Net Force on a Body Completely in a Fluid. Natural Convection Heat Transfer. Net Buoyancy Force and Temperature
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1 Natral Conection eat ranfer Net Force on a Bo Comletel in a Fli he net force alie to a bo comletel bmere in a fli i Bo F W F net bo bo boanc V bo fli fli V V bo bo W F boanc Fli q he bo can be a blk of hot fli ame fli bt at a hiher temeratre. F V net hot fli fli bo q Bt hot fli < fli. herefore F net i neatie an it mean that the hot fli will moe. Aance eat ranfer ME one Shaban Aance eat ranfer ME one Shaban Volmetric hermal Eanion Coefficient te of chane of a nit olme of material er nit temeratre chane at contant rere i calle olmetric thermal eanion coefficient. Aance eat ranfer ME where i the ecific olme of the material. q Since / an therefore -/ P P q For an ieal a R R P R P P P R R R P P R P one Shaban Net Boanc Force an emeratre If V i the olme of a blk of hot fli, the net war force alie to thi blk of fli i where Δ hot fli - fli i the enit ifference between the hot an col fli. q A correlation between thi force an temeratre ifference can be obtaine b in an aroimate ereion for. Δ Δ Δ Δ Aance eat ranfer ME F net ΔV F net ΔV q hi how that the larer the temeratre ifference between hot an col fli the larer the net war force an therefore the troner the natral conection. one Shaban 5 Pae
2 Pae 6 Aance eat ranfer ME one Shaban Natral Conection Bonar aer Eqation Conier natral conection bonar laer alon a ertical late with temeratre >. Since the bo force in the irection i X-, bonar laer eqation are,, α If there i no bo force in irection, /. herefore 7 Aance eat ranfer ME one Shaban Natral Conection Bonar aer Eqation herefore, bonar laer eqation are α 8 Aance eat ranfer ME one Shaban Normalize Bonar aer Eqation et introce normalize ariable where i the lenth of the late an i a characteritic elocit. q he -comonent momentm eqation can be normalize a follow. Re Re Gr 9 Aance eat ranfer ME one Shaban Grahof an leih Nmber Grahof nmber i efine a Gr where i a characteritic lenth of the eometr, an an are rface an free tream temeratre. q It can be hown that Grahof nmber i a meare of ratio of boanc force to ico force actin on the fli. q leih nmber i the roct of Grahof an Prantl nmber. Pr Gr
3 Similarit Soltion of Bonar aer Eqation Bonar laer eqation can be tranforme into two orinar ifferential eqation b introcin imilarit ariable Gr η an rereentin the elocit comonent in term of a tream fnction efine a Gr ψ, f η he two orinar ifferential eqation for f an are f ''' ff '' f ' '' Pr f' an the bonar conition are η η f f ' f ' Similarit Soltion of Bonar aer Eqation Nmerical oltion of thee eqation are hown below. Aance eat ranfer ME one Shaban Aance eat ranfer ME one Shaban Similarit Soltion of Bonar aer Eqation he local Nelt nmber ma be eree a '' h [ q / ] N k k where '' k Gr q k η η herefore h Gr N Gr Pr k η η Pr Pr / 5 Pr Pr Aerae Nelt nmber oer the entire lenth of the late,, i h Gr N Pr N k Interal Soltion of aminar Natral Conection Bonar aer he interal form of the momentm an ener eqation for natral conection bonar laer are α et ame the elocit an temeratre rofile are ien b U an U i a reference elocit ch a maimm elocit in the bonar laer. Aance eat ranfer ME one Shaban Aance eat ranfer ME one Shaban Pae
4 C Interal Soltion of aminar Natral Conection Bonar aer Sbtittin into the interal eqation ie U U 5 α U If we ame UC m an C n, it can be hown that m / an n / an / / / / 5 Gr 5 Gr / Pr an C Pr Pr Bonar laer thickne an Nelt nmber are Aance eat ranfer ME one Shaban 6.95 Pr / Pr.9 Gr an N.58 Pr.95 Pr Interal Soltion of rblent Natral Conection Bonar aer he interal form of the momentm an ener eqation for trblent natral conection bonar laer are et ame the elocit an temeratre rofile are ien b Amin UC m an C n, it can be hown that /5 Pr /5 N.95 / /5.9Pr Aance eat ranfer ME /7 U τ q c " an one Shaban /7 5 Emirical Correlation for Vertical Plate For a ertical late with lenth the characteritic lenth. he natral conection flow i laminar if < 9 an i trblent if > 9. Aerae Nelt nmber correlation for an iothermal ertical late are N N.59. Aance eat ranfer ME / < 9 < one Shaban 9 < < where h N an Gr Pr k α he followin correlation ma be e for the entire rane of / 6.87 N.85 9/6 8/ 7 [.9/ Pr ] 6 Nelt Nmber Correlation aboe ot orizontal Plate For a horizontal late the characteritic lenth i A /P where A an P are rface area an erimeter of the late. Aerae Nelt nmber correlation are N N.5.5 Aance eat ranfer ME / 7 7 one Shaban 7 Pae
5 Nelt Nmber Correlation ner ot orizontal Plate Nelt Nmber Correlation for Cliner Vertical Cliner D he characteritic lenth i A /P where A an P are rface area an erimeter of the late. Aerae Nelt nmber correlation are N 5.7 he characteritic lenth i. If D 5/Gr /, e ertical late correlation. If cliner i thin, i.e. D<5/Gr /, N q orizontal Cliner he characteritic lenth i D. Aerae Nelt nmber, if D <, i ien b N h 7 Pr k 5 Pr / 6.87 D D.6 9/6.559/ Pr 7 5Pr 56 6Pr D 8/ 7 D Aance eat ranfer ME one Shaban 8 Aance eat ranfer ME one Shaban 9 Natral Conection in Enclore Natral Conection in Enclore Conction imit con h W N con h k / con α h k con con k / Aance eat ranfer ME one Shaban Aance eat ranfer ME one Shaban Pae 5
6 Natral Conection in Enclore all Enclore imit Natral Conection in Enclore Shallow Enclore imit.e N.E.E /. /. /. /.5 /.9 /. > con con k / Aance eat ranfer ME one Shaban < /.E /.E.E.E.E.E5.E6.E7 Aance eat ranfer ME one Shaban Natral Conection in Enclore Bonar aer Reime / < <.5. h k N Aance eat ranfer ME C one Shaban. C.. /7 /7 / Pae 6
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