Chapter 6 Fundamental Concepts of Convection · 2012. 4. 24. · 6.1.4 Significance of the Boundary...

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Chapter 6 Fundamental Concepts of Convection

Transcript of Chapter 6 Fundamental Concepts of Convection · 2012. 4. 24. · 6.1.4 Significance of the Boundary...

  • Chapter 6

    Fundamental Concepts of Convection

  • 6.1 The Convection Boundary LayersVelocity boundary layer:surface shear stress: local friction coeff.:

    Thermal boundary layer:local heat flux:

    local heat transfer coeff.:

    τs = μ ∂u∂y y=0Cf ≡

    τsρu∞

    2 / 2

    ′ ′ q s = −kf ∂T∂y y=0 "0

    /f yss s

    k T yqhT T T T

    =

    ∞ ∞

    − ∂ ∂= =

    − −

    (6.2)

    (6.1)

    (6.3)

    (6.5)

  • Concentration boundary layer:local species flux:

    or (mass basis):

    local mass transfer coeff.:

    or (mass basis):

    "0,

    , , , ,

    /AB A yA sm

    A s A A s A

    D C yNh

    C C C C=

    ∞ ∞

    − ∂ ∂= =

    − −

    hm=−DAB ∂ρA / ∂y y=0

    ρA,s − ρA,∞

    (6.6)

    (6.9)

    " 2

    0

    [kmol/s m ]AA ABy

    CN Dy =

    ∂= − ⋅

    " 2

    0

    [kg/s m ]AA ABy

    n Dy

    ρ

    =

    ∂= − ⋅

  • 6.1.4 Significance of the Boundary LayersVelocity boundary layer: always exists for flow over any surfaceThermal boundary layer: exists if the surface and free stream temperature differConcentration boundary layer: exists if the surface concentration of a species differs from the free stream value

    The principal manifestations and boundary layer parameters areVelocity boundary layer: surface friction and friction coefficient CfThermal boundary layer: convection heat transfer and heat transfer convection coefficient hConcentration boundary layer: convection mass transfer and mass transfer convection coefficient hm

  • 6.2 Local and Average Convection Coefficients6.2.1 Heat TransferThe local heat flux q” may be expressed as (Newton's law of cooling)

    q”= h(Ts-T∞), h = heat transfer convection coeff.= f (fluid properties, surface geometry, flow conditions)

    The total heat transfer rate q may be obtained by integration, if Ts is uniform

    where .

    ( ),s shA T T∞= −

    ( )s s

    s s sA Aq q dA T T hdA∞′′= = −∫ ∫

    1s

    sAs

    h hdAA

    = ∫

    (6.10-12)

    (6.13)

  • 6.2.2 Mass TransferSimilar for mass transfer of species A, we have

    --local--average

    orwhere

    sm m sA

    h h dA= ∫--Mass transfer rate

    --Molar transfer rate

    " 2A A, A,

    A A, A,

    ( ) [kg/s m ]

    ( ) [kg/s]m s

    m s s

    n h

    n h A

    ρ ρ

    ρ ρ∞

    = − ⋅

    = −

    (6.15)

    (6.18)

    (6.16)

    (6.19)

    " 2A A, A,

    A A, A,

    ( ) [kmol/s m ]

    ( ) [kmol/s]m s

    m s s

    N h C C

    N h A C C∞

    = − ⋅

    = −

  • We can also write Fick’s law on a mass basis by multiplying Eq. 6.7 by MA to yield

    The value of CA,s or ρA,s can be determined by assuming thermodynamic equilibrium at the interface between the gas and the liquid or solid surface. Thus,

    " AA AB A, A,

    0

    AB A 0

    A, A,

    ( )

    /

    m sy

    ym

    s

    n D hy

    D yh

    ρ ρ ρ

    ρ

    ρ ρ

    ∞=

    =

    ∂= − = −

    − ∂ ∂=

    satA,

    ( )ss

    p TCT

    =R

    (6.20)

    (6.21)

    (6.22)

  • 6.2.3 The Problem of ConvectionThe local flux and/or the total transfer rate are of

    paramount importance in any convection problem. So, determination of these coefficients (local h or hm and average ) is viewed as the problem of convection.

    However, the problem is not a simple one, as

    EXs 6.1-6.3

    or mh h

    ,

    or (fluid properties, surface geometry, flow conditions)

    i.e., , , ,

    m

    f p f

    h h f

    k cρ μ

    =

  • 6.3 Laminar and Turbulent FlowCritical Reynolds no. where transition from laminar boundary layer

    to turbulent occurs:

    EX 6.4

    Rex,c =ρu∞xc

    μ = 5 ×105

  • 6.4 The Boundary Layer Equations (2-D, Steady)

    6.4.1 Boundary Layer Equations for Laminar FlowFor the steady, two-dimensional flow of an incompressible

    fluid with constant properties, the following equations are involved. (Read 6S.1 for detailed derivation.)continuity eq.: 0=

    ∂∂

    +∂∂

    yv

    xu

    (D.1)

  • momentum eq. (incompressible):

    energy equation:

    species equations:

    Xyu

    xu

    xp

    yuv

    xuu +⎟⎟

    ⎞⎜⎜⎝

    ∂∂

    +∂∂

    +∂∂

    −=⎟⎟⎠

    ⎞⎜⎜⎝

    ⎛∂∂

    +∂∂

    2

    2

    2

    2μρ

    Yy

    vx

    vyp

    yvv

    xvu +⎟⎟

    ⎞⎜⎜⎝

    ∂∂

    +∂∂

    +∂∂

    −=⎟⎟⎠

    ⎞⎜⎜⎝

    ⎛∂∂

    +∂∂

    2

    2

    2

    2μρy-dir.

    x-dir.

    A2

    2

    2

    2

    AB NyC

    xCD

    yCv

    xCu AAAA &+⎟⎟

    ⎞⎜⎜⎝

    ∂∂

    +∂

    ∂=

    ∂∂

    +∂

    ∂ (D.6)

    (D.2)

    (D.3)

    qyT

    xTk

    yTv

    xTucp &+Φ+⎟⎟

    ⎞⎜⎜⎝

    ∂∂

    +∂∂

    =⎟⎟⎠

    ⎞⎜⎜⎝

    ⎛∂∂

    +∂∂ μρ 2

    2

    2

    2

    2 22

    where 2u v u vy x x y

    μ μ⎧ ⎫⎡ ⎤⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂⎪ ⎪⎛ ⎞Φ = + + +⎢ ⎥⎨ ⎬⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠⎢ ⎥⎝ ⎠ ⎝ ⎠⎪ ⎪⎣ ⎦⎩ ⎭

    (D.4)

    (D.5)

    energy transport thru convection

    heat transport thru conduction

    heat generation

    viscous dissipation

  • Boundary layer approximations:

    Usual additional simplifications: incompressible, constant properties, negligible body forces, nonreacting, no energy generation

    Thermal boundary layerT Ty x

    ⎤∂ ∂>> ⎥∂ ∂ ⎦

    Velocity boundary layer, ,

    u vu u v vy x y x

    >> ⎤⎥∂ ∂ ∂ ∂ ⎥>>

    ∂ ∂ ∂ ∂ ⎥⎦

    Concentration boundary layerA AC Cy x

    ⎤∂ ∂>> ⎥∂ ∂ ⎦

  • Eqs. 6.27 is unchanged and the x-mom equation reduces to

    The energy equation reduces to

    and the species equation becomes

    For incompressible, constant property flow, Eq. (6.28) is uncoupled from (6.29) & (6.30). Eqs. (6.27) & (6.28) need to be solved first, for the velocity field u(x, y) and v(x, y), before (6.29) & (6.30) can be solved, i.e., the temperature and species fields are coupled to the velocity field.

    2

    2

    AByCD

    yCv

    xCu AAA

    ∂=

    ∂∂

    +∂

    ∂(6.30)

    (6.29)2

    2

    2

    ⎟⎟⎠

    ⎞⎜⎜⎝

    ⎛∂∂

    +∂

    ∂=

    ∂∂

    +∂∂

    yu

    cyT

    yTv

    xTu

    p

    ναusually negligible

    energy transport thru convection

    heat transport thru conduction

    2

    2

    1 dpu u uu vx y dx y

    νρ

    ∞∂ ∂ ∂+ = − +∂ ∂ ∂ (6.28)

  • 6.5 Boundary Layer Similarity: The Normalized Convection Transfer Equations

    6.5.1 Boundary Layer Similarity ParametersDefining and L is the characteristic length

    and V is the free stream velocity

    and

    we arrive at the convection transfer equations and their boundary conditions in nondimensional form, as shown in Table 6.1.

    Dimensionless groups:Reynolds no.: Prandtl no.:Schmidt no.:

    LVLRe v≡

    x* ≡ xL y* ≡yL

    u* ≡ uV v* ≡vV

    T* ≡T − TsT∞ − Ts

    CA * ≡CA − CA,s

    CA,∞ − CA,s

    vPr α≡

    AB

    vSc D≡

  • The final dimensionless governing eqs.: (6.35)-(6.37)--similar in form∂u *∂x *

    +∂v *∂y *

    = 0

    u* ∂u *∂x*

    + v* ∂u*∂y*

    = −dp*dx *

    + 1Re L∂ 2u *∂y*2

    u* ∂T *∂x *

    + v* ∂T *∂y*

    = 1ReL Pr∂ 2T *∂y*2

    u* ∂CA *∂x *

    + v* ∂CA *∂y*

    = 1ReL Sc∂ 2CA *∂y*2

    (6.35)

    (6.37)

    (6.36)

  • 6.5.2 Functional Form of the SolutionsFrom (6.35), we can write

    where dp*/dx* depends on the surface geometry.The shear stress and the friction coefficient at the surface

    are

    For a prescribed geometry, (6.45) is universally applicable.

    ⎟⎠⎞

    ⎜⎝⎛=

    **,*,*,*

    dxdpReyxfu L

    τ s = μ∂u∂y y=0

    =μVL

    ⎛ ⎝ ⎜

    ⎞ ⎠ ⎟

    ∂u*∂y* y*= 0

    C f =τ s

    ρV 2 / 2= 2Re L

    ∂u *∂y * y*=0

    )*,(2 LL

    RexfRe

    = (6.45, 46)

    (6.44)

  • From (6.36)

    Nusselt number can be defined as

    For a prescribed geometry, from (6.47)

    The spatially average Nusselt number is then

    Similarly for mass transfer,

    EX 6.5

    ⎟⎠⎞

    ⎜⎝⎛=

    **,,*,*,*

    dxdpPrReyxfT L

    h = −k fL

    (T∞ − Ts )(Ts − T∞ )

    ∂T *∂y* y*=0

    = +k fL

    ∂T *∂y * y*=0

    Nu ≡ hLk f= +

    ∂T *∂y* y*=0

    ( )PrRexfNu L ,*,=

    ( )PrRefk

    LhNu Lf

    ,==

    ( )ScRefD

    LhSh Lm ,AB

    ==

    (6.50)

    (6.48)

    (6.47)

    (6.49)

    (6.54)

  • 6.6 Physical Significance of the Dimensionless Parameters

    See Table 6.2.For heat transfer, the important dimensionless parameters are:Re, Nu, Pr, Bi, Fo, Pe (= RePr), GrFor mass transfer,Re, Sh, Sc, Bim, Fom, Le (= α/DAB)

    For boundary layer thicknesses,1, , for laminar boundary layer3

    for turbulent boundary layer (why?)

    n n nt

    t c c

    t c

    Pr Sc Le nδδ δδ δ δδ δ δ

    ≈ ≈ ≈ =

    ≈ ≈(6.55, 56, 58)

  • 6.7 Boundary Layer Analogies6.7.1 The Heat and Mass Transfer Analogy

    Since the dimensionless relations that govern the thermal and the concentration boundary layers are the same, the heat and mass transfer relations for a particular geometry are interchangeable.

  • With

    and

    and equivalent functions, f (x*, ReL),

    Then,

    and we have , n=1/3 for most applications

    EX 6.6

    ( ) nL PrRexfNu *,=

    ( ) nL ScRexfSh *,=

    // m ABn n n n

    h L DNu Sh hL kPr Sc Pr Sc

    = → =

    hhm

    = kDAB Le

    n = ρcpLe1−n

    (6.60)

  • 6.7.3 The Reynolds AnalogyFor dp*/dx* = 0 and Pr = Sc = 1, Eqs. 6.35-6.37 are of precisely the same form. Moreover, if dp*/dx*=0, the boundary conditions, 6.38-6.40 also have the same form. From Equations 6.45, 6.48, 6.52, it follows that

    Replacing Nu and Sh with Stanton number (St) and mass transfer Stanton number (Stm), as defined in (6.67) and (6.68), we have

    The modified Reynolds Analogy (or Chilton-Colburn Analogy) is

    Eqs. 6.70, 71 are appropriate for laminar flow when dp*/dx* ~ 0; for turbulent flow, conditions are less sensitive to pressure gradient.

    , mmp

    hh Nu ShSt StVc RePr V ReScρ

    ≡ = ≡ =

    ShNuReC Lf ==2

    / 2f mC St St= = -- Reynolds Analogy

    2/31/32

    fm m

    C ShSt Sc jReSc

    = ≡ =2/3 1/3 ;2f

    H

    C NuStPr jRePr

    = ≡ =

    (6.69)

    (6.70,71)

    (6.66)

  • 6.7.2 Evaporative CoolingFor evaporative cooling shown in Fig. 6.10,

    where

    If is absent, then

    Using the heat and mass transfer analogy, (6.64) becomes

    Note: Gas properties ρ, cp, Le should be evaluated at the arithmetic mean temperature of the thermal b. l., Tam= (Ts+T∞)/2.

    qconv" + qadd

    " = qevap"

    qevap" = nA

    " hfg

    h(T∞ − Ts ) = hfghm[ρA,sat (Ts ) − ρA,∞ ]

    A,sat A,[ ( ) ]ms fg shT T h Th ρ ρ∞ ∞

    ⎛ ⎞→ − = −⎜ ⎟⎝ ⎠

    (T∞ − Ts ) =MAhfg

    ℜρcpLe2/3 [

    pA,sat (Ts )Ts

    −pA,∞T∞

    ]

    qadd"

    EX 6.7

    (6.65)

    (6.64)

    (6.62)

  • h (W/m2K) Areal Rth,conv (Kcm2/W)___________________________________________________________________Natural Convection

    Air 2~25 5,000~400Oils 20~200 500~50Water 100~1,000 100~10

    Forced ConvectionAir 20~200 500~50Oils 200~2,000 50~5Water 1,000~10,000 10~1

    Microchannel Cooling 40,000 0.25Impinging Jet Cooling 2,400~49,300 4.1~0.20Heat Pipe 0.049*_________________________________________________________________________

    * based on THERMACORE 5 mm heat pipe having capacity of 20W with ΔT=5K.

    Representative Ranges of Convection Thermal ResistanceSpecial Topic—Heat Pipes

  • The Working Principle of Heat Pipes

    Evaporation SectionAdiabatic SectionCondensation Section

    The performance of a heat pipe is dictated by its thermal resistance Rth and maximum heat load, Qmax, which are mainly determined by the evaporation characteristics.

    --Based on phase-change heat transfer

  • containerWorking fluidwick

    管壁

    溝槽 燒結金屬 金屬網

    Advantages of Heat Pipessuperior heat spreading abilityfast thermal responselight weight and flexiblelow costsimple without active units

    A small amount of working fluid (mostly water) is filled in the heat pipe after it is evacuated.

    The Working Principle of Heat Pipes