TAMU CHEN 304 - L2 (6 pages)

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Texas A&M University
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Chen 304 - Chem Engr Fluid Ops
Chem Engr Fluid Ops Documents
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CBE 341 Lecture 2 9 15 Surface tension Molecules in a liquid feel a mutual attraction and thus have a natural tendency to minimize their surface area The obvious case is a liquid drop on a non wetting horizontal surface see figure below Small droplets become almost perfectly spherical minimizes surface area per volume However larger droplets become flatter as the gravitational forces acting on the fluid become more significant The fluid property that describes this phenomenon is called surface tension The surface tension at an interface between two immiscible fluids is a material property which has the units of energy per unit area or force per unit length Note s and g are the most commonly used symbols to denote surface tension Surface tension is also referred to as interfacial tension It is the energy required to create unit interfacial area which is a definition often used in thermodynamics The latter definition namely force per unit length is used in mechanics Because surface tension gives rise to forces at the interface it affects force balance at the interface To illustrate in a simple manner that these definitions are equivalent consider a planar interface of length L and width W see figure below Let us assume that the interface is homogeneous and no spatial variation of properties occurs We will begin by applying a mechanical definition for the surface tension The forces acting to maintain the interface are shown Now if we stretch this interface by moving the right side by a distance Dx then the work done gLex Dxex gLDx The amount of new interface created LDx The energy required to create unit interfacial area is given by Work done area created gDx L LDx g Thus we have recovered the definition of surface tension based on surface energy 13 gWey y x gLex gLex Blue region is the new area created gWey In this course I will not go into formal derivation of force balances at fluid fluid interfaces however I will simply state without proof some main results which are useful to know Even though we considered a planar interface in the previous page fluid fluid interfaces need not be planar instead they can assume complex three dimensional structures cylinders and spheres being the simplest examples In the presence of such a curved interface the pressure in the fluid on either side of the interface will differ from each other even in an equilibrium state Consider the surface in the above figure Now if the surface is planar flat the forces acting on either side of the interface are equivalent and DP 0 However if DP 0 the surface will be curved and the force bending the surface will be balanced by the surface tension Consider the liquid droplet in the figure below A force balance on half of the droplet yields Pi p r 2 Po p r 2 g 2p r where Pi is the pressure inside the droplet Po is the pressure outside the droplet and r is the radius 14 After rearrangement DP 2g r Note that in this derivation we sliced the droplet in half and performed a force balance on the control volume This control volume was selected for easy calculation of the forces however a smaller control volume which represents less than half of the spherical droplet could also be used figure below q r1 r q Half a Spherical Droplet Smaller Fraction of a Spherical Droplet Fg 2prg DPpr2 Fg 2pr1gcosq DPpr12 2g DP r DP 2gcosq r1 Performing the same procedure used to calculate DP for the half droplet we get DP 2g cos q r1 We recognize that r DP r1 with substitution we obtain the same expression cos q 2g r 15 This also shows that the pressure throughout a spherical droplet is homogenous Note that if the pressure is not homogenous within a droplet the droplet will warp away from a spherical geometry Analogously how do we arrive at this expression from the thermodynamic definition of surface tension Recall that the interfacial energy is given by Ei 4p r 2g This is the amount of work done to give the droplet its spherical shape Given that work dW Pg dV note that here Pg is the gauge pressure and Pg DP thus dW dEi Assuming the surface tension is constant dEi 8p rg dr Since for a spherical element dV 4p r 2 dr we obtain the following expression for P Pg DP dW dEi 8p rg dr 2g dV dV 4p r 2 dr r This is the same expression derived using a mechanistic approach Now for any surface there is a pair of orthogonal curves at any point on the surface and if the curves are of differential dimensions each curve may be approximated as the arc of a circle The radii of the two circles are called the principal radii of curvature R1 and R2 The curvature of a surface at any point is described in terms of R1 and R2 of the orthogonal curves corresponding to that point On a complex surface the radii may vary from point to point If the centers of the two circular arcs lie on the same side of the surface then both R1 and R2 assume the same sign If they lie on opposite sides then one of them is positive and the other is negative 16 To decide when a positive sign should be assigned and when a negative sign should be used we choose arbitrarily one side of the interface as inner side and the other as outer side Let the pressure in the inner fluid at the point of interest i e right next to the interface be denoted as Pinner Similarly let the pressure in the outer fluid at the point of interest be Pouter We define as capillary pressure Pc the difference between the two pressures PC DP Pinner Pouter The correct sign to assign for the radius of curvature depends on which side of the interface the center of the circular arcs lie If the center of the circular arc lies on the side of the inner fluid then the corresponding radius gets a positive sign If it lies on the side of the outer fluid it gets a negative sign Thus in the figure above if the inner fluid is chosen as that lying above the interface the radii R1 and R2 are both positive if you had chosen the fluid above the interface as outer fluid then both R1 and R2 are assigned negative signs The Young Laplace equation describing the relationship between the capillary pressure surface tension and the radii of curvature is 1 1 DP g R1 R2 17 7 One can then see that the pressure inside a spherical drop of radius R will exceed that immediately outside by a value of 2g R as R1 R2 R If we imagine a cylindrical fluid filament of radius R the pressure inside the filament will exceed that immediately outside by a value of g R as R1 R and R2 Finally if we have a planar interface the pressure will be the same on either


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