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Influence of initial heat generation on dynamic characteristics of transient boiling crisis of water
International Journal of Heat and Mass Transfer 53 (2010) 1851–1855
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Influence of initial heat generation on dynamic characteristics of transient boiling crisis of water V.I. Deev, K.V. Kutsenko, A.A. Lavrukhin *, V.S. Kharitonov Department of Thermophysics, National Research Nuclear University ‘‘MEPhI”, 31 Kashirskoe shosse, Moscow 115409, Russian Federation
a r t i c l e
i n f o
Article history: Received 22 September 2009 Received in revised form 19 October 2009 Accepted 22 December 2009 Available online 22 January 2010 Keywords: Power transients Water pool boiling Critical heat flux Physical models
a b s t r a c t The dynamic characteristics of transient boiling crisis were measured in the experiments with saturated water in a pool at atmospheric pressure. It was shown that initial heat generation in a heater strongly affects the process of transition from nucleate to film boiling under conditions of fast increase of heating power. A technique of calculation of critical heat flux, temperature drop and critical time interval is presented. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Under sharp increase of heating power at the initial stage of non-steady thermal process, there is always some time interval tcr during which, even after step power input above critical level, the rate of heat transfer from a heater into a liquid remains sufficiently high. Thus, if the duration of heating power pulse is less than tcr, there is an opportunity of heat transfer to liquid without significant overheating of the heated surface. Otherwise, the beginning of film boiling results in a sharp increase of heater temperature, which may lead to its destruction. Therefore, knowledge of value of tcr as well as main parameters, on which this value depends, is of big practical interest. The existence of heat flux from a heated wall q0 before step power input significantly changes thermal and hydrodynamic conditions nearby the heater and, therefore, influences on dynamic characteristics of transition to film boiling. At present, there is separate information on influence of initial heat generation in the heater on the value of critical time interval only for cryogenic liquids. Thus, for example, according to the paper [1], existence of preliminary heat load may result in reduction of critical time interval in liquid helium in one and a half times. 2. The experimental results The experimental apparatus and experimental technique are described fully enough in the papers [2,3]. A horizontal platinum * Corresponding author. Tel.: +7 495 3247328. E-mail addresses: [email protected], [email protected] (A.A. Lavrukhin). 0017-9310/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2010.01.004
wire 0.1 mm in diameter immersed in the vessel with saturated water under atmospheric pressure was used as a heater. Heat generation in the wire increased stepwise from an initial level of heat generation corresponding to heat load q0 to a set level qh. Experiments were conducted at several values q0 in the range of 0.014– 0.56 MW m2. In this range of initial heat loads, different modes of heat transfer were realized: natural convection if q0 < qeb, the onset of nucleate boiling when q0 qeb ( 0.2 MW m2), and developed nucleate boiling when q0 > qeb (here qeb is a heat flux when boiling of water on the surface of the heater begins). In any conditions, if, as a result of additional step power input, the total heat load of the heater exceeded critical heat flux at steady conditions qcr1, nucleate boiling before heat transfer crisis was always non-steady, and after some time a fast increase of the heated wall temperature began (heat transfer crisis at non-steady heating). Some experimentally obtained dynamic dependencies of wall superheat DT of the heater and the heat flux transferred to the liquid q on time t are shown in Figs. 1 and 2. The heat flux to the liquid as a function of time was calculated by using the experimental data and an equation of heat balance
d dðDTðtÞÞ qðtÞ ¼ qh ðtÞ ðcqÞh ; 4 dt
ð1Þ
where d is the diameter and (cq)h is the volumetric heat capacity of the heater. Experimental dynamic curves may be divided on two types. The first type corresponds to those curves that show strong fluctuations of temperature of the heater in metastable boiling stage preceding heat transfer crisis (Figs. 1a and 2a). For these situations, the heat flux q on the heated surface is on the average practically equal to the total heat load (q qh). In this case, parameters of transient
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Nomenclature a c d Dhv Ja k n P0 q qcr1 qh s DT DTcr1 t t1
thermal diffusivity specific heat capacity diameter of a wire latent heat of vaporization Jakob number, Ja = cqDT/(q00 Dhv) factor in Eq. (4) nucleation site density normalizing multiplier heat flux first critical heat flux at steady conditions heat load of a heater part of the surface occupied by vapor liquid (wall) superheat about saturation temperature first critical temperature drop at steady conditions time birth time of vapor bubble
Fig. 1. The variation of heater superheat DT and heat flux q on a heated surface with time t at q0 = 0.014 MW m2: } DT – experimental data; O q – calculation by using calculation using the model from the paper [3]. Eq. (1);
Greek symbols q density q00 vapor density u void fraction Subscripts 0 initial cal calculated value cr critical value d departure eb ebullition exp experimental value h heater mb metastable boiling
crisis qcr, DTcr were determined at the time moment tcr when the heater temperature began to increase sharply. It is to note that similar type of dynamic curves was also observed in experiments with step power input from a zero level [3] if heat load did not exceed critical heat flux at steady conditions qcr1 approximately in one and a half times. But if initial heat generation existed and condition q0 > qeb was kept, fluctuations of temperature of the heater took place at larger values of qh as well. Their existence is accounted for mixing of the liquid in the layer adjacent to the heating wall, caused by departure of vapor bubbles, which were on the heated surface before the moment of additional step power input. In the case of dynamic curves of second type, overheating of the heated surface monotonously increases, and heat flux q, transferred to the liquid, reaches distinct maximum value (Figs. 1b and 2b). This type of curves is observed at sufficiently high values of qh. Evidently, the probability of departure of vapor bubbles, existed before step power input to the heated wall, is proportional to the duration of the time interval tcr. Thus, the larger qh and, therefore, smaller time interval tcr, the less vapor bubbles depart from the heated surface before crisis comes, these circumstances, as a result, lead to smaller visible fluctuations of temperature of the heater. At high initial heat loads q0 > qeb, the parameters of transient boiling crisis qcr, DTcr, tcr may vary strongly, depending on structure and void fraction of boiling boundary layer at the moment just before step power input. This conclusion is confirmed by the results of the experiments in which successive photographs of the boiling process on the heated surface before increasing of power of heat generation were made. So, for example, the photographs shown in Fig. 3 demonstrate the pictures of boiling of saturated water at heat flux q0 = 0.56 MW m2. It is seen that at constant level of initial heat generation, part of the surface s0 occupied by vapor at different moments of time changes from 0.15 to 0.4. The complete set of the obtained experimental data that illustrates relationship between values qcr and DTcr for different values of q0 is shown in Fig. 4 with non-dimensional coordinates. Dashed line in the figure represents the curve published in the paper [3] for saturated water at q0 = 0. Solid line shows the results of calculations obtained by using known dependence [4] which describes relationship between superheat of the heated surface and heat flux for nucleate boiling mode in steady conditions. It is possible to note that the majority of the experimental data lies between these two
V.I. Deev et al. / International Journal of Heat and Mass Transfer 53 (2010) 1851–1855
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Fig. 2. The variation of heater superheat DT and heat flux q on a heated surface with calculation time t at q0 = 0.56 MW m2 (the marks are the same as in Fig. 1); by using Eqs. (1), (4) and (5).
curves. One can see that there is a tendency of decreasing of relative critical superheats of heated wall with increasing of value of q0 at fixed relative critical heat flux. 3. The technique of calculation of parameters of boiling crisis If value of q0 much less than qeb, calculation of characteristics of water ebullition, heat transfer crisis and dynamic curves q(t), DT(t) after step power input can be done using techniques presented in work [3]. Example of such a calculation is depicted in Fig. 1b by the solid lines. Now consider the case when a developed nucleate boiling regime existed on the heated surface before step power input. In this case, critical time interval tcr is entirely determined by duration of stage of metastable boiling, which is completed by formation of continuous vapor film on the heated surface. Thus, it is possible to consider that in this situation tcr equals the time interval from the step increase of heat load to the moment of merging of vapor bubbles growing on the heated surface to continuous film, or tcr = tmb. An approach described earlier in the papers [2,3] is applicable for calculations of tmb at high heat loads qh. According to this ap-
Fig. 3. Boiling of saturated water on wire heater at q0 = 0.56 MW m2: (a) s0 0.15; (b) s0 0.3; (c) s0 0.4.
proach, the number of bubbles forming on unit area of the surface of the heater per unit time is proportional to part of the surface not occupied by vapor. Then density n(t) of new sites of nucleation appearing after increase of power of heat generation may be described by the following equation:
nðtÞ ¼
Z
t
P 0 ð1 s1 ðt 1 Þ s0 Þ dt 1 ;
ð2Þ
0
where s0 is part of the heated surface occupied by vapor before the moment of step power input; s1(t1) is part of the surface additionally occupied by new vapor bubbles growing on the surface by the moment of time t1; P0 is a normalizing multiplier, the value of which is determined from the condition s1(tmb) + s0 = 1 at the moment of heat transfer crisis. Calculations equivalent to those done in [2] make it possible to get expression for density of centers of nucleation n(t):
nðtÞ ¼
1 s0 p t : sin pffiffiffi 2 2 tmb 2ðf ðJaÞ aÞ t mb
ð3Þ
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Fig. 5. Relationship between transient critical heat flux qcr and time interval tcr before the transition to film boiling: 1, 2, 3 – calculation by Eq. (4) at s0 = 0.2, 0.15, 0.1; 4 – calculation at s0 = 0 from the paper [3]; 5, 6 – calculation by Eq. (6) at u0 = 0, 0.5 (the marks are the same as in Fig. 4).
Fig. 4. Relationship between relative values of transient critical heat flux qcr/qcr1 and critical wall superheat DTcr/DTcr1: h q0 = 0.014; O 0.105; 4 0.39; } calculation by equation from the paper 0.56 MW m2 – experimental data; calculation for conditions of steady heating using the depen[4] at q0 = 0; dence [4].
Here f(Ja) is the module of vapor bubble growth taken from the paper [5]; the Jakob number Ja = cqDT/(Dhv q00 ) is calculated with using integral average superheat multiplied by 0.7 [3]; c, q, a are specific heat capacity, density, thermal diffusivity of water and q00 is vapor density along the saturation lines; Dhv – latent heat of vaporization. The heat balance equation for vapor bubbles generated during time interval 0 tmb gives the following equation:
Z 0
t mb
pffiffiffipffiffiffiffiffiffiffi p t dt ¼ kDhV q00 ð1 s0 Þf ðJaÞ a t mb ; qðtÞ 1 ð1 s0 Þcos 2 t mb ð4Þ
where k ffi 1.12. The heat flux q(t) transferred into the liquid may be set in the following form:
" qðtÞ ¼ ðqcr q0 Þ ðn 2Þ
t
t mb
3
2 # t t þ q0 ; þ ð3 2nÞ þn t mb t mb ð5Þ 0
mb where qcr = q(tmb) is transient critical heat flux; n ¼ qqð0Þt . cr q0 The selected polynomial expression (5) allows us to satisfy both the initial condition qð0Þ ¼ q0 and necessary condition of maximum of the heat flux at the moment of heat transfer crisis q0 ðtmb Þ ¼ 0. In the case of thin cylindrical heater, in order to find derivative q0 ð0Þ it is necessary to solve Eq. (1) with the assumption that qðtÞ ¼ DqT00 DTðtÞ; where DT0 is the initial superheat of the
heated surface. At qh(t) = const., we obtain for time derivative of h q0 Þq0 the heat flux to the liquid q0 ð0Þ ¼ ðq . ðcqÞ dDT h4
0
The solution of the system of Eqs. (1), (4) and (5), taking into consideration the relationship between values qcr and DTcr (see solid line in Fig. 4), allows us to calculate critical characteristics of boiling process as well as the dynamic curves q(t) and DT(t). Comparison of the results of calculations of dynamic curves q(t) and DT(t) at s0 = 0.15 with the data of one of the experiments at q0 = 0.56 MW m2 is shown in Fig. 2b. The experimental data that show the influence of initial heat load q0 on value of critical time interval tcr, are represented in Fig. 5 with coordinates qcr tcr. The results of calculations using
the technique described above at q0 = 0.56 MW m2 and different values of parameter s0 are drawn in the figure by dashed lines. It is possible to make a conclusion that the value of s0 strongly affects characteristics of transition to film boiling mode. This circumstance leads to significant scattering of the experimental data at set level of initial heat load q0. To calculate tmb at values q0 corresponding to developed nucleate boiling and step power inputs with moderate value qh P qcr1, it is possible to use equations developed in the paper [3], taking into account initial void fraction of the boundary layer u(0) = u0. Assuming that parameters u0 and s0 should be proportional each other, we find for the case when qh = const.:
0
1 u0 A; qcr ¼ qcr1 @ exp ttcr 1 exp
tcr td
ð6Þ
d
where td is the time of vapor bubble growth up to departure size. The results of calculations using Eq. (6) at different values of u0 are shown in Fig. 5. As can be seen from the figure, the dependence of qcr on tcr is quite conservative with regard to changes of value of initial void fraction u0.
4. Conclusion The obtained experimental data show that at small values of initial heat load (q0 < qeb) heat transfer by natural convection at the initial stage of process has hardly visible effect on the parameters of transient boiling crisis, while developed nucleate boiling with vapor bubbles departing from the surface of the heater at high initial heat loads (q0 > qeb) results in significant decrease of the critical time interval as well as of the critical heat flux. The technique of calculation of the parameters of transient boiling crisis at presence of initial heat load provides the satisfactory prediction of the experimental data. It is shown that part of the heated surface occupied by vapor before step power input strongly affects characteristics of transition to film boiling mode. Significant fluctuations of this value at set level of initial heat load lead to a wide scattering of the experimental data.
Acknowledgements The authors are grateful to Reviewer for helpful remarks. This work was financially supported under Federal Target Program ‘‘Scientific and scientific-pedagogical personnel of innovative Russia” in 2009–2013.
V.I. Deev et al. / International Journal of Heat and Mass Transfer 53 (2010) 1851–1855
References [1] V.K. Andreev, V.I. Deev, A.N. Savin, K.V. Kutsenko, I.N. Yusupova, Influence of initial heat generation on the transition to film boiling of helium under step input of heating power, in: Thermophysical Problems of Nuclear Engineering, Moscow, Energoatomizdat, 1987, pp. 63–65 (in Russian). [2] V.I. Deev, V.S. Kharitonov, K.V. Kutsenko, A.A. Lavrukhin, Transient boiling crisis of cryogenic liquids, Int. J. Heat Mass Transfer 47 (2004) 5477–5482.
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[3] V.I. Deev, Htay Lwin Oo, V.S. Kharitonov, K.V. Kutsenko, A.A. Lavrukhin, Critical heat flux modeling in water pool boiling during power transients, Int. J. Heat Mass Transfer 50 (2007) 3780–3787. [4] D.A. Labuntsov, Problems of heat transfer at nucleate boiling of liquids, Teploenergetika 9 (1972) 14–19 (in Russian). [5] D.A. Labuntsov, V.V. Yagov, To problem about growth rate of vapor bubbles at boiling, Trudy MEI 268 (1975) 3–15 (in Russian).
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Influence of initial heat generation on dynamic characteristics of transient boiling crisis of water V.I. Deev, K.V. Kutsenko, A.A. Lavrukhin *, V.S. Kharitonov Department of Thermophysics, National Research Nuclear University ‘‘MEPhI”, 31 Kashirskoe shosse, Moscow 115409, Russian Federation
a r t i c l e
i n f o
Article history: Received 22 September 2009 Received in revised form 19 October 2009 Accepted 22 December 2009 Available online 22 January 2010 Keywords: Power transients Water pool boiling Critical heat flux Physical models
a b s t r a c t The dynamic characteristics of transient boiling crisis were measured in the experiments with saturated water in a pool at atmospheric pressure. It was shown that initial heat generation in a heater strongly affects the process of transition from nucleate to film boiling under conditions of fast increase of heating power. A technique of calculation of critical heat flux, temperature drop and critical time interval is presented. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Under sharp increase of heating power at the initial stage of non-steady thermal process, there is always some time interval tcr during which, even after step power input above critical level, the rate of heat transfer from a heater into a liquid remains sufficiently high. Thus, if the duration of heating power pulse is less than tcr, there is an opportunity of heat transfer to liquid without significant overheating of the heated surface. Otherwise, the beginning of film boiling results in a sharp increase of heater temperature, which may lead to its destruction. Therefore, knowledge of value of tcr as well as main parameters, on which this value depends, is of big practical interest. The existence of heat flux from a heated wall q0 before step power input significantly changes thermal and hydrodynamic conditions nearby the heater and, therefore, influences on dynamic characteristics of transition to film boiling. At present, there is separate information on influence of initial heat generation in the heater on the value of critical time interval only for cryogenic liquids. Thus, for example, according to the paper [1], existence of preliminary heat load may result in reduction of critical time interval in liquid helium in one and a half times. 2. The experimental results The experimental apparatus and experimental technique are described fully enough in the papers [2,3]. A horizontal platinum * Corresponding author. Tel.: +7 495 3247328. E-mail addresses: [email protected], [email protected] (A.A. Lavrukhin). 0017-9310/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2010.01.004
wire 0.1 mm in diameter immersed in the vessel with saturated water under atmospheric pressure was used as a heater. Heat generation in the wire increased stepwise from an initial level of heat generation corresponding to heat load q0 to a set level qh. Experiments were conducted at several values q0 in the range of 0.014– 0.56 MW m2. In this range of initial heat loads, different modes of heat transfer were realized: natural convection if q0 < qeb, the onset of nucleate boiling when q0 qeb ( 0.2 MW m2), and developed nucleate boiling when q0 > qeb (here qeb is a heat flux when boiling of water on the surface of the heater begins). In any conditions, if, as a result of additional step power input, the total heat load of the heater exceeded critical heat flux at steady conditions qcr1, nucleate boiling before heat transfer crisis was always non-steady, and after some time a fast increase of the heated wall temperature began (heat transfer crisis at non-steady heating). Some experimentally obtained dynamic dependencies of wall superheat DT of the heater and the heat flux transferred to the liquid q on time t are shown in Figs. 1 and 2. The heat flux to the liquid as a function of time was calculated by using the experimental data and an equation of heat balance
d dðDTðtÞÞ qðtÞ ¼ qh ðtÞ ðcqÞh ; 4 dt
ð1Þ
where d is the diameter and (cq)h is the volumetric heat capacity of the heater. Experimental dynamic curves may be divided on two types. The first type corresponds to those curves that show strong fluctuations of temperature of the heater in metastable boiling stage preceding heat transfer crisis (Figs. 1a and 2a). For these situations, the heat flux q on the heated surface is on the average practically equal to the total heat load (q qh). In this case, parameters of transient
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Nomenclature a c d Dhv Ja k n P0 q qcr1 qh s DT DTcr1 t t1
thermal diffusivity specific heat capacity diameter of a wire latent heat of vaporization Jakob number, Ja = cqDT/(q00 Dhv) factor in Eq. (4) nucleation site density normalizing multiplier heat flux first critical heat flux at steady conditions heat load of a heater part of the surface occupied by vapor liquid (wall) superheat about saturation temperature first critical temperature drop at steady conditions time birth time of vapor bubble
Fig. 1. The variation of heater superheat DT and heat flux q on a heated surface with time t at q0 = 0.014 MW m2: } DT – experimental data; O q – calculation by using calculation using the model from the paper [3]. Eq. (1);
Greek symbols q density q00 vapor density u void fraction Subscripts 0 initial cal calculated value cr critical value d departure eb ebullition exp experimental value h heater mb metastable boiling
crisis qcr, DTcr were determined at the time moment tcr when the heater temperature began to increase sharply. It is to note that similar type of dynamic curves was also observed in experiments with step power input from a zero level [3] if heat load did not exceed critical heat flux at steady conditions qcr1 approximately in one and a half times. But if initial heat generation existed and condition q0 > qeb was kept, fluctuations of temperature of the heater took place at larger values of qh as well. Their existence is accounted for mixing of the liquid in the layer adjacent to the heating wall, caused by departure of vapor bubbles, which were on the heated surface before the moment of additional step power input. In the case of dynamic curves of second type, overheating of the heated surface monotonously increases, and heat flux q, transferred to the liquid, reaches distinct maximum value (Figs. 1b and 2b). This type of curves is observed at sufficiently high values of qh. Evidently, the probability of departure of vapor bubbles, existed before step power input to the heated wall, is proportional to the duration of the time interval tcr. Thus, the larger qh and, therefore, smaller time interval tcr, the less vapor bubbles depart from the heated surface before crisis comes, these circumstances, as a result, lead to smaller visible fluctuations of temperature of the heater. At high initial heat loads q0 > qeb, the parameters of transient boiling crisis qcr, DTcr, tcr may vary strongly, depending on structure and void fraction of boiling boundary layer at the moment just before step power input. This conclusion is confirmed by the results of the experiments in which successive photographs of the boiling process on the heated surface before increasing of power of heat generation were made. So, for example, the photographs shown in Fig. 3 demonstrate the pictures of boiling of saturated water at heat flux q0 = 0.56 MW m2. It is seen that at constant level of initial heat generation, part of the surface s0 occupied by vapor at different moments of time changes from 0.15 to 0.4. The complete set of the obtained experimental data that illustrates relationship between values qcr and DTcr for different values of q0 is shown in Fig. 4 with non-dimensional coordinates. Dashed line in the figure represents the curve published in the paper [3] for saturated water at q0 = 0. Solid line shows the results of calculations obtained by using known dependence [4] which describes relationship between superheat of the heated surface and heat flux for nucleate boiling mode in steady conditions. It is possible to note that the majority of the experimental data lies between these two
V.I. Deev et al. / International Journal of Heat and Mass Transfer 53 (2010) 1851–1855
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Fig. 2. The variation of heater superheat DT and heat flux q on a heated surface with calculation time t at q0 = 0.56 MW m2 (the marks are the same as in Fig. 1); by using Eqs. (1), (4) and (5).
curves. One can see that there is a tendency of decreasing of relative critical superheats of heated wall with increasing of value of q0 at fixed relative critical heat flux. 3. The technique of calculation of parameters of boiling crisis If value of q0 much less than qeb, calculation of characteristics of water ebullition, heat transfer crisis and dynamic curves q(t), DT(t) after step power input can be done using techniques presented in work [3]. Example of such a calculation is depicted in Fig. 1b by the solid lines. Now consider the case when a developed nucleate boiling regime existed on the heated surface before step power input. In this case, critical time interval tcr is entirely determined by duration of stage of metastable boiling, which is completed by formation of continuous vapor film on the heated surface. Thus, it is possible to consider that in this situation tcr equals the time interval from the step increase of heat load to the moment of merging of vapor bubbles growing on the heated surface to continuous film, or tcr = tmb. An approach described earlier in the papers [2,3] is applicable for calculations of tmb at high heat loads qh. According to this ap-
Fig. 3. Boiling of saturated water on wire heater at q0 = 0.56 MW m2: (a) s0 0.15; (b) s0 0.3; (c) s0 0.4.
proach, the number of bubbles forming on unit area of the surface of the heater per unit time is proportional to part of the surface not occupied by vapor. Then density n(t) of new sites of nucleation appearing after increase of power of heat generation may be described by the following equation:
nðtÞ ¼
Z
t
P 0 ð1 s1 ðt 1 Þ s0 Þ dt 1 ;
ð2Þ
0
where s0 is part of the heated surface occupied by vapor before the moment of step power input; s1(t1) is part of the surface additionally occupied by new vapor bubbles growing on the surface by the moment of time t1; P0 is a normalizing multiplier, the value of which is determined from the condition s1(tmb) + s0 = 1 at the moment of heat transfer crisis. Calculations equivalent to those done in [2] make it possible to get expression for density of centers of nucleation n(t):
nðtÞ ¼
1 s0 p t : sin pffiffiffi 2 2 tmb 2ðf ðJaÞ aÞ t mb
ð3Þ
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Fig. 5. Relationship between transient critical heat flux qcr and time interval tcr before the transition to film boiling: 1, 2, 3 – calculation by Eq. (4) at s0 = 0.2, 0.15, 0.1; 4 – calculation at s0 = 0 from the paper [3]; 5, 6 – calculation by Eq. (6) at u0 = 0, 0.5 (the marks are the same as in Fig. 4).
Fig. 4. Relationship between relative values of transient critical heat flux qcr/qcr1 and critical wall superheat DTcr/DTcr1: h q0 = 0.014; O 0.105; 4 0.39; } calculation by equation from the paper 0.56 MW m2 – experimental data; calculation for conditions of steady heating using the depen[4] at q0 = 0; dence [4].
Here f(Ja) is the module of vapor bubble growth taken from the paper [5]; the Jakob number Ja = cqDT/(Dhv q00 ) is calculated with using integral average superheat multiplied by 0.7 [3]; c, q, a are specific heat capacity, density, thermal diffusivity of water and q00 is vapor density along the saturation lines; Dhv – latent heat of vaporization. The heat balance equation for vapor bubbles generated during time interval 0 tmb gives the following equation:
Z 0
t mb
pffiffiffipffiffiffiffiffiffiffi p t dt ¼ kDhV q00 ð1 s0 Þf ðJaÞ a t mb ; qðtÞ 1 ð1 s0 Þcos 2 t mb ð4Þ
where k ffi 1.12. The heat flux q(t) transferred into the liquid may be set in the following form:
" qðtÞ ¼ ðqcr q0 Þ ðn 2Þ
t
t mb
3
2 # t t þ q0 ; þ ð3 2nÞ þn t mb t mb ð5Þ 0
mb where qcr = q(tmb) is transient critical heat flux; n ¼ qqð0Þt . cr q0 The selected polynomial expression (5) allows us to satisfy both the initial condition qð0Þ ¼ q0 and necessary condition of maximum of the heat flux at the moment of heat transfer crisis q0 ðtmb Þ ¼ 0. In the case of thin cylindrical heater, in order to find derivative q0 ð0Þ it is necessary to solve Eq. (1) with the assumption that qðtÞ ¼ DqT00 DTðtÞ; where DT0 is the initial superheat of the
heated surface. At qh(t) = const., we obtain for time derivative of h q0 Þq0 the heat flux to the liquid q0 ð0Þ ¼ ðq . ðcqÞ dDT h4
0
The solution of the system of Eqs. (1), (4) and (5), taking into consideration the relationship between values qcr and DTcr (see solid line in Fig. 4), allows us to calculate critical characteristics of boiling process as well as the dynamic curves q(t) and DT(t). Comparison of the results of calculations of dynamic curves q(t) and DT(t) at s0 = 0.15 with the data of one of the experiments at q0 = 0.56 MW m2 is shown in Fig. 2b. The experimental data that show the influence of initial heat load q0 on value of critical time interval tcr, are represented in Fig. 5 with coordinates qcr tcr. The results of calculations using
the technique described above at q0 = 0.56 MW m2 and different values of parameter s0 are drawn in the figure by dashed lines. It is possible to make a conclusion that the value of s0 strongly affects characteristics of transition to film boiling mode. This circumstance leads to significant scattering of the experimental data at set level of initial heat load q0. To calculate tmb at values q0 corresponding to developed nucleate boiling and step power inputs with moderate value qh P qcr1, it is possible to use equations developed in the paper [3], taking into account initial void fraction of the boundary layer u(0) = u0. Assuming that parameters u0 and s0 should be proportional each other, we find for the case when qh = const.:
0
1 u0 A; qcr ¼ qcr1 @ exp ttcr 1 exp
tcr td
ð6Þ
d
where td is the time of vapor bubble growth up to departure size. The results of calculations using Eq. (6) at different values of u0 are shown in Fig. 5. As can be seen from the figure, the dependence of qcr on tcr is quite conservative with regard to changes of value of initial void fraction u0.
4. Conclusion The obtained experimental data show that at small values of initial heat load (q0 < qeb) heat transfer by natural convection at the initial stage of process has hardly visible effect on the parameters of transient boiling crisis, while developed nucleate boiling with vapor bubbles departing from the surface of the heater at high initial heat loads (q0 > qeb) results in significant decrease of the critical time interval as well as of the critical heat flux. The technique of calculation of the parameters of transient boiling crisis at presence of initial heat load provides the satisfactory prediction of the experimental data. It is shown that part of the heated surface occupied by vapor before step power input strongly affects characteristics of transition to film boiling mode. Significant fluctuations of this value at set level of initial heat load lead to a wide scattering of the experimental data.
Acknowledgements The authors are grateful to Reviewer for helpful remarks. This work was financially supported under Federal Target Program ‘‘Scientific and scientific-pedagogical personnel of innovative Russia” in 2009–2013.
V.I. Deev et al. / International Journal of Heat and Mass Transfer 53 (2010) 1851–1855
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