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Transport Processes and Separation Process Principles (Geankoplis) CHE312 Dr. Othman Alothman Dr. Mohamed Kamel Omar Hadj-Kali 1 Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312 Chapter 3 Principles of Momentum Transfer and Applications Flow past immersed objects and packed an fluidized beds Measurement of flow of fluids Pumps and gas-moving equipment … 2 Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312 Flow Past Immersed Objects 2 courses Read and explain the introduction in one course 3 Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312 Flow Past Immersed Objects 3.1A Definitions of Drag Coefficient for Flow Past Immersed Objects Introduction In chapter 2, momentum transfer and the frictional losses for flow of fluids inside conduits and pipes were discussed. In chapter 3, the flow of fluids around solid immersed objects will be discussed. The flow of fluids outside immersed bodies occurs in many chemical applications such as: flow past spheres in settling, flow through packed beds in drying and filtration, flow past tubes in heat exchangers and so on. In chapter 2, the transfer of momentum perpendicular to the surface resulted in a tangential shear stress or drag on the smooth surface parallel to the direction of the flow. The force exerted by the fluid on the solid in the direction of the flow is called skin or wall drag. Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312 4 Flow Past Immersed Objects 3.1A Definitions of Drag Coefficient for Flow Past Immersed Objects Introduction For any surface in contact with a flowing fluid, skin friction will exist. In addition to skin drag, if the fluid has to change its direction to pass around a solid body such as sphere, significant additional frictional losses will occur and this is called form drag. For a fluid flowing parallel to a solid plate, the force dF on an element of the area dA of the plate is the wall shear stress times the area dA: dF w dA v0 F w dA dA Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312 5 Flow Past Immersed Objects 3.1A Definitions of Drag Coefficient for Flow Past Immersed Objects Introduction In many cases, the immersed body is a blunt-shaped solid with various angles. In approaching the body, v0 is uniform. Lines representing the path of the fluid elements around the body are called streamlines. The velocity at the stagnation point is zero, and the boundary layer start to grow at this point. The thin boundary layer is adjacent to the solid surface. The velocity at the edge of the boundary layer is the same as the bulk velocity adjacent to it. 6 Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312 Flow Past Immersed Objects 3.1A Definitions of Drag Coefficient for Flow Past Immersed Objects Introduction The tangential stress on the body because of the velocity gradient in the boundary layer is the skin friction. Outside the boundary layer, the fluid change direction to pass around the body accelerating near the front then decelerating. Thus, an additional force is exerted by the fluid on the body: This is the form drag. Separation of the boundary layer occurs and a wake covering the entire rear of the body occurs where large eddies are present and contribute to the form drag. Form drag for bodies can be minimized by streamlining the body which forces the separation point toward the rear of the body, reducing the size of the wake. 7 Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312 Flow Past Immersed Objects 3.1A Definitions of Drag Coefficient for Flow Past Immersed Objects Drag coefficient The geometry of the immersed solid is a main factor in determining the amount of total drag force exerted on the body. Similar to the f-NRe correlations for flow inside conduits, correlations for the drag coefficient- NRe for flow past an immersed body can be obtained. CD FD Ap v 2 Where, FD is the total drag force. 2 0 Ap is the area obtained by projecting the body on a plane perpendicular to the line of flow. For sphere: Ap 4 D 2 p For cylinder Ap LD p (axis perpendicular to the flow): Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312 8 Flow Past Immersed Objects 3.1A Definitions of Drag Coefficient for Flow Past Immersed Objects Drag coefficient So, solving the pervious equation for the total force: CD FD Ap v02 FD CD Ap 2 v 2 2 0 The Reynolds number for a given solid immersed in a flowing liquid is: N Re Dp v0 DpG0 G0 v0 9 Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312 Flow Past Immersed Objects 3.1B Flow past sphere, long cylinder and disk Correlations of drag coefficient CD vs. NRe depends mainly on the shape of the immersed body and its orientation. These correlations are shown in Figure 3.1-2 for spheres, long cylinders and disks. Note that the face of the disc and the axis of the cylinder are perpendicular to the direction of flow. These curves are determined experimentally. At laminar flow (NRe ≤ 1.0), the total drag force (FD) can be obtained from Stokes' law equation as follows: So: CD 24 CD N Re FD Ap v02 2 CD FD 3D p v0 3D v D 4 24 v 2 Dv 2 p p 0 2 0 p 0 Example 3.1.1 Discuss figure 3.1-2 Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312 Example 3.1.2 10 Flow in Packed and Fluidized Beds 3 courses 11 Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312 Flow in Packed Beds v’ Introduction The packed bed (or packed column) is found in a number of chemical processes including a fixed bed v catalytic reactor, filter bed, absorption and adsorption. The ration of diameter of the tower to packing diameter should be at least 8:1 to neglect wall effects Laminar flow in packed beds The void fraction, e, in a packed bed is defined as: volume of voids in bed e total volume of bed (voids plus solids) The specific surface of a particular is: av S p V p Where: Sp is the surface area of a particle and Vp its volume. Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312 12 Flow in Packed Beds (Laminar flow) v’ For a spherical particle: av = 6/Dp, (Sp=πDp2 Vp= πDp3/6) For a packed bed of non-spherical particles, the effective particle diameter Dp is: v D p 6 av The volume fraction of the particles in the bed is (1- ε) and, thus, the ratio of the total surface area in the bed to total volume of the bed, a: Sp 6 1 e a av 1 e Dp N Sp total surface area of particles av Vp N Vp total volume of particles total volume of particles 1 e total volume of bed Example 3.1.3 total surface area in the bed a av 1 e total volume of bed Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312 13 Flow in Packed Beds (Laminar flow) v’ = ev The average interstitial velocity in the bed (v) is related v ev to the superficial velocity (v’) by: The hydraulic radius rH for flow is modified to be: v cross - sectional area available for flow rH wetted perimeter void volume available for flow total wetted surface of solids volume of voids / volume of bed e wetted surface / volume of bed a Combining equations a av 1 e 6 1 e and Dp e rH a gives: rH e 61 e Dp 14 Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312 Flow in Packed Beds (Laminar flow) Since the equivalent diameter for a channel is v’ = ev D 4rH The Reynolds number for a packed bed is as follows: N Re 4rH v v 4 Dp v 61 e v ev For packed beds, Ergun defined NRe without 4/6: N Re, p Dp v D p G 1 e 1 e G v For laminar flow, the Hagen-Poiseuille equation can be expressed using rH to give: 32vL 32 v e L 72vL1 e p 2 2 D e 3 Dp2 4rH 2 15 Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312 Flow in Packed Beds (Laminar flow) v’ = ev Experimental data show that the constant should be 150 (the true L is larger because of the tortuous path and use of rH predicts too large v’). This leads to the Blake-Kozeny equation for laminar flow, e < 0.5, Dp and NRe,p < 10: 2 150vL 1 e p Dp2 e3 Flow in Packed Beds (Turbulent flow) For turbulent flow, we use the same procedure using equations: L v 2 p 4 f D 2 v ev D 4rH 3 f v L 1 e To obtain: p 3 D e p Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312 rH e 61 e Dp 2 16 Flow in Packed Beds (Turbulent flow) v’ = ev For highly turbulent flow, f should approach constant value. Another assumption is that all packed beds have the same relative roughness. Experimental data show that : 3 f 1.75 Hence, for turbulent flow (NRe,p>1000), the Burke-Plummer eqnation is used: 1.75 v L 1 e p Dp e3 2 Adding Blake-Kozeny equation (for laminar flow) and Burke-Plummer equation (for turbulent flow), Ergun proposed a general equation for low, intermediate and high Reynolds number (NRe,p) as: 150vL 1 e 1.75 v L 1 e p 2 3 Dp e Dp e3 2 Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312 2 17 Flow in Packed Beds (Turbulent flow) v’ = ev By dimensional analysis, the general equation of Ergun can be rewritten as: p D p e 3 150 1.75 2 G L 1 e N Re, p G v This equation can be used for gases where density is taken at the arithmetic average of the inlet and outlet pressures Example 3.1.4 Flow in Packed Beds (Shape factors) Particles in packed beds are often irregular in shape. The shape factor or sphericity (fs) is defined by: Surface area of sphere having the same volume as tha particle fs The actual surface area of the particle Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312 18 For a sphere: The surface area is Sp=πDp2 The volume is Vp= πDp3/6 Hence, for any particle: fs Dp2 S p Dp2 fs 6 av 3 Vp Dp 6 fs Dp Sp and For a sphere: fs 1 For a cylinder (L=D): fs 0.874 For a cube: fs 0.806 (Sp the surface area of the particle) a av 1 e 6 1 e fs D p (See table 3.1-1) Hw Mixtures of particles 19 Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312 Flow in Fluidized Beds Packed bed Fluidized bed Increased L Lmin velocity v’ v’min Two general types of fluidization in beds can occur: Particulate fluidization Bubbling fluidization 20 Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312 Flow in Fluidized Beds Minimum velocity and porosity At very low velocity, packed bed remains stationary Lmin When fluid velocity is increased, the pressure drop increases, according to Ergun (Eqn. 3.1-20). At a certain velocity, when the pressure drop force (i.e. ∆P*A) equalizes the gravitational force on the mass of v’min particle (i.e. m*g), the particles begin to move (fluidize). This velocity is called the minimum fluidization velocity (v’mf m/s) (based on the superficial velocity). At the minimum velocity, the porosity is called the minimum porosity of fluidization, εmf (See Table 3.1-2 for ε for some materials ) Similarly, the new height of the bed is Lmf in m. Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312 21 Flow in Fluidized Beds Relation between bed height L and porosity e The total volume of the solid particles is constant for a bed having a uniform cross-sectional area A and is: Therefore: V L A 1 e L1 A 1 e1 L2 A 1 e 2 L1 1 e 2 L2 1 e 1 Pressure drop and minimum fluidizing velocity At the onset of fluidization, the following is approximately true: p A Lmf A1 e mf p g p 1 e mf p g Lmf Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312 (SI) 22 Flow in Fluidized Beds Modifying the general equation of Ergun (substitute Dp with Dpfs) to correct for no spherical particles, this yield to: 2 2 p 150v 1 e 1.75 v 1 e 2 2 3 L fs D p e fs D p e3 This equation with the previous one can be combined to calculate the minimum fluidization velocity: 2 ' mf 3 2 s mf 2 p 1.75D v 2 fe ' 1501 e mf Dp vmf fe 2 3 s mf Defining a Reynolds number as: 1.75N Re, mf 2 We obtain: fe 3 s mf Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312 D3p p g N Re, mf 1501 e mf N Re, mf fe 2 3 s mf 2 0 ' D p vmf D 3p p g 2 0 23 Flow in Fluidized Beds 1.75N Re, mf 2 fe 3 s mf 1501 e mf N Re, mf fe 2 3 s mf D 3p p g 2 0 =0 =0 For small particles For large particles (NRe,mf < 20) (NRe,mf > 1000) fe 3 s mf If the terms emf and/or fs are not known, Wen and Yu proposed the following approximations: N Re, mf D p g 2 33.7 0.0408 2 3 p 1 e mf fe Substituting in the equation above implies: 1 14 2 3 s mf 12 11 33.7 0.001 N Re, mf 4000 Example 3.1.6 Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312 24 Flow in Fluidized Beds Expansion of fluidized beds For small particles We can estimate the variation of porosity D p v' and bed height L using equation a with the N Re, f 20 first term taken equal to zero. D p2 p gfs2 e 3 e3 v' K1 150 1 e 1 e This equation can be used with liquids to estimate e with e < 0.8 The maximum allowable velocity For fine solids and NRe,f < 0.4 For large solids and NRe,f > 1000 ' vt' 90vmf ' vt' 9vmf Example 3.1.7 Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312 25 Measurement of flow of fluids 1 course 26 Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312 Measurement of flow of fluids (Pitot tube) It is important to able to measure and control materials entering and leaving any processing plants. Many different types of devices are used to measure the flow of fluids . Including Pitot tube, venture meter, orifice meter and open-channel weirs. It is widely used to determine the Pitot tube airspeed on an aircraft and to measure air and gas velocities in industrial applications (AF447 (Rio-Paris) crash) It is used to measure the local velocity at given point in the flow stream h and not the average velocity in the pipe or conduit. 27 Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312 Measurement of flow of fluids (Pitot tube) The different between the stagnation pressure (at point 2) and the static pressure (measured by the static tube) represents the pressure rise associated with the deceleration of the fluid. For incompressible fluids, Bernoulli equation between point 1 and point 2 can be written as: Pitot tube v v p1 p2 0 2 2 2 1 Setting v2 = 0 2 2 v Cp 2 p1 p2 h Cp is a dimensionless coefficient to correct deviation from the Bernoulli equation and varies between 1.0 and 0.98. Example 3.2.1 Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312 28 Measurement of flow of fluids (Venturi meter) Venturi meter A venturi meter is usually inserted directly into a pipeline. Since the narrowing and the expansion are gradual, little friction loss is incurred. A manometer or is connected to the two pressure taps to measure the pressure difference between point 1 and 2. For incompressible fluids, Bernoulli equation between point 1 and point 2 can be written as: From the continuity equation: Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312 v12 v22 p1 p2 0 2 2 v1 D12 4 v2 D22 4 29 Measurement of flow of fluids (Venturi meter) Combining these equations and eliminating v1: v2 Cv 1 D2 D1 4 2 p1 p2 Cv is introduced to account for the small friction Venturi meter loss. Its value is between 0.98 and 1.0. for compressible gases , the adiabatic expansion(??) from p1 to p2 must be allowed for. m Cv A2Y 1 D2 D1 4 2 p1 p2 1 The same equation is used where m is the mass flowrate (kg/s) and Y the dimensionless expansion factor (shown in figure 3-2-3 for air). 30 Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312 Measurement of flow of fluids (Orifice meter) Orifice meter is similar to venturi meter, but presents some advantages are: Adjustable throat, Cheaper price, Needs less space. Orifice meter The equation for the orifice meter is similar to that of venturi and is: v0 C0 1 D0 D1 4 2 p1 p2 v0 and D0 are the velocity and diameter at the point 0 and C0 the dimensionless orifice coefficient C0 = 0.61 (for NRe > 2x104 and D0/ D1 < 0.5) 31 Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312 Measurement of flow of fluids (Orifice meter) As for venturi meter, for compressible gases, the adiabatic expansion(??) from p1 to p2 must be allowed for. m C0 A0Y 1 D0 D1 4 2 p1 p2 1 Orifice meter The pressure loss is much higher than in venture meter because of the eddies formed due to the jet expansion. This loss depends on D0/D1: 73% of (p1-p2) for D0/D1 = 0.50 56% of (p1-p2) for D0/D1 = 0.65 38% of (p1-p2) for D0/D1 = 0.80 Example 3.2.2 Dr. M.K. O. Hadj-Kali / Dr. O. Y. Alothman / ChE312 32