: • where Fk is the characteristic force, S is the friction surface area. This equation is general and it can be used for all flow processes. Used for a pipe: where Fk is the press force, S is the area of curved surface. Rearranged, we get a form of pressure loss:

22. Determination of Friction Factor with Dimensional Analysis The Funning’s friction factor is a function of Reynolds number, f=f(Re): Many important chemical engineering problems cannot be solved completely by theoretical methods. For example, the pressure loss from friction losses in a long, round, straight, smooth pipe depends on all these variables: the length and diameter of pipe, the flow rate of the liquid, and the density and viscosity of the liquid. If any one of these variables is changed, the pressure drop also changes. The empirical method of obtaining an equation relating these factors to pressure drop requires that the effect of each separate variable be determine in turn by systematically varying that variable while keeping all others constant. It is possible to group many factors into a smaller number of dimensionless groups of variables. The groups themselves rather than separate factors appear in the final equation. These method is called dimensional analysis, which is an algebric treatment of the symbols for units considered independtly of magnitude.

23. Determination of Pressure Difference by Dimensional Analysis Many important chemical engineering problems cannot be solved completely by theoretical methods. For example, the pressure loss from friction losses (or the pressure differencebetween two ends of a pipe) in a long, round, straight, smooth pipe a fluid is flowing depends on all these variables: pipe diameter d, pipe length , fluid velocity v, fluid density ,and fluid viscosity . p1 p2

24. The relationship may be written as: The form of the function is unknown, but since any function can be expanded as a power series, the function can be regarded as the sum of a number of terms each consisting of products of powers of the variables. The simplest form of relations will be where the function consists simply of a single term, when: The requirement of dimensional consistency is that the combined term on the right-hand side will have the same dimensions as that the on the left, i.e. it must have the dimensions of pressure. Each of the variables in equation (2) can be expressed in terms of mass, length, and time. Thus, dimensionally: i.e.:

25. The conditions of dimensional consistency must be met for the fundamentals of M, L, and Tand the indices of each of these variables can be equated. Thus: In Thus three equations and five unknowns result and the equations may be solved in terms of any two unknowns. Solving in terms of b and e: Substituting in the L equation:

26. Thus, substituting into equation (2): i.e. Let: Thus: b=1, and k and e have to determinate by experiments. For laminar flow k=64 and e=1 For turbulent flow k=0,0791 and e=0,25.

28. If a theoretical equation for this problem exist, it can be written in the general form. List of relevant parameters: If Eq.1. is a valid relationship, all terms in the function f must have the same dimensions as those of the left-hand side of the equation . Let the phrase the dimensions of be shown by the use of brackets. Then any term in the function must conform to the dimensional formula

29. M: 1 = c+d L: -2 = a+b -3c - d T: -2 = -b - d M: c=1-d T: b=2-d L: a=-2-b+3c+d=-2-2+d+3-3d+d a=-1-d

30. Fluid Flow in Pipes Goals: determination of friction losses of fluids in pipes or ducts, and of pumping power requirement. The resulting pressure (energy and head) loss is usually computed through the use of the modified Fanning friction factor: Used for a pipe: where Fk is the press force, S is the area of curved surface. Rearranged, we get a form of pressure loss: The Funning’s friction factor is a function of Reynolds number, f=f(Re):

31. Fluid Flow in Pipes Goals: determination of friction losses of fluids in pipes or ducts, and of pumping power requirement. The resulting pressure (energy and head) loss is usually computed through the use of the modified Fanning friction factor: Used for a pipe: where Fk is the press force, S is the area of curved surface. Rearranged, we get a form of pressure loss: The Funning’s friction factor is a function of Reynolds number, f=f(Re):

32. Calculation of Pumping Power Requirement The friction factors were determined with dimensional analysis for a smooth pipe : The pressure loss is directly calculated from Hagen-Poiseuille’s equation for laminar flow: . When the fluid flows in a duct which is not circle in cross-section then we have to use the hydraulic diameter, Dh: The pumping power requirement (pump power equation): Where P is the power (Watt), V is the quantity of flow (m3/s), Leq is the equivalent pipe length of fittings, η is the efficiency of the pump.

33. 6.2. Motion of Particles in Fluids.Flow Around Objects There are many processes that involve the motion of particles in fluids, or flow around objects: • Sedimentation • Liquid Mixing • Food Industry • Oil Reservoirs Flow around objects

34. Sedimentation The goal is the determination of drag force for the flow around an immersed object , and the determination of terminal velocity for sedimentation. In gravitational field an object reaches terminal velocity when the downward force of gravity (Archimedesian weight) equals the upward force of drag. Called the modified friction factor: where Fk=Fdis the drag force, S=Sp is the projected area of the particle, and f=fd is the drag coefficient. The drag force is Archimedesian weight in gravitational field is Principle of sedimentation

35. Terminal Velocity At the terminal velocity: if , For spherical objects: The drag coefficientis a function ofthe Re-number: For laminar flow drag coefficient can be calculated directly from Stokes’ law:

36. 6.3. Mixing of Liquids In the mixers, the fluids to be mixed are placed in containers or reactors and the stirrer is rotated. Most of the information that is available concerns the power requirements for the most commonly used liquid mixer – some form of paddle or propeller stirrer. The mixing of the liquid can be described with the problem of flow around immersed objects. The goal is the determination of the power consumption of agitators. Call the modified friction factor: The force changes continuously along the propeller; therefore the differential equation of force hasbeen written : Stirrer

37. Power Consumption of Stirrer Power consumption of stirrer (the power multiplied by velocity of the propeller) is The circumferential velocity of the propeller or paddle in distance x from the axis: and Where n is the rotational frequency of the propeller, revolutions per second. Integrate all of theprojected surface area of the stirrer: Stirrer

38. Power Consumption of Stirrer Let „a” be the geometrical simplex which is and The Power number (relating drag forces to inertial forces), Po=f*= f*(Restir) have to determine the function of Reynolds number with experiments. The results have been correlated in equations of form: Stirrer power equation: η is the efficiency

39. 7. Flow of Fluids in Complex SystemsFixed and Fluidized Beds 7.1Fluid flow through packed bed or porous media Goal: determination of pressure drop (friction losses) through packed bed or columns. 7.2. Fluidization. The goal is the determination of the minimum fluidization velocity. 7.3. Filtration The goal is the determination of differential equation of filtration.

40. 7.1. Flow through Porous Media or Packed Bed In many engineering systems, beds or packed columns, fluidization, filtration, are used in various processes. A typical packed bed is a cylindrical column that is filled with suitable spheres or other non-spherical packing material. Fluid flows between the particles in small diameter tortuous, winding channels. Packed bed

41. Pressure Drop for Packed Bed The goal is the determination of pressure drop through packed bed or columns. Called the modified friction factor: Using the modified friction factor for channels of packed bed similar to the flow in pipes: , . Porosity of bed: Interstitial velocity of fluid: Vois the superficial velocity Specific area for sphere:

42. Friction Coefficient for Packed Bed Definition of Reynolds number for packed bed: fp = fp (Rep), the results have been correlated in equations of form: The Ergun’s equation predicts the pressure drop (or flow) through porous media or packed columns quite well. Pressure drop:

43. 7.2. Fluidization When a fluid is passed upwards through a bed of particles the pressure loss in the fluid due to frictional resistance increases with increasing fluid flow. „A” point is reached when the upward drag force exerted by the fluid on the particles is equal to the apparent weight of particles in the bed. At this point the particles are lifted by the fluid, the separation of the particles increases, and the bed becomes fluidized. The goal is the determination of the minimum fluidization velocity. Δp Principle of fluidization Response to superfical velocities

44. Equation of Fluidization Net effect of gravity and buoyancy is: And the pressure: The pressure loss as a consequence of friction: . Pressure equals to the pressure loss: Principle of fluidization

45. Minimum Fluidization Velocity When the superficial velocity vo is equal to the minimum fluidization velocity vom we refer to the state of the bed as one of incipient fluidization: Called the Ergun equation: The minimum fluidization velocity can be calculated by the equation:

46. 7.3. Filtration Filtration is the separation of solids from liquids, by causing the mixture to flow through fine pores which are small enough to stop the solid particles but large enough to allow the liquid to pass. The particles are in the form of a fairly stable bed and the fluid has to pass through the tortuous channels formed by the pore spaces. In the tortuous channels the flow is always laminar. The goal is to give a function between the time of filtration and the volume of filtrate. How much time is necessary in the development of the given filtrate? Called the pressure drop of packed bed: Called the Blake-Kozeny equation: Principle of filtration

47. Equation of Filtration Combine the last two equations: Rearranged, we get the basic equation of filtration, Darcy equation: where α is termed the specific resistance, V is the volume of filtrate which has passed in time t, A is the total cross-sectional area of the filter cake, v is the superficial velocity of the filtrate, L is the cake thickness, ε is the voidage, μ is the viscosity of the filtrate, and Δp is the applied pressure difference. . . From the Darcy equation: The function, t = f(V), the duration that is necessary for the developing of the given filtrate is

48. 8. SUMMARY In simple and complex systems the resulting pressure (energy and head) losses are usually computed through the use of modified Fanning’s friction factors: 1. Flow of Fluids in Simple Systems Fluid Flow in Pipes (friction losses and pumping power requirement): Motion of Particles in Fluids (drag force and terminal velocity for sedimentation): Mixing of Liquids (power consumption of agitators):

49. 2. Flow of Fluids in Complex Systems Pressure Drop for Packed Bed: Fluidization (determination of the minimum fluidization velocity): Equation of Filtration (Darcy’s equation):

50. Thank you for your attention