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Wet steam flow energy analysis within thermo-compressors
Energy 47 (2012) 609e619
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Wet steam flow energy analysis within thermo-compressors Navid Sharifi a, Masoud Boroomand a, *, Ramin Kouhikamali b a b
Department of Aerospace Engineering, Amirkabir University of Technology, Tehran, Iran Department of Mechanical Engineering, Faculty of engineering, University of Guilan, Rasht, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 12 July 2012 Received in revised form 31 August 2012 Accepted 1 September 2012 Available online 4 October 2012
Thermo-compressors are widely used in industries for steam compression through a thermal process. Common methods of thermo-compressors analysis are based on the hypothesis of considering steam as a perfect gas. In this study, the deviation of thermo-compressor performance at wet steam conditions from the performance under the ideal gas assumption has been investigated. Firstly, a numerical method has been implemented to evaluate the formation of droplets due to condensation in a convergent edivergent nozzle. The results have been validated using existing experimental data for single nozzles. Afterwards, the verified numerical scheme has been applied to internal flow of the thermocompressor. The formation of droplets due to condensation effect and the resulting supersonic core in the thermo-compressor have been deeply investigated. Finally, the effect of wet steam assumption on the performance characteristics of thermo-compressors has been presented. Ó 2012 Elsevier Ltd. All rights reserved.
Keywords: Thermo-compressor Steam ejector Wet steam Performance parameters Energy analysis
1. Introduction Multi Effect Distillation (MED) systems could be improved by adding vapor compression devices. The vapor compression process could be performed mechanically or thermally. The desalination packages are known as MED-MVC (MED with mechanical vapor compression) or MED-TVC (MED with thermal vapor compression). The main advantages of the TVC usage are the reuse of the compressed vapor that reduces the required amount of motive steam, low capital and construction costs and the simplicity of the steam compressor. Some methods were presented to improve the performance and the efficiency of simple distillation units without thermo-compressor those are based on the multiple condensatione evaporation cycles [1]. However, thermo-compressors are one of the most important parts of desalination systems in which considerable quantities of energy loss is occurred and thus, needed to be designed carefully [2]. Moreover, the simplicity of this device, with no moving parts, gives a forward step compared against the mechanical vapor compressor. It was shown that a single stage mechanical compressor was not feasible for compressing water vapor because of high capital cost and technically challenging compressor design and development [3].
* Corresponding author. Tel.: þ98 21 64543280; fax: þ98 21 66959020. E-mail address: [email protected] (M. Boroomand). 0360-5442/$ e see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.energy.2012.09.003
Thermo-compressors compress large amounts of low-pressure steam through using the available pressurized steam as motive energy. The motive steam passed through a convergentedivergent nozzle of the thermo-compressor may be condensed at the exit section due to temperature decrease caused by supersonic conditions. The condensation phenomenon occurred in supersonic flow is studied deeply in “wet-steam” theory. Wet steam is typically considered as a multiphase mixture in which both vapor phase and liquid droplets exist. In this case, formation of liquid droplets due to pressure decline causes some complexities. Recent studies have been concentrated on the computation of such multiphase flows to determine the rate of droplets growth, condensation shock position, and energy losses in the last stages of steam turbines. The theory of steam nucleation in convergingediverging nozzles has been studied for several decades, and originally was focused on one-dimensional flow analysis in Laval nozzles, because of the well suited geometry for experimental purposes. Moore et al. used light scattering method to experimentally obtain the pressure distribution along the centerline of convergentedivergent nozzles [4]. Furthermore, the order of magnitude of the droplets diameter was reported in their study. In order to establish a reasonable numerical model, Jackson et al. [5] developed a general equation set for multidimensional, time variant, inviscid flow of a condensing vapor which is capable of predicting the effects of relative motion between the primary gas phase and the suspended liquid droplets. The complete set of conservation equations for gas-droplet
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multiphase flow was further introduced by Young [6]. These governing equations were frequently used in recent studies of wet steam flow. To assist in this objective a number of researchers developed wet steam theories to be used in understanding this phenomenon. The nucleation rate and droplets growth were investigated thoroughly for both steady and unsteady conditions [7,8]. Besides, two-dimensional calculations were developed for considering the real flow behavior in turbine cascades [9e11]. In order to explicitly track the particle motion, the Lagrangian approach was established based on inviscid time marching schemes. The EulerianeLagrangian approach was further introduced, whereby conservation equations for the mixture flow were solved in the Eulerian model, and the Lagrangian approach was used for calculating the specific properties of liquid droplets [12,13]. Compared with the Eulerian approach [14], this method had some advantages which were described in detail by White et al. [15]. Thermo-compressors are prevalently studied as a conventional supersonic ejector. The early ejector studies conducted by Keenan (1946), Fabri (1958), Taylor (1969), and Dutton (1976) were focused on the development of ejector design techniques [16e19]. Further studies developed design methods based on fundamental relations of gas dynamics under the assumptions of isentropic processes, inviscid flow and one-dimensional modeling [20e22]. More recently, computational fluid dynamics (CFD) has been extensively used for capturing the internal phenomena of ejectors [23e27]. However, these studies have generally used air [28e31] or other gases (for example, refrigerants [32e35]) as working fluid and a few of them focused on steam ejectors. Moreover, the ideal gas assumption for the vapor equation of state was used in the latter studies due to its simplicity [36e39]. Furthermore, some modifications were implemented based on the ideal steam hypothesis in order to improve the geometry of thermo-compressors through using the CFD methods [40e43]. The review of the previous works showed that numerous studies were conducted on numerical simulations of flow inside steam ejectors and thermo-compressors. However, according to the authors’ knowledge, all the above studies considered the flow as a dry gas and there is no specific study of flow behavior under the assumption of wet steam condition. The present investigation is devoted to study the effects of wet steam phenomenon on the fluid flow and heat transfer within thermo-compressors. In the current study, a numerical method of predicting flow properties through using wet steam theory is introduced. The assumption of homogeneously nucleating steam is firstly validated through using the experimental data from Moore et al. [4] and then applied to the computational domain of the studied thermocompressor. The influence of wetness conditions on the supersonic region of the internal flow is scrutinized and compared with that of the ideal steam assumption. Finally, the effects of homogenous steam condensation on the main characteristics of the thermo-compressor are investigated. From this point of view, the deviation of overall performance of the thermo-compressor from the ideal assumption is presented.
supersonic flow at the downstream of the nozzle exit plane. The low pressure region created locally is able to entrain the surrounding flow and drive it in the same direction. This entrained stream is often called “secondary flow” in contrast with “primary flow” which is related to the motive steam. Flow passing through the converging duct is choked in the constant area section (i.e. the thermo-compressor throat). The high pressure region at the downstream of diffuser and decelerating of mixed stream through the flow pathway, cause a normal shock which is often taken place in the constant area section. After the shock occurrence, the pressure of mixed flow is raised up and exited from the diffuser with higher pressure. Two characteristic parameters can be defined for the performance of a typical thermo-compressor. “entrainment ratio” (ER) is defined as the ratio of secondary mass flow rate to the primary one _ mot , while “compression ratio” (CR) is defined as the ratio of _ suc =m m discharge pressure to the suction pressure pdis/psuc. 2.2. Two-phase flow consideration It is obvious when the steam pressure in the mixing chamber is suddenly reduced, the corresponding temperature is decreased and some droplets might be created due to sub-cooling effect. The droplets appeared in the vapor are very small particles (103e 104 mm) and supposed to be spheres which are growing several orders of magnitude in a very short period of time. This concept may not be covered just by simple assumption of steam as a perfect gas, because this hypothesis has no capability to predict the real gas properties in the temperature regions lower than saturation level. Therefore, the assumption of perfect gas equation of state (i.e. P ¼ rRT) may not lead to proper results for such applications. 2.3. Wet steam theory The flowing steam which undergoes expansion through the nozzle causes the formation of the minute embryos of liquid droplets in the vapor which are called “nuclei” and this phenomenon is called “nucleation”. The distinct media which exist in the entire flow field consists of water vapor (gas phase), water droplets (liquid phase) and the mixture of two phases. Hence, in the following expressions, the subscripts g and l indicate the vapor and liquid phase properties, respectively. Moreover, parameters with no subscript indicate the mixture properties. The classic theory of nucleation which was proposed by VolmereFrenkeleZeldovich is used herein to calculate the number of liquid particles [44,45].
Jclassic ¼
A conventional thermo-compressor consists of four distinct parts as it can be observed in Fig. 1: Primary nozzle, mixing zone, constant area zone and the diffuser. Motive steam is externally provided for the thermocompressor and passes through a convergentedivergent nozzle (primary nozzle) for the purpose of accelerating to supersonic level. This phenomenon causes a major pressure drop due to high
!0:5
!
2 s r2g 4 p rcrit exp rl 3KB T
! (1)
where qc is the condensation coefficient which is generally taken as unity, KB is the Boltzmann constant, and h is a non-isothermal correction factor which is given by the following relation [46]:
2. Model description 2.1. Thermo-compressor identification
qc 2s 1 þ h pM 3 molc
h¼2
g1 gþ1
"
hlg RT
hlg 1 RT 2
# (2)
where hlg is the equilibrium latent heat, g is the ratio of specific heat capacities and R is the gas constant. The critical radius of nucleation is determined by the KelvineHelmholtz formula in the following form [9]:
rcrit ¼
2s rl RTlnðSÞ
(3)
N. Sharifi et al. / Energy 47 (2012) 609e619
611
Fig. 1. Schematic of a conventional thermo-compressor with different flows and geometrical zones.
It is supposed that the droplets formation happens only at this critical size for the tiny sphere diameter. S denotes the supersaturation ratio which is defined as:
p S ¼ psat
(4)
where psat is the saturation pressure at the corresponding temperature of the mixture. According to modified form of Gyamarthy’s formula, the growth rate of droplets is given by [9]:
G ¼
kg DTð1 rcrit =rÞ dr ¼ dt rl hg hl r þ 1:89ð1 nÞlg =Prg
(5)
The mean free path of vapor molecules lg is expressed as:
lg ¼
pffiffiffiffiffiffiffiffiffi 3mg R T 2p
(6)
n is a semi-empirical correction factor introduced to obtain precise agreements with experimental data [47]: " !# g þ 1 RTsat ðpÞ RTsat ðpÞ 1 2 qc n ¼ z g1 4qc hlg 2 hlg
(7)
The temperature difference required in the growth rate equation can be determined by the capillarity effects pertaining to small droplets [48]. It should be noticed, the droplet temperature Tl is slightly lower than saturation temperature at corresponding pressure Tsat(p), due to subcooling phenomenon occurred in the mixture [49].
Tl ¼ Tsat ðpÞ Tsub
rcrit r
(8)
The subcooling effect resulting from non-equilibrium condition of the mixture is defined as:
Tsub ¼ Tsat ðpÞ Tg
(9)
3. Numerical scheme 3.1. Governing equations The small droplets could be assumed to follow vapor-phase streamlines with no slip velocity. Hence, the conservation equations can be applied to the mixture of the vapor and droplets as a single-phase media. The main governing equations are the conservation equations of mass, momentum, and energy for the
mixture, which are then combined with the equations describing super-cooling condensation and droplets creation.
v r v ðr ui Þ ¼ 0 þ vt v xi
(10)
v ðr ui Þ v v p v sij r ui uj ¼ þ þ vt v xj v xi v xj
(11)
vðrEÞ v vp v vT v keff þ ðrui E þ ui pÞ ¼ us þ þ vt vxi vt vxi vxi vxi i ij
(12)
where r is the mixture density, ui is the mixture velocity and E ¼ e þ uiui/2 is the total specific energy of the mixture. Two additional conservation equations are needed to adequately model the liquid phase [50]. These two equations should be capable of representing the conservation of liquid phase and variations of droplets number during the condensation and evaporation processes. If N denotes the number of droplets per unit volume, the transport equation which models the variations of droplets number throughout the nucleation process is written as:
vðrNÞ v ðrui NÞ ¼ rJclassic þ vt vxi
(13)
In the case of the wet steam theory the total mass of droplets affects the overall mixture properties. This parameter is described as “wetness fraction” b and defined as the mass of liquid phase per unit mass of the mixture. Hence, the second transport equation that governs the mass fraction of liquid phase is expressed as:
vðrbÞ v ðrui bÞ ¼ U þ vt vxi
(14)
Total mass generation rate of the liquid phase U is composed from two separate terms due to condensation and growth rate of the liquid droplets:
U ¼ U1 þ U2
(15)
where U1 is the mass generation rate due to nucleation and U2 is the mass generation rate due to growth rate of droplets. These terms are expressed as:
U1 ¼
4 3 pr r h 3 crit l
U2 ¼ 4pr2 rl N
dr dt
(16)
(17)
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In order to estimate the required droplet diameter as a function of the mass of a single droplet (4prlr3/3 ¼ md), the values of b and N are used based on the following expression:
The reliable range of utilizing this relation is from 0.01 to 100 bar (for pressure) and 273 to 1000 K (for temperature). These ranges are coincided to the common ranges of pressure and temperature applied to the thermo-compressor boundaries. The required vapor density rg can be evaluated from the mixture density r as:
produce a system of algebraic equations which were solved based on the second order upwind method. A computational fluid dynamics solver, which could be linked to a supplementary homewritten code, was used to calculate wet steam related variables. The calculation procedure of this code is described as below. When mixture properties are needed, the main governing equations (10)e(12) are solved while the governing equations (13) and (14) are used to estimate the independent parameters required for evaluating the liquid droplets properties. At the beginning of each of the iterations, the required values of pressure, density, velocity and temperature are evaluated using old time step values. The vapor phase density is estimated from equation (20) and substituted into equation (19) to calculate real pressure of vapor phase. This value can be used to estimate the real value of saturation temperature Tsat(p) required in equations (8) and (9) through using equation (21). On the other hand, psat is also calculated at known temperature from equation (21). This value can be used to evaluate the supersaturation ratio (S) which is leading to determination of the values of critical radius (rcrit), nucleation rate (Jclassic) and total mass generation rate (U). Afterwards, the additional transport equations (13) and (14) can be solved to evaluate the independent variables of the wet steam model b and N. These values are further used to estimate the droplet parameters in the next iteration.
rg ¼ ð1 bÞr
4. Validation of numerical scheme
md ¼
rg N
b
(18)
1b
3.2. Equation of state The virial form of equations of state is more suitable for the purpose of computational usage. The virial equation of state for steam which was proposed by Vukalovich [51] is used herein. The validity of this equation was tested for extrapolation into subcooled regions [52].
p ¼ rg RT 1 þ B1 rg þ B2 r2g þ B3 r3g
(19)
(20)
In order to setup a complete procedure to anticipate the liquid and vapor phases properties, it is necessary to define the saturation pressure by an empirical relation. The equation proposed by Keenan and Keyes [53] is used to predict the saturation pressure as a function of saturation temperature, precisely:
" # 8 psat T Ts X T 273:15 i1 ai 0:65 s ¼ exp 0:01 crit pcrit Ts 100 i¼1
(21)
where Tcrit and pcrit are critical temperature (647.28 K) and pressure (22.087 MPa) of water, respectively and ai coefficients are constant values. 3.3. Turbulence modeling A two-variable turbulence model was applied to the whole domain of flow based on the keε turbulence hypothesis to simulate the turbulent characteristics of flow inside the thermo-compressor. The main governing equations for realizable keε model are expressed as below:
v v v ðrkui Þ ¼ ðrkÞ þ vt vxi vxj v v v ðrεui Þ ¼ ðrεÞ þ vt vxi vxj
"
#
m þ mt=s
k
"
m þ mt=s
ε
þ
vk þ Srck vxj
# vε þ þ Srcε vxj
(22)
(23)
The main reason of selecting realizable approach is the superiority of this model to standard one for predicting the behavior of the round jet flows in ejectors [36]. 3.4. Numerical method A control-volume-based technique employing a finite-volume discretization process according to density-based approach was utilized for the numerical simulations. The non-linear partial differential governing equations were implicitly linearized to
The formulation described in the last section was applied to the supersonic flow in the computational domain of the nozzle described by Moore et al. [4]. Since the general approach for verifying the wet steam results relies on the experimental reports of pressure distribution in the nozzles, in the current study quantitative validation of the numerical model was accomplished by using the experimental data reported by Moore et al. [4]. A convergentedivergent nozzle was designed based on the geometries adopted from Moore et al. report and the following conditions have been considered for inflow and outflow boundaries: Inflow: total pressure of 25 kPa, total temperature of 358 K, steam with no droplet containing. Outflow: low enough static pressure to adjust a supersonic outflow condition. Moreover, following assumptions were made for the numerical simulation: 1. The influent direction was considered to be normal to the inlet plane. 2. Symmetry conditions were enforced for all flow variables along the centerline. 3. No shock wave occurred within the diffuser part of the nozzle. 4. The adiabatic wall with no-slip condition was used. 5. Entering flow was considered to dry steam. There are two major parameters to be compared with the experimental measurements: First, the pressure distribution along the nozzle centerline, and second, the averaged droplet diameter near the exit plane of the nozzle. The variations of these parameters are represented in Fig. 2. As observed, the predicted static pressure along the nozzle centerline is in a reasonable agreement with the profile of experimental pressure distribution. Furthermore, a good compatibility with the experimental location of condensation shock was achieved. The size of generated droplets near the exit plane of the nozzle is well-predicted through the numerical procedure in comparison to
N. Sharifi et al. / Energy 47 (2012) 609e619 0.1
P / Ptot
0.6 0.06 0.5 0.04 0.4
0.2 -0.1
0.02
Condensation Shock Zone
0.3
0
0.1
0.2
0.3
0.08 1.5
0.06 1
0.04
0.5
0 -0.1
0 0.4
Nucleation Piont
Wetness Fraction
0.08
0.1 Wetness Fraction Wet Steam Mach Number Ideal Steam Mach Number
Mach Number
Numerical Pressure Ratio Experimental Pressure Ratio Numerical Droplet Diameter Experimental Droplet Diameter
0.7
2
Droplet Diameter (micron)
0.8
613
0.02
0
0.1
0.2
0.3
0 0.4
Axial Distance from the throat (m)
Axial distance from the throat (m)
Fig. 2. Comparisons between the simulation and experimental results in the nozzle: (Left) the distribution of pressure ratio on the nozzle centerline, (Right) the averaged droplets diameter near the nozzle exit plane.
Fig. 4. Numerical results of the wet steam calculations along the centerline of the Moore et al. nozzle: (Left) Mach number of ideal and wet steam flows, (Right) wetness fraction.
30
40 Log 10 of Nucleation Rate (1/m s) Subcooled Vapor Level (K)
25
35 30
20 25 15
20
10
15 10
5 5 0
-5 -0.1
0
0
0.1
0.2
0.3
Subcooled Vapor Level (K)
Log 10 of Nucleation Rate (1/m3s)
the experiments. However, the discrepancy between the numerical and experimental droplet sizes was lower than 15% which is acceptable enough for engineering problems especially for this order of magnitude of the droplet diameter. To demonstrate the current model is capable enough to predict some major phenomena resulting from the homogeneous nucleation, some further results have been provided. These results might not be experimentally validated but could be verified theoretically according to the main assumptions of the wet steam theory. Fig. 3 shows the variations of subcooling level and log 10 of nucleation rate, in order to compare the trends of both phenomena simultaneously. It is obvious that the vapor phase starts to condense (nucleation starting point) after a significant decline in the droplets temperature (positive subcooling value). As expected, the peak value of nucleation rate is reached at the same axial location of maximum subcooling level. The nucleation event causes some droplets to be generated and hence, the mass fraction of liquid phase is increased after the nucleation point. This phenomenon can be thoroughly understood from Fig. 4. The blue dashed line (in web version) in this illustration represents the variations of wetness fraction along the nozzle centerline. Increasing the wetness fraction after the nucleation point reveals that the production of liquid droplets is preceded by
-5 0.4
Axial distance from the throat (m) Fig. 3. Numerical results of the wet steam calculations along the centerline of the Moore et al. nozzle: (Left) nucleation rate, (Right) sub-cooling temperature level.
nucleation event. It is also obvious that the wetness fraction is smoothly increased toward the nozzle exit due to reducing the static temperature of the vapor. The additional effect which confirms the correct numerical results is the flow Mach number along the centerline of the nozzle which is depicted on the left vertical axis of Fig. 4. As expected, the trend of Mach number for the wet steam flow is lower than the corresponding value for the ideal steam after nucleation point, because the liquid droplets content cause to decline the vapor velocity and thus, the local Mach number is decreased through the diverging part of the nozzle. The reasonable agreements of the computational results with experiments prove the ability of this numerical model to provide valuable information on the flow pattern inside thermocompressors. 5. Geometry and boundary conditions Fig. 5 shows the configuration of the thermo-compressor (all sizes are in centimeters) with close-up mesh elements near the primary nozzle and around it. The entire computational domain is a region ranging from the inlet of the primary nozzle to the end of the diffuser, including the nozzle. The size of the computational domain is 14.0 m in the axial direction, and 0.91 m in radial direction. The test for grid-independence was performed in order to calculate the deviation of net mass flux before any further analysis. The grid was created based on the structured quadrilateral elements. The total cell number was firstly 35,100. Afterwards, different cell configurations were used in the test with the total cell numbers of 45,800 and 67,400. The results showed that the deviations for the net mass flow rate were respectively 0.51% and 0.43%. The boundaries of the computational domain were considered adiabatic walls (for the nozzle and the shell), two inlets (for motive and suction flow) and a single outlet (for discharge flow). The whole computational domain was assumed a single-phase media undergoing the boundary conditions defined in Table 1. These values were kept constant for both ideal and wet steam simulations. The working fluid considered at the boundaries was steam. At both inlet boundaries the total pressure and total temperature are specified. As observed in Table 1, the motive steam at mentioned condition is a superheated vapor and thus, contains no liquid droplet contents. Other unknown flow parameters are
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Fig. 5. Computational grid elements and dimensions of the thermo-compressor.
calculated from the interior by second-order extrapolation. At the exit plane (i.e. discharge flow), if the flow is subsonic, only the static pressure is required, whereas the total pressure, total temperature, and mass flow rate are extrapolated from the interior by secondorder extrapolation. If the exit flow is supersonic, all flow variables are extrapolated from the interior. 6. Results and discussion In this section, the numerical scheme derived for wet steam simulation was applied to the computational domain of the thermo-compressor. The general CFD method has been conducted with two different hypotheses: wet steam and ideal steam, in order to compare the flow properties along the centerline of the thermocompressor and to observe the significant discrepancies on the results of these two models.
temperature distribution for wet and ideal simulations with an additional horizontal line referring to the minimum limit of allowable temperature for liquid and gaseous phases of water. As Fig. 6 indicates, the minimum values of temperature in ideal steam simulation has been dropped beneath the freezing line (melting point), because the equation of state for a perfect gas has no sense of temperature decline lower than freezing limit and always generates a numerical value for the temperature corresponding to the pressure and density of the gas, regardless of any saturation effect. But, as expected, the minimum value of wet simulation curve has not dropped below the freezing line and whole range of static temperature located over the limit of 273.15 K. It can also be noticed that the flow temperature at the exit boundary is lower for wet steam compared to that for ideal steam. This effect can be explained by the latent heat capacity of water droplets that can absorb the released heat of temperature rising after the shock wave in the constant area section.
6.1. Flow temperature ideal steam wet steam
450 400 350
Tstatic (K)
As a matter of fact in the wet steam simulation the static temperature of fluid must not be lower than minimum range of achievable temperature for liquid or gaseous phase of water (i.e. 273.15 K). Therefore, an appropriate criterion to assess a correct flow simulation is the temperature distribution in the flow field. Since the minimum range of static temperature always occurs on the intersection point of oblique shock waves located on the axial centerline of the thermo-compressor, it is necessary to plot the temperature variations on the axial direction. Fig. 6 shows the
Freezing line 273.15 K
300 250 200
Table 1 Boundary conditions for different boundaries of the thermo-compressor.
Motive flow Suction flow Discharge flow
Temperature (K)
Pressure (kPa)
470 322 e
1000 10 26.3
150 100 0
2
4
6
8
10
12
14
Axial Distance (m) Fig. 6. Comparison between the static temperature of the wet steam and the ideal steam along the axial direction of the thermo-compressor.
N. Sharifi et al. / Energy 47 (2012) 609e619
6.2. Flow pressure
615
5
ideal steam wet steam
4.5
6.3. Flow Mach number The effect of containing infinitesimal droplets in the steam causes a decline in absolute averaged velocity of the flow. On the other hand, the temperature rise in the case of wet steam leads to pffiffiffiffiffiffiffiffiffiffi an increase in local speed of sound (i.e. g RT ). Hence, the flow Mach number is decreased significantly. According to Fig. 8, the values of local Mach number show that the maximum value of Mach number has been reduced from 4.9 (for ideal steam) to 3.6 (for wet steam). As observed in Fig. 8, the exact values of flow velocity at the outlet boundary are equal for both models. The direct result of this
100
ideal steam wet steam
Pstatic (kPa)
80
60
20
0
2
4
6
Shock Position
3 2.5 2 1.5 1 0.5 0
0
2
4
6
8
10
12
14
Axial Distance (m) Fig. 8. Comparison between the Mach number of the wet steam and the ideal steam along the axial direction of the thermo-compressor.
fact could be expressed as: “At the constant velocity and the fixed outlet area (i.e. A ¼ Const., V ¼ Const.), the mass flow rate of _ ¼ r AV), will be significantly increased in delivered steam (i.e. m the wet conditions”. At the downstream of the shock waves, the local Mach numbers for the wet and ideal steam models have been separated to different values which are dependent on both the downstream velocities and static temperatures. It is of interest to note that even though the local Mach number is greater than unity and flow is supersonic, the shock wave has been occurred (according to pressure rise location in Fig. 7) and the flow after the shock is still remaining supersonic. A very practical comparison between wet and ideal results may be related to the formation of supersonic jet core within the TVC. As illustrated in Fig. 9, the shape of exiting jet in the wet simulation is a little bit wider than the ideal jet core. Moreover, the angle of exiting supersonic flow right at the nozzle tip position is larger in wet conditions. This phenomenon implies a more realistic diamond wave pattern of supersonic flow for wet steam conditions. It reveals the presence of weak oblique shock waves rather than a strong normal shock in the throat of the thermo-compressor. Therefore, the hypothesis of normal shock occurrence in the constant area zone that has been mentioned in literature may not become a general assumption. Moreover, the axial position of the ideal steam shock wave inside the constant area zone is prior to wet steam shock wave. This observation gives rise to a practical result which means that the wet steam simulation could undergo a larger pressure at downstream, and hence results in a higher compression ratio. This fact may predict a better operational mode for the thermo-compressor with a higher safety factor. 6.4. Other parameters related to wet steam theory
Shock position
40
0
4 3.5
Mach No.
The major purpose of pressure study is to detect the shock position in the thermo-compressor. Static pressure distribution along the centerline of the thermo-compressor shows an abrupt jump in pressure fluctuations that can be recognized as shock position. When the nozzle is under-expanded, the supersonic flow outgoing from the primary nozzle forms a combination of oblique shocks and expansion waves called “diamond wave” pattern or “shock train” [54]. The direct effect of such phenomenon on the static pressure is the formation of some pressure fluctuations along the centerline which can be observed in Fig. 7. As indicated, the abrupt changes in pressure rise are weakened when the supersonic jet core contains some wetness. Therefore, the pressure loss due to passing through oblique shock waves is decreased. This phenomenon can be related to lower temperature fluctuations in wet simulation that can be observed in Fig. 6. Furthermore, shock position moves somewhat downstream, toward the exit boundary, and pressure recovery is strengthened in the case of wet simulation. The other important consequence of this discussion can be related to the flow density. As a matter of fact, the equal pressure at the exit boundary reveals a little bit greater values of downstream density for the wet steam compared to the ideal steam, because of higher temperature values for ideal steam simulation at this region. Furthermore, on account of higher density and higher static pressure of wet steam at the outlet of the thermo-compressor, the increased fluid inertia can strongly overcome the downstream back pressure. This means that the compression ratio might slightly increase in the wet steam conditions.
8
10
12
14
Axial Distance (m) Fig. 7. Comparison between the static pressure of the wet steam and the ideal steam along the axial direction of the thermo-compressor.
The variations of some important parameters which are normally discussed in the wet steam theory (such as nucleation rate, subcooling level, wetness fraction, and droplet size) have been shown previously for the flow inside a single nozzle. It is of interest to investigate these variables in the computational domain of the thermo-compressor. Fig. 10(a) shows the rate of nucleation phenomenon along the centerline of the TVC. It can be observed the nucleation peak has taken place in the primary nozzle and suddenly disappeared after reaching to its maximum limit. Therefore, the downstream flow no longer undergoes nucleation phenomenon and the rate of
616
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Fig. 9. Comparison between the supersonic jet core in the thermo-compressor for the wet and ideal steam conditions.
nucleation is terminated abruptly in remaining part of the TVC. This single occurrence of nucleation phenomenon provides a high wetness fraction distribution along the centerline of the TVC which can be observed in Fig. 10(b). As it is apparent, the wetness fraction is increased to a maximum level which corresponds to the point of exiting flow
from the nozzle. Afterwards, the fluctuations of wetness fraction are continued with a little amplitude along the TVC centerline prior to reach the shock wave position. After the shock, the wetness fraction is lowered gradually due to temperature raise caused by the shock wave. The large portion of wetness decreasing is taken place in the diffuser part which has an additional effect on evaporation of liquid droplets. The pressure rise along the diffuser part increases the saturation temperature of the liquid phase and thus, has a reverse effect on the evaporation of the water droplets. The variations of Tsat along the centerline of the TVC which is shown in Fig. 11(a), give further confirmation to this supposition. As a matter of fact, the variations of saturation temperature are originated from the pressure fluctuations in the centerline direction. Moreover, the higher level of Tsat indicates the larger amount of heat needed to evaporate the water droplets at a given temperature. By considering the six dominant points in Fig. 11, it can be seen that the maximum subcooling level (point “A”) is taken place at the maximum saturation temperature related to the maximum flow velocity. Points “B”, “C” and “D” are pertaining to the shock train fluctuation regions, while point “E” shows the exact position of the main shock wave with a negative value of subcooling temperature (i.e. the steam is locally superheated after the shock wave occurrence). The “EF” segment shows the droplets pathway through the diffuser with a little raise in Tsat while the negative subcooling value (i.e. superheating level) increases significantly. The major reason of this phenomenon can be related to the subsonic flow inside the diffuser which undergoes an abrupt pressure and temperature rise after the shock wave. The incremental change of saturation temperature (Tsat) is basically related to the pressure raise within the diffuser. It is expected to observe a descending trend for the absolute value of Tsub. But, the increasing rate of steam temperature (Tg) inside the diverging section of diffuser overcomes the increasing trend of saturation temperature (Tsat). Hence, the outgoing steam is negatively sub-cooled (i.e. superheated). 6.5. Characteristic parameters
Fig. 10. Variations of (a) the nucleation rate and (b) the wetness fraction along the centerline of the thermo-compressor.
By comparing the characteristics of the thermo-compressor with a fixed geometry under both wet and ideal steam models, the superiority of the wet steam model can be recognized and the discrepancies of dominant flow parameters may be clarified. The resulted mass flow rates for different boundaries of the thermo-compressor are depicted in Table 2, for both wet and ideal
N. Sharifi et al. / Energy 47 (2012) 609e619
617
ideal steam modeling. This is the second advantage of wet simulation which can be taken into account. Since the main goal of thermo-compressor utilization is to provide a reliable compression ratio, this result can be very useful for practical design purposes. The rate of increasing in CR was around %14.1 under specified boundary conditions. Therefore, by considering the dominant influence of droplet presence in the supersonic flow exiting the primary nozzle, both characteristic parameters of the TVC will be remarkably improved. 7. Concluding remarks In the current study, a new approach for investigating the internal flow of thermo-compressors was introduced based on the fundamental wet steam theory. The main idea of this work was to consider a homogenous steam condensation in the supersonic steam flow exiting from the convergentedivergent nozzle.
Fig. 11. Variations of (a) the saturation temperature and (b) the subcooling temperature along the centerline of the thermo-compressor.
steam modeling. As can be seen from the data shown in the table, the entrainment ratio is higher for the wet steam in comparison to the ideal steam simulation. This conclusion has the greatest importance from the viewpoint of operational conditions. Because, the higher amount of ER leads to the better TVC performance that implies a maximized capability of flow entraining with a fixed geometry thermo-compressor. In order to compare the CR of the two different models, the TVC discharge pressure was increased gradually up to the point where an abrupt drop in the rate of the suction flow is observed. This point can be considered as the maximum critical pressure of the device. The highest values of discharge pressure for wet and ideal simulations were recorded as 29.1 kPa and 27.3 kPa, respectively. Since the suction pressure remained constant during the pressure rise at the outlet boundary, it can be obviously concluded the overall compression ratio of the thermo-compressor is predicted higher in wet steam modeling than the corresponding value for
Table 2 Numerical values of the thermo-compressor characteristic parameters in different simulation methods.
Entrainment ratio Compression ratio
Wet steam
Ideal steam
0.89 2.96
0.78 2.73
A numerical method for calculating the thermodynamic properties of thermo-compressors internal flow in the context of the homogenous nucleation was developed. According to the authors’ knowledge, no previous study was implemented for evaluating the flow properties in thermocompressors under the assumption of wet steam theory. The present model has the potential for efficiently modeling the internal phenomena of thermo-compressors where droplets are generated under non-equilibrium conditions. Reasonable agreement of numerical scheme with experimental reports was obtained, especially for static pressure distribution and condensation shock position in the nozzle. The axial position of the shock wave inside the throat shifted to the discharge plane in the wet steam model that shows an extended margin for normal operation of the device. There is a further advancement in operating performance of thermo-compressors, since the entrainment ratio shows a substantial increasing under the wet steam assumption. This means that for a thermo-compressor with a fixed geometry, the capability of entraining the suction flow becomes higher for the wet steam case over the ideal steam. The numerical results show that the compression ratio of thermo-compressors is improved under the assumption of the wet steam. This fact evinces a conventional thermocompressor can undergo larger discharge pressures and produce higher compression ratios. Nomenclature
English letters B virial coefficients E total enthalpy G growth rate of droplets h static enthalpy J nucleation rate k thermal conductivity Kn Knudsen number Boltzmann’s constant KB l mean free path of vapor molecules m mass mass of one molecule of water Mmolc n total number of droplets N number of droplets per unit volume p static pressure Pr Prandtl number condensation coefficient qc
618
r rcrit R S Src t T u V x
N. Sharifi et al. / Energy 47 (2012) 609e619
droplet radius critical droplet radius gas constant super-saturation source term time static temperature velocity components mixture volume spatial dimension
Greek letters a constant coefficient for saturation curve b wetness fraction g specific heat ratio r density ε turbulent dissipation rate k turbulent kinetic energy m dynamic viscosity s liquid surface tension s stress tensor h non-isothermal correction factor n semi-empirical correction z empirical parameter U mass generation rate dij Kronecker delta function Subscripts crit critical properties d droplet dis discharge flow g vapor (gas) phase l liquid phase i, j tensor notations mot motive flow sat saturation properties sub subcooled properties suc suction flow t turbulent References [1] Adel M, Dayem A. Experimental and numerical performance of a multi-effect condensationeevaporation solar water distillation system. Energy 2006;31: 2710e27. [2] Ansari K, Sayyaadi H, Amidpour M. Thermoeconomic optimization of a hybrid pressurized water reactor (PWR) power plant coupled to a multi effect distillation desalination system with thermo-vapor compressor (MEDeTVC). Energy 2010;35:1981e96. [3] Li Q, Piechna J, Müller N. Numerical simulation of novel axial impeller patterns to compress water vapor as refrigerant. Energy 2011;36:2773e81. [4] Moore MJ, Walters PT, Crane RI, Davidson BJ. Predicting the fog drop size in wet steam turbines. In: Wet steam 4 conference. Inst. of Mechanical Engineers (UK), University of Warwick; 1973. C37/73. [5] Jackson R, Davidson BJ. An equation set for non-equilibrium two phase flow and an analysis of some aspects of choking, acoustic propagation and losses in low pressure wet steam. Int J Multiphas Flow 1983;9(5):491e510. [6] Young JB. The fundamental equations of gas-droplet multiphase flow. Int J Multiphas Flow 1995;21(2):175e91. [7] Bakhtar F, So KS. A study of nucleating flow of steam in a cascade of supersonic blading by the time-marching method. Int J Heat Fluid Flow 1991;12: 54e62. [8] White AJ, Young JB. Time marching method for the prediction of twodimensional unsteady flows of condensing steam. AIAA J Propuls Power 1993;9(4):579e87. [9] Young JB. Two-dimensional non-equilibrium wet-steam calculations for nozzles and turbine cascades. Trans ASME J Turbo 1992;114:569e78. [10] Bakhtar F, Mahpeykar MR, Abbas KK. An investigation of nucleating flows of steam in a cascade of turbine blading e theoretical treatment. ASME J Fluids Eng 1995;117:138e44.
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[18] Dutton JC, Mikkelsen CD, Addy AL. Theoretical and experimental analysis of the constant area supersonicesupersonic ejector. AIAA J 1976;20(10):1392e400. [19] Taylor D. Ejector design for a variety of application. AGARDograph; 1969. #163. [20] Christensen H. Application of gas-dynamic functions for steam ejector design. Heat Transf Eng 1983;4:83e105. [21] Dutton JC, Carroll BF. Optimal supersonic ejector designs. J Fluids Eng 1986; 108:414e20. [22] Huang BJ, Chang JM, Wang CP, Petrenko VA. A 1-D analysis of ejector performance. Int J Refrig 1999;22:354e64. [23] Ji MK, Utomob T, Woo JS, Lee YH, Jeong HM, Chung HS. CFD investigation on the flow structure inside thermo vapor compressor. Energy 2010;35: 2694e702. [24] Bartosiewic Y, Aidoun Z, Mercadier Y. Numerical assessment of ejector operation. Appl Therm Eng 2006;26:604e12. [25] Pianthong K, Seehanam W, Behnia M, Sriveerakul T, Aphornratana S. Investigation and improvement of ejector refrigeration system using computational fluid dynamics technique. Energ Convers Manag 2007;48:2556e64. [26] Park BH, Lim JH, Yoon W. Fluid dynamics in starting and terminating transients of zero-secondary flow ejector. Int J Heat Fluid Flow 2008;29:327e39. [27] Bartosiewicz Y, Aidoun Z, Desevaux P, Mercadier Y. Numerical and experimental investigations on supersonic ejectors. Int J Heat Fluid Flow 2005;26: 56e70. [28] Matsuo K, Sasaguchi K, Tasaki K, Mochizuki H. Investigation of supersonic air ejectors, part 1: performance in the case of zero-secondary flow. Bull JSME 1981;24(198):2090e7. [29] Matsuo K, Sasaguchi K, Tasaki K, Mochizuki H. Investigation of supersonic air ejectors, part 2: effects of throat-area-ratio on ejector performance. Bull JSME 1982;25(210):1898e905. [30] Desevaux P, Aeschbacher O. Numerical and experimental flow visualization of the mixing process inside an induced air ejector. Int J Turbo Jet Eng 2002;19:71e8. [31] Kaneko M, Men’shov I, Nakamura Y. Numerical simulation of non-equilibrium flow in high-enthalpy shock tunnel. Energy 2005;30:165e79. [32] Sun DW. Comparative study of the performance of an ejector refrigeration cycle operating with various refrigerants. Energ Convers Manag 1999;40: 873e84. [33] Yapıcı R, Ersoy HK. Performance characteristics of the ejector refrigeration system based on the constant area ejector flow model. Energ Convers Manag 2005;46:3117e35. [34] Korres CJ, Papaioannou AT, Lygerou V, Koumoutsos NG. Solar cooling by thermal compression: the dependence of the jet thermal compressor efficiency on the compression ratio. Energy 2002;27:795e805. [35] Sankarlal T, Mani A. Experimental studies on an ammonia ejector refrigeration system. Int Commun Heat Mass Transf 2006;33:224e30. [36] Sriveerakul T, Aphornratana S, Chunnanond K. Performance prediction of steam ejector using computational fluid dynamics: part 1. Validation of CFD results. Int J Therm Sci 2007;46(8):812e22. [37] Sriveerakul T, Aphornratana S, Chunnanond K. Performance prediction of steam ejector using computational fluid dynamics: part 2. Flow structure of a steam ejector influenced by operating pressures and geometries. Int J Therm Sci 2007;46(8):823e33. [38] Hemidi A, Henry F, Leclaire S, Seynhaeve JM, Bartosiewicz Y. CFD analysis of a supersonic air ejector. Part I: experimental validation of single-phase and two-phase operation. Appl Therm Eng 2009;29:1523e31. [39] Deberne N, Leone JF, Duque A, Lallemand A. A model for calculation of steam injector performance. Int J Multiphas Flow 1999;25:841e55. [40] Park IS, Park SM, Ha JS. Design and application of thermal vapor compressor for multi-effect desalination plant. Desalination 2005;182:199e208. [41] Park IS. Robust numerical analysis based design of the thermal vapor compressor shape parameters for multi-effect desalination plants. Desalination 2009;242:245e55. [42] Park IS. 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Contents lists available at SciVerse ScienceDirect
Energy journal homepage: www.elsevier.com/locate/energy
Wet steam flow energy analysis within thermo-compressors Navid Sharifi a, Masoud Boroomand a, *, Ramin Kouhikamali b a b
Department of Aerospace Engineering, Amirkabir University of Technology, Tehran, Iran Department of Mechanical Engineering, Faculty of engineering, University of Guilan, Rasht, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 12 July 2012 Received in revised form 31 August 2012 Accepted 1 September 2012 Available online 4 October 2012
Thermo-compressors are widely used in industries for steam compression through a thermal process. Common methods of thermo-compressors analysis are based on the hypothesis of considering steam as a perfect gas. In this study, the deviation of thermo-compressor performance at wet steam conditions from the performance under the ideal gas assumption has been investigated. Firstly, a numerical method has been implemented to evaluate the formation of droplets due to condensation in a convergent edivergent nozzle. The results have been validated using existing experimental data for single nozzles. Afterwards, the verified numerical scheme has been applied to internal flow of the thermocompressor. The formation of droplets due to condensation effect and the resulting supersonic core in the thermo-compressor have been deeply investigated. Finally, the effect of wet steam assumption on the performance characteristics of thermo-compressors has been presented. Ó 2012 Elsevier Ltd. All rights reserved.
Keywords: Thermo-compressor Steam ejector Wet steam Performance parameters Energy analysis
1. Introduction Multi Effect Distillation (MED) systems could be improved by adding vapor compression devices. The vapor compression process could be performed mechanically or thermally. The desalination packages are known as MED-MVC (MED with mechanical vapor compression) or MED-TVC (MED with thermal vapor compression). The main advantages of the TVC usage are the reuse of the compressed vapor that reduces the required amount of motive steam, low capital and construction costs and the simplicity of the steam compressor. Some methods were presented to improve the performance and the efficiency of simple distillation units without thermo-compressor those are based on the multiple condensatione evaporation cycles [1]. However, thermo-compressors are one of the most important parts of desalination systems in which considerable quantities of energy loss is occurred and thus, needed to be designed carefully [2]. Moreover, the simplicity of this device, with no moving parts, gives a forward step compared against the mechanical vapor compressor. It was shown that a single stage mechanical compressor was not feasible for compressing water vapor because of high capital cost and technically challenging compressor design and development [3].
* Corresponding author. Tel.: þ98 21 64543280; fax: þ98 21 66959020. E-mail address: [email protected] (M. Boroomand). 0360-5442/$ e see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.energy.2012.09.003
Thermo-compressors compress large amounts of low-pressure steam through using the available pressurized steam as motive energy. The motive steam passed through a convergentedivergent nozzle of the thermo-compressor may be condensed at the exit section due to temperature decrease caused by supersonic conditions. The condensation phenomenon occurred in supersonic flow is studied deeply in “wet-steam” theory. Wet steam is typically considered as a multiphase mixture in which both vapor phase and liquid droplets exist. In this case, formation of liquid droplets due to pressure decline causes some complexities. Recent studies have been concentrated on the computation of such multiphase flows to determine the rate of droplets growth, condensation shock position, and energy losses in the last stages of steam turbines. The theory of steam nucleation in convergingediverging nozzles has been studied for several decades, and originally was focused on one-dimensional flow analysis in Laval nozzles, because of the well suited geometry for experimental purposes. Moore et al. used light scattering method to experimentally obtain the pressure distribution along the centerline of convergentedivergent nozzles [4]. Furthermore, the order of magnitude of the droplets diameter was reported in their study. In order to establish a reasonable numerical model, Jackson et al. [5] developed a general equation set for multidimensional, time variant, inviscid flow of a condensing vapor which is capable of predicting the effects of relative motion between the primary gas phase and the suspended liquid droplets. The complete set of conservation equations for gas-droplet
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multiphase flow was further introduced by Young [6]. These governing equations were frequently used in recent studies of wet steam flow. To assist in this objective a number of researchers developed wet steam theories to be used in understanding this phenomenon. The nucleation rate and droplets growth were investigated thoroughly for both steady and unsteady conditions [7,8]. Besides, two-dimensional calculations were developed for considering the real flow behavior in turbine cascades [9e11]. In order to explicitly track the particle motion, the Lagrangian approach was established based on inviscid time marching schemes. The EulerianeLagrangian approach was further introduced, whereby conservation equations for the mixture flow were solved in the Eulerian model, and the Lagrangian approach was used for calculating the specific properties of liquid droplets [12,13]. Compared with the Eulerian approach [14], this method had some advantages which were described in detail by White et al. [15]. Thermo-compressors are prevalently studied as a conventional supersonic ejector. The early ejector studies conducted by Keenan (1946), Fabri (1958), Taylor (1969), and Dutton (1976) were focused on the development of ejector design techniques [16e19]. Further studies developed design methods based on fundamental relations of gas dynamics under the assumptions of isentropic processes, inviscid flow and one-dimensional modeling [20e22]. More recently, computational fluid dynamics (CFD) has been extensively used for capturing the internal phenomena of ejectors [23e27]. However, these studies have generally used air [28e31] or other gases (for example, refrigerants [32e35]) as working fluid and a few of them focused on steam ejectors. Moreover, the ideal gas assumption for the vapor equation of state was used in the latter studies due to its simplicity [36e39]. Furthermore, some modifications were implemented based on the ideal steam hypothesis in order to improve the geometry of thermo-compressors through using the CFD methods [40e43]. The review of the previous works showed that numerous studies were conducted on numerical simulations of flow inside steam ejectors and thermo-compressors. However, according to the authors’ knowledge, all the above studies considered the flow as a dry gas and there is no specific study of flow behavior under the assumption of wet steam condition. The present investigation is devoted to study the effects of wet steam phenomenon on the fluid flow and heat transfer within thermo-compressors. In the current study, a numerical method of predicting flow properties through using wet steam theory is introduced. The assumption of homogeneously nucleating steam is firstly validated through using the experimental data from Moore et al. [4] and then applied to the computational domain of the studied thermocompressor. The influence of wetness conditions on the supersonic region of the internal flow is scrutinized and compared with that of the ideal steam assumption. Finally, the effects of homogenous steam condensation on the main characteristics of the thermo-compressor are investigated. From this point of view, the deviation of overall performance of the thermo-compressor from the ideal assumption is presented.
supersonic flow at the downstream of the nozzle exit plane. The low pressure region created locally is able to entrain the surrounding flow and drive it in the same direction. This entrained stream is often called “secondary flow” in contrast with “primary flow” which is related to the motive steam. Flow passing through the converging duct is choked in the constant area section (i.e. the thermo-compressor throat). The high pressure region at the downstream of diffuser and decelerating of mixed stream through the flow pathway, cause a normal shock which is often taken place in the constant area section. After the shock occurrence, the pressure of mixed flow is raised up and exited from the diffuser with higher pressure. Two characteristic parameters can be defined for the performance of a typical thermo-compressor. “entrainment ratio” (ER) is defined as the ratio of secondary mass flow rate to the primary one _ mot , while “compression ratio” (CR) is defined as the ratio of _ suc =m m discharge pressure to the suction pressure pdis/psuc. 2.2. Two-phase flow consideration It is obvious when the steam pressure in the mixing chamber is suddenly reduced, the corresponding temperature is decreased and some droplets might be created due to sub-cooling effect. The droplets appeared in the vapor are very small particles (103e 104 mm) and supposed to be spheres which are growing several orders of magnitude in a very short period of time. This concept may not be covered just by simple assumption of steam as a perfect gas, because this hypothesis has no capability to predict the real gas properties in the temperature regions lower than saturation level. Therefore, the assumption of perfect gas equation of state (i.e. P ¼ rRT) may not lead to proper results for such applications. 2.3. Wet steam theory The flowing steam which undergoes expansion through the nozzle causes the formation of the minute embryos of liquid droplets in the vapor which are called “nuclei” and this phenomenon is called “nucleation”. The distinct media which exist in the entire flow field consists of water vapor (gas phase), water droplets (liquid phase) and the mixture of two phases. Hence, in the following expressions, the subscripts g and l indicate the vapor and liquid phase properties, respectively. Moreover, parameters with no subscript indicate the mixture properties. The classic theory of nucleation which was proposed by VolmereFrenkeleZeldovich is used herein to calculate the number of liquid particles [44,45].
Jclassic ¼
A conventional thermo-compressor consists of four distinct parts as it can be observed in Fig. 1: Primary nozzle, mixing zone, constant area zone and the diffuser. Motive steam is externally provided for the thermocompressor and passes through a convergentedivergent nozzle (primary nozzle) for the purpose of accelerating to supersonic level. This phenomenon causes a major pressure drop due to high
!0:5
!
2 s r2g 4 p rcrit exp rl 3KB T
! (1)
where qc is the condensation coefficient which is generally taken as unity, KB is the Boltzmann constant, and h is a non-isothermal correction factor which is given by the following relation [46]:
2. Model description 2.1. Thermo-compressor identification
qc 2s 1 þ h pM 3 molc
h¼2
g1 gþ1
"
hlg RT
hlg 1 RT 2
# (2)
where hlg is the equilibrium latent heat, g is the ratio of specific heat capacities and R is the gas constant. The critical radius of nucleation is determined by the KelvineHelmholtz formula in the following form [9]:
rcrit ¼
2s rl RTlnðSÞ
(3)
N. Sharifi et al. / Energy 47 (2012) 609e619
611
Fig. 1. Schematic of a conventional thermo-compressor with different flows and geometrical zones.
It is supposed that the droplets formation happens only at this critical size for the tiny sphere diameter. S denotes the supersaturation ratio which is defined as:
p S ¼ psat
(4)
where psat is the saturation pressure at the corresponding temperature of the mixture. According to modified form of Gyamarthy’s formula, the growth rate of droplets is given by [9]:
G ¼
kg DTð1 rcrit =rÞ dr ¼ dt rl hg hl r þ 1:89ð1 nÞlg =Prg
(5)
The mean free path of vapor molecules lg is expressed as:
lg ¼
pffiffiffiffiffiffiffiffiffi 3mg R T 2p
(6)
n is a semi-empirical correction factor introduced to obtain precise agreements with experimental data [47]: " !# g þ 1 RTsat ðpÞ RTsat ðpÞ 1 2 qc n ¼ z g1 4qc hlg 2 hlg
(7)
The temperature difference required in the growth rate equation can be determined by the capillarity effects pertaining to small droplets [48]. It should be noticed, the droplet temperature Tl is slightly lower than saturation temperature at corresponding pressure Tsat(p), due to subcooling phenomenon occurred in the mixture [49].
Tl ¼ Tsat ðpÞ Tsub
rcrit r
(8)
The subcooling effect resulting from non-equilibrium condition of the mixture is defined as:
Tsub ¼ Tsat ðpÞ Tg
(9)
3. Numerical scheme 3.1. Governing equations The small droplets could be assumed to follow vapor-phase streamlines with no slip velocity. Hence, the conservation equations can be applied to the mixture of the vapor and droplets as a single-phase media. The main governing equations are the conservation equations of mass, momentum, and energy for the
mixture, which are then combined with the equations describing super-cooling condensation and droplets creation.
v r v ðr ui Þ ¼ 0 þ vt v xi
(10)
v ðr ui Þ v v p v sij r ui uj ¼ þ þ vt v xj v xi v xj
(11)
vðrEÞ v vp v vT v keff þ ðrui E þ ui pÞ ¼ us þ þ vt vxi vt vxi vxi vxi i ij
(12)
where r is the mixture density, ui is the mixture velocity and E ¼ e þ uiui/2 is the total specific energy of the mixture. Two additional conservation equations are needed to adequately model the liquid phase [50]. These two equations should be capable of representing the conservation of liquid phase and variations of droplets number during the condensation and evaporation processes. If N denotes the number of droplets per unit volume, the transport equation which models the variations of droplets number throughout the nucleation process is written as:
vðrNÞ v ðrui NÞ ¼ rJclassic þ vt vxi
(13)
In the case of the wet steam theory the total mass of droplets affects the overall mixture properties. This parameter is described as “wetness fraction” b and defined as the mass of liquid phase per unit mass of the mixture. Hence, the second transport equation that governs the mass fraction of liquid phase is expressed as:
vðrbÞ v ðrui bÞ ¼ U þ vt vxi
(14)
Total mass generation rate of the liquid phase U is composed from two separate terms due to condensation and growth rate of the liquid droplets:
U ¼ U1 þ U2
(15)
where U1 is the mass generation rate due to nucleation and U2 is the mass generation rate due to growth rate of droplets. These terms are expressed as:
U1 ¼
4 3 pr r h 3 crit l
U2 ¼ 4pr2 rl N
dr dt
(16)
(17)
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In order to estimate the required droplet diameter as a function of the mass of a single droplet (4prlr3/3 ¼ md), the values of b and N are used based on the following expression:
The reliable range of utilizing this relation is from 0.01 to 100 bar (for pressure) and 273 to 1000 K (for temperature). These ranges are coincided to the common ranges of pressure and temperature applied to the thermo-compressor boundaries. The required vapor density rg can be evaluated from the mixture density r as:
produce a system of algebraic equations which were solved based on the second order upwind method. A computational fluid dynamics solver, which could be linked to a supplementary homewritten code, was used to calculate wet steam related variables. The calculation procedure of this code is described as below. When mixture properties are needed, the main governing equations (10)e(12) are solved while the governing equations (13) and (14) are used to estimate the independent parameters required for evaluating the liquid droplets properties. At the beginning of each of the iterations, the required values of pressure, density, velocity and temperature are evaluated using old time step values. The vapor phase density is estimated from equation (20) and substituted into equation (19) to calculate real pressure of vapor phase. This value can be used to estimate the real value of saturation temperature Tsat(p) required in equations (8) and (9) through using equation (21). On the other hand, psat is also calculated at known temperature from equation (21). This value can be used to evaluate the supersaturation ratio (S) which is leading to determination of the values of critical radius (rcrit), nucleation rate (Jclassic) and total mass generation rate (U). Afterwards, the additional transport equations (13) and (14) can be solved to evaluate the independent variables of the wet steam model b and N. These values are further used to estimate the droplet parameters in the next iteration.
rg ¼ ð1 bÞr
4. Validation of numerical scheme
md ¼
rg N
b
(18)
1b
3.2. Equation of state The virial form of equations of state is more suitable for the purpose of computational usage. The virial equation of state for steam which was proposed by Vukalovich [51] is used herein. The validity of this equation was tested for extrapolation into subcooled regions [52].
p ¼ rg RT 1 þ B1 rg þ B2 r2g þ B3 r3g
(19)
(20)
In order to setup a complete procedure to anticipate the liquid and vapor phases properties, it is necessary to define the saturation pressure by an empirical relation. The equation proposed by Keenan and Keyes [53] is used to predict the saturation pressure as a function of saturation temperature, precisely:
" # 8 psat T Ts X T 273:15 i1 ai 0:65 s ¼ exp 0:01 crit pcrit Ts 100 i¼1
(21)
where Tcrit and pcrit are critical temperature (647.28 K) and pressure (22.087 MPa) of water, respectively and ai coefficients are constant values. 3.3. Turbulence modeling A two-variable turbulence model was applied to the whole domain of flow based on the keε turbulence hypothesis to simulate the turbulent characteristics of flow inside the thermo-compressor. The main governing equations for realizable keε model are expressed as below:
v v v ðrkui Þ ¼ ðrkÞ þ vt vxi vxj v v v ðrεui Þ ¼ ðrεÞ þ vt vxi vxj
"
#
m þ mt=s
k
"
m þ mt=s
ε
þ
vk þ Srck vxj
# vε þ þ Srcε vxj
(22)
(23)
The main reason of selecting realizable approach is the superiority of this model to standard one for predicting the behavior of the round jet flows in ejectors [36]. 3.4. Numerical method A control-volume-based technique employing a finite-volume discretization process according to density-based approach was utilized for the numerical simulations. The non-linear partial differential governing equations were implicitly linearized to
The formulation described in the last section was applied to the supersonic flow in the computational domain of the nozzle described by Moore et al. [4]. Since the general approach for verifying the wet steam results relies on the experimental reports of pressure distribution in the nozzles, in the current study quantitative validation of the numerical model was accomplished by using the experimental data reported by Moore et al. [4]. A convergentedivergent nozzle was designed based on the geometries adopted from Moore et al. report and the following conditions have been considered for inflow and outflow boundaries: Inflow: total pressure of 25 kPa, total temperature of 358 K, steam with no droplet containing. Outflow: low enough static pressure to adjust a supersonic outflow condition. Moreover, following assumptions were made for the numerical simulation: 1. The influent direction was considered to be normal to the inlet plane. 2. Symmetry conditions were enforced for all flow variables along the centerline. 3. No shock wave occurred within the diffuser part of the nozzle. 4. The adiabatic wall with no-slip condition was used. 5. Entering flow was considered to dry steam. There are two major parameters to be compared with the experimental measurements: First, the pressure distribution along the nozzle centerline, and second, the averaged droplet diameter near the exit plane of the nozzle. The variations of these parameters are represented in Fig. 2. As observed, the predicted static pressure along the nozzle centerline is in a reasonable agreement with the profile of experimental pressure distribution. Furthermore, a good compatibility with the experimental location of condensation shock was achieved. The size of generated droplets near the exit plane of the nozzle is well-predicted through the numerical procedure in comparison to
N. Sharifi et al. / Energy 47 (2012) 609e619 0.1
P / Ptot
0.6 0.06 0.5 0.04 0.4
0.2 -0.1
0.02
Condensation Shock Zone
0.3
0
0.1
0.2
0.3
0.08 1.5
0.06 1
0.04
0.5
0 -0.1
0 0.4
Nucleation Piont
Wetness Fraction
0.08
0.1 Wetness Fraction Wet Steam Mach Number Ideal Steam Mach Number
Mach Number
Numerical Pressure Ratio Experimental Pressure Ratio Numerical Droplet Diameter Experimental Droplet Diameter
0.7
2
Droplet Diameter (micron)
0.8
613
0.02
0
0.1
0.2
0.3
0 0.4
Axial Distance from the throat (m)
Axial distance from the throat (m)
Fig. 2. Comparisons between the simulation and experimental results in the nozzle: (Left) the distribution of pressure ratio on the nozzle centerline, (Right) the averaged droplets diameter near the nozzle exit plane.
Fig. 4. Numerical results of the wet steam calculations along the centerline of the Moore et al. nozzle: (Left) Mach number of ideal and wet steam flows, (Right) wetness fraction.
30
40 Log 10 of Nucleation Rate (1/m s) Subcooled Vapor Level (K)
25
35 30
20 25 15
20
10
15 10
5 5 0
-5 -0.1
0
0
0.1
0.2
0.3
Subcooled Vapor Level (K)
Log 10 of Nucleation Rate (1/m3s)
the experiments. However, the discrepancy between the numerical and experimental droplet sizes was lower than 15% which is acceptable enough for engineering problems especially for this order of magnitude of the droplet diameter. To demonstrate the current model is capable enough to predict some major phenomena resulting from the homogeneous nucleation, some further results have been provided. These results might not be experimentally validated but could be verified theoretically according to the main assumptions of the wet steam theory. Fig. 3 shows the variations of subcooling level and log 10 of nucleation rate, in order to compare the trends of both phenomena simultaneously. It is obvious that the vapor phase starts to condense (nucleation starting point) after a significant decline in the droplets temperature (positive subcooling value). As expected, the peak value of nucleation rate is reached at the same axial location of maximum subcooling level. The nucleation event causes some droplets to be generated and hence, the mass fraction of liquid phase is increased after the nucleation point. This phenomenon can be thoroughly understood from Fig. 4. The blue dashed line (in web version) in this illustration represents the variations of wetness fraction along the nozzle centerline. Increasing the wetness fraction after the nucleation point reveals that the production of liquid droplets is preceded by
-5 0.4
Axial distance from the throat (m) Fig. 3. Numerical results of the wet steam calculations along the centerline of the Moore et al. nozzle: (Left) nucleation rate, (Right) sub-cooling temperature level.
nucleation event. It is also obvious that the wetness fraction is smoothly increased toward the nozzle exit due to reducing the static temperature of the vapor. The additional effect which confirms the correct numerical results is the flow Mach number along the centerline of the nozzle which is depicted on the left vertical axis of Fig. 4. As expected, the trend of Mach number for the wet steam flow is lower than the corresponding value for the ideal steam after nucleation point, because the liquid droplets content cause to decline the vapor velocity and thus, the local Mach number is decreased through the diverging part of the nozzle. The reasonable agreements of the computational results with experiments prove the ability of this numerical model to provide valuable information on the flow pattern inside thermocompressors. 5. Geometry and boundary conditions Fig. 5 shows the configuration of the thermo-compressor (all sizes are in centimeters) with close-up mesh elements near the primary nozzle and around it. The entire computational domain is a region ranging from the inlet of the primary nozzle to the end of the diffuser, including the nozzle. The size of the computational domain is 14.0 m in the axial direction, and 0.91 m in radial direction. The test for grid-independence was performed in order to calculate the deviation of net mass flux before any further analysis. The grid was created based on the structured quadrilateral elements. The total cell number was firstly 35,100. Afterwards, different cell configurations were used in the test with the total cell numbers of 45,800 and 67,400. The results showed that the deviations for the net mass flow rate were respectively 0.51% and 0.43%. The boundaries of the computational domain were considered adiabatic walls (for the nozzle and the shell), two inlets (for motive and suction flow) and a single outlet (for discharge flow). The whole computational domain was assumed a single-phase media undergoing the boundary conditions defined in Table 1. These values were kept constant for both ideal and wet steam simulations. The working fluid considered at the boundaries was steam. At both inlet boundaries the total pressure and total temperature are specified. As observed in Table 1, the motive steam at mentioned condition is a superheated vapor and thus, contains no liquid droplet contents. Other unknown flow parameters are
614
N. Sharifi et al. / Energy 47 (2012) 609e619
Fig. 5. Computational grid elements and dimensions of the thermo-compressor.
calculated from the interior by second-order extrapolation. At the exit plane (i.e. discharge flow), if the flow is subsonic, only the static pressure is required, whereas the total pressure, total temperature, and mass flow rate are extrapolated from the interior by secondorder extrapolation. If the exit flow is supersonic, all flow variables are extrapolated from the interior. 6. Results and discussion In this section, the numerical scheme derived for wet steam simulation was applied to the computational domain of the thermo-compressor. The general CFD method has been conducted with two different hypotheses: wet steam and ideal steam, in order to compare the flow properties along the centerline of the thermocompressor and to observe the significant discrepancies on the results of these two models.
temperature distribution for wet and ideal simulations with an additional horizontal line referring to the minimum limit of allowable temperature for liquid and gaseous phases of water. As Fig. 6 indicates, the minimum values of temperature in ideal steam simulation has been dropped beneath the freezing line (melting point), because the equation of state for a perfect gas has no sense of temperature decline lower than freezing limit and always generates a numerical value for the temperature corresponding to the pressure and density of the gas, regardless of any saturation effect. But, as expected, the minimum value of wet simulation curve has not dropped below the freezing line and whole range of static temperature located over the limit of 273.15 K. It can also be noticed that the flow temperature at the exit boundary is lower for wet steam compared to that for ideal steam. This effect can be explained by the latent heat capacity of water droplets that can absorb the released heat of temperature rising after the shock wave in the constant area section.
6.1. Flow temperature ideal steam wet steam
450 400 350
Tstatic (K)
As a matter of fact in the wet steam simulation the static temperature of fluid must not be lower than minimum range of achievable temperature for liquid or gaseous phase of water (i.e. 273.15 K). Therefore, an appropriate criterion to assess a correct flow simulation is the temperature distribution in the flow field. Since the minimum range of static temperature always occurs on the intersection point of oblique shock waves located on the axial centerline of the thermo-compressor, it is necessary to plot the temperature variations on the axial direction. Fig. 6 shows the
Freezing line 273.15 K
300 250 200
Table 1 Boundary conditions for different boundaries of the thermo-compressor.
Motive flow Suction flow Discharge flow
Temperature (K)
Pressure (kPa)
470 322 e
1000 10 26.3
150 100 0
2
4
6
8
10
12
14
Axial Distance (m) Fig. 6. Comparison between the static temperature of the wet steam and the ideal steam along the axial direction of the thermo-compressor.
N. Sharifi et al. / Energy 47 (2012) 609e619
6.2. Flow pressure
615
5
ideal steam wet steam
4.5
6.3. Flow Mach number The effect of containing infinitesimal droplets in the steam causes a decline in absolute averaged velocity of the flow. On the other hand, the temperature rise in the case of wet steam leads to pffiffiffiffiffiffiffiffiffiffi an increase in local speed of sound (i.e. g RT ). Hence, the flow Mach number is decreased significantly. According to Fig. 8, the values of local Mach number show that the maximum value of Mach number has been reduced from 4.9 (for ideal steam) to 3.6 (for wet steam). As observed in Fig. 8, the exact values of flow velocity at the outlet boundary are equal for both models. The direct result of this
100
ideal steam wet steam
Pstatic (kPa)
80
60
20
0
2
4
6
Shock Position
3 2.5 2 1.5 1 0.5 0
0
2
4
6
8
10
12
14
Axial Distance (m) Fig. 8. Comparison between the Mach number of the wet steam and the ideal steam along the axial direction of the thermo-compressor.
fact could be expressed as: “At the constant velocity and the fixed outlet area (i.e. A ¼ Const., V ¼ Const.), the mass flow rate of _ ¼ r AV), will be significantly increased in delivered steam (i.e. m the wet conditions”. At the downstream of the shock waves, the local Mach numbers for the wet and ideal steam models have been separated to different values which are dependent on both the downstream velocities and static temperatures. It is of interest to note that even though the local Mach number is greater than unity and flow is supersonic, the shock wave has been occurred (according to pressure rise location in Fig. 7) and the flow after the shock is still remaining supersonic. A very practical comparison between wet and ideal results may be related to the formation of supersonic jet core within the TVC. As illustrated in Fig. 9, the shape of exiting jet in the wet simulation is a little bit wider than the ideal jet core. Moreover, the angle of exiting supersonic flow right at the nozzle tip position is larger in wet conditions. This phenomenon implies a more realistic diamond wave pattern of supersonic flow for wet steam conditions. It reveals the presence of weak oblique shock waves rather than a strong normal shock in the throat of the thermo-compressor. Therefore, the hypothesis of normal shock occurrence in the constant area zone that has been mentioned in literature may not become a general assumption. Moreover, the axial position of the ideal steam shock wave inside the constant area zone is prior to wet steam shock wave. This observation gives rise to a practical result which means that the wet steam simulation could undergo a larger pressure at downstream, and hence results in a higher compression ratio. This fact may predict a better operational mode for the thermo-compressor with a higher safety factor. 6.4. Other parameters related to wet steam theory
Shock position
40
0
4 3.5
Mach No.
The major purpose of pressure study is to detect the shock position in the thermo-compressor. Static pressure distribution along the centerline of the thermo-compressor shows an abrupt jump in pressure fluctuations that can be recognized as shock position. When the nozzle is under-expanded, the supersonic flow outgoing from the primary nozzle forms a combination of oblique shocks and expansion waves called “diamond wave” pattern or “shock train” [54]. The direct effect of such phenomenon on the static pressure is the formation of some pressure fluctuations along the centerline which can be observed in Fig. 7. As indicated, the abrupt changes in pressure rise are weakened when the supersonic jet core contains some wetness. Therefore, the pressure loss due to passing through oblique shock waves is decreased. This phenomenon can be related to lower temperature fluctuations in wet simulation that can be observed in Fig. 6. Furthermore, shock position moves somewhat downstream, toward the exit boundary, and pressure recovery is strengthened in the case of wet simulation. The other important consequence of this discussion can be related to the flow density. As a matter of fact, the equal pressure at the exit boundary reveals a little bit greater values of downstream density for the wet steam compared to the ideal steam, because of higher temperature values for ideal steam simulation at this region. Furthermore, on account of higher density and higher static pressure of wet steam at the outlet of the thermo-compressor, the increased fluid inertia can strongly overcome the downstream back pressure. This means that the compression ratio might slightly increase in the wet steam conditions.
8
10
12
14
Axial Distance (m) Fig. 7. Comparison between the static pressure of the wet steam and the ideal steam along the axial direction of the thermo-compressor.
The variations of some important parameters which are normally discussed in the wet steam theory (such as nucleation rate, subcooling level, wetness fraction, and droplet size) have been shown previously for the flow inside a single nozzle. It is of interest to investigate these variables in the computational domain of the thermo-compressor. Fig. 10(a) shows the rate of nucleation phenomenon along the centerline of the TVC. It can be observed the nucleation peak has taken place in the primary nozzle and suddenly disappeared after reaching to its maximum limit. Therefore, the downstream flow no longer undergoes nucleation phenomenon and the rate of
616
N. Sharifi et al. / Energy 47 (2012) 609e619
Fig. 9. Comparison between the supersonic jet core in the thermo-compressor for the wet and ideal steam conditions.
nucleation is terminated abruptly in remaining part of the TVC. This single occurrence of nucleation phenomenon provides a high wetness fraction distribution along the centerline of the TVC which can be observed in Fig. 10(b). As it is apparent, the wetness fraction is increased to a maximum level which corresponds to the point of exiting flow
from the nozzle. Afterwards, the fluctuations of wetness fraction are continued with a little amplitude along the TVC centerline prior to reach the shock wave position. After the shock, the wetness fraction is lowered gradually due to temperature raise caused by the shock wave. The large portion of wetness decreasing is taken place in the diffuser part which has an additional effect on evaporation of liquid droplets. The pressure rise along the diffuser part increases the saturation temperature of the liquid phase and thus, has a reverse effect on the evaporation of the water droplets. The variations of Tsat along the centerline of the TVC which is shown in Fig. 11(a), give further confirmation to this supposition. As a matter of fact, the variations of saturation temperature are originated from the pressure fluctuations in the centerline direction. Moreover, the higher level of Tsat indicates the larger amount of heat needed to evaporate the water droplets at a given temperature. By considering the six dominant points in Fig. 11, it can be seen that the maximum subcooling level (point “A”) is taken place at the maximum saturation temperature related to the maximum flow velocity. Points “B”, “C” and “D” are pertaining to the shock train fluctuation regions, while point “E” shows the exact position of the main shock wave with a negative value of subcooling temperature (i.e. the steam is locally superheated after the shock wave occurrence). The “EF” segment shows the droplets pathway through the diffuser with a little raise in Tsat while the negative subcooling value (i.e. superheating level) increases significantly. The major reason of this phenomenon can be related to the subsonic flow inside the diffuser which undergoes an abrupt pressure and temperature rise after the shock wave. The incremental change of saturation temperature (Tsat) is basically related to the pressure raise within the diffuser. It is expected to observe a descending trend for the absolute value of Tsub. But, the increasing rate of steam temperature (Tg) inside the diverging section of diffuser overcomes the increasing trend of saturation temperature (Tsat). Hence, the outgoing steam is negatively sub-cooled (i.e. superheated). 6.5. Characteristic parameters
Fig. 10. Variations of (a) the nucleation rate and (b) the wetness fraction along the centerline of the thermo-compressor.
By comparing the characteristics of the thermo-compressor with a fixed geometry under both wet and ideal steam models, the superiority of the wet steam model can be recognized and the discrepancies of dominant flow parameters may be clarified. The resulted mass flow rates for different boundaries of the thermo-compressor are depicted in Table 2, for both wet and ideal
N. Sharifi et al. / Energy 47 (2012) 609e619
617
ideal steam modeling. This is the second advantage of wet simulation which can be taken into account. Since the main goal of thermo-compressor utilization is to provide a reliable compression ratio, this result can be very useful for practical design purposes. The rate of increasing in CR was around %14.1 under specified boundary conditions. Therefore, by considering the dominant influence of droplet presence in the supersonic flow exiting the primary nozzle, both characteristic parameters of the TVC will be remarkably improved. 7. Concluding remarks In the current study, a new approach for investigating the internal flow of thermo-compressors was introduced based on the fundamental wet steam theory. The main idea of this work was to consider a homogenous steam condensation in the supersonic steam flow exiting from the convergentedivergent nozzle.
Fig. 11. Variations of (a) the saturation temperature and (b) the subcooling temperature along the centerline of the thermo-compressor.
steam modeling. As can be seen from the data shown in the table, the entrainment ratio is higher for the wet steam in comparison to the ideal steam simulation. This conclusion has the greatest importance from the viewpoint of operational conditions. Because, the higher amount of ER leads to the better TVC performance that implies a maximized capability of flow entraining with a fixed geometry thermo-compressor. In order to compare the CR of the two different models, the TVC discharge pressure was increased gradually up to the point where an abrupt drop in the rate of the suction flow is observed. This point can be considered as the maximum critical pressure of the device. The highest values of discharge pressure for wet and ideal simulations were recorded as 29.1 kPa and 27.3 kPa, respectively. Since the suction pressure remained constant during the pressure rise at the outlet boundary, it can be obviously concluded the overall compression ratio of the thermo-compressor is predicted higher in wet steam modeling than the corresponding value for
Table 2 Numerical values of the thermo-compressor characteristic parameters in different simulation methods.
Entrainment ratio Compression ratio
Wet steam
Ideal steam
0.89 2.96
0.78 2.73
A numerical method for calculating the thermodynamic properties of thermo-compressors internal flow in the context of the homogenous nucleation was developed. According to the authors’ knowledge, no previous study was implemented for evaluating the flow properties in thermocompressors under the assumption of wet steam theory. The present model has the potential for efficiently modeling the internal phenomena of thermo-compressors where droplets are generated under non-equilibrium conditions. Reasonable agreement of numerical scheme with experimental reports was obtained, especially for static pressure distribution and condensation shock position in the nozzle. The axial position of the shock wave inside the throat shifted to the discharge plane in the wet steam model that shows an extended margin for normal operation of the device. There is a further advancement in operating performance of thermo-compressors, since the entrainment ratio shows a substantial increasing under the wet steam assumption. This means that for a thermo-compressor with a fixed geometry, the capability of entraining the suction flow becomes higher for the wet steam case over the ideal steam. The numerical results show that the compression ratio of thermo-compressors is improved under the assumption of the wet steam. This fact evinces a conventional thermocompressor can undergo larger discharge pressures and produce higher compression ratios. Nomenclature
English letters B virial coefficients E total enthalpy G growth rate of droplets h static enthalpy J nucleation rate k thermal conductivity Kn Knudsen number Boltzmann’s constant KB l mean free path of vapor molecules m mass mass of one molecule of water Mmolc n total number of droplets N number of droplets per unit volume p static pressure Pr Prandtl number condensation coefficient qc
618
r rcrit R S Src t T u V x
N. Sharifi et al. / Energy 47 (2012) 609e619
droplet radius critical droplet radius gas constant super-saturation source term time static temperature velocity components mixture volume spatial dimension
Greek letters a constant coefficient for saturation curve b wetness fraction g specific heat ratio r density ε turbulent dissipation rate k turbulent kinetic energy m dynamic viscosity s liquid surface tension s stress tensor h non-isothermal correction factor n semi-empirical correction z empirical parameter U mass generation rate dij Kronecker delta function Subscripts crit critical properties d droplet dis discharge flow g vapor (gas) phase l liquid phase i, j tensor notations mot motive flow sat saturation properties sub subcooled properties suc suction flow t turbulent References [1] Adel M, Dayem A. Experimental and numerical performance of a multi-effect condensationeevaporation solar water distillation system. Energy 2006;31: 2710e27. [2] Ansari K, Sayyaadi H, Amidpour M. Thermoeconomic optimization of a hybrid pressurized water reactor (PWR) power plant coupled to a multi effect distillation desalination system with thermo-vapor compressor (MEDeTVC). Energy 2010;35:1981e96. [3] Li Q, Piechna J, Müller N. Numerical simulation of novel axial impeller patterns to compress water vapor as refrigerant. Energy 2011;36:2773e81. [4] Moore MJ, Walters PT, Crane RI, Davidson BJ. Predicting the fog drop size in wet steam turbines. In: Wet steam 4 conference. Inst. of Mechanical Engineers (UK), University of Warwick; 1973. C37/73. [5] Jackson R, Davidson BJ. An equation set for non-equilibrium two phase flow and an analysis of some aspects of choking, acoustic propagation and losses in low pressure wet steam. Int J Multiphas Flow 1983;9(5):491e510. [6] Young JB. The fundamental equations of gas-droplet multiphase flow. Int J Multiphas Flow 1995;21(2):175e91. [7] Bakhtar F, So KS. A study of nucleating flow of steam in a cascade of supersonic blading by the time-marching method. Int J Heat Fluid Flow 1991;12: 54e62. [8] White AJ, Young JB. Time marching method for the prediction of twodimensional unsteady flows of condensing steam. AIAA J Propuls Power 1993;9(4):579e87. [9] Young JB. Two-dimensional non-equilibrium wet-steam calculations for nozzles and turbine cascades. Trans ASME J Turbo 1992;114:569e78. [10] Bakhtar F, Mahpeykar MR, Abbas KK. An investigation of nucleating flows of steam in a cascade of turbine blading e theoretical treatment. ASME J Fluids Eng 1995;117:138e44.
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