TAMU CHEN 304 - L1 (12 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 1 9 13 Concept of a fluid fluid accelerating as a rigid body buoyancy Definitions and concepts Elastic solid When an elastic solid is subjected to a small shear force it will deform slightly and assume a new static configuration The shear strain is defined as tan g g for small values of g Fn Ft Ft Fn If A is the area over which the normal and shear forces Fn and Ft respectively are acting then Normal stress Fn A Shear stress t Ft A The shear modulus is defined as Shear modulus Shear stress shear strain t g For a Hookean solid the shear modulus is constant independent of the magnitude of the shear strain or equivalently shear stress 1 Viscous fluid When a fluid element is subjected to the same set of forces as shown above the fluid element will also deform however it will not assume a static configuration at any particular value of strain g Instead the element will continue to deform indefinitely as long as the shear stress is acting on the fluid Hence it is not the deformation g but the dg rate of deformation g dt that is a measure of how a fluid responds to applied shear stress g is called the shear rate or the strain rate or the rate of deformation A typical velocity profile in the fluid element is sketched below The viscosity is defined as Shear stress shear rate t g For Newtonian fluids the viscosity is a constant independent of the shear stress or equivalently the shear rate In this course we will be mostly concerned with Newtonian fluids but will touch upon non Newtonian fluids briefly All gases and common liquids such as water liquid hydrocarbons alcohols glycerol and honey fall in the category of Newtonian fluids In chemical industry we also encounter a large variety of non Newtonian fluids Paints colloidal suspensions and polymer melts are common examples It suffices to say at the present time that for some fluids referred to as shear thinning fluids the viscosity decreases with increasing shear rate while for others it increases with shear rate shear thickening fluids 2 Units and dimensions M L and t denote units of mass length and time Shear or normal stress force area F L2 units M L t2 Shear strain rate units 1 t Viscosity common symbols or h F t L2 units M L t Note that viscosity has the same units as rV where r is the density V is velocity and denotes a length Kinematic viscosity n r units L2 t Note that kinematic viscosity has the same units of V SI Units of viscosity 1 N s m2 1 Pa s 1 pascal second CGS units of viscosity 1 dyne s cm2 1 Poise 1 Pa s 10 Poise Typical values to remember Viscosity of water 1 centipoise CP 10 2 Poise 1 mPa s mPa millipascal Viscosity of air 10 2 CP 10 5 Pa s Kinematic viscosity of gases In low density gases where viscous transport of momentum occurs through the Brownian motion of the molecules the kinematic viscosity scales as n v l where v is the random velocity of the molecules and l is the mean free path between collisions As molecules can execute random motion in three directions only a third of the motion will be in any one direction so a somewhat better estimate is n 1 v l A 3 more accurate prefactor has been derived for low density gases but a discussion of this derivation is not needed for this course The random also known as the Brownian velocity v scales as T 1 2 where T is the absolute temperature while l scales as T P where P is the pressure Thus the 3 kinematic viscosity of low density gases scale roughly as T 3 2 P Note that the density scales as P T so that the viscosity is independent of pressure and it varies as T 1 2 only Experiments by Maxwell confirmed that the viscosity of gases is indeed essentially independent of pressure over a wide range of pressure except when the pressure is so large that non ideal behavior sets in or when the pressure is so low that the mean free path becomes comparable to the dimensions of the container in which flow occurs The temperature dependence of viscosity of gases and liquids are distinctly different while it increases with temperature for gases it decreases with increasing temperature for liquids see below 4 Control volume Control surface Traction In the study of mechanics we will frequently consider a volume of fluid or solid or fluid solid mixture V enclosed by a closed surface and apply balance laws This volume V may be entirely arbitrary which is the case when we derive general balance equations or naturally suggested by the specific problem that we are trying to solve Such a volume is commonly referred to as Control Volume and denoted by V or C V The closed surface bounding this volume of material is commonly referred to as Control Surface and denoted by S or C S If no mass transfer occurs across the control surface the material bounded by the control volume is commonly referred to as a closed system in thermodynamics courses Heat transfer is permitted in closed systems and the system was allowed to do work on the surrounding the First law of Thermodynamics applied to closed systems is simply a relationship between the change in the internal energy of the system amount of heat transferred and the amount of work done In open systems mass exchange between the system and the surrounding is permitted The earlier courses extended the first law of thermodynamics to open systems as well Please review your notes from CBE 245 and CBE 246 to refresh your memory of the first law In the context of fluid mechanics we will consider both closed and open systems We will be concerned with forces acting on the material inside the control volume some of which such as the force due to gravity act directly on the material inside henceforth referred to as body force force is also imparted on the material inside the control volume through the control surface henceforth referred to as the surface force 5 Traction refers to force per unit area acting on the surface Consider a differential surface element dS as shown on the figure below The vector n in this figure denotes unit outward normal to the control surface at this surface element dS Let the total force s n dS V S exerted by the surrounding on the material inside the control volume through this surface dF dS 0 dS s lim element be dF The traction s is defined as In general s will have components both along the normal direction and in the tangent plane Normal component s n n Tangential component s s n n Frame of reference In mechanics we observe objects from different perspectives both to better understand and explain the manner


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