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Theoretical analysis of free-surface film flow on the rotary granulating disk in waste heat recovery process of molten slag
Applied Thermal Engineering 63 (2014) 387e395
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Theoretical analysis of free-surface film flow on the rotary granulating disk in waste heat recovery process of molten slag Dongxiang Wang, Xiang Ling*, Hao Peng Jiangsu Key Laboratory of Process Enhancement and New Energy Equipment Technology, School of Mechanical and Power Engineering, Nanjing University of Technology, No. 5 Xin Mo Fan Road, Nanjing 210009, PR China
h i g h l i g h t s A theoretic model describing the free-surface film flow on rotary disk is proposed. The proposed model is compared with existing data and CFD simulation result. Coriolis, centrifugal force and viscous drag govern the flow after hydraulic jump. The liquid flow after hydraulic jump is not effect by the hydraulic jump region.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 13 August 2013 Accepted 15 November 2013 Available online 26 November 2013
This paper proposed a simple theoretical model to characterize the continuous free-surface film flow on the rotary disk in waste heat recovery process of molten slag by the method of order of magnitude analysis. Liquid film thickness, mean radial velocity and degree of tangential slippage have been analyzed based on the proposed model. The model indicates that the free-surface film flow on the rotary disk is governed primarily by the balance between Coriolis and centrifugal forces and viscous drag after the hydraulic jump. The inertia force contributes primarily nearby the center of the rotary disk. It is revealed that the liquid film thickness distribution on a rotary disk is mainly determined by the volume flow rate, kinematic viscosity of molten slag and the rotational speed of disk. Compared with the existing results and CFD simulation data, the model shows good performance. As indicated by the CFD simulation results, the hydraulic jump region has no influence on the liquid flow after the hydraulic jump, hence has none effect on the granulation performance of molten slag. Crown Copyright Ó 2013 Published by Elsevier Ltd. All rights reserved.
Keywords: Film flow Free-surface Rotary disk Waste heat Molten slag
1. Introduction The International Energy Agency [1] reports that: “The energy intensity of most industrial processes is at least 50% higher than the theoretical minimum determined by the laws of thermodynamics”. Nowadays, under the context of the utmost concern for energy saving and emission reduction, adoption of environment-friendly and renewable technologies to recovery the waste heat in the metallurgy industry has attracted growing attention. With the technologies to recover the thermal and chemical energies of process off-gas being improved, the molten slag represents the last untapped waste heat source that may be used for energy saving in the rather energy-intensive metal industry [2].
* Corresponding author. Tel.: þ86 25 83587321; fax: þ86 25 83600956. E-mail address: [email protected] (X. Ling).
Molten slags carry a substantial amount of high quality thermal energy since the tapping temperature is up to 1000e1650 C [3e5]. As demonstrated from Fig. 1, ferrous slags account for over 90% of the available energy associated with slags and hot metal blast furnace (BF) slag alone constitutes 47% of this energy [2]. Therefore, in this paper we will focus on BF slag. At present, “wet granulation”, utilizing vast amounts of water to quench the molten slag for high cooling rate and producing granules with high glass phase for cement manufacture, is widely used for the treatment of BF slag in the iron and steel plant. However, the water treatment method reveals many obvious shortcomings in the practice: High consumption of water. For example, the granulation of 1 tonne BF slag by INBA process will consume 1.0e1.2 tonne water [6]. Pollutants emission. Sulfur dioxide, hydrogen sulfide and other sulfur compounds [7,8].
1359-4311/$ e see front matter Crown Copyright Ó 2013 Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.applthermaleng.2013.11.033
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D. Wang et al. / Applied Thermal Engineering 63 (2014) 387e395
Nomenclature Ek rc rin win hin b R h u v w z r p po U Re Ro G
Ekman number the critical radius of hydraulic jump [m] inlet radius [m] inlet velocity [m/s] inlet height [m] thickness of the disk [m] radius of the disk [m] liquid film thickness [m] radial velocity [m/s] tangential velocity [m/s] axial velocity [m/s] axial distance [m] radius distance [m] pressure [MPa] the ambient pressure [MPa] radial velocity [m/s] Reynolds number Rossby number mass flow rate [kg/min]
Poor energy efficiency [9,10]. Most of the sensible heat is wasted. Meanwhile, the water granulated process only produces large quantities of low-temperature steam or water for heat-supply in winter by a few plants [7]. Additional energy consumption. Such as drying requirement for the wet product. Dry slag granulation (DSG), an industry process utilizing a rotary disk or rotary cup to spray the molten slag into granules by centrifugal force, has received more and more attention. In particular, rotary disk granulation is capable of producing droplets with narrow size-distributed range and applicable to liquids of both low viscosity (e.g., water) and high viscosity (e.g., molten slag and organic fluids) [11]. Experimental study of the DSG process by CSIRO has identified that the slag granules are produced by three steps: free-surface film formation on the disk, ligament formation and granule formation, as shown in Fig. 2(a). The performance of the DSG primarily depends on the hydrodynamics of the flow on the rotary disk, while the flow is characterized by the free-surface film with a sharp gaseliquid interface changing continuously as indicated in Fig. 2(b). There are a large number of published literature concerning with the continuous liquid free-surface film flow on the rotary disk. The original mathematical model [12] considers a balance between the viscous drag of the fluid and the centrifugal force and provides a simple relation between fluid flux and the liquid film thickness. Rauscher et al. [13] and Lepehin et al. [14] obtained the approximate analytical solutions about the radial velocity and film thickness.
Fig. 1. The heat content of various kinds of metallurgical slags per year.
C1 C2 C3 C4 Q
integration constant integration constant integration constant integration constant volume of flow rate [m3/s]
Greek
U y r 3
m f
rotational speed [rad/s] kinematic viscosity [m2/s] density [kg/m3] aspect ratio dynamic viscosity [Pa s] degree of tangential slippage
Subscript in inlet c critical r radius Acronyms DSG dry slag granulation
Zhao et al. [15] presented three models to characterize the film flow on the rotary disk, the numerical model gives a more accurate approximation of the liquid film thickness. Unfortunately, no explicit relation is given. Sisoev et al. [16] derived the approximate evolution equations for the film thickness and volumetric flow rates by an integral method. Prieling et al. [17] analyzed the unsteady film flow on the rotary disk at large Ekman numbers by integral boundary layer method. In addition, non-dimensional method has also been given serious attention [16e19]. Wood [20] performed an experimental study to characterize the film flow on the rotary disk and showed existence of several wavy regimes in concentric zones across the radius of the disk. Leshev et al. [21] carried out an experimental and analytical investigation, the analytical result predicts film heights slightly lower than the experimental values near the center of the disk (typically the worst error is around 10%). Burns et al. [22] proposed a theoretical model based on experiment and compared with the Nusselt theory, they found that the Nusselt model could not be used accurately for the inertial flow conditions. Ghiasy et al. [23] first examined the temperature profiles and flow characteristics of liquid films on the rotary disk by infrared thermal imaging camera. Besides, there is still many literature concerning with the numerical simulation. Rice et al. [24] carried out a direct numerical simulation of the film flow on the rotary disk using the volume of fluid (VOF) method. Pan et al. [11] applied a two-dimensional multiphase CFD model to investigate the film thickness at the disk periphery and revealed that the surface tension has a negligible effect on the film thickness. It is evident that the simulate result is 10% smaller than the model of [12] on average. Recently, Bhatelia et al. [25] carried out both 2D and 3D CFD simulations of film flow on the rotary disk based on VOF approach. The predicted film thickness of 3D simulation was found to agree with the experimental data with the relative error of 10%, by contrast, the 2D model under-predicted the film thickness by 39%. The reason may be the assumption of laminar flow on the rotary disk, while Pan et al. [11] considered the flow is turbulent. It is essential to reliably predict the liquid film thickness on the rotary disk since most of the hydrodynamic properties like radial velocity, granulated particle size along with heat and mass transfer coefficients directly depend on that. Many investigations assumed the flow is laminar or utilized non-dimensional method and trial
D. Wang et al. / Applied Thermal Engineering 63 (2014) 387e395
389
Fig. 2. The slag granulation process and free-surface film flow on the rotary disk (a) a typical static image from high speed video recording of the slag granulation by a rotary disk in CSIRO’s dry slag granulation process [11], (b) schematic of the film thickness profile of molten slag on the rotary disk.
function of velocity to obtain an approximate analytic solution. Large variances always existed between the published literature, particularly in the CFD simulation result. There is still limited understanding of the film flow on a rotary disk. In this paper, a simple theoretical model of continuous free-surface film flow on the rotary disk was derived from the NaviereStokes equations based on the order of magnitude analysis. The proposed model has been compared with the existing data and CFD simulation results. Large attempt has been made to further understand the behavior of the film flow on a rotary disk involving the effect of operation parameters and physical properties on the film thickness.
u
vv vv uv v2 v 1 vv v2 v v þ w þ þ 2Uu ¼ y þ þ vr vz r vr 2 r vr vz2 r 2
! vw vw 1 vp v2 w v2 w þw ¼ þy þ 2 u r vz vr vz vr 2 vz
(2)
(3)
For the incompressible fluid, the equation of continuity is
1v vw ðruÞ þ ¼ 0 r vr vz
2. Theoretical analysis
!
(4)
2.1. Assumption and governing equation In the DSG process, the slag directly flows down to the center of a rapidly rotary disk under the force of gravity, followed by the radial and tangential acceleration toward the disk periphery prior to granulation. Consequently, as indicated in Fig. 2(b), a film thickness profile that is determined by the kinematic viscosity of the molten slag and the frictional and centrifugal forces will form on the disk, perhaps accompanied by a hydraulic jump at a critical radius rc in some special cases (e.g., with high inlet velocity). In order to theoretically simulate the thin film flow phenomenon encountered when molten slag is poured onto a rotary disk, the following assumptions have been made: The molten slag flow is uniform and continuous with constant volume flow rate. The slag is poured along the axis of disk center so that rotational symmetry is assumed. No slippage assumption in the interface between molten slag and the rotary disk. The disk is absolutely filled with slag so that the surface tension is negligible [11,26]. Neglecting the effects of heat transfer between the slag and the disk [11], the free-surface film flow on the disk can be determined by the NaviereStokes equations of momentum conservation and the equation of continuity, which in cylindrical coordinates can be written as
u
vu vu v2 1 vp v2 u 1 vu v2 u u 2 þ w 2Uv U r ¼ þy þ þ r vr vr vz r vr 2 r vr vz2 r 2
!
2.2. Boundary condition and model formation The first three terms on the left hand of Eq. (1) represent the fluid inertia force, the next two terms are the Coriolis and centrifugal forces, respectively. The terms on the right hand represent the pressure gradient and viscous drag. The Coriolis force also appears as the final term of Eq. (2). Now we attempt to determine which terms dominate the flow and systematically reduce the problem to a simpler one by the order of magnitude analysis. In the main flow direction, we supposed that U has the same magnitude as u. Meanwhile, r and R, z and h also posses the same magnitude, respectively. Wood [20] considered that the tangential velocity relative to the disk is far smaller than the radial velocity in a realistic condition, it is reasonable to assume: v/u w 3 . In the DSG process, what we are interested is the liquid film thickness at the periphery of disk, hence we assumed: 3 ¼ h/ R << 1 at large disk radius. Therefore we can get w/U w h/R from the equation of continuity. In conclusion, one can get as mentioned above: 3
¼
h w h << 1; w R U R
(5)
Consequently, the magnitude of all terms in Eqs. (1)e(3) can be defined as:
0
1
B 2 C vu vu v2 1 vp B v u 1 vu u v2 uC 2 C u þw 2Uv U r ¼ þyB þ þ 1 2 r vr B vr ffl{zfflfflfflfflfflfflffl vz vr 2 r vr r 2 vz2 C @ A RoEk Ek |fflfflfflfflfflfflffl ffl} 3 4rRe |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} 2 1 3
3 2 Re
(1)
v w3 ; u
uwU; rwR; zwh;
3
Re
32
(6)
390
D. Wang et al. / Applied Thermal Engineering 63 (2014) 387e395
0
1 vðzÞ ¼ C1 ezz ==; cosðzzÞ þ C2 ezz sinðzzÞ þ C3 ezz cosðzzÞ
B 2 C B v v 1 vv vv vv uv v v2 vC C u þw þ þ 2Uu ¼ yB þ þ B vr 2 r vr r 2 vz2 C 2 vr vz r @ |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} A Ek |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} 1
(7)
1 þ C4 ezz sinðzzÞ Ur 2
3
3 2 Re
32
uðzÞ ¼ C2 ezz cosðzzÞ C1 ezz sinðzzÞ C4 ezz cosðzzÞ
0
1 vw vw 1 vp @ v2 w v2 wA u þw ¼ þy þ 2 r vz vr ffl{zfflfflfflfflfflfflfflfflffl vzffl} vr 2 vz |fflfflfflfflfflfflfflfflffl 1 32
UR ; Ek ¼
y
y
h2 U
; Ro ¼
U
v2 u 2 ¼ 2Uv U r vz2
(17)
C3 ¼
1 2ezhðrÞ cos½zhðrÞ ezhðrÞ þ ezhðrÞ UrezhðrÞ 2 2 4 cos½zhðrÞ þ e2zhðrÞ 2 þ e2zhðrÞ
(18)
2
2
C2 ¼ C4 ¼
ZhðrÞ
(11)
Due to non-slip velocity condition, the boundary condition at the liquidesolid interface z ¼ 0 can be specified as
z ¼ 0 : u ¼ 0; v ¼ 0; w ¼ 0
(13)
At the free-surface z ¼ h(r), due to none surface tension and the kinematic and stress-free conditions, the boundary condition can be expressed as
vu vv ¼ 0; ¼ 0; p ¼ p0 z ¼ hðrÞ : vz vz
(14)
pffiffiffiffiffiffiffiffiffi Defining z ¼ U=y and integrating Eqs. (10) and (11) between 0 and h(r), we can obtain
Table 1 Some subsets of data demonstrated the magnitude of G, kg/min U, rad/s
R, mm
m,
r, h, Pa S kg/m3 mm
7.5 5 5 2.5 2.5 7.5 2.5 7.5 2.5 7.5 5 5
0.0375 0.025 0.025 0.025 0.05 0.05 0.0375 0.0375 0.0375 0.0375 0.025 0.05
0.7 0.35 1.05 0.7 0.7 0.7 0.35 0.35 1.05 1.05 0.7 0.7
2590 2590 2590 2590 2590 2590 2590 2590 2590 2590 1295 1295
Re
3
2
and
3
3
2
2
Re. 3
2
Re
0.535 53.12 0.000204 0.010812 0.379 99.98 0.000230 0.018979 0.584 21.63 0.000546 0.011803 0.374 25.33 0.000224 0.005669 0.237 39.97 0.000022 0.000898 0.347 81.90 0.000048 0.003945 0.212 89.37 0.000032 0.002856 0.326 174.36 0.000076 0.013177 0.346 18.25 0.000085 0.001554 0.51 37.15 0.000185 0.006871 0.66 28.71 0.000697 0.020008 0.395 47.97 0.000062 0.002994
(19)
2prdz
0
¼
(12)
Ur cos½zhðrÞsin½zhðrÞ 2 4 cos½zhðrÞ þ e2zhðrÞ 2 þ e2zhðrÞ
Since the volume flow rate of the molten slag at any radius remains constant equal to that from the inlet, the thickness of the slag on the disk as a function of the radius can be determined as
(10)
v2 w 1 vp ¼ r vz vz2
157.08 209.44 209.44 209.44 209.44 209.44 209.44 209.44 209.44 209.44 209.44 209.44
1 2ezhðrÞ cos½zhðrÞ ezhðrÞ þ ezhðrÞ UrezhðrÞ 2 2 4 cos½zhðrÞ þ e2zhðrÞ 2 þ e2zhðrÞ
Q ¼
v2 v y 2 ¼ 2Uu vz
y
C1 ¼
(9)
UR
In the following we will use the parameters employed in the reference [11] to demonstrate the magnitude of 3 2 and 3 2Re and only some subsets of data is shown in Table 1. It is indicated that neglecting terms of order 3 2 and 3 2Re will result in error of around 0.06% and 2%, respectively. However, what we are interested in is the film thickness at the disk periphery, the contribution of terms 2 3 Re would be even smaller, particularly easily less than 1% with the increasing radius. Consequently, we neglect terms of order 3 2 and 2 3 Re, the momentum conservation equations become as Eq. (10)e (12) which happens to coincide with the reference [15,21].
y
(16)
The integration constants C1, C2, C3 and C4 can be determined from the initial boundary conditions as following
Where Re is the Reynolds number (Re), E is the Ekman number (Ek) and Ro is the Rossby number (Ro).
Re ¼
þ C3 ezz sinðzzÞ
(8)
Re
3 2 Re
(15)
1 4e2zhðrÞ cos½zhðrÞsin½zhðrÞ e4zhðrÞ þ 1 pr 2 U 2 2z 4e2zhðrÞ cos½zhðrÞ þ e4zhðrÞ 2e2zhðrÞ þ 1
(20)
where Q is the volume flow rate and h(r) is the thickness of the liquid, h(r) as a function of radius r can be expressed by the volume flow rate Q, rotary speed U and z. However, we cannot obtain the explicit expression, the relation of r and h(r) can be expressed as
( r ¼
1 4e2zhðrÞ cos½zhðrÞsin½zhðrÞ e4zhðrÞ þ 1 pU 2 2zQ 4e2zhðrÞ cos½zhðrÞ þ e4zhðrÞ 2e2zhðrÞ þ 1
)1 2
(21)
3. Results and discussion 3.1. Liquid film thickness profile on the disk As indicated by Eqs. (10) and (11), the liquid film height profile on the rotary disk is governed primarily by the Coriolis and centrifugal forces along with the viscous drag, or rather, by the volume flow rate, the rotational speed of the disk and the kinematic viscosity as shown by Eq. (21). Due to the chemical composition of the BF slag of the SiO2eAl2O3eCaOeMgO system, the dynamic viscosity ranges from 0.08 to 2.16 Pa s [26], the density of the slag is set as 2500 kg/m3. Fig. 3(a)e(c) illustrates the variations in the liquid film height with the disk radius under different volume flow rate, rotational speed and kinematic viscosity based on Eq. (21), respectively. After flowing down to the center of the disk, the liquid film thickness decreases gradually while the liquid moves radially outward. At a certain disk radius, the thickness on the disk increases with increasing volume flow rate, decreasing rotational speed and increasing kinematic viscosity. At a radius of 0.075 m, the liquid film thickness increases from 1.59 104 to 4.01 104, 0.89 104 to 5.64 104 and 1.26 104 to 3.77 104 m with
D. Wang et al. / Applied Thermal Engineering 63 (2014) 387e395
increasing volume flow rate from 0.8 105 to 12.8 105 m3/s, decreasing rotational speed from 1005.31 to 62.83 rad/s and increasing kinematic viscosity from 0.32 104 to 8.64 104 m2/ s, respectively. Zhao et al. [27] found that the influence of kinematic viscosity on the liquid height is less significant. However, in this paper, we considered that the kinematic viscosity has a significant
(a)
391
effect on the liquid film thickness at high kinematic viscosity. For a volume flow rate 3.2 105 m3/s, rotational speed 209.44 rad/s and kinematic viscosity between 0.32 104 and 0.72 104 m2/s, the liquid film thickness at a radius of 0.075 m ranges from 1.26 104 to 1.65 104 m as shown in Fig. 3(c). Zhao et al. [15] indicated that this model can predict the hydraulic jump if we defined that the hydraulic jump happens at the radius where the radial velocity tends to zero or the liquid film thickness tends to infinity. It reveals in Fig. 3(a)e(c) that the hydraulic jump radius increases with increasing liquid volume flow rate, decreasing rotational speed and decreasing liquid kinematic viscosity. For the operation conditions in this paper, the hydraulic jump radius ranges from 0.00467 103 to 0.0187 103 m. A CFD simulation method will be employed to analyze the accuracy of the hydraulic jump location predicted by this model in the following chapter. 3.2. Mean radial velocity The granule performance is dominantly effect by the liquid film thickness at the periphery of the disk or the radial and tangential velocities relative to the disk. Although there is nonslippage at the liquidesolid interface, the radial and tangential velocities relative to the rotary disk increase with increasing axial distance. The mean radial and tangential velocities can be expressed as Eqs. (22) and (23), respectively.
ZhðrÞ
(b)
udz u ¼
0
hðrÞ
¼
Q 2prhðrÞ
(22)
ZhðrÞ vdz v ¼
(c)
0
hðrÞ
(23)
Fig. 4(a)e(c) show the variations in mean radial velocity with the disk radius under different volume flow rate, rotational speed and kinematic viscosity, respectively. When the liquid impinges on the disk, the axial momentum is transferred into the radial momentum, the mean radial velocity changes continuously with the radius. For a fixed operation condition, the mean radial velocity first increases rapidly up a certain value and then decreases steadily with increasing radius after the hydraulic jump. At any given radial position, the mean radial velocity increases with increasing volume flow rate, increasing rotational speed of disk and decreasing kinematic viscosity. 3.3. Degree of tangential slippage
Fig. 3. Liquid film thickness with respect to the disk radius with different (a) volume flow rates, (b) disk rotation speeds, (c) kinematic viscosities.
The maximum liquid film thickness at a radius of 0.075 m is smaller than 5.64 104 m for all the operation condition in this paper, therefore the relative tangential velocity of the liquid flow on the rotary disk is very small, typically at large disk radius. Consequently, it is convenient to describe the deviation of the tangential velocity of the liquid from that of the disk by the degree of tangential slippage f ¼ jvj=r U, which is the ratio between the mean tangential velocity of the liquid relative to the disk and the absolute velocity of the disk [27]. The degree of slippage increases with increasing volume flow rate, decreasing disk rotational speed and decreasing kinematic viscosity at any certain radius indicated in Fig. 5(a)e(c). Just after the hydraulic jump, the degree of slippage is very high, up to 0.5. However, the degree of slippage decreases
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D. Wang et al. / Applied Thermal Engineering 63 (2014) 387e395
first rapidly and then relatively slowly, finally levels off with increasing radius. At a radius of 0.075 m, the degree of slippage decreases from 0.0044 to 1.11 104, from 0.0016 to 2.47 104 and from 0.0028 to 3.16 104 with increasing volume flow rate from 0.8 105 to 12.8 105 m3/s, decreasing rotational speed
from 1005.31 to 62.83 rad/s and increasing kinematic viscosity from 0.32 104 to 8.64 104 m2/s, respectively. 3.4. Comparison with the existing data Some theoretical or experimental studies concerning with the film flow on a rotary disk are available in the published literature.
Fig. 4. Mean radial velocity with respect to the disk radius with different (a) volume flow rates, (b) disk rotation speeds, (c) kinematic viscosities.
Fig. 5. Degree of tangential slippage with respect to the disk radius with different (a) volume flow rates, (b) disk rotational speeds, (c) kinematic viscosities.
D. Wang et al. / Applied Thermal Engineering 63 (2014) 387e395 Table 2 Some models concerning with the liquid film thickness on a rotary disk. Name
Equation
Emslie et al. [12] Charwat et al. [25] Lepehin et al. [14] Zhao et al. [27]
h ¼
3Q y 2 2pr 2 U
h ¼ 2
1
3Q y 2 2pr 2 U
3
1 3
Qy 2 r5 U
1
15
h ¼ 0:886Q 0:348 y0:328 U r < rc, h ¼ r > rc, h ¼
0:676 0:70 r
rin y 0:424105w qinffiffiffi r y 1 0:702 Uln 1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:702Q 0:702Q pffiffiffiffiffi 1 , rc ¼ pffiffiffiffiffi r
0:739p yU
0:739p Uy
Table 2 shows some theoretical models or dimensionless relations of the liquid film thickness on a rotary disk. Fig. 6(a) and (b) shows the variations in liquid film thickness with the disk radius for different models under the operation condition of Q ¼ 3.2 105 m3/s, y ¼ 0.000216 m2/s and U ¼ 209.44 rad/s. It reveals that with increasing radius, the liquid film thicknesses predicted by all the models decrease rapidly first
-3
Liquid film thickness [x10 m]
0.80
(a)
0.75
Present model Emslie et al. Charwat et al. Lepehin et al. Zhao et al.
0.70 0.65
393
and then relatively slowly. When r < 0.018 m, the film thickness calculated by the models have intersection points, the film thickness predicted by Zhao’s model is the maximum while that predicted by Emslie’s is the minimum. While r > 0.018 m, Lepehin’s model shows the peak value. Once r > 0.028 m, the film thickness predicted by the models divide into two groups, the present model proposed in this paper, Emslie’s and Lepehin’s model almost coincide with each other, especially at large disk radius, while the remaining models reveal similar performance. At a radius of 0.3 m, the film thicknesses predicted by the present model, Emslie’s and Lepehin’s model are 9.39 105, 9.39 105 and 9.47 105 m, respectively. The maximum relative error is 0.84%. However, the thicknesses predicted by Charwat’s and Zhao’s model are 3.92 105 and 3.27 105 which is nearly 3 times smaller than the formers. Fig. 7 illustrates the variations in the liquid film thickness predicted by the present model with that measured in the experiment [21]. It is revealed that good coincidences are present for aqueous solutions of glycerol, the maximum errors are 7.01% at the disk radius of 0.0297 m and 12.4% at the disk radius of 0.0297 m for 30% aqueous solution of glycerol and 50% aqueous solution of glycerol, respectively. However, when r > 0.06 m, the maximal errors for the two solutions are less than 1%. As for water, the height predicted by the present model is in good agreement with the experimental data with the maximal error is 6.3% at r ¼ 0.079 m when r > 0.034 m. In conclusion, the present model proposed in this paper can predict the liquid film height on the rotary disk with sufficient precision at the large disk radius.
0.60
3.5. Comparison with the CFD simulation
0.55
Compared with other simulation method, the VOF model shows particular advantages in interface tracking, therefore the model was used to simulate the free-surface film flow on the rotary disk in this paper. We also assumed the film flow is turbulent as reference [11] and adopted the same setting except the mesh size. In this paper, the mesh sizes of air region and molten slag region were set as 0.05 mm and 0.02 mm through the extensive simulations of grid independence, respectively. As the same with reference [11], the density of BF slag was set as 2590 kg/m3, the dynamic viscosity was 0.7 Pa s, the surface tension was set as 0.478 N/m. However, unlike reference [11], the density and dynamic viscosity of the air were set
0.50 0.45 0.40 0.012
0.015
0.018
0.021
0.024
0.027
Radius [m] 0.40
0.30 0.25 0.20 0.15 0.10 0.05
0.9 Present model (a) (b) (c) Leshev et al. (a) (b) (c)
0.8 3
-3
Liquid film thickness [x10 m]
Present model Emslie et al. Charwat et al. Lepehin et al. Zhao et al.
Liquid film height [x10 m]
(b)
0.35
0.7 0.6 0.5 0.4 0.3 0.2 0.1
0.00 0.04
0.08
0.12
0.16
0.20
0.24
0.28
Radius [m] Fig. 6. Comparison of liquid film thickness predicted by the present model with that calculated by the exist model at Q ¼ 3.2 105 m3/s, y ¼ 0.000216 m2/s and U ¼ 209.44 rad/s for: (a) The radius ranging from 0 to 0.028 m; (b) The radius ranging from 0.028 to 0.3 m.
0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 Radius [m] Fig. 7. Comparison of liquid film thickness predicted by the present model with that measured in the experiment [21] at Q ¼ 5.33 106 m3/s and U ¼ 10.47 rad/s for: (a) water (y ¼ 1.007 106 m2/s); (b) 30% aqueous solution of glycerol (y ¼ 6.25 106 m2/s); (c) 50% aqueous solution of glycerol (y ¼ 1.063 105 m2/s).
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D. Wang et al. / Applied Thermal Engineering 63 (2014) 387e395
Fig. 8. Liquid film thickness predicted by the present model and CFD simulation with different inlet velocities at Q ¼ 4.5045 105 m3/s, U ¼ 209.44 rad/s, y ¼ 0.00027 m2/s and r ¼ 2590 kg/m3, (a) comparison of film thickness by the present model with the CFD simulation, (b) film thickness profile by the CFD simulation with win ¼ 3.585 m/s, (c) film thickness profile by the CFD simulation with win ¼ 0.896 m/s.
Fig. 9. The radial velocity predicted by the present model and CFD simulation with different inlet velocities at Q ¼ 4.5045 105 m3/s, U ¼ 209.44 rad/s, y ¼ 0.00027 m2/s and r ¼ 2590 kg/m3, (a) comparison of mean radial velocity by the present model with the CFD simulation, (b) the radial velocity distribution by the CFD simulation with win ¼ 3.585 m/ s, (c) the radial velocity distribution by the CFD simulation with win ¼ 0.896 m/s.
as 0.2395 kg/m3 and 5.58 105 Pa s at the temperature of 1200 C. It reveals in Figs. 8 and 9 that the proposed model shows good agreement with the CFD simulation result at large radius. For a given operation condition, the liquid film thickness on the rotary disk was not influenced by the inlet velocity at large radius, especially after the hydraulic jump as indicated in Fig. 8. It is concluded that the liquid film profile on the rotary disk is mainly determined by the Coriolis and centrifugal forces at large radius, particularly after the hydraulic jump, which coincides with the proposed model. The inertia force contributes primarily nearby the disk center. In addition, the width of the hydraulic jump region is very small and this region has no influence on the flow far away from the hydraulic jump. Consequently, it will have none effect on the granulation performance of molten slag. Meanwhile, when the inlet velocity is smaller than 0.896 m/s in this paper, no hydraulic jump happens. However, it must be pointed out that this model cannot predict the hydraulic jump radius accurately as indicated in Fig. 8.
heat recovery process of molten slag. Some flow parameters, such as liquid film thickness, the mean radial velocity and degree of tangential slippage, have been analyzed based on the proposed model. The model has also been compared with the existing relations, experimental results, CFD simulation data and shows good performance. It is revealed that the continuous free-surface film flow on the rotary disk is governed primarily by a balance between the Coriolis and centrifugal forces and the viscous drag at large disk radius, particularly after the hydraulic jump. The inertia force contributes primarily nearby the center of the disk. The model indicates that the liquid film thickness distribution is mainly determined by the volume flow rate, kinematic viscosity of the molten slag and the rotational speed of disk. As indicated by the CFD simulation results, the proposed model cannot predicted the hydraulic jump radius accurately. The hydraulic jump region is very small and has no influence on the flow after the hydraulic jump. Therefore it will have non effect on the granulation performance of molten slag.
4. Conclusion
Acknowledgements
A simplified theoretical model has been proposed by the method of order of magnitude analysis to characterize the continuous free-surface film flow of molten slag on a rotary disk in waste
The authors gratefully acknowledge the financial support provided by the Research Fund for the Doctoral Program of Higher Education (RFDP) (Grant No. 20103221110005), the Graduate
D. Wang et al. / Applied Thermal Engineering 63 (2014) 387e395
Innovation Project of Jiangsu Province (Grant No. CXZZ13_0435) and the Jiangsu Key Laboratory of Process Enhancement and New Energy Equipment Technology. References [1] International Energy Agent, Energy Technology Perspective 2006: Scenario and Strategies to 2050, International Energy Agency, Pairs, 2006. [2] M. Barati, S. Esfahani, T.A. Utigard, Energy recovery from high temperature slags, Energy 36 (2011) 5440e5449. [3] M. Allibert, H. Gaye, J. Geiseler, D. Janke, B.J. Keene, D. Kriner, Slag Atlas, VerlagStahleisen, Germany, 1995. [4] M.P. Sudbury, D.J. Kemp, The Properties and Uses of Granulated Non Ferrous Slags, Minerals, metals and materials society, San Antonio, United states, 2006. [5] J.X. Liu, Q.B. Yu, P. Li, W.Y. Du, Cold experiments on ligament formation for blast furnace slag granulation, Appl. Therm. Eng. 40 (2012) 351e357. [6] H. Guo, S.H. Zhou, Discussion about heat recovery technology of blast furnace slag, in: The Proceeding of Ironmaking Technology Conference and Ironmaking Academic Annual Meeting, 2010. [7] H. Zhang, H. Wang, X. Zhu, Y.J. Qiu, K. Li, R. Chen, Q. Liao, A review of waste heat recovery technologies towards molten slag in steel industry, Appl. Energy 112 (2013) 956e966. [8] Y. Jin, Z.B. Tian, The thermodynamic analysis of SO2 and H2S emissions from INBA system of blast furnace, in: The 5th International Conference on Bioinformatics and Biomedical Engineering, WuHan, China, 2011. [9] D.X. Wang, X. Ling, H. Peng, Performance analysis of double organic Rankine cycle for discontinuous low temperature waste heat recovery, Appl. Therm. Eng. 48 (2012) 63e71. [10] D.X. Wang, X. Ling, H. Peng, L. Liu, L.L. Tao, Efficiency and optimal performance evaluation of organic Rankine cycle for low grade waste heat power generation, Energy 50 (2013) 343e352. [11] Y.H. Pan, P.J. Witt, B. Kuan, D.S. Xie, Effect of flow and operating parameters on the spreading of a viscous liquid on a rotary disk, in: Ninth International Conference on CFD in the Minerals and Process Industries CSIRO, Melbourne, Australia, 2012. [12] A.G. Emslie, F.T. Bonner, L.G. Peck, Flow of a viscous liquid on a rotating disk, J. Appl. Phys. 29 (1958) 858e862. [13] J.W. Rauscher, R.E. Kelly, J.D. Cole, An asymptotic solution for the laminar flow of thin film on a rotating disk, J. Appl. Mech. 40 (1973) 43e47.
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[14] G.I. Lepehin, G.V. Riabchuk, Temperature Distribution in Film of Viscous Liquid with Heating on a Rotating Disk, Rheology in Processes and Apparatus of Chemical Technology, Works of Polytechnic Institute of Volgograd, Volgograd, 1975. [15] Y.Y. Zhao, A.L. Dowson, M.H. Jacobs, Modelling of liquid flow after a hydraulic jump on a rotating disk prior to centrifugal atomization, Modell. Simul. Mater. Sci. Eng. 8 (2000) 55e65. [16] G.M. Sisoev, O.K. Matar, C.J. Lawrence, Modelling of film flow over a rotary disk, J. Chem. Technol. Biotechnol. 78 (2003) 151e155. [17] D. Prieling, H. Steiner, Unsteady thin film flow on rotary disks at large Ekman numbers using an integral boundary layer method, Int. J. Heat Mass Transfer 65 (2013) 10e22. [18] T.G. Myers, M. Lombe, The importance of the Coriolis force on axisymmetric horizontal rotating thin film flows, Chem. Eng. Process. 45 (2006) 90e98. [19] S. Basu, B.M. Cetegen, Analysis of hydrodynamics and heat transfer in a thin liquid film flowing over a rotating disk by the integral method, J. Heat Transfer 128 (2006) 217e225. [20] W.P. Woods, The Hydrodynamics of Thin Liquid Films Flowing over a Rotating Disk (PhD thesis), University of Newcastle upon Tyne, UK, 1995. [21] I. Leshev, G. Peev, Film flow on a horizontal rotating disk, Chem. Eng. Process. 42 (2003) 925e929. [22] J.R. Burns, C. Ramshaw, R.J. Jachuck, Measurement of liquid film thickness and the determination of spin-up radius on a rotating disk using an electrical resistance technique, Chem. Eng. Sci. 58 (2003) 2245e2253. [23] D. Ghiasy, K.V.K. Boodhoo, M.T. Tham, Thermographic analysis of thin liquid films on a rotating disk: approach and challenges, Appl. Therm. Eng. 44 (2012) 39e49. [24] J. Rice, A. Faghri, B. Cetegen, Analysis of a free-surface film from a controlled liquid impinging jet over a rotating disk including conjugate effects with and without evaporation, Int. J. Heat Mass Transfer 48 (2005) 5192e5204. [25] T.J. Bhatelia, R.P. Utikar, V.K. Pareek, M.O. Tade, Characterizing liquid film thickness in rotary disk reactors, in: Seventh International Conference on CFD in the Minerals and Process Industries CSIRO, Melbourne, Australia, 2009. [26] T. Lida, H. Sakai, Y. Kita, K. Shigeno, An equation for accurate prediction of the viscosities of blast furnace type slags from chemical composition, ISIJ Int. 40 (Suppl.) (2000) S110eS114. [27] Y.Y. Zhao, M.H. Jacobs, A.L. Dowson, Liquid flow on a rotating disk prior to centrifugal atomization and spray deposition, Metall. Mater. Trans. B 29 (1998) 1357e1369.
Contents lists available at ScienceDirect
Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Theoretical analysis of free-surface film flow on the rotary granulating disk in waste heat recovery process of molten slag Dongxiang Wang, Xiang Ling*, Hao Peng Jiangsu Key Laboratory of Process Enhancement and New Energy Equipment Technology, School of Mechanical and Power Engineering, Nanjing University of Technology, No. 5 Xin Mo Fan Road, Nanjing 210009, PR China
h i g h l i g h t s A theoretic model describing the free-surface film flow on rotary disk is proposed. The proposed model is compared with existing data and CFD simulation result. Coriolis, centrifugal force and viscous drag govern the flow after hydraulic jump. The liquid flow after hydraulic jump is not effect by the hydraulic jump region.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 13 August 2013 Accepted 15 November 2013 Available online 26 November 2013
This paper proposed a simple theoretical model to characterize the continuous free-surface film flow on the rotary disk in waste heat recovery process of molten slag by the method of order of magnitude analysis. Liquid film thickness, mean radial velocity and degree of tangential slippage have been analyzed based on the proposed model. The model indicates that the free-surface film flow on the rotary disk is governed primarily by the balance between Coriolis and centrifugal forces and viscous drag after the hydraulic jump. The inertia force contributes primarily nearby the center of the rotary disk. It is revealed that the liquid film thickness distribution on a rotary disk is mainly determined by the volume flow rate, kinematic viscosity of molten slag and the rotational speed of disk. Compared with the existing results and CFD simulation data, the model shows good performance. As indicated by the CFD simulation results, the hydraulic jump region has no influence on the liquid flow after the hydraulic jump, hence has none effect on the granulation performance of molten slag. Crown Copyright Ó 2013 Published by Elsevier Ltd. All rights reserved.
Keywords: Film flow Free-surface Rotary disk Waste heat Molten slag
1. Introduction The International Energy Agency [1] reports that: “The energy intensity of most industrial processes is at least 50% higher than the theoretical minimum determined by the laws of thermodynamics”. Nowadays, under the context of the utmost concern for energy saving and emission reduction, adoption of environment-friendly and renewable technologies to recovery the waste heat in the metallurgy industry has attracted growing attention. With the technologies to recover the thermal and chemical energies of process off-gas being improved, the molten slag represents the last untapped waste heat source that may be used for energy saving in the rather energy-intensive metal industry [2].
* Corresponding author. Tel.: þ86 25 83587321; fax: þ86 25 83600956. E-mail address: [email protected] (X. Ling).
Molten slags carry a substantial amount of high quality thermal energy since the tapping temperature is up to 1000e1650 C [3e5]. As demonstrated from Fig. 1, ferrous slags account for over 90% of the available energy associated with slags and hot metal blast furnace (BF) slag alone constitutes 47% of this energy [2]. Therefore, in this paper we will focus on BF slag. At present, “wet granulation”, utilizing vast amounts of water to quench the molten slag for high cooling rate and producing granules with high glass phase for cement manufacture, is widely used for the treatment of BF slag in the iron and steel plant. However, the water treatment method reveals many obvious shortcomings in the practice: High consumption of water. For example, the granulation of 1 tonne BF slag by INBA process will consume 1.0e1.2 tonne water [6]. Pollutants emission. Sulfur dioxide, hydrogen sulfide and other sulfur compounds [7,8].
1359-4311/$ e see front matter Crown Copyright Ó 2013 Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.applthermaleng.2013.11.033
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D. Wang et al. / Applied Thermal Engineering 63 (2014) 387e395
Nomenclature Ek rc rin win hin b R h u v w z r p po U Re Ro G
Ekman number the critical radius of hydraulic jump [m] inlet radius [m] inlet velocity [m/s] inlet height [m] thickness of the disk [m] radius of the disk [m] liquid film thickness [m] radial velocity [m/s] tangential velocity [m/s] axial velocity [m/s] axial distance [m] radius distance [m] pressure [MPa] the ambient pressure [MPa] radial velocity [m/s] Reynolds number Rossby number mass flow rate [kg/min]
Poor energy efficiency [9,10]. Most of the sensible heat is wasted. Meanwhile, the water granulated process only produces large quantities of low-temperature steam or water for heat-supply in winter by a few plants [7]. Additional energy consumption. Such as drying requirement for the wet product. Dry slag granulation (DSG), an industry process utilizing a rotary disk or rotary cup to spray the molten slag into granules by centrifugal force, has received more and more attention. In particular, rotary disk granulation is capable of producing droplets with narrow size-distributed range and applicable to liquids of both low viscosity (e.g., water) and high viscosity (e.g., molten slag and organic fluids) [11]. Experimental study of the DSG process by CSIRO has identified that the slag granules are produced by three steps: free-surface film formation on the disk, ligament formation and granule formation, as shown in Fig. 2(a). The performance of the DSG primarily depends on the hydrodynamics of the flow on the rotary disk, while the flow is characterized by the free-surface film with a sharp gaseliquid interface changing continuously as indicated in Fig. 2(b). There are a large number of published literature concerning with the continuous liquid free-surface film flow on the rotary disk. The original mathematical model [12] considers a balance between the viscous drag of the fluid and the centrifugal force and provides a simple relation between fluid flux and the liquid film thickness. Rauscher et al. [13] and Lepehin et al. [14] obtained the approximate analytical solutions about the radial velocity and film thickness.
Fig. 1. The heat content of various kinds of metallurgical slags per year.
C1 C2 C3 C4 Q
integration constant integration constant integration constant integration constant volume of flow rate [m3/s]
Greek
U y r 3
m f
rotational speed [rad/s] kinematic viscosity [m2/s] density [kg/m3] aspect ratio dynamic viscosity [Pa s] degree of tangential slippage
Subscript in inlet c critical r radius Acronyms DSG dry slag granulation
Zhao et al. [15] presented three models to characterize the film flow on the rotary disk, the numerical model gives a more accurate approximation of the liquid film thickness. Unfortunately, no explicit relation is given. Sisoev et al. [16] derived the approximate evolution equations for the film thickness and volumetric flow rates by an integral method. Prieling et al. [17] analyzed the unsteady film flow on the rotary disk at large Ekman numbers by integral boundary layer method. In addition, non-dimensional method has also been given serious attention [16e19]. Wood [20] performed an experimental study to characterize the film flow on the rotary disk and showed existence of several wavy regimes in concentric zones across the radius of the disk. Leshev et al. [21] carried out an experimental and analytical investigation, the analytical result predicts film heights slightly lower than the experimental values near the center of the disk (typically the worst error is around 10%). Burns et al. [22] proposed a theoretical model based on experiment and compared with the Nusselt theory, they found that the Nusselt model could not be used accurately for the inertial flow conditions. Ghiasy et al. [23] first examined the temperature profiles and flow characteristics of liquid films on the rotary disk by infrared thermal imaging camera. Besides, there is still many literature concerning with the numerical simulation. Rice et al. [24] carried out a direct numerical simulation of the film flow on the rotary disk using the volume of fluid (VOF) method. Pan et al. [11] applied a two-dimensional multiphase CFD model to investigate the film thickness at the disk periphery and revealed that the surface tension has a negligible effect on the film thickness. It is evident that the simulate result is 10% smaller than the model of [12] on average. Recently, Bhatelia et al. [25] carried out both 2D and 3D CFD simulations of film flow on the rotary disk based on VOF approach. The predicted film thickness of 3D simulation was found to agree with the experimental data with the relative error of 10%, by contrast, the 2D model under-predicted the film thickness by 39%. The reason may be the assumption of laminar flow on the rotary disk, while Pan et al. [11] considered the flow is turbulent. It is essential to reliably predict the liquid film thickness on the rotary disk since most of the hydrodynamic properties like radial velocity, granulated particle size along with heat and mass transfer coefficients directly depend on that. Many investigations assumed the flow is laminar or utilized non-dimensional method and trial
D. Wang et al. / Applied Thermal Engineering 63 (2014) 387e395
389
Fig. 2. The slag granulation process and free-surface film flow on the rotary disk (a) a typical static image from high speed video recording of the slag granulation by a rotary disk in CSIRO’s dry slag granulation process [11], (b) schematic of the film thickness profile of molten slag on the rotary disk.
function of velocity to obtain an approximate analytic solution. Large variances always existed between the published literature, particularly in the CFD simulation result. There is still limited understanding of the film flow on a rotary disk. In this paper, a simple theoretical model of continuous free-surface film flow on the rotary disk was derived from the NaviereStokes equations based on the order of magnitude analysis. The proposed model has been compared with the existing data and CFD simulation results. Large attempt has been made to further understand the behavior of the film flow on a rotary disk involving the effect of operation parameters and physical properties on the film thickness.
u
vv vv uv v2 v 1 vv v2 v v þ w þ þ 2Uu ¼ y þ þ vr vz r vr 2 r vr vz2 r 2
! vw vw 1 vp v2 w v2 w þw ¼ þy þ 2 u r vz vr vz vr 2 vz
(2)
(3)
For the incompressible fluid, the equation of continuity is
1v vw ðruÞ þ ¼ 0 r vr vz
2. Theoretical analysis
!
(4)
2.1. Assumption and governing equation In the DSG process, the slag directly flows down to the center of a rapidly rotary disk under the force of gravity, followed by the radial and tangential acceleration toward the disk periphery prior to granulation. Consequently, as indicated in Fig. 2(b), a film thickness profile that is determined by the kinematic viscosity of the molten slag and the frictional and centrifugal forces will form on the disk, perhaps accompanied by a hydraulic jump at a critical radius rc in some special cases (e.g., with high inlet velocity). In order to theoretically simulate the thin film flow phenomenon encountered when molten slag is poured onto a rotary disk, the following assumptions have been made: The molten slag flow is uniform and continuous with constant volume flow rate. The slag is poured along the axis of disk center so that rotational symmetry is assumed. No slippage assumption in the interface between molten slag and the rotary disk. The disk is absolutely filled with slag so that the surface tension is negligible [11,26]. Neglecting the effects of heat transfer between the slag and the disk [11], the free-surface film flow on the disk can be determined by the NaviereStokes equations of momentum conservation and the equation of continuity, which in cylindrical coordinates can be written as
u
vu vu v2 1 vp v2 u 1 vu v2 u u 2 þ w 2Uv U r ¼ þy þ þ r vr vr vz r vr 2 r vr vz2 r 2
!
2.2. Boundary condition and model formation The first three terms on the left hand of Eq. (1) represent the fluid inertia force, the next two terms are the Coriolis and centrifugal forces, respectively. The terms on the right hand represent the pressure gradient and viscous drag. The Coriolis force also appears as the final term of Eq. (2). Now we attempt to determine which terms dominate the flow and systematically reduce the problem to a simpler one by the order of magnitude analysis. In the main flow direction, we supposed that U has the same magnitude as u. Meanwhile, r and R, z and h also posses the same magnitude, respectively. Wood [20] considered that the tangential velocity relative to the disk is far smaller than the radial velocity in a realistic condition, it is reasonable to assume: v/u w 3 . In the DSG process, what we are interested is the liquid film thickness at the periphery of disk, hence we assumed: 3 ¼ h/ R << 1 at large disk radius. Therefore we can get w/U w h/R from the equation of continuity. In conclusion, one can get as mentioned above: 3
¼
h w h << 1; w R U R
(5)
Consequently, the magnitude of all terms in Eqs. (1)e(3) can be defined as:
0
1
B 2 C vu vu v2 1 vp B v u 1 vu u v2 uC 2 C u þw 2Uv U r ¼ þyB þ þ 1 2 r vr B vr ffl{zfflfflfflfflfflfflffl vz vr 2 r vr r 2 vz2 C @ A RoEk Ek |fflfflfflfflfflfflffl ffl} 3 4rRe |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} 2 1 3
3 2 Re
(1)
v w3 ; u
uwU; rwR; zwh;
3
Re
32
(6)
390
D. Wang et al. / Applied Thermal Engineering 63 (2014) 387e395
0
1 vðzÞ ¼ C1 ezz ==; cosðzzÞ þ C2 ezz sinðzzÞ þ C3 ezz cosðzzÞ
B 2 C B v v 1 vv vv vv uv v v2 vC C u þw þ þ 2Uu ¼ yB þ þ B vr 2 r vr r 2 vz2 C 2 vr vz r @ |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} A Ek |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} 1
(7)
1 þ C4 ezz sinðzzÞ Ur 2
3
3 2 Re
32
uðzÞ ¼ C2 ezz cosðzzÞ C1 ezz sinðzzÞ C4 ezz cosðzzÞ
0
1 vw vw 1 vp @ v2 w v2 wA u þw ¼ þy þ 2 r vz vr ffl{zfflfflfflfflfflfflfflfflffl vzffl} vr 2 vz |fflfflfflfflfflfflfflfflffl 1 32
UR ; Ek ¼
y
y
h2 U
; Ro ¼
U
v2 u 2 ¼ 2Uv U r vz2
(17)
C3 ¼
1 2ezhðrÞ cos½zhðrÞ ezhðrÞ þ ezhðrÞ UrezhðrÞ 2 2 4 cos½zhðrÞ þ e2zhðrÞ 2 þ e2zhðrÞ
(18)
2
2
C2 ¼ C4 ¼
ZhðrÞ
(11)
Due to non-slip velocity condition, the boundary condition at the liquidesolid interface z ¼ 0 can be specified as
z ¼ 0 : u ¼ 0; v ¼ 0; w ¼ 0
(13)
At the free-surface z ¼ h(r), due to none surface tension and the kinematic and stress-free conditions, the boundary condition can be expressed as
vu vv ¼ 0; ¼ 0; p ¼ p0 z ¼ hðrÞ : vz vz
(14)
pffiffiffiffiffiffiffiffiffi Defining z ¼ U=y and integrating Eqs. (10) and (11) between 0 and h(r), we can obtain
Table 1 Some subsets of data demonstrated the magnitude of G, kg/min U, rad/s
R, mm
m,
r, h, Pa S kg/m3 mm
7.5 5 5 2.5 2.5 7.5 2.5 7.5 2.5 7.5 5 5
0.0375 0.025 0.025 0.025 0.05 0.05 0.0375 0.0375 0.0375 0.0375 0.025 0.05
0.7 0.35 1.05 0.7 0.7 0.7 0.35 0.35 1.05 1.05 0.7 0.7
2590 2590 2590 2590 2590 2590 2590 2590 2590 2590 1295 1295
Re
3
2
and
3
3
2
2
Re. 3
2
Re
0.535 53.12 0.000204 0.010812 0.379 99.98 0.000230 0.018979 0.584 21.63 0.000546 0.011803 0.374 25.33 0.000224 0.005669 0.237 39.97 0.000022 0.000898 0.347 81.90 0.000048 0.003945 0.212 89.37 0.000032 0.002856 0.326 174.36 0.000076 0.013177 0.346 18.25 0.000085 0.001554 0.51 37.15 0.000185 0.006871 0.66 28.71 0.000697 0.020008 0.395 47.97 0.000062 0.002994
(19)
2prdz
0
¼
(12)
Ur cos½zhðrÞsin½zhðrÞ 2 4 cos½zhðrÞ þ e2zhðrÞ 2 þ e2zhðrÞ
Since the volume flow rate of the molten slag at any radius remains constant equal to that from the inlet, the thickness of the slag on the disk as a function of the radius can be determined as
(10)
v2 w 1 vp ¼ r vz vz2
157.08 209.44 209.44 209.44 209.44 209.44 209.44 209.44 209.44 209.44 209.44 209.44
1 2ezhðrÞ cos½zhðrÞ ezhðrÞ þ ezhðrÞ UrezhðrÞ 2 2 4 cos½zhðrÞ þ e2zhðrÞ 2 þ e2zhðrÞ
Q ¼
v2 v y 2 ¼ 2Uu vz
y
C1 ¼
(9)
UR
In the following we will use the parameters employed in the reference [11] to demonstrate the magnitude of 3 2 and 3 2Re and only some subsets of data is shown in Table 1. It is indicated that neglecting terms of order 3 2 and 3 2Re will result in error of around 0.06% and 2%, respectively. However, what we are interested in is the film thickness at the disk periphery, the contribution of terms 2 3 Re would be even smaller, particularly easily less than 1% with the increasing radius. Consequently, we neglect terms of order 3 2 and 2 3 Re, the momentum conservation equations become as Eq. (10)e (12) which happens to coincide with the reference [15,21].
y
(16)
The integration constants C1, C2, C3 and C4 can be determined from the initial boundary conditions as following
Where Re is the Reynolds number (Re), E is the Ekman number (Ek) and Ro is the Rossby number (Ro).
Re ¼
þ C3 ezz sinðzzÞ
(8)
Re
3 2 Re
(15)
1 4e2zhðrÞ cos½zhðrÞsin½zhðrÞ e4zhðrÞ þ 1 pr 2 U 2 2z 4e2zhðrÞ cos½zhðrÞ þ e4zhðrÞ 2e2zhðrÞ þ 1
(20)
where Q is the volume flow rate and h(r) is the thickness of the liquid, h(r) as a function of radius r can be expressed by the volume flow rate Q, rotary speed U and z. However, we cannot obtain the explicit expression, the relation of r and h(r) can be expressed as
( r ¼
1 4e2zhðrÞ cos½zhðrÞsin½zhðrÞ e4zhðrÞ þ 1 pU 2 2zQ 4e2zhðrÞ cos½zhðrÞ þ e4zhðrÞ 2e2zhðrÞ þ 1
)1 2
(21)
3. Results and discussion 3.1. Liquid film thickness profile on the disk As indicated by Eqs. (10) and (11), the liquid film height profile on the rotary disk is governed primarily by the Coriolis and centrifugal forces along with the viscous drag, or rather, by the volume flow rate, the rotational speed of the disk and the kinematic viscosity as shown by Eq. (21). Due to the chemical composition of the BF slag of the SiO2eAl2O3eCaOeMgO system, the dynamic viscosity ranges from 0.08 to 2.16 Pa s [26], the density of the slag is set as 2500 kg/m3. Fig. 3(a)e(c) illustrates the variations in the liquid film height with the disk radius under different volume flow rate, rotational speed and kinematic viscosity based on Eq. (21), respectively. After flowing down to the center of the disk, the liquid film thickness decreases gradually while the liquid moves radially outward. At a certain disk radius, the thickness on the disk increases with increasing volume flow rate, decreasing rotational speed and increasing kinematic viscosity. At a radius of 0.075 m, the liquid film thickness increases from 1.59 104 to 4.01 104, 0.89 104 to 5.64 104 and 1.26 104 to 3.77 104 m with
D. Wang et al. / Applied Thermal Engineering 63 (2014) 387e395
increasing volume flow rate from 0.8 105 to 12.8 105 m3/s, decreasing rotational speed from 1005.31 to 62.83 rad/s and increasing kinematic viscosity from 0.32 104 to 8.64 104 m2/ s, respectively. Zhao et al. [27] found that the influence of kinematic viscosity on the liquid height is less significant. However, in this paper, we considered that the kinematic viscosity has a significant
(a)
391
effect on the liquid film thickness at high kinematic viscosity. For a volume flow rate 3.2 105 m3/s, rotational speed 209.44 rad/s and kinematic viscosity between 0.32 104 and 0.72 104 m2/s, the liquid film thickness at a radius of 0.075 m ranges from 1.26 104 to 1.65 104 m as shown in Fig. 3(c). Zhao et al. [15] indicated that this model can predict the hydraulic jump if we defined that the hydraulic jump happens at the radius where the radial velocity tends to zero or the liquid film thickness tends to infinity. It reveals in Fig. 3(a)e(c) that the hydraulic jump radius increases with increasing liquid volume flow rate, decreasing rotational speed and decreasing liquid kinematic viscosity. For the operation conditions in this paper, the hydraulic jump radius ranges from 0.00467 103 to 0.0187 103 m. A CFD simulation method will be employed to analyze the accuracy of the hydraulic jump location predicted by this model in the following chapter. 3.2. Mean radial velocity The granule performance is dominantly effect by the liquid film thickness at the periphery of the disk or the radial and tangential velocities relative to the disk. Although there is nonslippage at the liquidesolid interface, the radial and tangential velocities relative to the rotary disk increase with increasing axial distance. The mean radial and tangential velocities can be expressed as Eqs. (22) and (23), respectively.
ZhðrÞ
(b)
udz u ¼
0
hðrÞ
¼
Q 2prhðrÞ
(22)
ZhðrÞ vdz v ¼
(c)
0
hðrÞ
(23)
Fig. 4(a)e(c) show the variations in mean radial velocity with the disk radius under different volume flow rate, rotational speed and kinematic viscosity, respectively. When the liquid impinges on the disk, the axial momentum is transferred into the radial momentum, the mean radial velocity changes continuously with the radius. For a fixed operation condition, the mean radial velocity first increases rapidly up a certain value and then decreases steadily with increasing radius after the hydraulic jump. At any given radial position, the mean radial velocity increases with increasing volume flow rate, increasing rotational speed of disk and decreasing kinematic viscosity. 3.3. Degree of tangential slippage
Fig. 3. Liquid film thickness with respect to the disk radius with different (a) volume flow rates, (b) disk rotation speeds, (c) kinematic viscosities.
The maximum liquid film thickness at a radius of 0.075 m is smaller than 5.64 104 m for all the operation condition in this paper, therefore the relative tangential velocity of the liquid flow on the rotary disk is very small, typically at large disk radius. Consequently, it is convenient to describe the deviation of the tangential velocity of the liquid from that of the disk by the degree of tangential slippage f ¼ jvj=r U, which is the ratio between the mean tangential velocity of the liquid relative to the disk and the absolute velocity of the disk [27]. The degree of slippage increases with increasing volume flow rate, decreasing disk rotational speed and decreasing kinematic viscosity at any certain radius indicated in Fig. 5(a)e(c). Just after the hydraulic jump, the degree of slippage is very high, up to 0.5. However, the degree of slippage decreases
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first rapidly and then relatively slowly, finally levels off with increasing radius. At a radius of 0.075 m, the degree of slippage decreases from 0.0044 to 1.11 104, from 0.0016 to 2.47 104 and from 0.0028 to 3.16 104 with increasing volume flow rate from 0.8 105 to 12.8 105 m3/s, decreasing rotational speed
from 1005.31 to 62.83 rad/s and increasing kinematic viscosity from 0.32 104 to 8.64 104 m2/s, respectively. 3.4. Comparison with the existing data Some theoretical or experimental studies concerning with the film flow on a rotary disk are available in the published literature.
Fig. 4. Mean radial velocity with respect to the disk radius with different (a) volume flow rates, (b) disk rotation speeds, (c) kinematic viscosities.
Fig. 5. Degree of tangential slippage with respect to the disk radius with different (a) volume flow rates, (b) disk rotational speeds, (c) kinematic viscosities.
D. Wang et al. / Applied Thermal Engineering 63 (2014) 387e395 Table 2 Some models concerning with the liquid film thickness on a rotary disk. Name
Equation
Emslie et al. [12] Charwat et al. [25] Lepehin et al. [14] Zhao et al. [27]
h ¼
3Q y 2 2pr 2 U
h ¼ 2
1
3Q y 2 2pr 2 U
3
1 3
Qy 2 r5 U
1
15
h ¼ 0:886Q 0:348 y0:328 U r < rc, h ¼ r > rc, h ¼
0:676 0:70 r
rin y 0:424105w qinffiffiffi r y 1 0:702 Uln 1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:702Q 0:702Q pffiffiffiffiffi 1 , rc ¼ pffiffiffiffiffi r
0:739p yU
0:739p Uy
Table 2 shows some theoretical models or dimensionless relations of the liquid film thickness on a rotary disk. Fig. 6(a) and (b) shows the variations in liquid film thickness with the disk radius for different models under the operation condition of Q ¼ 3.2 105 m3/s, y ¼ 0.000216 m2/s and U ¼ 209.44 rad/s. It reveals that with increasing radius, the liquid film thicknesses predicted by all the models decrease rapidly first
-3
Liquid film thickness [x10 m]
0.80
(a)
0.75
Present model Emslie et al. Charwat et al. Lepehin et al. Zhao et al.
0.70 0.65
393
and then relatively slowly. When r < 0.018 m, the film thickness calculated by the models have intersection points, the film thickness predicted by Zhao’s model is the maximum while that predicted by Emslie’s is the minimum. While r > 0.018 m, Lepehin’s model shows the peak value. Once r > 0.028 m, the film thickness predicted by the models divide into two groups, the present model proposed in this paper, Emslie’s and Lepehin’s model almost coincide with each other, especially at large disk radius, while the remaining models reveal similar performance. At a radius of 0.3 m, the film thicknesses predicted by the present model, Emslie’s and Lepehin’s model are 9.39 105, 9.39 105 and 9.47 105 m, respectively. The maximum relative error is 0.84%. However, the thicknesses predicted by Charwat’s and Zhao’s model are 3.92 105 and 3.27 105 which is nearly 3 times smaller than the formers. Fig. 7 illustrates the variations in the liquid film thickness predicted by the present model with that measured in the experiment [21]. It is revealed that good coincidences are present for aqueous solutions of glycerol, the maximum errors are 7.01% at the disk radius of 0.0297 m and 12.4% at the disk radius of 0.0297 m for 30% aqueous solution of glycerol and 50% aqueous solution of glycerol, respectively. However, when r > 0.06 m, the maximal errors for the two solutions are less than 1%. As for water, the height predicted by the present model is in good agreement with the experimental data with the maximal error is 6.3% at r ¼ 0.079 m when r > 0.034 m. In conclusion, the present model proposed in this paper can predict the liquid film height on the rotary disk with sufficient precision at the large disk radius.
0.60
3.5. Comparison with the CFD simulation
0.55
Compared with other simulation method, the VOF model shows particular advantages in interface tracking, therefore the model was used to simulate the free-surface film flow on the rotary disk in this paper. We also assumed the film flow is turbulent as reference [11] and adopted the same setting except the mesh size. In this paper, the mesh sizes of air region and molten slag region were set as 0.05 mm and 0.02 mm through the extensive simulations of grid independence, respectively. As the same with reference [11], the density of BF slag was set as 2590 kg/m3, the dynamic viscosity was 0.7 Pa s, the surface tension was set as 0.478 N/m. However, unlike reference [11], the density and dynamic viscosity of the air were set
0.50 0.45 0.40 0.012
0.015
0.018
0.021
0.024
0.027
Radius [m] 0.40
0.30 0.25 0.20 0.15 0.10 0.05
0.9 Present model (a) (b) (c) Leshev et al. (a) (b) (c)
0.8 3
-3
Liquid film thickness [x10 m]
Present model Emslie et al. Charwat et al. Lepehin et al. Zhao et al.
Liquid film height [x10 m]
(b)
0.35
0.7 0.6 0.5 0.4 0.3 0.2 0.1
0.00 0.04
0.08
0.12
0.16
0.20
0.24
0.28
Radius [m] Fig. 6. Comparison of liquid film thickness predicted by the present model with that calculated by the exist model at Q ¼ 3.2 105 m3/s, y ¼ 0.000216 m2/s and U ¼ 209.44 rad/s for: (a) The radius ranging from 0 to 0.028 m; (b) The radius ranging from 0.028 to 0.3 m.
0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 Radius [m] Fig. 7. Comparison of liquid film thickness predicted by the present model with that measured in the experiment [21] at Q ¼ 5.33 106 m3/s and U ¼ 10.47 rad/s for: (a) water (y ¼ 1.007 106 m2/s); (b) 30% aqueous solution of glycerol (y ¼ 6.25 106 m2/s); (c) 50% aqueous solution of glycerol (y ¼ 1.063 105 m2/s).
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D. Wang et al. / Applied Thermal Engineering 63 (2014) 387e395
Fig. 8. Liquid film thickness predicted by the present model and CFD simulation with different inlet velocities at Q ¼ 4.5045 105 m3/s, U ¼ 209.44 rad/s, y ¼ 0.00027 m2/s and r ¼ 2590 kg/m3, (a) comparison of film thickness by the present model with the CFD simulation, (b) film thickness profile by the CFD simulation with win ¼ 3.585 m/s, (c) film thickness profile by the CFD simulation with win ¼ 0.896 m/s.
Fig. 9. The radial velocity predicted by the present model and CFD simulation with different inlet velocities at Q ¼ 4.5045 105 m3/s, U ¼ 209.44 rad/s, y ¼ 0.00027 m2/s and r ¼ 2590 kg/m3, (a) comparison of mean radial velocity by the present model with the CFD simulation, (b) the radial velocity distribution by the CFD simulation with win ¼ 3.585 m/ s, (c) the radial velocity distribution by the CFD simulation with win ¼ 0.896 m/s.
as 0.2395 kg/m3 and 5.58 105 Pa s at the temperature of 1200 C. It reveals in Figs. 8 and 9 that the proposed model shows good agreement with the CFD simulation result at large radius. For a given operation condition, the liquid film thickness on the rotary disk was not influenced by the inlet velocity at large radius, especially after the hydraulic jump as indicated in Fig. 8. It is concluded that the liquid film profile on the rotary disk is mainly determined by the Coriolis and centrifugal forces at large radius, particularly after the hydraulic jump, which coincides with the proposed model. The inertia force contributes primarily nearby the disk center. In addition, the width of the hydraulic jump region is very small and this region has no influence on the flow far away from the hydraulic jump. Consequently, it will have none effect on the granulation performance of molten slag. Meanwhile, when the inlet velocity is smaller than 0.896 m/s in this paper, no hydraulic jump happens. However, it must be pointed out that this model cannot predict the hydraulic jump radius accurately as indicated in Fig. 8.
heat recovery process of molten slag. Some flow parameters, such as liquid film thickness, the mean radial velocity and degree of tangential slippage, have been analyzed based on the proposed model. The model has also been compared with the existing relations, experimental results, CFD simulation data and shows good performance. It is revealed that the continuous free-surface film flow on the rotary disk is governed primarily by a balance between the Coriolis and centrifugal forces and the viscous drag at large disk radius, particularly after the hydraulic jump. The inertia force contributes primarily nearby the center of the disk. The model indicates that the liquid film thickness distribution is mainly determined by the volume flow rate, kinematic viscosity of the molten slag and the rotational speed of disk. As indicated by the CFD simulation results, the proposed model cannot predicted the hydraulic jump radius accurately. The hydraulic jump region is very small and has no influence on the flow after the hydraulic jump. Therefore it will have non effect on the granulation performance of molten slag.
4. Conclusion
Acknowledgements
A simplified theoretical model has been proposed by the method of order of magnitude analysis to characterize the continuous free-surface film flow of molten slag on a rotary disk in waste
The authors gratefully acknowledge the financial support provided by the Research Fund for the Doctoral Program of Higher Education (RFDP) (Grant No. 20103221110005), the Graduate
D. Wang et al. / Applied Thermal Engineering 63 (2014) 387e395
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