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Pseudocritical heat transfer inside vertical tubes
183
Pseudocritical
Heat Transfer inside Vertical Tubes
R. BELLINGHAUSEN Lehrstuhl
fiir
and U. RENZ
Wiirmeiibertragung
und Klimatechnik,
R WTH
Aachen
(F.R.G.)
Dedicated
to Prof. Dr.-Zng. K. Stephan on the occasion of his 6@th birthday
(Received
June 11, 1990)
Abstract Heat transfer to fluids with pressures slightly above the critical is strongly influenced by the variation of the fluid properties occurring in the so-called pseudocritical region. The heat transfer can either be improved or reduced, depending on the flow conditions. The improvement is due to the enhanced heat transport caused by the increased heat capacity of the fluid. In the case of reduced heat transfer the turbulence is weakened by gravitational effects. In the present paper the heat transfer due to the complex interaction between gravity and turbulence in the so-called pseudocritical region is predicted by a numerical solution of the conservation equations for a turbulent pipe flow. The numerical results are compared with published experimental data for an electrically heated vertical pipe with the fluid R12.
1. Introduction Heat transfer to fluids with pressures slightly above the critical state, that is, between 1.O
Ckm.
Length
x
Fig. 1. Heat transfer coefficient in the pseudocritical region: curve a, low heat flux, low mass flux; b, high heat flux, high mass flux.
critical temperatures can be regarded as an extrapolation of the vapour pressure curve into the supercritical region (see refs. 11 and 12). Other fluid properties like viscosity, thermal conductivity and density also vary widely in those temperature and pressure regions. It should be noted that large density gradients in the flow result in gravitational effects which are found to be most important for the heat transfer. 2. Calculation
method
The calculations of the present paper are based on the parabolic conservation equations for mass, momentum and energy in combination with the k, e-turbulence model for low turbulent Reynolds numbers by Jones and Launder [ 13,141. The conservation equations can be written in a general form:
Eng. Process., 28 (1990) 183-186
0 Elsevier Sequoia/Printed
in The Netherlands
184 TABLE
1. Terms of the
general conservation equation (the model constants are taken from Kawamura [IS])
1
I
Pr
Pr,
1
=lc
1
0,
with the general variable 4, the transport property ~7 and the source term S defined as shown in Table 1. In the central regions of the pipe flow an upper limit for the turbulent viscosity, as proposed by Odenthal [ 16, 171, is used: 2 g 150
B The solution procedure published by Patankar and Spalding [ 181 is applied. Detailed descriptions are given in refs. 18 and 19. A computational grid with 121 radial grid lines is used. In the axial direction the step width is about half a millimeter, which is equivalent to one per cent of the tube diameter. In the entrance and in the pseudocritical region considerably smaller steps are applied, down to one hundredth of a millimeter.
3. Results
100
50
Relativelength
8.6
kW/m'
q
l0.0
kW/m’
A
n
‘-“1
XL-Xl
Fig. 2. Wall temperature profiles along the pipe axis.
of calculations X-
In an earlier investigation by the authors, qualitatively good agreement with experimental data was found [ 11, 121. It has been shown that the improved heat transfer is due to an increased heat capacity, whereas in the case of reduced heat transfer the decreasing density in the heated wall layers leads to an expansion and acceleration of this part of the flow. In the case of upward flow this effect is amplified by gravity, resulting in M-shaped velocity profiles. The velocity gradients in the turbulent wall layer become smaller and hence the turbulence production will be damped. However, this laminarization effect of the layers near the wall was clearly overpredicted and the heat transfer showed quantitatively bad agreement with the experimental data. A comparison between predicted and measured wall temperature profiles along the pipe is presented in Fig. 2. Similar results were published by Schnurr [20,21] and Cornelissen [22]. An analysis of the turbulence distribution in the wall layer indicates that the kinetic energy of turbulence is totally damped by gravity effects (see Fig. 3). The k, s-model is not able to reproduce enough turbulence further downstream and the flow remains laminar.
I.= X= XX’ X:
L-
10-S IO -6
3m
0
5m 6m 7m xL=13.43m
i
4m
4-_s
IQ-4 distance tmm the wall
m y
Fig. 3. Radial profiles of kinetic energy of turbulence with the standard turbulence model.
In order to avoid a complete laminarization a minimum value for the turbulent energy must be introduced. The minimum value is determined empirically from a limiting turbulent viscosity value by comparison of predictions and measurements: ttt,“‘“tJ VI,in
(31
185 4. Conclusion
x=Jm
0
x=lm
0
The reduction of heat transfer in vertical tubes in the so-called pseudocritical region can be explained by the steep density decrease near the heated wall resulting in gravity-driven buoyancy effects, especially in upward-oriented flows. Under turbulent flow conditions these buoyancy effects tend to relaminarize the flow near the wall. This behaviour can be predicted quite well by a numerical solution of the conservation equations using the k, &-turbulence model with a minimum value of the turbulent viscosity in the core of the Bow.
Fig. 4. Radial profiles of kinetic energy of turbulence with limiting turbulent viscosity.
Nomenclature Cl,
c2,
300
I FIbi g k ::
. ,,
m
‘i”” ( 56.7 kW/mz 0.2
0.3
0.1
0.5
0.6
0.7
v
v
0.8
0.9
1.0
Rtlalive length 2
Fig. 5. Wall temperature profiles along the pipe axis with limiting turbulent viscosity.
from which it follows that kmin
constants
of turbulence
model
c39 c4
=
(%&)“2
(4)
This limitation is not used in the area of the laminar sublayer, which is located at a dimensionless distance y + -C 5 corresponding to approximately 0.01 mm in the actual situation. Recalculated radial profiles of the turbulent energy, equivalent to those in Fig. 3, are shown in Fig. 4 using the above-mentioned limitation. It can be seen that the production of turbulence in the near-wall layers is enhanced and there are small maxima near the wall in the pseudocritical region (x = 5 m). Further, it can be seen from Fig. 5 that the axial wall temperature profiles in the pseudocritical region compare better now with the experimental data. The temperatures outside the pseudocritical region, however, are now slightly too high, and the numerical instabilities caused by the highly variable fluid properties are still apparent.
Pr Pr, P PC I, 4w R Re r S T TC TF TW u, 0 U, X XL XT
Y
inner pipe diameter functions of turbulence model acceleration due to gravity heat transfer coefficient total enthalpy kinetic energy of turbulence mass flux density Prandtl number turbulent Prandtl number pressure critical pressure heat flux density pipe radius Reynolds number radial coordinate source term temperature critical temperature fluid temperature wall temperature velocity components shear stress velocity axial coordinate pipe length heated length distance from wall dissipation of kinetic energy viscosity laminar viscosity at pipe inlet turbulent viscosity viscosity at wall density density at wall general Prandtl/Schmidt number general variable
Acknowledgement The authors gratefully acknowledge the financial support of the Deutsche Forschungsgemeinschaft.
186
References 2
I K. Yamagata,
2 3
4
5
6
7
8
9
10 11
K. Nishikawa, S. Hasegawa, T. Fuji and S. Yoshida, Forced convective heat transfer to supercritical water flowing in tubes, ht. J. Heat Mass Transfer, 15 (1972) 2575 2593. M. E. Shitsman, Impairment of the heat transmission at supercritical pressures, High Temp. (USSR), I (1963) 237-244. W. Kastner, W. Kohler, W. Krltzer and D. Hein, Wlrmetibergang im Post Dryout-Bereich in vertikalen und horizontalen Rohren bei gleichfiirmiger Beheizung, KWU-Abschlu~bericht zum Fiiderungsuorhaben BMFT ET 1409 A, 1983. H. S. Swenson, J. R. Carver and C. R. Kakarala, Heat transfer to supercritical water in smooth-bore tubes, J. Heat Transfer, 87 (1965) 477-484. J. D. Jackson and W. B. Hall, Forced convection heat transfer to fluids at supercritical pressure, Turbulent Forced Convection in Channels and Bundles: Theory and Application to Heat Exchangers and Nuclear Reactors, Istanbul, Turkey, 1978, Vol. 2, NATO Advanced Study Inst. Book, Hemisphere. Washington, DC, 1979, pp. 563-611. B. S. Shiralkar and P. Griffith, Deterioration in heat transfer to flmds at supcrcritical pressure and high heat fluxes, J. Heat Transfer, 91 (1969) 27-36. M. Adelt and J. Mikielewicz, Heat transfer in a channel at supercritical pressure, ht. J. Heat Mass Transfer, 24 (1981) 1667p 1674. R. D. Wood and J. M. Smith, Heat transfer in the critical regiontemperature and velocity profiles in turbulent flow, AfChE J., f0 (1964) 180 186. P. J. Giarratano and M. C. Jones, Deterioration in heat transfer to supercritical helium at 2.5 atmosphere, ht. J. Heat Mass Transfer, 105 (1975) 649-653. M. Scheidt, Untersuchung zum Wgrmeiibergang bei iiberkritischen Driicken, Dissertation, Univ. Hannover, F.R.G., 1983. R. Bellinghausen and U. Renz, The prediction of heat transfer and pressure loss of refrigerants near the critical state, ICHMT XVIII Symp. Heat and Mass Transfer in Cryoengineering and
3
4
5
16
17
18
19 20
21
22
Refrigeration, Int. Centre Heat and Mass Transfer, Dubrovnik, Yugoslavia, 1986. U. Renz and R. Bellinghausen, heat transfer in a vertical pipe at supercritical pressure, 8th Int. Heat Transfer Conf., Hemisphere, Washington, DC, 1986, pp. 957-962. W. P. Jones and U. Renz, Condensation from a turbulent stream onto a vertical surface, ht. J. Heat Mass Transfer, 17 (1974) 1019-1028. W. P. Jones and B. E. Launder, The prediction of laminarization with a two-equation model of turbulence, Int. J. Heat Mass Transfer, 15(1972) 3Oll314. H. Kawamura, Analysis on laminarization of heated gas using a two-equation model of turbulence, btd Symp. Turbulent Shear Flow, Imperial College, London, U.K., 1979, pp. 1816-1821; Nippon Genshiryoku Kenkyusho, [Rep.]JAERI-M 1989, JAERIM- 7844, React. Eng. Div. Annu. Rep., 108-11. H.-P. Odenthal, Berechnung des Wlrme- und Stoffaustausches bei partieller Kondensation in turbulent durchstriimten Kondensatorrohren, Dissertation, RWTH Aachen, F.R.G., 1983. U. Renz and H.-P. Odenthal, Numerical prediction of heat and mass transfer during condensation from a turbulent vapour/ gas-stream onto a vertical liquid film, in P. J. Marto and P. G. Kroeger (eds.), 18th Nat. Heat Transfer ConfI. San Diego, CA, U.S.A., 1979, ASME, New York, pp. 27-33. S. V. Patankar and D. B. Spalding, Heat and Mass Transfer in Boundary Layers, Vol. 1, Intertext Books, London, 2nd edn., 1970, p. 21. R. Bellinghausen, Die Filmkondensation bei hohen Dampfgeschwindigkeiten, Dissertation, RWTH Aachen, F.R.G., 1990. N. M. Schnurr, V. S. Sastry and A. B. Shaping, A numerical analysis of heat transfer to fluids near the thermodynamic critical point including the thermal entrance region, J. Heat Transfer, 98 (1976) 609-615. N. M. Schnurr, Numerical predictions of heat transfer to supercritical helium in turbulent flow through circular tubes, J. Heat Transfer, 99 (1977) 580-585. M. Cornelissen, The thermal stability of superconductors cooled by a turbulent flow of supercritical helium, Dissertation, Univ. Delft, The Netherlands, 1984.
Pseudocritical
Heat Transfer inside Vertical Tubes
R. BELLINGHAUSEN Lehrstuhl
fiir
and U. RENZ
Wiirmeiibertragung
und Klimatechnik,
R WTH
Aachen
(F.R.G.)
Dedicated
to Prof. Dr.-Zng. K. Stephan on the occasion of his 6@th birthday
(Received
June 11, 1990)
Abstract Heat transfer to fluids with pressures slightly above the critical is strongly influenced by the variation of the fluid properties occurring in the so-called pseudocritical region. The heat transfer can either be improved or reduced, depending on the flow conditions. The improvement is due to the enhanced heat transport caused by the increased heat capacity of the fluid. In the case of reduced heat transfer the turbulence is weakened by gravitational effects. In the present paper the heat transfer due to the complex interaction between gravity and turbulence in the so-called pseudocritical region is predicted by a numerical solution of the conservation equations for a turbulent pipe flow. The numerical results are compared with published experimental data for an electrically heated vertical pipe with the fluid R12.
1. Introduction Heat transfer to fluids with pressures slightly above the critical state, that is, between 1.O
Ckm.
Length
x
Fig. 1. Heat transfer coefficient in the pseudocritical region: curve a, low heat flux, low mass flux; b, high heat flux, high mass flux.
critical temperatures can be regarded as an extrapolation of the vapour pressure curve into the supercritical region (see refs. 11 and 12). Other fluid properties like viscosity, thermal conductivity and density also vary widely in those temperature and pressure regions. It should be noted that large density gradients in the flow result in gravitational effects which are found to be most important for the heat transfer. 2. Calculation
method
The calculations of the present paper are based on the parabolic conservation equations for mass, momentum and energy in combination with the k, e-turbulence model for low turbulent Reynolds numbers by Jones and Launder [ 13,141. The conservation equations can be written in a general form:
Eng. Process., 28 (1990) 183-186
0 Elsevier Sequoia/Printed
in The Netherlands
184 TABLE
1. Terms of the
general conservation equation (the model constants are taken from Kawamura [IS])
1
I
Pr
Pr,
1
=lc
1
0,
with the general variable 4, the transport property ~7 and the source term S defined as shown in Table 1. In the central regions of the pipe flow an upper limit for the turbulent viscosity, as proposed by Odenthal [ 16, 171, is used: 2 g 150
B The solution procedure published by Patankar and Spalding [ 181 is applied. Detailed descriptions are given in refs. 18 and 19. A computational grid with 121 radial grid lines is used. In the axial direction the step width is about half a millimeter, which is equivalent to one per cent of the tube diameter. In the entrance and in the pseudocritical region considerably smaller steps are applied, down to one hundredth of a millimeter.
3. Results
100
50
Relativelength
8.6
kW/m'
q
l0.0
kW/m’
A
n
‘-“1
XL-Xl
Fig. 2. Wall temperature profiles along the pipe axis.
of calculations X-
In an earlier investigation by the authors, qualitatively good agreement with experimental data was found [ 11, 121. It has been shown that the improved heat transfer is due to an increased heat capacity, whereas in the case of reduced heat transfer the decreasing density in the heated wall layers leads to an expansion and acceleration of this part of the flow. In the case of upward flow this effect is amplified by gravity, resulting in M-shaped velocity profiles. The velocity gradients in the turbulent wall layer become smaller and hence the turbulence production will be damped. However, this laminarization effect of the layers near the wall was clearly overpredicted and the heat transfer showed quantitatively bad agreement with the experimental data. A comparison between predicted and measured wall temperature profiles along the pipe is presented in Fig. 2. Similar results were published by Schnurr [20,21] and Cornelissen [22]. An analysis of the turbulence distribution in the wall layer indicates that the kinetic energy of turbulence is totally damped by gravity effects (see Fig. 3). The k, s-model is not able to reproduce enough turbulence further downstream and the flow remains laminar.
I.= X= XX’ X:
L-
10-S IO -6
3m
0
5m 6m 7m xL=13.43m
i
4m
4-_s
IQ-4 distance tmm the wall
m y
Fig. 3. Radial profiles of kinetic energy of turbulence with the standard turbulence model.
In order to avoid a complete laminarization a minimum value for the turbulent energy must be introduced. The minimum value is determined empirically from a limiting turbulent viscosity value by comparison of predictions and measurements: ttt,“‘“tJ VI,in
(31
185 4. Conclusion
x=Jm
0
x=lm
0
The reduction of heat transfer in vertical tubes in the so-called pseudocritical region can be explained by the steep density decrease near the heated wall resulting in gravity-driven buoyancy effects, especially in upward-oriented flows. Under turbulent flow conditions these buoyancy effects tend to relaminarize the flow near the wall. This behaviour can be predicted quite well by a numerical solution of the conservation equations using the k, &-turbulence model with a minimum value of the turbulent viscosity in the core of the Bow.
Fig. 4. Radial profiles of kinetic energy of turbulence with limiting turbulent viscosity.
Nomenclature Cl,
c2,
300
I FIbi g k ::
. ,,
m
‘i”” ( 56.7 kW/mz 0.2
0.3
0.1
0.5
0.6
0.7
v
v
0.8
0.9
1.0
Rtlalive length 2
Fig. 5. Wall temperature profiles along the pipe axis with limiting turbulent viscosity.
from which it follows that kmin
constants
of turbulence
model
c39 c4
=
(%&)“2
(4)
This limitation is not used in the area of the laminar sublayer, which is located at a dimensionless distance y + -C 5 corresponding to approximately 0.01 mm in the actual situation. Recalculated radial profiles of the turbulent energy, equivalent to those in Fig. 3, are shown in Fig. 4 using the above-mentioned limitation. It can be seen that the production of turbulence in the near-wall layers is enhanced and there are small maxima near the wall in the pseudocritical region (x = 5 m). Further, it can be seen from Fig. 5 that the axial wall temperature profiles in the pseudocritical region compare better now with the experimental data. The temperatures outside the pseudocritical region, however, are now slightly too high, and the numerical instabilities caused by the highly variable fluid properties are still apparent.
Pr Pr, P PC I, 4w R Re r S T TC TF TW u, 0 U, X XL XT
Y
inner pipe diameter functions of turbulence model acceleration due to gravity heat transfer coefficient total enthalpy kinetic energy of turbulence mass flux density Prandtl number turbulent Prandtl number pressure critical pressure heat flux density pipe radius Reynolds number radial coordinate source term temperature critical temperature fluid temperature wall temperature velocity components shear stress velocity axial coordinate pipe length heated length distance from wall dissipation of kinetic energy viscosity laminar viscosity at pipe inlet turbulent viscosity viscosity at wall density density at wall general Prandtl/Schmidt number general variable
Acknowledgement The authors gratefully acknowledge the financial support of the Deutsche Forschungsgemeinschaft.
186
References 2
I K. Yamagata,
2 3
4
5
6
7
8
9
10 11
K. Nishikawa, S. Hasegawa, T. Fuji and S. Yoshida, Forced convective heat transfer to supercritical water flowing in tubes, ht. J. Heat Mass Transfer, 15 (1972) 2575 2593. M. E. Shitsman, Impairment of the heat transmission at supercritical pressures, High Temp. (USSR), I (1963) 237-244. W. Kastner, W. Kohler, W. Krltzer and D. Hein, Wlrmetibergang im Post Dryout-Bereich in vertikalen und horizontalen Rohren bei gleichfiirmiger Beheizung, KWU-Abschlu~bericht zum Fiiderungsuorhaben BMFT ET 1409 A, 1983. H. S. Swenson, J. R. Carver and C. R. Kakarala, Heat transfer to supercritical water in smooth-bore tubes, J. Heat Transfer, 87 (1965) 477-484. J. D. Jackson and W. B. Hall, Forced convection heat transfer to fluids at supercritical pressure, Turbulent Forced Convection in Channels and Bundles: Theory and Application to Heat Exchangers and Nuclear Reactors, Istanbul, Turkey, 1978, Vol. 2, NATO Advanced Study Inst. Book, Hemisphere. Washington, DC, 1979, pp. 563-611. B. S. Shiralkar and P. Griffith, Deterioration in heat transfer to flmds at supcrcritical pressure and high heat fluxes, J. Heat Transfer, 91 (1969) 27-36. M. Adelt and J. Mikielewicz, Heat transfer in a channel at supercritical pressure, ht. J. Heat Mass Transfer, 24 (1981) 1667p 1674. R. D. Wood and J. M. Smith, Heat transfer in the critical regiontemperature and velocity profiles in turbulent flow, AfChE J., f0 (1964) 180 186. P. J. Giarratano and M. C. Jones, Deterioration in heat transfer to supercritical helium at 2.5 atmosphere, ht. J. Heat Mass Transfer, 105 (1975) 649-653. M. Scheidt, Untersuchung zum Wgrmeiibergang bei iiberkritischen Driicken, Dissertation, Univ. Hannover, F.R.G., 1983. R. Bellinghausen and U. Renz, The prediction of heat transfer and pressure loss of refrigerants near the critical state, ICHMT XVIII Symp. Heat and Mass Transfer in Cryoengineering and
3
4
5
16
17
18
19 20
21
22
Refrigeration, Int. Centre Heat and Mass Transfer, Dubrovnik, Yugoslavia, 1986. U. Renz and R. Bellinghausen, heat transfer in a vertical pipe at supercritical pressure, 8th Int. Heat Transfer Conf., Hemisphere, Washington, DC, 1986, pp. 957-962. W. P. Jones and U. Renz, Condensation from a turbulent stream onto a vertical surface, ht. J. Heat Mass Transfer, 17 (1974) 1019-1028. W. P. Jones and B. E. Launder, The prediction of laminarization with a two-equation model of turbulence, Int. J. Heat Mass Transfer, 15(1972) 3Oll314. H. Kawamura, Analysis on laminarization of heated gas using a two-equation model of turbulence, btd Symp. Turbulent Shear Flow, Imperial College, London, U.K., 1979, pp. 1816-1821; Nippon Genshiryoku Kenkyusho, [Rep.]JAERI-M 1989, JAERIM- 7844, React. Eng. Div. Annu. Rep., 108-11. H.-P. Odenthal, Berechnung des Wlrme- und Stoffaustausches bei partieller Kondensation in turbulent durchstriimten Kondensatorrohren, Dissertation, RWTH Aachen, F.R.G., 1983. U. Renz and H.-P. Odenthal, Numerical prediction of heat and mass transfer during condensation from a turbulent vapour/ gas-stream onto a vertical liquid film, in P. J. Marto and P. G. Kroeger (eds.), 18th Nat. Heat Transfer ConfI. San Diego, CA, U.S.A., 1979, ASME, New York, pp. 27-33. S. V. Patankar and D. B. Spalding, Heat and Mass Transfer in Boundary Layers, Vol. 1, Intertext Books, London, 2nd edn., 1970, p. 21. R. Bellinghausen, Die Filmkondensation bei hohen Dampfgeschwindigkeiten, Dissertation, RWTH Aachen, F.R.G., 1990. N. M. Schnurr, V. S. Sastry and A. B. Shaping, A numerical analysis of heat transfer to fluids near the thermodynamic critical point including the thermal entrance region, J. Heat Transfer, 98 (1976) 609-615. N. M. Schnurr, Numerical predictions of heat transfer to supercritical helium in turbulent flow through circular tubes, J. Heat Transfer, 99 (1977) 580-585. M. Cornelissen, The thermal stability of superconductors cooled by a turbulent flow of supercritical helium, Dissertation, Univ. Delft, The Netherlands, 1984.