Scale Analysis of Internal Natural Convection Dr. Om

Scale Analysis of Internal Natural Convection Dr. Om

Scale Analysis of Internal Natural Convection Dr. Om Prakash Singh Asst. Prof., IIT Mandi www.omprakashsingh.com Boundary layer thickens

Boundary layer thickens Two-dimensional rectangular enclosure with isothermal sidewalls. Governing equations The equations governing the conservation of mass, momentum, and energy at every point in the cavity are

Note that in writing these equations, we modeled the fluid as Boussinesqincompressible, in other words, = constant everywhere except in the body force term of the y momentum equation, where it is replaced by [1(TT0)].TT0)].)]. Scale analysis of governing equations Immediately after t = 0)]., the fluid bordering each sidewall is motionless: This means that near the sidewall, the energy equation (TT0)].5.4) expresses a balance between thermal inertia and conduction normal to the wall,

This equality of scales follows from recognizing T, t, and T as the scales of changes in T, t, and x in eq. (TT0)].5.4) and velocity has not yet developed (TT0)].u=v=0)].) Hence, The layer T rises along the heated wall. Scale analysis of governing equations

The velocity scale of this motion v is easier to see if we first eliminate the pressure P between the two momentum equations (TT0)].5.2) and (TT0)].5.3): This new equation contains three basic groups of terms: inertia terms on the lefthand side and four viscous diffusion terms plus the buoyancy term on the righthand side. It is easy to show that the three terms that dominate each basic group are Scale analysis of governing equations In terms of representative scales, the momentum balance (TT0)].5.8) reads

The driving force in this balance is the buoyancy effect (TT0)].g T)/T, which is not zero. T, which is not zero. It is important to determine whether the buoyancy effect is balanced by friction or inertia. Dividing eq. (TT0)].5.9) through the friction scale and recalling that 2T t yields t yieldst yields Therefore, for fluids with Prandtl number of order 1 or greater, the correct momentum balance at t = 0)]. + is between buoyancy and friction,

Scale analysis of governing equations We conclude that the initial vertical velocity scale is This velocity scale is valid for fluids such as water and oils (TT0)].Pr > 1) and is marginally valid for gases (TT0)].Pr t yields 1). The heat conducted from the sidewall into the fluid layer T is no longer spent solely on thickening the layer: Part of this heat input is carried away by the layer T rising with velocity v. Thus, in the energy equation, we see a competition among three distinct

effects: Scale analysis of governing equations As t increases, the convection effect increases [v t yields t, eq. (TT0)].5.11)], while the effect of inertia decreases in importance. There comes a time tf when the energy equation expresses a balance between the heat conducted from the wall and the enthalpy carried away vertically by the buoyant layer,

At such a time, the layer thickness is where RaH is the Rayleigh number based on the enclosure height, Scale analysis of governing equations In addition to thermal layers of thickness T, f, the sidewalls develop viscous (TT0)].velocity) wall jets. The thickness of these jets v from the momentum balance (TT0)].5.7) for the region of thickness x t yields v outside the thermal layer. In this region, the buoyancy

effect is minor, and we have a balance between inertia and viscous diffusion, Hence, In the steady state, t > tf, the fluid near each sidewall is characterized by a twolayer structure: 1. a thermal boundary layer of thickness T, f and 2. a thicker wall jet v,f t yields Pr1/T, which is not zero. 2T,f. The development of this structure is shown in Fig. 5.2.

Fig. 5.2. Development of two-layer structure near the warm wall. End

Recently Viewed Presentations

  • John 14:7-11 I. The Challenge: Jesus prove you

    John 14:7-11 I. The Challenge: Jesus prove you

    John 20:30-31 (NKJV) "Jesus did many other miraculous signs in the presence of his disciples, which are not recorded in this book. But these are written that you may believe that Jesus is the Christ, the Son of God, and...
  • CAUTIONS AND LIMITATIONS  Not for use in  Failure

    CAUTIONS AND LIMITATIONS Not for use in Failure

    DESCRIPTION. For one-time escape use only - The W65 Self-Rescuer is a one-time device and must be discarded after use. Never use the respirator for other than escape through areas containing carbon monoxide in otherwise breathable air; do not use...
  • Protecting Yourself from Fraud including Identity Theft

    Protecting Yourself from Fraud including Identity Theft

    Prevent business practices that are anticompetitive, deceptive or unfair to consumers. Federal Trade Commission. Consumer Financial Protection Bureau. Federal Drug Administration. Federal Communications Commission. Most Common Government Protection Agencies .
  • Podmínky pro absolvování

    Podmínky pro absolvování

    Mikrobiologické a příbuzné vědy Stručná historie mikrobiologie 2.st. p.n.l. - Lucretius: nemoci jsou způsobeny neviditelnými organismy 1665 - Robert Hooke, kniha „Micrographia" - buňka 1673 - Antony van Leeuwenhoek sestrojil mikroskop a poprvé pozoroval bakterie Ferdinand Cohn (1828-1898) - klasifikace...
  • When: Tuesday, January 23rd at 6:30 PM -7:30

    When: Tuesday, January 23rd at 6:30 PM -7:30

    Where: Carver Auditorium, 850 Hungerford Drive What: Join us for a panel discussion with various perspectives from across the state and county level on the Kirwan Commission and the Operating Budget Our Panelists Ms. Nancy King, State Senator Mr. Craig...
  • Chapter Six, Section Two

    Chapter Six, Section Two

    Sources of LawsStatutory Law. Statutory . Law - laws from statutes. Any law that is passed by a legislative body is called a . statute, or statutory . law. Statutory . law can be passed by Congress, state legislatures or...
  • Probationary Constable Workplace Assessment Programme Field ...

    Probationary Constable Workplace Assessment Programme Field ...

    Workplace Assessment. Read the programme. After a couple of months discuss programme with your FTO / Supervisor. Pick a standard and work at it to become competent and collect necessary evidence
  • Harmony Search - kocseaa.org

    Harmony Search - kocseaa.org

    Types of Optimization Algorithms Existing Nature-Inspired Algorithms Existing Meta-Heuristic Algorithms Algorithm from Music Phenomenon Algorithm from Jazz Improvisation Analogy Comparison Factors Good Harmony & Bad Harmony Procedures of Harmony Search HS Operators Random Playing Memory Considering Pitch Adjusting Ensemble Considering...