Chapter 3 Unidirectional transport - Chemical Engineering ...

More documents

Recommendations

Info

18 CHAPTER 3. UNIDIRECTIONAL TRANSPORTmodification that the plate has an oscillatory velocity U = U 0 cos (ωt). Thedifferential equation for the velocity field is given by equation 3.25,∂u x∂t= ν ∂2 u x∂z 2 (3.72)The boundary conditions in this case, analogous to 3.7, areu x = U cos (ωt) at z = 0u x = 0 at z = L (3.73)The solution procedure is simplified if the boundary condition is recast asfollows. The ‘complex’ velocity field u † x is defined as a velocity field thatsatisfies the same differential equation as 3.72,∂u † x∂tbut which satisfies the boundary conditions= ν ∂2 u † x∂z 2 (3.74)u † x = U exp (ıωt) at z = 0u † x= 0 at z = L (3.75)where ı is the square root of −1. It is apparent that the solution to thedifferential equation 3.74, with the boundary condition 3.75, is the real partof the solution to the differential equation 3.72 with the boundary condition3.73. Since dealing with exponential functions is easier than dealing withsines and cosines, it is more convenient to solve equation 3.74 with boundaryconditions 3.75 for u † x , and then take the real part of the solution to obtainthe solution u x of equation 3.72.The non-dimensional velocity u †∗x is defined as u†∗ x = (u† x /U), while thescaled z and time coordinates are defined as z ∗ = (z/L) and t ∗ = (tν/L 2 ).The momentum conservation equation and boundary conditions in terms ofthis scaled velocity are∂u †∗x∂t = ∂2 u †∗x(3.76)∗ ∂z ∗2u †∗x = exp (ω∗ t ∗ ) at z = 0= 0 at z = L (3.77)u †∗x
3.1. SOLUTIONS OF THE DIFFUSION EQUATION 19z−−>Infinityu*=0~xz+ ∆ zzzz=0u*=1~xux=U cos( ω t)Figure 3.4: Oscillatory flow at a flat surface.xwhere the scaled frequency is given by ω ∗ = (ωL 2 /ν).The differential equation 3.76 for the velocity field is a linear differentialequation, since all terms in the equation contain only the first power of u †∗x .This first order differential equation is driven by a wall which is oscillatorywall velocity with scaled frequency ω ∗ . When a linear system is driven bywall motion of frequency ω ∗ , the response of the system also has the samefrequency ω ∗ . (This is not true if the system is non-linear, since forcing of acertain frequency will generate response at different harmonics of this basefrequency). Therefore, the time dependence of the velocity field in the fluidcan be considered to be of the formu †∗x = ũ∗ x (z∗ ) exp (ıω ∗ t ∗ ) (3.78)When this form is inserted into the differential equation 3.76, and divided byexp (ıω ∗ t ∗ ), the resulting equation is an ordinary differential equation for ũ ∗ x .The boundary conditions for ũ ∗ x (3.77) becomeıω ∗ ũ ∗ x = ∂2 ũ ∗ x∂z ∗2 (3.79)ũ ∗ x = 1 at z = 0
Page 1 and 2: Chapter 3Unidirectional transportIn
Page 3 and 4: 3.1. SOLUTIONS OF THE DIFFUSION EQU
Page 14 and 15: 14 CHAPTER 3. UNIDIRECTIONAL TRANSP

Chapter 3 Unidirectional transport - Chemical Engineering ...

Create successful ePaper yourself

Delete template?

Save as template?