A New Approach for the Modelization of Water and Steam Mixing at High Pressure Conditions 1876 6102 © 2016 The Authors Published by Elsevier Ltd This is an open access article under the CC BY NC ND li[.]
Available online at www.sciencedirect.com ScienceDirect Energy Procedia 101 (2016) 34 – 41 71st Conference of the Italian Thermal Machines Engineering Association, ATI2016, 14-16 September 2016, Turin, Italy A new approach for the modelization of water and steam mixing at high pressure conditions Eugenia Rossi di Schio*, Beatrice Pulvirenti, Michele Celli Alma Mater Studiorum-Università di Bologna, Dipartimento di Ingegneria Industriale DIN Viale Risorgimento 2, I-40136 Bologna, Italia Abstract In the present paper a zero-dimensional thermodynamic model of liquid water and steam mixing is presented The model provides the temperature distribution during the mixing process, as an exponential function of time A comparison with experimental data is performed, showing an excellent agreement In the experimental set up, steam from an adiabatic high pressure storage is mixed, in a mixing chamber, together with liquid water Measures of the temperature both on the inner and outer surface of the mixing chamber are taken A comparison between the experimental data and the thermodynamic model proposed shows that the heat transfer between the mixing chamber and the environment is negligible Moreover, a discussion on the time dependency of the experimental pressure and of the theoretical internal energy of the steam is presented 2016The TheAuthors Authors Published by Elsevier © 2016 Published by Elsevier Ltd Ltd This is an open access article under the CC BY-NC-ND license Peer-review under responsibility of the Scientific Committee of ATI 2016 (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the Scientific Committee of ATI 2016 Keywords: mixing ; experiment ; high pressure ; zero-dimensional thermodinamic model Introduction Mixing of water and pressurized steam frequently appears as necessary in many technological applications and, since 1942 [1], many patents have been deposited in order to develop the devices and to increase the energetic efficiency of the devices The subject has been investigated in the scientific literature as well In particular, the topic deserves attention especially with reference to the tank filling processes As is well known, in these processes, the two key parameters * Corresponding author Tel.: +39 051 2093294; fax: +39 051 2093296 E-mail address: eugenia.rossidischio@unibo.it 1876-6102 © 2016 The Authors Published by Elsevier Ltd This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the Scientific Committee of ATI 2016 doi:10.1016/j.egypro.2016.11.005 Eugenia Rossi di Schio et al / Energy Procedia 101 (2016) 34 – 41 to control the filling are the temperature and pressure profile of the gas which flows in the vessel In the literature on tank filling processes, some papers deal with a thermodynamic analysis of refueling of a gaseous hydrogen fuel tank [2-6] In [2], the gaseous hydrogen is treated as an ideal or a non-ideal gas and the refueling process is analyzed based on adiabatic, isothermal, or diathermal condition of the tank A constant feed-rate is assumed in the analysis The thermodynamic state of the feed stream also remains constant during refueling In [3], a zero-dimensional thermodynamic real gas simulation model is presented; ideal gas and real gas simulations are compared and the entropy balance of the filling process is formulated Calculated results are validated with measurements In [4-5], attention is paid to the maximum gas temperature within the vessels during the process Different filling strategies in terms of pressure and temperature of the gas injected into the cylinder and their effects on key parameters like maximum temperature, state of charge, and energy cooling demand are investigated The authors describe the results of CFD, and show that the most convenient filling strategy from the cooling energy point of view is identified: an almost linear pressure rise and pre-cooling in the second half of the process In all the above mentioned papers very few details on the mathematical model adopted are given On the contrary, details on the mathematical model adopted are given in [6], where a simplified model is described for simulating gas blowdown through a thin and long tube connecting a high-pressure gas-filled reservoir to a vacuum vessel During the depressurization process, the flow is assumed to be quasi-steady and approximated as one-dimensional The transient compressible flow is solved analytically and the solution is compared with experimental results As described in [6], the blowdown or simply the depressurization of large pressure vessels may occur in process plants either accidentally or voluntarily Consequently, and also for the sake of security, the system behavior must be understood However, the related scientific literature is not abundant In the present paper, a first part of a wider study dealing with the mathematical model of liquid water and steam mixing is presented In detail, the mathematical model describes water steam blowdown from a high-pressure steam storage to a low pressure mixing chamber filled by steam and liquid water and air as well Recently, in order to design, realize and develop a commercial device that works under steam high pressure conditions, the authors have used a commercial boiler to produce water steam to be mixed together with liquid water at different pressure and temperature The problem is non trivial, since air is present together with liquid water in the mixing chamber Indeed, the usual thermodynamical approach gives only a rough estimation of the phenomenon In the present paper a new approach to the mathematical model formulation is presented, through a zerodimensional thermodynamic analysis Then, a comparison with the experimental measurements of the pressure and temperature obtained in the mixing chamber of the realized technological prototype is presented, showing an excellent agreement Figure - Sketch of the experimental setup 35 36 Eugenia Rossi di Schio et al / Energy Procedia 101 (2016) 34 – 41 Nomenclature A constant defined in Eq.(5) [s-1] B constant defined in Eq.(15) [J/kg] c constant defined in Eq.(3) [W/kg] cv specific heat at constant volume [J/(kg K)] D diameter of the mixing chamber [m] g modulus of the gravitational acceleration [m/s2] Gr Grashof number h specific enthalpy [J/kg] H height of the mixing chamber [m] m mass [kg] m0 initial mass in the HPS [kg] Nu Nusselt number p pressure [bar] Q heat power [W] r radius of the mixing chamber [m] R pressure loss [m2/s2] Ra Rayleigh number S section of the duct between HPS and mixing chamber [m2] Sl lateral surface of the cylindrical mixing chamber [m2] t time [s] T temperature [K] u specific internal energy [J/kg] U convection coefficient [W/(m2K)] x title of the saturated steam in the HPS W mean velocity in the duct between HPS and mixing chamber [m/s] z mean height of the duct section [m] E local loss coefficient for the valve U water density [kg/m3] V Stefan-Boltzmann constant [W/(m2K4)] subscripts d differerential l liquid s steam Description of the experimental setup Recently, in order to design, realize and develop a commercial device that works under steam high pressure conditions, the authors have used a commercial boiler to produce water steam to be mixed together with liquid water at different pressure and temperature This experimental apparatus has been employed in order to measure pressure and especially temperature, and the experimental measures have been compared with the predictions of the analytical model presented in the next section In Figure 1, a schematic overview of the experimental apparatus is given In detail, the steam is produced by employing a commercial boiler, having capacity of liters and keeping the water and steam inside it at an initial pressure of bar The boiler is treated as a high pressure storage (HPS): in fact, it accounts for an “infinite” quantity of steam with reference to the steam effectively employed in the mixing chamber The technical characteristics of the HPS are given in Table 37 Eugenia Rossi di Schio et al / Energy Procedia 101 (2016) 34 – 41 Table - Technical characteristics of the HPS supply capacity power at the boiler steam temperature in the boiler steam pressure extra capacity sizes weight gross weight 230V / 380V lt from 3000 to 7500 W 190°C from to bar 10 lt 48x36xh100 cm 31 kg 33,5 kg The water steam is driven to the mixing chamber, due to the pressure difference In fact, at the beginning of the mixing process the valve is opened and the pressure in the mixing chamber is bar Attention is paid to the analysis of unsteady mixing At the beginning, he mixing chamber contains 0.6 kg of liquid water at 90°C It is a vertical cylinder (with no inclination with respect to the vertical direction) made of stainless steel INOX AISI 316L (thermal conductivity 15 W/(m K)), having radius 10 cm, height 27 cm and thickness cm The walls have undergone electropolishing and passivation processes The HPS and the mixing chamber are connected through an electrically driven, usually closed valve ODE KT130ZT30-TG At t=0, the valve is open and saturated vapour starts to flow from the HPS to the mixing chamber As sketched in Figure 1, the pressure is measured though a pressure meter Pm and the temperature is measured through two K-type thermocouples, Tca and Tcb, placed inside the mixing chamber and outside it respectively Both thermocouples are placed at the mixing chamber lateral surface of the cylinder, at a height of 2.5 cm from the base of the cylinder The thermocouple Tca measures the temperature of the mixing between liquid water and steam, while the thermocouple Tcb measures the boundary temperature In the experimental setup, no precise boundary condition is designed for the mixing chamber Indeed, the boundary is not adiabatic However, the experimental measures have been performed in rapid succession and the boundary of the mixing chamber has shown to be mostly isothermal and adiabatic during the mixing time In fact, before proceeding with the thermodynamic performances analysis of the system, let us evaluate the power dispersion of the mixing chamber with respect to its environment The air surrounding the chamber is at temperature Tair =20 ˚C while the surface of the chamber is assumed to be isothermal at Tcb =110 ˚C The value Tcb is obtained by taking the mean value of the chamber surface temperature during the mixing transient The air properties necessary for the following analysis are taken at the reference temperature T0=65 ˚C We start evaluating the heat losses due to natural convection The chamber is a cylinder of external radius r=9 cm and height H=24 cm The surface of a vertical cylinder can be treated, from [7], as a vertical plate when the diameter of the cylinder is sufficiently large so that the curvature effects are negligible The diameter is sufficiently large if Dt 35H Gr1/ (1) where D is the cylinder diameter and Gr is the Grashof number Since the diameter doubles the threshold quantity given in the last equation, we will treat the cylinder as a vertical plate The Rayleigh number relative to the system is Ra ~ O(108) The Nusselt number of a vertical plate characterized by a Rayleigh number of that order of magnitude is defined, from [8], by Nu 0.59 Ra1/4 We obtain a convective heat transfer coefficient U=6.5 W/(m2K) and thus a power dispersion of (2) 38 Eugenia Rossi di Schio et al / Energy Procedia 101 (2016) 34 – 41 Q=Sl U ∆T=3.6 W (3) where Sl is the cylinder lateral surface and ∆T= Tcb – Tair The radiative component of the heat losses can be evaluated, as first approximation, by the power transferred by a convex surface completely sorrounded by another surface, namely Q=Sl σ (Tcb 4-Tair4)=4.9 W (4) We can conclude that the heat losses are negligible with respect to the heat exchanged inside the mixing chamber Mathematical model Let us first consider the high pressure storage HPS As described in the previous section, the HPS is filled by a mass of saturated water and steam mixing, having title x, at a pressure of bar Indeed, the energy balance applied to the steam that exits the HPS gives: dms d (ul xud ) d ( xud ) dm du du (5) m # ms s ud ms d hs m dt dt dt dt dt dt where hs is the specific enthalpy of steam, ul and us are the specific energy of liquid and steam (ud = ul – us) at the HPS Equation (5) can be rewritten as dms ms dud dt hs ud dt (6) Since, as seen in the experimental measures, the pressure is a decreasing function of time, let us assume that dud c , dt where c is a constant An integration of Eq (6) gives Đ Ã c m0 Exp ă t m0 Exp At , â hs ud where the constant parameter A has been introduced Indeed, the mass flow rate at the HPS outlet is ms ms A m0 Exp A t (7) (8) (9) Let us now consider the energy balance at the mixing chamber inlet: m s hs mv cv dT dt (10) In eq.(10), the heat transfer from the mixing chamber to the external environment is neglected On account of eqs (8) and (9), eq (10) becomes m s hs A m0 hs dt Exp A t mv cv mv cv An integration of eq (11) yields dT T T0 ms hs ê1 Exp At ẳ mv cv ¬ (11) (12) 39 Eugenia Rossi di Schio et al / Energy Procedia 101 (2016) 34 – 41 Results and discussion With reference to our experiments, since the pressure in the HPS the initial pressure is p=6 bar, one has: hs=2756.1 kJ/kg, ud=1897.1 kJ/kg, cv=3.4839 kJ/(kg K), mv=0.6 kg, m0=0.028 kg and dud/dt=600 kJ/(kg s) In the experiment, more than 40 measures of the temperature time distribution have been performed In Figure 2, a comparison between the experiments and the prediction from the model is reported The figure shows that in the first 10 seconds there is a very good agreement between the experimental data and the model Figure - Comparison between experiments and model predictions The mass flow rate predicted by the model is shown in Fig The mass flow rate at the beginning is 3.6 g/s Figure shows that after s the mass flow rate is halved Let us now apply the mechanical energy balance between the sections immediately before and after the valve shown in Figure 1, and let us call and the two sections respectively W22 W12 p p1 g ( z z1 ) R , U where p1 and p2 are the values of the effective pressure in the two sections (then p2 ~ 0) Since W1=W2= W and z1= z2, eq.(13) becomes m s2 W2 , # 2U S with local loss coefficient E ~ in the valve and with S cross section of the duct On account of eq.(9), eq (14) yields p1 p1 UR UE A2 m02 Exp A t , 2U S (13) (14) (15) 40 Eugenia Rossi di Schio et al / Energy Procedia 101 (2016) 34 – 41 and p1abs p1 p1 atm p1 atm A2 m02 Exp 2 At 2U S (16) In eq.(16) the absolute pressure has been introduced, by summing the effective pressure and the atmospheric pressure Figure – Mass flow rate for steam Figure - Differential internal energy of water ud versus pressure 41 Eugenia Rossi di Schio et al / Energy Procedia 101 (2016) 34 – 41 If § p · u d # B ln ăă áá , â p0 eqs (16) and (17) yield dud # 2AB dt (17) (18) A comparison between eq (18) and the initial hypothesis described in eq (7) allows one to conclude that ud hs B (19) According to the values choose for the experiments, here B=2326.6 kJ/kg The behavior of ud as a function of p (expressed in bar) is reported in Figure 4, with reference to water The figure well describes the behavior as predicted through eq (17) Conclusions The present paper deals with the investigation of mixing of high pressure water steam and low pressure liquid water The analysis deserves attention because this phenomenon is involved in many technological applications In the present paper the first section describes an experimental setup in which water steam, exiting a High Pressure Storage, is mixed in a mixing chamber with low pressure liquid water In the experiments, measures of the pressure and especially of the temperature distribution are performed A new mathematical model is introduced in order to describe the mixing by employing a zero-dimensional thermodyamic approach In detail, a law that describes how temperature in the mixing chamber decreases with time is obtained A comparison between the analytical prediction and the experimental results is performed, showing an excellent agreement Finally, a discussion on the hypothesis assumed during the formulation of the mathematical model is reported and a further cross validation with the experimental data is presented References [1] Napier A.T., Steam and water mixing device, US Patent 2,335,250, 1943 [2] Jiann C Yang, A thermodynamic analysis of refueling of a hydrogen tank, international journal of hydrogen energy 34 (2009) 6712–6721 [3] M Striednig, S Brandstätter, M Sartory, M Klell, Thermodynamic real gas analysis of a tank filling process, international journal of hydrogen energy 39 (2014) 8495-8509 [4] D Melideo, D Baraldi , CFD analysis of fast filling strategies for hydrogen tanks and their effects on key-parameters, international journal of hydrogen energy 40 (2015) 735 745 [5] D Melideo, D Baraldi , Erratum to “CFD analysis of fast filling strategies for hydrogen tanks and their effects on key-parameters” [Int J Hydrogen Energy 40 (2015) 735-745], Int J Hydrogen Energy 40 (2015) 6260-6268 [6] S Charton, V Blet, J.P Corriou, A Simplified model for real gas expansion between two reservoirs connected by a thin tube, Chemical Engineering Science 51 (1996) 295-308 [7] T Cebeci, Laminar Free Convection Heat Transfer from the Outer Surface of a Vertical Slender Circular Cylinder, Proceedings Fifth International Heat Transfer Conference paper NCI (1974) 15-19 [8] Çengel, Yunus A.,Heat transfer A practical approach, McGraw-Hill ( 2003 2nd Ed) ... is analyzed based on adiabatic, isothermal, or diathermal condition of the tank A constant feed-rate is assumed in the analysis The thermodynamic state of the feed stream also remains constant... presented In detail, the mathematical model describes water steam blowdown from a high- pressure steam storage to a low pressure mixing chamber filled by steam and liquid water and air as well Recently,... previous section, the HPS is filled by a mass of saturated water and steam mixing, having title x, at a pressure of bar Indeed, the energy balance applied to the steam that exits the HPS gives: