Chapter Basic Conservation Equations for Laminar Convection Abstract In this chapter, the basic conservation equations related to laminar fluid flow conservation equations are introduced. On this basis, the corresponding conservation equations of mass, momentum, and energ for stead laminar forced convection boundar laer are obtained..1 Continuit Equation The conceptual basis for the derivation of the continuit equation of fluid flow is the mass conservation law. The control volume for the derivation of continuit equation is shown in Fig..1 in which the mass conservation principle is stated as ṁ increment = ṁ in ṁ out..1 where ṁ increment expresses the mass increment per unit time in the control volume, ṁ in represents the mass flowing into the control volume per unit time, and ṁ out is the mass flowing out of the control volume per unit time. The dot notation signifies a unit time. d m out Fig..1 Control volume for derivation of the continuit equations m in dx d D.Y. Shang, Theor of Heat Transfer with Forced Convection Film Flows, Heat and Mass Transfer, DOI 10.1007/978-3-64-1581-_, C Springer-erlag Berlin Heidelberg 011 1
Basic Conservation Equations for Laminar Convection In the control volume, the mass of fluid flow is given b dx d d, and the mass increment per unit time in the control volume can be expressed as ṁ increment = dx d d.. τ The mass flowing per unit time into the control volume in the x direction is given b w x d d. The mass flowing out of the control volume in a unit time in the x direction is given b w x + w x dx d d. Thus, the mass increment per unit timeinthex direction in the control volume is given b w x dx d d. Similarl, the mass increments in the control volume in the and directions per unit time are given b w d dx d and w d dx d, respectivel. We thus obtain wx ṁ out ṁ in = + w With. and.3,.1 becomes in Cartesian coordinates: or in the vector notation + w dx d d..3 τ + w x + w + w = 0,.4 τ + W = 0,.5 where W = iw x + jw + kw is the fluid velocit. For stead state, the vector and Cartesian forms of the continuit equation are given b and w x + w + w = 0,.6 w x + w = 0,.7 respectivel for three- and two- dimensional continuit equations.
. Momentum Equation Navier Stokes Equations 3. Momentum Equation Navier Stokes Equations The control volume for derivation of the momentum equation of fluid flow is shown in Fig... Take an enclosed surface A that includes the control volume. Meanwhile, take F m as mass force per unit mass fluid, F m,v as the total mass force in the control volume, τ n as surface force per unit mass fluid flow at per area of surface, τ n,a as surface force in the control volume, Ġ increment as momentum increment of the per unit mass fluid flow at unit time, and G increment as momentum increment of the fluid flow per unit time in the volume. According to the momentum law, the momentum increment of the fluid flow per unit time in the control volume equals the sum of total mass force and surface forced in the same volume, as G increment = F m,v + τ n,a..8 F m,v, τ n,a, and G increment in the control volume are expressed as, respectivel, F m,v = τ n,a = F m d,.9 τ n da,.10 G increment = A D W d..11 According to tensor calculation, the right side of.10 is expressed as τ n da = τd,.1 A where τ is divergence of the shear force tensor. With.9,.10,.11, and.1,.8 is rewritten as D W d = F m d + τd,.13 Fig.. Control volume for derivation of momentum equations G x, in dx F x d d G x, out
4 Basic Conservation Equations for Laminar Convection i.e. D W F τ d = 0..14 Therefore, we have D W = F+ τ..15 This is the Navier Stokes equations of fluid flow. For Cartesian coordinates,.15 can be expressed as Dw x Dw Dw = τ xx = τ x = τ x + τ x + τ + τ + τ x + g x,.16 + τ + g,.17 + τ + g,.18 where τ xx = p + 3 wx τ = p + 3 wx τ = p + 3 wx + w + w + w + w + w + w w τ x = τ x = w τ = τ = wx τ x = τ x = + + w + w +, + w, + w,,,, g x, g, and g are gravit accelerations in x,, and directions, respectivel. Then,.16 to.18 become Dw x + w x + wx + w + wx + w 3 + w + w + g x,.19
. Momentum Equation Navier Stokes Equations 5 Dw Dw + 3 + 3 wx wx + w wx wx + w + w + + w + w + + w w + w + w + g,.0 w + w + w + g.1 For stead state, the momentum equations.19.1 are given as follows, respectivel, wx w x + w + w x w + w x w x + w + w = p + w x + wx + w + wx + w 3 wx + w + w + g x,. w w x + w w + w w + w w x + w + w = p + wx + w + w + w + w 3 wx + w + w + g,.3 w w x + w w + w w + w w x + w + w = p + wx + w + w + w + 3 wx + w + w Let us compare term wx w x + w + w + w w + g..4 w with term w x w x + in.. In general, derivatives,, and are much larger than the derivatives x, x, and x, respectivel. In this case, the term w x w x + w + w is omitted, and. is rewritten as generall
6 Basic Conservation Equations for Laminar Convection wx w x + w + w x w + w x + wx + w + wx + w 3 wx + w + w + g x..5 Similarl, in general,.3 and.4 are rewritten as respectivel w w x + w + w 3 w w x + w + w + w w + + w + w wx + w + w w + w w + w + w + w 3 wx + w + w wx + w + g,.6 wx + w + g.7.3 Energ Equation The control volume for derivation of the energ equation of fluid flow is shown in Fig..3. Take an enclosed surface A that includes the control volume. According to the first law of thermodnamics, we have the following equation: W out ΔE d Fig..3 Control volume for derivation of the energ equations of fluid flow Q in dx d
.3 Energ Equation 7 Ė = Q + Ẇ out,.8 where Ė is energ increment in the sstem per unit time, Q is heat increment in the sstem per unit time, and Ẇ out denotes work done b the mass force and surface force on the sstem per unit time. The energ increment per unit time in the sstem is described as Ė = D W e + W d,.9 where τ denotes time, is the fluid kinetic energ per unit mass, W is fluid velocit, and the smbol e represents the internal energ per unit mass. The work done b the mass force and surface force on the sstem per unit time is expressed as Ẇ out = F W d + A τ n W da,.30 where F is the mass force per unit mass and τ n is surface force acting on unit area. The heat increment entering into the sstem per unit time through thermal conduction is described b using Fourier s law as follows: Q = λ t da,.31 A n where n is normal line of the surface, and here the heat conduction is considered onl. With.9 to.31,.8 is rewritten as D where A e + W D τ n W da = d = F W d + A e + W d = n τ W da = A A D τ n W da + λ t da,.3 A n e + W n τ W da = v d,.33 τ W d,.34 λ t A n da = λ td..35 With.33,.34, and.35,.3 is rewritten as
8 Basic Conservation Equations for Laminar Convection Then D e + W D d = F W d + τ W d + λ td. v.36 e + W = F W + τ W + λ t,.37 where τ denotes tensor of shear force. Equation.37 is the energ equation. Through tensor and vector analsis,.37 can be further derived into the following form: De =τ ε+ λ t..38 Equation.38 is another form of the energ equation. Here, τ ε is the scalar quantit product of force tensor τ and deformation rate tensor ε, and represents the work done b fluid deformation surface force. The phsical significance of.38 is that the internal energ increment of fluid with unit volume during the unit time equals the sum of the work done b deformation surface force of fluid with unit volume and the heat entering the sstem. With the general Newtonian law, the tensor of shear force is expressed as τ =ε p + W I,.39 3 where I is unit tensor. According to.39 the following equation can be obtained: Then,.38 can be rewritten as where = 3 described as { = τ ε = p W W + ε..40 3 De = p W + + λ t,.41 W + ε is viscous thermal dissipation, which is further wx + w w + w + + w w + + wx + + w } div W,.4 3
.3 Energ Equation 9 where div W = + w + w. According to continuit equation.5, we have W = 1 D = D 1. With the above equation,.41 is changed into the following form: De + p D 1 = + λ t..43 According to thermodnamics equation of fluid, we have Dh = De + p D 1 + Dp..44 Then, Equation.43 can be expressed as the following enthalp form: or Dh = Dp + + λ t,.45 Dc p t = Dp where h = c p t, while c p is specific heat. In Cartesian form, the energ.46 can be rewritten as + + λ t,.46 c p t τ = DP + c p t c p t c p t + w x + w + w λ t + λ t + λ t +..47 For stead state, the three-dimensional Cartesian form of the energ equation.47 is changed into w x c p t + w c p t + w c p t = + λ t λ t + λ t +..48
30 Basic Conservation Equations for Laminar Convection Above equation is usuall approximatel rewritten as c p t c p t c p t w x + w + w = + λ t λ t + λ t +,.49 where the temperature-dependence of densit is ignored. For stead state, the two-dimensional Cartesian form of the energ equation.48 is changed into w x c p t + w c p t Above equation is usuall approximatel rewritten as = λ t + λ t +..50 c p t c p t w x + w = λ t t + λ t +..51 With the two-dimensional form, the viscous thermal dissipation is { wx = + where w wx + + w } div W,.5 3 div W = + w..4 Governing Partial Differential Equations of Laminar Forced Convection Boundar Laers with Consideration of ariable Phsical Properties.4.1 Principle of the Quantitative Grade Analsis The general governing partial differential equations can be transformed to the related boundar laer equations. In fact, the onl difference between the governing equations of free and forced convection is that buoant force is onl considered in the related momentum equation of free convection. Before the quantitative grade analsis, it is necessar to define its analtical standard. A normal quantitative grade is regarded as, i.e. unit quantit grade, a ver small quantitative grade is regarded as {δ}, even ver small quantitative grade
.4 Governing Partial Differential Equations of Laminar Forced Convection 31 is regarded as {δ }, and so on. Then, =, {δ} {δ} =, {δ} = {δ 1 }, {δ } ={δ }. Furthermore, according to the theor of laminar forced boundar laer, the quantities of the velocit component w x and the coordinate x can be regarded as unit, i.e. {w x }= and {x} =. The quantities of the velocit component w and the coordinate should be regarded as δ, i.e. {w }={δ} and {} ={δ}. However, the quantit of temperature t can be regarded as {t} =, the quantit grade of } =, but the quan- { the pressure gradient p can be regarded as unit, i.e. p tit grade of the pressure gradient p { p } ={δ}. is onl regarded as small quantit grade, i.e. According to the quantitative grade of the general fluid, the quantitative grade of densit, thermal conductivit λ, absolute viscosit, and specific heat c p are {} =, {λ} ={δ }, {} ={δ }, and {c p }=. In addition, for gravit acceleration, we have {g x }={0}, and {g }= for forced convection..4. Continuit Equation Based on the.7, the stead-state two-dimensional continuit equation is given b w x + w = 0.53 where variable fluid densit is considered. According to the above principle for the quantit analsis, now, we take the stead-state two-dimensional continuit equation.53 as example to do the quantit analsis as follows: w x + w = 0 + {δ} = 0 {δ} i.e. + = 0.53a The above ratios of quantit grade related to the terms of.53 shows that both of the two terms of.53 should be kept, and.53 can be regarded as the continuit equation of the two-dimensionless stead-state forced convection with laminar two-dimensional boundar laer..4.3 Momentum Equations Navier Stokes Equations The momentum equations.5 and.6 are rewritten as for stead twodimensional convection
3 Basic Conservation Equations for Laminar Convection w x + w w w x + w + w 3 wx + 3 w x + + w wx + w wx + w wx + w + g x,.54 + w + g..55 The quantit grades of the terms of.54 and.55 are expressed as follows, respectivel: w x + w + w x + wx + w 3 wx + w + g x {1 +{δ} = + {δ} {δ } + {δ} {δ {δ} + {δ} δ + {δ} {δ} +{0}.54a w w x + w w + wx + w + w 3 wx + w + g {δ}+{δ} ={δ}+{δ}+{δ 3 } +{δ} {δ}+ +{0}.55a Here, for forced convection, the volume forces g x and g are regarded as ero. Then, the quantit grades of.54a is further simplified as follows, respectivel: w x + w + 3 wx + + w wx + w
.4 Governing Partial Differential Equations of Laminar Forced Convection 33 {1 + =+{δ }++{δ } {δ }+{δ }.54b Observing the quantit grades in.54a it is found that the terms w in term wx + w, 3 wx + w ignored. Then,.54a is simplified as follows: w x + w, are ver small and can be + wx..56 Comparing the quantit grades of.55a with those of.54b, it is found that the quantit grades of.55a are ver small. Then,.55 can be ignored, and onl.56 is taken as the momentum equation of stead-state forced convection with two-dimensional boundar laer..4.4 Energ Equations The quantit grades of the terms of.51 for laminar two-dimensional energ equation is expressed as c p t w x c p t + w = +{δ} {δ} λ t = + δ λ t + + δ {δ} {δ}.51a The quantit grade of.51a is simplified to c p t c p t w x + w = λ t + λ t + + = δ }+.51b With the quantit grade analsis from.51b, it is seen that the term λ t is ver small compared with other terms, and then can be omitted from the equation. Then, the energ equation for stead-state forced convection with two-dimensional laminar boundar laer is simplified to c p t c p t w x + w = λ t +..57 Now, we anale the viscous thermal dissipation. According to.5, is the following equation for the laminar stead state convection:
34 Basic Conservation Equations for Laminar Convection { wx = + w wx + + w wx 3 With quantit grade analsis, the right side of.5a is expressed as { wx w = + {δ } + wx + + w wx 3 {δ} + {δ} {δ} + {δ} + {δ} {δ} + w }..5a + w } The quantit grade of the above equation is equivalent to {δ } ++ {δ } +{δ } +.5b With the quantit grade comparison, it is found that onl the term wx ma be kept. Then, for the stead state forced convection with two-dimensional boundar laer, the viscous thermal dissipation is simplified to wx =..5c With.5c,.57 is changed to the following equation c p t c p t w x + w = λ t wx +.58 as the stead-state energ equation with laminar forced convection for twodimensional boundar laer. In a later chapter, I will investigate the effect of the viscous thermal dissipation on heat transfer of laminar forced convection. Now the basic governing partial differential equations for description of mass, momentum, and energ conservation of two-dimensional laminar stead-state forced convection boundar laers are shown as follows with consideration viscous thermal dissipation and variable phsical properties: w x + w = 0,.53 w x + w + wx,.56 c p t c p t w x + w = λ t wx +..58
.4 Governing Partial Differential Equations of Laminar Forced Convection 35 Suppose a bulk flow with the velocit w x, beond the boundar laer,.56 is simplified to the following form: i.e. w x,, = 1 dp dx, Then.56 is changed to w x + w dp dx = w, x,., = w x, + wx.56a Obviousl, if the main stream velocit w x, is constant, i.e., = 0,.56a is identical to the following equations: w x + w = wx..56b For rigorous solutions of the governing equations, the fluid temperaturedependent properties, such as densit, absolute viscosit, specific heat c p, and thermal conductivit λ will be considered in the successive chapters of this book. The laminar forced convection with two-dimensional boundar laer belongs to two-point boundar value problem, which is the basis of three-point boundar value problem for film condensation. Obviousl, the basic governing partial differential equations for mass, momentum, and energ conservation of two-dimensional laminar stead-state forced convection boundar laers with considering viscous thermal dissipation but ignoring the variable phsical properties are obtained as based on.53,.56, and.58: + w = 0.59 w x + w = 1 p + ν w x.60 with 1 dp dx = w, x, t w x + w t = ν t Pr + ν/c p wx..61
36 Basic Conservation Equations for Laminar Convection Now, the three-dimensional basic conservation equations for laminar convection and Two-dimensional basic conservation equations for laminar forced convection boundar laer can be summaried in Tables.1 and., respectivel..5 Summar Table.1 Three-dimensional basic conservation equations for laminar convection with consideration of variable phsical properties Mass equation w x + w + w = 0.6 Momentum equation Energ equation wx w x + w + + wx + w w w x + w w + w + w w + w w w w x + w w + w w + w + w + + 3 3 wx + wx + w w x + + w + w wx + w wx 3 + w wx + w + g x.5 + w + w + g.6 + w wx + w + w c p t c p t c p t w x + w + w = λ t + λ t + { = +g.7 λ t +.49 wx + w + w wx + + w w + + w w + + w } x div W,.4 3
.5 Summar 37 Table. Two-dimensional basic conservation equations for laminar forced convection boundar laer With consideration of variable phsical properties Mass equation w x + w = 0.7 Momentum equation w x + w + wx dp dx = w, x,.56 Energ equation c p t c p t w x + w = λ t wx +.58 wx for consideration of viscous thermal dissipation With ignoring variable phsical properties Mass equation + w = 0.59 Momentum equation w x + w = 1 p + ν w x 1 dp dx = w x,,.60 Energ equation t w x + w t = ν t Pr + ν/c p for consideration of viscous thermal dissipation wx.61 wx
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