External Flow: Flow over Bluff Objects (Cylinders, Spheres, Packed Beds) and Impinging Jets

Transcription

1 External Flow: Flow over Bluff Object (Cylinder, Sphere, Packed Bed) and Impinging Jet he Cylinder in Cro Flow - Condition depend on pecial feature of boundary layer development, including onet at a tagnation point and eparation, a well a tranition to turbulence. - Stagnation point: Location of zero velocity ( u 0) and maximum preure. - Followed by boundary layer development under a favorable preure gradient ( dp / dx 0 ) and hence acceleration of the free tream flow ( du / dx 0 ). - A the rear of the cylinder i approached, the preure mut begin to increae. Hence, there i a minimum in the preure ditribution, p(x), after which boundary layer development occur under the influence of an advere preure gradient ( dp / dx 0, du / dx 0). - Separation occur when the velocity gradient du / dy reduce to zero y0

2 And i accompanied by flow reveral and a downtream wake. - Location of eparation depend on boundary layer tranition. V V Re - What feature differentiate boundary development for the flat plate in parallel flow from that for flow over a cylinder? - Force impoed by the flow i due to the combination of friction and form drag. he dimenionle form of the drag force i C F A ( V f / ) Figure 7.8

3 - Heat ranfer Conideration - he Local Nuelt Number: 5 - How doe the local Nuelt number vary with for Re 10? What condition are aociated with maxima and minima in the variation? - he Average Nuelt Number ( Nu h / k ): Zukauka (197) Nu m n Pr C Re Pr Pr 1/4 0.7 Pr Re 10 where all propertie except Pr are evaluated at. Churchill and Berntein (1977) propoed for all Re Pr 0. Nu Re 1/ Pr 1/ 3 1 (0.4 / Pr) / 3 1/ 4 Re / 8 4 / 5

4 Cylinder of Noncircular Cro Section: 1/ 3 Nu C Re m Pr C,m able 7.3

5 Flow Acro ube Bank - A common geometry for two-fluid heat exchanger. Aligned and Staggered Array: Aligned: V S S V max Staggered: V S S V max if S or, V S S V max if S

6 - Flow Condition: How do convection coefficient vary from row-to-row in an array? How do flow condition differ between the two configuration? Why hould an aligned array not be ued for S /S L < 0.7? Average Nuelt Number for an Iothermal Array: Zukauka (197) Nu C C,m able 7.7 able 7.8 C m 0.36 C Re,max Pr Pr/ Pr All propertie are evaluated at i o / except for Pr. 1/ 4

7 - Fluid Outlet emperature ( ): o P i o C S VN N h exp N N L N What may be aid about a o N? - otal Heat Rate: ha lm q N( L) A o i o i lm ln ) ( ) ( - Preure rop:

8 p V N X f L max X, f Figure 7.13 and 7.14

9 he Sphere and Packed Bed Flow over a phere - Boundary layer development i imilar to that for flow over a cylinder, involving tranition and eparation. Whitaker (197) recommend Nu 1/ / /4 0.4Re 0.06Re Pr / 0.71Pr Re ( / ) 3. All propertie except are evaluated at. - What are the limiting value of the Nuelt number and the convection coefficient for low flow over mall phere. -C Figure Ga Flow through a Packed Bed -Flow i characterized by tortuou path through a bed of fixed particle. -Large urface area per unit volume render configuration deirable for the tranfer and torage of thermal energy. -For a packed bed of phere: j H.06 Re void fraction q ha p, tlm A p, t total urface area of particle - o i ha p, t exp VAc, bc p

10 A c, b cro-ectional area of bed

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12 Ga Jet Impingement - Characterized by large convection coefficient and ued for cooling and heating in numerou manufacturing, electronic and aeronautic application. - Flow and Heat ranfer for a Circular or Rectangular Jet: Significant Feature: - Mixing and velocity profile development in the free jet. - Stagnation point and zone. - Velocity profile development in the wall jet. - Local Nuelt number ditribution: - Average Nuelt number: hh Nu f (Re, Pr, Ar, H / h ) k V e h Re, A r from Fig Correlation Section 7.7.

13 Jet Array - What i the nature and effect of jet interaction and dicharge condition? - Nuelt number correlation for array of circular and lot jet Section 7.7..

14 Problem 7.63: Cooling of extruded copper wire by convection and radiation. KNOWN: Velocity, diameter, initial temperature and propertie of extruded wire. emperature and velocity of air. emperature of urrounding. FIN: (a) ifferential equation for temperature ditribution (x) along the wire, (b) Exact olution for negligible radiation and correponding value of temperature at precribed length ( x = L = 5m) of wire, (c) Effect of radiation on temperature of wire at precribed length. Effect of wire velocity and emiivity on temperature ditribution. ASSUMPIONS: (1) Negligible variation of wire temperature in radial direction, () Negligible effect of axial conduction along the wire, (3) Contant propertie, (4) Radiation exchange between mall urface and large encloure,

15 (5) Motion of wire ha a negligible effect on the convection coefficient (V e << V). PROPERIES: 3 Copper: 8900 kg/m, 400 J / kg K, Air: k W/m K, 310 m /, Pr C p ANALYSIS: (a) Applying conervation of energy to a tationary control urface, through which the wire move, teady-tate condition exit and E E 0. in out Hence, with inflow due to advection and outflow due to advection, convection and radiation, V A C V A C ( d) dq e c p e c p conv dq rad / 4C d dx h 0 Ve εσ p 4 4 h d 4 εσ ur (1) dx VeCp Alternatively, if the control urface i fixed to the wire, condition are tranient and the energy balance i of the form, 4 4 dx h εσ d dt p ur 4 h εσ C E 4 out E t dx Cp 4 4 ividing the left- and right-hand ide of the equation by dx/dt and Ve dx/dt, repectively, Eq. (1) i obtained. (b) Neglecting radiation, eparating variable and integrating, Eq. (1) become ur ur, or d dt i d ρ 4h V C e p 0 x dx ln i 4hx ρvec p

16 4hx (i )exp( ) () ρv C with V 5 m/ m Re 833, m / the Churchill-Berntein correlation yield e p Nu 1/ 0.6(833) (0.69) 0.3 / 3 1 (0.4 / 0.69) 1/ 3 1/ k W/m K m Hence, applying Eq. () at x = L, 5 / 8 4 / 5 h Nu W/m K C (575 C)exp 8900 kg/m 0. m/0.005 m400 J/kg K W/m K 5 m o 3 c) Numerically integrating from x = 0 to x = L = 5.0m, we obtain o 309 C Hence, radiation make a dicernable contribution to cooling of the wire. Parametric condition reveal the following ditribution Wire temperature, (C) Wire temperature, (C) itance from extruder exit, x(m) itance from extruder exit, x(m) Ve=0.5 m / Ve=0. m / Ve=0.1 m / ep=0.8 ep=0.55 ep=0 he peed with which the wire i drawn from the extruder ha a ignificant influence on the temperature ditribution. he temperature decay decreae with increaing V e due to the increaing effect of advection on energy tranfer in the x direction. he effect of the urface emiivity i le pronounced, although, a

17 expected, the temperature decay become more pronounced with increaing. COMMENS: (1) A critical parameter in wire extruion procee i the coiling temperature, that i, the temperature at which the wire may be afely coiled for ubequent torage or hipment. he larger the production rate (V e ), the longer the cooling ditance needed to achieve a deired coiling temperature. () Cooling may be enhanced by increaing the cro-flow velocity, and the pecific effect of V may alo be explored.

18 Problem 7.78 Meaurement of combution ga temperature with a pherical thermocouple junction. KNOWN: Velocity and temperature of combution gae. iameter and emiivity of thermocouple junction. Combutor temperature. FIN: (a) ime to achieve 98% of maximum thermocouple temperature rie for negligible radiation, (b) Steady-tate thermocouple temperature, (c) Effect of ga velocity and thermocouple emiivity on meaurement error. ASSUMPIONS: (1) Validity of lumped capacitance analyi, () Contant propertie, (3) Negligible conduction through lead wire, (4) Radiation exchange between mall urface and a large encloure (part b and c). PROPERIES: hermocouple: , k = 100 W/mK, c = 385 J/kgK, = 890

19 kg/m 3 ; Gae: k = 0.05 W/mK, = m /, Pr = ANALYSIS: (a) If the lumped capacitance analyi may be ued, it follow from Equation 5.5 that ρvc i ρc t ln ln(50) ha 6h Neglecting the vicoity ratio correlation for variable property effect, ue of V = 5 m/ with the Whitaker correlation yield h 1/ /3 0.4 Nu (0.4Re 0.06Re )Pr, k with V 5 m/(0.001m) Re ν 5010 m / /3 1/ W/m K h m (100) W/m K ince Bi = ued. (r /3)/k = , the lumped capacitance method may be h o m 890 kg/m 385 J/kg K 638 W/m K t ln(50) 6.83 (b) Performing an energy balance on the junction, q conv = q rad. Hence, evaluating radiation exchange from Equation 1.7 and with = 0.5, ha 4 4 ε A σ( ) W/m K 1000 K 38 W/m K (400) K 936K Parametric calculation to determine the effect of V and yield the following reult: c

20 emperature, (K) 950 emperature, (K) Velocity, V(m/) Emiivity Emiivity, epilon = 0.5 Velocity, V = 5 m/ Since the temperature recorded by the thermocouple junction increae with increaing V and decreaing, the meaurement error, -, decreae with increaing V and decreaing. he error i due to net radiative tranfer from the junction (which depree ) and hence hould decreae with decreaing. For a precribed heat lo, the temperature difference ( - ) decreae with decreaing convection reitance, and hence with increaing h(v). COMMENS: o infer the actual ga temperature (1000 K) from the meaured reult (936 K), correction would have to be made for radiation exchange with the cold urrounding. What meaure may be taken to reduce the error aociated with radiation effect?