A Unified Theory of Transient Instabilities, Convection and Turbulence
Ka Kheng, Tan
[email protected],
[email protected] Department of Chemical and Environmental Engineering, University Putra Malaysia, Serdang, Selangor Malaysia
Serdang, Selangor A unified principle of transient instability induced by diffusion of heat, mass and Malaysia momentum will be proposed. The paper will re-visit with critical remarks on the basic principles of linear stability analysis (LSA) and its application to convection 43400UPM, Serdang, Selangor, Malaysia induced by diffusion processes. It was found that the onset of transient instability Email:
[email protected] leading to transient convection is governed by a common principle. They all can be predicted by a transient stability parameter appropriate to the process of diffusion, such as Rayleigh number in buoyancy induced instability caused by transient heat and mass diffusion, Taylor number in momentum diffusion induced instability, and the Biot number dictates them. The post-instability flow is characterized by convection plumes, which are generally mushroom-shaped. At high diffusion flux the plumes may rapidly detach and crash to form turbulent eddies The theory of transient instability has been verified accurately by comprehensive experiments. The stable diffusion times that set limit to Fourier’s law of heat conduction and Fick’s law of mass diffusion are determined by the theory of transient instability. Therefore, the critical times and critical mass of diffusion for causing convection can now be predicted accurately. .
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INTRODUCTION Lord Rayleigh’s (1916) seminal treatise on thermal instability for steady-state heat conduction is characterized by a linear temperature profile, which is different from the non-linear one induced by unsteady-state heat conduction in a deep fluid. The latter result in transient instability and convection plumes that are different from those of rolls caused by steady-state heat conduction. Tan and Thorpe (1992, 1996, 1999a, b and c) showed that onsets of instability and buoyancy convection are governed by the Biot number of the interface, its corresponding critical wavenumber and Rayleigh numbe. They defined a new transient Rayleigh number that agrees with theoretical values from linear stability analysis (Jeffreys, 1928, Low, 1929, Sparrow et al., 1964) for the same Biot number and physical boundary conditions. At high diffusion flux the convection plumes may rapidly detach from the interface and crash to form myriads of turbulent eddies. PRINCIPLE OF TRANSIENT INSTABILITY The criterion of thermal instability provided by the LSA is based on the stability of a perturbed plane 2D wave whereas the structure of plumes in transient convection is 3D, and the temperature profile is non-linear. Strictly the LSA cannot predict the onset of convection, however, its point of neutral instability may predict the initial stage of convection with the Biot number and boundary conditions, Table 1. Table 1. Theoretical Biot, Rayleigh and Wave numbers
Bi
Top surface free a~ c
RaC
Top surface solid a~
RaC
c
Bottom surface at fixed temperature 0*
2.09
669.0
2.55
1295.8
1
2.30
770.6
22.75
1398.5
**
2.68
1100.7
3.12
1707.8
Bottom surface at constant heat flux 0*
0
320.0
0
720.0
1
1.64
513.8
1.94
974.2
**
2.21
816.7
2.55
1295.8
* Corresponds to constant heat flux. ** Corresponds to fixed surface temperature. From Sparrow et al. (1964)
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The definition of the conventional Rayleigh number, Ra gd
3
T , shows that
thermal diffusion in a thick layer of fluid or deep fluid will lead to infinitesimally small Tc at the onset of convection. This is unrealistic as there must be a finite value of Tc that will induce instability and cause convection. Indeed, the onset of instability and convection in deep fluid occurs in the thin thermal boundary layer with the subsequent evolution of thermal plumes. This localised phenomenon has been observed in thermal experiments of Spangenberg and Rowland (1961) and Foster (1965), Sparrow et al. (1971), in mass diffusion of Blair & Quinn (1969) and in momentum diffusion of Kirchner & Chen (1967) and Chen & Kirchner (1970).
Consequently Tan (2000) proposed a unified principle of transient
instability induced by the diffusion of heat, mass and momentum. The onset of instability is caused by the local adverse gradient of the driving force, and is followed by the formation of narrow plumes initially and later mushroom-shaped convection plumes. This results in momentum transfer that relieves the unstable boundary layer from its potential total collapse. Tan & Thorpe (1992, 1996, 1999a, b, & c) have defined a new transient Rayleigh number, Ra gz 4 (T / z ) t / , that incorporated the mode and rate of heat transport for the prediction of the onset of instability and convection. The mode of heat transport is characterised by a thermal boundary condition that determines the Biot number and its corresponding critical wavenumber. The maximum transient Ra is determined from
d(Ra)t dz 0 , and is located at zmax, hence the transient Ramax will have the same value as the theoretical Rac of the LSA for the same Biot number and physical conditions. A new transient
Taylor
number
is
similarly
defined
from
the
conventional
one
as
Ta z 5 (u / z ) t2 / 2 R0 , and its maximum value determined as for transient Rayleigh number. The equivalence of the transient instability theory with the LSA theory is due to the effective penetrative depths in unsteady-state processes being similar to the narrow-gap plate for thermal instability or small-gap Taylor-Couette system. Typically the transient penetration depth for thermal instability is less than 10 mm, they are only about 2 mm for mass and fast momentum transfer for the brief stable diffusion. A new transient Biot number for thermal instability at a gas-liquid was also defined,
Bi k g / kl l / g ( c p k ) g /( c p ) l , so that the appropriate temperature profile may be used to determine the local gradient for the transient Ra. For instance, the fixed surface temperature (FST) boundary that correspond to Bi = will have the temperature profile:
T Ts /T To erf z / 2 t , and the constant heat flux (CHF) boundary that corresponds
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to Bi = 0, it is T0 T 2q 0 t (ierfc ( z / 2 t )) / k . The transient Biot number for mass diffusion is similarly defined as BiD Ds / Dc H * . Some values of transient BiD and Rac are shown in Table 2. The more soluble the gas-liquid system, the lower the Biot number and hence its corresponding Rayleigh number. The critical times, tc, can also be predicted from the theoretical value of critical Rayleigh number from the transient Rayleigh number. The resultant transient Rayleigh numbers for heat and mass transfer and transient Taylor number for momentum transfer are summarized in Table 3. Table 2. Transient BiD, Rac and critical times for interfacial gas absorption in water Oxygen
CO2
SO2
NH3
BiD*
2780
90
0.78
0.05
Rac
1100
1080
760
669
tc s
5007
120
6.4
3.1
From Tan & Thorpe (1999d) Table 3. Transient Rayleigh number and transient Taylor number FST/FSC/FSV Heat transfer
Mass transfer
Ramax
CHF/CMF/CMF
4.89 g t Tc
g (1.70 t )3 Tc
Ramax
4.89 g o c * Dt 3
3.02 gq ot 2 k g (1.39 t )3 Ts
Ramax
Ra max
3.02 g o j o t
Momentum transfer
1.46U i2 tc Tamax 2 R0
3
Tamax
2
2.68 g o Dt 3 cs ce
0.798 t 2 R0
5
2
0
0.626U i2 t c
3
2 R0
Form RESULTS AND DISCUSSIONS 4
The verification of the theory transient instability is first provided by Foster’s (1965) evaporative cooling experiments, and experiments of Davenport and King (1974) of bottom heating of several organic liquids and those of Foster (1969) for water. In the evaporative cooling experiments, the Biot number was found to be close to zero as the air is insulating and the corresponding critical Rayleigh number has a value of 669. The transient Rayleigh numbers of Foster’s (1965) experiments are near to the theoretical value of 669 for CHF boundary, Figure 1. While the bottom-heating experiments of Davenport and King (1974) are shown to have a transient Rayleigh number of about 1296 for organic liquids, Figure 2. The sizes of the convection plumes can also be predicted rather accurately with the theoretical prediction, c 9.40 t c . Similarly the critical times for the experiments may be predicted accurately with the known value of critical Rayleigh number, 669, and the transient Rayleigh number shown in Table 3. In the case of gas diffusion in water at an interface, the critical transient Rayleigh number may be predicted easily with the Biot number in Table 2 and the equation of transient Rayleigh number for CHF boundary since the sulphur dioxide, ammonia and ethyl ether are very soluble in water. The critical transient Rayleigh numbers are found to lie close to 669, Figure 4. More interestingly, the critical times can also be predicted accurately as in the case of sulphur dioxide diffusing in water, Figure 5. This critical time is the limit to Fick’s law of diffusion, it can also be used to predict the critical mass that may lead to the onset of convection. The sizes of convection plumes are also predicted with reasonable accuracy with equation c 9.40 Dtc as shown in Figure 6. The transient instability induced by momentum diffusion for impulsively started cylinder is analogous to the thermal instability in horizontal fluid with an upper free-surface boundary. Hence the expected critical transient Taylor number for a fixed surface velocity (FSV) boundary is 1100, which is well corroborated by data of Chen and Kirchner (1971) who rotated a thin cylindrical rod in a large tank of water, Figure 7.
The average critical
dimensionless wavenumber was found to be 2.9, which is close to the theoretical value of 2.7, Figure 8. The critical sizes of the toroidal plumes may be predicted with the theoretical wavenumber and the equation c 8.08 t c . It should be noted that the unsteady-state experiments are not perfect replica of the LSA theory as the virtual outer boundary away from the rod is not a truly free surface with zero-shear, because there always exists laminar shear between adjacent layers of moving fluid. Hence, deviation from theory is expected, but
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this is less significant for fast speed rotation where the penetration depth will be very shallow for the emerging boundary layer to be affected by the wall of the tank. The foregoing analyses show that the onset of convection induced by unsteady-state heat conduction, mass diffusion and an impulsively started rotating cylinder in deep fluids is governed by a common principle, that it depends on the transient Biot number and the corresponding transient Taylor number and transient Taylor number respectively.
The
emerging convection plumes are determined by the same critical wavenumber for the same boundary conditions. The comprehensive verification of the principle of transient instability by experiments has established a unified theory of transient instability for unsteady-state heat conduction, mass diffusion and momentum diffusion. The critical timers of stable heat conduction and mass diffusion have been predicted accurately.
Figure 1: Top cooling of water, Bi = 0 and Rac = 669. (Tan & Thorpe, 1996)
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Figure 2: Bottom heating of various organic compounds. (Tan & Thorpe, 1999)
Figure 3: Plume sizes from top-cooling of water, c 9.40 t c . (Tan & Thorpe,1999)
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Figure 4: Critical transient Ra induced by diffusion of gases
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Predicted times, s
100 80 60 40 Sulphur Dioxide Ammonia Ethyl ether
20 0 0
20
40
60
80
100
120
Measured times, s Figure 5: Critical times of gas diffusion in water, c 9.40 Dtc . (Tan &Thorpe, 1999c)
8
..
Critical transient Taylor numbers
Figure 6: Sizes of plumes produced by the diffusion of SO2 in water
10000
1000
100
0
10 Critical time, s
20
Figure 7: Transient Taylor number for impulsively started cylinder
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Critical wavenumber
5 4 3 2 1 0
10 Critical time,s
20
Figure 8: Critical wavenumber as toroidal plumes induced by momentum
Figure 9. Boundary layer instability on flat plates
CONCLUDING REMARKS The principle of transient instability induced by unsteady-state heat conduction, mass diffusion and an impulsively started rotating cylinder in deep fluids is found to be common to 10
the three classes of diffusion. Good comprehensive verification of the unified theory of transient instability by experiments has been firmly established. The critical timers of stable heat conduction and mass diffusion have been predicted accurately. They represent the theoretical time limits to the validity of Fourier’s law of heat conduction and Fick’s law of diffusion where convection may occur.
NOMENCLATURE ãc
Dimensionless wave number
a
Wave number [m-1]
cb c0 c* D d g
Bulk concentration [kmol/m3] Initial concentration [kmol/m3] Gas-liquid interfacial equilibrium concentration [kmol/m3] Diffusion coefficient [m2/s] Depth of fluid layer [m] Acceleration due to gravity [m2/s]
H* j˚ k
Dimentionless Henry's constant, H* = H/RT Interfacial mass flux [kmol/m2 s] Thermal conductivity of fluid [W/m C]
m˚ q˚ R0
Relative molecular mass of solute (constant) heat flux [W/m2] Radius of cylinder [m]
tc
Critical time for onset of convection [s]
T
Temperature [C]
Ts
Surface temperature at time t [C]
T
Temperature difference between top and bottom surface [C]
Ts
Temperature difference of the surface of the porous media [C]
Ui
Surface velocity of cylinder [m/s]
v˚ z
Partial molar volume [m3/kmol] Penetration depth of fluid layer in vertical direction [m]
Greek Symbols
Volumetric coefficient of thermal Expansion [K-1]
˚
molar density coefficient for gas difusion in liquid [kg/kmol]
Thermal diffusivity [m2/s]
Wavelength [m] 11
Kinematics viscosity [m2/s]
Viscosity [Pa.s]
Density [kg/m3]
shear stress [Pa]
Abbreviations CHF
Constant heat flux boundary condition
CHF
Constant mass flux boundary condition
CHF
Constant momentum flux boundary condition
FSC
fixed surface concentration boundary condition
FST
fixed surface temperature boundary condition
FSV
fixed surface velocityboundary condition
LSA
Linear stability analysis
Subscripts c
critical
o
initial condition
max
maximum
f
fluid
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Foster, T. D., (1965). “Onset of convection in a layer of fluid cooled from above.” Phys. Fluids, 8, October, 1770...
6. Foster, T. D., (1969). “Onset of manifest convection in a layer of fluid with a timedependent surface temperature.” Phys. Fluids, 12, December, 2482. 12
7. Jeffreys, H.,(1928). “Some cases of instability in fluid motion.” Proc. Roay. Soc., A118, 195 8. Low, A.R.,(1929). “Instability of viscous fluid motion.” Nature,115, 299. 9.Pearson, J.R.A.,(1958). “On convection cells induced by surface tension.” J. Fluid Mech., 4, 489. 10. Spangenberg, W. G. and Rowland, W. R., (1961).
Convective circulation in water
induced by evaporative cooling. Phys. Fluids, 4, June, 743. 11. Sparrow, E.M., Goldstein, R.J. & Jonsson, V.K. (1964)
“Thermal instability in a
horizontal fluid layer: effect of boundary conditions and nonlinear temperature profile.” J. Fluid Mech., 18, 5130 12. Sparrow, E. M., Husar, R. B. and Goldstein, R. J., (1970). “Observations and other characteristics of thermals.” J. Fluid Mech., 41, 793. 13. Tan, K. K. and Thorpe, R. B., (1992). Gas diffusion into viscous and non-Newtonian liquids.” Chem. Eng. Sci, 47, 3565. 14. Tan, K. K. & Thorpe, R. B. (1996). “The onset of convection caused by buoyancy during transient heat conduction in deep fluids.”. Chem. Eng. Sci. 51, 4127-4136 15. Tan, K. K. & Thorpe, R. B. (1999a). “The onset of convection driven by buoyancy effects caused by various modes of transient heat conduction. Part 1: Transient Rayleigh Numbers.” Chem. Eng Sci, 54, 225-238 16. Tan, K. K., & Thorpe, R. B. (1999b). “The onset of convection driven by buoyancy effects caused by various modes of transient heat conduction. Part II: The sizes of plumes.”. Chem. Eng. Sci, 54, 239-244 17. Tan, K. K., & Thorpe, R. B. (1999c). “The onset of convection driven by buoyancy caused by gas diffusion.”. Chem. Eng. Sci, 54, 239-244 18. Tan, K. K. (2000). “A unified principle of transient instability and convection induced by diffusion of heat, mass and momentum.” Presented in the afternoon seminar at the Department of Chemical Engineering, University of Cambridge, 10th June, 2000.
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