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Fluid temperature fluctuations accompanying turbulent heat transfer in a pipe
Chemical Engineering Science, 1963, Vol. 18, pp. 307-311. Pergamon Press Ltd., Oxford. Printed in Great Britain.
Fluid temperature fluctuations accompanying turbulent heat transfer in a pipe S. TANIMOTO* and T. J. HANRMTY Department of Chemistry and Chemical Engineering, University of Illinois, Urbana, Illinois. (Received 11 June 1962) Abstract-The fluctuating temperatures in a fluid which accompany turbulent heat transfer were studied for fully developed turbulent flow of air in a pipe using small heat fluxes at the wall. In the fully developed region of the temperature field the root-mean-square temperature fluctuation, B’, at the centre of the pipe equals about 0.042 (TO - TB) and inCreaSeS t0 a maXimUm Of about O-10(To - TB) close to the pipe wall, where TOis the wall temperature and TB is the mixed average temperature. The maximum occurs at y+ = 25-30. Temperature fluctuations close to the wall are of lower frequency than temperature fluctuations in the pipe centre. The temperature fluctuations are markedly similar, both in their magnitude and in their variation with distance from the wall, to measurements of velocity fluctuations. _
described in terms of a root-mean-square temperature 0 = ($)‘P
INTRODUCTION
THE heating of a turbulent fluid gives rise to a temperature field which is varying with time. The temperature at any location may be defined as the sum of a time average temperature, T; and a fluctuating temperature, f?: T=T+8
(2)
and a frequency distribution function, F,, defined as $ =
(1)
An experimental
study has been carried out by HANRATTYin this laboratory [ 1,2] of the distribution of time average temperatures for fully developed turbulent flow of air in a pipe using small wall heat fluxes. Uniform wall-heat flux was employed in the heat-transfer section, which was long enough to allow the temperature profile to become fully developed. Radial temperature profiles were measured over a Reynolds number range of 18,000 to 71,000 and at locations in the 3.084 in. i.d. pipe of from 0 to 11.5 ft from the inlet to the heat-transfer section. A specially designed probe mechanism which entered the system through the pipe outlet and which could be located accurately at any radial or longitudinal position was used in this study. This paper reports on a study of the fluctuating temperature field in the system used by JOHNKand HANRA~Y to measure average temperatures. The unsteady component of the temperature field is
fluctuating
s
wFo dn 0
(3)
measurements are compared with measurements of the fluctuations of the velocity component in the direction of mean flow made in a pipe by These
JOHNK and
LAUFER [3] and by SANDBORN [4].
DFWRIPTION OF EXPERIMENTS
The temperature fluctuation measurements were made with hot-wire anemometer probes and circuits designed by the Flow Corporation. The main part of the anemometer probe consisted of O+IOO15in. tungsten wire O-044 in. long, soldered to the ends of two bent 1 in. long pins which were mounted in a 12 in. stem of l/8 in. diameter tubing. A small brass piece with a pointed tip was attached to the stem to prevent the wire probe from touching the wall of the pipe. When the tip of the brass piece touched the wall, the resistance wire was OXMJ5 in. from the wall and this was used as a reference point for radial temperature traverses. During a
* Present address: Ube Industries, Ltd., Japan. 307
S. TANIMOTO and T. J. HANIWTY
6-
--
NR~~~I,OOO
(LAUFER)
,_
u”B 4-
FIG.
1.
Comparison of temperature and velocity fluctuation measurements in the fully developed temperature field.
run this reference point was located by electrical contact. The distance of the probe tip from the wall at this reference point was determined in a small length of pipe of the same diameter as the test section. The pipe was mounted in a vertical position with the traversing mechanism inside. A light was transmitted down the pipe and the position of the tips of the pins holding the resistance wire was viewed with a cathetometer through a mirror mounted at a 45” angle at the bottom of the pipe. The anemometer equipment consisted of a Model HWB Anemometer modified to allow for low enough probe currents (5 mA) that it responded to temperature fluctuations and not to velocity fluctuations and a TBM Random Signal Voltmeter. The anemometer is reported by the manufacturer as correct &-1 dB from 2 c/s to 100 kc. The amplifiers were calibrated directly rather than using the square-wave circuit in the instrument. The ratio of the output to the input was measured when a sine wave of frequency much smaller than the compensation frequency was introduced to the input of the anemometer. A network with 1000 Q input impedance and 2 CIoutput impedance matched the impedance of the signal generator and the input of the amplifier. The characteristics of the bandpass filter used in measuring frequency spectra and
the methods used in treating these measurements are described in another paper [5]. &SJLTS
Measurements of 8’ at a distance of 45.3 pipe diameters from the inlet to the heat-transfer section, where the temperature field is fully developed, are presented in Figs. 1 and 2. The data at each Reynolds number were taken on two separate runs and the spread of the data for any one run was less
FIG. 2. Comparison of temperature andVeiocity fluctuation measurements in the fully developed temperature field.
308
Fltid temperature fluctuations accompanying turbulent heat transfer in a pipe
Y
2-
Ti-
F,u, i%
10-k 8: 6: 4-
VEL
F,Uer
SPECTRA
-
N,3.=41.000
SANDBORN
Ga IO+a_: 64;
0
+-
2-
A
0
TEMP
4
Y
SPECTRA = 0.91
y+
= 0.0045
I
10-Z
NR, = 39.000
q950
1
y+=4.7
8
IO-'
,
I,,,
,,I
na
,
IO0
,
.,,,,,
,
IO
UB
FIG. 3.
Comparison of temperature and velocity frequency spectra.
than is indicated in the graphs. The value of W/(Ts - T,) has a value of about O-042 at the pipe centre and increases to a maximum of about 0.1 close to the pipe wall. The maximum occurs at a value of y+ = 25-30 and it is closer to the pipe wall the larger the Reynolds number. A plot of e’l(T, - T,,) vs. y/u correlates on a single curve the data for both Reynolds numbers from v/a = 0.2 to Y/U = 1.0. As shown in Fig. 2 a plot V/t*iV,, vs. yu*/v tends to bring the data at the two Reynolds numbers together at small values of y. In this plot the Prandtl number has been incorporated into the ordinate on the supposition that this might be a reasonable dimensionless group to compare numerically with ~‘/a* close to the wall. This supposition is based on the fact that very close to the wall (To-T) t*N,,
=-
if u*
(4)
. The comparison of 8’ with the measurement of u’ by LAUFER in the dimensionless forms used ‘in Figs. 1 and 2 shows striking similarities in the two sets of measurements. At a Reynolds number of 41,000 the value of z//U, has a value of about 309
0.044 at the centre of the pipe and increases to.a maximum of about O-15 close to the wall. This maximum occurs at yu*/v = H-20. It has a value of u’/u* of about 26 which is to be compared with the maximum of O’/t*N,, of about 24 obtained at a Reynolds number of 39,000. Frequency distribution functions for the temperature fluctuations at X = 45.3 are shown in Fig. 3 for a Reynolds number of 39,000 and in Fig. 4 for a Reynolds number of 11,000. Both Figs. show that the temperature fluctuations are of lower frequency closer to the wall. As indicated in Fig. 3 the frequency spectra of the velocity fluctuations measured by SANDBORNshow the same trend. In the beginning of the heat-transfer section the temperature fluctuation field may be broken into three regions by examining the recorded signal: a quiet zone in the centre of the pipe, a zone of fluctuating temperature close to the wall and an intermediate zone which is intermittently quiet and fluctuating. Farther into the heat-transfer section there is no quiet zone and still farther into the heattransfer section the region of intermittency is not in evidence. Measurements of temperature fluctuations in the heat-transfer entry section are compared with measurements in the fully developed region (X = 45.3) in Fig. 5. The profiles of the temperature fluctuation measurements at the different values of X appear to approach the fully developed curve at small enough values of y. The approximate boundary between the intermittent zone and the wall zone is shown for X = 1.9 and X = 4.9 in Fig. 5. Even though the definition of this boundary was somewhat arbitrary, it does appear that the sharp break from the fully developed data does not coincide with the boundary of the intermittent region. It is also interesting to note that although the differences between the average temperature fields at X = 12.6 and X = 45.3 are not very significant when plotted as T,, - T/t* the temperature fluctuation measurements plotted as W/t* show marked differences. It appears that temperature fluctuation measurements are a more sensitive test as to when a temperature field is fully developed. Acknowledgement-Financial support for this project was received from E. I. duPont Company.
S. TANU&YIQand T. J. ~TIY
IO0 8 6h
13
a
NR~ = 11,000
6
l
10; 6 4
- 0.922 330
0 - 0.275
92
a - 0.107 0 - 0.021
36
+ - 0.0045
69 1.5
na US
Fax 4. Temperature
frequency spectra at NRE = 11,000.
NOTATION
t*
Radius of the pipe Heat capacity of the fluid Frequency distribution function for temperature fluctuations Frequency distribution function for velocity fluctuations Thermal conductivity of the fluid &p/k = Prandtl number hUB/V = Reynolds number Frequency c/s Heat flow through the wall per unit area per unit time Temperature of the fluid Time averaged temperature Wall temperature Mixed average temperature
310
0
PGU%
Local time average velocity Mixed average velocity Velocity fluctuation in the direction of mean flow a”: l/(70/p) = friction velocity X Longitudinal distance from the beginning of the heat transfer section x X/2a Distance from the wall Y; YU*lV Temperature fluctuation 0 (@)1/Z P Viscosity Y pJp = kinematic viscosity P Density Shear stress at the wall 70
UB
Fluid temperature
fluctuations accompanying
turbulent heat transfer in a pipe
NRp= 11,000
I .o 6 6 8 p-4
L
2
‘0.1 I 6 6 4F
FIG. 5.
Temperature
fluctuation measurements
in the thermal entry region.
REFERENCES
[~]~JoHNK R. E. and HANRAIW T. J., Chem. Engng. Sci. 1962 17 867. [2] _JOHNKR. E. and J~ANRATIY T. J., Chem. Engng. Sci. 1962 17 881. [3] LAUFER J., Nat, Advis. Comm. Aeronaut. Tech. Rep. 1174 1954. [4] SANDBORNV. A., Nat. Advis. C’omm. Aeronaut. Tech. Note 3266 1955. [5] LILLELEIITL. U. and HANIIA~~~ T. J., .I. Fluid Mech. 1961 11 65. R&an&-Les fluctuations de temp&ature darts un fluide qui accompagnent le transfer% de chaleur turbulent sont ttudi&es pour un ecoulement d’air dans un tuyau en regime completement turbulent et en utilisant de faibles flux de chaleur issus de la paroi. Darts la region stable du champ de temperature la racine carr6e moyenne des fluctuations de temperature I, au centre du tuyau est approximativement &ale a 0,042 (To - TB) et augmente jusqu’a un maximum d’environ 0,lO (To - 2%) pr&s de la paroi du tuyau. T0 represente la temp6rature de la paroi, et TB la temperature moyenne du fluide. Le maximum se prod& a y+ = 25 - 30. Les fluctuations de temp6rature pres de la paroi sont moins frequentes que lea fluctuations au centre du tube. Les fluctuations de temperature sont nettement similaires aux variations de vitesse du fluide ii la fois dam leur amplitude et leur variation en fonction de la distance il la paroi. Zasammenfassung-Am Beispiel striimender Luft in einem geheizten Rohr wurden die Temperaturschwankungen als Folge des turbulenten W%rmeaustausches bei kleinen Warrnestromdichten untersucht. Nach voller Entwickhmg des Temperaturprofils betrlgt die mittlere quadratische Abweichung der Temperaturschwankungen in der Rohrachse ungefti 0,042 (To - TB) und steigt bis zu einem Maximum von ungefti 0,lO (To - TB) [To = Wandtemperatur, TB = mittlere Temperatur des striimenden Mediums]; das Maximum riickt mit steigender Reynolds-Zahl n%her zur Rohrwand. Die Temperaturschwankungen, die inbezug auf Amplitude und Eintluss der Radialkoordinate vie1 Aehnlichkeit mit den Geschwindigkeitsschwankungen aufweisen, zeigen an der Roluwand eine. kleinere Frequenz als in der Rohrmitte.
311
Fluid temperature fluctuations accompanying turbulent heat transfer in a pipe S. TANIMOTO* and T. J. HANRMTY Department of Chemistry and Chemical Engineering, University of Illinois, Urbana, Illinois. (Received 11 June 1962) Abstract-The fluctuating temperatures in a fluid which accompany turbulent heat transfer were studied for fully developed turbulent flow of air in a pipe using small heat fluxes at the wall. In the fully developed region of the temperature field the root-mean-square temperature fluctuation, B’, at the centre of the pipe equals about 0.042 (TO - TB) and inCreaSeS t0 a maXimUm Of about O-10(To - TB) close to the pipe wall, where TOis the wall temperature and TB is the mixed average temperature. The maximum occurs at y+ = 25-30. Temperature fluctuations close to the wall are of lower frequency than temperature fluctuations in the pipe centre. The temperature fluctuations are markedly similar, both in their magnitude and in their variation with distance from the wall, to measurements of velocity fluctuations. _
described in terms of a root-mean-square temperature 0 = ($)‘P
INTRODUCTION
THE heating of a turbulent fluid gives rise to a temperature field which is varying with time. The temperature at any location may be defined as the sum of a time average temperature, T; and a fluctuating temperature, f?: T=T+8
(2)
and a frequency distribution function, F,, defined as $ =
(1)
An experimental
study has been carried out by HANRATTYin this laboratory [ 1,2] of the distribution of time average temperatures for fully developed turbulent flow of air in a pipe using small wall heat fluxes. Uniform wall-heat flux was employed in the heat-transfer section, which was long enough to allow the temperature profile to become fully developed. Radial temperature profiles were measured over a Reynolds number range of 18,000 to 71,000 and at locations in the 3.084 in. i.d. pipe of from 0 to 11.5 ft from the inlet to the heat-transfer section. A specially designed probe mechanism which entered the system through the pipe outlet and which could be located accurately at any radial or longitudinal position was used in this study. This paper reports on a study of the fluctuating temperature field in the system used by JOHNKand HANRA~Y to measure average temperatures. The unsteady component of the temperature field is
fluctuating
s
wFo dn 0
(3)
measurements are compared with measurements of the fluctuations of the velocity component in the direction of mean flow made in a pipe by These
JOHNK and
LAUFER [3] and by SANDBORN [4].
DFWRIPTION OF EXPERIMENTS
The temperature fluctuation measurements were made with hot-wire anemometer probes and circuits designed by the Flow Corporation. The main part of the anemometer probe consisted of O+IOO15in. tungsten wire O-044 in. long, soldered to the ends of two bent 1 in. long pins which were mounted in a 12 in. stem of l/8 in. diameter tubing. A small brass piece with a pointed tip was attached to the stem to prevent the wire probe from touching the wall of the pipe. When the tip of the brass piece touched the wall, the resistance wire was OXMJ5 in. from the wall and this was used as a reference point for radial temperature traverses. During a
* Present address: Ube Industries, Ltd., Japan. 307
S. TANIMOTO and T. J. HANIWTY
6-
--
NR~~~I,OOO
(LAUFER)
,_
u”B 4-
FIG.
1.
Comparison of temperature and velocity fluctuation measurements in the fully developed temperature field.
run this reference point was located by electrical contact. The distance of the probe tip from the wall at this reference point was determined in a small length of pipe of the same diameter as the test section. The pipe was mounted in a vertical position with the traversing mechanism inside. A light was transmitted down the pipe and the position of the tips of the pins holding the resistance wire was viewed with a cathetometer through a mirror mounted at a 45” angle at the bottom of the pipe. The anemometer equipment consisted of a Model HWB Anemometer modified to allow for low enough probe currents (5 mA) that it responded to temperature fluctuations and not to velocity fluctuations and a TBM Random Signal Voltmeter. The anemometer is reported by the manufacturer as correct &-1 dB from 2 c/s to 100 kc. The amplifiers were calibrated directly rather than using the square-wave circuit in the instrument. The ratio of the output to the input was measured when a sine wave of frequency much smaller than the compensation frequency was introduced to the input of the anemometer. A network with 1000 Q input impedance and 2 CIoutput impedance matched the impedance of the signal generator and the input of the amplifier. The characteristics of the bandpass filter used in measuring frequency spectra and
the methods used in treating these measurements are described in another paper [5]. &SJLTS
Measurements of 8’ at a distance of 45.3 pipe diameters from the inlet to the heat-transfer section, where the temperature field is fully developed, are presented in Figs. 1 and 2. The data at each Reynolds number were taken on two separate runs and the spread of the data for any one run was less
FIG. 2. Comparison of temperature andVeiocity fluctuation measurements in the fully developed temperature field.
308
Fltid temperature fluctuations accompanying turbulent heat transfer in a pipe
Y
2-
Ti-
F,u, i%
10-k 8: 6: 4-
VEL
F,Uer
SPECTRA
-
N,3.=41.000
SANDBORN
Ga IO+a_: 64;
0
+-
2-
A
0
TEMP
4
Y
SPECTRA = 0.91
y+
= 0.0045
I
10-Z
NR, = 39.000
q950
1
y+=4.7
8
IO-'
,
I,,,
,,I
na
,
IO0
,
.,,,,,
,
IO
UB
FIG. 3.
Comparison of temperature and velocity frequency spectra.
than is indicated in the graphs. The value of W/(Ts - T,) has a value of about O-042 at the pipe centre and increases to a maximum of about 0.1 close to the pipe wall. The maximum occurs at a value of y+ = 25-30 and it is closer to the pipe wall the larger the Reynolds number. A plot of e’l(T, - T,,) vs. y/u correlates on a single curve the data for both Reynolds numbers from v/a = 0.2 to Y/U = 1.0. As shown in Fig. 2 a plot V/t*iV,, vs. yu*/v tends to bring the data at the two Reynolds numbers together at small values of y. In this plot the Prandtl number has been incorporated into the ordinate on the supposition that this might be a reasonable dimensionless group to compare numerically with ~‘/a* close to the wall. This supposition is based on the fact that very close to the wall (To-T) t*N,,
=-
if u*
(4)
. The comparison of 8’ with the measurement of u’ by LAUFER in the dimensionless forms used ‘in Figs. 1 and 2 shows striking similarities in the two sets of measurements. At a Reynolds number of 41,000 the value of z//U, has a value of about 309
0.044 at the centre of the pipe and increases to.a maximum of about O-15 close to the wall. This maximum occurs at yu*/v = H-20. It has a value of u’/u* of about 26 which is to be compared with the maximum of O’/t*N,, of about 24 obtained at a Reynolds number of 39,000. Frequency distribution functions for the temperature fluctuations at X = 45.3 are shown in Fig. 3 for a Reynolds number of 39,000 and in Fig. 4 for a Reynolds number of 11,000. Both Figs. show that the temperature fluctuations are of lower frequency closer to the wall. As indicated in Fig. 3 the frequency spectra of the velocity fluctuations measured by SANDBORNshow the same trend. In the beginning of the heat-transfer section the temperature fluctuation field may be broken into three regions by examining the recorded signal: a quiet zone in the centre of the pipe, a zone of fluctuating temperature close to the wall and an intermediate zone which is intermittently quiet and fluctuating. Farther into the heat-transfer section there is no quiet zone and still farther into the heattransfer section the region of intermittency is not in evidence. Measurements of temperature fluctuations in the heat-transfer entry section are compared with measurements in the fully developed region (X = 45.3) in Fig. 5. The profiles of the temperature fluctuation measurements at the different values of X appear to approach the fully developed curve at small enough values of y. The approximate boundary between the intermittent zone and the wall zone is shown for X = 1.9 and X = 4.9 in Fig. 5. Even though the definition of this boundary was somewhat arbitrary, it does appear that the sharp break from the fully developed data does not coincide with the boundary of the intermittent region. It is also interesting to note that although the differences between the average temperature fields at X = 12.6 and X = 45.3 are not very significant when plotted as T,, - T/t* the temperature fluctuation measurements plotted as W/t* show marked differences. It appears that temperature fluctuation measurements are a more sensitive test as to when a temperature field is fully developed. Acknowledgement-Financial support for this project was received from E. I. duPont Company.
S. TANU&YIQand T. J. ~TIY
IO0 8 6h
13
a
NR~ = 11,000
6
l
10; 6 4
- 0.922 330
0 - 0.275
92
a - 0.107 0 - 0.021
36
+ - 0.0045
69 1.5
na US
Fax 4. Temperature
frequency spectra at NRE = 11,000.
NOTATION
t*
Radius of the pipe Heat capacity of the fluid Frequency distribution function for temperature fluctuations Frequency distribution function for velocity fluctuations Thermal conductivity of the fluid &p/k = Prandtl number hUB/V = Reynolds number Frequency c/s Heat flow through the wall per unit area per unit time Temperature of the fluid Time averaged temperature Wall temperature Mixed average temperature
310
0
PGU%
Local time average velocity Mixed average velocity Velocity fluctuation in the direction of mean flow a”: l/(70/p) = friction velocity X Longitudinal distance from the beginning of the heat transfer section x X/2a Distance from the wall Y; YU*lV Temperature fluctuation 0 (@)1/Z P Viscosity Y pJp = kinematic viscosity P Density Shear stress at the wall 70
UB
Fluid temperature
fluctuations accompanying
turbulent heat transfer in a pipe
NRp= 11,000
I .o 6 6 8 p-4
L
2
‘0.1 I 6 6 4F
FIG. 5.
Temperature
fluctuation measurements
in the thermal entry region.
REFERENCES
[~]~JoHNK R. E. and HANRAIW T. J., Chem. Engng. Sci. 1962 17 867. [2] _JOHNKR. E. and J~ANRATIY T. J., Chem. Engng. Sci. 1962 17 881. [3] LAUFER J., Nat, Advis. Comm. Aeronaut. Tech. Rep. 1174 1954. [4] SANDBORNV. A., Nat. Advis. C’omm. Aeronaut. Tech. Note 3266 1955. [5] LILLELEIITL. U. and HANIIA~~~ T. J., .I. Fluid Mech. 1961 11 65. R&an&-Les fluctuations de temp&ature darts un fluide qui accompagnent le transfer% de chaleur turbulent sont ttudi&es pour un ecoulement d’air dans un tuyau en regime completement turbulent et en utilisant de faibles flux de chaleur issus de la paroi. Darts la region stable du champ de temperature la racine carr6e moyenne des fluctuations de temperature I, au centre du tuyau est approximativement &ale a 0,042 (To - TB) et augmente jusqu’a un maximum d’environ 0,lO (To - 2%) pr&s de la paroi du tuyau. T0 represente la temp6rature de la paroi, et TB la temperature moyenne du fluide. Le maximum se prod& a y+ = 25 - 30. Les fluctuations de temp6rature pres de la paroi sont moins frequentes que lea fluctuations au centre du tube. Les fluctuations de temperature sont nettement similaires aux variations de vitesse du fluide ii la fois dam leur amplitude et leur variation en fonction de la distance il la paroi. Zasammenfassung-Am Beispiel striimender Luft in einem geheizten Rohr wurden die Temperaturschwankungen als Folge des turbulenten W%rmeaustausches bei kleinen Warrnestromdichten untersucht. Nach voller Entwickhmg des Temperaturprofils betrlgt die mittlere quadratische Abweichung der Temperaturschwankungen in der Rohrachse ungefti 0,042 (To - TB) und steigt bis zu einem Maximum von ungefti 0,lO (To - TB) [To = Wandtemperatur, TB = mittlere Temperatur des striimenden Mediums]; das Maximum riickt mit steigender Reynolds-Zahl n%her zur Rohrwand. Die Temperaturschwankungen, die inbezug auf Amplitude und Eintluss der Radialkoordinate vie1 Aehnlichkeit mit den Geschwindigkeitsschwankungen aufweisen, zeigen an der Roluwand eine. kleinere Frequenz als in der Rohrmitte.
311