# CFD modeling of moisture transfer in porous material subject to dynamic change of boundary temperature.

INTRODUCTION

Porous material has been widely utilized for thermal and sound insulation. The pore structure of the insulation material allows for penetration of the air that contains water vapor. Subject to change of boundary temperature, the water vapor may condense and form the liquid water within the material. The accumulated liquid can cause numerous adverse impacts, such as degradation of insulation performance, microbe growth, and corrosion, etc. In porous insulation material, fluid flow, energy transport, species transfer, and moisture phase change are coupled together, which requires sophisticated modeling to predict the temporal moisture content of each phase in space.

There have been many research works addressing moisture transfer in porous media. The current computing models differ in space from one dimension (Talukdar et al, 2007), two dimensions (Erriguible et al, 2006), to three dimensions (Mohan and Talukdar, 2010), and differ in time from steady (like the dew-point method, ASHRAE, 2013) to transient type. Most of the computational models are transient because dynamic change of moisture content is highly concerned. The moisture considered may be in a single separate phase (Litavcova et al, 2014), two phases (Erriguible et al, 2006), or three phases (Kong and Zhang, 2013). If two or three phases of moisture are handled, the conversion between phases may not be omitted. The gaseous phase may include both inert air and water vapor (as treated by most models), or just a single species of water vapor for simplicity (Janssen et al, 2007). Most studies disregard convective transport of species because the porosity in common situation is small enough to inhibit macroscopic-scale flow motion. Darcy's Law is applied to represent air motion velocity and diffusion flux of water vapor (Erriguible et al, 2006; Li, et al, 2004). For the liquid water, movement of the free water is also approximated by Darcy's Law, while Fick's Law is for diffusion of the adsorbate liquid (Erriguible et al, 2006). However, for porous material that has large porosity the bulk flow motion may be significant, and the Navier-Stokes equations may have to be solved instead of the simiplified Darcy's Law. Subject to dynamic change of boundary temperature, the induced buoyancy force can be meaningful. The temporal buoyancy force varying with temperature must be taken into account.

This investigation proposes a transient two-dimensional computational fluid dynamics (CFD) model to predict the natural dispersion of water vapor and its phase change in a sponge block (flexible polyurethane foam (PUF) with the density 27.06 kg/[m.sup.3] (1.69 lb/[ft.sup.3]) and the porosity 97%) when subject to dynamic change of the psychrometric condition. The temporal buoyancy force is represented by the Boussinesq model. An experimental test was conducted to obtain the measurement data for model validation. Some critical parameters that have great impacts to modeling accuracy are analyzed.

CFD MODELING PRINCIPLES

To well predict the temporal moisture in porous media, dispersion of water vapor, energy transport, and the phase change of moisture are modeled together. The following assumptions are adopted to simplify the problem:

1. Two-dimensional space

2. Identical velocity at the same location regardless of gas component

3. Stationary liquid water and ice if available and only with a small amount (volumetric fraction<0.1%)

4. Phase change of moisture occurring only when air is saturated

5. Homogeneous porosity distribution

6. Thermodynamic balance at any time

The volumetric-based Navier-Stokes equations are applied to solve for flows permeating into the porous insulation material. The additional source terms including the viscous resistance term and the inertial resistance term are added into the momentum equations as,

[[partial derivative]([phi][rho]V)/[partial derivative]t] + div ([phi][rho]VV) = -[phi][nabla]p + [nabla]([phi][??]) - [phi][rho]g[beta][DELTA]T - ([[mu]/[alpha]] + [[rho][C.sub.2]/2] [absolute value of V])V (1)

where [phi] is porosity, [rho] is fluid density, kg/[m.sup.3] (lb/[ft.sup.3]), V is fluid velocity vector, m/s (ft/s), t is time, s, p is pressure Pa (lb/[ft.sup.2]), [??] is shear stress tensor, Pa(lb/[ft.sup.2]), g is the gravitational acceleration, m/[s.sup.2] (ft/[s.sup.2]), [beta] is thermal expansion coefficient, [K.sup.-1] ([R.sup.-1]), [DELTA]T is temperature difference with respect to a reference temperature,[degrees]C ([degrees]F), [mu] is dynamic viscosity, Pa x s (lb/ft x s), 1/[alpha] is viscous resistance coefficient, [m.sup.-2] ([ft.sup.-2]), and [C.sub.2] is inertial resistance coefficient, [m.sup.-1] ([ft.sup.-1]) and

[alpha] = [[D.sup.2.sub.P]/150] [[[phi].sup.3]/[(1 - [phi]).sup.2]] and [C.sub.2] = [3.5/[D.sub.P]][(1 - [phi])/[[phi].sup.3]) (2)

where [D.sub.P] is equivalent mean diameter of the solid skeleton particles in the porous media, m (ft). Subject to dynamic change of boundary temperature, the buoyancy force approximated by the Boussinesq model (the third term on the right hand side of Eq. (1)) can be temporal.

According to assumption #3 above, only the moisture transfer for the water vapor is considered. The water vapor is treated as a passive tracer gas carried by air and the governing equation is,

[[partial derivative][[rho].sub.v]/[partial derivative]t] + div(V[[rho].sub.v]) = div([D.sub.eff]grad[[rho].sub.v]) + [S.sub.v] (3)

where [[rho].sub.v] is water vapor density, kg/[m.sup.3] (lb/[ft.sup.3]), [D.sub.eff] is effective diffusion coefficient, [m.sup.2]/s ([ft.sup.2]/s), and [S.sub.v] is source or sink of the water vapor due to moisture phase change. Based on assumption #4, [S.sub.v] is nonzero only when the air is saturated. Hence, the source or sink term holds value during the moisture phase change process.

If RH = 1 and T > 0[degrees]C (32[degrees]F), the evaporation or condensation of moisture is modeled by,

[S.sub.v] = -[[rho].sub.1] [[partial derivative][[phi].sub.1]/[partial derivative]t]

where [[rho].sub.1] is the liquid water density, kg/[m.sup.3] (lb/[ft.sup.3]), and [[phi].sub.1] is the volumetric fraction of the liquid.

During the evaporation or condensation of the moisture, the species transfer is closely coupled with the energy equation that can be expressed as,

[[rho].sub.m][C.sub.p,m] + [[partial derivative]T/[partial derivative]t] + [[rho].sub.g][C.sub.p,g] div(VT) = div ([k.sub.eff]gradT) + [S.sub.T] (5)

where [[rho].sub.m] is the density of the mixture that Includes skeleton of Insulation material, air, water vapor and liquid water, kg/[m.sup.3] (lb/[ft.sup.3]), [C.sub.p,m] is the specific heat of the mixture, kJ/(kg x K) (Btu/(lb x[degrees]F)), [[rho].sub.g] is the density of the gas mixture, kg/[m.sup.3] (lb/[ft.sup.3]), T is temperature,[degrees]C([degrees]F), [k.sub.eff] is the effective heat conduction coefficient of the mixture, W/K x m (Btu/(ft x h x[degrees]F)), and [S.sub.T] is the heat source or sink term during moisture phase change which has the following expression,

[S.sub.T] = [[rho].sub.1] [[partial derivative][[phi].sub.1]/[partial derivative]t] [h.sub.gl] (6)

where [h.sub.gl] is the latent heat between water vapor and liquid water, kJ/kg (Btu/lb).

In the above equations, the mixture properties are defined as,

[[rho].sub.m][C.sub.p,m] = [[phi].sub.g][[rho].sub.g][C.sub.p,g] + [[phi].sub.s][[rho].sub.s][C.sub.p,s] (7)

[k.sub.eff] = [[phi].sub.g][k.sub.g] + [[phi].sub.s][k.sub.s] (8)

[D.sub.eff] = [D.sub.va][[phi].sub.g]/J (9)

where variables with subscript of "s" represent properties for skeletons of the porous material, [D.sub.va] is diffusion coefficient between water vapor and air, [m.sup.2]/s ([ft.sup.2]/s), and J is tortuosity factor to account for tortuous paths in the porous material that may inhibit gas diffusion (Stephen et al, 2005).

By coupling the above equations into a CFD software by means of user-defined functions, the gas flow, temperature, transport of water vapor and moisture phase change can be solved.

A DEMONSTRATION CASE AND SOLUTION STRATEGIES

The above numerical model was applied to resolve the condensation of water vapor within a sponge block when the sponge block was shifted to a high temperature and humidity condition. The dimensions of the sponge block are 0.294 m x 0.212 m x 0.044 m (0.965 ft x 0.696 ft x 0.144 ft). The sponge block was initially stored indoors and the mass and thermal balance was reached between the block and the surrounding air that was at 17.5[degrees]C (63.5[degrees]F) and 30% relative humidity. Then the block was put into a simplified psychrometric chamber that maintained a higher temperature and humidity as shown in Figure 1(a). In the chamber, there 'was a heater beneath to control the inside air temperature and a water pan to stabilize the interior humidity. The sponge block was hung to the hook of a precision digital balance, which threaded a small opening of the chamber with a string. The resolution of the balance is 0.01 g (2.20 x [10.sup.-5] lb) with an accuracy of [+ or -] 0.04 g (8.82 x [10.sup.-5] lb). The temporal mass change of the sponge block was continuously monitored to indicate the mass transfer and phase change of the moisture therein.

[FIGURE 1 OMITTED]

Figure 1(b) shows the monitored air temperature and the calculated water vapor density according to the directly measured air temperature and relative humidity. The resolution of the temperature measurement is 0.1[degrees]C (0.18 ([degrees]F)) with an accuracy of [+ or -] 0.2[degrees]C (0.36 ([degrees]F)), and for relative humidity 0.1% RH with an accuracy of [+ or -] 1.7% RH. Although before putting the sponge block into the chamber, the chamber was maintained at around 34[degrees]C (93.2 ([degrees]F)), the monitored air temperature decreased first and then increased due to the opening of the chamber door. The water vapor density also decreased first and then steadily increased to the nearly steady value. The air temperature and water vapor density in Figure 1(b) serve as boundary parameters for the numerical modeling.

Considering symmetry along the center surface, numerical modeling of the right half section of the sponge block is sufficient. Because both the top and bottom surfaces are well sealed by plastic films, the exchange of water vapor can only occur through the vertical right surface. The first-kind boundary conditions as shown in Figure 1(b) were specified to the right boundary. Due to a large temperature difference at the initial stage, significant buoyancy force was induced which resulted in meaningful bulk air movement through the sponge block. Hence, the thermal buoyancy force was activated.

Table 1 outlines some property parameters adopted in the CFD modeling. An effective porosity was employed when approximating the fluid flow resistance because not all pores were open and well connected between each other for gas flow. A tortuosity factor of 1.9 was applied to account for the tortuous paths within the sponge block (Stephen et al, 2005).

A two-dimensional geometric model was created by CFD preprocessing software. Uniform square grid meshes with a size of 1 mm were generated in space. The total grid number in the porous region is 6468. All variables were discretized by the second-order upwind scheme. The solution time step for energy and moisture equations is 5 s, which was checked already achieving the time-step independency. A maximum of 3000 iterations were executed over space at each time step.

COMPARISON OF THE SOLUTION RESULTS WITH THE MEASUREMENT

Figure 2 presents comparison of the temporal mass change provided by the CFD modeling and the measurement. It can be seen that the CFD modeling has obtained results in excellent agreement with the measurement. The mass increasing rate is large at the initial stage and then reduces as time advances, which is consistent with common sense. Because the water vapor density is so small, the moisture gain is attributed to the liquid water accumulated in the sponge block resulting from water vapor condensation. We concluded that the proposed modeling is able to well predict fluid flow, heat and mass transfer, as well as phase change of moisture.

[FIGURE 2 OMITTED]

To disclose more information on condensation of water vapor and the accompanying heat and mass transfer, Figure 3 presents distribution of gas velocity, temperature, water vapor density, and volume fraction of liquid water at t = 5 min, 10 min, and 20 min, respectively. Due to thermal buoyancy, counterclockwise flow was formed in the right half of the sponge block, although velocity magnitudes were kept in small values. Resulting from flow motion, the upper right part of the sponge block was heated earlier than the bottom left part. And the contour of water vapor density resembles that of the temperature, since water vapor is close to saturation in the whole process. The water vapor condensed more in the upper left part, because the water vapor traced the flow motion of the air mixture to permeate into the sponge block. At locations where temperature was below the dew point, vapor condensation occurred. As time advances, the temperature difference in different parts minimized. Hence the buoyancy-driven flow became weaker. The distribution of water vapor density and temperature appeared more uniform. Then the water vapor condensation rate decreased and the temporal moisture gain increased more gently.

IMPACTS OF SOME KEY PROPERTY PARAMETERS TO MODELING ACCURACY

In the numerical modeling some key parameters or numerical processes were adopted. These include consideration of fluid flow or not (i.e., including both convection and diffusion, or just considering pure diffusion), adopting effective porosity, taking into account tortuosity, and specifying a correct effective heat conduction coefficient, etc. In the following, impact of each of the above factors to the numerical solution results was analyzed.

[FIGURE 3 OMITTED]

Before going further, let us define the abbreviated legends first, as shown in Table 2. Figure 4(a) compares the predicted temporal mass change when including or excluding the convection flow. It can be seen that when considering only the pure diffusion mass transfer, the predicted mass gain is slightly less than that of the measurement. This shows in this case the water vapor transfer is dominated by diffusion instead of convection. Nevertheless, considering the convection flow does help to better match the experimental data.

Figure 4(b) presents comparison of the predicted temporal mass gain when disregarding the effective porosity, i.e., by treating all pores in the sponge block as well open and connected. It can be seen that when including the convective flow, the predicted temporal mass increased faster than that of excluding the convective flow. This is because with a large porosity the convective flow and the associated mass transfer can be meaningful. According to Eq. (9), a larger porosity also leads to a larger effective mass diffusion coefficient. Hence, both predicted mass gain exceed that of the measurement. However, at the final stage both methods predicted temporal mass closer to that by the measurement. This shows that adopting the effective porosity is critical to the temporal mass change rate but not too much for the final mass gain.

[FIGURE 4 OMITTED]

Figure 4(c) outlines the results when adopting a smaller porosity and disregarding the effective porosity. It can be seen that the porosity is very sensitive to the predicted temporal mass gain. This is because the porosity determines the mass ratio between the solid skeleton and the gases. A smaller porosity means less gaseous mass but more solid skeleton. Due to larger thermal mass of the solid skeleton, it is thus understandable that more water vapor can be condensed within the sponge block with a smaller porosity. By comparing Figure 4(b) and Figure 4(c), it can be seen that the convective flow for a porosity of 0.95 is not as important as that for a porosity of 0.97. This implies that it is not quite meaningful to consider the convective flow if the porosity (or effective porosity) is less than 0.95.

Figure 4(d) compares the results when disregarding the tortuosity factor, i.e. by adopting J = 1 in the modeling. Without considering the tortuosity factor, the effective diffusion coefficient increases according to Eq. (9), which makes the diffusion dominate the water vapor transfer. Hence, there is no difference at all when including or excluding the convective flow. A larger effective diffusion coefficient also leads to the overestimation of the temporal mass gain, because the heating of the sponge block was mainly fulfilled by the water vapor condensation rather than by the heat conduction. This shows that adopting the tortuosity factor is also critical to the modeling accuracy. Figure 4(d) also presents the predicted temporal mass gain when adopting a larger heat conduction coefficient. It can be seen that with increase of a heat conduction coefficient the predicted temporal mass gain decreases. This is because a larger heat conduction coefficient means stronger heat conduction, and hence the condensation rate decreases due to faster temperature rise of the sponge block by the heat conduction. It shows that a correct heat conduction coefficient is also of great importance.

CONCLUSION

A numerical model was established to predict moisture transfer in porous insulation material. For accuracy the Navier-Stokes equations instead of simplified Darcy's equations were solved for gaseous flow field. The temporal buoyancy force was accounted for by the Boussinesq density model based on the dynamically updated temperature field. The developed model can predict distribution of gas flow, temperature, water vapor density, liquid water fraction, and temporal mass of liquid water accumulation in the sponge block. After comparing the predicted temporal mass gain that indicates the water vapor condensation rate with the measurement, it shows that the developed model is able to accurately predict the moisture transfer in the porous material. The analysis of some key parameters on the modeling accuracy showed that inclusion of the convective flow is only meaningful when porosity is large (not smaller than 0.95) or when diffusion mass transfer is suppressed. Specification of a correct porosity, effective porosity, tortuosity factor, and heat conduction coefficient is critical to the numerical modeling accuracy.

REFERENCES

ASHRAE. 2013. ASHRAE Handbook Fundamentals. Atlanta: American Society of Heating, Refrigerating and Air-Conditioning Engineers.

Erriguible, A, Bernada, P, Couture, F, and M. Roques. 2006. Simulation of convective drying of a porous medium with boundary conditions provided by CFD, Institution of Chemical Engineers, 84(A2): 113-123.

Janssen, H., Blocken, B., and J. Carmeliet. 2007. Conservative modeling of the moisture and heat transfer in building components under atmospheric excitation. International Journal of Heat and Mass Transfer, 50:1128-1140.

Kong, F., and Q. Zhang. 2013. Effect of heat and mass coupled transfer combined with freezing process on building exterior envelope. Energy and Buildings, 62: 486-495.

Li, F., Li, Y., Liu, Y., and Z. Luo. 2004. Numerical simulation of coupled heat and mass transfer in hygroscopic porous materials considering the infulence of atmospheric pressure. Numerical Heat Transfer, Part B, 45:249-262.

Litavcova, E., Korjenic, A., Korjenic, S., Pavlus, M., Sarhadov, I., Seman, J., and T. Bednar. 2014. Diffusion of moisture into building materials: A model for moisture transport. Energy and Buildings, 68:558-561.

Mohan, V.P.C., and P. Talukdar. 2010. Three dimensional numerical modeling of simultaneous heat and moisture transfer in a moist object subjected to convective drying. International Journal of Heat and Mass Transfer, 53:4638-4650.

Olutimayin, S.O., and C.J. Simonson. 2005. Measuring and modeling vapor boundary layer growth during transient diffusion heat and moisture transfer in cellulose insulation. International Journal of Heat and Mass Transfer, 48: 3319-3330.

Talukdar, P., Osanyintola, O.F., Olutimayin, S.O., and C.J. Simonson. 2007. An experimental data set for benchmarking 1D, transient heat and moisture transfer models of hygroscopic building materials. Part II: Experimental, numerical and analytical data. International Journal of Heat and Mass Transfer, 50:4915-4926.

Lei Chen

Chao-Hsin Lin, PhD, PE

Fellow ASHRAE Member ASHRAE

Jiusheng Yin

Shugang Wang, PhD

Tengfei Zhang, PhD

Member ASHRAE

Lei Chen is a PhD student, Shugang Wang is a professor, Tengfei Zhang is a professor in the Civil Engineering School, Dalian University of Technology (DUT), China, Chao-Hsin Lin is a technical fellow, Jiusheng Yin is a project manager of the Boeing Company, USA.

Porous material has been widely utilized for thermal and sound insulation. The pore structure of the insulation material allows for penetration of the air that contains water vapor. Subject to change of boundary temperature, the water vapor may condense and form the liquid water within the material. The accumulated liquid can cause numerous adverse impacts, such as degradation of insulation performance, microbe growth, and corrosion, etc. In porous insulation material, fluid flow, energy transport, species transfer, and moisture phase change are coupled together, which requires sophisticated modeling to predict the temporal moisture content of each phase in space.

There have been many research works addressing moisture transfer in porous media. The current computing models differ in space from one dimension (Talukdar et al, 2007), two dimensions (Erriguible et al, 2006), to three dimensions (Mohan and Talukdar, 2010), and differ in time from steady (like the dew-point method, ASHRAE, 2013) to transient type. Most of the computational models are transient because dynamic change of moisture content is highly concerned. The moisture considered may be in a single separate phase (Litavcova et al, 2014), two phases (Erriguible et al, 2006), or three phases (Kong and Zhang, 2013). If two or three phases of moisture are handled, the conversion between phases may not be omitted. The gaseous phase may include both inert air and water vapor (as treated by most models), or just a single species of water vapor for simplicity (Janssen et al, 2007). Most studies disregard convective transport of species because the porosity in common situation is small enough to inhibit macroscopic-scale flow motion. Darcy's Law is applied to represent air motion velocity and diffusion flux of water vapor (Erriguible et al, 2006; Li, et al, 2004). For the liquid water, movement of the free water is also approximated by Darcy's Law, while Fick's Law is for diffusion of the adsorbate liquid (Erriguible et al, 2006). However, for porous material that has large porosity the bulk flow motion may be significant, and the Navier-Stokes equations may have to be solved instead of the simiplified Darcy's Law. Subject to dynamic change of boundary temperature, the induced buoyancy force can be meaningful. The temporal buoyancy force varying with temperature must be taken into account.

This investigation proposes a transient two-dimensional computational fluid dynamics (CFD) model to predict the natural dispersion of water vapor and its phase change in a sponge block (flexible polyurethane foam (PUF) with the density 27.06 kg/[m.sup.3] (1.69 lb/[ft.sup.3]) and the porosity 97%) when subject to dynamic change of the psychrometric condition. The temporal buoyancy force is represented by the Boussinesq model. An experimental test was conducted to obtain the measurement data for model validation. Some critical parameters that have great impacts to modeling accuracy are analyzed.

CFD MODELING PRINCIPLES

To well predict the temporal moisture in porous media, dispersion of water vapor, energy transport, and the phase change of moisture are modeled together. The following assumptions are adopted to simplify the problem:

1. Two-dimensional space

2. Identical velocity at the same location regardless of gas component

3. Stationary liquid water and ice if available and only with a small amount (volumetric fraction<0.1%)

4. Phase change of moisture occurring only when air is saturated

5. Homogeneous porosity distribution

6. Thermodynamic balance at any time

The volumetric-based Navier-Stokes equations are applied to solve for flows permeating into the porous insulation material. The additional source terms including the viscous resistance term and the inertial resistance term are added into the momentum equations as,

[[partial derivative]([phi][rho]V)/[partial derivative]t] + div ([phi][rho]VV) = -[phi][nabla]p + [nabla]([phi][??]) - [phi][rho]g[beta][DELTA]T - ([[mu]/[alpha]] + [[rho][C.sub.2]/2] [absolute value of V])V (1)

where [phi] is porosity, [rho] is fluid density, kg/[m.sup.3] (lb/[ft.sup.3]), V is fluid velocity vector, m/s (ft/s), t is time, s, p is pressure Pa (lb/[ft.sup.2]), [??] is shear stress tensor, Pa(lb/[ft.sup.2]), g is the gravitational acceleration, m/[s.sup.2] (ft/[s.sup.2]), [beta] is thermal expansion coefficient, [K.sup.-1] ([R.sup.-1]), [DELTA]T is temperature difference with respect to a reference temperature,[degrees]C ([degrees]F), [mu] is dynamic viscosity, Pa x s (lb/ft x s), 1/[alpha] is viscous resistance coefficient, [m.sup.-2] ([ft.sup.-2]), and [C.sub.2] is inertial resistance coefficient, [m.sup.-1] ([ft.sup.-1]) and

[alpha] = [[D.sup.2.sub.P]/150] [[[phi].sup.3]/[(1 - [phi]).sup.2]] and [C.sub.2] = [3.5/[D.sub.P]][(1 - [phi])/[[phi].sup.3]) (2)

where [D.sub.P] is equivalent mean diameter of the solid skeleton particles in the porous media, m (ft). Subject to dynamic change of boundary temperature, the buoyancy force approximated by the Boussinesq model (the third term on the right hand side of Eq. (1)) can be temporal.

According to assumption #3 above, only the moisture transfer for the water vapor is considered. The water vapor is treated as a passive tracer gas carried by air and the governing equation is,

[[partial derivative][[rho].sub.v]/[partial derivative]t] + div(V[[rho].sub.v]) = div([D.sub.eff]grad[[rho].sub.v]) + [S.sub.v] (3)

where [[rho].sub.v] is water vapor density, kg/[m.sup.3] (lb/[ft.sup.3]), [D.sub.eff] is effective diffusion coefficient, [m.sup.2]/s ([ft.sup.2]/s), and [S.sub.v] is source or sink of the water vapor due to moisture phase change. Based on assumption #4, [S.sub.v] is nonzero only when the air is saturated. Hence, the source or sink term holds value during the moisture phase change process.

If RH = 1 and T > 0[degrees]C (32[degrees]F), the evaporation or condensation of moisture is modeled by,

[S.sub.v] = -[[rho].sub.1] [[partial derivative][[phi].sub.1]/[partial derivative]t]

where [[rho].sub.1] is the liquid water density, kg/[m.sup.3] (lb/[ft.sup.3]), and [[phi].sub.1] is the volumetric fraction of the liquid.

During the evaporation or condensation of the moisture, the species transfer is closely coupled with the energy equation that can be expressed as,

[[rho].sub.m][C.sub.p,m] + [[partial derivative]T/[partial derivative]t] + [[rho].sub.g][C.sub.p,g] div(VT) = div ([k.sub.eff]gradT) + [S.sub.T] (5)

where [[rho].sub.m] is the density of the mixture that Includes skeleton of Insulation material, air, water vapor and liquid water, kg/[m.sup.3] (lb/[ft.sup.3]), [C.sub.p,m] is the specific heat of the mixture, kJ/(kg x K) (Btu/(lb x[degrees]F)), [[rho].sub.g] is the density of the gas mixture, kg/[m.sup.3] (lb/[ft.sup.3]), T is temperature,[degrees]C([degrees]F), [k.sub.eff] is the effective heat conduction coefficient of the mixture, W/K x m (Btu/(ft x h x[degrees]F)), and [S.sub.T] is the heat source or sink term during moisture phase change which has the following expression,

[S.sub.T] = [[rho].sub.1] [[partial derivative][[phi].sub.1]/[partial derivative]t] [h.sub.gl] (6)

where [h.sub.gl] is the latent heat between water vapor and liquid water, kJ/kg (Btu/lb).

In the above equations, the mixture properties are defined as,

[[rho].sub.m][C.sub.p,m] = [[phi].sub.g][[rho].sub.g][C.sub.p,g] + [[phi].sub.s][[rho].sub.s][C.sub.p,s] (7)

[k.sub.eff] = [[phi].sub.g][k.sub.g] + [[phi].sub.s][k.sub.s] (8)

[D.sub.eff] = [D.sub.va][[phi].sub.g]/J (9)

where variables with subscript of "s" represent properties for skeletons of the porous material, [D.sub.va] is diffusion coefficient between water vapor and air, [m.sup.2]/s ([ft.sup.2]/s), and J is tortuosity factor to account for tortuous paths in the porous material that may inhibit gas diffusion (Stephen et al, 2005).

By coupling the above equations into a CFD software by means of user-defined functions, the gas flow, temperature, transport of water vapor and moisture phase change can be solved.

A DEMONSTRATION CASE AND SOLUTION STRATEGIES

The above numerical model was applied to resolve the condensation of water vapor within a sponge block when the sponge block was shifted to a high temperature and humidity condition. The dimensions of the sponge block are 0.294 m x 0.212 m x 0.044 m (0.965 ft x 0.696 ft x 0.144 ft). The sponge block was initially stored indoors and the mass and thermal balance was reached between the block and the surrounding air that was at 17.5[degrees]C (63.5[degrees]F) and 30% relative humidity. Then the block was put into a simplified psychrometric chamber that maintained a higher temperature and humidity as shown in Figure 1(a). In the chamber, there 'was a heater beneath to control the inside air temperature and a water pan to stabilize the interior humidity. The sponge block was hung to the hook of a precision digital balance, which threaded a small opening of the chamber with a string. The resolution of the balance is 0.01 g (2.20 x [10.sup.-5] lb) with an accuracy of [+ or -] 0.04 g (8.82 x [10.sup.-5] lb). The temporal mass change of the sponge block was continuously monitored to indicate the mass transfer and phase change of the moisture therein.

[FIGURE 1 OMITTED]

Figure 1(b) shows the monitored air temperature and the calculated water vapor density according to the directly measured air temperature and relative humidity. The resolution of the temperature measurement is 0.1[degrees]C (0.18 ([degrees]F)) with an accuracy of [+ or -] 0.2[degrees]C (0.36 ([degrees]F)), and for relative humidity 0.1% RH with an accuracy of [+ or -] 1.7% RH. Although before putting the sponge block into the chamber, the chamber was maintained at around 34[degrees]C (93.2 ([degrees]F)), the monitored air temperature decreased first and then increased due to the opening of the chamber door. The water vapor density also decreased first and then steadily increased to the nearly steady value. The air temperature and water vapor density in Figure 1(b) serve as boundary parameters for the numerical modeling.

Considering symmetry along the center surface, numerical modeling of the right half section of the sponge block is sufficient. Because both the top and bottom surfaces are well sealed by plastic films, the exchange of water vapor can only occur through the vertical right surface. The first-kind boundary conditions as shown in Figure 1(b) were specified to the right boundary. Due to a large temperature difference at the initial stage, significant buoyancy force was induced which resulted in meaningful bulk air movement through the sponge block. Hence, the thermal buoyancy force was activated.

Table 1 outlines some property parameters adopted in the CFD modeling. An effective porosity was employed when approximating the fluid flow resistance because not all pores were open and well connected between each other for gas flow. A tortuosity factor of 1.9 was applied to account for the tortuous paths within the sponge block (Stephen et al, 2005).

A two-dimensional geometric model was created by CFD preprocessing software. Uniform square grid meshes with a size of 1 mm were generated in space. The total grid number in the porous region is 6468. All variables were discretized by the second-order upwind scheme. The solution time step for energy and moisture equations is 5 s, which was checked already achieving the time-step independency. A maximum of 3000 iterations were executed over space at each time step.

COMPARISON OF THE SOLUTION RESULTS WITH THE MEASUREMENT

Figure 2 presents comparison of the temporal mass change provided by the CFD modeling and the measurement. It can be seen that the CFD modeling has obtained results in excellent agreement with the measurement. The mass increasing rate is large at the initial stage and then reduces as time advances, which is consistent with common sense. Because the water vapor density is so small, the moisture gain is attributed to the liquid water accumulated in the sponge block resulting from water vapor condensation. We concluded that the proposed modeling is able to well predict fluid flow, heat and mass transfer, as well as phase change of moisture.

[FIGURE 2 OMITTED]

To disclose more information on condensation of water vapor and the accompanying heat and mass transfer, Figure 3 presents distribution of gas velocity, temperature, water vapor density, and volume fraction of liquid water at t = 5 min, 10 min, and 20 min, respectively. Due to thermal buoyancy, counterclockwise flow was formed in the right half of the sponge block, although velocity magnitudes were kept in small values. Resulting from flow motion, the upper right part of the sponge block was heated earlier than the bottom left part. And the contour of water vapor density resembles that of the temperature, since water vapor is close to saturation in the whole process. The water vapor condensed more in the upper left part, because the water vapor traced the flow motion of the air mixture to permeate into the sponge block. At locations where temperature was below the dew point, vapor condensation occurred. As time advances, the temperature difference in different parts minimized. Hence the buoyancy-driven flow became weaker. The distribution of water vapor density and temperature appeared more uniform. Then the water vapor condensation rate decreased and the temporal moisture gain increased more gently.

IMPACTS OF SOME KEY PROPERTY PARAMETERS TO MODELING ACCURACY

In the numerical modeling some key parameters or numerical processes were adopted. These include consideration of fluid flow or not (i.e., including both convection and diffusion, or just considering pure diffusion), adopting effective porosity, taking into account tortuosity, and specifying a correct effective heat conduction coefficient, etc. In the following, impact of each of the above factors to the numerical solution results was analyzed.

[FIGURE 3 OMITTED]

Before going further, let us define the abbreviated legends first, as shown in Table 2. Figure 4(a) compares the predicted temporal mass change when including or excluding the convection flow. It can be seen that when considering only the pure diffusion mass transfer, the predicted mass gain is slightly less than that of the measurement. This shows in this case the water vapor transfer is dominated by diffusion instead of convection. Nevertheless, considering the convection flow does help to better match the experimental data.

Figure 4(b) presents comparison of the predicted temporal mass gain when disregarding the effective porosity, i.e., by treating all pores in the sponge block as well open and connected. It can be seen that when including the convective flow, the predicted temporal mass increased faster than that of excluding the convective flow. This is because with a large porosity the convective flow and the associated mass transfer can be meaningful. According to Eq. (9), a larger porosity also leads to a larger effective mass diffusion coefficient. Hence, both predicted mass gain exceed that of the measurement. However, at the final stage both methods predicted temporal mass closer to that by the measurement. This shows that adopting the effective porosity is critical to the temporal mass change rate but not too much for the final mass gain.

[FIGURE 4 OMITTED]

Figure 4(c) outlines the results when adopting a smaller porosity and disregarding the effective porosity. It can be seen that the porosity is very sensitive to the predicted temporal mass gain. This is because the porosity determines the mass ratio between the solid skeleton and the gases. A smaller porosity means less gaseous mass but more solid skeleton. Due to larger thermal mass of the solid skeleton, it is thus understandable that more water vapor can be condensed within the sponge block with a smaller porosity. By comparing Figure 4(b) and Figure 4(c), it can be seen that the convective flow for a porosity of 0.95 is not as important as that for a porosity of 0.97. This implies that it is not quite meaningful to consider the convective flow if the porosity (or effective porosity) is less than 0.95.

Figure 4(d) compares the results when disregarding the tortuosity factor, i.e. by adopting J = 1 in the modeling. Without considering the tortuosity factor, the effective diffusion coefficient increases according to Eq. (9), which makes the diffusion dominate the water vapor transfer. Hence, there is no difference at all when including or excluding the convective flow. A larger effective diffusion coefficient also leads to the overestimation of the temporal mass gain, because the heating of the sponge block was mainly fulfilled by the water vapor condensation rather than by the heat conduction. This shows that adopting the tortuosity factor is also critical to the modeling accuracy. Figure 4(d) also presents the predicted temporal mass gain when adopting a larger heat conduction coefficient. It can be seen that with increase of a heat conduction coefficient the predicted temporal mass gain decreases. This is because a larger heat conduction coefficient means stronger heat conduction, and hence the condensation rate decreases due to faster temperature rise of the sponge block by the heat conduction. It shows that a correct heat conduction coefficient is also of great importance.

CONCLUSION

A numerical model was established to predict moisture transfer in porous insulation material. For accuracy the Navier-Stokes equations instead of simplified Darcy's equations were solved for gaseous flow field. The temporal buoyancy force was accounted for by the Boussinesq density model based on the dynamically updated temperature field. The developed model can predict distribution of gas flow, temperature, water vapor density, liquid water fraction, and temporal mass of liquid water accumulation in the sponge block. After comparing the predicted temporal mass gain that indicates the water vapor condensation rate with the measurement, it shows that the developed model is able to accurately predict the moisture transfer in the porous material. The analysis of some key parameters on the modeling accuracy showed that inclusion of the convective flow is only meaningful when porosity is large (not smaller than 0.95) or when diffusion mass transfer is suppressed. Specification of a correct porosity, effective porosity, tortuosity factor, and heat conduction coefficient is critical to the numerical modeling accuracy.

REFERENCES

ASHRAE. 2013. ASHRAE Handbook Fundamentals. Atlanta: American Society of Heating, Refrigerating and Air-Conditioning Engineers.

Erriguible, A, Bernada, P, Couture, F, and M. Roques. 2006. Simulation of convective drying of a porous medium with boundary conditions provided by CFD, Institution of Chemical Engineers, 84(A2): 113-123.

Janssen, H., Blocken, B., and J. Carmeliet. 2007. Conservative modeling of the moisture and heat transfer in building components under atmospheric excitation. International Journal of Heat and Mass Transfer, 50:1128-1140.

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Mohan, V.P.C., and P. Talukdar. 2010. Three dimensional numerical modeling of simultaneous heat and moisture transfer in a moist object subjected to convective drying. International Journal of Heat and Mass Transfer, 53:4638-4650.

Olutimayin, S.O., and C.J. Simonson. 2005. Measuring and modeling vapor boundary layer growth during transient diffusion heat and moisture transfer in cellulose insulation. International Journal of Heat and Mass Transfer, 48: 3319-3330.

Talukdar, P., Osanyintola, O.F., Olutimayin, S.O., and C.J. Simonson. 2007. An experimental data set for benchmarking 1D, transient heat and moisture transfer models of hygroscopic building materials. Part II: Experimental, numerical and analytical data. International Journal of Heat and Mass Transfer, 50:4915-4926.

Lei Chen

Chao-Hsin Lin, PhD, PE

Fellow ASHRAE Member ASHRAE

Jiusheng Yin

Shugang Wang, PhD

Tengfei Zhang, PhD

Member ASHRAE

Lei Chen is a PhD student, Shugang Wang is a professor, Tengfei Zhang is a professor in the Civil Engineering School, Dalian University of Technology (DUT), China, Chao-Hsin Lin is a technical fellow, Jiusheng Yin is a project manager of the Boeing Company, USA.

Table 1. Adopted property parameters in the CFD modeling Item Value Solid Volume 0.03 skeleton fraction, [[phi].sub.s] Density 1000 kg/[m.sup.3] [[rho].sub.s] (62.42 lb/[ft.sup.3]) Heat conduction 0.05 W/(m x K) (0.03 coefficient, Btu/(ft x h x [k.sub.s] [degrees]F)) Specific heat, 1.4 kJ/(kg x K) (0.3 [C.sub.p,s] Btu/(lb x [degrees]F)) Liquid Density 1000 kg/[m.sup.3] water [[rho].sub.1] (62.42 lb/[ft.sup.3]) Volume 0.97 fraction, [[phi].sub.g] Effective 0.8 porosity, [[phi].sub.g_ effective] Gas Diffusion 2.55 x [10.sup./5] coefficient ([m.sup.2]/s) (2.74 between water x [10.sup./4] vapor and air, [ft.sup.2]/s) [D.sub.va] Tortuosity 1.9 factor J Heat conduction 0.0242 W/K x m (0.0140 coefficient, Btu/(ft x h x [k.sub.g] [degrees]F)) Latent heat 2537 kJ/kg (1090.71 between liquid Btu/lb) water and water vapor, [h.sub.gl] Table 2. Description of the abbreviated legends Abbreviation Description Abbreviation Description CI Convection EP80% Effective included porosity: 80% DO Diffusion only J=1 Tortuosity factor: 1 P95% Porosity: 95% J=1.9 Tortuosity P97% Porosity: 97% factor: 1.9

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Title Annotation: | computational fluid dynamics |
---|---|

Author: | Chen, Lei; Lin, Chao-Hsin; Yin, Jiusheng; Wang, Shugang; Zhang, Tengfei |

Publication: | ASHRAE Transactions |

Article Type: | Report |

Date: | Jul 1, 2014 |

Words: | 3664 |

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