The Effect of Density Variation on Heat Transfer in the Critical Region

Transcription

1 The Effect of Density Vaiation on Heat Tansfe in the Citical Region YIH-YUN HSU 1 J. M. SMITH Nothwesten Univesity, Evanston, 111. The heat-tansfe coefficient between fluid and tube wall in tubulent flow depends upon the physical and themal popeties of the fluid. When density changes acoss the diamete of the tube ae lage (fo example, when the fluid is nea the citical point), the vaiable density can affect the tansfe of momentum and heat. Equations ae developed fo pedicting the magnitude of this effect on the heat-tansfe coefficient. Deissle's [5] 2 expessions fo the ed diffusivity ae employed in solving the equations fo heat and momentum tansfe. Fo flow in vetical tubes lage density vaiations can also affect the heat tansfe by inducing natual convection. By consideing the influence of bo foces on the shea stess, equations ae deived to pedict the effect of natual convection on the heat-tansfe coefficient fo tubulent flow. The esults indicate that the effect is significant only fo elatively high Gashof numbes and low Reynolds numbes. Such conditions may be encounteed inflow of a fluid nea its themonamic citical point. The deived equations ae applied fo cabon dioxide flow in the citical egion unde the conditions fo which expeimental data wee measued by Binge and Smith [2], Because of the high Reynolds and low Gashof membes, natual convection is not significant. Howeve, the effect of the lage density vaiations is found to be significant, and the pedicted esults agee well with the expeimental data. H e a t tansfe to fluids in tubulent flow in tubes is affected by vaiations in the physical and themal popeties of the fluid. When the fluid is nea its themonamic citical point, the change in popeties with tempeatue is paticulaly lage; fo example, the specific heat at constant pessue ises to an infinite value at the citical point. The conventional pocedues [3, 13] fo handling popety vaiations in equations fo heal^tansfe coefficients ae not applicable in the citical egion. Deissle's analogy [4], which is based upon a moe fundamental viewpoint, but neglects two effects of density changes, has been found [2] satisfactoy fo modeate popety vaiations. Howeve, when density changes ae lage, as, fo example, nea the 1 Pesent addess: NASA Laboatoy, Cleveland, Ohio. 2 Numbes in backets designate Refeences at end of pape. Contibuted by the Heat Tansfe Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS and pesented at the ASME- AIChE Heat Tansfe Confeence, Buffalo, N. Y., August 15-17, Manuscipt eceived at ASME Headquates, Apil 20, Pape No. 60 HT-8. citical point, these effects may not be negligible. The fist is the influence of density vaiations on the adial tansfe of momentum and heat between adjacent elements of fluid. The second is due to the vaiation in the foce of gavity on the fluid acoss the tube adius. This natual-convection contibution is consideed hee only fo vetical tubes. The objectives of this pape ae to develop equations fo pedicting the magnitude of each of these two effects on heat tansfe in tubulent flow in tubes. Effect of Density Vaiation on the Tansfe of Momentum and Heat Fo a fluid in the citical egion, the density deceases apidly with inceasing tempeatue. Theefoe in a flow with tempeatue and velocity gadients, the momentum in the axial diection of two elements of fluid at diffeent adii will be diffeent, not only because of the diffeences in velocity, but also because of -Nomenclatuea = const in Equation (17), dimensionless CT = specific heat at const pessue, Btu(lb deg F) F = density vaiation facto, defined by Equations (7) o (13), dimensionless K k L n _ 2o. Nu f. = fiction facto =, dimen- P.C's sionless g = acceleation due to gavity, fth 2 G = Gashof numbe - (Pb ~ Po) Mo 2 p0 3 g, dimensionless h = heat-tansfe coefficient, Btu(h sq ft deg F) A P P. q di- a const = 0.36, dimensionless + dimensionless adius themal conductivity, Btu(h ft deg F) ( P, ) ( Mo ) Pandtl mixing length, defined by 2pbub Equation (1), ft Re = Reynolds numbe, a const = 0.124, dimensionless Mo mensionless Nusselt numbe,, dimension- 1, K tempeatue, deg R less t = tempeatue, deg F pessue dop pe unit length of tube, lb(sq ft h 2 ) Pandtl numbe, Cp0p.0k0, dimensionless heat flux, Btuh sq ft tube adius, ft ' dimensionless tempeatue 0 V tj U = mean velocity in the axial (x) diection, fth (Continued on next page) 176 MAY Tansactions of the ASME Copyight 1961 by ASME Downloaded Fom: on Tems of Use:

2 diffeences in density. This situation may be expessed mathematically by stating with Pandtl's concept of a mixing length L fo the tansfe of momentum in tubulent flow. If the density is constant whee T = wl d(pv) du du TI = wlp = em,p emi = wl Hee e - is the conventional ed diffusivity of momentum. If the density is vaiable, Equation (1) may be witten: T du T Whp em = wl 1 + U and an ed diffusivity fo vaiable density defined by the expession : P M p[~). du\ In tems of the conventional ed diffusivity emi Equations (3) and (5) indicate that the moe geneal diffusivity em is given by: whee em = «m,(l + Fm) F = U dp <2(ln p) du d(ln U) In Equation (6) Fm may be consideed a coection tem which is to be applied to the constant-density diffusivity. Deissle [5] has developed the following equations fo emi: y+ < 26 emi = n*uy( 1 - e-" 2^") ( du y \ ) y+ > 26 emi = IV (amy \ *) (1) (2) (3) (4) (5) (6) (7) (8) (9) The expession fo flow away fom the tube wall, y+ > 26, is von Kaman's elationship [10], Deissle used n = and K 0.36, based upon velocity measuements fo the adiabatic flow of ai. These equations will be used in Equation (6) to define diffusivities em fo the geneal, vaiable density case. In dimensionless fom the expessions become, fo y + < 26,.,= (1 + FJn 2 ^ U * y [ 1 - e^'f^ww] (10) and fo y+> 26 ( du+ V V + ) d 2 U + Y \ +* ) (11) Simila consideations fo the themal ed diffusivity e, lead to equations analogous to Equations (6) and (7), namely, e, = e (l + Ft) d(lnp) F, = d(in CPT) (12) (13) It will be supposed that em = e, = e, a easonable assumption at high Peclet numbes. Then F, and Fm can both be evaluated fom Equation (7). In dimensionless fom this expession may be witten: F L m = 1 F, t = <2(ln p) + d(ln U+) (14) It is possible to evaluate F fom the vaiation of density with tempeatue by an appoximate pocedue. It is noted fom the computed tempeatue and velocity pofiles (fo example, see Fig. 1) that the slopes of the two cuves ae appoximately the same, and that this equality would be moe exact if the slopes of In t + and In U + wee consideed. Thus as an appoximation: dqn U+) ^ d(in t+) * (15) Utilizing Equation (15) and the definition of the dimensionless tempeatue, Equation (14) becomes: V = Nomenclatueu+ = dimensionless velocity -'A slope of dimensionless velocity pofile, du + + velocity component pependicula to the diection of flow, fth distance along axis of tube, ft distance fom the tube wall, ft dimensionless distance v ( Po ) ( ) a constant in equation (17), dimensionless p =, dimensionless kpo T o1 0 e, = total ed diffusivity of momentum, defined by Equation (6), ft 2 h m( = ed diffusivity of momentum fo constant density, defined by Equation (2), ft 2 h e( = total ed diffusivity of heat, defined by Equation (12), ft 2 h el{ = ed diffusivity of heat fo constant density, ft 2 h ji = viscosity, lbh ft p = density, lbft 3 T = shea stess, lb ft(h 2 sq ft) Subscipts b = denotes bulk mean value i = denotes constant density m = denotes momentum o = denotes value of quantity at the tube wall t = denotes heat Supescipt + = denotes dimensionless fom Jounal of Heat Tansfe MAY Downloaded Fom: on Tems of Use:

3 10 0» WALL f* DISTANCE MRAMETER Fig. 1 Tempeatue and velocity pofiles ((3 = ; to deg F) cabon dioxide no (a) dominating the heat-tansfe ate; and (ft) having a negligible effect on the esult. Jackson, et al. [9], consideed the effect of gavitational attaction fom an ove-all viewpoint by empiically modifying the usual tubulent flow equations fo heat tansfe. Hanatty, et al. [7], have shown that the velocity in the diection of foced flow could, in the sevee case, be so affected by natual convection that a evesal in flow diection would occu. Ostach [11] and Pigfod [12] analj'tically teated the effect of natual convection in the lamina flow poblem. An appoximate analytical solution of the poblem of tubulent flow in tubes can be developed by examining the effect of bo foce on the distibution of shea stess in the fluid. This is done in the following paagaphs. Conside a ing-shaped, diffeential volume of diamete ( y) and height Ax. A foce balance on a unit aea pependicula to the diection of flow yields: d ~ V (21) F = i(lnp) \1 - Pt + ) _P_ Po d(lnp) d(ln T) (16) The vaiation of density with tempeatue can be epesented by the equation: (17) whee a and a ae constants. By utilizing diffeent sets of values fo a and a, the density can be accuately expessed b} Equation (17) ove a wide tempeatue ange. Combining Equations (16) and (17) and noting that 3i + is usually small with espect to unity leads to the following simple expession: F = a3f. + With this equation fo F, Equations (10) and (11) povide a means fo evaluating the ed diffusivity fo vaiable density flow. Then the tempeatue and velocity pofiles acoss the tube can be obtained by integating the equations fo momentum and enegy tansfe: du T = (p. + pe) dl 2 = -(ft + CpPe) (19) (20) The solution of this equation, with the bounday condition = 0 aty = 0, is: - y 0 + ~ i A P\ pg + ^-J (V - ) (22) - ' Jo y - f Futhe, a foce balance on the entie coss section of the tube gives: AP 2T -7 Ax = PUB whee the bulk mean density is defined by the integal: f Pb = ; I 2 p(>' - v) Jo ( 18) Combining Equations (22) and (23) yields: T (23) (24) - y i < + (P - Pb)g('J - ) (25) o(v ~ ) Jo Reducing Equation (25) to dimensionless fom, and intoducing the Gashof numbe (defined in the Nomenclatue) gives the fom: To V +3 +-y+j0 pb - p0 - y+)+ (26) The integation pocedue equies appoximation because of the numbe of vaiables in the equations. Howeve, Deissle [5] has developed a pocedue which gives heat-tansfe coefficients in good ageement with expeimental data fo constant and modeately vaiable popeties [2, 6], The pocedue includes assuming that the vaiation of viscosity and specific heat with tempeatue can be epesented by equations analogous to Equation (17). This same method has been employed in this pape to compute pofiles fo conditions of lage density vaiations. The esults ae pesented in a late section. Effect of Natual Convection Acivos [1 ] attacked the poblem of natual convection supeimposed upon foced convection by using von Kaman's momentum integal method ove a bounday laye. He solved the poblem on this basis fo the exteme cases of natual convection Equations (25) o (26) satisfy the following thee conditions fo shea stess distibution in flow inside a tube: (а) (б) (c) T T = 0 ~ V if p = a const if y = 0 if y = Two points ae of inteest in Equation (26). Fist, the goup (P Pb)(Pb ~ Po) is a function of tempeatue only (at constant pessue). Second, the fist tem on the ight is the linea shea stess distibution fo constant density, and the second is the contibution due to vaiable density. In tems of the Reynolds numbe (see Equation 35) and fiction facto, Equation (26) may be expessed: 178 MAY Tansactions of the ASME Downloaded Fom: on Tems of Use:

4 Ta + f Jo y+ < 26 G Re,';' + 16 V2 V - y + Jo P - Pb Pb - Po 0+ -!+ ) + (27) The fiction facto is not vey sensitive to the Reynolds numbe in tubulent flow. Hence the impotance of the second tem on the ight side of the equation will be detemined by GRe0 3. If G Re, the effect of bo foce is negligible. If G is lage, the shape of the shea foce pofile will change and in the exteme case, TT0 will become negative. This means that a maximum in the velocity pofile would exist between y + = 0 and y = 4. The citeia fo the existence of such a maximum ib consideed in efeence [8]. Velocity and Tempeatue Pofiles Fo flow close to the wall, Equation (10) fo e, and Equation (19) detemine the velocity distibution. In dimensionless fom, the esult may be witten: To + (1 + F)n*u + y+ll - e~ n ' u+ v + '"">'"i i ] Ho Po (28) Fo constant density, the coection facto F is zeo, and TT is given by the fist linea tem of Equation (26). Deissle [4] has computed velocity pofiles on this basis and concluded that assuming TT0 = 1 gives esults in close ageement with the linea shea foce distibution. To take into account the effect of density changes, F is evaluated fom Equation (18) and Equation (26) is used to epesent TT as a function of Gashof numbe. Since the popety vaiations ae a function of tempeatue, it is necessay to solve simultaneously Equation (28) and the analogous expession fo the tempeatue distibution. Fo flow at y + > 26, Equation (11) fo em can be used to evaluate the effect of F on the velocity pofile, but not the effect of Gashof numbe. This is because the von Kaman equation fo emi will not pedict the potential maximum in the velocity at y+ < +. Hence fo this egion Pandtl's expession fo diffusivity will be employed: du e, = 2! 2 (29) Then fom Equation (19), in dimensionless fom, the slope of the velocity pofile may be expessed: du + Mo m + + 4(1 + F) K*y + * Po To 2 K*y+! (1 + F) Po Except in a naow egion nea 0 = 0, M «2 K+y + j l (1 + F) L Po To With this estiction, Equation (30) becomes: Ky + L p T0 \1 + F) _ A 'A VA (30) (31) This simplified equation educes to the equied value of v + = 0 at TT0 = 0 and will give values of v + close to zeo as TT appoaches zeo. Hence Equation (31) is used to compute the velocity at all y + geate than 26. Equations simila to Equation (28) and (31) wee developed fo computing the tempeatue pofile. Fo flow nea the wall, Equation (10) was used fo the diffusivity, while at y + > 26 Pandtl's expession [Equation (29)] was employed. Specific velocity and tempeatue pofiles wee detemined by these equations fo cabon dioxide in tubulent flow at 1200 psia (citical pessue = 1071 psia). This system was chosen because expeimental heat-tansfe data ae available as well as physical popety infomation [2], Since density vaiations ae geatest when the fluid is in the egion of the tansposed citical tempeatue, pofiles wee computed fo these conditions. The tansposed citical tempeatue (tempeatue fo which the specific heat is a maximum) fo cabon dioxide at 1200 psia is 98 deg F. The following wall tempeatues and heat fluxes (as epesented by fl values) insued that 98 cleg F was between the wall and bulk mean tempeatues: Wall tempeatue to, deg F (3 (See Nomenclatue fo definition) Fig. 1 illustates the computed pofiles fo t = 110 deg F and j3 = unde conditions such that the effect of natual con GO IOO looo 2000 Dimensionless Distance fom Wall, y* IOO IOOO Dimensionless Distance fom Wall.y* Fig. 2 Effect of bulk density on velocity and tempeatue pofiles Fig. 3 Effect of bulk density on velocity and tempeatue pofiles Jounal of Heat Tansfe MAY Downloaded Fom: on Tems of Use:

5 vection is neglected. That is, Equations (28) and (31) wee employed, along with the coesponding expessions fo the tempeatue, but TT0 was assumed equal to unity. The calculations wee caied out on an IBM-650 compute. The physical popeties of cabon dioxide used wee the same as those pesented by Binge [2], Fo compaison, the cuves neglecting density vaiations, F = 0, ae also included. The esults show that including the density vaiation facto F flattens both the tempeatue and velocity cuves. The pofiles at othe l0 and j8 values exhibit simila esults. Figs. 2 and 3 show pofiles fo the same conditions (t0 = 110 deg F, 0 = ), except the effect of natual convection is taken into account. That is, TT 0 in Equations (28) and (31) was not taken equal to unity but evaluated fom Equation (26). Fig. 2 is based upon a value of 10~ 2 fo the paamete G(?-+) 3 and includes esults fo thee mean densities. Compaison of the cuves in Figs. 1 and 2 shows that the effect of natual convection is to flatten futhe the velocity pofile. Fig. 3 applies when the Gashof numbe is inceased such that G( + ) 3 = In this case, natual convection is sufficiently impotant that the velocity cuves exhibit a maximum at y + < Heat-Tansfe Results Fom the velocity and tempeatue pofiles, bulk mean values, uh + and tb +, can be deived by integation of the equations: and ub 2_ * ~ Jo ( J > ) Jo \C,j>J t+ U + ( + - y + )^ Jo P \CPopJ (32) (33) Fom these mean values, the Reynolds and Nusselt numbes (with popeties evaluated at the wall tempeatue) ae detemined as follows: 2 h 2+P0 Nu = = K Re = 2UbPo Mo 1 ], = 2 +Uh+ : Y 5 R HI II ICAL ALU! s E TI \ REYNOLD NUMBER, Re. (34) (35) Fig. 4 shows the esultant Nusselt vesus Reynolds numbe elationship based upon the pofiles in Fig. 1. Thee computed Fig. 4 Compaison of theoetical and expeimental values of Nusselt numbe Nu<, vesus Reynolds numbe Re (3 = , f0 = 110 deg F) cabon dioxide lines ae indicated. The top line includes the effect of density vaiation on momentum and heat tansfe (F ^ 0); the second neglects this effect (F = 0), and the lowest line indicates the esult expected when all popety changes ae neglected (0 = 0). The expeimental points plotted in Fig. 4 coespond to the measuements of Binge [2]. It is seen that taking into account the effect of density vaiations impoves the ageement with the expeimental data. The influence of natual convection would be to incease the Nusselt numbe above the cuve fo F ^ 0 in Fig. 4. Howeve, it will be obseved late that the Gashof numbe, at Binge's opeating conditions, was too low fo natual convection to be significant. Plots simila to Fig. 4 (given in efeence [8]) at t = 130 deg F, 3 = 0.006, and t = 110 deg F, 3 = also show impoved ageement with the data when F is not neglected. These computed and expeimental esults ae based upon heating the gas; i.e., heat is tansfeed fom the tube wall into the flowing steam. Unde these conditions, F is positive, and the effect is to incease the Nusselt numbe. If the gas is cooled, F becomes negative, and the Nusselt numbe and the heat-tan8fe coefficient ae educed. Fig. 5 pesents the Nusselt vesus Reynolds numbe lines evaluated fom the velocity and tempeatue pofiles given in Figs. 2 and 3. Also included is the esult fo G( + ) 3 = 0. This line, which neglects natual convection, is the same as the line fo F ^ 0 at the top of Fig. 4. Fig. 5 cannot be employed to evaluate eadily the effect of natual convection, because the paamete G( + ) 3 is a function of Reynolds numbe as well as the Gashof goup. A moe useful method of pesentation is Nu vesus Re with G alone as a paamete. The esults ae shown in this fom in Fig. 6. The line fo G = 0 is again the same as the line in Fig. 4 fo F ^ 0. It is of inteest to conside the possible influence of natual convection on the cabon dioxide data published by Binge [2], The measuements wee made by heating cabon dioxide at 1200 psia in a vetical tube, 0.18-in. ID. Fo a wall tempeatue of 110 deg F,? = (bulk tempeatue 95 deg F), the Gashof numbe is: Pi - P» G = M» looo z m a 50 z to _ (16.5)(32.2)(3600) 2 (7.5 X lo" 3 )' 0.05 = 1.1 X 10 6 f, V, - CARBON DIOXIDE 1200 ptio l.» IIO'F () - O.OOI * * Reynolds* Numbe, Re,, Fig. 5 Nusselt numbe as a function of Reynolds numbe with G( + ) 3 as a paamete 180 MAY Tansactions of the ASME Downloaded Fom: on Tems of Use:

6 I~O ,,0. ~ ~ ~ LL,J V ' 1---\0-- V ' c i... ~!..--::::;;; - ~ "o~~ ~,,\\... ~~ ~...,O,...--~,...\J;::~ 'l ~...\(;) V k?' b - EXP. IlATA(Z ) CARBON DIOXIDE 1200 plio ~ - "O. ; I 10.0 ;) 4 6 a 10 2 l 4, 10" R,ynoldS' Numb. I R 9 T. W. Jackson, et ai., TRANS. ASME, vol. 80, 1958, pp Th. von Kaman, NACA, Technical Memo. 611, S. Ostach NACA, T echnical Note 3144, R. L. Pigfod, AIChE Heat Tansfe Symposium, St. Louis, Mo., Decembe 13-16, E. N. Seide and G. E. Tate, Indtlstial and Enoineeting Chem- 1'sty, vol. 28, 1936, p Fig.6 Nusselt numbe vesus Reynolds numbe with Gashof numbe as paamele The density and viscosity values wee obtained fom Binge [2]. Refeence to Fig. 6 shows that natual convection would not effect the Nusselt numbe unde these conditions, unless Reo was less than about 20,000. Binge's data wee obtained at Reo geate than 45,000. The expeimental values ae shown by the cicled points in Fig. 6 and epesent the same data as indicated in Fig. 4. Conclusions The poposed method of pedicting the influence of density vaiations on the tansfe of momentum and heat indicates a significant effect if the fluid is in the citical egion. The esult fo heating is to flatten the velocity and tempeatue pofiles and incease the heat-tansfe coefficient at a given Reynolds numbe. The limited expeimental data available indicate that the poposed method is an impovement ove the constant-density appoach. Additional measnements ae desiable to solidify this conclusion. The analysis poposed fo the influence of natual convection shows that this effect also flattens the velocity pofile and inceases the heat-tansfe coefficient. Again, the effect would be significant only when thee is a lage vaiation in density acoss the tube adius, as when the fluid is nea its citical point. Expeimental data ae needed at elatively low Reynolds numbes and high values of the Gashof goup in ode to test the analysis pesented. Acknowledgment This investigation was caied out unde the sponsoship of the Office of Odnance Reseach, Contact No. DA-ll-022-0RD Refeences 1 A. Acivos, AlChE Jounal, vol. 4, 1958, p R. P. Binge and J. M. Smith, AIChE Jounal, vol. 3, 1957, p A. P. Colbun, Tans. AlChE, vol. 29, 1933, p R. G. Deissle, TRANS. ASME, vol. 76, 1954, pp R. G. D eissle, NACA, Technical Note 2138, R. G. Deissle, NACA, Technical Note 3145, T. J. Hanatty, E. M. Rosen, and R. L. Kabel, Industial and Enoineeino Chemisty, vol. 50, 1958, p Yih-yull Hsu and J. M. Smith, Inteim epot No.2, OOR Contact DA-ll-022-0RD-2674, July I- Decembe 31, Jounal of Heat Tansfe MAY Downloaded Fom: on Tems of Use: