Forced onvection Heat raner onvection i the mechanim o heat traner through a luid in the preence o bul luid motion. onvection i claiied a natural (or ree) and orced convection depending on how the luid motion i initiated. In natural convection, any luid motion i caued by natural mean uch a the buoyancy eect, i.e. the rie o warmer luid and all the cooler luid. Wherea in orced convection, the luid i orced to low over a urace or in a tube by eternal mean uch a a pump or an. Mechanim o Forced onvection onvection heat traner i complicated ince it involve luid motion a well a heat conduction. he luid motion enhance heat traner (the higher the velocity the higher the heat traner rate). he rate o convection heat traner i epreed by Newton law o cooling: q Q conv conv h W / m ha W he convective heat traner coeicient h trongly depend on the luid propertie and roughne o the olid urace, and the type o the luid low (laminar or turbulent). V Q conv V Zero velocity at the urace. Q cond Solid hot urace, Fig. 1: Forced convection. It i aumed that the velocity o the luid i zero at the wall, thi aumption i called nolip condition. A a reult, the heat traner rom the olid urace to the luid layer adjacent to the urace i by pure conduction, ince the luid i motionle. hu, M. Bahrami ENS 388 (F09) Forced onvection Heat raner 1
q conv q q conv luid y h cond y0 h luid y y0 W / m. K he convection heat traner coeicient, in general, varie along the low direction. he mean or average convection heat traner coeicient or a urace i determined by (properly) averaging the local heat traner coeicient over the entire urace. Velocity Boundary ayer onider the low o a luid over a lat plate, the velocity and the temperature o the luid approaching the plate i uniorm at U and. he luid can be conidered a adjacent layer on top o each other. Fig. : Velocity boundary layer. Auming no lip condition at the wall, the velocity o the luid layer at the wall i zero. he motionle layer low down the particle o the neighboring luid layer a a reult o riction between the two adjacent layer. he preence o the plate i elt up to ome ditance rom the plate beyond which the luid velocity U remain unchanged. hi region i called velocity boundary layer. Boundary layer region i the region where the vicou eect and the velocity change are igniicant and the invicid region i the region in which the rictional eect are negligible and the velocity remain eentially contant. he riction between two adjacent layer between two layer act imilar to a drag orce (riction orce). he drag orce per unit area i called the hear tre: V y y0 N / m where μ i the dynamic vicoity o the luid g/m. or N./m. Vicoity i a meaure o luid reitance to low, and i a trong unction o temperature. he urace hear tre can alo be determined rom: M. Bahrami ENS 388 (F09) Forced onvection Heat raner
U N / m where i the riction coeicient or the drag coeicient which i determined eperimentally in mot cae. he drag orce i calculated rom: U FD A N he low in boundary layer tart a mooth and treamlined which i called laminar low. At ome ditance rom the leading edge, the low turn chaotic, which i called turbulent and it i characterized by velocity luctuation and highly diordered motion. he tranition rom laminar to turbulent low occur over ome region which i called tranition region. he velocity proile in the laminar region i approimately parabolic, and become latter in turbulent low. he turbulent region can be conidered o three region: laminar ublayer (where vicou eect are dominant), buer layer (where both laminar and turbulent eect eit), and turbulent layer. he intene miing o the luid in turbulent low enhance heat and momentum traner between luid particle, which in turn increae the riction orce and the convection heat traner coeicient. Non dimenional Group In convection, it i a common practice to non dimenionalize the governing equation and combine the variable which group together into dimenionle number (group). elt number: non dimenional heat traner coeicient h q q where δ i the characteritic length, i.e. D or the tube and or the lat plate. elt number repreent the enhancement o heat traner through a luid a a reult o convection relative to conduction acro the ame luid layer. Reynold number: ratio o inertia orce to vicou orce in the luid Re inertia orce vicou orce conv cond V V At large Re number, the inertia orce, which are proportional to the denity and the velocity o the luid, are large relative to the vicou orce; thu the vicou orce cannot prevent the random and rapid luctuation o the luid (turbulent regime). M. Bahrami ENS 388 (F09) Forced onvection Heat raner 3
he Reynold number at which the low become turbulent i called the critical Reynold number. For lat plate the critical Re i eperimentally determined to be approimately Re critical = 10. andtl number: i a meaure o relative thicne o the velocity and thermal boundary layer molecular diuivity o momentum p molecular diuivity o heat where luid propertie are: ma denity : ρ, (g/m 3 ) peciic heat capacity : p (J/g K) dynamic vicoity : µ, (N /m ) inematic vicoity : ν, µ / ρ (m /) thermal conductivity :, (W/m K) thermal diuivity : α, /(ρ p ) (m /) hermal Boundary ayer Similar to velocity boundary layer, a thermal boundary layer develop when a luid at peciic temperature low over a urace which i at dierent temperature. Fig. 3: hermal boundary layer. he thicne o the thermal boundary layer δ t i deined a the ditance at which: 0.99 he relative thicne o the velocity and the thermal boundary layer i decribed by the andtl number. For low andtl number luid, i.e. liquid metal, heat diue much ater than momentum low (remember = ν/α<<1) and the velocity boundary layer i ully contained within the thermal boundary layer. On the other hand, or high andtl number luid, i.e. oil, heat diue much lower than the momentum and the thermal boundary layer i contained within the velocity boundary layer. M. Bahrami ENS 388 (F09) Forced onvection Heat raner 4
Flow Over Flat Plate he riction and heat traner coeicient or a lat plate can be determined by olving the conervation o ma, momentum, and energy equation (either approimately or numerically). hey can alo be meaured eperimentally. It i ound that the elt number can be epreed a: h Re where, m, and n are contant and i the length o the lat plate. he propertie o the luid are uually evaluated at the ilm temperature deined a: aminar Flow he local riction coeicient and the elt number at the location or laminar low over a lat plate are, h 0.664 Re 0.33 Re 1/ 1/ m n 0.6 where i the ditant rom the leading edge o the plate and Re = ρv / μ. he averaged riction coeicient and the elt number over the entire iothermal plate or laminar regime are: h 1.38 Re 0.664 Re 1/ 1/ 0.6 aing the critical Reynold number to be 10, the length o the plate cr over which the low i laminar can be determined rom urbulent Flow Re cr 10 V he local riction coeicient and the elt number at location or turbulent low over a lat iothermal plate are: cr M. Bahrami ENS 388 (F09) Forced onvection Heat raner
, h 0.09 Re 0.096 Re 1/ 4 / 10 Re 0.6 60 10 10 Re 10 he averaged riction coeicient and elt number over the iothermal plate in turbulent region are: h 0.04 Re 0.03 Re 1/ 4 / 10 Re 0.6 60 10 ombined aminar and urbulent Flow 10 Re 10 I the plate i uiciently long or the low to become turbulent (and not long enough to diregard the laminar low region), we hould ue the average value or riction coeicient and the elt number. cr 1 0 cr 1 h h, 0,, a min ar a min ar d d cr h cr,, urbulent,, urbulent where the critical Reynold number i aumed to be 10. Ater perorming the integral and impliication, one obtain: 1/ h 0.03 Re 0.04 14 Re Re 4 / 81 10 Re 0.6 60 10 d d 10 Re 10 he above relationhip have been obtained or the cae o iothermal urace, but could alo be ued approimately or the cae o non iothermal urace. In uch cae aume the urace temperature be contant at ome average value. For iolu (uniorm heat lu) plate, the local elt number or laminar and turbulent low can be ound rom: h 0.43Re h 0.0308 Re 0. 0.8 aminar (iolu plate) urbulent (iolu plate) Note the iolu relationhip give value that are 36% higher or laminar and 4% or turbulent low relative to iothermal plate cae. M. Bahrami ENS 388 (F09) Forced onvection Heat raner 6
Eample 1 Engine oil at 60 low over a m long lat plate whoe temperature i 0 with a velocity o m/. Determine the total drag orce and the rate o heat traner per unit width o the entire plate. oil = 60 V = m/ Q =0 = m We aume the critical Reynold number i 10. he propertie o the oil at the ilm temperature are: he Re number or the plate i: 40 3 86 g / m 0.144 W /( m. K) 80 6 410 m / Re = V / ν = 4.1310 4 which i le than the critical Re. hu we have laminar low. he riction coeicient and the drag orce can be ound rom: F D 1.38 Re V A 0. 0.0063 0.0063 he elt number i determined rom: 3 g m m m 86 / / 1.N h hen, W h. m K Q ha 0664 Re 11040W 0. 1918 M. Bahrami ENS 388 (F09) Forced onvection Heat raner
Flow acro ylinder and Sphere he characteritic length or a circular tube or phere i the eternal diameter, D, and the Reynold number i deined: V D Re he critical Re or the low acro phere or tube i 10. he approaching luid to the cylinder (a phere) will branch out and encircle the body, orming a boundary layer. Fig. 4: ypical low pattern over phere and treamlined body and drag orce. At low Re (Re < 4) number the luid completely wrap around the body. At higher Re number, the luid i too at to remain attached to the urace a it approache the top o the cylinder. hu, the boundary layer detache rom the urace, orming a wae behind the body. hi point i called the eparation point. o reduce the drag coeicient, treamlined bodie are more uitable, e.g. airplane are built to reemble bird and ubmarine to reemble ih, Fig. 4. In low pat cylinder or phere, low eparation occur around 80 or laminar low and 140 or turbulent low. V FD D AN N AN : rontal area where rontal area o a cylinder i A N = D, and or a phere i A N = πd / 4. M. Bahrami ENS 388 (F09) Forced onvection Heat raner 8
he drag orce acting on a body i caued by two eect: the riction drag (due to the hear tre at the urace) and the preure drag which i due to preure dierential between the ront and rear ide o the body. A a reult o tranition to turbulent low, which move the eparation point urther to the rear o the body, a large reduction in the drag coeicient occur. A a reult, the urace o gol ball i intentionally roughened to induce turbulent at a lower Re number, ee Fig.. Fig. : Roughened gol ball reduce D. he average heat traner coeicient or cro low over a cylinder can be ound rom the correlation preented by hurchill and Berntein: 1/ / 8 hd 0.6 Re Re yl 0.3 1 / 3 1/ 4 1 0.4 8,000 where luid propertie are evaluated at the ilm temperature = ( + ) /. For low over a phere, Whitaer recommended the ollowing: Sph 4 / 0.4Re 1 / 0.06 Re / 3 0.4 / 1/ 4 hd / M. Bahrami ENS 388 (F09) Forced onvection Heat raner 9
which i valid or 3. < Re < 80,000 and 0. < < 380. he luid propertie are evaluated at the ree tream temperature, ecept or μ which i evaluated at urace temperature. he average elt number or low acro circular and non circular cylinder can be ound rom able 10 3 engel boo. Eample he decorative platic ilm on a copper phere o 10 mm diameter i cured in an oven at. Upon removal rom the oven, the phere i ubjected to an air tream at 1 atm and 3 having a velocity o 10 m/, etimate how long it will tae to cool the phere to 3. P = 1 atm. V = 10 m/ = 3 opper phere D = 10 mm i = = 3 Aumption: 1. Negligible thermal reitance and capacitance or the platic layer.. Spatially iothermal phere. 3. Negligible Radiation. opper at 38 K ρ = 8933 g / m 3 = 399 W / m.k p = 38 J / g.k Air at 96 K μ = 181.6 10 N. / m v = 1.36 10 6 m / = 0.08 W / m.k = 0.09 μ = 19.8 10 N. / m he time required to complete the cooling proce may be obtained rom the reult or a lumped capacitance. V t ha P i ln p D i ln 6h Whitaer relationhip can be ued to ind h or the low over phere: Sph where Re = ρvd / μ = 610. Hence, 0.4Re 1 / 0.06 Re / 3 0.4 / 1/ 4 hd / M. Bahrami ENS 388 (F09) Forced onvection Heat raner 10
Sph h hd / 0.4(610) D 1 W / m K 1/ he required time or cooling i then t 0.06(610) / 3 3 8933g / m 38J / g. K 0.01m 61W / m. K (0.09) 0.4 181.6 10 19.810 3 ln 69. ec 3 3 1/ 4 4.4 M. Bahrami ENS 388 (F09) Forced onvection Heat raner 11