Module 1 : The equation of continuity. Lecture 1: Equation of Continuity
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1 1 Module 1 : The equaton of contnuty Lecture 1: Equaton of Contnuty
2 2 Advanced Heat and Mass Transfer: Modules 1. THE EQUATION OF CONTINUITY : Lectures 1-6 () () () (v) (v) Overall Mass Balance Momentum Balance Energy Balance Specal Mass Balance Equaton for the fluxes 2. DIFFUSIE HEAT AND MASS TRANSFER: Lectures 7-20 () () () (v) (v) Steady and Unsteady/One and Multple Dmensons Mass Transfer wth Chemcal Reacton Perturbaton Technques Movng Boundary Problems Smultaneous Heat and Mass Transfer 3. CONECTIE HEAT AND MASS TRANSFER: Lectures () () () (v) Flow Insde Ducts Dsperson Lamnar Boundary Layers Mass Transfer wth Chemcal Reactons
3 3 (v) (v) Asymptotc Methods Smultaneous Momentum, Heat and Mass Transfer (v) Natural Convecton 4. MULTICOMPONENT TRANSPORT: Lectures () () () (v) Bnary Systems Mut-component Flux Equatons Thermal Dffuson Dmensonal Analyss 5. MASS TRANSFER IN TURBULANT FLOWS: Lectures () () () Tme Averagng and Eddy scosty Unversal elocty Mass Transfer n Turbulent Ppe Flow Reference Books 1. Brd, R.B., Stewart, W.E. and Lghtfoot, E.N., Transport Phenomenon, Wley (1960). 2. Carslaw, H.S. and Jaeger, J.C., Conducton of heat n Solds, (2 nd ed) Oxford (1975). 3. Slattery, J., Momentum, Energy and Mass Transfer n Contnua, (2 nd ed) Krueger (1981). Transport Processes
4 4 Goals of the Course To relate mathematcal symbols to physcal realty To revew several classc problems To show examples of how to approach the unknown THE EQUATION OF CONTINUITY The Contnuum Approxmaton Feld varables (e.g. velocty) at a pont are spatal averages over a small volume around that pont, where has to be such that l << 1/3 << D, (1.1a) where, l s a characterstc mcroscopc length scale, whch can be of molecular dmensons or the dstance between molecules n a gas or the partcle sze n a sold/flud two-phase system, and D s a characterstc macroscopc length scale. For example, for flow n a ppe, D can be the ppe dameter. The contnuum approxmaton consders the fluds to be contnuous. Thus, the flud propertes such as temperature, pressure, densty and velocty of the flud are taken to be well defned at nfntely small ponts (.e. at mcroscopc level), defnng a reference element of volume, let s call ths volume RE, at the geometrc order of the dstance between the two adjacent molecules of flud. Propertes are assumed to vary contnuously from one pont to another, and are averaged over the volume RE. The fact that the flud s made up of dscrete molecules s gnored.
5 5 Euleran and Lagrangan coordnates Euleran Coordnate: n ths system the ndependent varables are x, y, z and t or x (=1, 2, 3) and t. Ths s a fxed coordnate system. The basc conservaton equaton are n the Euleran frame, R = R (x, t). In the Lagrangan frame, attenton s fxed on a partcular mass of flud as t flows, R = R (x o, t), where the coordnate x o specfes whch flud element s beng consdered. Materal Dervatve Consder a varable α such that α = α ( x, t) (1.1b) Then the total dfferental of α can be expressed as δα = δt + δx (1.1c) Dvson by a tme dfferental δt leads to the followng expresson: δα δt δx = + (1.1d) δt After takng the lmt δt 0, we obtan for the materal dervatve Dα Dt = + ν (1.1e) where, ν = s the flud velocty n drecton, n the lmt of δt 0 Dα s called the Materal Dervatve or Lagrangan Dervatve n tme and Dt Dervatve n tme. s the Euleran
6 6 The Materal dervatve or Lagrangan tme dervatve represents n total change n α as seen by an observer who s movng wth a partcular flud element. In the Lagrangan frame, we observe the partcle for a tme δt as t flows. The poston of a partcle changes by δx whle α changes as δα. Tme dervatves The tme dervatve s a dervatve of a functon wth respect to tme. It mples the rate of change of value of a functon wth respect to tme t. Partal tme dervatve Total tme dervatve (at a pont) By dvdng by dt, the total dfferental can be wrtten as total tme dervatve dc dt dx dy dc dt y dt z dt = (1.1f) Ths expresson represents the change n the tme of the functon c.e., (/) as we move about wth arbtrary veloctes n the coordnate drectons.e., (dx/dt, dy/dt and dz/dt). Substantal Dervatve or Materal Dervatve If we constran the moton to follow the moton of the ndvdual flud partcles, we obtan the Substantal Dervatve or Materal Dervatve (also known as Convectve Dervatve) gven by Dc Substantal tme Dervatveor Convectve Dervatve: = + U dt x + U y + U y z z (1.1g)
7 7 where U x, U y and U z are the components of the local flud velocty U n x, y and z drectons, respectvely. The Mass Contnuty Equaton The contnuty equaton s an overall mass balance about a control volume. Consder a volume element of volume fxed n space as shown n fgure below. Here the volume s bounded by a surface S wth outward unt normal vector n Fg. 1.1 Control olume for Mass Contnuty equaton ( Accumulat on of mass nsde ) = ( The Net nflux of mass through surface S) v ρ d = ρu n ds s (1.2) Here, ρ s the mass densty. The mnus sgn (-) n front of the ntegral s because of the choce of n pontng outwards. The Dvergence Theorem (Gauss) for a vector feld A gves
8 8 ( ) d = ( A n)ds S Fg.1.2. Gauss Dvergence theorem A (1.3a) Where s gradent (a vector) and can be expressed as = + j + k (1.3b) y z In equaton (1.3a), the left hand sde s the volume ntegral over the volume, the rght hand sde s the surface ntegral over the boundary of the volume. The closed manfold d s qute generally the boundary of orented by outward-pontng normals and n s the outward pontng unt normal feld of the boundary d. For A = ρu, equaton (1.2) becomes ( U ) d = ( ρu n)ds ρ (1.3c) S Substtute equaton (1.3c) on the rght hand sde of equaton (1.2) we get, ρ d = ( ρu )d (1.3d) or
9 9 ρ + U (1.3e) ( ρ ) d = 0 Equaton (1.3e) s known as Mass Contnuty equaton. Snce ths equaton must hold for arbtrary, Mass Contnuty Equaton becomes ρ + or ( ρ ) = 0 U (1.4a) Dρ + U Dt ρ( ) = 0 (1.4b) From equaton (1.1g), we know that Dρ Dt ρ = + U ( ρ) (1.4c) Incompressble fluds: Incompressble fluds are those fluds that do not exhbt any varaton n ρ densty ether n space or tme. Therefore for ncompressble fluds ρ = 0 and = 0. If ρ s constant (for ncompressble fluds) n space and tme, then the equaton of contnuty for ncompressble fluds becomes U =0 (1.4d) Equaton (1.4b) can also be wrtten as: 1 D Dt = ( U ) (1.4e) where, 1 = and ρ 1 D Dt s the rate of dlaton of the flud.
10 10 Applcaton: We can apply the prncple of contnuty to ppes wth cross sectons whch changes along ther length. See Fg 1.3 below. Fg.1.3. flud flowng through convergent-dvergent secton of the ppe A lqud s flowng from left to rght and the ppe s narrowng n the same drecton. By the contnuty prncple, the mass flow rate must be the same at each secton.
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