Question 2 Consider the two-dimensional laminar flow of a Newtonian, incompressible fluid of density p and viscosity u o

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Question 2 Consider The Two Dimensional Laminar Flow Of A Newtonian Incompressible Fluid Of Density P And Viscosity U O 1 (245.03 KiB) Viewed 52 times

Question 2 Consider The Two Dimensional Laminar Flow Of A Newtonian Incompressible Fluid Of Density P And Viscosity U O 2 (273 KiB) Viewed 52 times

Question 2 Consider the two-dimensional laminar flow of a Newtonian, incompressible fluid of density p and viscosity u over a horizontal flat plate. v, e, is the fluid approach velocity, ez being a unit vector in the flow direction, L is the plate length, and 8, is the thickness of the velocity boundary layer. The fluid contains a dilute solute A, whose concentration far from the plate is Cao and on the plate is cal: The diffusivity of A in the fluid is a constant equal to DA. Close to the plate, there also exists a concentration boundary layer (i.e., a thin region where the concentration of A changes) of thickness &c. y CA = CAO Concentration Boundary Layer v = Voex CA = CAO Velocity Boundary Layer v = Voex 8, CA = CA1 L Figure 2: Developments of the velocity and concentration boundary layers for the flow along a horizontal flat plate, for A = 8/8, > 1. The concentration of A at the surface of the plate is can, whilst the concentration of A in the approaching fluid is co- V = Voex Velocity Boundary Layer v = Voex 5. CA = CAO. Concentration Boundary Layer CA = CAO 8 CA = CA1 x Figure 3: Developments of the velocity and concentration boundary layers for the flow along a horizontal flat plate, for A = 8/8, < 1. The concentration of A at the surface of the plate is cal, whilst the concentration of A in the approaching fluid is CAO In general, 8 differs from 8,- for some fluids this difference being of various orders of magnitude. Figs. 2 and 3 picture the cases in which A= 8/8, is far larger and far smaller than unity, respectively. Here, we will focus on these limiting cases. Note. In reality, both boundary layers (the velocity boundary layer and the concentration boundary layer) are very thin, their thicknesses being far smaller than L. Thus, Figs. 2 and 3 are inaccurate, for they grossly exaggerate the boundary layer thicknesses. However, this was done on purpose to emphasize the difference in thickness between the two boundary layers. This difference plays a crucial role in the problem that we are addressing in this exercise, so this is what the figures stress.

Answer all the questions in the reported order. a) Starting from the general mass balance equation for component A in Eulerian form (please, refer to the literature), show that the equivalent equation in Lagrangian form reads: DiCA = DAV.VCA (2.1) where Dě represents the substantial (or material, or Lagrangian) derivative operator, and ca is the molar concentration of A. Report clearly all the passages. b) Write Eq. 2.1 in steady-state conditions using the Cartesian coordinates x and y shown in Figs. 2 and 3. Then, scale the equation using these variables: СА - CA1 CA= CAO CA1 V. ; U = Urs Uy ; y = Vys 2 c= L y ; y = (2.2) where for now the velocity scales are undefined. Finally, with order of magnitude arguments, show that the scaled mass balance equation can be simplified as follows: CA д°CA (°4,50 = Dot acA uy ay + az 2 L (2.3) DA ay? c) Explain why, in Eq. 2.1, L and 8 are the appropriate length scales in the r and y directions, respectively. In particular, explain why we cannot used, as length scale in the y direction. [4/50] d) Consider the case where A > 1. Using the definition of velocity scale and Fig. 2, estimate the values of V.3,3 and Vy,s. Remember that the values of the (velocity) scales depend on the flow region on which you are focusing. After, using Eq. 2.3 and the expression for 8, holding for the velocity boundary layer, with order of magnitude arguments prove that: 8c L 1 Sc Rei 1 and Δω Sc with €= 1/2 and w= 1/2 (2.4) where: pvol Sc= PDA (2.5) and Re = e) Now consider the case where A <1. Using the definition of velocity scale and Fig. 3, estimate the value of Vz, that is appropriate for the concentration boundary layer. Then, using the continuity equation, find the corresponding value of vy,s. Finally, with Eq. 2.3 and the expression for 8y, show that the functional forms in Eq. 2.4 still hold and obtain the values of e and w. f) The results obtained in parts d) and e) are valid for any value of L (provided that Re, > 1). Hence, for a generic value of 2, we can write: 8(x) 1 puo with Re = (2.6) Sc Remote р Using this result, estimate JA(0,0), that is, the molar flux of A on the plate at a generic distance x from its leading edge. Then, estimate the molar flow rate of A over the entire plate. Finally, using this result, prove that for the mean) mass transfer coefficient: (ka~(2) (2)se Rej Sc (2.7) g) Based on the results obtained in part d), do you think that a concentration boundary layer can be present even in flows where a velocity boundary layer is absent? Justify your answer.