1 Unit operations of chemical engineering Fourth year Chemical Engineering Department College of Engineering AL-Qadesyia University Lecturer:
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3 Syllabus 1) Boundary layer theory 2) Transfer of heat, mass and momentum(the concept of universal velocity profile) 3) Transfer of heat, mass and momentum (Reynolds analogy) 4) Non-Newtonian fluids 5) Mixing 6) Fluid flow through packed column 7) Flow of particles though fluids 8) Fluidization 9) Filtration 10) Sedimentation 11) Centrifuge 12) Selective topics of unit operation application in renewable energy References 1) Unit Operations of Chemical Engineering,Seventh Edition, Prepared by Julian C. Smith,Peter Harriott,2009 2) Principles Of Unit Operation, second edition, By Alan S. Foust,1980 3) Chemical Engineering, Volume 1, Sixth edition, ByJ, M. Coulson and J. F. Richardson,1999 4) Chemical Engineering, Volume 2, fifth edition, ByJ, M. Coulson and J. F. Richardson,2002.
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5 CHAPTER (ONE) The Boundary Layer When a fluid flows over a surface that part of the stream which is close to the surface suffers a significant retardation, and a velocity profile develops in the fluid. The velocity gradients are steepest close to the surface and become progressively smaller with distance from the surface. It is convenient to divide the flow into two parts for practical purposes: 1) A boundary layer close to the surface in which the velocity increases from zero at the surface itself to a near constant stream velocity at its outer boundary. 2) A region outside the boundary layer in which the velocity gradient in a direction perpendicular to the surface is negligibly small and in which the velocity is everywhere equal to the stream velocity. The thickness of the boundary layer may be arbitrarily defined as the distance from the surface at which the velocity reaches some proportion (such as 0.9, 0.99, 0.999 ) of the undisturbed stream velocity. The flow conditions in the boundary layer are of considerable interest to chemical engineers because these influence, not only the drag effect of the fluid on the surface, but also the heat or mass transfer rates where a temperature or a concentration gradient exists. 1-1) BOUNDARY LAYER THEORY APPLIED TO A THIN PLATE It is convenient first to consider the flow over a thin plate inserted parallel to the flow of a fluid with a constant stream velocity u s. It will be assumed that the plate is sufficiently wide for conditions to be constant across any finite width(w) of the plate which is being considered. Furthermore, the extent of the fluid in a direction perpendicular to the surface is considered as sufficiently large for the velocity of the fluid remote from the surface to be unaffected and to remain constant at the stream velocity u s. The development of the boundary layer over flat surface is illustrated in Figure (1-1 )
6 Figure (1-1) Development of the boundary layer On the assumption that there is no slip at the surface, the fluid velocity at all points on the surface, where y = 0, will be zero. At some position a distance x from the leading edge, the velocity will increase from zero at the surface to approach the stream velocity u s asymptotically. Near the leading edge of the surface where the boundary layer thickness is small, the flow will be streamline, or laminar, and the shear stresses will arise solely from viscous shear effects. When the boundary layer thickness exceeds a critical value, the streamline flow ceases to be stable and turbulence sets in.the important flow parameter is the Reynolds number Re δ (= u s δρ/µ). Because δ can be expressed as a function of x(the distance from the leading edge of the surface) the usual criterion is taken as the value of the Reynolds number Re x (= u s xρ/µ). If the location of the transition point is at a distance x c from the leading edge, then [Re xc = 10 5 ]. When the flow in the boundary layer is turbulent, streamline flow persists in a thin region close to the surface called the laminar sub-layer. This region is of particular importance because, in heat or mass transfer, it is where the greater part of the resistance to transfer lies. High heat and mass transfer rates therefore depend on the laminar sub-layer being thin. Separating the laminar sub-layer from the turbulent part of the boundary layer is the buffer layer, in which the contributions of the viscous effects and of the turbulent eddies are of comparable magnitudes.
7 1-1-A)The momentum equation It will be assumed that a fluid of density (ρ) and viscosity (µ) flows over a plane surface and the velocity of flow outside the boundary layer is u s. A boundary layer of thickness δ forms near the surface, and at a distance y from the surface the velocity of the fluid is reduced to a value u x.as shown in figure(1-2) Figure (1-2) Element of boundary layer The equilibrium is considered of an element of fluid bounded by the planes 1-2 and 3-4 at distances x and x + dx respectively from the leading edge; the element is of length (l) in the direction of flow and is of depth w in the direction perpendicular to the plane 1-2-3-4. The distance (l) is greater than the boundary layer thickness δ, and conditions are constant over the width (w). The velocities and forces in the X- direction are now considered. At plane 1-2, mass rate of flow through a strip of thickness dy at distance y from the surface = ρu x w dy The total flow through plane 1-2 ---------(1-1) The rate of transfer of momentum through the elementary strip =ρu x wdyu x = wρu x 2 dy The total rate of transfer of momentum through plane 1-2:
8 ---------------- (1-2) In passing from plane 1-2 to plane 3-4, the mass flow changes by: and the momentum flux changes by: ----------------- (1-3) -------------- (1-4) where M ii is the momentum flux across the plane 3-4. A mass flow of fluid equal to the difference between the flows at planes 3-4 and 1-2 must therefore occur through plane 2-4. Since plane 2-4 lies outside the boundary layer, the fluid crossing this plane must have a velocity u s in the X-direction. Thus the rate of transfer of momentum through plane 2-4 out of the element is: ------------- (1-5) It will be noted that the derivative is negative, which indicates a positive outflow of momentum from the element. Thus, the net momentum flux out of the element M ex is given by: Then, since u s is assumed not to vary with x: ---------- (1-6)
9 The net force acting F is just the retarding force attributable to the shear stress at the surface only and Thus: F=R 0 wdx Equating the net momentum flux out of the element to the net retarding force and simplifying gives: -------------- (1-7) This expression, known as the momentum equation, may be integrated provided that the relation between u x and y is known. If the velocity of the main stream remains constant at u s and the density (ρ) may be taken as constant, the momentum equation then becomes: Where R o =-µ( u x / y) y=0 R o is the shear stress in the fluid at the surface 1-1-B) The streamline portion of the boundary layer ----------- (1-8) In the streamline boundary layer the only forces acting within the fluid are pure viscous forces and no transfer of momentum takes place by eddy motion. Assuming that the relation between u x and y can be expressed approximately by: ----------------- (1-9) The coefficients a, b, c and u o may be evaluated because the boundary conditions which the relation must satisfy are known as shown in figure (1-3).
10 Figure (1-3) Velocity distribution in streamline boundary layer It is assumed here that the fluid in contact with the surface is at rest and therefore u o must be zero. Furthermore, all the fluid close to the surface is moving at very low velocity and therefore any changes in its momentum as it flows parallel to the surface must be extremely small. Thus the shear stress R o in the fluid near the surface must approach a constant value. Since R o = -µ(du x /dy) y=0, du x /dy must also be constant at small values of y and: At the distant edge of the boundary layer it is assumed that the velocity just equals the main stream velocity and that there is no discontinuity in the velocity profile. Thus, when y=δ: Now with u o = 0, equation (1-9) becomes: And At y=0 thus b=0
11 At y=δ And thus a=-3cδ 2 Hence The equation for the velocity profile is therefore: ------------- (1-10) Equation (1-10) corresponds closely to experimentally determined velocity profiles in a laminar boundary layer. This relation applies over the range 0 < y <δ. When y > δ, then: u x = u s The integral in the momentum equation (1-8) can now be evaluated for the streamline boundary layer by considering the ranges 0 < y < δ and δ < y < l separately. Thus: -------------------------------- (1-11) In addition ----------- (1-12) Substitution from equations (1-11) and (1-12) in equation (1-8), gives:
12 (Since δ =0 when x=0) --------------- (1-13) Thus δ And: ----------- (1-14) 1-1-C)Shear stress at the surface The shear stress in the fluid at the surface is given by: The shear stress R acting on the surface itself is equal and opposite to the shear stress in the fluid at the surface; that is, R = -R o. thus: ---------------- (1-15)
13 Equation (1-15) gives the point values of R and (R/ρu 2 )at x = x. In order to calculate the total frictional force acting at the surface, it is necessary to multiply the average value of R between x = 0 and x = x by the area of the surface. The average value of (R/ρu 2 ) denoted by the symbol (R/ρu 2 ) m is then given by; -------- (1-16) HOMEWORK: If the velocity profile in the laminar region may be represented approximately by a sine function, calculate the boundary-layer thickness in terms of distance from the leading edge of the surface. 1-1-D)The turbulent boundary layer I) The turbulent portion Equation (1-10) does not fit velocity profiles measured in a turbulent boundary layer and an alternative approach must be used. In the simplified treatment of the flow conditions within the turbulent boundary layer the existence of the buffer layer is neglected and it is assumed that the boundary layer consists of a laminar sub-layer, in which momentum transfer is by molecular motion alone, outside which there is a turbulent region in which transfer is effected entirely by eddy motion (Figure1-4). The approach is based on the assumption that the shear stress on the surface can be calculated from the simple power law developed by Blasius.
14 Figure (1-4) Turbulent boundary layer Blasius has given the following approximate expression for the shear stress at a plane smooth surface over which a fluid is flowing with a velocity u s, for conditions where Re x < 10 7 : ---------------- (1-17) Thus, the shear stress is expressed as a function of the boundary layer thickness δ and it is therefore implicitly assumed that a certain velocity profile exists in the fluid. As a first assumption, it may be assumed that a simple power relation exists between the velocity and the distance from the surface in the boundary layer, or: Hence -------------------- (1-18) - (1-19)
15 If the velocity profile is the same for all stream velocities, the shear stress must be defined by specifying the velocity u x at any distance y from the surface. The boundary layer thickness, determined by the velocity profile, is then no longer an independent variable so that the index of δ in equation (1-19) must be zero or: Thus ----------------- (1-20) Equation (1-20) is sometimes known as the Prandtl seventh power law. Differentiating equation (1-20) with respect to y gives: --------- (1-21) This relation is not completely satisfactory in that it gives an infinite velocity gradient at the surface, where the laminar sub-layer exists, and a finite velocity gradient at the outer edge of the boundary layer. This is in contradiction to the conditions which must exist in the stream. However, little error is introduced by using this relation for the whole of the boundary layer in the momentum equation because, firstly both the velocities and hence the momentum fluxes near the surface are very low, and secondly it gives the correct value of the velocity at the edge of the boundary layer. Accepting equation for the limits 0 < y < δ, the integral in equation (1-8) becomes: -------------------------------- (1-22)
16 From the Blasius equation: Substituting from equations (1-17) and (1-22) into (1-8) Putting the constant equal to zero, implies that δ= 0 when x = 0, that is that the turbulent boundary layer extends to the leading edge of the surface. An error is introduced by this assumption, but it is found to be small except where the surface is only slightly longer than the critical distance x c for the laminar-turbulent transition ------------------------ (1-23) II) The laminar sub-layer If at a distance x from the leading edge the laminar sub-layer is of thickness δ b and the total thickness of the boundary layer is δ, the properties of the laminar sub-layer can be found by equating the shear stress at the surface as given by the Blasius equation (1-17) to that obtained from the velocity gradient near the surface.
17 It has been noted that the shear stress and hence the velocity gradient are almost constant near the surface. Since the laminar sub-layer is very thin, the velocity gradient within it may therefore be taken as constant. Thus the shear stress in the fluid at the surface, Equating this to the value obtained from equation 1-17 gives If the velocity at the edge of the laminar sub-layer is u b, that is, if u x =u b, when y = δ b -------------- (1-24) The velocity at the inner edge of the turbulent region must also be given by the equation for the velocity distribution in the turbulent region. -------------------------- (1-25)
18 From equation (1-23) ------------------ (1-26) The thickness of the laminar sub-layer is given by: ---------------------- (1-27) Thus δ b x 0.1 that is, δ b it increases very slowly as x increases. Further, δ b αu s -0.9 and therefore decreases rapidly as the velocity is increased, and heat and mass transfer coefficients are therefore considerably influenced by the velocity. III) Shear stress at the surface The shear stress at the surface, at a distance x from the leading edge, is given by:
19 The mean value is: --------------------------- (1-28) ---------- (1-29) The total shear force acting on the surface is found by adding the forces acting in the streamline (x < x c ) and turbulent (x > x c ) regions. This can be done provided the critical value Re Xc, is known. A more accurate value for the mean value of (R/ρu 2 s) m over the whole surface can be obtained by using the expression for streamline conditions over the range from x = 0 to x = x c (where x c is the critical distance from the leading edge) and the expression for turbulent conditions in the range x = x c to x= x, -- (1-30) ---- (1-31)
20 Example (1-1) Calculate the thickness of the boundary layer at a distance of 150 mm from the leading edge of a surface over which oil, of viscosity 0.05 N s/m 2 and density 1000 kg/m 3 flows with a velocity of 0.3 m/s. Solution Example(1-2) Air is flowing at a velocity of 5 m/s over a plane surface. Derive an expression for the thickness of the laminar sub-layer and calculate its value at a distance of 1 m from the leading edge of the surface. Assume that within the boundary layer outside the laminar sub-layer, the velocity of flow is proportional to the one-seventh power of the distance from the surface and that the shear stress R at the surface is given by: ( ) ( ) Where ρ is the density of the fluid (1.3 kg/m 3 for air), µ is the viscosity of the fluid (17 x 10-6 N s/m 2 for air), u s is the stream velocity (m/s), and x is the distance from the leading edge (m). Solution The shear stress in the fluid at the surface: R =µu x /y From the equation given: ( ) ( )
21 If the velocity at the edge of the laminar sub-layer is u b, u x = u b at y=δ b The velocity distribution is given by:
22 1-2)BOUNDARY LAYER THEORY APPLIED TO PIPE FLOW 1-2-A)Entry conditions When a fluid flowing with a uniform velocity enters a pipe, a boundary layer forms at the walls and gradually thickens with distance from the entry point. Since the fluid in the boundary layer is retarded and the total flow remains constant, the fluid in the central stream is accelerated. At a certain distance from the inlet, the boundary layers, which have formed in contact with the walls, join at the axis of the pipe, and, from that point onwards, occupy the whole cross-section and consequently remain of a constant thickness. Fully developed flow then exists. If the boundary layers are still streamline when fully developed flow commences, the flow in the pipe remains streamline. On the other hand, if the boundary layers are already turbulent, turbulent flow will persist as shown in Figure 1-5. Figure (1-5) Conditions at entry to pipe An approximate experimental expression for the inlet length L e for laminar flow is: ------------------- (1-32) where d is the diameter of the pipe and Re is the Reynolds group with respect to pipe diameter, and based on the mean velocity of flow in the pipe. This expression is only approximate, and is inaccurate for Reynolds numbers in the region of 2500 because the boundary layer thickness increases very rapidly in this region. An average value of Le at a Reynolds number of 2500 is about 100d.
23 1-2-B)Application of the boundary-layer theory The velocity distribution and frictional resistance have been calculated from purely theoretical considerations for the streamline flow of a fluid in a pipe to be: The boundary layer theory can now be applied in order to calculate, approximately, the conditions when the fluid is turbulent. For this purpose it is assumed that the boundary layer expressions may be applied to flow over a cylindrical surface and that the flow conditions in the region of fully developed flow are the same as those when the boundary layers first join. The thickness of the boundary layer is thus taken to be equal to the radius of the pipe and the velocity at the outer edge of the boundary layer is assumed to be the velocity at the axis. The velocity of the fluid may be assumed to obey the Prandtl oneseventh power law, given by equation 1-20. If the boundary layer thickness δ is replaced by the pipe radius r, this is then given by: -------------------------- (1-33) The shear stress at the walls is given by the Blasius equation (1-17) as: Writing u = 0.817u s, and d = 2r: This equation is more usually written: --- (1-34) --------------------------- (1-35) The discrepancy between the coefficients in equations 1-34 and 1-35 is attributable to the fact that the effect of the curvature of the pipe wall has not been taken into account in applying the equation for flow over a
24 plane surface to flow through a pipe. In addition, it takes no account of the existence of the laminar sub-layer at the walls. Equation 1-35 is applicable for Reynolds numbers up to 10 5. The velocity at the edge of the laminar sub-layer is given by: ------------------------- (1-36) The thickness of the laminar sub-layer is given by: -------------------- (1-37) Example (1-3) Calculate the thickness of the laminar sub-layer when benzene flows through a pipe 50 mm in diameter at 2 1/s. What is the velocity of the benzene at the edge of the laminar sub-layer? Assume that fully developed flow exists within the pipe and that for benzene, ρ=870 kg/m 3 andµ = 0.7 mn s/m 2.
25 Solution The mass flow rate of benzene = (2 x 10-3 x 870) = 1.74 kg/s 1-3) THE BOUNDARY LAYER FOR HEAT TRANSFER Where a fluid flows over a surface which is at a different temperature, heat transfer occurs and a temperature profile is established in the vicinity of the surface. A number of possible conditions may be considered. At the outset, the heat transfer rate may be sufficient to change the temperature of the fluid stream significantly or it may remain at a substantially constant temperature. Furthermore, a variety of conditions may apply at the surface. Thus the surface may be maintained
26 at a constant temperature, particularly if it is in good thermal conduct with a heat source or sink of high thermal capacity. Alternatively, the heat flux at the surface may be maintained constant, or conditions may be intermediate between the constant temperature and the constant heat flux conditions 1-3-A)The heat balance The procedure here is similar to that adopted previously for momentum. A heat balance, as opposed to a momentum balance, is taken over an element which extends beyond the limits of both the velocity and thermal boundary layers. In this way, any fluid entering or leaving the element through the face distant from the surface is at the stream velocity u s and stream temperature θ s. A heat balance is made therefore on the element shown in Figure (1-6) in which the length l is greater than the velocity boundary layer thickness δ and the thermal boundary layer thickness δ t. Figure (1-6) The thermal boundary layer The rate of heat transfer through an element of width w of the plane 1-2, of thickness dy at a distance y from the surface is: ------------------ (1-38) The total rate of transfer of heat through the plane, 1-2 is then: -------------- (1-39) In the distance dx this heat flow changes by an amount given by:
27 --------------- (1-40) Since the plane, 2-4, lies outside the boundary layers, the heat leaving the element through the plane as a result of this flow is: ------------------- (1-41) Where θ s is the temperature outside the thermal boundary layer. The heat transferred by thermal conduction into the element through plane, 1-3 is: ----------------------- (1-42) If the temperature θ s of the main stream is unchanged, a heat balance on the element gives: ------(1-43) ------- (1-44) where D H (= k/c p ρ) is the thermal diffusivity of the fluid. I)Heat transfer for streamline flow over a plane surface constant surface temperature The flow of fluid over a plane surface, heated at distances greater than x 0 from the leading edge, is now considered. As shown in Figure1-7 the velocity boundary layer starts at the leading edge and the thermal boundary layer at a distance x 0 from it. If the temperature of the heated portion of the plate remains constant, this may be taken as the datum temperature. It is assumed that the temperature at a distance y from the surface may be represented by a polynomial of the form:
28 --------------------- (1-45) Figure (1-7) Thermal boundary later- streamline flow the heat transferred per unit area and unit time q o is given by: the heat transfer rate into and out of the element must be the same, or: At the outer edge of the thermal boundary layer, the temperature is θ s and the temperature gradient (dθ/dy) = 0 if there is to be no discontinuity in the temperature profile. if the thickness of the thermal boundary layer is δ t, the temperature distribution is given by: -------------------- (1-46) ---------------- (1-47) It is assumed that the velocity boundary layer is everywhere thicker than the thermal boundary layer, so that δ >δ t. Thus the velocity distribution everywhere within the thermal boundary layer is given by equation1-10.
29 The integral in equation 1-44 clearly has a finite value within the thermal boundary layer, although it is zero outside it. When the expression for the temperature distribution in the boundary layer is inserted, the upper limit of integration must be altered from l to δ t --------------- (1-48) where σ=δ t /δ Since δt<δ, the second term is small compared with the first, and: ----------------- (1-49) Substituting from equations1-47 and 1-49 in equation 1-44 gives: It has already been shown that: ------------------ (1-50)
30 where the Prandtl number Pr=c p µ/k ------ (1-51) If the whole length of the plate is heated, x 0 =0 and: ------------------------ (1-52) In this derivation, it has been assumed that σ < 1. For all liquids other than molten metals, Pr > 1 and hence, from equation 1-52, σ < 1,For gases, Pr < 0.6, so that σ > 1.18.Thus only a small error is introduced when this expression is applied to gases.
31 If h is the heat transfer coefficient, then: --------------------------- (1-53) -------------------------- (1-54) If the surface is heated over its entire length, so that x 0 =0, then: --------------- (1-55) The mean value of (h) between x = 0 and x = x is given by: Hence h m =2h The mean value of the heat transfer coefficient between x = 0 and x = x is equal to twice the point value at x = x. The mean value of the Nusselt group is given by: -------------------- (1-56)
32 II)Heat transfer for streamline flow over a plane surface constant surface heat flux Another important case is where the heat flux, as opposed to the temperature at the surface, is constant; this may occur where the surface is electrically heated. Then, the temperature difference θ s -θ 0 will increase in the direction of flow (x-direction) as the value of the heat transfer coefficient decreases due to the thickening of the thermal boundary layer. The equation for the temperature profile in the boundary layer becomes: ------------------ (1-57) ------------- (1-58) The value of the integral in the energy balance (equation 1-44) is again given by equation 1-49 [substituting (θ s -θ 0 ) for θ s ]. The heat flux q 0 at the surface is now constant, and the right-hand side of equation 1-44 may be expressed as (-q o /ρc p ). Thus, for constant surface heat flux, equation 1-44 becomes: ------- (1-59) Equation 1-59 cannot be integrated directly, however, because the temperature driving force (θ s -θ 0 ) is not known as a function of location x on the plate. The solution of equation 1-59 involves a quite complex procedure and takes the following form: -------------------- (1-60) The average value of the Nusselt group (Nux)m is obtained by integrating over the range x = 0 to x = x, giving: ------------- (1-61)
33 1-4)THE BOUNDARY LAYER FOR MASS TRANSFER If a concentration gradient exists within a fluid flowing over a surface, mass transfer will take place, and the whole of the resistance to transfer can be regarded as lying within a diffusion boundary layer in the vicinity of the surface. For low concentrations of the diffusing component, the effects of bulk flow will be small and the mass balance equation for component A is: --------------------- (1-62) where C A and C AS are the molar concentrations of A at a distance y from the surface and outside the boundary layer respectively, and l is a distance at right angles to the surface which is greater than the thickness of any of the boundary layers. Equation 1-62 is obtained in exactly the same manner as equation 1-44 for heat transfer. Again, the form of the concentration profile in the diffusion boundary layer depends on the conditions which are assumed to exist at the surface and in the fluid stream. For the conditions corresponding to those used in consideration of the thermal boundary layer, that is constant concentrations both in the stream outside the boundary layer and at the surface, the concentration profile is of similar form to that given by equation 1-57: ----------------- (1-63) Where δ D is the thickness of the concentration boundary layer, C A is the concentration of A at y = y, C AO is the concentration of A at the surface (y = 0), and C AS is the concentration of A outside the boundary layer
34 Substituting from equation 1-63 to evaluate the integral in equation 1-62, assuming that mass transfer takes place over the whole length of the surface (x 0 = 0), by analogy with equation 1-52 gives: ------------ (1-64) where Sc =µ/ρd is the Schmidt number. Equation 1-64 is applicable provided that Sc > 1. The point values of the Sherwood number Sh x and mass transfer coefficient h D are then given by: --------------------- (1-65) The mean value of the coefficient between x = 0 and x = x is then given by: ----------------------- (1-66)