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Preliminary exergy analysis of static flash of pure water
International Journal of Heat and Mass Transfer 86 (2015) 377–387
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Preliminary exergy analysis of static flash of pure water Dan Zhang ⇑, Junjie Yan, Yingwen Liu, BingChao Zhao State Key Laboratory of Multiphase Flow in Power Engineering, School of Energy & Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, PR China
a r t i c l e
i n f o
Article history: Received 24 September 2014 Received in revised form 1 March 2015 Accepted 6 March 2015
Keywords: Static flash Exergy destruction Exergy efficiency Boiling temperature difference Steam-carrying effect
a b s t r a c t The waterfilm was selected as control volume and exergy transfer within it during static flash of pure water at different flash speeds was analyzed. Exergy balance during differential time was set up and exergy destruction was also deduced. It suggested that static flash was an irreversible process and boiling temperature difference (BTD) played a major role for exergy destruction. In order to measure the effectiveness of static flash on exergy transfer during entire flash duration time, exergy efficiency of flash (EE) was introduced as the fraction of the delivered exergy to the total released exergy from unit mass of initial waterfilm. Exergy efficiency of flash steam (EEstm) was defined as the fraction of exergy contained in latent heat of flash steam to the total released exergy from unit mass of initial waterfilm. Both efficiencies were evaluated and analyzed according to our former experimental results with initial temperature ranging between 46.5 and 104.6 °C, superheat between 1.78 and 43.9 K, initial height of waterfilm between 0.10 and 0.30 m, flash speed between 4.8 104 and 2.18 s1. Results suggested that, first, EE varies between 0.86 and 0.99, and EEstm varied between 0.037 and 0.99 in current experimental range. Second EEstm could be improved by increasing initial temperature of waterfilm while at the same time reducing superheat, or initial height of waterfilm, or flash speed. At last, on basis of an empirical formula for BTD fitted from experimental results, a pair of calculation formulae for EE and EEstm were set up within acceptable error range. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Flash defines the phenomenon that when liquid at certain temperature is exposed to sudden pressure drop below its saturation pressure corresponding to its temperature, it suddenly evaporates, leading its temperature to drop significantly and generating massive flash steam. At the same time, some superheated liquid is entrained away by the flash steam through steam-carrying effect. Flash can be classified into static flash or dynamic flash according to whether the superheated liquid has macroscopic velocity during flash. Static flash includes flash from static waterfilm, flash from static droplet and so on. Dynamic flash includes flash from horizontally moving waterfilm (named as circulatory flash in our previous works), flash from water jet, and so on. Study in this paper focuses only on flash from static waterfilm, so this kind of flash, in following, is named as static flash for short. According to the working fluid, flash can also be classified into flash of pure water, flash of aqueous salt solution, and so on. Flash is a complex phenomenon including both heat and mass transfer. Macroscopically,
⇑ Corresponding author. Tel./fax: +86 (29) 82668705. E-mail address: [email protected] (D. Zhang). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.03.022 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.
heat transfer is caused by evaporation only, while mass transfer is caused by both evaporation and steam-carrying effect. Due to its good performance on heat transfer, flash is widely used in energy recycle, such as geothermal power plant [1]. Besides, flash can also be used to separate nonvolatile solute and volatile solvent from their solution, such as desalination [2] and thin film deposition [3]. Because steam is easier to be transported than liquid, and the latent heat of steam is much larger than the specific heat of liquid water, the latent heat contained in flash steam is the most concerned energy in industrial flash systems. For example, in geothermal power plant, flash is used to convert thermal energy from low-temperature heat source into the latent heat of flash steam that is used to drive steam turbine and deliver work output. While in each unit of MSF (multi-stage flash), the generated flash steam is condensed at upper part of flash chamber and its latent heat is used to heat original seawater in order to improve thermal performance of the entire system. Therefore, mechanism of flash and its thermodynamic properties, especially properties of flash steam, receive wide attention. Static flash was the simplest form of flash evaporation and thus was always selected as prototype to examine its mechanism. Miyatake et al. [4,5] presented experimental study on static flash of pure water with superheat varying between 3 and 5 K. They
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Nomenclature A BTD c D e EE EEstm Ex ex ex,H ex,U FDE FS FUE h H 0 hfg hfg Hr m NEF p Q s s0fg t T Tfg u v
cross-section area of flash chamber (m2) boiling temperature difference (K) specific heat (kJ kg1 K1) orifice diameter (mm) error (–) exergy efficiency of flash (–) exergy efficiency of flash steam (–) exergy (kJ) specific exergy (kJ kg1) flow exergy (kJ kg1) nonflow exergy (kJ kg1) fraction of destroyed exergy (–) flash speed (s1) fraction of utilized exergy (–) specific enthalpy (kJ kg1) height of waterfilm (m) equivalent specific latent heat of vaporization for superheated water (kJ kg1) specific latent heat of vaporization (kJ kg1) relative waterfilm height (–) mass (kg) non-equilibrium fraction (–) pressure (MPa) heat (kJ) specific entropy (kJ kg1 K1) equivalent specific entropy of vaporization for superheated water (kJ kg1 K1) temperature (°C) absolute temperature (K) equivalent heat absorption temperature (K) specific internal energy (kJ kg1) specific volume (m3 kg1)
found that waterfilm temperature dropped quickly at start, then the drop speed slowed down and the temperature equalized at certain value in the end. Thus, they divided flash into fast evaporation stage and gradual evaporation stage. Besides, they also introduced non-equilibrium fraction (NEF, will be clearly introduced in Section.2) to measure completion degree of flash, and suggested that 1-NEF was a measure of energy conversion efficiency for flash. On basis of these concepts, they examined heat transfer properties of flash during fast evaporation stage and found that higher superheat or lower initial height of waterfilm caused flash to take place faster and evaporate more. Saury et al. [6] examined energy conversion during static flash of pure water with superheat ranging between 1 and 35 K. Results suggested that the sensible heat released in the temperature drop of waterfilm could be considered to all change into the latent heat of flash steam. Saury et al. [7] also changed depressurization rate of flash chamber through an adjusting valve installed between flash and vacuum chambers. Result suggested that depressurization rate had nearly nothing to do with final evaporated mass. Kim and Lior [8] also examined static flash and revealed several critical transition points in temperature evolution of waterfilm. Liu et al. [9] performed experiments on flash from aqueous NaCl droplet, and found evaporation rate decreased with increasing environment pressure. Gopalakrishna et al. [10] did experiment of static flash of aqueous NaCl solution with Superheat ranging between 0.5 and 10 K, concentration of NaCl between 0 and 0.035 (mass fraction), and finally proposed a calculation formula for final evaporated mass. In recent years, our research team also did a series of experimental study for static flash of pure water [11] and aqueous NaCl solution [12] at different flash speeds [13]. In our experiments,
V Wi
volume (m3) boundary work (kJ)
Greek symbols DH height drop of waterfilm (m) DT superheat (K) q density (kg m3) s time (s) Subscript 0 bm cal CV dp e exp f fit im l r rf s sc sh stm tg v
start of flash benchmark for exergy calculation calculated value control volume dividing point between fast and gradual evaporation stages equilibrium experimental flash chamber fitting value integral mean liquid relative reference saturation steam-carrying superheated steam tangent point vacuum chamber
waterfilm concentration varied between 0 and 0.15 (mass fraction), superheat between 1.7 and 53.9 K. Flash speed was defined as mean decreasing rate of NEF during fast evaporation stage, and in experiment this speed was adjusted by adding throttle orifice plate with different orifice diameters (5–80 mm) between flash and vacuum chambers. In our research, theoretical height drop of waterfilm during flash was defined as the drop that calculated by heat balance. It was found that actual height drop of waterfilm measured in experiment was always far greater than the theoretical height drop, suggesting that some superheated liquid was directly entrained away by flash steam without taking part in evaporation. This phenomenon was named as steam-carrying effect. Steam-carrying ratio was also proposed as the mass ratio of entrained liquid to flash steam. Experimental results indicated that this ratio varied between 1.7 and 65.4 in current range, thus this effect cannot be neglected. Therefore, properties of boiling heat transfer and steam-carrying effect were studied together by our research team. Results suggested that, increasing flash speed or initial height of waterfilm significantly intensified steam-carrying effect, but had weak influences on boiling heat transfer. While increasing superheat could intensify the two aspects simultaneously. At last, a fitting formula for NEF evolution and a semi-empirical calculation model for steam-carrying effect at different flash speeds were proposed [14], [15]. Besides, according to the first law of thermodynamics, energy conversion during static flash was also analyzed with steam-carrying effect taken into consideration [16]. Studies on dynamic flash was also prevailing. Sami Mutair [17] investigated flash from superheated water jet. Hanshik Chung et al. [18] examined flash of pure water from horizontally moving waterfilm. Our research team also carried out experimental study
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2. Literature synthesis Analysis in this paper is presented in light of several basic concepts proposed by former scholars or our research team. They are also the start points of following deductions and thus are briefly introduced first. During static flash, both temperature of waterfilm (tl) and pressure of flash chamber (pf) quickly drop with time. Fig. 1(a) [12] compares typical evolutions of tl and saturation temperature (ts) corresponding to pf. It suggests that ts at start quickly drops to below tl to certain value, makes the waterfilm to be superheated under pf and thus starts flash boiling, leading tl begin to drop. As shown in Fig. 1(a), ts is always lower than tl during flash in order to maintain waterfilm at superheated state. The difference between tl and ts is defined as boiling temperature difference (BTD, Eq. (1)) which varies with time. The cause of BTD is being studied now and may be related with nucleation of vapor bubble (Fig. 1(a)) [12,26].
BTD ¼ t l t s ðpf Þ
ð1Þ
Finally, pf equalizes at certain value (pfe), the saturation temperature corresponding to pfe is suggested to be theoretical equilibrium temperature of waterfilm (tse, Eq. (2)) by Miyatake et al. [4,5]. tse represents the lowest temperature that waterfilm can reach in theory.
t se ¼ t s ðpfe Þ
ð2Þ
98 FES
t
GES
l
t ←p
96
s
f
Intergral domain of BTD
im
t
BTD(τ)
t / oC
94 92
, Eq.(6)
l,cal
ΔT=6.4K, H =0.30m 0
↓
D=20mm, BTD=1.7K
90
↑
88
tse←pfe
84
τdp
86
0
20
40
60
τ/ s
80
100
120
(a) Boiling temperature difference (Ref.[12]).
1 FES
0.8
NEF NEF , Eq.(5)
GES
fit
A
ΔT=6.4K, H =0.30m 0
D=20mm, τ =21.20s dp
LST, Largest slop tangent
NEF
0.6 NEF
dp
0.2
0 0
τ
dp
0.4
LST
on flash from horizontally moving waterfilm of pure water and aqueous NaCl solution [19,20], and compared their boiling heat transfer properties with that of static flash [11]. In addition, there were also many works on industrial flash systems. Miyatake et al. [21] and Jin and Low [22] did researches on multi-stage flash (MSF) system. N Kahraman et al. carried out exergy analysis of MSF [23], and found that the largest exergy destruction occurred during evaporation in each flash unit. An improvement for custom MSF system was also proposed and analyzed by our research team [24]. Ho Kon Tiat et al. [25] optimized two-stage flash evaporators. Former works revealed basic mechanism of heat and mass transfer during flash and started research on its commercial use, but limitations still existed. First, energy released from superheated waterfilm was considered to all changed into the latent heat of flash steam, without considering the energy loss caused by steam-carrying effect. This loss was considerable with increasing superheat or flash speed and cannot be neglected. Second, energy transfer in former studies were mainly analyzed according to the first law of thermodynamics. These analysis did not consider the maximum possible work that could be delivered during flash at given environment. Therefore, exergy analysis for flash should be carried out. Third, static flash in previous works mainly took place freely; the temperature drop and height drop of waterfilm were out of control. In order to develop its commercial application, controllable flash should be developed and its properties at different flash speeds, especially its effectiveness on exergy transfer at different flash speeds, should be examined. Therefore, in this paper, exergy transfer during static flash of pure water at different flash speed is analyzed with waterfilm selected as control volume (because the phase change during flash primarily takes place within it). Exergy balance is set up and exergy destruction is also deduced. According to experimental data, the effectiveness of static flash on exergy transfer, especially on that transferred into the latent heat of flash steam during flash duration time are both evaluated, and their dependences on initial conditions, such as initial temperature and height of waterfilm, superheat, flash speed were also analyzed and formulized.
B 20
40
60
τ/s
80
100
120
(b) Dividing point between fast/gradual evaporation stages. (Ref.[15]) Fig. 1. Boiling temperature difference and dividing point between fast/gradual evaporation stages. FES, fast evaporation stage. GES, gradual evaporation stage.
Superheat is defined as the difference between initial temperature of waterfilm and tse (Eq. (3)). It is a measure of unstable energy contained in waterfilm and is also viewed as driving force for flash.
DT ¼ t l0 tse
ð3Þ
Non-equilibrium fraction (NEF), is also introduced by Miyatake et al. [4,5] as Eq. (4) to measure completion degree of actual flash. According to experimental results, an empirical formula of NEF during static flash of pure water is proposed by our research team as Eq. (5) [13], upon which temperature evolution of waterfilm can be calculated as Eq. (6). The reference temperature is defined as Eq. (7).
NEFðsÞ ¼
t l ðsÞ t se t l ðsÞ t se ¼ t l0 t se DT
" rffiffiffiffiffiffiffiffia # 2 H0 ql cl NEF fit ðsÞ ¼ erf 2 ks where k ¼ 1:106 DDTH2 0
- 0:535
ð4Þ
ð5Þ
, a2 ¼ 0:0011 þ 0:3400lnDT þ 0:0202D
2
0:0002D .
tl;cal ðsÞ ¼ DT NEF fit ðsÞ þ t se trf ¼
1 ðtl0 þ t se Þ 2
ð6Þ ð7Þ
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Flash is divided into fast evaporation stage and gradual evaporation stage according to the decreasing rate of NEF [15]. The duration of fast evaporation stage is defined as flash duration time ([0,sdp]) and can be calculated by Eq. (8)
sdp;cal ¼ stg
NEFðstg Þ NEF 0 ðstg Þ
stg ¼
where
2b1
b1 2a2 b
,
ð8Þ
a¼
H0 2
qffiffiffiffiffiffia2 qcp
,
k
b ¼ a22 ,
NEF 0 ðsÞ ¼ p2ffiffipffi
abea2 s2b sb1 . Flash speed (FS) is defined as mean decreasing rate of NEF during flash duration time [0,sdp] as Eq. (9). FS can also be directly calculated through Eq. (10) on basis of above empirical formula of NEF (Eq. (5)).
FS ¼
1 NEF dp
ð9Þ
sdp
FScal ¼
1 NEF fit ðsdp;cal Þ
ð10Þ
sdp;cal
Besides, the height of waterfilm also drops significantly during flash, which is caused by both evaporation and steam-carrying effect. If defining relative height of waterfilm as Eq. (11), the height drop of waterfilm during flash can be calculated as Eq. (12) [15].
Hr ¼
H H0
DHrdp;cal ¼
ð11Þ H0 Hdp cl DT ¼ hfg H0 "
4 l 3 stm
ð12Þ
b1 ¼ 0:002D2 0:5043D þ 62:083, b2 is listed in Table 1 of Ref. [15]. 3. Experimental system and uncertainty analysis 3.1. Experimental system Research in this paper is carried out on basis of the same experimental system (Fig. 2) and the same steps clearly stated in our former works [13,14]. The system contains high and low pressure parts. The former includes heater and flash chamber. The flash chamber is a 0.20 0.20 0.50 m rectangular tank. Its front and rear faces are made of tempered glass for observation. The low pressure part includes vacuum chamber, vacuum pump and auxiliary condensing system. Two parts are connected by an Table 1 Experimental range of main parameters and uncertainty analysis. Parameter xi
Experimental range
Absolute uncertainty dxi
Minimal measured value ximin
Maximal uncertainty dxi/ximin
T (°C) H (m)
46.5–104.6 0.10, 0.15, 0.20, 0.25, 0.30 20–1000 8.68 103– 0.118 1.78–43.9 0.86–9.82 4.8 104– 2.18 0.86–0.99 0.037–0.99
0.2 5.0 104
46.5 0.10
4.30 103 0.005
0.0125 1.13 103
20 8.68 103
6.25 104 0.130
– – –
– – –
0.134 0.134 0.189
– –
– –
0.276 0.272
DT (K) BTDim FS (s1) EE EEstm
Uncertainty analysis for all directly and indirectly measured values in experiment is carried out according to method of constant odds—product form proposed by Moffat [27], and the results as well as experimental ranges of main parameters are listed in Table 1.
4.1. Exergy analysis of static flash
where ln C ¼ b1 þ b2 DT 27:6204 expðDT 0:5 Þ þ 10:3227 ln H0 , and
p (MPa)
3.2. Uncertainty analysis
4. Results and analysis
6 # q c l DT ð1 NEF dp;cal Þ 1 þ C H0 FScal hfg q
s (s)
80 mm-diameter electromagnetic valve. In order to adjust flash speed, four thin orifice plates with different orifice diameters (5, 10, 20, 40 mm) are respectively installed at the inlet of electromagnetic valve (section A-A in Fig. 2) in experiments. In trials, heated pure water was filled from heater into flash chamber to reach required initial height of waterfilm. While vacuum pump and auxiliary condensing system were turned on to reduce pressure of vacuum chamber to designed value. Then, the electromagnetic valve was opened and flash suddenly took place. During flash, the quickly decaying temperature of waterfilm, pressure of flash chamber and vacuum chamber were real-timely measured. After flash, final equilibrium height of waterfilm was measured by a cathetometer attached outside the front glass of flash chamber. According to these measured data, exergy analysis of static flash could be carried out. The detailed structure of each component, information of measurement equipments, as well as data processing (including following uncertainty analysis) were the same with our previous works [13,14] and thus are not repeated here.
Exergy analysis of static flash is carried out with the waterfilm selected as control volume (Fig. 3(a)) because the phase change primarily takes place within it. Besides, following assumption are also adopted, (1) flash chamber keeps adiabatic; (2) specific volume of waterfilm (dvl = 0) keeps constant and static pressure inside waterfilm is neglected, thus the pressure of waterfilm is equal to the pressure of flash chamber (pf, Eq. (13)); (3) kinetic and potential energy of steam and entrained liquid are neglected; (4) the state that steam and entrained liquid mix together has no effect on their respective value of entropy; (5) duration of fast evaporation stage is defined as flash duration time, and the height drop of waterfilm is thought to completely take place during that time.
pl ¼ pf
ð13Þ
Considering that waterfilm keeps superheated during flash, its thermal properties, such as internal energy, enthalpy, entropy, are frequently used in following discussion but cannot be found in tabulate data, so calculations of these properties should be firstly discussed. 4.1.1. Thermo-properties of superheated water During flash, waterfilm is superheated water because its temperature tl is always higher than the saturation temperature (ts) corresponding to the pressure of flash chamber (pf) (Fig. 1(a)). Considering that specific internal energy depends only on temperature and the specific volume of water is thought to keep constant, the specific internal energy and entropy of superheated water are equal to that of saturated water at the same temperature tl. (Eqs. (14) and (15)). In this paper, properties of superheated water is labeled with subscript ‘‘sh’’, while symbols without that subscript stand for properties of saturated state.
ul;sh ðtl ; pf Þ ¼ ul ðt l Þ
ð14Þ
sl;sh ðt l ; pf Þ ¼ sl ðt l Þ
ð15Þ
D. Zhang et al. / International Journal of Heat and Mass Transfer 86 (2015) 377–387
(1). heater
(8). cooling water pump
(14). adjusting valve
(2). flash chamber
(9). thermometer
(15). drain valve
(3). electromagnetic valve
(10). barometer
(16). cathetometer
(4). vacuum chamber
(11). pressure transducer
(17). adjusting valve
(5). condenser (6). vacuum pump (7). cooling water tank
(18). data acquisition (12). thermocouples (13). filling valve Fig. 2. Experimental system of static flash.
Fig. 3. Exergy analysis for static flash.
system (19). hi-speed camera
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D. Zhang et al. / International Journal of Heat and Mass Transfer 86 (2015) 377–387
The specific enthalpy of superheated water (hl,sh) can be estimated by Eq. (16), or marked at point 2 in h-s diagram shown as Fig. 3(b). If further assuming that flash steam is saturated under flash chamber pressure pf (point 5 in Fig. 3(b)), equivalent enthalpy and entropy of vaporization for superheated water can be respectively expressed as Eqs. (17) and (18) (Fig. 3(b)).
hl;sh ðt l ; pf Þ ¼ ul ðtl Þ þ pf v l
ð16Þ
0
hfg ¼ hstm ðt s Þ hl;sh ðt l ; pf Þ ¼ hstm ðt l BTDÞ ul ðt l Þ pf ðt l BTDÞv l
ð17Þ
s0fg ¼ sstm ðt s Þ sl;sh ðt l ; pf Þ ¼ sstm ðtl BTDÞ sl ðt l Þ
ð18Þ
4.1.2. Exergy balance During flash, mass decrease of the control volume (waterfilm) is caused by both evaporation and steam-carrying effect, thus its mass balance during differential time is expressed as Eq. (19).
dm ¼ ðdmstm þ dmsc Þ
ð19Þ
exergy destruction, the first three items in the first line of Eq. (25) stand for the all delivered exergy during static flash. In Eq. (25), specifically, ex,U,l,sh is nonflow exergy of superheated water and can be expressed as Eq. (26) according to Eqs. (14) and (15). It also indicates that nonflow exergy of superheated water is equal to that of saturated water at the same temperature (tl). ex,H,l,sh stands for flow exergy of superheated water and is expressed as Eq. (27). The difference between flow and nonflow exergy of superheated water can be easily testified to be Eq. (28). ex,H,stm stands for flow exergy of saturated steam. It can be expressed according to definition of flow exergy (behind first equal sign of Eq. (29)) and can also be expressed by ex,H,l,sh and equivalent specific enthalpy 0 ðhfg Þ and entropy ðs0fg Þ of vaporization for superheated water (behind second equal sign of Eq. (29)). As shown in the second line of Eq. (25), ex,H,stm includes two parts, first is the exergy contained in its latent heat that is widely utilized in industrial flash systems; the second part is that contained in saturated water after its condensation. For saturated flash steam at temperature ts, the exergy contained in its latent heat could be deduced as Eq. (30). Eqs. (26)–(30) are very useful in following deduction.
X X T bm dQ ðdW i pbm dV CV Þ þ ex;H dmin 1 T in X ex;H dmout dEx;destroyed
Strictly speaking, thermal properties of waterfilm and flash steam as well as entrained liquid all change with time, because both temperature of waterfilm, pressure of flash chamber (pf) quickly decay during flash. In order to simplify discussion, it is assumed that during any differential time (ds), these properties all keep constant. Thus, energy balance for this control volume (the waterfilm) during ds can be expressed as Eq. (20), in which the positive direction of work transfer is from control volume. With the departure of flash steam and entrained liquid, control volume shrinks for dVcv (Fig. 3(a)). Considering that flash steam expands as soon as overflowing from the control volume (waterfilm), and its pressure gradually drops from pf at steam–liquid interface to pressure of vacuum chamber (pv) (Fig. 3(a)). Thus, the pressure at steam–liquid interface can still be viewed as pf, and the boundary work (dWi) can be expressed as Eq. (21). Substituting it as well as Eqs. (16) and (17) and Eq. (19) into Eq. (20), energy balance can be simplified step by step as Eq. (22). From its last line, the steam production during ds can be reshaped as Eq. (23).
dEx;CV ¼
0 ¼ dðmul;sh Þ þ hstm dmstm þ hl;sh dmsc þ dW i
ð20Þ
ex;H;stm ¼ hstm hbm T bm ðsstm sbm Þ ¼ ex;H;l;sh þ hfg T bm s0fg
ð29Þ
dW i ¼ pf ðdV CV Þ ¼ pf dðmv l Þ ¼ pf v l dm
ð21Þ
T bm hfg ex;H;fg ¼ ex;H;stm ex;H;l ¼ 1 Ts
ð30Þ
0 ¼ dðmul;sh Þ þ hstm dmstm þ hl;sh dmsc þ pf v l dm
ð24Þ
out
dðmex;U;l;sh Þ ¼ ex;H;stm dmstm ex;H;l;sh dmsc ðdW i pbm dV CV Þ dEx;destroyed |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} all delivered exergy during static flash
¼ ex;H;fg dmstm ex;H;l dmstm ex;H;l;sh dmsc ðpf pbm Þv l dm |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} Industrial concerned
dEx;destroyed
ð25Þ
ex;U;l;sh ¼ ðul;sh ubm Þ T bm ðsl;sh sbm Þ þ pbm ðv l v bm Þ ðul ubm Þ T bm ðsl sbm Þ ¼ ex;U;l
ð26Þ
ex;H;l;sh ¼ hl;sh hbm T bm ðsl sbm Þ
ð27Þ
ex;H;l;sh ex;U;l;sh ¼ ðpf pbm Þv l
ð28Þ 0
0
¼ dðmul Þ þ ðul þ pf v l þ hfg Þdmstm þ ðul þ pf v l Þdmsc þ pf v l dm 0
0
¼ dðmul Þ ul dm pf v l dm þ hfg dmstm þ pf v l dm ¼ mdul þ hfg dmstm
400
ð22Þ 380
ð23Þ
Exergy balance for control volume is expressed as Eq. (24), in which the positive direction of work transfer is also from the control volume [28]. In this paper, the tripod point of pure water is selected as benchmark, or environment, for exergy calculation and is labeled with subscript ‘‘bm’’. Considering there is no mass or heat exchange with environment, and substituting Eq. (21) into exergy balance, it can be simplified as Eq. (25), in which ex,U is nonflow exergy and ex,H is flow exergy [28]. According to the first line of Eq. (25), exergy is released during flash due to the decrease of both mass and exergy of waterfilm. This released exergy converts into 4 parts, including exergy contained in flash steam, exergy contained in entrained liquid, work done against surrounding, as well as exergy destruction. Except
360 fg
m 0 dul hfg
T /K
dmstm ¼
340 o
Tl=343K, (70 C) 320
o
Tl=373K, (100 C) o
T =393K, (120 C) l
300 0
2
4
6
8
10
BTD / K Fig. 4. Equivalent heat absorption temperature versus BTD under different temperatures of waterfilm.
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D. Zhang et al. / International Journal of Heat and Mass Transfer 86 (2015) 377–387
4.1.3. Exergy destruction Exergy destroyed in differential time (ds) is a measure of irreversibility and can be rearranged from the first line of Eq. (25) as Eq. (31). Substituting Eqs. 29, 19, 28 and (23) into it successively, exergy destruction is simplified step by step as Eq. (32), and finally a short expression (last line of Eq. (32)) is reached.
dEx;destroyed ¼ dðmex;U;sh Þ ex;H;stm dmstm ex;H;l;sh dmsc ðpf pbm Þv l dm ð31Þ 0
dEx;destroyed ¼ dðmex;U;l;sh Þ ðex;H;l;sh þ hfg T bm s0fg Þdmstm ex;H;l;sh dmsc ðpf pbm Þv l dm ¼ dðmex;U;l;sh Þ ex;H;l;sh ðdmstm þ dmsc Þ
BTDim ¼
Z sdp 1
sdp
BTDðsÞds
ð35Þ
0
In current experimental range, BTDim varies between 0.86 and 9.8 K. Fig. 5 further displays that BTDim almost keeps constant with rising initial temperature of waterfilm (tl0), but significantly increases with shrinking orifice diameters (D). Fig. 6 displays that BTDim increases with rising superheats (DT) or with reducing initial height of waterfilm (H0). According to these variation laws, BTDim is fitted by Eq. (36). Fig. 7 further displays that 88.6% of relative errors between experimental and fitting BTDim fall into the range of ±40% in current range.
BTDim;fit ¼ a1 þ a2 DT þ a3 H0
ð36Þ
0
þ ðhfg T bm s0fg Þdmstm ðpf pbm Þv l dm where a1 ¼ 35:677 þ 0:786, a2 ¼ 0:072 expð0:0184DÞ, a3 ¼ 14:339 D expð0:046DÞ. The cause of BTD and its dependence on initial conditions are complex and are being researched now. It may be related to bubble nucleation, bubbles rupture, overflowing speed of flash steam, as well as drop speed of flash chamber pressure. So the relation between BTDim and initial conditions in this paper is temporarily set up through this empirical formula. This does not affect following discussion.
¼ mdex;U;l;sh ex;U;l;sh dm þ ex;H;l;sh dm 0
ðhfg T bm s0fg Þdmstm ðpf pbm Þv l dm ¼ mdex;U;l;sh þ ðex;H;l;sh ex;U;l;sh Þdm 0
ðhfg T bm s0fg Þdmstm ðpf pbm Þv l dm ¼ mdex;U;l;sh þ ðpf pbm Þv l dm ! m 0 hfg T bm s0fg 0 dul ðpf pbm Þv l dm hfg
#
0 1 sfg ¼ T bm m 0 dul T l hfg
0
hfg
dul
10
ð32Þ
If further defining equivalent heat absorption temperature (Tfg) as Eq. (33), exergy destruction during ds can be further simplified as Eq. (34). Tfg can be directly calculated through Eqs. (17) and (18) and is displayed in Fig. 4, from which two conclusions can be drawn. First, Tfg is always lower than Tl. It suggests that flash evaporation during ds can be viewed as heat transfer between finite temperature difference between Tl and Tfg, during which some exergy is destroyed. Specifically, heat mdul is rejected by part of waterfilm at temperature Tl, this heat is absorbed by the other part of waterfilm (dmstm) at equivalent heat absorption temperature Tfg to overcome h0 fg, and change itself into flash steam. Second, combined with the fact that dul < 0, exergy destruction is positive (Eq. (34)), indicating that static flash is an irreversible process. Fig. 4 further indicates that, with the decreasing of BTD, Tfg gradually tends to Tl. In ideal case that BTD = 0, Tfg equals Tl, making exergy destruction to be zero and flash to be a reversible process. This suggests that BTD is the main cause for irreversibility of static flash.
1, D=80mm 2, D=20mm 3, D=10mm 4, D=5mm Eq.(36) ΔT=18K H0=0.3m
8
6
im
"
s0fg
BTD / K
¼ m½dul T bm ds þ mdul T bm m
4
2
0 80
85
90
95 o t / C
100
105
110
l0
Fig. 5. Mean boiling temperature difference versus initial temperature of waterfilm under different orifice diameters.
5
0
1 1 dul > 0 ¼ T bm m T l T fg ðt l ; BTDÞ
4
ð33Þ
1 2
ð34Þ
3
3
im
dEx;destroyed
hfg s0fg
BTD / K
T fg ðt l ; BTDÞ ¼
2
1, H0=0.1m 2, H0=0.2m
4.2. Mean boiling temperature difference Above discussion suggests that boiling temperature difference (BTD) plays a major role in exergy destruction during static flash. Fig. 1(a) indicates that it is not a constant but varies with time. In order to evaluate the overall level of BTD and simplify following discussion, mean boiling temperature difference (BTDim) is introduced as integral mean of BTD during flash duration time [0,sdp] (Eq. (35) (shaded area in Fig. 1(a)).
3, H0=0.3m
1
Eq.(36) t0=80oC
0 0
D=20mm
5
10
15
20 25 ΔT / K
30
35
40
45
Fig. 6. Mean boiling temperature difference versus superheat under different initial heights of waterfilm.
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D. Zhang et al. / International Journal of Heat and Mass Transfer 86 (2015) 377–387
to completely take place during [0,sdp] , the relative height of waterfilm, in this paper, is approximately thought to drop linearly from Hr = 1 to Hr = Hrdp during [0,sdp] as Eq. (39). Substituting it into Eqs. (37) and (38), a pair of approximate expressions for EE and EEstm are deduced as Eqs. (40) and (41) respectively, which are recruited as their experimental measurement formulae.
10 88.6% in range of ±40.0%
40.0% →
BTD
im,fit
6
Hr ðsÞ 1 ← -40.0%
4
2
4 BTD
6
im,exp
8
R dp h
10
0
EEstm
/K
Fig. 7. Relative error between fitting and experimental mean boiling temperature difference.
4.3. Exergy efficiency of flash and flash steam Exergy destruction during differential time is deduced as Eq. (34). In order to measure effectiveness of static flash on exergy transfer during flash duration time ([0,sdp]), exergy efficiency of flash (EE) is defined as fraction of the delivered exergy (terms except exergy destruction in the first line of Eq. (25)) to total released exergy from unit mass of initial waterfilm during entire flash duration time as Eq. (37). But in industrial flash systems, exergy contained in latent heat of flash steam is the most concerned, therefore, as shown in Eq. (38), exergy efficiency of flash steam (EEstm) is also introduced as the fraction of exergy transferred into the latent heat of flash steam to total released exergy from unit mass of initial waterfilm during flash duration time.
EE ¼ 1
¼1
¼1
EEstm ¼
¼
0
dEx;destroyed DEx;U;l m0
T bm
R sdp 0
h
ql AH 1 ql AH0 T l
i T1 dul ½tl ðsÞ fg
m
ex;U;l;0 mdp0 ex;U;l;dp h i Rs 1 T bm 0 dp Hr ðsÞ T 1ðsÞ T ½t ðsÞ;BTDð sÞ dul ½t l ðsÞ
1 m0
¼
R sdp
1 m0
l
fg l
ex;U;l;0 Hrdp ex;U;l;dp R sdp 0
R dp 0
0
EE 1
2
0 0
DHrdp
sdp
R dp h
ð37Þ
s
ð39Þ
i h i DH 1 s rdp s T bm T 1ðsÞ T ½t ðs1Þ;BTD dul ½t l ðsÞ im dp l fg l ex;U;l ðt l0 Þ Hrdp ex;U;l ðt l;dp Þ
i DH h ðt s Þ dul ½t l ðsÞ 1 s rdp s 1 TTbms h0 ðtfg;BTD dp im Þ fg l
ð40Þ
ð41Þ
ex;U;l ðt l0 Þ Hrdp ex;U;l ðt l;dp Þ
The denominators of Eqs. (40) and (41) are the same and they represent total released exergy during flash. The released exergy depends on both temperature drop and height drop of waterfilm during [0,sdp]. The first square brackets in numerator of the two equations are also the same. They stand for the mass of superheated waterfilm that takes part in flash evaporation at every instant. In numerator of Eq. (40), the second square brackets stand for fraction of exergy destroyed in unit drop of specific internal energy (du), thus it is named as fraction of destroyed exergy (FDE, Eq. (42)). Similarly, the second square brackets in the numerator of Eq. (41) stands for fraction of exergy that is delivered into the latent heat of flash steam, and is named as fraction of utilized exergy (FUE, Eq. (43)). These two factors are both dimensionless and can be directly calculated in light of Eqs. (33) and (17). Fig. 8 displays that, first, both FDE and FUE are negative, suggesting that it is during the decreasing of specific internal energy that some exergy is destroyed and some is transferred into latent heat of flash steam. Second, the absolute value of FDE decreases with rising waterfilm temperature or with reducing BTDim; while the absolute value of FUE increases with either of them, suggesting that reducing BTDim or rising waterfilm temperature at which flash takes place can minimize irreversibility of flash and improve the quality of flash steam. Besides, Eqs. (40)–(43) also indicates that EE and EEstm do not only depend on the height drop or temperature drop of waterfilm during flash, but also depend on the paths along which the height and temperature of waterfilm drop.
ex;H;fg ðts Þdmstm DEx;U;l m0
ql AH ql AH0
1 TTbms
hfg ðts Þ h0fg
0
dul ½t l ðsÞ
m
ex;U;l;0 mdp0 ex;U;l;dp R dp h ðt s Þ 0 Hr ðsÞ 1 TTbms h0 ðtfg ;BTDÞ dul ½t l ðsÞ fg
l
ex;U;l;0 Hrdp ex;U;l;dp
-0.05 FDE, BTD =0K im
FDE, BTDim=5K
-0.1
ð38Þ
Strictly speaking, Eqs. (37) and (38) can hardly be integrated for two reasons. First, BTD varies with time. Second, Fig. 2 displays photos of flash process taken by high speed camera (cited from Ref. [16]). It suggests that, during flash, flash steam has no time to diffuse out the bulk of waterfilm but fully blends with liquid to form foam-like mixture. These make the height of liquid (Hr) cannot be real-timely measured. Therefore, calculations for the two efficiencies are simplified in following ways. (1) BTD is thought to keep constant at its integral mean BTDim during flash. (2) Considering that the height of waterfilm can only be measured at the initial and final equilibrium state and height drop is assumed
FDE, FUE
/ K, Eq.(36)
8
FDE, BTD =10K im
FUE, BTDim=0K
-0.15
FUE, BTDim=5K FUE, BTDim=10K
-0.2
T =273.2 K bm
-0.25 -0.3 70
80
90
100
110
120
t / oC l
Fig. 8. FDE and FUE versus temperature of waterfilm under different BTDs.
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D. Zhang et al. / International Journal of Heat and Mass Transfer 86 (2015) 377–387
FDE ¼ T bm
1 1 T l ðsÞ T fg ½t l ðsÞ; BTDim
ð42Þ
hfg ðts Þ T bm FUE ¼ 1 T s h0fg ðt l ; BTDim Þ
ð43Þ
According to Eqs. (40) and (41), EE in current experimental range varies between 0.86 and 0.99, while EEstm ranges between 0.037 and 0.99. Their dependences on initial parameters are displayed in Fig. 9 and Fig. 10. Fig. 9 displays that EE slightly decreases with rising initial temperatures of waterfilm (tl0), but EEstm increases with it. Previous study concludes that increasing tl0 reduces relative height drop of waterfilm (DHrdp) [16]. This effect has two results, first, it elevates the relative height of waterfilm at the end of flash duration time (Hrdp), reducing total released exergy during flash. Second, lower DHrdp increases the overall level of relative waterfilm height (Hr) during flash, and thus increases the mass of waterfilm that take part in the finite-temperature-difference heat transfer at every instant. Therefore the total destroyed exergy during flash increases (numerator in Eqs. (40)), making EE drops. Oppositely, more waterfilm taking part in flash boiling at every instant increases steam production. At the same time, increasing tl0 also elevates the overall level of steam temperature and thus improves FUE. In other word,
1
1
2 G1
3 G2
0.8
EE, EE
stm
3
0.6 1, EE, tl0 =80o C 2, EE, tl0 =90o C
2
3, EE, tl0 =100o C
0.4
1, EEstm , tl0 =80o C 2, EEstm , tl0 =90o C 3, EEstm , tl0 =100o C
1
0.2
Eq.(44) Eq.(45) H0 =0.2m D=80mm
0 0
20
10
30 ΔT / K
40
Fig. 9. Exergy efficiency of flash and steam versus superheat under different initial temperatures of waterfilm.
1 3
1
2
D=5mm
0.8
D=10mm
EE, EE
stm
1 D=20mm
0.6
2 D=40mm
1, EE, H0 =0.1m
0.4
3
2, EE, H0 =0.2m 3, EE, H0 =0.3m 1, EEstm , H0 =0.1m
D=80mm
2, EEstm , H0 =0.2m
0.2
3, EEstm , H0 =0.3m Eq.(44) Eq.(45)
EEcal 1
tl0 =90o C, Δ T=18K
0 -3 10
higher tl0 elevates both the quantity and quality of steam (numerator in Eq. (41)), therefore improves EEstm ultimately. Fig. 9 also displays that both EE and EEstm drops with increasing superheat (DT). Former study suggests [13,15] that increasing DT intensifies both heat and mass transfer, reduces both waterfilm temperature (tdp) and relative height (Hrdp, or increasing DHrdp) at sdp. These effects increase total exergy released during flash. For exergy destruction, increasing DHrdp reduces the overall level of Hr, or the mass of waterfilm that takes part in flash boiling at every instant. But on the other side, increasing DT enlarges BTD, and reduces the overall level of waterfilm temperature, thus significantly enlarges FDE, finally makes exergy destruction does not only increase, but increases faster than total released exergy, leading EE to drop. Take tl0 = 100 °C in Fig. 9 for example, when superheat increases from 24.4 K at point G1 to 43.4 K at point G2, the integral mean of Hr during [0,sdp] (represents overall level of Hr) drops from 0.90 to 0.74, for 17.7%, while the integral mean of FDE drops from 19.9 to 33.3, leading total exergy destruction increases from 1.41 to 3.59 for as large as 155%. This increment is faster than that of total released exergy which increases from 27.0 to 47.0, for only 74.1%, and thus causes EE finally to drop. But for flash steam, the reduced overall level of Hr and tl does not only reduces steam production, but also reduces the overall level of steam temperature. Therefore, larger DT reduces both quantity and quality of flash steam, making EEstm drop significantly. Fig. 10 displays the two efficiencies versus flash speed (FS) under different initial waterfilm heights (H0). It suggests that EE slightly increases with accelerating FS or with rising H0, while EEstm significantly drops with either of them. Pervious study indicates that faster FS or higher H0 slightly influences NEFtp (or tdp), but significantly increases DHrdp through intensifying steam-carrying effect. The enhanced DHrdp does not only enlarges total exergy released, but also minimizes overall level of Hr during flash, reducing the mass of waterfilm that takes part in the irreversible boiling heat transfer. In addition, pervious study also suggests that FS is accelerated mainly by enlarging orifice diameter (D) [13]. Enlarging D or elevating H0 could reduce BTDim and thus reduces exergy destruction at every instant, making EE increase. But on the other side, the drop of the overall level of Hr reduces steam production, leading EEstm drops. In brief, although the total released exergy can be enlarged by increasing DT, or elevating H0, or accelerating FS, the enlarged exergy is mainly transferred into the entrained liquid rather than into flash steam. Therefore, thermal performance of industrial flash systems could be improved by elevating initial waterfilm temperature, while at the same time reducing superheat, or initial height of waterfilm or flash speed. At last, EE and EEstm could also be directly calculated through Eqs. (44) and (45) by replacing the experimental value of relative height drop of waterfilm (DHrdp), flash duration time (sdp), temperature of waterfilm (tl) and BTDim in Eqs. (40) and (41) with their fitting or calculation formulae (Eqs. (12), (8), (6) and (36)). The operating conditions of Eqs. (44) and (45) are the same with the experimental range listed in Table 1 Solid lines in Figs. 9 and 10 displays that calculated EE and EEstm match well with their experimental results. Figs. 11 and 12 further display that 98.8% of relative error between experimental and calculated EE in current experimental range fall into ±10%, and 92.6% of that of EEstm fall into ±40%.
10
-2
-1
10 FS / s-1
0
10
T bm
ih R dp h DH 1 s rdp;cal s T 1 T 0
Fig. 10. Exergy efficiency of flash and steam versus flash speed under different initial heights of waterfilm.
EEstm;cal
l;cal
1 fg ½t l;cal ;BTDim;fit
i
dul ðt l;cal Þ
ex;U;l ðtl0 Þ ð1 DHrdp;cal Þex;U;l ðtl;cal;dp Þ
1
10
dp;cal
R dp h 0
i DH h ðt s;cal Þ 1 s rdp;cal s 1 TT bm h0 ðt fg ;BTD dul ðt l;cal Þ dp;cal s;cal im;fit Þ fg l;cal ex;U;l ðtl0 Þ ð1 DHrdp;cal Þex;U;l ðt l;cal;dp Þ
ð44Þ
ð45Þ
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D. Zhang et al. / International Journal of Heat and Mass Transfer 86 (2015) 377–387
1
98.8% in range of ±10.0%
10.0% → 0.6
← -10.0%
cal
EE , Eq.(44)
0.8
0.4
0.2
0 0
0.2
0.4
0.6 EEexp
0.8
1
Fig. 11. Relative error between experimental and calculated exergy efficiency of flash.
1
92.6% in range of ±30.0%
released exergy from unit mass of initial waterfilm during flash duration time was defined as exergy efficiency of flash steam (EEstm, Eq. (38)). Results suggested that, EEstm varied between 0.037 and 0.99 in current range. It could be improved by increasing initial temperature of waterfilm while at the same time reducing superheat, or initial height of waterfilm, or flash speed. At last, on basis of an empirical formula for BTD fitted from experimental data, a pair of calculation formulae for EE and EEstm were set up within acceptable error range. Strictly, study in this paper is just a preliminary exergy analysis on static flash. Because, first, it depends on a series of assumptions stated in Section 4.1 and 4.3. Second, the effectiveness is evaluated on basis of former experimental results at macro level, from which the cause of BTD can hardly be revealed. Now, experimental study on flash evaporation at micro level is being carried out, and instantaneous steam-carrying ratio is also being examined. After these studies, the empirical formula of BTD and some assumptions above, such as the linearly drop of waterfilm height, can be removed and the mechanism of exergy transfer during flash will be better understood.
Conflict of interest None declared.
, Eq.(45)
0.6
EE
0.8
0.4
stm,cal
30.0% →
Acknowledgement This work is supported by National Natural Science Foundation of China (51306148/51125027) and the Fundamental Research Funds for the Central Universities.
← -30.0%
0.2
References 0 0
0.2
0.4
0.6 EEstm,exp
0.8
1
Fig. 12. Relative error between experimental and calculated exergy efficiency of flash steam.
where t s;cal ¼ tl;cal BTDim . 5. Conclusion Exergy transfer during static flash of pure water at different flash speeds was analyzed in this study. First, waterfilm was selected as control volume and the exergy balance within it during differential time was set up, from which exergy destruction was also deduced with help of mass and energy balance. Results suggested that static flash was an irreversible process and could be viewed as heat transfer between finite temperature difference. The boiling temperature difference (BTD) played a major role for exergy destruction. Second, according to our former experimental results (the experimental range of main parameters were listed in Table 1.), the effectiveness of static flash on exergy transfer during entire flash duration time was also evaluated. Exergy efficiency of flash (EE) was introduced as the fraction of the delivered exergy to total released exergy from unit mass of initial waterfilm during flash duration time (Eq. (37)). Results suggested that, it varies between 0.86 and 0.99 in current experimental range, and could be improved by increasing flash speed or initial height of waterfilm, or by minimizing initial temperature of waterfilm or superheat. Considering the exergy contained in latent heat of flash steam was widely utilized in industrial flash systems, its fraction in total
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Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Preliminary exergy analysis of static flash of pure water Dan Zhang ⇑, Junjie Yan, Yingwen Liu, BingChao Zhao State Key Laboratory of Multiphase Flow in Power Engineering, School of Energy & Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, PR China
a r t i c l e
i n f o
Article history: Received 24 September 2014 Received in revised form 1 March 2015 Accepted 6 March 2015
Keywords: Static flash Exergy destruction Exergy efficiency Boiling temperature difference Steam-carrying effect
a b s t r a c t The waterfilm was selected as control volume and exergy transfer within it during static flash of pure water at different flash speeds was analyzed. Exergy balance during differential time was set up and exergy destruction was also deduced. It suggested that static flash was an irreversible process and boiling temperature difference (BTD) played a major role for exergy destruction. In order to measure the effectiveness of static flash on exergy transfer during entire flash duration time, exergy efficiency of flash (EE) was introduced as the fraction of the delivered exergy to the total released exergy from unit mass of initial waterfilm. Exergy efficiency of flash steam (EEstm) was defined as the fraction of exergy contained in latent heat of flash steam to the total released exergy from unit mass of initial waterfilm. Both efficiencies were evaluated and analyzed according to our former experimental results with initial temperature ranging between 46.5 and 104.6 °C, superheat between 1.78 and 43.9 K, initial height of waterfilm between 0.10 and 0.30 m, flash speed between 4.8 104 and 2.18 s1. Results suggested that, first, EE varies between 0.86 and 0.99, and EEstm varied between 0.037 and 0.99 in current experimental range. Second EEstm could be improved by increasing initial temperature of waterfilm while at the same time reducing superheat, or initial height of waterfilm, or flash speed. At last, on basis of an empirical formula for BTD fitted from experimental results, a pair of calculation formulae for EE and EEstm were set up within acceptable error range. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Flash defines the phenomenon that when liquid at certain temperature is exposed to sudden pressure drop below its saturation pressure corresponding to its temperature, it suddenly evaporates, leading its temperature to drop significantly and generating massive flash steam. At the same time, some superheated liquid is entrained away by the flash steam through steam-carrying effect. Flash can be classified into static flash or dynamic flash according to whether the superheated liquid has macroscopic velocity during flash. Static flash includes flash from static waterfilm, flash from static droplet and so on. Dynamic flash includes flash from horizontally moving waterfilm (named as circulatory flash in our previous works), flash from water jet, and so on. Study in this paper focuses only on flash from static waterfilm, so this kind of flash, in following, is named as static flash for short. According to the working fluid, flash can also be classified into flash of pure water, flash of aqueous salt solution, and so on. Flash is a complex phenomenon including both heat and mass transfer. Macroscopically,
⇑ Corresponding author. Tel./fax: +86 (29) 82668705. E-mail address: [email protected] (D. Zhang). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.03.022 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.
heat transfer is caused by evaporation only, while mass transfer is caused by both evaporation and steam-carrying effect. Due to its good performance on heat transfer, flash is widely used in energy recycle, such as geothermal power plant [1]. Besides, flash can also be used to separate nonvolatile solute and volatile solvent from their solution, such as desalination [2] and thin film deposition [3]. Because steam is easier to be transported than liquid, and the latent heat of steam is much larger than the specific heat of liquid water, the latent heat contained in flash steam is the most concerned energy in industrial flash systems. For example, in geothermal power plant, flash is used to convert thermal energy from low-temperature heat source into the latent heat of flash steam that is used to drive steam turbine and deliver work output. While in each unit of MSF (multi-stage flash), the generated flash steam is condensed at upper part of flash chamber and its latent heat is used to heat original seawater in order to improve thermal performance of the entire system. Therefore, mechanism of flash and its thermodynamic properties, especially properties of flash steam, receive wide attention. Static flash was the simplest form of flash evaporation and thus was always selected as prototype to examine its mechanism. Miyatake et al. [4,5] presented experimental study on static flash of pure water with superheat varying between 3 and 5 K. They
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Nomenclature A BTD c D e EE EEstm Ex ex ex,H ex,U FDE FS FUE h H 0 hfg hfg Hr m NEF p Q s s0fg t T Tfg u v
cross-section area of flash chamber (m2) boiling temperature difference (K) specific heat (kJ kg1 K1) orifice diameter (mm) error (–) exergy efficiency of flash (–) exergy efficiency of flash steam (–) exergy (kJ) specific exergy (kJ kg1) flow exergy (kJ kg1) nonflow exergy (kJ kg1) fraction of destroyed exergy (–) flash speed (s1) fraction of utilized exergy (–) specific enthalpy (kJ kg1) height of waterfilm (m) equivalent specific latent heat of vaporization for superheated water (kJ kg1) specific latent heat of vaporization (kJ kg1) relative waterfilm height (–) mass (kg) non-equilibrium fraction (–) pressure (MPa) heat (kJ) specific entropy (kJ kg1 K1) equivalent specific entropy of vaporization for superheated water (kJ kg1 K1) temperature (°C) absolute temperature (K) equivalent heat absorption temperature (K) specific internal energy (kJ kg1) specific volume (m3 kg1)
found that waterfilm temperature dropped quickly at start, then the drop speed slowed down and the temperature equalized at certain value in the end. Thus, they divided flash into fast evaporation stage and gradual evaporation stage. Besides, they also introduced non-equilibrium fraction (NEF, will be clearly introduced in Section.2) to measure completion degree of flash, and suggested that 1-NEF was a measure of energy conversion efficiency for flash. On basis of these concepts, they examined heat transfer properties of flash during fast evaporation stage and found that higher superheat or lower initial height of waterfilm caused flash to take place faster and evaporate more. Saury et al. [6] examined energy conversion during static flash of pure water with superheat ranging between 1 and 35 K. Results suggested that the sensible heat released in the temperature drop of waterfilm could be considered to all change into the latent heat of flash steam. Saury et al. [7] also changed depressurization rate of flash chamber through an adjusting valve installed between flash and vacuum chambers. Result suggested that depressurization rate had nearly nothing to do with final evaporated mass. Kim and Lior [8] also examined static flash and revealed several critical transition points in temperature evolution of waterfilm. Liu et al. [9] performed experiments on flash from aqueous NaCl droplet, and found evaporation rate decreased with increasing environment pressure. Gopalakrishna et al. [10] did experiment of static flash of aqueous NaCl solution with Superheat ranging between 0.5 and 10 K, concentration of NaCl between 0 and 0.035 (mass fraction), and finally proposed a calculation formula for final evaporated mass. In recent years, our research team also did a series of experimental study for static flash of pure water [11] and aqueous NaCl solution [12] at different flash speeds [13]. In our experiments,
V Wi
volume (m3) boundary work (kJ)
Greek symbols DH height drop of waterfilm (m) DT superheat (K) q density (kg m3) s time (s) Subscript 0 bm cal CV dp e exp f fit im l r rf s sc sh stm tg v
start of flash benchmark for exergy calculation calculated value control volume dividing point between fast and gradual evaporation stages equilibrium experimental flash chamber fitting value integral mean liquid relative reference saturation steam-carrying superheated steam tangent point vacuum chamber
waterfilm concentration varied between 0 and 0.15 (mass fraction), superheat between 1.7 and 53.9 K. Flash speed was defined as mean decreasing rate of NEF during fast evaporation stage, and in experiment this speed was adjusted by adding throttle orifice plate with different orifice diameters (5–80 mm) between flash and vacuum chambers. In our research, theoretical height drop of waterfilm during flash was defined as the drop that calculated by heat balance. It was found that actual height drop of waterfilm measured in experiment was always far greater than the theoretical height drop, suggesting that some superheated liquid was directly entrained away by flash steam without taking part in evaporation. This phenomenon was named as steam-carrying effect. Steam-carrying ratio was also proposed as the mass ratio of entrained liquid to flash steam. Experimental results indicated that this ratio varied between 1.7 and 65.4 in current range, thus this effect cannot be neglected. Therefore, properties of boiling heat transfer and steam-carrying effect were studied together by our research team. Results suggested that, increasing flash speed or initial height of waterfilm significantly intensified steam-carrying effect, but had weak influences on boiling heat transfer. While increasing superheat could intensify the two aspects simultaneously. At last, a fitting formula for NEF evolution and a semi-empirical calculation model for steam-carrying effect at different flash speeds were proposed [14], [15]. Besides, according to the first law of thermodynamics, energy conversion during static flash was also analyzed with steam-carrying effect taken into consideration [16]. Studies on dynamic flash was also prevailing. Sami Mutair [17] investigated flash from superheated water jet. Hanshik Chung et al. [18] examined flash of pure water from horizontally moving waterfilm. Our research team also carried out experimental study
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2. Literature synthesis Analysis in this paper is presented in light of several basic concepts proposed by former scholars or our research team. They are also the start points of following deductions and thus are briefly introduced first. During static flash, both temperature of waterfilm (tl) and pressure of flash chamber (pf) quickly drop with time. Fig. 1(a) [12] compares typical evolutions of tl and saturation temperature (ts) corresponding to pf. It suggests that ts at start quickly drops to below tl to certain value, makes the waterfilm to be superheated under pf and thus starts flash boiling, leading tl begin to drop. As shown in Fig. 1(a), ts is always lower than tl during flash in order to maintain waterfilm at superheated state. The difference between tl and ts is defined as boiling temperature difference (BTD, Eq. (1)) which varies with time. The cause of BTD is being studied now and may be related with nucleation of vapor bubble (Fig. 1(a)) [12,26].
BTD ¼ t l t s ðpf Þ
ð1Þ
Finally, pf equalizes at certain value (pfe), the saturation temperature corresponding to pfe is suggested to be theoretical equilibrium temperature of waterfilm (tse, Eq. (2)) by Miyatake et al. [4,5]. tse represents the lowest temperature that waterfilm can reach in theory.
t se ¼ t s ðpfe Þ
ð2Þ
98 FES
t
GES
l
t ←p
96
s
f
Intergral domain of BTD
im
t
BTD(τ)
t / oC
94 92
, Eq.(6)
l,cal
ΔT=6.4K, H =0.30m 0
↓
D=20mm, BTD=1.7K
90
↑
88
tse←pfe
84
τdp
86
0
20
40
60
τ/ s
80
100
120
(a) Boiling temperature difference (Ref.[12]).
1 FES
0.8
NEF NEF , Eq.(5)
GES
fit
A
ΔT=6.4K, H =0.30m 0
D=20mm, τ =21.20s dp
LST, Largest slop tangent
NEF
0.6 NEF
dp
0.2
0 0
τ
dp
0.4
LST
on flash from horizontally moving waterfilm of pure water and aqueous NaCl solution [19,20], and compared their boiling heat transfer properties with that of static flash [11]. In addition, there were also many works on industrial flash systems. Miyatake et al. [21] and Jin and Low [22] did researches on multi-stage flash (MSF) system. N Kahraman et al. carried out exergy analysis of MSF [23], and found that the largest exergy destruction occurred during evaporation in each flash unit. An improvement for custom MSF system was also proposed and analyzed by our research team [24]. Ho Kon Tiat et al. [25] optimized two-stage flash evaporators. Former works revealed basic mechanism of heat and mass transfer during flash and started research on its commercial use, but limitations still existed. First, energy released from superheated waterfilm was considered to all changed into the latent heat of flash steam, without considering the energy loss caused by steam-carrying effect. This loss was considerable with increasing superheat or flash speed and cannot be neglected. Second, energy transfer in former studies were mainly analyzed according to the first law of thermodynamics. These analysis did not consider the maximum possible work that could be delivered during flash at given environment. Therefore, exergy analysis for flash should be carried out. Third, static flash in previous works mainly took place freely; the temperature drop and height drop of waterfilm were out of control. In order to develop its commercial application, controllable flash should be developed and its properties at different flash speeds, especially its effectiveness on exergy transfer at different flash speeds, should be examined. Therefore, in this paper, exergy transfer during static flash of pure water at different flash speed is analyzed with waterfilm selected as control volume (because the phase change during flash primarily takes place within it). Exergy balance is set up and exergy destruction is also deduced. According to experimental data, the effectiveness of static flash on exergy transfer, especially on that transferred into the latent heat of flash steam during flash duration time are both evaluated, and their dependences on initial conditions, such as initial temperature and height of waterfilm, superheat, flash speed were also analyzed and formulized.
B 20
40
60
τ/s
80
100
120
(b) Dividing point between fast/gradual evaporation stages. (Ref.[15]) Fig. 1. Boiling temperature difference and dividing point between fast/gradual evaporation stages. FES, fast evaporation stage. GES, gradual evaporation stage.
Superheat is defined as the difference between initial temperature of waterfilm and tse (Eq. (3)). It is a measure of unstable energy contained in waterfilm and is also viewed as driving force for flash.
DT ¼ t l0 tse
ð3Þ
Non-equilibrium fraction (NEF), is also introduced by Miyatake et al. [4,5] as Eq. (4) to measure completion degree of actual flash. According to experimental results, an empirical formula of NEF during static flash of pure water is proposed by our research team as Eq. (5) [13], upon which temperature evolution of waterfilm can be calculated as Eq. (6). The reference temperature is defined as Eq. (7).
NEFðsÞ ¼
t l ðsÞ t se t l ðsÞ t se ¼ t l0 t se DT
" rffiffiffiffiffiffiffiffia # 2 H0 ql cl NEF fit ðsÞ ¼ erf 2 ks where k ¼ 1:106 DDTH2 0
- 0:535
ð4Þ
ð5Þ
, a2 ¼ 0:0011 þ 0:3400lnDT þ 0:0202D
2
0:0002D .
tl;cal ðsÞ ¼ DT NEF fit ðsÞ þ t se trf ¼
1 ðtl0 þ t se Þ 2
ð6Þ ð7Þ
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Flash is divided into fast evaporation stage and gradual evaporation stage according to the decreasing rate of NEF [15]. The duration of fast evaporation stage is defined as flash duration time ([0,sdp]) and can be calculated by Eq. (8)
sdp;cal ¼ stg
NEFðstg Þ NEF 0 ðstg Þ
stg ¼
where
2b1
b1 2a2 b
,
ð8Þ
a¼
H0 2
qffiffiffiffiffiffia2 qcp
,
k
b ¼ a22 ,
NEF 0 ðsÞ ¼ p2ffiffipffi
abea2 s2b sb1 . Flash speed (FS) is defined as mean decreasing rate of NEF during flash duration time [0,sdp] as Eq. (9). FS can also be directly calculated through Eq. (10) on basis of above empirical formula of NEF (Eq. (5)).
FS ¼
1 NEF dp
ð9Þ
sdp
FScal ¼
1 NEF fit ðsdp;cal Þ
ð10Þ
sdp;cal
Besides, the height of waterfilm also drops significantly during flash, which is caused by both evaporation and steam-carrying effect. If defining relative height of waterfilm as Eq. (11), the height drop of waterfilm during flash can be calculated as Eq. (12) [15].
Hr ¼
H H0
DHrdp;cal ¼
ð11Þ H0 Hdp cl DT ¼ hfg H0 "
4 l 3 stm
ð12Þ
b1 ¼ 0:002D2 0:5043D þ 62:083, b2 is listed in Table 1 of Ref. [15]. 3. Experimental system and uncertainty analysis 3.1. Experimental system Research in this paper is carried out on basis of the same experimental system (Fig. 2) and the same steps clearly stated in our former works [13,14]. The system contains high and low pressure parts. The former includes heater and flash chamber. The flash chamber is a 0.20 0.20 0.50 m rectangular tank. Its front and rear faces are made of tempered glass for observation. The low pressure part includes vacuum chamber, vacuum pump and auxiliary condensing system. Two parts are connected by an Table 1 Experimental range of main parameters and uncertainty analysis. Parameter xi
Experimental range
Absolute uncertainty dxi
Minimal measured value ximin
Maximal uncertainty dxi/ximin
T (°C) H (m)
46.5–104.6 0.10, 0.15, 0.20, 0.25, 0.30 20–1000 8.68 103– 0.118 1.78–43.9 0.86–9.82 4.8 104– 2.18 0.86–0.99 0.037–0.99
0.2 5.0 104
46.5 0.10
4.30 103 0.005
0.0125 1.13 103
20 8.68 103
6.25 104 0.130
– – –
– – –
0.134 0.134 0.189
– –
– –
0.276 0.272
DT (K) BTDim FS (s1) EE EEstm
Uncertainty analysis for all directly and indirectly measured values in experiment is carried out according to method of constant odds—product form proposed by Moffat [27], and the results as well as experimental ranges of main parameters are listed in Table 1.
4.1. Exergy analysis of static flash
where ln C ¼ b1 þ b2 DT 27:6204 expðDT 0:5 Þ þ 10:3227 ln H0 , and
p (MPa)
3.2. Uncertainty analysis
4. Results and analysis
6 # q c l DT ð1 NEF dp;cal Þ 1 þ C H0 FScal hfg q
s (s)
80 mm-diameter electromagnetic valve. In order to adjust flash speed, four thin orifice plates with different orifice diameters (5, 10, 20, 40 mm) are respectively installed at the inlet of electromagnetic valve (section A-A in Fig. 2) in experiments. In trials, heated pure water was filled from heater into flash chamber to reach required initial height of waterfilm. While vacuum pump and auxiliary condensing system were turned on to reduce pressure of vacuum chamber to designed value. Then, the electromagnetic valve was opened and flash suddenly took place. During flash, the quickly decaying temperature of waterfilm, pressure of flash chamber and vacuum chamber were real-timely measured. After flash, final equilibrium height of waterfilm was measured by a cathetometer attached outside the front glass of flash chamber. According to these measured data, exergy analysis of static flash could be carried out. The detailed structure of each component, information of measurement equipments, as well as data processing (including following uncertainty analysis) were the same with our previous works [13,14] and thus are not repeated here.
Exergy analysis of static flash is carried out with the waterfilm selected as control volume (Fig. 3(a)) because the phase change primarily takes place within it. Besides, following assumption are also adopted, (1) flash chamber keeps adiabatic; (2) specific volume of waterfilm (dvl = 0) keeps constant and static pressure inside waterfilm is neglected, thus the pressure of waterfilm is equal to the pressure of flash chamber (pf, Eq. (13)); (3) kinetic and potential energy of steam and entrained liquid are neglected; (4) the state that steam and entrained liquid mix together has no effect on their respective value of entropy; (5) duration of fast evaporation stage is defined as flash duration time, and the height drop of waterfilm is thought to completely take place during that time.
pl ¼ pf
ð13Þ
Considering that waterfilm keeps superheated during flash, its thermal properties, such as internal energy, enthalpy, entropy, are frequently used in following discussion but cannot be found in tabulate data, so calculations of these properties should be firstly discussed. 4.1.1. Thermo-properties of superheated water During flash, waterfilm is superheated water because its temperature tl is always higher than the saturation temperature (ts) corresponding to the pressure of flash chamber (pf) (Fig. 1(a)). Considering that specific internal energy depends only on temperature and the specific volume of water is thought to keep constant, the specific internal energy and entropy of superheated water are equal to that of saturated water at the same temperature tl. (Eqs. (14) and (15)). In this paper, properties of superheated water is labeled with subscript ‘‘sh’’, while symbols without that subscript stand for properties of saturated state.
ul;sh ðtl ; pf Þ ¼ ul ðt l Þ
ð14Þ
sl;sh ðt l ; pf Þ ¼ sl ðt l Þ
ð15Þ
D. Zhang et al. / International Journal of Heat and Mass Transfer 86 (2015) 377–387
(1). heater
(8). cooling water pump
(14). adjusting valve
(2). flash chamber
(9). thermometer
(15). drain valve
(3). electromagnetic valve
(10). barometer
(16). cathetometer
(4). vacuum chamber
(11). pressure transducer
(17). adjusting valve
(5). condenser (6). vacuum pump (7). cooling water tank
(18). data acquisition (12). thermocouples (13). filling valve Fig. 2. Experimental system of static flash.
Fig. 3. Exergy analysis for static flash.
system (19). hi-speed camera
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The specific enthalpy of superheated water (hl,sh) can be estimated by Eq. (16), or marked at point 2 in h-s diagram shown as Fig. 3(b). If further assuming that flash steam is saturated under flash chamber pressure pf (point 5 in Fig. 3(b)), equivalent enthalpy and entropy of vaporization for superheated water can be respectively expressed as Eqs. (17) and (18) (Fig. 3(b)).
hl;sh ðt l ; pf Þ ¼ ul ðtl Þ þ pf v l
ð16Þ
0
hfg ¼ hstm ðt s Þ hl;sh ðt l ; pf Þ ¼ hstm ðt l BTDÞ ul ðt l Þ pf ðt l BTDÞv l
ð17Þ
s0fg ¼ sstm ðt s Þ sl;sh ðt l ; pf Þ ¼ sstm ðtl BTDÞ sl ðt l Þ
ð18Þ
4.1.2. Exergy balance During flash, mass decrease of the control volume (waterfilm) is caused by both evaporation and steam-carrying effect, thus its mass balance during differential time is expressed as Eq. (19).
dm ¼ ðdmstm þ dmsc Þ
ð19Þ
exergy destruction, the first three items in the first line of Eq. (25) stand for the all delivered exergy during static flash. In Eq. (25), specifically, ex,U,l,sh is nonflow exergy of superheated water and can be expressed as Eq. (26) according to Eqs. (14) and (15). It also indicates that nonflow exergy of superheated water is equal to that of saturated water at the same temperature (tl). ex,H,l,sh stands for flow exergy of superheated water and is expressed as Eq. (27). The difference between flow and nonflow exergy of superheated water can be easily testified to be Eq. (28). ex,H,stm stands for flow exergy of saturated steam. It can be expressed according to definition of flow exergy (behind first equal sign of Eq. (29)) and can also be expressed by ex,H,l,sh and equivalent specific enthalpy 0 ðhfg Þ and entropy ðs0fg Þ of vaporization for superheated water (behind second equal sign of Eq. (29)). As shown in the second line of Eq. (25), ex,H,stm includes two parts, first is the exergy contained in its latent heat that is widely utilized in industrial flash systems; the second part is that contained in saturated water after its condensation. For saturated flash steam at temperature ts, the exergy contained in its latent heat could be deduced as Eq. (30). Eqs. (26)–(30) are very useful in following deduction.
X X T bm dQ ðdW i pbm dV CV Þ þ ex;H dmin 1 T in X ex;H dmout dEx;destroyed
Strictly speaking, thermal properties of waterfilm and flash steam as well as entrained liquid all change with time, because both temperature of waterfilm, pressure of flash chamber (pf) quickly decay during flash. In order to simplify discussion, it is assumed that during any differential time (ds), these properties all keep constant. Thus, energy balance for this control volume (the waterfilm) during ds can be expressed as Eq. (20), in which the positive direction of work transfer is from control volume. With the departure of flash steam and entrained liquid, control volume shrinks for dVcv (Fig. 3(a)). Considering that flash steam expands as soon as overflowing from the control volume (waterfilm), and its pressure gradually drops from pf at steam–liquid interface to pressure of vacuum chamber (pv) (Fig. 3(a)). Thus, the pressure at steam–liquid interface can still be viewed as pf, and the boundary work (dWi) can be expressed as Eq. (21). Substituting it as well as Eqs. (16) and (17) and Eq. (19) into Eq. (20), energy balance can be simplified step by step as Eq. (22). From its last line, the steam production during ds can be reshaped as Eq. (23).
dEx;CV ¼
0 ¼ dðmul;sh Þ þ hstm dmstm þ hl;sh dmsc þ dW i
ð20Þ
ex;H;stm ¼ hstm hbm T bm ðsstm sbm Þ ¼ ex;H;l;sh þ hfg T bm s0fg
ð29Þ
dW i ¼ pf ðdV CV Þ ¼ pf dðmv l Þ ¼ pf v l dm
ð21Þ
T bm hfg ex;H;fg ¼ ex;H;stm ex;H;l ¼ 1 Ts
ð30Þ
0 ¼ dðmul;sh Þ þ hstm dmstm þ hl;sh dmsc þ pf v l dm
ð24Þ
out
dðmex;U;l;sh Þ ¼ ex;H;stm dmstm ex;H;l;sh dmsc ðdW i pbm dV CV Þ dEx;destroyed |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} all delivered exergy during static flash
¼ ex;H;fg dmstm ex;H;l dmstm ex;H;l;sh dmsc ðpf pbm Þv l dm |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} Industrial concerned
dEx;destroyed
ð25Þ
ex;U;l;sh ¼ ðul;sh ubm Þ T bm ðsl;sh sbm Þ þ pbm ðv l v bm Þ ðul ubm Þ T bm ðsl sbm Þ ¼ ex;U;l
ð26Þ
ex;H;l;sh ¼ hl;sh hbm T bm ðsl sbm Þ
ð27Þ
ex;H;l;sh ex;U;l;sh ¼ ðpf pbm Þv l
ð28Þ 0
0
¼ dðmul Þ þ ðul þ pf v l þ hfg Þdmstm þ ðul þ pf v l Þdmsc þ pf v l dm 0
0
¼ dðmul Þ ul dm pf v l dm þ hfg dmstm þ pf v l dm ¼ mdul þ hfg dmstm
400
ð22Þ 380
ð23Þ
Exergy balance for control volume is expressed as Eq. (24), in which the positive direction of work transfer is also from the control volume [28]. In this paper, the tripod point of pure water is selected as benchmark, or environment, for exergy calculation and is labeled with subscript ‘‘bm’’. Considering there is no mass or heat exchange with environment, and substituting Eq. (21) into exergy balance, it can be simplified as Eq. (25), in which ex,U is nonflow exergy and ex,H is flow exergy [28]. According to the first line of Eq. (25), exergy is released during flash due to the decrease of both mass and exergy of waterfilm. This released exergy converts into 4 parts, including exergy contained in flash steam, exergy contained in entrained liquid, work done against surrounding, as well as exergy destruction. Except
360 fg
m 0 dul hfg
T /K
dmstm ¼
340 o
Tl=343K, (70 C) 320
o
Tl=373K, (100 C) o
T =393K, (120 C) l
300 0
2
4
6
8
10
BTD / K Fig. 4. Equivalent heat absorption temperature versus BTD under different temperatures of waterfilm.
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4.1.3. Exergy destruction Exergy destroyed in differential time (ds) is a measure of irreversibility and can be rearranged from the first line of Eq. (25) as Eq. (31). Substituting Eqs. 29, 19, 28 and (23) into it successively, exergy destruction is simplified step by step as Eq. (32), and finally a short expression (last line of Eq. (32)) is reached.
dEx;destroyed ¼ dðmex;U;sh Þ ex;H;stm dmstm ex;H;l;sh dmsc ðpf pbm Þv l dm ð31Þ 0
dEx;destroyed ¼ dðmex;U;l;sh Þ ðex;H;l;sh þ hfg T bm s0fg Þdmstm ex;H;l;sh dmsc ðpf pbm Þv l dm ¼ dðmex;U;l;sh Þ ex;H;l;sh ðdmstm þ dmsc Þ
BTDim ¼
Z sdp 1
sdp
BTDðsÞds
ð35Þ
0
In current experimental range, BTDim varies between 0.86 and 9.8 K. Fig. 5 further displays that BTDim almost keeps constant with rising initial temperature of waterfilm (tl0), but significantly increases with shrinking orifice diameters (D). Fig. 6 displays that BTDim increases with rising superheats (DT) or with reducing initial height of waterfilm (H0). According to these variation laws, BTDim is fitted by Eq. (36). Fig. 7 further displays that 88.6% of relative errors between experimental and fitting BTDim fall into the range of ±40% in current range.
BTDim;fit ¼ a1 þ a2 DT þ a3 H0
ð36Þ
0
þ ðhfg T bm s0fg Þdmstm ðpf pbm Þv l dm where a1 ¼ 35:677 þ 0:786, a2 ¼ 0:072 expð0:0184DÞ, a3 ¼ 14:339 D expð0:046DÞ. The cause of BTD and its dependence on initial conditions are complex and are being researched now. It may be related to bubble nucleation, bubbles rupture, overflowing speed of flash steam, as well as drop speed of flash chamber pressure. So the relation between BTDim and initial conditions in this paper is temporarily set up through this empirical formula. This does not affect following discussion.
¼ mdex;U;l;sh ex;U;l;sh dm þ ex;H;l;sh dm 0
ðhfg T bm s0fg Þdmstm ðpf pbm Þv l dm ¼ mdex;U;l;sh þ ðex;H;l;sh ex;U;l;sh Þdm 0
ðhfg T bm s0fg Þdmstm ðpf pbm Þv l dm ¼ mdex;U;l;sh þ ðpf pbm Þv l dm ! m 0 hfg T bm s0fg 0 dul ðpf pbm Þv l dm hfg
#
0 1 sfg ¼ T bm m 0 dul T l hfg
0
hfg
dul
10
ð32Þ
If further defining equivalent heat absorption temperature (Tfg) as Eq. (33), exergy destruction during ds can be further simplified as Eq. (34). Tfg can be directly calculated through Eqs. (17) and (18) and is displayed in Fig. 4, from which two conclusions can be drawn. First, Tfg is always lower than Tl. It suggests that flash evaporation during ds can be viewed as heat transfer between finite temperature difference between Tl and Tfg, during which some exergy is destroyed. Specifically, heat mdul is rejected by part of waterfilm at temperature Tl, this heat is absorbed by the other part of waterfilm (dmstm) at equivalent heat absorption temperature Tfg to overcome h0 fg, and change itself into flash steam. Second, combined with the fact that dul < 0, exergy destruction is positive (Eq. (34)), indicating that static flash is an irreversible process. Fig. 4 further indicates that, with the decreasing of BTD, Tfg gradually tends to Tl. In ideal case that BTD = 0, Tfg equals Tl, making exergy destruction to be zero and flash to be a reversible process. This suggests that BTD is the main cause for irreversibility of static flash.
1, D=80mm 2, D=20mm 3, D=10mm 4, D=5mm Eq.(36) ΔT=18K H0=0.3m
8
6
im
"
s0fg
BTD / K
¼ m½dul T bm ds þ mdul T bm m
4
2
0 80
85
90
95 o t / C
100
105
110
l0
Fig. 5. Mean boiling temperature difference versus initial temperature of waterfilm under different orifice diameters.
5
0
1 1 dul > 0 ¼ T bm m T l T fg ðt l ; BTDÞ
4
ð33Þ
1 2
ð34Þ
3
3
im
dEx;destroyed
hfg s0fg
BTD / K
T fg ðt l ; BTDÞ ¼
2
1, H0=0.1m 2, H0=0.2m
4.2. Mean boiling temperature difference Above discussion suggests that boiling temperature difference (BTD) plays a major role in exergy destruction during static flash. Fig. 1(a) indicates that it is not a constant but varies with time. In order to evaluate the overall level of BTD and simplify following discussion, mean boiling temperature difference (BTDim) is introduced as integral mean of BTD during flash duration time [0,sdp] (Eq. (35) (shaded area in Fig. 1(a)).
3, H0=0.3m
1
Eq.(36) t0=80oC
0 0
D=20mm
5
10
15
20 25 ΔT / K
30
35
40
45
Fig. 6. Mean boiling temperature difference versus superheat under different initial heights of waterfilm.
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D. Zhang et al. / International Journal of Heat and Mass Transfer 86 (2015) 377–387
to completely take place during [0,sdp] , the relative height of waterfilm, in this paper, is approximately thought to drop linearly from Hr = 1 to Hr = Hrdp during [0,sdp] as Eq. (39). Substituting it into Eqs. (37) and (38), a pair of approximate expressions for EE and EEstm are deduced as Eqs. (40) and (41) respectively, which are recruited as their experimental measurement formulae.
10 88.6% in range of ±40.0%
40.0% →
BTD
im,fit
6
Hr ðsÞ 1 ← -40.0%
4
2
4 BTD
6
im,exp
8
R dp h
10
0
EEstm
/K
Fig. 7. Relative error between fitting and experimental mean boiling temperature difference.
4.3. Exergy efficiency of flash and flash steam Exergy destruction during differential time is deduced as Eq. (34). In order to measure effectiveness of static flash on exergy transfer during flash duration time ([0,sdp]), exergy efficiency of flash (EE) is defined as fraction of the delivered exergy (terms except exergy destruction in the first line of Eq. (25)) to total released exergy from unit mass of initial waterfilm during entire flash duration time as Eq. (37). But in industrial flash systems, exergy contained in latent heat of flash steam is the most concerned, therefore, as shown in Eq. (38), exergy efficiency of flash steam (EEstm) is also introduced as the fraction of exergy transferred into the latent heat of flash steam to total released exergy from unit mass of initial waterfilm during flash duration time.
EE ¼ 1
¼1
¼1
EEstm ¼
¼
0
dEx;destroyed DEx;U;l m0
T bm
R sdp 0
h
ql AH 1 ql AH0 T l
i T1 dul ½tl ðsÞ fg
m
ex;U;l;0 mdp0 ex;U;l;dp h i Rs 1 T bm 0 dp Hr ðsÞ T 1ðsÞ T ½t ðsÞ;BTDð sÞ dul ½t l ðsÞ
1 m0
¼
R sdp
1 m0
l
fg l
ex;U;l;0 Hrdp ex;U;l;dp R sdp 0
R dp 0
0
EE 1
2
0 0
DHrdp
sdp
R dp h
ð37Þ
s
ð39Þ
i h i DH 1 s rdp s T bm T 1ðsÞ T ½t ðs1Þ;BTD dul ½t l ðsÞ im dp l fg l ex;U;l ðt l0 Þ Hrdp ex;U;l ðt l;dp Þ
i DH h ðt s Þ dul ½t l ðsÞ 1 s rdp s 1 TTbms h0 ðtfg;BTD dp im Þ fg l
ð40Þ
ð41Þ
ex;U;l ðt l0 Þ Hrdp ex;U;l ðt l;dp Þ
The denominators of Eqs. (40) and (41) are the same and they represent total released exergy during flash. The released exergy depends on both temperature drop and height drop of waterfilm during [0,sdp]. The first square brackets in numerator of the two equations are also the same. They stand for the mass of superheated waterfilm that takes part in flash evaporation at every instant. In numerator of Eq. (40), the second square brackets stand for fraction of exergy destroyed in unit drop of specific internal energy (du), thus it is named as fraction of destroyed exergy (FDE, Eq. (42)). Similarly, the second square brackets in the numerator of Eq. (41) stands for fraction of exergy that is delivered into the latent heat of flash steam, and is named as fraction of utilized exergy (FUE, Eq. (43)). These two factors are both dimensionless and can be directly calculated in light of Eqs. (33) and (17). Fig. 8 displays that, first, both FDE and FUE are negative, suggesting that it is during the decreasing of specific internal energy that some exergy is destroyed and some is transferred into latent heat of flash steam. Second, the absolute value of FDE decreases with rising waterfilm temperature or with reducing BTDim; while the absolute value of FUE increases with either of them, suggesting that reducing BTDim or rising waterfilm temperature at which flash takes place can minimize irreversibility of flash and improve the quality of flash steam. Besides, Eqs. (40)–(43) also indicates that EE and EEstm do not only depend on the height drop or temperature drop of waterfilm during flash, but also depend on the paths along which the height and temperature of waterfilm drop.
ex;H;fg ðts Þdmstm DEx;U;l m0
ql AH ql AH0
1 TTbms
hfg ðts Þ h0fg
0
dul ½t l ðsÞ
m
ex;U;l;0 mdp0 ex;U;l;dp R dp h ðt s Þ 0 Hr ðsÞ 1 TTbms h0 ðtfg ;BTDÞ dul ½t l ðsÞ fg
l
ex;U;l;0 Hrdp ex;U;l;dp
-0.05 FDE, BTD =0K im
FDE, BTDim=5K
-0.1
ð38Þ
Strictly speaking, Eqs. (37) and (38) can hardly be integrated for two reasons. First, BTD varies with time. Second, Fig. 2 displays photos of flash process taken by high speed camera (cited from Ref. [16]). It suggests that, during flash, flash steam has no time to diffuse out the bulk of waterfilm but fully blends with liquid to form foam-like mixture. These make the height of liquid (Hr) cannot be real-timely measured. Therefore, calculations for the two efficiencies are simplified in following ways. (1) BTD is thought to keep constant at its integral mean BTDim during flash. (2) Considering that the height of waterfilm can only be measured at the initial and final equilibrium state and height drop is assumed
FDE, FUE
/ K, Eq.(36)
8
FDE, BTD =10K im
FUE, BTDim=0K
-0.15
FUE, BTDim=5K FUE, BTDim=10K
-0.2
T =273.2 K bm
-0.25 -0.3 70
80
90
100
110
120
t / oC l
Fig. 8. FDE and FUE versus temperature of waterfilm under different BTDs.
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D. Zhang et al. / International Journal of Heat and Mass Transfer 86 (2015) 377–387
FDE ¼ T bm
1 1 T l ðsÞ T fg ½t l ðsÞ; BTDim
ð42Þ
hfg ðts Þ T bm FUE ¼ 1 T s h0fg ðt l ; BTDim Þ
ð43Þ
According to Eqs. (40) and (41), EE in current experimental range varies between 0.86 and 0.99, while EEstm ranges between 0.037 and 0.99. Their dependences on initial parameters are displayed in Fig. 9 and Fig. 10. Fig. 9 displays that EE slightly decreases with rising initial temperatures of waterfilm (tl0), but EEstm increases with it. Previous study concludes that increasing tl0 reduces relative height drop of waterfilm (DHrdp) [16]. This effect has two results, first, it elevates the relative height of waterfilm at the end of flash duration time (Hrdp), reducing total released exergy during flash. Second, lower DHrdp increases the overall level of relative waterfilm height (Hr) during flash, and thus increases the mass of waterfilm that take part in the finite-temperature-difference heat transfer at every instant. Therefore the total destroyed exergy during flash increases (numerator in Eqs. (40)), making EE drops. Oppositely, more waterfilm taking part in flash boiling at every instant increases steam production. At the same time, increasing tl0 also elevates the overall level of steam temperature and thus improves FUE. In other word,
1
1
2 G1
3 G2
0.8
EE, EE
stm
3
0.6 1, EE, tl0 =80o C 2, EE, tl0 =90o C
2
3, EE, tl0 =100o C
0.4
1, EEstm , tl0 =80o C 2, EEstm , tl0 =90o C 3, EEstm , tl0 =100o C
1
0.2
Eq.(44) Eq.(45) H0 =0.2m D=80mm
0 0
20
10
30 ΔT / K
40
Fig. 9. Exergy efficiency of flash and steam versus superheat under different initial temperatures of waterfilm.
1 3
1
2
D=5mm
0.8
D=10mm
EE, EE
stm
1 D=20mm
0.6
2 D=40mm
1, EE, H0 =0.1m
0.4
3
2, EE, H0 =0.2m 3, EE, H0 =0.3m 1, EEstm , H0 =0.1m
D=80mm
2, EEstm , H0 =0.2m
0.2
3, EEstm , H0 =0.3m Eq.(44) Eq.(45)
EEcal 1
tl0 =90o C, Δ T=18K
0 -3 10
higher tl0 elevates both the quantity and quality of steam (numerator in Eq. (41)), therefore improves EEstm ultimately. Fig. 9 also displays that both EE and EEstm drops with increasing superheat (DT). Former study suggests [13,15] that increasing DT intensifies both heat and mass transfer, reduces both waterfilm temperature (tdp) and relative height (Hrdp, or increasing DHrdp) at sdp. These effects increase total exergy released during flash. For exergy destruction, increasing DHrdp reduces the overall level of Hr, or the mass of waterfilm that takes part in flash boiling at every instant. But on the other side, increasing DT enlarges BTD, and reduces the overall level of waterfilm temperature, thus significantly enlarges FDE, finally makes exergy destruction does not only increase, but increases faster than total released exergy, leading EE to drop. Take tl0 = 100 °C in Fig. 9 for example, when superheat increases from 24.4 K at point G1 to 43.4 K at point G2, the integral mean of Hr during [0,sdp] (represents overall level of Hr) drops from 0.90 to 0.74, for 17.7%, while the integral mean of FDE drops from 19.9 to 33.3, leading total exergy destruction increases from 1.41 to 3.59 for as large as 155%. This increment is faster than that of total released exergy which increases from 27.0 to 47.0, for only 74.1%, and thus causes EE finally to drop. But for flash steam, the reduced overall level of Hr and tl does not only reduces steam production, but also reduces the overall level of steam temperature. Therefore, larger DT reduces both quantity and quality of flash steam, making EEstm drop significantly. Fig. 10 displays the two efficiencies versus flash speed (FS) under different initial waterfilm heights (H0). It suggests that EE slightly increases with accelerating FS or with rising H0, while EEstm significantly drops with either of them. Pervious study indicates that faster FS or higher H0 slightly influences NEFtp (or tdp), but significantly increases DHrdp through intensifying steam-carrying effect. The enhanced DHrdp does not only enlarges total exergy released, but also minimizes overall level of Hr during flash, reducing the mass of waterfilm that takes part in the irreversible boiling heat transfer. In addition, pervious study also suggests that FS is accelerated mainly by enlarging orifice diameter (D) [13]. Enlarging D or elevating H0 could reduce BTDim and thus reduces exergy destruction at every instant, making EE increase. But on the other side, the drop of the overall level of Hr reduces steam production, leading EEstm drops. In brief, although the total released exergy can be enlarged by increasing DT, or elevating H0, or accelerating FS, the enlarged exergy is mainly transferred into the entrained liquid rather than into flash steam. Therefore, thermal performance of industrial flash systems could be improved by elevating initial waterfilm temperature, while at the same time reducing superheat, or initial height of waterfilm or flash speed. At last, EE and EEstm could also be directly calculated through Eqs. (44) and (45) by replacing the experimental value of relative height drop of waterfilm (DHrdp), flash duration time (sdp), temperature of waterfilm (tl) and BTDim in Eqs. (40) and (41) with their fitting or calculation formulae (Eqs. (12), (8), (6) and (36)). The operating conditions of Eqs. (44) and (45) are the same with the experimental range listed in Table 1 Solid lines in Figs. 9 and 10 displays that calculated EE and EEstm match well with their experimental results. Figs. 11 and 12 further display that 98.8% of relative error between experimental and calculated EE in current experimental range fall into ±10%, and 92.6% of that of EEstm fall into ±40%.
10
-2
-1
10 FS / s-1
0
10
T bm
ih R dp h DH 1 s rdp;cal s T 1 T 0
Fig. 10. Exergy efficiency of flash and steam versus flash speed under different initial heights of waterfilm.
EEstm;cal
l;cal
1 fg ½t l;cal ;BTDim;fit
i
dul ðt l;cal Þ
ex;U;l ðtl0 Þ ð1 DHrdp;cal Þex;U;l ðtl;cal;dp Þ
1
10
dp;cal
R dp h 0
i DH h ðt s;cal Þ 1 s rdp;cal s 1 TT bm h0 ðt fg ;BTD dul ðt l;cal Þ dp;cal s;cal im;fit Þ fg l;cal ex;U;l ðtl0 Þ ð1 DHrdp;cal Þex;U;l ðt l;cal;dp Þ
ð44Þ
ð45Þ
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D. Zhang et al. / International Journal of Heat and Mass Transfer 86 (2015) 377–387
1
98.8% in range of ±10.0%
10.0% → 0.6
← -10.0%
cal
EE , Eq.(44)
0.8
0.4
0.2
0 0
0.2
0.4
0.6 EEexp
0.8
1
Fig. 11. Relative error between experimental and calculated exergy efficiency of flash.
1
92.6% in range of ±30.0%
released exergy from unit mass of initial waterfilm during flash duration time was defined as exergy efficiency of flash steam (EEstm, Eq. (38)). Results suggested that, EEstm varied between 0.037 and 0.99 in current range. It could be improved by increasing initial temperature of waterfilm while at the same time reducing superheat, or initial height of waterfilm, or flash speed. At last, on basis of an empirical formula for BTD fitted from experimental data, a pair of calculation formulae for EE and EEstm were set up within acceptable error range. Strictly, study in this paper is just a preliminary exergy analysis on static flash. Because, first, it depends on a series of assumptions stated in Section 4.1 and 4.3. Second, the effectiveness is evaluated on basis of former experimental results at macro level, from which the cause of BTD can hardly be revealed. Now, experimental study on flash evaporation at micro level is being carried out, and instantaneous steam-carrying ratio is also being examined. After these studies, the empirical formula of BTD and some assumptions above, such as the linearly drop of waterfilm height, can be removed and the mechanism of exergy transfer during flash will be better understood.
Conflict of interest None declared.
, Eq.(45)
0.6
EE
0.8
0.4
stm,cal
30.0% →
Acknowledgement This work is supported by National Natural Science Foundation of China (51306148/51125027) and the Fundamental Research Funds for the Central Universities.
← -30.0%
0.2
References 0 0
0.2
0.4
0.6 EEstm,exp
0.8
1
Fig. 12. Relative error between experimental and calculated exergy efficiency of flash steam.
where t s;cal ¼ tl;cal BTDim . 5. Conclusion Exergy transfer during static flash of pure water at different flash speeds was analyzed in this study. First, waterfilm was selected as control volume and the exergy balance within it during differential time was set up, from which exergy destruction was also deduced with help of mass and energy balance. Results suggested that static flash was an irreversible process and could be viewed as heat transfer between finite temperature difference. The boiling temperature difference (BTD) played a major role for exergy destruction. Second, according to our former experimental results (the experimental range of main parameters were listed in Table 1.), the effectiveness of static flash on exergy transfer during entire flash duration time was also evaluated. Exergy efficiency of flash (EE) was introduced as the fraction of the delivered exergy to total released exergy from unit mass of initial waterfilm during flash duration time (Eq. (37)). Results suggested that, it varies between 0.86 and 0.99 in current experimental range, and could be improved by increasing flash speed or initial height of waterfilm, or by minimizing initial temperature of waterfilm or superheat. Considering the exergy contained in latent heat of flash steam was widely utilized in industrial flash systems, its fraction in total
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