460
INDUSTRIAL A N D ENGINEERING CHEMISTRY
Vol. 16, No. 5
Forced Convection of Heat in Gases and Liquids-II’~z
B I D = K (p/pDv)a (6) which is the simpler expression assumed for both gases and liquids in the previous paper.3 The writer has recently found that the same expression was previously published By Chester W. Rice in the valuable contribution to the subject of heat transfer GENERAL ELECTRIC Co., SCHENECTADY. N. Y. by McAdams and Frost.4 For thin films and moderate temperature differences, we STUDY of a fund of valuable work on the heat transfer take to liquids flowing in pipes, which has recently come to W , = ( A / B ) k,,,. At (watts) (7) the writer’s attention, showed that the simple as- Substituting Equation 5 in 7 we obtain the following general sumptions of his previous paper3 failed to correlate the new expression for heat transfer: data on liquids with the data on gases. In the present W , = (Ak t/Ko D) ( w p / K ) B (pDv/p)a (watts) (8) paper a more general expression for the film thickness in forced convection is developed, which appears to correlate Equation 8 is the same as that obtained by Boussinesq,6 satisfactorily the available data on convection in both gases Nusselt,G and Davis.’ and liquids. For an ideal gas, the factor ,ucp/k is independent of temperaThe film theory naturally leads to a relation between skin ture and pressure over a wide range and does not vary greatly friction and heat transfer which enables us to calculate the from one gas to another. As an approximation we may forced convection from the available data on surface friction. therefore omit this factor when dealing with perfect gases A study of the problem from this point of view is made which and obtain seems to throw considerable new light on the complex problem W , = (AkAt/KD)(pDv/p)a (watts) (9) of turbulent fluid motion. This relation was tested over a wide range of conditions in GENERAL EXPRESSION FOR EFFECTIVE FILMTHICKNEBS different gases in the previous paper13with good agreemdnt. Therefore, all the work in the previous paper on ideal gases IN FORCED CONVECTION is in agreement with the present findings and requires no Assume that the ratio of the effective film thickness to the modification. Because the few rough tests in liquids apsize of the body is given by peared to fall in line with the gases, the general applicability of the equations to all fluids was assumed. In the light of BID = K Opap’ De vn:G K B At? (numeric) (1) the present work we find that in the case of liquids and where B = effective film thickness (L) mixtures of a light with a heavy gas the previous equaD = a characteristic linear dimension ( L ) tions will require revisipn. K O = numerical constant p = density of fluid In liquids, the factor Ic/,ucp varies rapidly with the temq , = viscosity of fluid (L”lMT-l) perature and may be widely different for various liquids. v = velocity of fluid (LT-’) For air, k / p c p = 1.35; for water a t 16’ C. the value is apc - specific heat per gram a t constant pressure (HM-l8-1) proximately 0.135, or one-tenth that for air. For water a t heat conductivity of flui! (HT-1 L-18-1 ) Ai = temperature difference, C. (8-l) 80” C. it is approximately 0.42, or three times that a t 16” C. Substituting the dimensions of the quantities in terms of It is therefore evident that the effect of this factor can readily be determined by comparing forced convection tests in air mass, length, and time, in Equation 1 with those in water at various temperatures under given flow conditions. FORCED HEATCONVECTION IN PIPES From the above we obtain the following: A representative portion of the available data by the ByL, 0 = - 3 a - ~ + ~ + n L @ various investigators has been collected in Table I. M, O = a + y - - m The data by Soennekenn have been taken from Stanton1s9 T, 0 = - y - n - P (3) H,O=m+P valuable papers on heat transmission. Stanton (I), (11), 8, 0 = - m - P + r and (IV) have been taken from the same papers. Stanton (111) is from the original source.1° Solving (3) in terms of a and p, The data by Clement and Garland,’l Panne11,12and Jordan13 y = -a- P are from the originals. The mean fluid temperature, t,, which 2 = a is used for determining the approximate mean temperature n =cy (4) f f L = -f9 difference (Column VI, Table 1) is taken as the arithmetic r = O mean of the inlet and exit fluid temperatures, except in the From physical consideration we know that the film thick- case of Jordan’s tests. Here the rise in air temperature was ness should decrease with increasing velocity. We there- considerable and Jordan calculated the effective mean temfore change the sign of (Y and substitute the relations (4)in perature of the fluid on certain reasonable assumptions. (1) and obtain ‘THIS JOURNAL,14, 1101 (1922). 6 Comp1. rend., 132, 1382 (1901); I b i d . , 133, 257 (1001); J. math. Dur. B I D = K O( k / p c , ) a (p/pDv)a (numeric) (5) a p p l . , 81 (1905). For the case of ideal gases we may omit the first factor 8 “Mitteilungen uber Forschungsarbeiten,” Heft 80-96, 1910. 7 Phzl. Mag., 44, 944 (1922). as an approximation and obtain Equation 6:
A
1 Received
January 19, 1924. 2 An abstract of this paper was presented a t the Cincinnati, Ohio, meeting of the American Physical Society, December, 1923. (See J. Am. Inst. Elec. Eng., December, 1923.) The companion paper on “Free Convection of Heat in Gases and Liquids-11” was presented a t the midwinter convention of the American Institute of Electrical Engineers, February 5, 1924. 8 Presented before the American Institute of Electrical Engineers, Swampscott, Mass., June 27, 1993.
6 “Die Warmeubergang von R o h r w h d e n an strdmender Wasser,” die Koniglichen Technischen Hochschule zu Munchen. 1910. 0 Technical Report Advisory Committee for Aeronautics (Great Britain), 1916-1917, pp. 16’31. 10 Trans. Roy. Sac. London, 190A, 67 (1897). Bulletin 40. 11 Unzv. of Ilianozs Bull., 7 (1909). 12 Technical Report of Advisory Committee for Aeronautics (Great Britain), 1916-1917, p. 22. 18 Proc. l n s t Mech. Eng. (London), 1909, Parts I11 and IV. p 1317.
May, 1924
I N D UXTRIAL A N D ENGINEERING CHEiMISTRY TABf E - I
461
INDUXTRIAL A N D ENGI NEERING CHEMISTRY
462
value has been assumed in most of the calculations, as tabulated in Column XII.. The value of 7c used for water corresponds to the values ranging from 0.347 to 0.388 B. t. u. x feet per hour X degrees Fahrenheit X square feet. If the factor k/pc, in Equation 5 could be neglected when dealing with liquids, a logarithmic plot of BID vs. p/pDv would find all the points falling on a single curve, as was found to be the case for gases in the previous paper.s When
Vol. 16, No. 6
In addition to the data in Table I the mean line determined from Nusselt's data as given in the previous paper3 has been shown dotted in Fig. 2. In a similar manner the approximate position of McAdams and Frost's4equation for liquids has been shown by the dot-dash line. Their equation in our units becomes W, = (AkA6/24.7D)(pD (watts) (10) 9mean value of k/pcp = 0.25 has been assumed to represent the average conditions for the tests upon which their equation is based. Inspection of Fig. 2 shows that the data are approximately represented by a line of slope CY = 5 / 6 , although the agreement between the various observers is not good. I n Column XV, Table I, the value of K Ohas been calculated from Equation 5, assuming P = I / z and a = 6/6, which gives an average value K O = 63.5. The value of K Odetermined from the position of the average s / ~power curve, which attempts to average in the data by Nusselt and McAdams and Frost, gives K J = 60. Substituting these approximate numerical values in Equation 8, we obtain for the forced convection of heat in smooth pipes W , = ( A k A t / 6 0 D)( p c , / k ) ' h ( p D v / ~ ) ~ / / s(watts) (11) Obviously, this equation can only be applied as an approximation where the flow is well above the critical value. Actually the exponents P and a probably vary with the degree of turbulence, so that no single mean values will cover the whole range. CALCULATION OF FORCED HEATCONVECTION FROM DATA ON
FIG.1
-
this is done we find the points widely scattered. If p = a in Equation 5, as assumed by Nusselt,B a similar plot of B / D vs. k/c,pDv would bring the points on a single curve. Here again the result is not satisfactory. In the first method of plotting the effect of viscosity enters directly, while in the second the viscosity does not enter. A comparison of the two methods of plotting shows that the effect of viscosity should enter in some intermediate manner. To study the effect of viscosity alone, wc/At vs. for Soenneken's data has been plotted on logarithmic paper in Fig. 1 from Columns XI and XV of Table I. Inspection of Equation 8 shows that, except,for the relatively small variations of k with temperature, the convection per degree Centigrade will be a function of the viscosity alone for constant velocity. The best average slope for the three velocities gives we/At proportional to (1/p)0.445.There are now two alternative assumptions which may be used in correlating the available data. The firSt is that and a in Equation 5 Itre related-i. e., /3 = f ( a ) ,which is the kind of assumption made by Nusselt. The second is that ,6 and a are entirely independent. Additional experiments or some further theoretical considerations seem to be needed to settle this question. For the present, therefore, we have adopted the second assumption and taken p = '/z as a sufficiently close approximation. On this assumption we then plot (B/D)(pc,/k)'/e vs. p / p D u on logarithmic paper, and should find that all the points for both gases and liquids a t various temperatures fall on a single smooth curve. This has been done in Fig. 2. I n the present work all the fluid variables have been taken a t the average $lm temperature, for simplicity, In the previous work3 King's data on the forced convection from cylinders a t different temperature risesindicated that the appropriate value was the geometric mean of the surface and ambient fluid temperature. Additional data for the heat transfer in pipes appear to be needed to settle this point. For the present the simple average has been used.
SUKFACE FRICTION
Osborne Reynolds14 was the first to suggest a connection between heat transfer and mechanical resistance to flow. Since then Stanton,lo Perry,16 Lanchester,'B Stanton," Taylor,18 StantonJg Pannell,12 Rayleigh,lg RoydslZo and Davis2I have discussed different aspects of the problem from this point of view. I n the following discussion a somewhat different point of view is suggested: In the film theory of mechanical resistance to flow R = vfi/BE (dynes per sq. cm.) (12) where v = mean velocity of fluid relative to the surface, cm. per second .u = viscosity of fluid, c. g. s. units BE = effective film thickness, cm. The method of dimensions then gives the following general expression for the effective film thickness: B E = K D (p/pDv)"' (cm.1 (13) where K = numerical constant to be determined by experiment D = characteristic linear dimension of surface m = numerical exponent, depending on the degree of turbulence, t o be determined by experiment If we substitute (13) in (12) we have for the resistance per unit area R = ( V ~ / K D( p) D v / p ) m (dynes per sq. cm.) (14) The usual expression is R = Cpvz ( p D v / p ) " (dynes per sq. cm.) (15) We may transform the first expression into the second by multiplying and dividing (14) by pDv/fi and taking 1/K = c and m-l=n. Let us now look for a relation between the heat transfer and mechanical resistance to flow. The actual mean thickness of the film that separates the surface of the body from the turbulent fluid will be given by the following equation: 14 Proceedwps Manchester Literary and Philosophical Society, 1874 ; for a reprint see Reference 15. 16 "The Steam Engine," p. 588. The' Mscmillan Co., Ltd., London, 1909. 16 Technical Report of Advisory Committee for Aeronautics (Great Britain), 1912-13, p. 40. 17 I b i d . , Appendix, p. 45. 18 I b i d . , Reports and Memoranda, 373 (1916-1917). 19 I b i d . , 1917-1918, p. 16. 30 "Heat Transmission by Radiation, Conduction and Convection," Constable &'Co.. London, 1931. Phil. Mag., 41, 899 (1921).
May, 1924
IND U8TRIAL A N D ENGINEERING CHEMISTRY
bR = vi (a.(16) ) where q = velocity of fluid a t the outer surface of the film Rl = resistance due t o existence of film We do not know how much of the total resistance is due to skin friction, Rf, or the tangential velocity at the surface of the film vi. We will therefore assume . MV mean (17) R/ = N R total (18) From observation of mean velocitv and total resistance we therefore actually determine BR = NbR/M = Vmp/Rt In the case of heat transfer we have taken for thin films and moderate temperature differences W . = kAt/B (watts per sq. cm. transfer) (20) Here B is the effective film thickness to account for the total heat transfer, assuming that the total temperature drop from the surface to the fluid is consumed in the effective film, B. From Pannell'slz experiments on the distribution OF temperature across a pipe during heat transfer, we know that only a portion of the total teinperature drop is consumed by the film. It also seems probable that a portion of the heat will be transferred from the surface of the pipe t o the central portion by an eddying motion which tears away a section of the film and carries it bodily away from the surface into the central part of the pipe. We will therefore assume that the actual mean film thickness during heat transfer is b = kef/Wf (cm.1 (21) where 0, =temperature of outer film surface referred to the pipe surface as zero, in 'C., and wf = portion of heat transferred from the surface by conduction through the actual film. We do not know wf or Of,and will therefore assume 0, = M'e mean (" C.) (22) = N'w t o t d (watts) (23) From observations of total heat transfer per unit area, wt, and mean temperature difference, e , what we actually have determined in Column XI11 of Table I is B N' b/M' = ke,/wt (24) We will now make the following assumptions with regard t o mechanical resistance and heat transfer: ba = b (25) This states that the actual mean film thicknesses are equal in both cases. M = M'
This aswmes that the distributions of temperature and velocity are identical across the pipe. In this connection the remarks by RayleighZ2and Stanton,23 and Pannell'sla experiments are of interest. The genera1 expression for the effective film thickness given by the method of dimensions for mechanical resistance is BR = NbR/M = K D (p/pDv)'" (27)
--
**
Technical Report of Advisory Committee for Aeronautics (Great Britain), 1917-1918, p. 15; Sci. Pepers, 6, 486 (1911-1919). a* Technical Report of Advisory Committee for Aeronantia (Great Britain), 1917-1917, appendix.
463
For the case of heat transfer we obtain B = N'b/M' = K O ( k / p C p ) @ (p/pDv)a (28) Taking the ratio of heat t o resistance and making the Assumptions 25 and 26, we obtain N'IN = (Ko/K) (k/pcp)o (p/pDv)a-m (29) If we further assume t h a t K O= K (30) m = a (31) we have N ' / N = (kpc,)@ (32) We now obtain from Equations 19 and 24 with the aid of Assumptions 25 and 26 wt = ( N / N 1 ) kernR t / v , ~ (watts per sq. cm.) (33) By making use of Equation 29 and Assumptions 30 and 31 we obtain wt = ( p c p / k ) @kemRt/v,p (watts per sq. cm.) (34) If we multiply and divide Equation 34 by cp, we obtain wt = ( k / p c , ) ' - @ cpRtern/v, (watts per sq. cm.) (35)
Equation 35 is seen to conform to Reynolds' equation as given by S t a n t ~ n . ~ Lees24 has shown that the observed resistance to flow in smooth pipes may be represented by the following empirical equation: Rt = p v i [0.0009
+ 0 . 0 7 6 3 / ( ~ , D p / y ) ~(dyne; ~ ~ 5 ] persq. cm.)(36)
From the available data on heat transfer a fair approximation If we substitute Equation is obtained by taking 0 = 36 in 35 we obtain in our usual notation the following convenient equation for forced convection in smooth pipes: wi = ( k / p ~ , ) ~ / 2 p ~ [0.0009 ~~At
+(watts 0.0763/(~Dp//.~)~.~~] per sq. cm.) (37)
To see the extent of the agreement of the foregoing assumptions with the available experiments, we have given in Table I, ColumnXVIII, the ratio of the effective film thickness in mechanical resistance to the pipe diameter, as a function of p / p D v from Column XVI. The values were read from a logarithmic plot of B R / D vs. p / p D v , which was obtained by applying Equation 12 to the data by Stanton and Panne11.26 Column X I X gives the ratio of effective film thickness for heat, from Column XIV, t o that for resistance. From Equations 19 and 24 combined with Assump tions 25 and 26 we see that this column represents " I N . If Equation 29 and Assumptions 30 and 31 were strictly true, we should iind N 1 / N = ( k / p c p ) P . From the work on heat transfer we found that the exponent was approximately 0 = 1/2. On these assumptions the ratio given in Column XX should be unity, while the average value for the data in Table I gives the ratio 1.08. If we include Nusselt's data the agreement is still closer. At the bottom of Table I some data by StantonQon the resistance and heat transfer in a roughened pipe indicates that here also the rate of heat transfers and the mechanical resistance to flow go hand in hand. It will be observed that these and the corresponding data on the smooth pipe show a poor agreement at the lower velocities. In the case of the rough pipe the resistance is more than proportional to the square of the velocity and the heat transfer more than proportional to the velocity. A similar condition will also be noted with respect to the heat transfer in the oase of the corresponding smooth pipe, Stanton (IV). A condition of this kind is always found near a critical velocity point and is probably due t o an oscillation from one type of flow t o the other. Under certain conditions this region of instability may extend over a rather wide range, as is a g parently the case in these experiments. Another check on the present theory may be obtained from a study of the data by Stanton, Marshall, and Bryantz6on the measurements of the actual velocities close t o the surface of 24
Pyot. Roy. Soc. (London), 91, 46 (1915-1916). Roy. SOC. London, 214A, 220 (1914). Pvoc. Roy. SOC.(London), 97,413 (1920).
flTrans.
**
Vol. 16, No. 5
INDUSTRIAL A N D ENGINEERING CHEMISTRY
464
TABLE 11-FORCED CONVECTION OF HEATIN GASESAND LIQUIDS (Data from Stanton, Marshall, and Bryant%) Air in 12.7 cm. diameter pipe. Assumed values: p = 0.000172,p
I Total Resistance Dynes per Sq.Cm. Rt
30 20 15 10 6
4
I1
I11
Mean
Velocity Cm.
8:: Urn
3760 3000 2580 2060 1550 1240
los, pDvm 3.08 3.86 4.50 5.63 7.48 9.35
IV Velocit a t A m Surface Cm. per Sec.
V
Uf
Actual Film Thickness Cm. bR
1280 980 840 650 470 350
0.0137 0.0142 0.0149 0.0152 0.0162 0.0170
VI I
Vf -
Urn
M
0.340 0.326 0.326 0.316 0,304 0.282
VI11
VI1 S P
X
IX
f
or
1o-svf bR
IO-*Rt
93.5 69.0 56.4 42.7 29.0 20.6
174 116 87.3 58.2 34.9 23.3
P
pipes carrying a stream of air. The data obtained with the 12.7-cm. drain pipe were used. I n their Table VI11 the velocities a t several fixed distances from the wall of the pipe are given for different values of total flow resistance. The resistance has been plotted against the velocity for each fixed distance from the wall on logarithmic paper and smooth curves have been drawn through the points. The velocities corresponding to fixed values of the resistance have then been read for different distances from the wall from the family of curves. The mian distances have been changed to effective distances according to the data of their Table VII. The velocity has then been plotted against effective distance from the wall of the pipe for different fixed values of resistance on logarithmic paper in Fig. 3. It is observed from Fig. 3 that close to the wall of the pipe the velocity is proportional to the distance from the wall; accordingly a family of 45-degree lines has been drawn through the points in this region. For the larger distances the points show a decided and abrupt departure from the 45-degree lines or stream line law. The adjacent turbulent region has been approximately represented by another family of parallel lines shown in the figure. The location of these lines is fairly well determined for the first three high-velocity curves where two points are available. The intersection of the two lines gives the actual mean film thickness and corresponding tangential velocity a t the outer boundary of the film. The values read from Fig. 3 are given in Columns IV and V of Table 11. The mean velocities given in Column I1 have been calculated from the re4 sistances of Column I by Equation 36 given by L e e ~ , ~assuming p = 0.000172 c. g. s. and p = 0.00117 gram per cc. In Column XI the ratio of effective film thickness to pipe diameter is calculated from the heat transfer Equation 5, taking KO= 60, P = l / 2 , and a = s/~.I n Columns XII, XIII, and XIV the Assumptions 25, 26, and 29 are worked out for each velocity. We observe that the ratio of N 1 / Nis approximately equal to ( k / p ~ c a ) ' / which ~ results from Assumptions 30 and 31. Some other points of interest should be noted-for example, from Column VI we see that the ratio of maximum film velocity to mean velocity changes very slowly with the mean velocity. Taylor1* has estimated the value of this ratio from the Lorentz criterion as 0.38, which is not greatly different from the values of Column VI. We may also observe from Column IX that for high velocities -approximately half the resistance energy is consumed by the fdm and the other half by eddy motion in the central portion of the pipe. As the velocity is reduced a greater proportion of the energy is consumed by viscous shear a t the surface.
Actual Value bR
R/
Ft
N 0.538 0.595 0.645 0.735 0.832 0.885
D
0.00108 0.00112 0.00117 0.00120 0,00128 0.00134
-
0.00117 XI B
ES. D Eq. 5 or N'b
DM 0.00177 0.00213 0.00241 0.00292 0.00365 0.00444
XI11
XI1
-
Assume
M'
then N'b
M
D
0.000602 0.000695 0 000785 0.000923 0.00111 0.00125
0.558 0.620 0.670 0.770 0.868 0.934
N'
N 1.04 1.04
1.04 1.05
W, = (AkAt/KoD) ( p c , / k ) @ (pDv/p)a (watts) (38) where A = area of surface, sq. cm. K = heat conductivity of fluid Et average film temperature, watts per cm. per C . At = diffoerencebetween surface and ambient temperature, C. K O = numerical coefficient, to be determined by experiment, for the system of similar bodies under consideration D = characteristic linear dimewion of body, cm. p = viscosity of fluid at average film temperature, grams per cm. per second t p = specific heat a t constant pressure for fluid at average film temperature, joules per gram per O C. P = experimental exponent p = density of fluid at average film temperature, grams per cc. v = mean velocity of fluid relative to surface, cm. per sec. a = experimental exponent
For ideal gases we may omit the second factor as a fair approximation and use the simpler expression given in Equation 9, which is W , = ( A k A t / K D ) (pDv/p)a
(watts)
(39)
CYLINDERS IN IDEAL GASES
For a cylinder in a stream of gas flowing a t right angles to the axis the values obtained in the previous paperg still hold. We therefore substitute a = '/z and K = 2.2 in (39) and obtain for thin films where
D
L
= =
W , = ( ~ L / 2 . 2(pDv/p)1/2kAt ) (watts) diameter of cylinder, cm. length of cylinder, cm.
FORCED CONVECTION FROM LARGB SURFACES I n most engineering calculations we are concerned with high velocities and large bodies, so that the film thickness will usually be small compared with the size. We will also limit the discussion to moderate temperature differences. Under these conditions we apply the general Equation 8, which is
-
Assume b bR then N'
XIV
Fro. 3
,
(40)
I N D U S T R I A L A N D ENGINEERING CHEMISTRY
May, 1924
20.
lo.
IO. zo /a
/8. 9. 9. la
9.
16.
8. 8.
/.6
8.
144
7:
z
L4
7
/2. 6. 6. A2
6.
/o.
j :
5 5
8. A
LO
4. .8
6. 3. 3.
.6
4 2. 2. 4 1. I:
465
4. 3. 2,
1 -2 L
.Q--. 0- 0-
0-0
0
/O
20
30
50
40 FIG.
If we substitute the approximate values expressing the properties of air as a function of temperature^,^ we obtain for air W , = 0.0018 L m 4 t / T : ; ' , 3 (watts) (41) where p = absolute air pressure in atmospheres TSvg. = average absolute temperature of surface and ambient air, ' Kelvin
HEATTRANSFER 137 PIPES, GASES,-4ND LIQUIDS
For this case the present work shows that ~ 7 , 7 1 2 may take /3 = l/2, a = 5/6, and K O= 60 in Equation 38 as a fair average, We thus obtain for the heat transfer in smooth pipes when {,hevelocity is well above the critical value W , = (AkA?+/60D ) ( p c p / k ) l / Z (pDv/p)6/6 (watts) (42) A more general equation, which takes into account the variation of the exponent a with the degree of turbulence, was obtained in Equation 37 from the mechanical resistance to flow combined with Lees' equation. This gives for gases and liquids
60
70
80
90
/OO%&
4
all but the high velocity points obtained by Stanton (V), for the reasons already stated. For ideal gases, Equation 46 becomes ' W , = 0.00635 ApukAt/p (watts) (47) For air, Equation 47 becomes approximately W, = 0.0031 ApvAt/T,,,. (watts) (48)
LARGEPLANE SURFACES
I n making approximate calculations of the heat transfer from large surfaces, it is convenient to assume that the convection per unit area is independent of the total surface area or size factor. Under these conditions the general Equation 38 becomes W , = ( A v 4 t / K o ) (kp/p) ( p c p / k ) ' / z (watts) (49) if we make the further assumption that p = l/2. To determine the value of the constant K Owe will assume that the conditions are similar to those existing in a large pipe through which fluid is flowing. Since for very large pipes or high W , = ApcpvAt ( k / p c p ) ' / Z [0.0009 -I- 0.0763/(vDp//.~)~.~~] velocities the resistance per uni$ area is proportional to the (watts) (43) square of the velocity, the rate of heat transfer will be proFor ideal gases, Equation 43 becomes portional to the first power of the velocity. w,= 1.16ApcpvAt [0.0009 Jr 0.0763/ ( ~ o p / p ) ~(watts) . ~ ~ ] (44) For smooth surfaces we will assume the conditions represented by a 45-degree line in Fig. 2 drawn through B/D For air, Equation 44 becomes approximately ( , u c ~ / k ) l /=~ 0.00350 and ( p / p D v ) = 0.00001, which give W , = 0.417 A p v ~ t / T , , , . [0.0009 0.0012 T:fj.5/ K O= 350 for use in Equation 49. (vDp)o.35] (watts) (45) For ideal gases and smooth surfaces, we then obtain apFor rough pipes (surface like fine sandpaper) we will assume proximately a = 1, 6 = l/2, and KO= 136 in Equation 38, and obtain W , = 0.00246 A p v k A t / p (watts) (50) W , = (Apvk A t / 1 3 6 p ) (pcp/k)'/z (watts) (46) For air this gives I n making the fbregoing assumptions we have disregarded W , = 0.0012 ApvAt/T,,,. (watts) (51)
+
Vol. 16, No. 5
INDUSTRIAL A N D ENGINEERING CHEMISTRY
466
For rough surfaces we may use Equations 46, 47, and 48. Another estimate may be obtained from the tests on the rough wire-wound heater of the previous paper,3 which gave for ideal gases on forcing the slope to unity W , = 0.010 ApvKAt/p (watts) (52) I n other words, we may expect from two to four times the heat transfer from rough surfaces as from smooth ones.
FORCED HEATCONVECTION IN MIXTURES The heat transfer in mixtures is covered by the same general equations that apply to liquids and gases. I n the previous paper3 the case of mixtures of a light with a heavy gas was investigated and an unexpected maximum was found in the forced-convection for a mixture of approximately 60 per cent hydrogen and 40 per cent oxygen by volume. On the present theory it is seen from Equation 49 that the forced convection coefficient now contains the additional factor ( p ~ b / k ) ' / ~the , complete expression being ( k P / P ) (PcP/k)1/2
(53)
For mixtures of a light with a heavy gas the second factor is not a constant and therefore requires evaluation. In Fig. 4 the data have been reproduced for mixtures of hydrogen and oxygen. The specific heat curve has been calculated from the relation C;p'Pf+C;pj'P"
(joules per gram per O C.) (54) P' p " PI1 where cp = specific heat per gram a t constant pressure of hydro. , gen c p = specific heat per gram a t constant pressure of oxygen pl: = density of hydrogen, grams per cc. p = density of oxygen, grams per cc. P' = percentage hydrogen by volume i 100 P " = (1 - P') = percentage oxygen by volume + 100 Cp
=
pl
+
Inspection of the figure shows that on the present theory the forced convection coefficient retains the maximum value for 60 per cent hydrogen but is less pronounced. We may now compare the forced convection coefficients for air, pure hydrogen, and commercial hydrogen, after making use of the approximate relations connecting the fluid properties with the temperature and pressure as in the previous paper. For air (pcp/k)'/2( k p l p ) = 0.863 (0.475 p / T )
(55)
For pure hydrogen (pc,/k)'/z
(kp/p) =
0.863 (0.464 P / T )
3-When dealing with ideal gases the first factor in Equation 58 can be neglected, since it does not vary greatly from one gas to another. Under these conditions we obtain B = K D (p/pDv)a
where
We = (AKAt R / v p ) ( p c p / K ) B (watts) where R = mechanical resistance, dynes per sq. cm.
8-We
(60)
(62)
may write Equation 62 in the following form: We = (Ac,AtR/v) (K/pcP)l-@(watts)
(63)
which is seen to conform to Reynolds' law as given by Stanton. 9-If we substitute Lees' equation for the resistanoe to flow in smooth pipes in Equation 63 and assume P = l/a, we obtain for gases and liquids W, = (k/pCp)1/2ApCpV At [0.0009
+ O.O763/(~Dp/p) (watts) (64) O.aS]
T l i s equation takes into account the variation of the resistance with the degree of turbulence, and therefore should be applicable over the entire range above the critical velocity. For ideal gases we may take with sufficient approximation
+ O . O 7 6 3 / ( v D p / , ~ ~(watts) ) ~ ~ ~ ~(65) ] W , = (0.417 A@Aat/TaVg.)[0.0009 + 0.0012 T::gY/ W , = 1.16 Apc,vAt [0.0009
For air, Equation 65 becomes approximately
( V D ~ ) (watts) ~ . ~ ~ ] (66) = absolute air pressure in atmospheres where p Tavg, = average of surface and ambient absolute temperatures, ' Kelvin CY
B = .KoD (k/pcp)@( P / P D v ) ~ (em.) (58) where K O = experimental constant depending on the system of similar bodies under consideration D = characteristic linear dimensions of body, cm. k = heat konductivity a t average film temperature, watts per cm. per C. p = viscoqity of fluid a t average film temperature, grams per cm. per second
W, = ( d k A t / K o D ) ( p c p / k ) @(pDv/,u)a (watts) A = area of surface, sq. cm. A1 = temperature difference, C.
5-For ideal gases the factor (pcp/k)B may be neglected with sufficient approximation so that W, = ( A A A t j K D ) (pDv/p)a (watts) (61) This relation was tested over a wide range of conditions in different gases in the previous paper, with good agreement. 6-For smooth pipes well above the critical velocity the available data in gases and liquids are approximately satisfied by taking P = l/2, a = 6/a, and KO= 60 in Equation 60. 7-The film theory naturally suggests a relation between the mechanical resistance to flow and the heat transfer. A study of the heat transfer and mechanical resistance for pipe leads to the following relation:
SUXMARY AND CONCLUSIONS 1-The more general expression for the effective film thickness, here developed, is superior to the expression of the previous paper in that it permits the correlation of the heat transfer in gases and liquids. 2-The general expression for the effective film thickness obtained by the method of dimensions is
(59)
O
(57)
Thus the forced convection for high velocities over plane surfaces will be approximately equal in air and pure hydrogen and approximately 13per cent greater in commercial hydrogen.
(cm.1
which is the expression assumed for both gases and liquids in the previous paper.3 4-For thin films and moderate temperature differences we obtain for gases and liquids
(56)
For commercial hydrogen (95 per cent Hz, 5 per cent Nz) (pc,/k)'/z (kp/p) = 0.75 (0.602 P / T )
cP = specific heat a t constant pressure f,or average am temperature, joules per gram per C. B = experimental exponent p = density of fluid a t average film temperature, grams per cc. Y = mean velocity of fluid relative to the surface, cm. per second CY = experimental exponent
10-For rough pipes (fine sandpaper) we will assume and KO= 136 in Equation 60. = 1, = For ideal gases we then obtain We = 0.00635 ApvkAt/p
(watts)
(67)
W , = 0.0031 ApvAt/T.,,.
(watts)
(68)
For air we have
11-For a cylinder in ideal gases the values obtained in the previous paper still hold. We therefore take CY = ' 1 2 and K = 2.2 in Equation 61 and obtain for thin films where D
=
W , = (AkAt/2.2D) (pDv/p)l/z (watts) diameter of cylinder, cm.
(69)
For air we obtain
O
where L
W , = 0.0018 L l ; / g u At/Ts:z (watts) = length of cylinder, cm.
(70)
I N 0 USTRIAL A N D ENGINEERING CHEMISTRY
May, 1924
. 12-For approximate calculations of the heat transfer from large smooth surfaces we take a! = 1, /3 = I/*, and KO = 350 in Equation 60. This gives for ideal gases and smooth surfaces We = 0.00246 ApokAt/p (watts) (71) and for air We = 0.0012 A@At/T,,,. (watts) (72) For moderately rough surfaces we may employ Equations 67 and 68, and for very rough surfaces in ideal gases We = 0.010 ApwkAt/p (watts) (73) 13-The heat transfer in mixtures is covered by the same general equations that apply to liquids and gases. 14-For mixtures of a light with a heavy gas the forced convection a t high velocities over plane surfaces may be greater than in either gas alone. For air, (pc,/k)L/a ( k p / p ) = 0.41 p/Tsvg. Pure hydrogen, ( p c / k ) I / 9 ( k p / p ) = 0.40 p/T,,,. 95% hydrogen 5
