Chater 8 Internal Forced Convection
8.1 Hydrodynamic Considerations 8.1.1 Flow Conditions may be determined exerimentally, as shown in Figs. 7.1-7.2. Re D ρumd μ where u m is the mean fluid velocity over the tube cross section. The critical Re D corresonding to the onset of turbulence is Re, 2300 Dc The entrance length for laminar flow is x fd,h D lam 0.05 Re D (8.1) (8.2) (8.3) The entrance length for turbulent flow is aroximately inde. of Re 10 x fd,h D turb 60 (8.4)
8.1.2 The mean Velocity ρurxda (, ) A c 2 r0 c um = = u(, r x) rdr 2 ρ A 0 c r (8.8) 0 8.1.3 Velocity Profile in the Fully Develoed Region The velocity rofile of a fully develoed laminar flow is arabolic as shown in (8.13)-(8.15). 2 1 d 2 r ur () = r 0 1 (8.13) 4μ dx r0 8.1.4 Pressure Gradient and Friction Factor in Fully Develoed Flow Friction factor for laminar flow: f = 64 (8.19) ReD Friction factor for turbulent flow: (8.20) or (8.21) or Fig. 8.3. Pressure dro: ρu ρu Δ = = = 2D 2D 2 2 x ( 2 1) (8.22a) 2 2 m m d f dx f x x 1 x1
8.2 Thermal Considerations The thermal entrance length for laminar flow is x D fd, t lam 0.05Re Pr The thermal entrance length for turbulent flow is x fd,t D turb 10 D (8.23) (8.24)
8.2.1 The Mean Temerature T m = A ρuc For incomressible flow in a circular tube with constant c, T m = 2 u m r 0 2 c 8.2.2 Newton's Law of Cooling m& c TdA r 0 0 q s " = h(t s T m ) where h is the local convection heat transfer coefficient. c utrdr (8.25) (8.26) (8.27) This relation alies for different wall conditions, e.g., T s = C or q = C. Since T m usually varies along the tube, the variation of T m must be accounted for.
8.2.3 Fully Develoed Conditions The thermally fully develoed condition is when the relative shae of the rofile no longer changes and is stated as T s (x) T (r, x) (8.28) x T s (x) T m (x) = 0 fd,t for cases with either a uniform surface heat flux or a uniform surface temerature. Since the term in the bracket of (8.29) is indeendent of x, its r-derivative is also indeendent of x. So, T/ r r T T T T Ts T r= r0 = = h f x s m r= r s m 0 k () Hence, in the thermally fully develoed flow of a fluid with constant roerties, the local convection coefficient is a constant. (Fig. 8.5) (8.29)
Eq. (8.28) imlies T Tm ( T Tm) " = or = 0, qs = constant x x x fd, t fd, t fd, t s m fd, t T m (x) is a very imortant variable for internal flows and may be obtained by alying an overall energy balance to the flow. fd, t T ( Ts T) Tm =, Ts = constant x ( T T ) x EX 8.1 8.3 The Energy Balance 8.3.1 General Considerations Energy conservation for a differential control volume of Fig. 8.6 leads to dq conv = mc & dt m " dtm qsp P ht s T m dx = mc & = mc & or ( ) (8.36) (8.37) (8.32) (8.33)
8.3.2 Constant Surface Heat Flux qp T ( x) = T + x, q = constant " s " m m, i mc & s (8.40) EX. 8.2
8.3.3 Constant Surface Temerature Define ΔT = T s -T m, Eq. 8.37 becomes dt dx m d( ΔT ) = dx Integration leads to ΔT ln ΔT o i = PL mc & = From energy balance, m& P c L 1 0 L hδt Substituting for from (8.41a), we obtain hdx = PL mc & Ts Tm( x) Then, ex Px = h, T s = constant Ts T m, i mc & q conv = mc & ( Tm, o Tm, i ) = mc & [( Ts Tm, i ) ( Ts T, m& c ΔTo ΔTi qconv = hasδtlm, ΔTlm, Ts = constant ln( / ΔT) o This means for constant T s, the total q conv is roortional to h and the log mean temerature difference ΔT lm. i h L m o )] = mc & ( ΔT i ΔT (8.41a) o (8.42) ) (8.44)
In many alications, the temerature of an external fluid, rather than the tube surface temerature, is fixed (Fig. 8.8). The revious formulae can still be alied (with modification adoting the concet of thermal resistance), as indicated by Eqs. (8.45) and (8.46). Δ T T o Tmo, UA = = s ex Ti T T m, i mc (8.45a) Δ & q = UAΔT s lm (8.45b) T T Δ (8.46a) ΔTlm q = (8.46b) R Δ To = mo, = ex 1 Ti T T m, i mc & Rtot EX 8.3 tot
8.4 Laminar Flow in Circular Tubes: Thermal Analysis 8.4.1 The Fully Develoed Region For the case of uniform surface heat flux, the exact correlation is Nu D hd = = k " 4.36, qs constant For the case of constant surface tem., with (8.53) ( 2 T/ 2 x)<<( 2 T/ 2 r) Nu D = 3.66, T = constant s (8.55) EXs. 8.4, 8.5
8.4.2 The Entry Region Thermal entry length roblem (8.56) Combined entry length roblem (8.57)
Fully Develoed Pie Flow For Pr < 1 entrance region fully-develoed region velocity boundary layer thermal boundary layer
8.5 Convection Correlations: Turbulent Flow in Circular Tubes Using the modified Reynolds analogy, there are (8.58)-(8.60). (Note the exonent of Re D is 4/5). For higher accuracies, use (8.61)-(8.62). EX 8.6 8.6 Convection Correlations: Noncircular Tubes and the Concentric Tube Annulus Hydraulic diameter: D h 4A c (8.66) P It is this diameter that should be used in calculating arameters such as Re D and Nu D. For turbulent flow, it is reasonable to use the correlations of Section 8.5 for Pr > 0.7. However, the coefficient is resumed to be an average over the erimeter. For laminar flow, the use of circular tube correlations is less accurate, articularly with cross sections characterized by shar corners. For laminar fully develoed conditions: use Table 8.1
Fully Develoed Laminar Flow in Noncircular Tubes Hydraulic diameter: D h 4A c P hdh Nu = = constant k f Re = constant D h as D h h and ΔP
8.7 Heat Transfer Enhancement Heat transfer enhancement may be achieved by increasing the convection coefficient (inducing turbulence or secondary flow) and/or by increasing the convection surface area. See Figs. 8.12 and 8.13 To induce secondary flow or turbulence
8.8 Microscale Internal Flow Many new technologies involve microscale internal flow with D h 100μm. For gases, the results of Chaters 6 through 8 are not exected to aly when D h /λ mf 10. For liquids, some features of revious results, e.g., friction factor (8.19), ressure dro (8.22a), and transition criterion (8.2) remain alicable. 8.9 Convection Mass Transfer Omitted.