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The effect of the Prandtl number on temperature profiles for heat transfer in turbulent pipe flow
Chemical Engineering Science, 1967, Vol. 22, pp. 1701-1711. Pergamon Press Ltd., Oxford.
Printed in Great Britain.
The effect of the Prandtl number on temperature profdes for heat transfer in turbulent pipe flow R. A. GOWENt and J. W. SMITH Department
of Chemical Engineering and Applied Chemistry, University of Toronto
(Received 25 April 1967; in revised form 12 June 1967)
Abstract-There are few experimental results that support the theoretical discussions on the effect of the Prandtl number on temperature profiles in turbulent pipe flow. This has been particularly true for fluids having a Prandtl number greater than unity. Radial temperature profiles and heat transfer coefficients have been measured for the turbulent flow of air (Npr=0*7) and aqueous ethylene glycol (N~,=14.3) in a smooth tube over a Reynolds number range between 10,000 and 50,OOO. These and other temperature profiles in the turbulent core for fluids having Prandtl numbers between 0.026 and 14.3 have been presented in the form
t+= Aslog,y++Bs . For the higher Prandtl number fluids the constant B8 appears to be directly proportional to NP~ and independent of NR= whereas As is reasonably insensitive to both. Heat transfer coefficients calculated from the temperature profiles are in excellent agreement with the experimental results.
RELATIVELY few experimental measurements have been made to determine the temperature profiles for heat transfer in turbulent pipe flow and most of these studies have been confined to gases [ 1,2] or to liquid metals [3, 41. Temperature profiles with higher Prandtl number fluids are very scarce and only recently they have been measured in water by SMITH,GOWEN and WAshluND [5] and to a limited extent by BECKWITH and FAHIEN [6] and TRUCHASSON [7]. This paper presents the results of a systematic study of the effects of a wide range of Prandtl numbers on temperature profiles. Data on temperature profiles and heat transfer coefficients have been obtained for turbulent pipe flow of air (Np,=O*7) and 30% aqueous ethylene glycol (NE+= 14.3 at 80”F, [S]). The experimental equipment used in the present
investigation is described in detail in [5l and is illustrated in Figs. 1, 2 and 3. The heat transfer section consisted of a 2,058 in. brass pipe 433 in. long. Heat was supplied electrically with an electric resistance ribbon wrapped around the outer wall of the test section. The temperature profiles were measured with a travelling thermocouple probe whose radial position was controlIed to within O@Ol in. by means of a micrometer spindle. In all runs temperature profiles were measured with a probable accuracy of + 0.001 in. and +O* 1°F. The results cover a Reynolds number range from 10,000 to 50,000 and a Prandtl number range from 0.7 to 14.3. The film viscosity was used to calculate the Prandtl number of all fluids. The film temperature was defined as
t R. Shriver Associates, Denville, New Jersey. 1701
TFilm =-*T,+T’ 2
R. A.
GOWEN
and J. W. SMITH
Thermocoupte
Heal Tronsfer Section
Epoxy
Resin
Thermocouple . ..
SIDE FIN.
1. Schematic
diagram of experimental system.
flow
VIEW
FQo. 3. Detailed diagram of the tip of the traversing thermocouple.
I&am-s
AND
Drscuss~o~
Developed turbulent velocity profile measurements in pipe flow in a region outside the viscous sublayer are well described by the universal velocity profile, that is,
u+=2*5Iny++5*5. irq Carriage
hermocwptes II of Barr
for
on Tube
Nictwome Ribbon Rerittarce
Heater
The universal velocity profile was originally developed for a region very close to the wall and hence it is sometimes called the “law of the wall”. Surprisingly enough the law represents the velocity profile for over 90 per cent of the pipe radius and deviates only near and at the pipe centre. A velocity defect law has also been developed to describe the outer region of the turbulent boundary layer and it it given by, %
Fro. 2.
Diagram
Thermocou#e
of the heat transfer Dimensions in.
--u+= -2*51ny/R-0.18.
(2)
If there is an analogy between heat and momentum transfer, developed turbulent temperature profiles should be expressed in a similar form and hence,
PI
rik
(1)
t+ =A,Iny+
+B,
(3)
-A,lny/R+C.
(4)
and section.
&-t+=
1702
The effect of the Prandtl number on remperarure profiles for heat transfer in turbulent pipe flow
Equation (3) is known as the universal temperature profile and Eq. (4) as the temperature defect law. The Prandtl number is an additional parameter describing the effect of the thermal properties of the fluid in the transfer of heat. While Eqs. (1) and (3) are analogous, the dependency of the constants A, and B,on Np, is not generally known. The results of the present experiments are shown in Figs. 5-10 as temperature profiles in the form of Eqs. (3) and (4) for air, water and 30 % aqueous ethylene glycol. Figure 4 shows the Stanton number as a function of Reynolds number on logarithmic co-ordinates for all three fluids. The data of KOKOREV[4] are plotted in Fig. 11 to show the universal form of the temperature profile for mercury (iVp,=0.026).
NRe- IO-’
Fro. 4. Stanton number as a function of Reynolds number.
20 I8 16 14 12 IO 8 6
20
50
100
200
500
Y*
FIG. 5. Radial temperature profiles in air for smooth
tube S-l.
6
FIG. 6. Temperature
defect law with air.
1703
R. A. G~WEN
and J. W. Shnrrrr
6
4
> 1
I 2
I 4
III 6 610
I 2.
I 4
III 6810”
I 2
Y+ FIG. 7. Universal temperature profile with water.
FIG. 8. Temperature defect law with water.
1704
4
6
6 IO’
The effect of the Prandtl number on temperature
profiles for heat transfer in turbulent pipe flow
93 92 91 so 89 88 4
07
-
86 85 84 83 82
FIG. 9. Radial temperature profiles in aqueous ethylene glycol for smooth tube S-l.
12
I
P I
II
ml
I
III1
I
I.1 b
IIII
q
1
IO 9
0’
I II I
-0
I I
A
N R* 13700 21100 I5700 18100 14500 16400
WATERNP,
GLYCOL = 14.3
. 1
1
1
1
I
8 7 6 5
l
T4 .-J 3 2
0 -I
.02
.05
.I0
.20
.50
FIG. 10. Temperature defect law with aqueous ethylene glycol.
1705
I.0
R.yA. GOWENand J. W. Shnru 12 II ICI 9 0 z-7 6 5 4 3 200
500
1000
2000
5000
10000
Y* FIG. 11.
Universal temperature profile with mercury.
From the results, it can be seen that Eqs. (3) and (4) are nearly identical experimentally. This has also been shown to be true in [I] and [5]. The experimentally determined values of A, and B, from Eq. (3) are tabulated in Table 1. Results from [I, 2, 5 and 61 are given in addition to those of the present work. Unfortunately, not all data [3] can be converted into the universal form because the heat fluxes are not given.
Figure 7 shows the universal temperature profile 2’ vs. logy’ for water. However, the profile is not universal in the low Reynolds number range because the value of the t+ intercept at logy+ =0 depends on the Reynolds number. Rearrangement of the variables can easily show why this happens. t+
4,
TABLE 1. EXPERIMENTAL VALUES OF A8 AND B,
NP~
Reference
A8
BS
A3dpu*C,
t+ _
C--T PU*CP Tw-Te
C
h
assume 0.026
I41
2.04
-7.25
0.7
121
2.18
3.8
-0.4
0.705
111
2.2
3.3
-0.4
2.18
3.0
-0.4
0.7
This work
TB=TE
5.7
I51
2.58
34.5
-0.4
6.0
El
2.55
28.0
-0.4
14.3
This work
2.52
76*3
-0.7
then CF 2
J t+“--&l-e). St
The constant A, shows no strong dependence on the Prandtl number even though it is somewhat lower for air and mercury than it is for either water or aqueous ethylene glycol. The intercept constant for the temperature defect law also appears to be insensitive to NPr and this is to be expected since, in effect, the wall conditions have been removed. This point has been discussed in [5].
Because the friction factor reaches the fully turbulent region sooner than the Stanton number the J(CF/2)/Ns, ratio behaves in an abnormal fashion for a Reynolds number less than 20,000. As a consequence the universal temperature profile exhibits a Reynolds number dependence. When both Ns, and C, reach the fully turbulent region the NRe dependence is seen to diminish and the law is truly universal. However, for the range of NRe studied the slope on the logarithmic axis remained constant at 2.59f0.13. It should be emphasized that this difference in behaviour of Ns, and Cr is not due to
1706
The effect of the Prandtl number on temperature
undeveloped temperature or velocity profiles and longer momentum or thermal entry lengths would not change the pattern. Most textbooks on fluid dynamics report that 40 to 50 pipe diameters are necessary to develop a turbulent velocity profile and 45& were used in the present study, the last 19 of which were in the heated section. The entry length that is needed to establish fully developed temperature profiles has been discussed by SLEICHER and TRIBUS[9] who reviewed entrance length studies of seven investigators and conclude that in the Reynolds number range 10,OUOto 100,000, 5 to 10 diameters with a developed velocity profile are sufficient. MILLS[lo] andABBBBCHTand CHURCHILL [l l] in later studies confirmed this requirement. The constants in the universal temperature profile can be predicted from the velocity profile and a knowledge of the eddy diffusivities of heat and momentum transfer. Thermal and momentum eddy diffusivities can be conveniently expressed by the following equations, r/r,=(l+~&)du+/dy+
(5)
4/4,=(1/N,,+E,/v)dt+ldy+.
(6)
The transfer of heat and momentum in the viscous sublayer is predominately molecular and therefore the eddy motion may be neglected. Also, this region is so close to the wall that;
protiles for heat transfer in turbulent pipe flow
is assumed equal to unity then Eqs. (5) and (6) may be combined to give,
(11) Since the velocity profile in the buffer zone is represented by u+ =Slny+-3.05 the solution of Eq. (11) is, +5Np,
Equation (13) has been derived by SQUIRE[ 121using similar assumptions. In the turbulent core, the molecular properties may be neglected in comparison with the eddy diffusivities but the shear stress and the heat flux cannot be assumed constant and equal to those at the wall. Unlike the shear stress the radial heat flux distribution is not a linear function of the radial distance; however they may be assumed equal to one another without introducing sign&ant error [5J Dividing Eqs. (6) by (5) and again assuming the eddy diffusivities of heat and momentum to be equal yields,
t+ =Np,y+ u+=y+.
(14)
and simplifying
(8)
(9)
The shear stress and the heat flux may also be assumed equal in the buffer zone (5
U+
=25lny++5*5
for y+>30
t&, = 5 ln(5Np,+ 1) + 5N,
&=5ln30-3.05
[from Eq. (1311
[from Eq. (12)]
then integrating Eq. (15) yields, t+=2*5hy++5hI
NP; = %&a
_dt+ldy+ du +/dy +
(7)
and hence solutions of Eqs. (5) and (6) for 0 < y+ < 5 are given by;
(13)
5
,
t/r,=Lj/&=l
(12)
(10) 1707
+835+5N,.
(16)
R. A. GOWENand J. W. SMITH Therefore, in the turbulent core (y + > 30) the intercept constant is given by,
B,=51n(5N~oo+l)+8"55+ 5Ne,.
•
z/zw dt+/dy + en q/(twdu+/dY+"
(17)
SQtnm~ [12] assumed that the shear stress and heat flux remained constant in the turbulent core but since the right-hand sides of Eqs. (5) and (6) are equated the same results as Eqs. (16) and (17) are obtained. Since fluids with low Prandtl numbers have very high thermal conductivities, heat transfer by conduction cannot be neglected in the turbulent core. ROGERS and MAYHEW [13] have theoretically evaluated the temperature profiles in the turbulent core for fluids with a Prandtl number less than 0" 1. Their equation accounts for conductive heat transfer but the result was obtained by also assuming that the shear stress and heat flux remained constant [Eq. (7)]. They proposed the following equation,
t+=1251n (
Neglecting the molecular properties and solving Eq. (5) and (6) for N~,, yields, N~,, = g M
By experiment, in the turbulent core, the gradients are, dt + _As dy + y+
(20)
du + 2"5
(21)
dy + y+ hence
Nte _r/~w As. 01e~,.2"5
(22)
Figure 12 shows the radial variation of the turbulent Prandtl number for the Prandtl number range that was studied. The turbulent Prandtl number increases with the Prandtl number but remains relatively insensitive to the Reynolds number and the radial position. The solution of the temperature profile in the turbulent core without assuming unity for the turbulent Prandtl number yields, in place of Eq. (16),
x+y/R(l-y/R) "~
+!"25.1n~((2y/R-1) + W'~ W kk(2y/R- 1 ) - W] (60/Re*-I)-W)] (60/Re* - 1) + + 5[Nr,+ln(5Ne,+ 1)]
(19)
(18)
where
W = ~/(1 +
where fl= AJ2.5. The intercept constant B~ is given by,
20/NI,,)Re*
x = 5[(Ne,Re*) Bs=51nI5N~o0+1 ) + 8"55fl+ 5Net •
Re* =Ru*/v. Usually in the absence of experimental data, the turbulent Prandtl number is assumed equal to unity but once the temperature profiles have been measured experimentally it can easily be calculated. 1708
Comparison of the predicted constants experimental values is shown in Table teresting to note that as the Prandtl increased, Bs may be approximated by
(24)
Bs with the 2. It is innumber is 5.5 Nvr.
The effect of the Prandtl number on temperature profiles for heat transfer. in turbulent pipe flow
64 -
---
Npl= 5.7,143
+
Nh=0*7
0.2
0.2
0 FIG.
0.4
___
Experimental
Predicted
B8
B8
and subtraction of Eq. (26) from (25) and removing y + B yields, t+,=B,+(u+,-5.5)
Eq. (17) Eq. (18) Eq. (24) 5.5 Npr (WIV= 1) (Rogers) 0.026
-7.25
-7.72
-13.2
-9.3
~+JJ=JG/2ws,
0.143
0.7 5.7
3.0 34.5
2.56 36.9
-2.93 37.5
1.53 37.2
3.85 31.4
14.3
76.3
84.5
78.9
84.5
78.6
2 J G
substituting
into Eq. (27) gives,
(28)
Heat transfer coeficients
Heat transfer coefficients may be determined by integrating the turbulent temperature profiles. A much simpler method of prediction can be obtained if it is assumed that the respective distances from the wall at which the velocity equals ug and the temperature Ts are the same [14, 151. Then,
u+,=2*5lny+,+5*5
(27)
since by definition
l&=
t+,=A,lny+,+B,
1
12. Radial variation of the turbulent Prandtl number.
TABLE 2. INTERCEPT CONSTANT B,; EXPERIMENTAL AND PREDICTED VALUES
Npr
0.8
0.6
(25)
(26)
Ns’=BS+g:5.5,, where
+8*55+5N
Pr *
Equation (28) with B assumed equal to unity is shown as a dotted line in Fig. 1. The agreement with the experimental data is excellent.
1709
R. A. GOWEN and J. W. &urn CONCLUSIONS
The temperature profiles in the turbulent core of pipe flow are well represented by a universal profile of the form, t+ =A,lny+
+B,
where +&55+5N
Pr*
The value of constant A, appears to be relatively insensitive to the Prandtl number but intercept constant B, is strongly influenced by the thermal properties of the fluid. Stanton numbers in turbulent pipe flow can be predicted from the temperature profiles with the following equation,
UB
fanning friction factor heat transfer coefficient, B.t.u./hr-ft2”F thermal conductivity, B.t.u./hr-ft2”F/ft Prandtl number = C&k Du,lv Stanton number = h/pu,C, radial heat flux, B.t.u./hr-ft2 Ru*/v, friction Reynolds number temperature, “F bulk temperature, “F centre-line temperature, “F (T, - T)u*pC I&,, dimensionless temperature mean bulk flow velocity, ft/hr velocity, ft/hr d(r,g,/p) = friction velocity, ft/hr u/u*, dimensionless velocity ;i{c ~;WJJRe*
l4:
u+ W
X
Y Y+
radiaydistance from wall, ft yu*/v, dimensionless distance
Greek letters
B A,W
JGP~
sY, sH eddy diffusivity, ft/2hr p absolute viscosity, lb,,,/ft-hr v kinematic viscosity, ft2/hr p density, lb/ft 3 z,, z shear stress, lb/ft2
Nst = B, + (4(2/Cr) - 5.5) ’ NOTATION
constant in the universal temperature profile constant in the universal temperature profile constant in the velocity and temperature defect laws heat capacity, B.t.u./lb,“F pipe diameter, ft
Subscripts
bulk centre-line pertaining to heat transfer A4 pertaining to momentum transfer s pertaining to a smooth surface t pertaining to temperature wall condition W
hFFBENCE.9 JOHNKR. E. and h~~~r-ry T. J., Chem. L?ngng Sci. 1962 17 867-879. Darss=R. G. a.ndErmC. S., N.A.C.A. TN2629, 1952. ISAKOPP S. E. and DRBW T. B., Proceedings of the General Discassion on Heat Transfer, A.S.M.E. 11-13 September, 1951. KOKOREVL. S. and RY-V V. N., Znt. Chem. ZL%gng 1962 2 514-519. Shmx J. W., &WEN R. A. and WASMUNDB., Eddy Ditfusivities and Temperature Profiles for Turbulent Heat Transfer to Water in Pipes (presented at the 59th Annual A.1.Cb.E. Meeting in Detroit, and submitted to A.1.Cb.E. Journal). BECK‘WITHW. F. and FAH[EN R., U.S.A.E.C. report IS 734, 1963. TaUCHASSONC., Chem. Engng Sci. 1964 19 305-317. B~RNARDOE. and &4N C. S., N.A.C.A. WR E136, 1945. SLEICHERC. A. and Tm.m M., Recent Advances in Ht Mass Transfer, McGraw-Hill 1961. MULLSA. F., J. Mech. Engng Sci. 1962 4 63-77. ~BRECHT P. H. and ~H~BCHILL S. W., A.Z.Ch.E. JI 1960 6 268-273. Sqvrrc~ H. B., Proc. Gen. Discussion on Heat Transfer, London, 11-13, Sept. 1951 (A.S.M.E.). ROGWS G. F. C. and MAYHEW Y. R., Int. J. Ht Mass Transfer, Vol. I, pp. 55-67. 1960. RANNIE W. D., J. Aero Sci. 1956 23 485+89. VONWN T., Trans. A.S.M.E. 1939 61 705-710.
1710
The effect of the Prandtl number on temperature
profiles for heat transfer in turbulent pipe flow
R&sum&II existe peu de resultats experimentaux qui soutiennent ies discussions theoriques sur l’effet du nombre de Prandtl relatif aux courbes de temperatures dans un courant turbulent a l’int&ieur d’un tuyau. Ceci s’est revele particulierement vrai pour les fluides ayant un nombre de Prandtl plus grand que l’unit6. On a mesure les courbes de temperature radiale et les coefficients de transfert de chaleur pour un ecoulement turbulent d’air (Npr=0,7) et d’ethylene aqueux (Npr= 14,3) B I’interieur d’un tube aux parois limes, et dam une gamme du nombre de Reynolds variant de 10.000 a 50.000. Cellesci et d’autres courbes de temperature relev6es dans le foyer turbulent pour des fluides ayant des nombres de Prandtl entre 0,026 et 14,3 ont et6 present&es sous la forme de: t+=Aslog,y++l.l* Pour les-fluides dont le nombre de Prandtl est sup&ieur, la constante Bs apparalt comme &ant directement proportionnelle a Npr et independante de NR~ tandis que As est raisonnablement insensible aux deux. Les coefficients de transfert de chaleur calcules ii partir des courbes de temperature sont tout a fait en accord avec r&.ultats experimentaux. Zusammenfassung-Es gibt wenige Versuchsergebnisse, die die t heoretischen Diskussionen liber die Auswirkung der Prandtl-Zahl auf Temperaturprofile in RBhren-Wirbelstrbmungen best%tigen. Dies gilt hesonders fiir Fliissigkeiten bei einer Prandtl-Zahl grosser als 1. Radialtemperaturprofile und Wiirmelibergangskoeffizienten fur die Wirhelstriimung von Luft (Npr-0,7) und w&&gem Glykol (Athylenglykol, Npl=14,3) in einer glatten Rohre wurden bei Reynolds-Zahlen von 10 Ooo bis 50 000 gemessen. Diese und andere Temperaturprofile im Wirbelkern wurden fur Fliissigkeiten mit Prandtl-Zahlen zwischen 0,026 und 14,3 in der Formel
dargestellt. Ftir Fliissigkeiten mit hoheren Prandtl-Zahlen scheint die Konstante B, direkt proportional zu NP~und von Nnc unabhangig zu sein, wahrend As ziemlich unabhangig von beiden ist. Die aus den Temperaturprofilen errechneten Wlrmeiibergangskoeffizienten stimmen mit den Versuchsergebnissen ausgezeichnet i&rein.
1711
Printed in Great Britain.
The effect of the Prandtl number on temperature profdes for heat transfer in turbulent pipe flow R. A. GOWENt and J. W. SMITH Department
of Chemical Engineering and Applied Chemistry, University of Toronto
(Received 25 April 1967; in revised form 12 June 1967)
Abstract-There are few experimental results that support the theoretical discussions on the effect of the Prandtl number on temperature profiles in turbulent pipe flow. This has been particularly true for fluids having a Prandtl number greater than unity. Radial temperature profiles and heat transfer coefficients have been measured for the turbulent flow of air (Npr=0*7) and aqueous ethylene glycol (N~,=14.3) in a smooth tube over a Reynolds number range between 10,000 and 50,OOO. These and other temperature profiles in the turbulent core for fluids having Prandtl numbers between 0.026 and 14.3 have been presented in the form
t+= Aslog,y++Bs . For the higher Prandtl number fluids the constant B8 appears to be directly proportional to NP~ and independent of NR= whereas As is reasonably insensitive to both. Heat transfer coefficients calculated from the temperature profiles are in excellent agreement with the experimental results.
RELATIVELY few experimental measurements have been made to determine the temperature profiles for heat transfer in turbulent pipe flow and most of these studies have been confined to gases [ 1,2] or to liquid metals [3, 41. Temperature profiles with higher Prandtl number fluids are very scarce and only recently they have been measured in water by SMITH,GOWEN and WAshluND [5] and to a limited extent by BECKWITH and FAHIEN [6] and TRUCHASSON [7]. This paper presents the results of a systematic study of the effects of a wide range of Prandtl numbers on temperature profiles. Data on temperature profiles and heat transfer coefficients have been obtained for turbulent pipe flow of air (Np,=O*7) and 30% aqueous ethylene glycol (NE+= 14.3 at 80”F, [S]). The experimental equipment used in the present
investigation is described in detail in [5l and is illustrated in Figs. 1, 2 and 3. The heat transfer section consisted of a 2,058 in. brass pipe 433 in. long. Heat was supplied electrically with an electric resistance ribbon wrapped around the outer wall of the test section. The temperature profiles were measured with a travelling thermocouple probe whose radial position was controlIed to within O@Ol in. by means of a micrometer spindle. In all runs temperature profiles were measured with a probable accuracy of + 0.001 in. and +O* 1°F. The results cover a Reynolds number range from 10,000 to 50,000 and a Prandtl number range from 0.7 to 14.3. The film viscosity was used to calculate the Prandtl number of all fluids. The film temperature was defined as
t R. Shriver Associates, Denville, New Jersey. 1701
TFilm =-*T,+T’ 2
R. A.
GOWEN
and J. W. SMITH
Thermocoupte
Heal Tronsfer Section
Epoxy
Resin
Thermocouple . ..
SIDE FIN.
1. Schematic
diagram of experimental system.
flow
VIEW
FQo. 3. Detailed diagram of the tip of the traversing thermocouple.
I&am-s
AND
Drscuss~o~
Developed turbulent velocity profile measurements in pipe flow in a region outside the viscous sublayer are well described by the universal velocity profile, that is,
u+=2*5Iny++5*5. irq Carriage
hermocwptes II of Barr
for
on Tube
Nictwome Ribbon Rerittarce
Heater
The universal velocity profile was originally developed for a region very close to the wall and hence it is sometimes called the “law of the wall”. Surprisingly enough the law represents the velocity profile for over 90 per cent of the pipe radius and deviates only near and at the pipe centre. A velocity defect law has also been developed to describe the outer region of the turbulent boundary layer and it it given by, %
Fro. 2.
Diagram
Thermocou#e
of the heat transfer Dimensions in.
--u+= -2*51ny/R-0.18.
(2)
If there is an analogy between heat and momentum transfer, developed turbulent temperature profiles should be expressed in a similar form and hence,
PI
rik
(1)
t+ =A,Iny+
+B,
(3)
-A,lny/R+C.
(4)
and section.
&-t+=
1702
The effect of the Prandtl number on remperarure profiles for heat transfer in turbulent pipe flow
Equation (3) is known as the universal temperature profile and Eq. (4) as the temperature defect law. The Prandtl number is an additional parameter describing the effect of the thermal properties of the fluid in the transfer of heat. While Eqs. (1) and (3) are analogous, the dependency of the constants A, and B,on Np, is not generally known. The results of the present experiments are shown in Figs. 5-10 as temperature profiles in the form of Eqs. (3) and (4) for air, water and 30 % aqueous ethylene glycol. Figure 4 shows the Stanton number as a function of Reynolds number on logarithmic co-ordinates for all three fluids. The data of KOKOREV[4] are plotted in Fig. 11 to show the universal form of the temperature profile for mercury (iVp,=0.026).
NRe- IO-’
Fro. 4. Stanton number as a function of Reynolds number.
20 I8 16 14 12 IO 8 6
20
50
100
200
500
Y*
FIG. 5. Radial temperature profiles in air for smooth
tube S-l.
6
FIG. 6. Temperature
defect law with air.
1703
R. A. G~WEN
and J. W. Shnrrrr
6
4
> 1
I 2
I 4
III 6 610
I 2.
I 4
III 6810”
I 2
Y+ FIG. 7. Universal temperature profile with water.
FIG. 8. Temperature defect law with water.
1704
4
6
6 IO’
The effect of the Prandtl number on temperature
profiles for heat transfer in turbulent pipe flow
93 92 91 so 89 88 4
07
-
86 85 84 83 82
FIG. 9. Radial temperature profiles in aqueous ethylene glycol for smooth tube S-l.
12
I
P I
II
ml
I
III1
I
I.1 b
IIII
q
1
IO 9
0’
I II I
-0
I I
A
N R* 13700 21100 I5700 18100 14500 16400
WATERNP,
GLYCOL = 14.3
. 1
1
1
1
I
8 7 6 5
l
T4 .-J 3 2
0 -I
.02
.05
.I0
.20
.50
FIG. 10. Temperature defect law with aqueous ethylene glycol.
1705
I.0
R.yA. GOWENand J. W. Shnru 12 II ICI 9 0 z-7 6 5 4 3 200
500
1000
2000
5000
10000
Y* FIG. 11.
Universal temperature profile with mercury.
From the results, it can be seen that Eqs. (3) and (4) are nearly identical experimentally. This has also been shown to be true in [I] and [5]. The experimentally determined values of A, and B, from Eq. (3) are tabulated in Table 1. Results from [I, 2, 5 and 61 are given in addition to those of the present work. Unfortunately, not all data [3] can be converted into the universal form because the heat fluxes are not given.
Figure 7 shows the universal temperature profile 2’ vs. logy’ for water. However, the profile is not universal in the low Reynolds number range because the value of the t+ intercept at logy+ =0 depends on the Reynolds number. Rearrangement of the variables can easily show why this happens. t+
4,
TABLE 1. EXPERIMENTAL VALUES OF A8 AND B,
NP~
Reference
A8
BS
A3dpu*C,
t+ _
C--T PU*CP Tw-Te
C
h
assume 0.026
I41
2.04
-7.25
0.7
121
2.18
3.8
-0.4
0.705
111
2.2
3.3
-0.4
2.18
3.0
-0.4
0.7
This work
TB=TE
5.7
I51
2.58
34.5
-0.4
6.0
El
2.55
28.0
-0.4
14.3
This work
2.52
76*3
-0.7
then CF 2
J t+“--&l-e). St
The constant A, shows no strong dependence on the Prandtl number even though it is somewhat lower for air and mercury than it is for either water or aqueous ethylene glycol. The intercept constant for the temperature defect law also appears to be insensitive to NPr and this is to be expected since, in effect, the wall conditions have been removed. This point has been discussed in [5].
Because the friction factor reaches the fully turbulent region sooner than the Stanton number the J(CF/2)/Ns, ratio behaves in an abnormal fashion for a Reynolds number less than 20,000. As a consequence the universal temperature profile exhibits a Reynolds number dependence. When both Ns, and C, reach the fully turbulent region the NRe dependence is seen to diminish and the law is truly universal. However, for the range of NRe studied the slope on the logarithmic axis remained constant at 2.59f0.13. It should be emphasized that this difference in behaviour of Ns, and Cr is not due to
1706
The effect of the Prandtl number on temperature
undeveloped temperature or velocity profiles and longer momentum or thermal entry lengths would not change the pattern. Most textbooks on fluid dynamics report that 40 to 50 pipe diameters are necessary to develop a turbulent velocity profile and 45& were used in the present study, the last 19 of which were in the heated section. The entry length that is needed to establish fully developed temperature profiles has been discussed by SLEICHER and TRIBUS[9] who reviewed entrance length studies of seven investigators and conclude that in the Reynolds number range 10,OUOto 100,000, 5 to 10 diameters with a developed velocity profile are sufficient. MILLS[lo] andABBBBCHTand CHURCHILL [l l] in later studies confirmed this requirement. The constants in the universal temperature profile can be predicted from the velocity profile and a knowledge of the eddy diffusivities of heat and momentum transfer. Thermal and momentum eddy diffusivities can be conveniently expressed by the following equations, r/r,=(l+~&)du+/dy+
(5)
4/4,=(1/N,,+E,/v)dt+ldy+.
(6)
The transfer of heat and momentum in the viscous sublayer is predominately molecular and therefore the eddy motion may be neglected. Also, this region is so close to the wall that;
protiles for heat transfer in turbulent pipe flow
is assumed equal to unity then Eqs. (5) and (6) may be combined to give,
(11) Since the velocity profile in the buffer zone is represented by u+ =Slny+-3.05 the solution of Eq. (11) is, +5Np,
Equation (13) has been derived by SQUIRE[ 121using similar assumptions. In the turbulent core, the molecular properties may be neglected in comparison with the eddy diffusivities but the shear stress and the heat flux cannot be assumed constant and equal to those at the wall. Unlike the shear stress the radial heat flux distribution is not a linear function of the radial distance; however they may be assumed equal to one another without introducing sign&ant error [5J Dividing Eqs. (6) by (5) and again assuming the eddy diffusivities of heat and momentum to be equal yields,
t+ =Np,y+ u+=y+.
(14)
and simplifying
(8)
(9)
The shear stress and the heat flux may also be assumed equal in the buffer zone (5
U+
=25lny++5*5
for y+>30
t&, = 5 ln(5Np,+ 1) + 5N,
&=5ln30-3.05
[from Eq. (1311
[from Eq. (12)]
then integrating Eq. (15) yields, t+=2*5hy++5hI
NP; = %&a
_dt+ldy+ du +/dy +
(7)
and hence solutions of Eqs. (5) and (6) for 0 < y+ < 5 are given by;
(13)
5
,
t/r,=Lj/&=l
(12)
(10) 1707
+835+5N,.
(16)
R. A. GOWENand J. W. SMITH Therefore, in the turbulent core (y + > 30) the intercept constant is given by,
B,=51n(5N~oo+l)+8"55+ 5Ne,.
•
z/zw dt+/dy + en q/(twdu+/dY+"
(17)
SQtnm~ [12] assumed that the shear stress and heat flux remained constant in the turbulent core but since the right-hand sides of Eqs. (5) and (6) are equated the same results as Eqs. (16) and (17) are obtained. Since fluids with low Prandtl numbers have very high thermal conductivities, heat transfer by conduction cannot be neglected in the turbulent core. ROGERS and MAYHEW [13] have theoretically evaluated the temperature profiles in the turbulent core for fluids with a Prandtl number less than 0" 1. Their equation accounts for conductive heat transfer but the result was obtained by also assuming that the shear stress and heat flux remained constant [Eq. (7)]. They proposed the following equation,
t+=1251n (
Neglecting the molecular properties and solving Eq. (5) and (6) for N~,, yields, N~,, = g M
By experiment, in the turbulent core, the gradients are, dt + _As dy + y+
(20)
du + 2"5
(21)
dy + y+ hence
Nte _r/~w As. 01e~,.2"5
(22)
Figure 12 shows the radial variation of the turbulent Prandtl number for the Prandtl number range that was studied. The turbulent Prandtl number increases with the Prandtl number but remains relatively insensitive to the Reynolds number and the radial position. The solution of the temperature profile in the turbulent core without assuming unity for the turbulent Prandtl number yields, in place of Eq. (16),
x+y/R(l-y/R) "~
+!"25.1n~((2y/R-1) + W'~ W kk(2y/R- 1 ) - W] (60/Re*-I)-W)] (60/Re* - 1) + + 5[Nr,+ln(5Ne,+ 1)]
(19)
(18)
where
W = ~/(1 +
where fl= AJ2.5. The intercept constant B~ is given by,
20/NI,,)Re*
x = 5[(Ne,Re*) Bs=51nI5N~o0+1 ) + 8"55fl+ 5Net •
Re* =Ru*/v. Usually in the absence of experimental data, the turbulent Prandtl number is assumed equal to unity but once the temperature profiles have been measured experimentally it can easily be calculated. 1708
Comparison of the predicted constants experimental values is shown in Table teresting to note that as the Prandtl increased, Bs may be approximated by
(24)
Bs with the 2. It is innumber is 5.5 Nvr.
The effect of the Prandtl number on temperature profiles for heat transfer. in turbulent pipe flow
64 -
---
Npl= 5.7,143
+
Nh=0*7
0.2
0.2
0 FIG.
0.4
___
Experimental
Predicted
B8
B8
and subtraction of Eq. (26) from (25) and removing y + B yields, t+,=B,+(u+,-5.5)
Eq. (17) Eq. (18) Eq. (24) 5.5 Npr (WIV= 1) (Rogers) 0.026
-7.25
-7.72
-13.2
-9.3
~+JJ=JG/2ws,
0.143
0.7 5.7
3.0 34.5
2.56 36.9
-2.93 37.5
1.53 37.2
3.85 31.4
14.3
76.3
84.5
78.9
84.5
78.6
2 J G
substituting
into Eq. (27) gives,
(28)
Heat transfer coeficients
Heat transfer coefficients may be determined by integrating the turbulent temperature profiles. A much simpler method of prediction can be obtained if it is assumed that the respective distances from the wall at which the velocity equals ug and the temperature Ts are the same [14, 151. Then,
u+,=2*5lny+,+5*5
(27)
since by definition
l&=
t+,=A,lny+,+B,
1
12. Radial variation of the turbulent Prandtl number.
TABLE 2. INTERCEPT CONSTANT B,; EXPERIMENTAL AND PREDICTED VALUES
Npr
0.8
0.6
(25)
(26)
Ns’=BS+g:5.5,, where
+8*55+5N
Pr *
Equation (28) with B assumed equal to unity is shown as a dotted line in Fig. 1. The agreement with the experimental data is excellent.
1709
R. A. GOWEN and J. W. &urn CONCLUSIONS
The temperature profiles in the turbulent core of pipe flow are well represented by a universal profile of the form, t+ =A,lny+
+B,
where +&55+5N
Pr*
The value of constant A, appears to be relatively insensitive to the Prandtl number but intercept constant B, is strongly influenced by the thermal properties of the fluid. Stanton numbers in turbulent pipe flow can be predicted from the temperature profiles with the following equation,
UB
fanning friction factor heat transfer coefficient, B.t.u./hr-ft2”F thermal conductivity, B.t.u./hr-ft2”F/ft Prandtl number = C&k Du,lv Stanton number = h/pu,C, radial heat flux, B.t.u./hr-ft2 Ru*/v, friction Reynolds number temperature, “F bulk temperature, “F centre-line temperature, “F (T, - T)u*pC I&,, dimensionless temperature mean bulk flow velocity, ft/hr velocity, ft/hr d(r,g,/p) = friction velocity, ft/hr u/u*, dimensionless velocity ;i{c ~;WJJRe*
l4:
u+ W
X
Y Y+
radiaydistance from wall, ft yu*/v, dimensionless distance
Greek letters
B A,W
JGP~
sY, sH eddy diffusivity, ft/2hr p absolute viscosity, lb,,,/ft-hr v kinematic viscosity, ft2/hr p density, lb/ft 3 z,, z shear stress, lb/ft2
Nst = B, + (4(2/Cr) - 5.5) ’ NOTATION
constant in the universal temperature profile constant in the universal temperature profile constant in the velocity and temperature defect laws heat capacity, B.t.u./lb,“F pipe diameter, ft
Subscripts
bulk centre-line pertaining to heat transfer A4 pertaining to momentum transfer s pertaining to a smooth surface t pertaining to temperature wall condition W
hFFBENCE.9 JOHNKR. E. and h~~~r-ry T. J., Chem. L?ngng Sci. 1962 17 867-879. Darss=R. G. a.ndErmC. S., N.A.C.A. TN2629, 1952. ISAKOPP S. E. and DRBW T. B., Proceedings of the General Discassion on Heat Transfer, A.S.M.E. 11-13 September, 1951. KOKOREVL. S. and RY-V V. N., Znt. Chem. ZL%gng 1962 2 514-519. Shmx J. W., &WEN R. A. and WASMUNDB., Eddy Ditfusivities and Temperature Profiles for Turbulent Heat Transfer to Water in Pipes (presented at the 59th Annual A.1.Cb.E. Meeting in Detroit, and submitted to A.1.Cb.E. Journal). BECK‘WITHW. F. and FAH[EN R., U.S.A.E.C. report IS 734, 1963. TaUCHASSONC., Chem. Engng Sci. 1964 19 305-317. B~RNARDOE. and &4N C. S., N.A.C.A. WR E136, 1945. SLEICHERC. A. and Tm.m M., Recent Advances in Ht Mass Transfer, McGraw-Hill 1961. MULLSA. F., J. Mech. Engng Sci. 1962 4 63-77. ~BRECHT P. H. and ~H~BCHILL S. W., A.Z.Ch.E. JI 1960 6 268-273. Sqvrrc~ H. B., Proc. Gen. Discussion on Heat Transfer, London, 11-13, Sept. 1951 (A.S.M.E.). ROGWS G. F. C. and MAYHEW Y. R., Int. J. Ht Mass Transfer, Vol. I, pp. 55-67. 1960. RANNIE W. D., J. Aero Sci. 1956 23 485+89. VONWN T., Trans. A.S.M.E. 1939 61 705-710.
1710
The effect of the Prandtl number on temperature
profiles for heat transfer in turbulent pipe flow
R&sum&II existe peu de resultats experimentaux qui soutiennent ies discussions theoriques sur l’effet du nombre de Prandtl relatif aux courbes de temperatures dans un courant turbulent a l’int&ieur d’un tuyau. Ceci s’est revele particulierement vrai pour les fluides ayant un nombre de Prandtl plus grand que l’unit6. On a mesure les courbes de temperature radiale et les coefficients de transfert de chaleur pour un ecoulement turbulent d’air (Npr=0,7) et d’ethylene aqueux (Npr= 14,3) B I’interieur d’un tube aux parois limes, et dam une gamme du nombre de Reynolds variant de 10.000 a 50.000. Cellesci et d’autres courbes de temperature relev6es dans le foyer turbulent pour des fluides ayant des nombres de Prandtl entre 0,026 et 14,3 ont et6 present&es sous la forme de: t+=Aslog,y++l.l* Pour les-fluides dont le nombre de Prandtl est sup&ieur, la constante Bs apparalt comme &ant directement proportionnelle a Npr et independante de NR~ tandis que As est raisonnablement insensible aux deux. Les coefficients de transfert de chaleur calcules ii partir des courbes de temperature sont tout a fait en accord avec r&.ultats experimentaux. Zusammenfassung-Es gibt wenige Versuchsergebnisse, die die t heoretischen Diskussionen liber die Auswirkung der Prandtl-Zahl auf Temperaturprofile in RBhren-Wirbelstrbmungen best%tigen. Dies gilt hesonders fiir Fliissigkeiten bei einer Prandtl-Zahl grosser als 1. Radialtemperaturprofile und Wiirmelibergangskoeffizienten fur die Wirhelstriimung von Luft (Npr-0,7) und w&&gem Glykol (Athylenglykol, Npl=14,3) in einer glatten Rohre wurden bei Reynolds-Zahlen von 10 Ooo bis 50 000 gemessen. Diese und andere Temperaturprofile im Wirbelkern wurden fur Fliissigkeiten mit Prandtl-Zahlen zwischen 0,026 und 14,3 in der Formel
dargestellt. Ftir Fliissigkeiten mit hoheren Prandtl-Zahlen scheint die Konstante B, direkt proportional zu NP~und von Nnc unabhangig zu sein, wahrend As ziemlich unabhangig von beiden ist. Die aus den Temperaturprofilen errechneten Wlrmeiibergangskoeffizienten stimmen mit den Versuchsergebnissen ausgezeichnet i&rein.
1711