NUMERICAL SIMULATION OF WATER-VAPOUR CONDENSATION BY MEANS OF A FLOW ORIENTED SCHEME
D.P. Karadimou,
[email protected] N.C. Markatos*,
[email protected]
National Technical University of Athens, Iroon Polytechniou 9, Zografou Campus, 15780, Greece
Abstract In this study the numerical simulation of the water-vapour condensation by means of the flow oriented discretization scheme SUPER is presented. A three-dimensional two-phase flow Eulerian model is developed within the CFD computer program PHOENICS, which considers the phases as interpenetrating continua. The study focused on the heat and mass transfer interaction between humid air and liquid droplets in a real scale indoor space for two different humidity conditions. The model takes into account momentum interaction by means of interphase friction and calculates the humid air properties distribution inside the room. The numerical results are presented in terms of the absolute humidity ratio (kg H2O/kg of dry air) distribution on the cold surface of the walls. It is concluded that there is a qualitative agreement between the numerical solution and the expected humidity formation. Water droplets are forming on the wall surface as soon as the temperature reduces below the dew point of the water vapour, as expected.
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2 NOMENCLATURE AND UNITS AED d wd
aerodynamic diameter diameter of droplets
m m
F f ,int
frictional force
Newton / m3
Fb h m
buoyancy force
Newton / m3
enthalpy
J / m3
rate of mass transfer
mwv
mass of water vapour
kg / m3 sec kg
p
vapour pressure saturation vapour pressure
Pa Pa
atmospheric pressure interphase heat transport
Pa
qint Ri
volume fraction of each phase
m3 / m3
Rg
m3 / m3
Ra
volume fraction of gaseous phase volume fraction of water droplets constant of dry air
R
gas constant
RH
ha
relative humidity density of humid air
kJ / moleo K dimensionless
da
density of dry air
kg / m3
T
atmospheric temperature first phase temperature
o
K
o
K
second phase temperature
o
K
volume of each cell
m3 dimensionless dimensionless
ps P
Rwd
T1 T2 Vcell Y YGR
kg water vapour /kg dry air absolute humidity or kg H2O /kg dry air
J / m3
m3 / m3
kJ / kg o K
kg / m3
Subscripts
i 1 2 b da ha int wd Abbreviations CFD IPSA PHOENICS
index for phases first phase second phase buoyancy dry air humid air interphase water droplets
Computational Fluid Dynamics Interphase Slip Algorithm Parabolic Hyperbolic or Elliptic Numerical Integration Code Series
INTRODUCTION The prediction and control of the indoor moisture distribution, particularly of wall condensation, is very important for a healthy and energy-efficient environment. The majority of the available models are based on the
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3 mass balance between gained and lost moisture. The simplest model predicts the indoor humidity as a function of occupancy and ventilation rate. Other models have considered more time-dependent mechanisms of moisture transport in building cavities with and without condensation. Diasty et al. (1992) described humidity dynamics within an enclosure by a linear differential equation. Teodosiou et al. (2003) and Stavrakakis et al. (2011) solved a convection-diffusion transport equation of a scalar quantity, so that the humidity of atmospheric air is taken into account for the prediction and optimization of thermal comfort conditions. Additionally Ma et al. (2013) solved a convection-diffusion transport equation of water-vapour, treated as a passive contaminant. Methods for calculating surface condensation are based on the well-known mass transfer equations (Diasty et al., 1992; Lu., 2002; Lu., 2003; Isetti et al., 1988; Liu et al., 2004; Pu et al., 2012) and the assumption that the indoor air is well-mixed. Mimouni et al. (2011) developed a two-phase flow model for the numerical simulation of wall steam condensation. Two-phase approach, presented in this study, allows the prediction of water-vapour condensation phenomena, calculating pressure, temperature and absolute humidity distribution, as well as transient heat and mass transfer to the water-phase. The physical problem considered The geometry examined is a real-scale hypothetical room without natural ventilation. The dimensions of the room are: width (X) x height (Y) x length (Z) = 4.0 x 4.0 x 8.0 m. At the initial time the room is filled with humid air at rest. The humid air consists of dry air and water-vapour and is considered as the primary gaseous phase. The temperature ranges between 290-303 oK and as it drops below the dew point of humid air, mass transfer and phase change to water droplets takes place. The water droplets are considered as the secondary dispersed-liquid phase. The two separate phases exchange momentum, mass and energy and condensation takes place on the wall surface and in the interior space of the room in regions of low temperature.
MATHEMATICAL MODELLING A two-phase flow Euler-Euler mathematical model is developed, wherein the humid air and water droplets are being treated as separate phases. The mathematical model consists of the time-dependent Navier-Stokes (NS) equations and the continuity equation for a 3D, two-phase, fluid flow. The assumptions made for the problem are the following: i) both phases, gaseous phase (humid air) and liquid phase (water droplets), are treated as incompressible fluids, ii) each phase is a continuum, so that the derivatives are uniquely defined, iii) Newtonian fluids, interdispersed and exchanging momentum, mass and energy, iv) phase change takes place, v) homogeneous condensation process takes place at equilibrium, vi) spherical monodisperse droplets of one size group (mean AED 10μm), vii) laminar viscosity is assumed constant and equal to the dry air viscosity, viii) no moisture storage capacity of the wall surfaces is taken into account. The general form of all the governing differential conservation equations is (Spalding, 1978; Markatos, 1989): Ri ii div Ri i ui i Ri ,i gradi S ,i t
(1)
The water-vapour mass condensed to water droplets is described by Eq. (2) (Padfield, 2014): mwv 0.062 1.0E 05 p dry air Vcell R1
(2)
The interphase heat transport is described by Eq. (3) (Markatos, 1986b): qint Cq,int T1 T2
(3)
where Cq,int the interphase thermal transfer coefficient and T1 , T2 the bulk temperatures of each phase. The frictional force F f ,int per unit volume at the humid air-water droplets interphase, due to the differing phase velocities is (Ishii and Mishima, 1984):
F f ,int 0.5CD Apr g Vg Vw Vg Vw C f ,int Vg Vw
3
(4)
4 The buoyancy force Fb per unit volume in the vertical direction follows the Boussinesq approximation (Markatos, 1986a). The properties of humid air are described by the following equations (Padfield, 2014; Perry and Green, 1999): a) The concentration of water-vapour, usually defined as kg water vapour / kg dry air , is described by the following equation: kg water vapour / kg dry air 0.018 p / 0.029P p (5) b) The relative humidity (RH) is described by the following ratio: RH
p ps
(6)
c) The saturation vapour pressure ( ps ) obeys the following equation:
ps 610.78 expT / T 238.317.2694
(7)
where T is the temperature in degrees Celsius d) The enthalpy of humid air ( kJ / kg ) follows Eq. (8):
h 1.007 T 0.026 b (2501 1.84 T )
(8)
where b the water content in kg water vapour / kg dry air Initial and boundary conditions The initial temperature of humid air inside the room is 303 oK and on the surface of the cold walls 290 oK. Both phases have a constant density based on the pressure, temperature and relative humidity conditions inside the room. Specific heat of water droplets equals the value (4190.0 J / kg K ) of the dew point temperature. The initial pressure inside the room is equal to the atmospheric pressure. Two cases of initial relative humidity conditions (90% and 60%) are tested (Table 1) (ASHRAE handbook, 1983). Table 1: Dew point temperature Humidity of air Dew point temperature
90% 301.2 oΚ
60% 294.4 oΚ
NUMERICAL SIMULATION The equations are discretized by the finite volume method and solved by the IPSA algorithm (Markatos, 1986b) all embodied in the CFD code PHOENICS (Spalding, 1981), properly modified to include the extra physical processes. A fully implicit method is employed for the numerical solution. The convection terms of all the conservation equations are discretized by the SUPER numerical scheme (Karadimou and Markatos, 2012). The diffusion terms are discretized by the central-differencing scheme. The first-order fully implicit scheme is used for time discretization. The real time of the condensation procedure simulated is 100 sec, enough time for the water vapour to meet the saturation conditions and to start changing its phase. The dew point temperature is defined as an input in the algorithm.
RESULTS AND DISCUSSION In the following sections the grid- and time step independent numerical results of the humidity formation on the surface of the walls are presented, for two different initial humidity conditions (Table 1) at real time 100 sec. The results are presented in terms of the absolute humidity ratio (kg H2O/kg of dry air) and the relevant temperature distribution of humid air.
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Figure 1: Absolute humidity distribution (kg H2O/kg of dry air) at the X-Y plane of the room at time 100 sec (initial relative humidity 90%).
Figure 2: Temperature distribution at the X-Y plane of the room at time 100 sec (initial relative humidity 90%). For the case of initial relative humidity 90% the dew point temperature is 301.2K. It is observed (Fig. 1) that the process of condensation at time 100 sec has taken place near the floor and the surrounding walls, where the temperature is below the dew point (Fig. 2).
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Figure 3: Absolute humidity distribution (kg H2O/kg of dry air) at the X-Y plane of the room at time 100 sec (initial relative humidity 60%).
Figure 4: Temperature distribution at the X-Y plane of the room at time 100 sec (initial relative humidity 60%). For the case of initial relative humidity 60% the dew point temperature is 294.4 K, thus the process of condensation starts to take place at a later time, and as expected at time 100 sec, the water phase covers less part of the walls surface and the interior of the room. Numerical results of the water-vapour condensation agree well with the middle stages of the condensation procedure. The process of condensation starts to take place when the temperature of humid air drops below its dew point. Humidity is formed for the first time on the floor, where the temperature is lower and decreases at an earlier time below the dew point. As the temperature further drops water phase is formed and expanded to cover
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7 larger part of the walls surface. Finally, as the heat, momentum and mass transfer between the two phases continues, water droplets fill the interior of the room.
CONCLUSIONS A (3D) two-phase flow Euler-Euler mathematical model for the prediction of humid air-distribution and condensation in a non-ventilated room is developed. The two phases (humid air and droplets) exchange momentum, mass and energy and the process of condensation procedure takes place when the temperature drops below the dew point. The equations are discretized by the finite volume method and the convection terms of all the conservation equations are discretized by the flow-oriented numerical scheme SUPER. The numerical simulation of the water vapour condensation leads to a reasonable numerical humidity distribution on the surface of the walls and the interior of the room.
REFERENCES ASHRAE handbook, 1983, Thermodynamic Properties of Dry Air and Water and S.I Psychrometric Charts, pp. 360. El Diasty, R., Fazio, P., and Budaiwi, I., 1992, Modelling of indoor air humidity: the dynamic behaviour within an enclosure, Energy and Buildings, Vol. 19, pp. 61-73. Isetti, C., Laurenti, L., and Ponticiello, A., 1988, Predicting vapour content of the indoor air and latent loads for air-conditioned environments: Effect of moisture storage capacity of the walls, Energy and Buildings, Vol. 12, pp. 141-148. Ishii, M., and Mishima, K., 1984, Two-fluid model and hydrodynamic constitutive relations, Nuclear Engineering and Design, Vol. 82, pp. 107-126. Karadimou, D.P., Markatos, N.C., 2012, A novel flow oriented discretization scheme for reducing false diffusion in three dimensional (3D) flows: An application in the indoor environment, Atmospheric Environment, Vol. 61, pp. 327-339. Liu, J., Aizawa, H., Yoshino, H., 2004, CFD prediction of surface condensation on walls and its experimental validation, Building and Environment, Vol. 39, pp. 905-911. Lu, X., 2002, Modelling of heat and moisture transfer in buildings II. Applications to indoor thermal and moisture control, Energy and Buildings, Vol. 34, pp. 1045-1054. Lu, X., 2003, Estimation of indoor moisture generation rate from measurement in buildings, Building and Environment, Vol. 38, pp. 665-675. Ma, X., Li, X., Shao, X., Jiang, X., 2013, An algorithm to predict the transient moisture distribution for wall condensation under a steady flow field, Building and Environment, Vol. 67, pp. 56-68. Markatos, N.C., 1986a, The mathematical modelling of turbulent flows, Applied Mathematical Modelling, Vol. 10. Markatos, N.C., 1986b, Modelling of two-phase transient flow and combustion of granular propellants, International Journal of Multiphase Flow, Vol. 12, pp. 913-933. Markatos, N.C., 1989, Computational fluid flow capabilities and software, Ironmaking and Steelmaking, Vol. 16, pp. 266-273. Mimouni, S., Foissac, A., Lavieville, J., 2011, CFD modeling of wall steam condensation by a two-phase flow approach, Nuclear Engineering and Design, Vol. 241, pp. 4445-4455. Padfield, T., 2014, Conservation Physics, An online textbook in serial form: http://www.conservationphysics.org/atmcalc/atmoclc2.pdf Perry, R.H., Green, D.W., 1999, Perry’ s chemical engineers’ handbook, McGraw-Hill, New York. Pu, L., Xiao, F., Li, Y., Ma, Z., 2012, Effects of initial mist conditions on simulation accuracy of humidity distribution in an environmental chamber, Building and Environment, Vol. 47, pp. 217-222. Spalding, D.B., 1978, Numerical Computation of Multiphase Flow and Heat-transfer. A Contribution to Recent Advances in Numerical Methods in Fluids, in Taylor C. & Morgan K, eds. Pineridge Press, pp. 139-167. Spalding, D.B., 1981, A general purpose computer program for multi-dimensional one or two-phase flow, Mathematics and Computers in Simulation XIII, pp. 267-276. Stavrakakis, G.M., Karadimou, D.P., Zervas, P.L., Sarimveis, H., Markatos, N.C., 2011, Selection of window sizes for optimizing occupational comfort and hygiene based on computational fluid dynamics and neural networks. Building and Environmnet, Vol. 46, pp. 298-314. Teodosiu, C., Hohota, R., Rusaouen, G., Woloszyn, M., 2003, Numerical prediction of indoor air humidity and its effects on indoor environment, Building and Environment, Vol. 38, pp. 655-664.
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