A new approach for the modelization of water and steam mixing at high pressure conditions

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[.]

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

Energy Procedia 101 ( 2016 ) 34 – 41

ScienceDirect

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

© 2016 The Authors Published by Elsevier Ltd

Peer-review under responsibility of the Scientific Committee of ATI 2016

Keywords: mixing ; experiment ; high pressure ; zero-dimensional thermodinamic model

1 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

© 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.

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 zero-dimensional 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 1 - Sketch of the experimental setup

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]

c v 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]

S l 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/(m2

K4)]

subscripts

d differerential

l liquid

2 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 5 liters and keeping the water and steam inside it at an initial pressure of 6 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 1

Table 1 - Technical characteristics of the HPS

power at the boiler from 3000 to 7500 W steam temperature in the boiler 190°C

steam pressure from 6 to 7 bar

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 1 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 1 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 P m and the temperature is measured

through two K-type thermocouples, T ca and T cb, 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 T ca measures the temperature of the mixing between liquid water and steam,

while the thermocouple T cb 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

T air =20 ˚C while the surface of the chamber is assumed to be isothermal at T cb =110 ˚C The value T cb 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 T 0=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

(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

(2)

We obtain a convective heat transfer coefficient U=6.5 W/(m2K) and thus a power dispersion of

1/ 4

35H

D

Gr

t

1/ 4

0.59

Q =S l U ∆T=3.6 W (3)

where S l is the cylinder lateral surface and ∆T= T cb – T air

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

We can conclude that the heat losses are negligible with respect to the heat exchanged inside the mixing chamber

3 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 6 bar Indeed, the energy balance applied

to the steam that exits the HPS gives:

(5)

where h s is the specific enthalpy of steam, u l and u s are the specific energy of liquid and steam (u d = u l – u s) at the HPS Equation (5) can be rewritten as

(6) Since, as seen in the experimental measures, the pressure is a decreasing function of time, let us assume that

(7)

where c is a constant An integration of Eq (6) gives

(8)

where the constant parameter A has been introduced Indeed, the mass flow rate at the HPS outlet is

Let us now consider the energy balance at the mixing chamber inlet:

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

An integration of eq (11) yields



1

dt



,

c

dt

du d

s

c

s

dt

dT c m

h

ms s v v

c m

h m A dt c m

h m

dT

v v s v

v s



v v

m h

m c

 ª¬   º¼

4 Results and discussion

With reference to our experiments, since the pressure in the HPS the initial pressure is p=6 bar, one has:

h s=2756.1 kJ/kg,

u d=1897.1 kJ/kg,

c v=3.4839 kJ/(kg K),

m v=0.6 kg,

m0=0.028 kg and

du d /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 2 - Comparison between experiments and model predictions

The mass flow rate predicted by the model is shown in Fig 3 The mass flow rate at the beginning is 3.6 g/s

Figure 3 shows that after 5 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 1 and 2 the two sections respectively

(13)

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

(14) with local loss coefficient E~ 1 in the valve and with S cross section of the duct

On account of eq.(9), eq (14) yields

(15)

, 0 )

(

2

1 2 1 2

2

1

2

2 W g z z  p p R

W

U

, 2

2 2 1

S

m W

R

U E

U

 2  ,

2

0

2

S

m

A

U

and

(16)

In eq.(16) the absolute pressure has been introduced, by summing the effective pressure and the atmospheric pressure

Figure 3 – Mass flow rate for steam

Figure 4 - Differential internal energy of water u versus pressure

2 2 0

2

A m

S

U

If

(17) eqs (16) and (17) yield

(18)

A comparison between eq (18) and the initial hypothesis described in eq (7) allows one to conclude that

(19)

According to the values choose for the experiments, here B=2326.6 kJ/kg

The behavior of u d 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)

5 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 4 (1974) 15-19

[8] Çengel, Yunus A.,Heat transfer A practical approach, McGraw-Hill ( 2003 2nd Ed)

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p

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u d

2AB

dt

du d

#

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d h

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