Chapter 6 Fundamental Concepts of Convection

Chapter 6 Fundamental Concepts of Convection

6.1 The Convection Boundary Layers Velocity boundary layer: τ surface shear stress: s = μ u local friction coeff.: C f y y=0 τ s ρu / (6.) (6.1) Thermal boundary layer: local heat flux: q s = k T f local heat transfer coeff.: y y=0 q " kf T / y s y= 0 h = = T T T T s s (6.3) (6.5)

Concentration boundary layer: local species flux: " CA NA = DAB [kmol/s m ] y y= 0 " ρ or (mass basis): A na = DAB [kg/s m ] y y= 0 " N D C / y As, local mass transfer coeff.: hm = = C C C C AB A y= 0 As, A, As, A, (6.6) (6.9) or (mass basis): h m = D ABρ A / y y=0 ρ A,s ρ A,

6.1.4 Significance of the Boundary Layers Velocity boundary layer: always exists for flow over any surface Thermal boundary layer: exists if the surface and free stream temperature differ Concentration boundary layer: exists if the surface concentration of a species differs from the free stream value The principal manifestations and boundary layer parameters are Velocity boundary layer: surface friction and friction coefficient C f Thermal boundary layer: convection heat transfer and heat transfer convection coefficient h Concentration boundary layer: convection mass transfer and mass transfer convection coefficient h m

6. Local and Average Convection Coefficients 6..1 Heat Transfer The local heat flux q may be expressed as (Newton's law of cooling) q = h(t s -T ), h = heat transfer convection coeff. = f (fluid properties, surface geometry, flow conditions) The total heat transfer rate q may be obtained by integration q = qda = ( T T ), if T s is uniform A s s hda s s As (6.10-1) = ha ( T ), s 1 where h = hdas. A A s s (6.13) s T

6.. Mass Transfer Similar for mass transfer of species A, we have N = h ( C C ) [kmol/s m ] where or " A m A, s A, N = h A ( C C ) [kmol/s] A m s A, s A, n n h = m A m s s h da = h ( ρ ρ ) [kg/s m ] " A m A, s A, = h A ( ρ ρ ) [kg/s] A m s A, s A, --local --average --Molar transfer rate (6.15) (6.16) (6.18) --Mass transfer rate (6.19)

We can also write Fick s law on a mass basis by multiplying Eq. 6.7 by M A to yield " ρa na = DAB = hm( ρa, s ρa, ) y h m = y= 0 D ρ / y AB A y= 0 ρ ρ A, s A, (6.0) (6.1) The value of C A,s or ρ A,s can be determined by assuming thermodynamic equilibrium at the interface between the gas and the liquid or solid surface. Thus, C A, s p ( T ) sat s = R T (6.)

6..3 The Problem of Convection The local flux and/or the total transfer rate are of paramount importance in any convection problem. So, determination of these coefficients (local h or h m and average h or h m ) is viewed as the problem of convection. However, the problem is not a simple one, as h or h = f m (fluid properties, surface geometry, flow conditions) i.e., ρμ,, k, c f p, f EXs 6.1-6.3

6.3 Laminar and Turbulent Flow Critical Reynolds no. where transition from laminar boundary layer to turbulent occurs: Re x,c = ρu x c μ = 5 10 5 EX 6.4

6.4 The Boundary Layer Equations (-D, Steady) 6.4.1 Boundary Layer Equations for Laminar Flow For the steady, two-dimensional flow of an incompressible fluid with constant properties, the following equations are involved. (Read 6S.1 for detailed derivation.) u x v y continuity eq.: + = 0 (D.1)

momentum eq. (incompressible): energy equation: species equations: X y u x u x p y u v x u u + + + = + μ ρ Y y v x v y p y v v x v u + + + = + μ ρ y-dir. x-dir. A AB N y C x C D y C v x C u A A A A & + + = + (D.6) (D.) (D.3) q y T x T k y T v x T u c p & Φ + + + = + μ ρ where u v u v y 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: u >> v u u v v Velocity boundary layer >>,, y x y x T y T >> x Thermal boundary layer CA CA >> y x Concentration boundary layer Usual additional simplifications: incompressible, constant properties, negligible body forces, nonreacting, no energy generation

Eqs. 6.7 is unchanged and the x-mom equation reduces to u u 1 dp u u + v = + ν x y ρ dx y (6.8) The energy equation reduces to T u x T + v y energy transport thru convection and the species equation becomes C A C A CA u + v = DAB x y y T = α y ν u + c p y heat transport thru conduction usually negligible (6.9) (6.30) For incompressible, constant property flow, Eq. (6.8) is uncoupled from (6.9) & (6.30). Eqs. (6.7) & (6.8) need to be solved first, for the velocity field u(x, y) and v(x, y), before (6.9) & (6.30) can be solved, i.e., the temperature and species fields are coupled to the velocity field.

6.5 Boundary Layer Similarity: The Normalized Convection Transfer Equations 6.5.1 Boundary Layer Similarity Parameters Defining x* x and y* y L is the characteristic length L L u* u and v* V V v V is the free stream velocity T* T T s T and T C A * C C A A,s s C A, C A,s we arrive at the convection transfer equations and their boundary conditions in nondimensional form, as shown in Table 6.1. Dimensionless groups: Reynolds no.: ReL Prandtl no.: Pr v Schmidt no.: Sc α v D VL v AB

The final dimensionless governing eqs.: (6.35)-(6.37)--similar in form u* u * x* u* T * x * u * x * + v * y * = 0 u* + v* y* = dp* dx * + Re 1 u * L y* + v* T * y* = 1 Re L Pr T * y* u* C A * x * + v* C A * y* = 1 C A * Re L Sc y* (6.35) (6.36) (6.37)

6.5. Functional Form of the Solutions From (6.35), we can write u* = f x*, y*, Re L, dp dx where dp*/dx* depends on the surface geometry. * * The shear stress and the friction coefficient at the surface are τ s = μ u = μv y L u* y=0 y* y*= 0 C f = τ s ρv / = Re u * = f ( x*, ReL) (6.45, 46) L y * y*=0 Re For a prescribed geometry, (6.45) is universally applicable. L (6.44)

From (6.36) T* = f x*, y*, Re h = k f (T T s ) L (T s T ) EX 6.5 L, Pr, dp dx T * =+ k f T * y* L y*=0 y * y*=0 Nusselt number can be defined as Nu hl =+ T * k f y* y*=0 For a prescribed geometry, from (6.47) Nu = f ( x *, ReL, Pr) The spatially average Nusselt number is then hl Nu = = f ( ReL, Pr) k f Similarly for mass transfer, h L Sh = = f Re D * * ( Sc) m L, AB (6.47) (6.48) (6.49) (6.50) (6.54)

6.6 Physical Significance of the Dimensionless Parameters See Table 6.. For heat transfer, the important dimensionless parameters are: Re, Nu, Pr, Bi, Fo, Pe (= RePr), Gr For mass transfer, Re, Sh, Sc, Bi m, Fo m, Le (= α/d AB ) For boundary layer thicknesses, δ n δ n δ n 1 Pr, Sc, t Le n = for laminar boundary layer δt δc δc 3 (6.55, 56, 58) δ δ δ for turbulent boundary layer (why?) t c

6.7 Boundary Layer Analogies 6.7.1 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*, Re L ), Then, and we have Nu = f *, Sh = f *, n ( x Re L ) Pr ( ) n x Re L Sc Nu Sh hl k hl/ D = = n n n n Pr Sc Pr Sc h h m = / m AB k D AB Le n = ρc p Le 1 n (6.60), n=1/3 for most applications EX 6.6

6.7.3 The Reynolds Analogy For 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, 6.5, it follows that Re C L f = Nu = Sh (6.66) Replacing Nu and Sh with Stanton number (St) and mass transfer Stanton number (St m ), as defined in (6.67) and (6.68), we have Cf /= St = Stm -- Reynolds Analogy (6.69) h Nu hm Sh St =, Stm = ρvcp RePr V ReSc The modified Reynolds Analogy (or Chilton-Colburn Analogy) is C f C /3 Nu f /3 Sh = StPr j ; H = = StmSc jm = 1/3 1/3 (6.70,71) RePr ReSc Eqs. 6.70, 71 are appropriate for laminar flow when dp*/dx* ~ 0; for turbulent flow, conditions are less sensitive to pressure gradient.

6.7. Evaporative Cooling For evaporative cooling shown in Fig. 6.10, where " q add " q conv " q evap + q add If is absent, then " = n A " h fg " = q evap h(t T s ) = h fg h m [ρ A,sat (T s ) ρ A, ] hm T Ts = hfg [ A,sat ( Ts) A, ] h ρ ρ (6.64) Using the heat and mass transfer analogy, (6.64) becomes M A h fg (T T s ) = Rρc p Le /3 [ p A,sat (T s ) T p A, s T ] (6.65) Note: Gas properties ρ, c p, Le should be evaluated at the arithmetic mean temperature of the thermal b. l., T am = (T s +T )/. EX 6.7 (6.6)

Special Topic Heat Pipes Representative Ranges of Convection Thermal Resistance h (W/m K) Areal R th,conv (Kcm /W) Natural Convection Air ~5 5,000~400 Oils 0~00 500~50 Water 100~1,000 100~10 Forced Convection Air 0~00 500~50 Oils 00~,000 50~5 Water 1,000~10,000 10~1 Microchannel Cooling 40,000 0.5 Impinging Jet Cooling,400~49,300 4.1~0.0 Heat Pipe 0.049* * based on THERMACORE 5 mm heat pipe having capacity of 0W with ΔT=5K.

The Working Principle of Heat Pipes --Based on phase-change heat transfer Evaporation Section Adiabatic Section Condensation Section The performance of a heat pipe is dictated by its thermal resistance R th and maximum heat load, Q max, which are mainly determined by the evaporation characteristics.

A small amount of working fluid (mostly water) is filled in the heat pipe after it is evacuated. 管壁 container Working fluid wick 溝槽燒結金屬金屬網 Advantages of Heat Pipes superior heat spreading ability fast thermal response light weight and flexible low cost simple without active units