Progress in Computational Fluid Dynamics, Volume 2, Nos. 2/3/4, 2002
144
A numerical investigation of laminar convection of supercritical carbon dioxide in vertical mini/micro tubes S. M. Liao and T. S. Zhao* Department of Mechanical Engineering, The Hong Kong University of Science & Technology, Clear Water Bay, Kowloon, Hong Kong, China *Corresponding author: E-mail:
[email protected]; Tel.: +852 2358-8647; Fax: +852 2358-1543 Abstract: Laminar convective heat transfer of supercritical carbon dioxide flowing in miniature vertical tubes was investigated numerically. Typical velocity profiles, temperature profiles, Nusselt numbers, and skin-friction coefficients for circular tubes having diameters of 0.5, 0.7, 1.4 and 2.16 mm, under both cooling and heating conditions, with and without gravity, were obtained. It is shown that for the supercritical carbon dioxide tube flow, hydrodynamically and thermally fully developed flow regions can be reached only when the fluid and the pipe wall are under the thermal equilibrium condition, which usually occurs at a location a long way downstream from the tube inlet (x/d > 1000). In addition, it has been revealed that buoyancy effects are significant for all cases considered, even for small tubes and at high Reynolds numbers. The results of this paper are of significance for the design of high efficiency compact supercritical carbon dioxide heat exchangers. Keywords: carbon dioxide, forced convection, numerical solution, supercritical. Reference to this article should be made as follows: Liao, S. M. and Zhao, T. S. (2002) ‘A numerical investigation of laminar convection of supercritical carbon dioxide in vertical mini/micro tubes’, Progress in Computational fluid Dynamics, Vol. 2, Nos. 2/3/4, pp. 144–152.
NOMENCLATURE
A cp Cf d e g, G h k L
m
Nu p Pr r,x R Re T u, v U, V
tube cross section area (m2) specific heat (J/K-kg) skin-friction coefficient tube diameter (m) enthalpy (J/kg) acceleration of gravity (m/s2) heat transfer coefficient (W/m2-K) thermal conductivity (W/m-K) tube length (m) mass flow rate (kg/s) Nusselt number pressure, bar Prandtl number cylindrical coordinates (m) tube radius (m) Reynolds number temperature (°C or K) velocities in x and r directions (m/s) dimensionless velocities in x and r directions
Greek symbols φ variables u, v and e µ dynamic viscosity (kg/m-s) Θ dimensionless temperature ρ density (kg/m3) τ shear stress (N/m2) Subscripts b bulk fluid in at the fluid inlet pc pseudocritical w wall
1 INTRODUCTION
It has been suggested that the use of carbon dioxide (CO2) as a refrigerant may provide a safe, economical, and cost–effective solution to environmental problems, because this natural fluid has a zero ODP (ozone depleting potential) and a zero effective GWP (global warming
INVESTIGATION OF LAMINAR CONVECTION
145
potential) [1]. Since in many refrigeration systems, such as automobile air-conditioners and heat pumps, the heat rejection temperatures are above the critical temperature of CO2 (31.1 °C), the systems using CO2 as a refrigerant will have to operate in the transcritical cycle. As such, the heat rejection takes place above the critical pressure (74–120 bar) in a so-called gas cooler (corresponding to the condenser in the conventional subcritical systems). To meet the need for low weight and small volume heat exchangers in automobile air-conditioning systems and to handle the high pressures of CO2 without excessive wall thickness, the gas cooler can be made of microchannel tubes. A gas cooler made of aluminum multichannel tubes with an individual channel diameter of 0.79 mm has been reported [2]. One of the most important characteristics of supercritical fluids near the critical point is that their physical properties exhibit rapid variations with the change of temperature, especially near the pseudocritical point (the temperature at which the specific heat reaches a peak for a given pressure). The variation of specific heat cp, density ρ , dynamic viscosity µ for CO2 at 80 bar and 100 bar is shown in Figure 1. These physical property curves are plotted based on the data in NIST Refrigerants Database REFPROP [3]. The changes in the fluid properties may make a large impact on the characteristics of convective heat transfer in tubes. Thus, heat transfer in supercritical fluids generally becomes more complex than in constantproperty fluids. Since the momentum and energy equations are coupled and nonlinear, they can only be solved numerically. In addition to the application in CO2 transcritical refrigerating systems, heat transfer in supercritical fluids is also relevant to many other industrial applications, for example, supercritical fluid extractions, supercritical power plants, cooling of superconducting machines, and power transmission cables, etc. Because of the wide range of applications, forced convection of supercritical fluids, such as water, carbon dioxide, nitrogen, hydrogen, and helium, in channels has been extensively studied both experimentally [4–7] and numerically [8–11]. Most of the previous investigations have been concerned with turbulent ρ 1000
µ -5
9x10
cp 30
-5
600
6x10
400
-5
10
0
0
3x10
µ [Pa-s],
p= 80 bar
µ [Pa-s],
p=100 bar
cp [kJ/K-kg], p=100 bar 3 ρ [kg/m ], p=100 bar
20
Figure 1
0 20
40
60 80 Temperature [°C]
100
FORMULATION
Consider a steady, laminar flow in a vertical tube (with diameter d and length L) cooled or heated at a constant wall temperature Tw. The governing conservation equations of mass and momentum in a cylindrical coordinate system (x, r) are given by:
∂ ( ρ u ) 1 ∂ (r ρ v) + = 0, ∂x r ∂r ∂ ( ρ uu ) ∂ ( ρ vu ) + ∂x ∂r ∂u ∂p ∂ ∂u 1 ∂ r (µ ) , = − ρ G − + (µ ) + ∂x ∂x ∂x r ∂ r ∂ r ∂ ( ρ uv) ∂ ( ρ vv) + ∂x ∂r v ∂v ∂p ∂ ∂v 1 ∂ r (µ ) − µ 2 , + (µ ) + ∂r ∂r ∂x ∂x r ∂ r r
(1)
(2)
(3)
where u, v are the fluid velocities in x and r directions separately; p represents the pressure of the fluid; G is equal to g, − g or 0 for upward, downward or zero gravity flow separately and g represents the acceleration of gravity. Neglecting the heat dissipative term, the terms related to the pressure gradient, and the internal heat source term, the energy equation in terms of enthalpy e can be expressed as
200
0
2
=−
cp [kJ/K-kg], p= 80 bar 3 ρ [kg/m ], p= 80 bar
800
flows in tubes with diameters larger than 2.0 mm. The purpose of this work was to numerically study laminar convective heat transfer of supercritical CO2 in mini/micro tubes, with special emphasis on gaining an understanding of the fluid flow and heat transfer behavior of CO2, with pressures and temperatures falling in the ranges of the typical operating conditions of gas coolers used in transcritical refrigeration systems. In solving the problem, the SIMPLER algorithm [12] and a latest supercritical CO2 property database [3] were used. The velocity profiles, temperature profiles, heat transfer coefficients, Nusselt numbers and skin-friction coefficients for supercritical CO2 laminar flow in vertical mini/micro tubes, under both cooling and heating conditions, with and without gravity, were obtained. The buoyancy effects on the flow and heat transfer were examined. An understanding of the variable property effects and buoyancy effects, on laminar forced convective heat transfer of supercritical CO2 flowing in mini/micro tubes, is of significance for the design of high efficiency compact supercritical CO2 heat exchangers.
120
Physical properties of supercritical carbon dioxide.
∂ ( ρ ue) ∂ ( ρ ve) ∂ k ∂e 1 ∂ k ∂ e + = ( )+ ) , (4) r ( ∂x c p ∂x r ∂ r cP ∂ r ∂x ∂r
S. M. LIAO AND T. S. ZHAO
146 where k presents the conductivity of fluid. For a constant wall temperature problem, the boundary conditions are: at the wall :
r = R, u = 0, v = 0, e = e (Tw ) , (5)
at the inlet :
x = 0, e = e (Tin ) , u = uin , v = 0, (6)
at the outlet :
∂e ∂u x = L, = 0, = 0, (7) ∂x x = L ∂x x = L
∂u ∂e at the centerline : r = 0, = 0, = 0, (8) ∂r r = 0 ∂r r = 0
where the subscripts in and w represent the quantities at the tube inlet and the tube wall, respectively.
3 NUMERICAL METHOD
A numerical solution to Equations (1–4) subject to the boundary conditions (5–8) was carried out based on a control-volume method detailed by Patankar [12]. In this procedure, the domain is discretized by a series of control volumes, each containing a grid point. The differential equations are expressed in an integral form over the control volume, and a power law variation is assumed in each coordinate direction, leading to a system of algebraic equations that can be solved in an iterative manner. Pressure–velocity interlinkages are handled by the SIMPLER formulation [12]. Physical properties, density, dynamic viscosity, specific heat and thermal conductivity, in the above governing equations, are treated as variables and are calculated from REFPROP (NIST Database) [3]. The computation started with guessing the pressure field and solving the momentum equations to obtain the velocity 1.5 d=0.5 mm, uin=0.706 m/s, cooling
x/d=272, upward U=u/uin
1.0 x/d=272, zero gravity x/d=696, zero gravity 0.5 x/d=272, downward x/d=696, downward 0
0
0.2
x/d=696, upward 0.4
r/R
0.6
0.8
1.0
Figure 2 Velocity profiles for upward, downward and zero gravity flows, under the cooling condition, for d = 0.5 mm at x/d = 272 and 696.
field. The continuity equation was then used to obtain the corrected pressure field, which was used as a new guess. At each time step a converged solution was checked by examining the residuals of the dependent variables as well as examining the invariance of spot-checked values. For most of the calculations, a grid of 200 × 100 (axial × radial) was used. Grid dependence of the solution was checked by refining the radial and axial grid system. The converged solution is obtained when the following convergence criteria are satisfied for the dependent variables:
φ i +1 − φ i ≤ 10−3 ; φ = u , v, and e. φi
(9)
4 RESULTS AND DISCUSSION
In the following, we shall present the numerical solutions to the problem of heat transfer of supercritical CO2 flowing through vertical circular tubes with diameters of 0.5, 0.7, 1.4 and 2.16 mm and a length of 1000.0 mm at 80 and 100 bar under both cooling and the heating conditions. Unless otherwise specified the default inlet temperature is 120 °C and the default wall temperature is 25 °C for the cooling flow, whereas for the heating flow the inlet and the wall temperatures are kept at 25 and 90°C, respectively. 4.1 Velocity profiles and temperature profiles
Figure 2 shows the axial velocity profiles, in the dimensionless form defined as U = u / uin ,
(10)
at different locations along the tube length (x/d = 272 and 696) for upward, downward and zero gravity cooling flows in a tube having a diameter of 0.5 mm for m = 1.77 × 10−5 kg/s (the corresponding inlet Reynolds number Rein = 2060) and p = 80 bar. The magnitude of the axial velocity U is seen to decrease downstream for all the flow directions. This simply arises from the fact that the density of CO2 progressively increases along the axial direction as the tube is cooled, while the mass flow rate is fixed. It is also found that the shape of the velocity profiles keeps changing with the axial locations, suggesting that a hydrodynamically fully developed flow can never be reached when supercritical carbon dioxide in the tube is cooled. The buoyancy effect on the velocity profiles can also be seen from Figure 2. The velocity gradients adjacent to the wall for the downward cooling (buoyancy-assisted) flow become larger than those for the zero gravity cooling flow. However, for the upward cooling (buoyancyopposed) flow, the velocity gradients adjacent to the wall are smaller than those for the zero gravity cooling flow. Figure 3 presents the temperature profiles, in the dimensionless form defined as
INVESTIGATION OF LAMINAR CONVECTION 16
d=0.5 mm, cooling
x/d=272, upward
0.20
147
12
x/d=272, downward 0.15
x/d=800, upward
10 U=u/uin
Θ=(T-Tw)/(Tin-Tw)
d=0.7 mm, uin=0.083 m/s, heating
14
x/d=272, zero gravity
x/d=696, upward
0.10
x/d=696, zero gravity
x/d=800, zero gravity
8
x/d=800, downward
6 x/d=269, downwaod 4
0.05
0 x/d=269, upward
x/d=696, downward 0
x/d=269, zero gravity
2
0
0.2
0.4
r/R
0.6
0.8
1.0
Figure 3 Temperature profiles for upward, downward and zero gravity flows under the cooling condition for d = 0.5 mm at x/d = 272 and 696.
Θ=
T − Tw , Tin − Tw
(11)
at different locations along the tube length (x/d = 272 and 696) for upward, downward and zero gravity cooling flows in the tube having a diameter of 0.5 mm for m = 1.77 × 10−5 kg/s (the corresponding inlet Reynolds number Rein = 2060) and p = 80 bar. As the fluid is cooled, the magnitude of the temperature is seen to decrease downstream for all the flow directions. It is interesting to note from Figure 3 that the temperature profiles over the upstream portion (x/d = 272) of the tube exhibits a significant ‘jump’. This is because the temperature over the tube cross-section passes through the pseudocritical point. It can be seen from Figure 3 that the shape of the temperature profiles keep changing with the axial locations, suggesting that a thermally fully developed flow can never be reached when supercritical carbon dioxide in the tube is cooled. Furthermore, Figure 3 indicates that the buoyancy effect on the temperature profiles is much more complicated than those on the velocity profiles. It is noted that the temperature gradient adjacent to the wall for the downward cooling (buoyancy-assisted) flow in the upstream (x/d = 272) is larger than that for the zero gravity cooling flow, whereas it becomes smaller in the downstream of the tube (x/d = 696). This is due to the fact that heat transfer is enhanced in the downward cooling (buoyancy-assisted) flow, and thus temperatures drop faster along the tube length as compared with the zero gravity flow. On the contrary, the temperature gradient adjacent to the wall for the upward cooling (buoyancyopposed) flow in the upstream (x/d = 272) is smaller than that for the zero gravity cooling flow, whereas it becomes larger downstream of the tube (x/d = 696). Figure 4 shows the dimensionless axial velocity profiles U at different locations along the tube length (x/d = 269 and 800) for upward, downward and zero
-2
0
0.2
0.4
r/R
0.6
0.8
1.0
Figure 4 Velocity profiles for upward, downward and zero gravity flows under the heating condition for d = 0.7 mm at x/d = 269 and 800.
gravity heating flows in a tube having a diameter of 0.7 mm for m = 2.48 × 10−5 kg/s (the corresponding inlet Reynolds number Rein = 658) and p = 80 bar. Contrary to the case of the cooling flow, Figure 4 indicates that the magnitude of the axial velocity, U, for the heating flow increases downstream for all the flow directions, as heating leads to a reduction in the fluid density. It can also be seen from Figure 4 that the shape of the velocity profiles keep changing with the axial locations. This fact suggests that a hydrodynamically fully developed flow can never be reached for the supercritical carbon dioxide flow in a tube heated at a constant temperature. Contrary to the case for the cooling flow, the upward flow is buoyancy-assisted and the downward flow is buoyancy-opposed, under the heating condition. It can be noted from Figure 4 that, as compared with the zero gravity heating flow, the velocity gradients adjacent to the wall are larger for the upward heating flow, but smaller for the downward heating flow. The velocity gradients adjacent to the wall for the downward heating flow may become negative when the temperature difference between the bulk fluid and the tube wall is large. The buoyancy force leads to a M-shaped velocity profile for the upward heating (buoyancy-assisted) flow, when the temperature difference between the bulk fluid and the tube wall becomes sufficiently large. Along the heated tube length, the M-shaped velocity profile changes toward the normal velocity profiles of constant property fluids when the temperature difference between the bulk fluid and the tube wall becomes smaller, in particular when the pseudocritical point is not within the temperature profile of the tube section. Figure 5 presents the dimensionless temperature profiles at different locations along the tube length (x/d = 269 and 800) for upward, downward and zero gravity heating flows in a tube having a diameter of 0.7 mm for m = 2.48 × 10−5 kg/s (the corresponding inlet Reynolds number Rein = 658) and p = 80 bar. As the fluid is heated along the tube, the temperature increases. For this reason the magnitude of the dimensionless temperature,
S. M. LIAO AND T. S. ZHAO
148 100
1.5
d=0.5mm, cooling
d=0.7 mm, heating
1.0
x/d=269, upward Cf×Reb
Θ=(T-Tw)/(Tin-Tw)
80
x/d=269, zero gravity 0.5
x/d=269, downward
x/d=800, downward
80 64
60
48 40
downward
32
zero graivity upward
20
x/d=800, upward
96
16
x/d=800, zero gravity 0
0
0
0.2
0.4
r/R
0.6
0.8
1.0
0
500
1000 x/d
1500
0 2000
Figure 5 Temperature profiles for upward, downward and zero gravity flows under the heating condition for d = 0.7 mm at x/d = 269 and 800.
Figure 6 Cf × Reb along the tube length for upward, downward and zero gravity flows, under the cooling condition, for d = 0.5 mm.
defined in Equation (11), is seen to decrease downstream for all the flow conditions. Like the cases for the cooling flow, it is interesting to note from Figure 5 that the radial temperature gradient changes significantly at the pseudocritical point if this point is within the temperature profile of the tube section. It can also be seen from Figure 5 that the shape of the temperature profiles keep changing with the axial locations. This fact suggests that a thermally fully developed flow can never be reached for the supercritical carbon dioxide flow in a tube heated at a constant temperature. When compared with the zero gravity flow, the upward heating (buoyancy-assisted) flow exhibits higher temperature gradients adjacent to the wall in the upstream section (x/d = 269), but smaller ones in the downstream section (x/d = 800). Conversely, for the downward heating (buoyancy-opposed) flow, the temperature gradients adjacent to the wall upstream (x/d = 269) are smaller than those for the zero gravity, but become larger downstream (x/d = 800).
with A representing the area of the tube cross-section. The local Reynolds number is defined as
4.2 Skin-friction coefficients
The local skin-friction coefficient Cf is defined as: Cf =
1 2
τw ∂u = −µw ρb ub2 ∂r w
(
1 2
ρb ub2 ) ,
(12)
where τ w is the wall shear stress, ub and ρb represent the bulk velocity and the fluid density evaluated at the bulk fluid temperature Tb; and Tb is obtained from the crosssectional mean enthalpy eb defined as:
eb
∫ ρ uedA , = ∫ ρ udA A
A
(13)
Reb =
ρb ub d , µb
(14)
Figure 6 shows the value of C f × Reb along the tube length for upward, downward and zero gravity cooling flows for d = 0.5 mm for m = 1.77 × 10−5 kg/s (the corresponding inlet Reynolds number Rein = 2060) and p = 80 bar. It is seen from Figure 6 that the variations of C f × Reb with the tube length for all the flow conditions undergo significant fluctuations as compared with the constant-property fluid flow. The fluctuations are primarily caused by the variations of the fluid properties along the tube length as the fluid temperature changes. However, it is interesting to note that the values of C f × Reb for all the three flow conditions converge at C f × Reb = 16 , the solution for the constant-property fluid flow, as x/d is sufficiently large (x/d > 1000), where the fluid temperature and the tube wall temperature become identical and the heat transfer between the fluid and the tube wall ceases. It is also found from Figure 6 that over the entrance region (x/d < 1000), the local friction coefficients for both downward and upward flow deviate tremendously from that for the constant property fluid flow, whereas there are only small discrepancies between the zero gravity flow and the constant property fluid flow. Apparently, the large discrepancies in the friction coefficients for both upward and downward flow are caused by the buoyancy effect, as evident from the velocity profiles shown in Figure 2. Figure 7 presents the variations of C f × Reb along the tube length for upward, downward and zero gravity flows under the heating condition for d = 0.7 mm, m = 2.48 × 10−5 kg/s (the corresponding inlet Reynolds number Rein = 658 ), and p = 80 bar. Generally, similar
INVESTIGATION OF LAMINAR CONVECTION
96
80
80
Cf×Reb
64
64 upward
48
zero gravity
32
32
downward
16
16
0
0
-16
d= 0.5 mm, cooling
500
x/d
1000
upward
0
0
hd . kb
1500
2000
d=0.7 mm, heating
400 2
upward zero gravity
downward
(15) 0
where the local Nusselt number is defined as Nu b =
1000 x/d
600
200
k w ∂T h= , Tw − Tb ∂r w
500
Figure 8 Heat transfer coefficient distributions along the tube length for upward, downward and zero gravity flows, under the cooling condition, for d = 0.5 mm and p = 80 bar.
h [W/m -K]
The local heat transfer coefficient h is given by
600
200
behaviors for the cooling flow described above are also found for the heating flow. It is noted that for downward flow the friction coefficients over the entrance (x/d < 700) exhibit negative values, because the velocities near the wall over this region are negative, as evident from Figure 4. 4.3 Heat transfer coefficients and Nusselt numbers
800
400
1500
Figure 7 Cf × Reb along the tube length for upward, downward and zero gravity flows, under the heating condition, for d = 0.7 mm.
zero gravity
1000
-16 0
downward
1200
2
96
48
1400
112
d=0.7 mm, heating
h [W/m -K]
112
149
(16)
Figure 8 shows the heat transfer coefficient distributions along the tube length for upward, downward and zero gravity cooling flows for d = 0.5 mm, m = 1.77 × 10−5 kg/s (the corresponding inlet Reynolds number Rein = 2060) and p = 80 bar. It is noted from Figure 8 that there exists a peak of the heat transfer coefficients for all the flow conditions. The heat transfer coefficient peak for each flow condition occurs at the point where the bulk fluid temperature is close to the pseudocritical point, because the specific heat cp reaches its peak at the corresponding pseudocritical temperature, as shown in Figure 1. Like the distribution of the friction coefficients along the tube length, the heat transfer coefficients for all the flow conditions converge at a constant value toward the exit of the tube (x/d > 1200), when the fluid temperature and the wall temperature become identical. It can also be seen that the heat transfer coefficients over the entrance region (x/d < 1200) are different for the three different flow conditions. It is noticeable that the heat transfer coefficients for the downward cooling flow exhibit higher values than those
0
500
x/d
1000
1500
Figure 9 Heat transfer coefficient distributions along the tube length for upward, downward and zero gravity flows, under the heating condition, for d = 0.7 mm and p = 80 bar.
for the upward cooling and the zero gravity flows, because for the downward cooling flow, heat transfer is enhanced by buoyancy. Figure 9 presents the heat transfer coefficient distributions along the tube length for upward, downward and zero gravity flows under the heating condition for d = 0.7 mm, m = 2.48 × 10−5 kg/s (the corresponding inlet Reynolds number Rein = 658), and p = 80 bar. Similar to the flows under the cooling condition, the heat transfer coefficients for all the three flow conditions approach a constant value, as the fluid temperature approaches the wall temperature sufficiently far downstream from the inlet (x/d > 900). Over the entrance region (x/d > 900), the heat transfer coefficients for the upward flow are higher than those for the downward and the zero gravity flow, as heat transfer is enhanced by buoyancy for the upward flow under the heating condition. Figure 10 shows the variations of the heat transfer coefficient with the bulk fluid temperature for upward, downward and zero gravity cooling flows for d = 0.5 mm
S. M. LIAO AND T. S. ZHAO
150 1400
p=80 bar, zero gravity
1000
20
p=80 bar, upward 800
p=100 bar, downward p=100 bar, zero gravity
600
d=2.16 mm
Nub
2
downward, cooling
p=80 bar, downward
1200
h [W/m -K]
30
d=0.5 mm, cooling
d=1.4 mm
p=100 bar, upward
10
400
d=0.7 mm
200
d=0.5 mm 0
30
40
50
60 Tb [°C]
70
80
90
Figure 10 Variations in the heat transfer coefficient h with the bulk mean temperature for upward, downward and zero gravity flows, under the cooling condition, for d = 0.5 mm and p = 80 and 100 bar. Reb Prb 2500
12
2000 8 Prb Reb
1500 4 1000
500
0
0.95
1.05
Tb/Tpc
1.15
1.25
Figure 11 Reb and Prb Number vs. Tb/Tpc at m / d = 0.0355 kg/m-s at p = 80 bar.
and p = 80 bar and 100 bar. The mass flow rates for both p = 80 and 100 bar were set to be 1.77 × 10−5 kg/s and the corresponding inlet Reynolds numbers Rein are 2060 for p = 80 bar and 1954 for p = 100 bar. It can be seen that for p = 80 bar the heat transfer coefficient peaks occur at the corresponding pseudocritical temperature (Tpc = 34.6 °C at p = 80 bar) for all the three flow conditions. However, as the pressure is increased to p = 100 bar, the heat transfer coefficient peaks become insignificant or completely disappear near the corresponding pseudocritical temperature (Tpc = 45.0 °C at p = 100 bar). This is primarily because the peak value of the specific heat cp, at the corresponding pseudocritical temperature, becomes smaller with the increase of pressure, as evident from Figure 1. It can also be noted from Figure 10 that among the upward, downward and zero gravity cooling flow conditions, the heat transfer coefficient for the buoyancy-assisted downward cooling flow is the largest and that for the buoyancy-opposed
0 0.95
1.05
Tb/Tpc
1.15
1.25
Figure 12 Effects of tube diameter on Nusselt numbers Nub for m / d = 0.0355 kg/m-s and p = 80 bar for the downward cooling flow.
upward cooling flow is the smallest for both p = 80 and 100 bar. This fact suggests that the buoyancy effect on heat transfer coefficients of supercritical CO2 is significant even with very small diameter tubes. In order to investigate the effect of tube diameter on the heat transfer rate, at different flow directions under both the cooling and the heating conditions, the variations of the Nusselt numbers Nub with the bulk fluid temperature for four tube diameters (d = 0.50, 0.70, 1.40, and 2.16 mm) were numerically examined. To GIve a meaningful comparison, both the Reynolds number Reb and the Prandtl number Prb have to be kept constant for a given bulk mean temperature of the fluid. To this end, the calculations were conducted at a fixed ratio of the mass flow rate to the tube diameter at m / d = 0.0355 kg/m-s. The variations of both Reb and the Prb with the dimensionless bulk fluid temperature Tb/Tpc of the fluid under this condition are shown in Figure 11, where Tpc represents the pseudocritical temperature of CO2. Note that the unit of temperature is K when calculating the ratio Tb/Tpc. Figure 12 illustrates the effect of the tube diameters on the Nusselt number (Nub) at m / d = 0.0355 kg/m-s and p = 80 bar for the downward cooling flow. It can be clearly seen that for the downward cooling (buoyancy-assisted) flow the Nusselt numbers increase significantly as the tube diameters increase from 0.5 to 2.16 mm. This fact indicates that buoyancy cannot be ignored for convection of supercritical CO2 even with rather small tubes. It is also noted from Figure 12 that for d = 1.4 and 2.16 mm, the Nusselt numbers near the pseudocritical point are much larger than the value of Nu = 3.66 for constant property fluids cooled at constant wall temperature in laminar flow. To further elaborate on the buoyancy effect on heat transfer, we present Nusselt numbers (Nub) for the zero gravity cooling flow at m / d = 0.0355 kg/m-s and p = 80 bar for various tube diameters in Figure 13. It is clear from Figure 13 that over the downstream of the tube (corresponding to low bulk temperatures in Figure 13) the
INVESTIGATION OF LAMINAR CONVECTION
151
7 18
6
d=0.7 mm
d=1.4 mm d=0.7 mm
Nub
d=1.4 mm
Nub
12
4 d=2.16 mm
3
d=0.5 mm
9 6
2
3
1 zero gravity, cooling 0 0.95
zero gravity, heating
15
d=0.5 mm
5
d=2.16 mm
1.00
1.05
1.10 Tb/Tpc
1.15
1.20
0 0.95
1.25
1.00
1.05
Tb/Tpc
1.10
1.15
1.20
Figure 13 Effects of tube diameter on Nusselt numbers Nub for m / d = 0.0355 kg/m-s and p = 80 bar for the zero gravity cooling flow.
Figure 15 Effects of tube diameter on Nusselt numbers Nub at m / d = 0.0355 kg/m-s and p = 80 bar for zero gravity heating flow.
Nusselt numbers are almost identical for different tube diameters, proving that the tube diameter-dependent Nusselt numbers shown in Figure 12 are attributed to the buoyancy effect. It should be recognized that in the entrance region (or for high bulk temperatures in Figure 13) Nusselt numbers are affected by the entrance effect in addition to the Reynolds number. For this reason, Nusselt numbers in the entrance region are different for different tube diameters. Similar phenomena can be observed in the heating flow. Figure 14 illustrates the effect of the tube diameters on the Nusselt number (Nub) at m / d = 0.0355 kg/m-s and p = 80 bar for the upward heating flow. For the upward heating (buoyancy-assisted) flow, the Nusselt numbers increase significantly as the tube diameters increase from 0.5 to 2.16 mm. However, when the gravity was set to be zero, as shown in Figure 15, the effect of tube diameters on Nusselt numbers disappears. Again, differences in Nusselt numbers for different tube diameters in the entrance region are caused by the entrance effect.
The above-described phenomena suggest that buoyancy plays an important role in laminar convective heat transfer of supercritical carbon dioxide. Figure 16 shows the effect of the wall temperature Tw (12 and 25 °C) on Nusselt numbers for upward, downward and zero gravity cooling flows at d = 0.5 mm and p = 80 bar. The mass flow rate is 1.77 × 10−5 kg/s and the inlet Reynolds number Rein = 2060. It is clear from Figure 16 that Nusselt numbers for the lower wall temperature Tw = 12 °C are significantly higher than those for the higher wall temperature Tw = 25 °C for all the flow directions when the bulk fluid temperature is higher than the pseudocritical temperature (Tb/Tpc > 1), implying that the wall temperature Tw has an important influence on heat transfer rates for supercritical fluids. This is because different wall temperatures Tw give different velocity and temperature profiles, even if other conditions are kept the same. When the bulk fluid temperature approaches the wall temperature, the variable property effect on heat transfer
60
12 d=0.5 mm, p=80 bar, cooling
upward, heating 50 d=2.16 mm
Tw=12°C, downward
9
Tw=12°C, zero gravity
Nub
Nub
40 30
6
Tw=12°C, upward
d=1.4 mm 20
d=0.7 mm d=0.5 mm
3
10 0 0.95
Tw=25°C, zero gravity Tw=25°C, downward
1.00
1.05
Tb/Tpc
1.10
1.15
1.20
Figure 14 Effects of tube diameter on Nusselt numbers Nub for m / d = 0.0355 kg/m-s and p = 80 bar for upward heating flow.
0 0.90
0.95
1.00
1.05 Tb/Tpc
Tw=25°C, upward 1.10
1.15
1.20
Figure 16 Effects of the wall temperature Tw on Nusselt numbers Nub for upward, downward and zero gravity flows, under the cooling condition, for d = 0.5 mm and p = 80 bar.
S. M. LIAO AND T. S. ZHAO
152 becomes negligible and the Nusselt number approaches the value of the constant property solution (Nub = 3.66). An important observation that can be made from the local Nusselt numbers presented in Figures 12–16 is that, because of the temperature-dependent properties of supercritical CO2, the Nusselt numbers exhibit rather large variations with the bulk fluid temperature (or along the axial locations), even at a distance far downstream from the tube entrance. Apparently, this peculiar behavior is different from the case for the constant-property laminar tube flow, in which the Nusselt number reaches a constant (3.66 for constant wall temperature) at the thermally fully developed region.
5 CONCLUDING REMARKS
The velocity profiles, temperature profiles, heat transfer coefficients, Nusselt numbers and skin-friction coefficients for supercritical CO2 laminar flow in vertical mini/micro tubes, under both cooling and heating conditions, with and without gravity, have been obtained. It is shown that due to the effect of large variations of thermophysical properties of CO2 near the critical point, the heat transfer and fluid flow characteristics of supercritical CO2 are rather different from those of the constant-property fluids. The results show that for the supercritical CO2 tube flow, hydrodynamically and thermally fully developed flow regions can be reached only when the fluid and the pipe wall are under the thermal equilibrium condition, which usually occurs at a location a long way downstream from the tube inlet (x/d > 1000). In addition, it has been revealed that buoyancy plays an important role in laminar forced convection of supercritical CO2 flowing even for small tubes. The results presented in this paper are of significance for the design of high efficiency compact supercritical CO2 heat exchangers.
3 McLinden, M., Klein, S.A., Lemmon, E.W. and Peskin, A.P. (1998) ‘NIST thermodynamic and transport properties of refrigerants and refrigerant mixtures-REFPROP’, Version 6.01, National Institute of Standards and Technology, USA. 4 Krasnoshchekov, E.A., Kuraeva, I.V. and Protopopov, S.V. (1970) ‘Local heat transfer of carbon dioxide at supercritical pressure under cooling conditions’, Teplofizika Vysokikh Temperatur, Vol. 7, No. 5, pp. 922–930. 5 V.L. Baskov, I.V. Kuraeva, V.S. Protopopov, (1977) ‘Heat transfer with the turbulent flow of a liquid at supercritical pressure in tubes under cooling conditions’, Teplofizika Vysokikh Temperatur, Vol. 15, No. 5, pp. 96–102. 6 Jackson, J.D., Hall, W.B., Fewster, J., Watson, A. and Watts, M.J. (1975) ‘Heat transfer to supercritical pressure fluids’, U.K.A.E.A. A.E.R.E.-R 8158, Design Report 34. 7 Liao, S.M. and Zhao, T.S. (2002) ‘Measurements of heat transfer coefficients from supercritical carbon dioxide flowing in horizontal mini/micro channels’, ASME Journal of Heat transfer, in press. 8 Lee, S.H. and Howell, J.R. (1996) ‘Laminar forced convection at zero gravity to water near the critical region’, Journal of Thermophysics and Heat Transfer, Vol. 10, No. 3, pp. 504–510. 9 Lee, S.H. and Howell, J.R. (1998) ‘Turbulent developing convective heat transfer in a tube for fluids near the critical point’, International Journal of Heat and Mass Transfer, Vol. 41, No. 10, pp. 1205–1218. 10 Petrov, N.E. and Popov, V.N. (1985) ‘Heat transfer and resistance of carbon dioxide cooled in the supercritical region’, Thermal Engineering, Vol. 32, No. 3, pp. 131–134. 11 Zhou, N. and Krishnan, A. (1995) ‘Laminar and turbulent heat transfer in flow of supercritical CO2, Proceedings of the of the 30th National Heat Transfer Conference’, Vol. 5, ASME, Portland, OR, pp. 53–63. 12 Patankar, S.V. (1980) Numerical Heat Transfer and Fluid Flow, McGraw-Hill.
ACKNOWLEDGEMENTS
This work was supported by the Hong Kong RGC Earmarked Research Grant HKUST 6046/99E.
REFERENCES 1 Lorentzen, G. and Pettersen, J. (1993) ‘A new efficient and environmentally benign system for car air-conditioning’, International Journal of Refrigeration, Vol. 16, No. 1, pp. 4–12. 2 Pettersen, J., Hafner, A. and Skaugen, G. (1998) ‘Development of compact heat exchangers for CO2 airconditioning systems’, International Journal of Refrigeration, Vol. 21, No. 3, pp. 180–193.
BIOGRAPHICAL NOTES S. M. Liao obtained his B.S. and M.S. degrees in thermal engineering from Tsinghua University. He is currently a Ph. D candidate at the Hong Kong University of Science and Technology and an associate professor at the Central South University in China. T. S. Zhao obtained his Ph. D degree in mechanical engineering from the University of Hawaii in the United States. He is currently an associate professor at the Hong Kong University of Science and Technology. Professor Zhao's research activities include microscale heat transfer and phase-change heat transfer in capillary structures, as well as advanced air conditioning and refrigeration systems.
