- Email: [email protected]
An experiment on liquid solidification in thermal entrance region of a circular tube
IN HE~TANDMASS~RANSFER Vol. 4, pp. 437 - 444, 1977
~ P r e s s Printed in Great Britain
AN EXPERIMENT ON LIQUID SOLIDIFICATION IN THERMAL ENTRANCE REGION OF A CIRCULAR TUBE
Hsing-Lung Liu and G. J. Hwang Power Mechanical Engineering National Tsing Hua University Hsinchu, Taiwan, China
(OuL.,~nicated by J.P. HartD~=tt and W.J. Minkowycz)
ABSTRACT This note presents an experimental investigation on the effect of liquid solidification in a circular tube upon laminar heat transfer. A test section of small inner tube diameter was employed to decrease the free convection effects in this experiment. A series of data on the inlet temperature, the wall temperature, the mixed mean temperature at the exit and the mass flow rate were obtained to calculate the dimensionless heat transfer rate and the Nusselt number. A reasonable agreement between the present data and the result in the previous literature was observed
INTRODUCTION Zerkle and Sunderland [lJ studied the effect of liquid solidification in a tube upon laminar heat transfer by using a test section with diameter 38.1 mm and obtained experimental heat transfer rate which showed a discrepancy of 150% at ~ = 0.03 in comparison with their theoretical result.
It was concluded that
the difference might be caused by the effects of free convection. D e p e w and Zenter [2J investigated experimentally the same problem in a circular tube of 20 mm diameter.
Discrepancy due to the
effects of free convection decreases and 100% difference between theory and experiment was observed at ~ = 0.03. Hwang and Sheu [3J presented a theoretical and experimental investigation of liquid solidification in combined hydrodynamic and thermal entrance region of a circular tube with a uniform 437
438
H.-L. Liu and G.J. Hwang
wall temperature. employed.
Vol. 4, No. 6
A test section of 9.5 mm diameter was
Reasonable agreement between their experimental data
and theoretical results was observed.
Based on the reasonable
results obtained in the study [3] in combined hydrodynamic and thermal entrance region, this note presents results of a experimental study of the effect of liquid solidificatlon upon laminar heat transfer in thermal entrance region of a circular tube.
A test section of small inner tube diameter was used in
this experiment to improve the results of references [i, 2]. EXPERIMENTAL APPARATUS The entire experimental system is shown schematically in Figure i, which consisted primarily of a constant head tank, an inlet tube, a horizontal test section, an outlet tube, a refrigerator,
a water pump, a thermometer,
iron-constantan
thermocouples and one digltal temperature indicator. constant head tank was made of plastic plates.
The
The vertlcal
distance between the overflow tubing and the outlet near the bottom was about 340 mm which gave a sufficient flow rate for the experiment.
A net was installed in the tank to prevent
disturbances and air bubbles from the water reservoir.
A
thermometer was also located in the tank to indicate the inlet temperature,
.
.
~
.
T
o
.
.
IT E M P E R A T U ~
tOVERF~OW ,,~
iINOlCA~O~|
CONSTAN~ l W
|
n
"'"
(la)
~"
FIG. I Schematic of Experimental Apparatus
VOl. 4, No. 6
LIQUID SOLIDIFIC~TICN IN A ~JBE
439
An inlet tube of length 950 mm was connected to the constant head tank to ensure fully developed laminar flow in the test section for Re ° below i000.
The test section was made of copper
tubing with diameter 9.5 mm ID, 12.5 mm OD and length 400 mm and was jacketed by a copper tube with 32 mm ID and 35 mm OD.
The
inside diameter of the test section employed in the present work was about a quarter of the one used by Zerkle and Sunderland EI~, it was believed that this would reduce the effects of free convection to about two orders of magnitude.
A divergent-
convergent glass tube with a net in it was connected to the other end of test section for measuring the mixed mean temperature of the outlet temperature,
T • The water flow rate was measured by m using a graduated cylinder and a stop watch. A control valve was employed to regulate the flow rate from zero to the turbulent flow regime. A solution of 65 percent water and 35 percent alcohol was used to cool the test section.
The solution was stored and
cooled in a half ton refrigerator and pumped to the annular space of the test section. Ten thermocouples were installed initially on the inner tube of the test section for test run No. 1 to No. 9 without the effect of ice-formation.
To avoid irregular measurements at two
ends of the test section, eight thermocouples were then located at positions as indicated in Fig. ib for test run No. i0 to No. 18 with the effect of ice-formation.
In this experiment, the
indication of thermocouples and thermometer were calibrated at 0°C.
in an ice-water mixture and at room temperature within an
error of ~ 0.15°C.
RESULTS AND DI$CUS~IO N The present experimental data are shown in Table 1 and for both cases of no ice-formation and ice-formation.
In this table,
the computations of the mean wall temperature are T w = E(T 1 + T 2) + 4(T 3 + T 4) + 2(T 5 + T 6) + 4(T 7 + T 8) + (T9 + TI0) ] /24 for ten-point thermocouple measurement, and
440
H.-L. Liu and G.J. Hwang
Vol. 4, No. 6
8
T=
(~
Ti)/8
i=l for eight-point thermocouple measurement.
It is noted that the
ten-point computation formula is obtained by using the Simpson's rule and the average temperature difference between T w and the measured values, T i is ~ 1.44°C.
Re o is the inlet Reynolds
number based on the radius of the inner tube and the physical properties p , ~, Pr and ~ are evaluated at the arithmetical average of T o and T w. q* = (TO - % )
The dimensionless heat transfer rate is
/ (TO - Tw) without ice-formation,
and
q* = (To - Tm) / (TO - Tf) with ice-formation, and the dimensionless axial coordinate is = X / (Pr Re o R w) The values of q* and ~ are translated into Nu and Gz by using the relations Nu = 2q* / (2
- q*
) and Gz = 4 / ~, respectively.
Error Analysis for the parameters used in the experiment shows that the average error for the computation of ~
is ~ 3.9 percent
and for q* is ~ 5.4% based on the eighteen runs tabuled in TABLE
I.
Figure 2 shows variations in the dimensionless heat transfer rate q* with the dimensionless axial position ~ with and without the effects of ice-formation.
The results obtained by Zerkle and
Sunderland [lJ and Depew and Zenter [23 are also plotted in this figure.
Due to free convection effectsp
~t is seen that the
previous experimental data [1, 2J lie considerably above the theoretical curve.
Because of the small Reynolds number or the
relatively weak forced convection at large ~ , the discrepancy become pronounced as
~ increases.
The present experimental data
agree reasonably with the theoretical curve for 0.02 S ~ S 0.2 and an average difference of 0.044 is observed in this figure. Figure 3 depicts the Nusselt number versus the Graetz number.
It is seen in this figure the experimental results of
Zerkle and Sunderlana [lJ and Depew and Zenter [2J deviate considerably from the theoretical curve [13. Sunderland
Zerkle and
[13 used a test section of diameter 38 nun.
The
difference was 150 percent at Gz = 120 based on the theoretical
Vol. 4, No. 6
LIQUID SOLIDIFICATIflq IN A %~JHE
T A B L E
i.
Experimental Data
Tw oC
Tm
To
oC
oC
g/sec
1
5.05
16.9
20.3
6.83
0.0258
2
5.20
15.9
20.3
4.67
3
4.98
14.7
19.2
4
4.50
13.3
5
3.32
6
Run no.
441
m
~
q*
Gz
Nu
0.224
155.04
9,78
0.0374
0.291
115.27
9.12
3.57
0j0488
0.317
81.97
7.72
19.2
2.33
0.0746
0.401
53.62
6.72
10.5
19.2
1.33
0.1288
0.547
31,06
5.85
2.60
9.1
19,2
1.01
0.1708
0,608
23.42
5.11
7
2.10
8.1
19.2
0.82
0.2064
0.649
19.38
4.65
8
1.68
13.0
17.7
5,00
0.0344
0.294
116.28
10.02
9
1,40
11.2
17.7
2.83
0.0602
0,399
66,45
8.28
i0
-
3.37
20.5
27.2
6.10
0.0292
0,246
133.99
9.61
Ii
-
4.16
18.7
27.2
4.20
0.0424
0.313
94.34
8.75
12
- 6.37
15.8
27.2
2.80
0.0634
0.419
63.09
8.36
13
- 9.90
14.4
27.2
1.78
0.0994
0,471
40.24
6.19
14
- 4.90
20.2
25.7
8.20
0.0213
0.214
187.79
11,25
15
- 6,55
17.5
25.7
4.13
0.0425
0.319
94.12
8.93
16
-10.73
10.2
25.7
1.06
0.1648
0.603
24.27
5.24
17
- 2.11
14.5
26.0
1.90
0.0909
0.442
44.01
6.24
18
- 3.05
13.2
26.0
1.40
0.1282
0.492
31,20
5.09
result.
Depew and Zenter [2J employed a circular test section of
diameter 20 mm and obtained experimental data with I00 percent difference at Gz = 120 with theoretical curve.
It is found that
the present experimental data only show an average difference of 20 percent higher than the theoretical values.
Besides, due to
free convection effects the data of previous work [i, 2J scatter randomly in comparison with the present data and the theoretical curve.
442
H.-L. Liu and G.J. Hw~ng
Vol. 4, No. 6
0.0
/
• DEPEW AND ZENTER X ZERKLE AND SUNDERLAND • PRESENT
0.7
!
/ I
0.6 0.5
x X x
0.4 q*
x •
• XAA
•
•e
t
;/
x~ 0.3
A I~,~
¸
THEORY
0.2 0.i ,
2
i
4
i ,i,,l
I
6 8 102
2
,
,
4
i iilil
6 8101
i
2
I
i
4
FIG. 2 Comparison of Experimental and Theoretical Heat Transfer Rates It is noted that the Grashof number which indicated the effects of free convection is of order of 106 in Zerkle and Sunderland's experiment [i~ and is of order of 105 in Depew and Zenter's experiment [2~.
In the present work the Grashof number
is only or order of 104 .
This means that in the present
experimental set up, the effects of free convection are not significant.
CONCLUSION The note presents experlmental data in thermal entrance region of a circular tube with the effects of ice-formation. Reasonable agreement with theoretical curve was observed by minimizing the effects of free convection by using a test section with small diameter.
At Gz = 120, the results of Zerkle and
Sunderland EI~ and Depew and Zenter [2~ show deviations of 150
Vol. 4, NO. 6
LIQUID SCLIDIFICATICN IN A ~TJHE
24 22
i
• DEPEW & ZENTER X ZERKLE AND SUNDERLAND • PRESENT
20
x ~
443
m
/I
/i
X
/
18
#
Ax 16 14 NU
t
12 •
.:" /
i0
":/ ./"
8
6 4
~THEORY
• ~ Y ,
i01 2
,I,,,,I
,
I il,i,,l
I
3 4 68 102 2 3 4 6 8103 2 3 Gz FIG. 3 Nu versus Gz
Percent, respectively, from the theoretical curve [i].
It is
found that the present experimental data only show an average difference of 20 percent high than the theoretical values. A.CKNOWLEDGEMENT The authors would like to thank the technical staff in the machine shop of National Tsing Hua University for constructing the exPerimental apparatus and Miss Chy~ng Wu for typing the manuscript.
N.ONZNCLATURE g Gr
gravitational accelerationj Grashof number, 8ga 3 [(T O + % )
Gz
Graetz number, 4 /~
/ 2 - %3
/ v2;
444
H.-L. Liu and G.J. Hwang
Vol. 4, N o
Nu
Nusselt number, 2hR
Pr
Prandtl nlnnber, v / ~;
q*
dimensionless heat transfer rate;
Re °
Renoylds number, UmR w / v;
R
radius of tube;
w
T
/K ;
temperature;
X
axial coordinate; thermal diffusivity;
8
expansion coefficient;
v
kinematic viscosity; dimensionless axial coordinate; SUBSCRIPTS
f
interface;
i
location of thermocouples,
m
mixed mean value;
o
inlet condition;
w
wall;
i = I, 2, ... , i0;
REFERENCES
i,
R. D. ZERKLE and J. E. SUNDERLAND,
"The Effect of Liquid
Solidification in a Tube upon Laminar - Flow Heat Transfer and Pressure Drop", Jr. Heat Transfer, Trans. ASME 90, 183-190, 1968. 2.
C. A. DEPEW and R. C. ZENTER,
"Laminar Flow Heat Transfer
and Pressure Drop with Freezing at the Wall",
Int. Jr. Heat
Mass Transfer, 12, 1710-1714, 1969. 3.
G. J. HWANG and J. P. SHEU, "Liquid Solidification in Combined Hydrodynamic and Thermal Entrance Region of a Circular Tube", The Can. Jr. Chem. Eng. 54, 66-71, 1976.
~ P r e s s Printed in Great Britain
AN EXPERIMENT ON LIQUID SOLIDIFICATION IN THERMAL ENTRANCE REGION OF A CIRCULAR TUBE
Hsing-Lung Liu and G. J. Hwang Power Mechanical Engineering National Tsing Hua University Hsinchu, Taiwan, China
(OuL.,~nicated by J.P. HartD~=tt and W.J. Minkowycz)
ABSTRACT This note presents an experimental investigation on the effect of liquid solidification in a circular tube upon laminar heat transfer. A test section of small inner tube diameter was employed to decrease the free convection effects in this experiment. A series of data on the inlet temperature, the wall temperature, the mixed mean temperature at the exit and the mass flow rate were obtained to calculate the dimensionless heat transfer rate and the Nusselt number. A reasonable agreement between the present data and the result in the previous literature was observed
INTRODUCTION Zerkle and Sunderland [lJ studied the effect of liquid solidification in a tube upon laminar heat transfer by using a test section with diameter 38.1 mm and obtained experimental heat transfer rate which showed a discrepancy of 150% at ~ = 0.03 in comparison with their theoretical result.
It was concluded that
the difference might be caused by the effects of free convection. D e p e w and Zenter [2J investigated experimentally the same problem in a circular tube of 20 mm diameter.
Discrepancy due to the
effects of free convection decreases and 100% difference between theory and experiment was observed at ~ = 0.03. Hwang and Sheu [3J presented a theoretical and experimental investigation of liquid solidification in combined hydrodynamic and thermal entrance region of a circular tube with a uniform 437
438
H.-L. Liu and G.J. Hwang
wall temperature. employed.
Vol. 4, No. 6
A test section of 9.5 mm diameter was
Reasonable agreement between their experimental data
and theoretical results was observed.
Based on the reasonable
results obtained in the study [3] in combined hydrodynamic and thermal entrance region, this note presents results of a experimental study of the effect of liquid solidificatlon upon laminar heat transfer in thermal entrance region of a circular tube.
A test section of small inner tube diameter was used in
this experiment to improve the results of references [i, 2]. EXPERIMENTAL APPARATUS The entire experimental system is shown schematically in Figure i, which consisted primarily of a constant head tank, an inlet tube, a horizontal test section, an outlet tube, a refrigerator,
a water pump, a thermometer,
iron-constantan
thermocouples and one digltal temperature indicator. constant head tank was made of plastic plates.
The
The vertlcal
distance between the overflow tubing and the outlet near the bottom was about 340 mm which gave a sufficient flow rate for the experiment.
A net was installed in the tank to prevent
disturbances and air bubbles from the water reservoir.
A
thermometer was also located in the tank to indicate the inlet temperature,
.
.
~
.
T
o
.
.
IT E M P E R A T U ~
tOVERF~OW ,,~
iINOlCA~O~|
CONSTAN~ l W
|
n
"'"
(la)
~"
FIG. I Schematic of Experimental Apparatus
VOl. 4, No. 6
LIQUID SOLIDIFIC~TICN IN A ~JBE
439
An inlet tube of length 950 mm was connected to the constant head tank to ensure fully developed laminar flow in the test section for Re ° below i000.
The test section was made of copper
tubing with diameter 9.5 mm ID, 12.5 mm OD and length 400 mm and was jacketed by a copper tube with 32 mm ID and 35 mm OD.
The
inside diameter of the test section employed in the present work was about a quarter of the one used by Zerkle and Sunderland EI~, it was believed that this would reduce the effects of free convection to about two orders of magnitude.
A divergent-
convergent glass tube with a net in it was connected to the other end of test section for measuring the mixed mean temperature of the outlet temperature,
T • The water flow rate was measured by m using a graduated cylinder and a stop watch. A control valve was employed to regulate the flow rate from zero to the turbulent flow regime. A solution of 65 percent water and 35 percent alcohol was used to cool the test section.
The solution was stored and
cooled in a half ton refrigerator and pumped to the annular space of the test section. Ten thermocouples were installed initially on the inner tube of the test section for test run No. 1 to No. 9 without the effect of ice-formation.
To avoid irregular measurements at two
ends of the test section, eight thermocouples were then located at positions as indicated in Fig. ib for test run No. i0 to No. 18 with the effect of ice-formation.
In this experiment, the
indication of thermocouples and thermometer were calibrated at 0°C.
in an ice-water mixture and at room temperature within an
error of ~ 0.15°C.
RESULTS AND DI$CUS~IO N The present experimental data are shown in Table 1 and for both cases of no ice-formation and ice-formation.
In this table,
the computations of the mean wall temperature are T w = E(T 1 + T 2) + 4(T 3 + T 4) + 2(T 5 + T 6) + 4(T 7 + T 8) + (T9 + TI0) ] /24 for ten-point thermocouple measurement, and
440
H.-L. Liu and G.J. Hwang
Vol. 4, No. 6
8
T=
(~
Ti)/8
i=l for eight-point thermocouple measurement.
It is noted that the
ten-point computation formula is obtained by using the Simpson's rule and the average temperature difference between T w and the measured values, T i is ~ 1.44°C.
Re o is the inlet Reynolds
number based on the radius of the inner tube and the physical properties p , ~, Pr and ~ are evaluated at the arithmetical average of T o and T w. q* = (TO - % )
The dimensionless heat transfer rate is
/ (TO - Tw) without ice-formation,
and
q* = (To - Tm) / (TO - Tf) with ice-formation, and the dimensionless axial coordinate is = X / (Pr Re o R w) The values of q* and ~ are translated into Nu and Gz by using the relations Nu = 2q* / (2
- q*
) and Gz = 4 / ~, respectively.
Error Analysis for the parameters used in the experiment shows that the average error for the computation of ~
is ~ 3.9 percent
and for q* is ~ 5.4% based on the eighteen runs tabuled in TABLE
I.
Figure 2 shows variations in the dimensionless heat transfer rate q* with the dimensionless axial position ~ with and without the effects of ice-formation.
The results obtained by Zerkle and
Sunderland [lJ and Depew and Zenter [23 are also plotted in this figure.
Due to free convection effectsp
~t is seen that the
previous experimental data [1, 2J lie considerably above the theoretical curve.
Because of the small Reynolds number or the
relatively weak forced convection at large ~ , the discrepancy become pronounced as
~ increases.
The present experimental data
agree reasonably with the theoretical curve for 0.02 S ~ S 0.2 and an average difference of 0.044 is observed in this figure. Figure 3 depicts the Nusselt number versus the Graetz number.
It is seen in this figure the experimental results of
Zerkle and Sunderlana [lJ and Depew and Zenter [2J deviate considerably from the theoretical curve [13. Sunderland
Zerkle and
[13 used a test section of diameter 38 nun.
The
difference was 150 percent at Gz = 120 based on the theoretical
Vol. 4, No. 6
LIQUID SOLIDIFICATIflq IN A %~JHE
T A B L E
i.
Experimental Data
Tw oC
Tm
To
oC
oC
g/sec
1
5.05
16.9
20.3
6.83
0.0258
2
5.20
15.9
20.3
4.67
3
4.98
14.7
19.2
4
4.50
13.3
5
3.32
6
Run no.
441
m
~
q*
Gz
Nu
0.224
155.04
9,78
0.0374
0.291
115.27
9.12
3.57
0j0488
0.317
81.97
7.72
19.2
2.33
0.0746
0.401
53.62
6.72
10.5
19.2
1.33
0.1288
0.547
31,06
5.85
2.60
9.1
19,2
1.01
0.1708
0,608
23.42
5.11
7
2.10
8.1
19.2
0.82
0.2064
0.649
19.38
4.65
8
1.68
13.0
17.7
5,00
0.0344
0.294
116.28
10.02
9
1,40
11.2
17.7
2.83
0.0602
0,399
66,45
8.28
i0
-
3.37
20.5
27.2
6.10
0.0292
0,246
133.99
9.61
Ii
-
4.16
18.7
27.2
4.20
0.0424
0.313
94.34
8.75
12
- 6.37
15.8
27.2
2.80
0.0634
0.419
63.09
8.36
13
- 9.90
14.4
27.2
1.78
0.0994
0,471
40.24
6.19
14
- 4.90
20.2
25.7
8.20
0.0213
0.214
187.79
11,25
15
- 6,55
17.5
25.7
4.13
0.0425
0.319
94.12
8.93
16
-10.73
10.2
25.7
1.06
0.1648
0.603
24.27
5.24
17
- 2.11
14.5
26.0
1.90
0.0909
0.442
44.01
6.24
18
- 3.05
13.2
26.0
1.40
0.1282
0.492
31,20
5.09
result.
Depew and Zenter [2J employed a circular test section of
diameter 20 mm and obtained experimental data with I00 percent difference at Gz = 120 with theoretical curve.
It is found that
the present experimental data only show an average difference of 20 percent higher than the theoretical values.
Besides, due to
free convection effects the data of previous work [i, 2J scatter randomly in comparison with the present data and the theoretical curve.
442
H.-L. Liu and G.J. Hw~ng
Vol. 4, No. 6
0.0
/
• DEPEW AND ZENTER X ZERKLE AND SUNDERLAND • PRESENT
0.7
!
/ I
0.6 0.5
x X x
0.4 q*
x •
• XAA
•
•e
t
;/
x~ 0.3
A I~,~
¸
THEORY
0.2 0.i ,
2
i
4
i ,i,,l
I
6 8 102
2
,
,
4
i iilil
6 8101
i
2
I
i
4
FIG. 2 Comparison of Experimental and Theoretical Heat Transfer Rates It is noted that the Grashof number which indicated the effects of free convection is of order of 106 in Zerkle and Sunderland's experiment [i~ and is of order of 105 in Depew and Zenter's experiment [2~.
In the present work the Grashof number
is only or order of 104 .
This means that in the present
experimental set up, the effects of free convection are not significant.
CONCLUSION The note presents experlmental data in thermal entrance region of a circular tube with the effects of ice-formation. Reasonable agreement with theoretical curve was observed by minimizing the effects of free convection by using a test section with small diameter.
At Gz = 120, the results of Zerkle and
Sunderland EI~ and Depew and Zenter [2~ show deviations of 150
Vol. 4, NO. 6
LIQUID SCLIDIFICATICN IN A ~TJHE
24 22
i
• DEPEW & ZENTER X ZERKLE AND SUNDERLAND • PRESENT
20
x ~
443
m
/I
/i
X
/
18
#
Ax 16 14 NU
t
12 •
.:" /
i0
":/ ./"
8
6 4
~THEORY
• ~ Y ,
i01 2
,I,,,,I
,
I il,i,,l
I
3 4 68 102 2 3 4 6 8103 2 3 Gz FIG. 3 Nu versus Gz
Percent, respectively, from the theoretical curve [i].
It is
found that the present experimental data only show an average difference of 20 percent high than the theoretical values. A.CKNOWLEDGEMENT The authors would like to thank the technical staff in the machine shop of National Tsing Hua University for constructing the exPerimental apparatus and Miss Chy~ng Wu for typing the manuscript.
N.ONZNCLATURE g Gr
gravitational accelerationj Grashof number, 8ga 3 [(T O + % )
Gz
Graetz number, 4 /~
/ 2 - %3
/ v2;
444
H.-L. Liu and G.J. Hwang
Vol. 4, N o
Nu
Nusselt number, 2hR
Pr
Prandtl nlnnber, v / ~;
q*
dimensionless heat transfer rate;
Re °
Renoylds number, UmR w / v;
R
radius of tube;
w
T
/K ;
temperature;
X
axial coordinate; thermal diffusivity;
8
expansion coefficient;
v
kinematic viscosity; dimensionless axial coordinate; SUBSCRIPTS
f
interface;
i
location of thermocouples,
m
mixed mean value;
o
inlet condition;
w
wall;
i = I, 2, ... , i0;
REFERENCES
i,
R. D. ZERKLE and J. E. SUNDERLAND,
"The Effect of Liquid
Solidification in a Tube upon Laminar - Flow Heat Transfer and Pressure Drop", Jr. Heat Transfer, Trans. ASME 90, 183-190, 1968. 2.
C. A. DEPEW and R. C. ZENTER,
"Laminar Flow Heat Transfer
and Pressure Drop with Freezing at the Wall",
Int. Jr. Heat
Mass Transfer, 12, 1710-1714, 1969. 3.
G. J. HWANG and J. P. SHEU, "Liquid Solidification in Combined Hydrodynamic and Thermal Entrance Region of a Circular Tube", The Can. Jr. Chem. Eng. 54, 66-71, 1976.