Simulation of Moisture Uptake and Transport in a Bed of Urea

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Ind. Eng. Chem. Res. 2008, 47, 7888–7896

Simulation of Moisture Uptake and Transport in a Bed of Urea Particles Xiao-Dong Nie,† Richard W. Evitts,*,‡ and Robert W. Besant† Department of Mechanical Engineering and Department of Chemical Engineering, UniVersity of Saskatchewan, 57 Campus DriVe, Saskatoon, Saskatchewan, Canada S7N 5A9

Granular urea is a commonly used fertilizer, and it is subject to caking when exposed to small quantities of moisture. In this paper, coupled heat and moisture transport equations are used to predict one-dimensional temperature and moisture content distributions in a bed of bulk granular urea particles when humid air flows uniformly through the bed. The moisture sorption and transport processes consider two computational domainsswater vapor diffusion inside each particle and water vapor convection and diffusion in the interstitial air space in the urea particle bed. For energy transport, the temperature is assumed to be uniform inside each particle, but convective heat transfer and conduction between the urea particles and the interstitial air outside particles occur throughout the bed. Comparisons between simulations and data show agreement within the experimental uncertainties for low Reynolds number conditions, where both internal particle and external bed sorption processes are important for porous urea particles in bulk storage. 1. Introduction Granular urea fertilizer is comprised of particles which typically have a particle diameter of 1.8-4.0 mm and have a porous internal structure. When bulk granular fertilizers such as urea are stored and transported, they are often exposed to humid air. Urea particles readily adsorb moisture on the external and internal surfaces when the ambient air relative humidity is higher than 60%, and subsequent drying of these particles will cause crystal growth on both surfaces. Often crystal bridges form between the particles. The formation of the bridges between particles is a fertilizer handling problem because it can lead to the formation of large clumps or cake. There have been several research investigations on heat and mass transfer in porous materials including beds of granular fertilizer. Zili and Nasrallah1 presented a theoretical formulation of forced convective drying of granular products and solved the system of equations using the finite volume method. Li and Fan2 numerically investigated the transient heat and moisture transfer behavior in clothing insulation made of fibrous batting and covering fabrics. They combined the moisture adsorption process with liquid condensation movement and considered the effect of condensate on the porosity and permeability of the batting. Travkin and Catton3 reviewed several studies that used volume averaging theory to investigate heat and mass transport in a variety of porous media. Zhou4 measured the properties of granular potash and developed a model using the local volume average method for transient moisture diffusion coupled with heat conduction in a potash bed. Peng et al.5 developed a thermodynamic model of water vapor interactions and a theoretical/numerical model of adsorption and dissolution processes in a potash bed. Chen et al.6,7 considered water movement and accumulation in a potash bed, taking into account liquid film transport by capillarity and gravity as well as vapor diffusion through the interstitial air spaces inside a granular potash bed. Research has also been done on particles that have a porous structure. For example, silica gel is a well-known granular desiccant used to remove moisture from confined spaces. The * To whom correspondence should be addressed. Tel.: 1-306-9664766. Fax: +1-306-966-4777. E-mail: [email protected]. † Department of Mechanical Engineering. ‡ Department of Chemical Engineering.

very large internal surface area inside each silica gel particle gives it a very large moisture content capacity. Pesaran and Mills8 simulated heat and mass transfer of airborne moisture into a thin bed of silica gel particles with an inlet air flow parallel to the bed surface. They accounted for different internal diffusion processes but without considering the moisture diffusion through the interparticle spaces. They concluded that surface diffusion is dominant for microporous silica gel while both surface diffusion and Knudsen diffusion are important for macroporous silica gel. Sun and Besant9 simulated and measured the convection of moist air through a bed of silica gel particles and showed that the internal diffusion coefficient inside a spherical particle is much smaller than the diffusion coefficient in the air between the particles. Chen et al.10,11 examined porous potash particles consisting of a solid core or seed particle that is surrounded by a layer of microscopic particles. They found that for small moisture contents these porous potash particles were less likely to cake than the solid crystal form of potash fertilizer at the same moisture content. They attributed this reduced caking as being due to moisture transport into the internal particle surfaces away from the external surface of each particle. In this paper moisture and heat transfer through a bed of urea particles is presented. Urea is a very important nitrogen-based fertilizer, and it has several other important uses, including ones in the pharmaceutical and plastics industries. Research has been performed on the manufacture of urea particles, and basic research has been done on its interaction with water.12,13 However, little research has been published on the moisture uptake and caking of urea. Urea is very susceptible to product degradation caused by moisture adsorption, crystal growth, and caking, even at moisture contents as low as 0.25% (w/w). A urea particle also has a porous internal structure. Scanning electron microscopic (SEM) images and BET measurements show that urea particles, which are generally nearly spherical in shape, are quite porous with a large internal surface area compared to their external surface area. Figure 1 shows one such SEM image of (a) the interior fractured surface of a urea particle and (b) the external surface magnified by a factor of 10 compared to a.14 Due to the physical differences in the particles as well as their chemical composition, the mechanism of moisture transport in a urea bed is different from that in a granular solid crystal

10.1021/ie701744g CCC: $40.75  2008 American Chemical Society Published on Web 09/18/2008

Ind. Eng. Chem. Res., Vol. 47, No. 20, 2008 7889

Figure 1. SEM images of urea particles (dp ) 1.8-4.0 mm): (a) fracture surface; (b) outer surface.9

Figure 2. Schematic diagram of (a) the overall bed considered and (b) the internal domain (particle) and the external domain (gas).

potash particle bed. In the presence of the humid air, a thin liquid film, consisting of a concentrated solution of the solute, forms on the surface of crystalline solid potash particles. This water film, even at very small moisture contents (e.g., less than 0.1% (w/w)), tends to coat the entire outer surface of the solid particles; however, due to surface tension there may be a larger thickness of solution near the contact points between particles. For urea particles, some moisture may be adsorbed onto the outer surface of the particles and some water may penetrate into the internal surfaces of each particle. Water transport by vapor diffusion through the internal air spaces of each urea particle and surface diffusion on the internal surfaces are different than the transport via convection and diffusion through the external interstitial air space in a bed of particles. In addition, the important properties of a urea particle bed often do not vary in the same manner with moisture content as they do in the bed of solid crystals such as potash fertilizer. For example, experimental data show the effective thermal conductivity of a potash bed depends on moisture content,2 while the effective thermal conductivity of a urea bed is insensitive to small variations in moisture content. Urea is thought to behave in a manner somewhat similar to silica gel, but it has very different physical and chemical characteristics. Nie et al. presented experimental data on the moisture uptake and removal characteristics of a bed of urea particles and measurements of urea particle physical properties.14 In this paper, the objective is to develop a numerical physical model for moisture and energy transfer in a bed of porous urea particles and compare the model with the experimental data to validate the numerical model. 2. Theoretical Model 2.1. Model Assumptions and Development. To aid in the physical modeling, it is assumed that the walls of the urea

particle bed are adiabatic and impermeable and the urea particle bed only varies in temperature and particle-layer-average moisture content uniaxially, in the direction of airflow. That is, only time and the axial flow direction need to be included as the independent variables for the bed. Figure 2 shows a packed bed of urea particles with a flow length or depth of 160 mm and known initial moisture content and temperature that is subject to a step change of the inlet of air flow rate, temperature, and relative humidity. The following assumptions were used to develop the model, which is comprised of two domains, one for the gaseous void space in the particle bed and a second domain for the filled space, i.e., each individual particle embedded into the whole bed. These two domains are coupled at the surface interface of each particle (dashed circle line in Figure 2b), which is assumed to have a uniform temperature and external surface water vapor density. The water vapor convective transport in the air flowing through the urea bed, caused by a forced air pressure gradient and diffusion due to vapor pressure gradient, is one-dimensional. The gaseous interstitial mixture is assumed to be an ideal gas mixture which can be volume-averaged because it is at a quasiequilibrium condition. Heat diffusion, water vapor diffusion, and adsorption/desorption processes are assumed to vary in the radial direction inside each urea particle, and the internal surface area is assumed to be larger than the external surface area of each particle. It is assumed that the external particle surface adsorption/desorption processes are much more rapid than the internal diffusion processes. It is also assumed that the urea particles are spherical and have homogeneous internal physical properties that do not change significantly except for the internal diffusion coefficient, which changes during the adsorption/ desorption processes. The gas in the interconnected pore spaces inside the urea particles is also assumed to behave ideally and is in equilibrium with the urea particle at any time and radial

7890 Ind. Eng. Chem. Res., Vol. 47, No. 20, 2008

Figure 3. Comparison of simulation and experimental relative humidity data for an adsorption process.

Figure 4. Comparison of simulation and experimental relative humidity data for a desorption process.

position. The local total vapor pressure inside the internal pore space is constant and equal to atmospheric pressure. Air flow inside each particle is only sufficient enough to result in a constant total gas pressure within the particle. Heat generation or consumption from chemical reactions such as urea decomposition is considered to be negligible, but the heat of dissolution and adsorption/desorption are included. In Figure 2a,b the two computational domains are schematically illustrated. The external domain (gas phase) labeled “e” includes the air with water vapor in the volume outside of the particles. The interior domain (particle phase) “i” includes the solid particles, particle internal pore space and contents, and adsorbed liquid on the internal and external surfaces of the particles. These two domains are coupled at the air- particle/ liquid interface shown in Figure 2b, where convective heat and water vapor mass transfer occur between the domains. Humid air is transported through the bed by diffusion and forced convection, and water is adsorbed on the outer particle surfaces before it diffuses inside and deposits on the internal pore surfaces.

Each particle is taken as an individual porous medium. Property variables such as temperature and moisture content or water vapor density in the internal domain are dependent on time,t, the position of the particle within the bed in the direction of flow,x, and the radial position within the particle,r, while the property variables in the external domain only depend on time and the position in the bed in the flow direction,x. Mass and energy transport models are developed for each domain using the local volume average (LVA) theory of porous media.15-17 Inside each urea particle in the internal domain, it is assumed that there is no temperature gradient since the Biot number (Bi ) ha(R/3)/kp < 0.1) is very small; therefore, each particle can be considered thermally as a lumped system. Hence, only the temperature gradient in the axial air flow direction is considered, and the temperature gradient inside each particle is ignored. Step changes between one particle and another particle at different values of x can exist, and this will give rise to some heat conduction when the particles are in contact. Thus, heat transport in the axial direction is only considered in the external domain.


Figure 5. Simulated air temperature versus time for an adsorption process at different bed depths.

Figure 6. Simulated air temperature versus time for a desorption process at different bed depths.

Because the heat generation only arises in the particle domain, the difference of heat capacity and thermal conductivity between the particle and gas phase is not negligible, and local thermal equilibrium cannot be assumed between the two domains. Instead, the assumption of nonlocal thermal equilibrium is used to develop the heat transport model between the two domains, and it is assumed that each phase is continuous and local phase thermal equilibrium is applicable.15 Due to its much slower diffusion process and a larger internal surface area compared with the external surface area, the internal diffusion of water vapor inside each individual particle must be included in the model. Therefore, the mass transport models are developed separately for the water vapor inside a particle in the internal domain and water vapor in the interstitial particle bed air space of the external domain. 2.2. Governing Equations. For the coupled mass transport between the internal and external domains, only the behavior of water vapor needs to be considered. Water vapor in the inlet air passes through the urea particle bed together with the air from the inlet to the outlet in the axial direction. Meanwhile,

convective mass transfer occurs between water vapor in the interstitial air space of the bed and the filled urea particle surface. Due to the porous structure inside each urea particle, the convective water vapor mass flux occurring at the particle surfaces will result in adsorption and condensation on the outer surfaces of urea particles, as well as penetration into the internal particle pores by diffusion, where it will be adsorbed to form a condensate on the internal pore surface of each urea particle. In the external domain, the transient water vapor transport within the interstitial air space of the particle bed follows the water mass balance equation.

(

)

∂Fv,e ∂(εg,eFv,e) ∂ ∂ + (u F ) ) D + m ˙ e (1) ∂t ∂x D v,e ∂x e,eff ∂x where the mass flux of dry air is a constant. The effective external diffusion coefficient isDe,eff ) Dv,aεg,e/τe, where Dv,a ) 2.5 × 10-5 m2/s, the porosity is measured to be εg,e ) 0.36, and the turtuosity is assumed to be τe ) 1.1. While water vapor passes through the bed, convective mass transfer occurs between the water vapor in the interstitial space and the urea particle


Figure 7. Simulated transient particle temperature versus time for an adsorption process at different bed depths.

Figure 8. Simulated transient particle temperature versus time for a desorption process at different bed depths.

surface, m ˙ e, and this is regarded as a source/sink term for water vapor flow to or from the external domain. In the internal domain, water vapor diffusion inside each spherical particle is governed by

which describes the mass convection of water vapor passing through the area boundary of each particle,Av,e.

(

where Fv,i(x,r)R,t) is the interface vapor density at the liquid-gas interface on each particle and hm is the mass convection coefficient for water vapor at this same interface. This term is required at every point and any time in the external domain eq 1 but is only a boundary condition for eq 2 in the internal domain. In the gaseous phase, the energy equation in the interstitial air space between particles is

)

∂Fv,i ∂(εg,iFv,i) 1 ∂ 2 ) 2 r Di,eff - m ˙i ∂t ∂r r ∂r

(2)

The effective internal diffusion coefficient,Di,eff, is a function of moisture content which varies as the moisture content changes with time as described in section 2.3. In eq 2, m ˙ i is the water vapor phase change rate inside the internal domain on both the internal pore surfaces of each particle and on the external surface of each particle. An aqueous liquid or adsorbed phase continuity equation is also required. m ˙ i ) Fl

∂εl,i ∂t

(3)

Mass transfer of water vapor between the internal and external domains is coupled at the interface of each particle by eq 4,

m ˙ e ) - hm[Fv,e(x, t) - Fv,i(x, r ) R, t)]Av,e

(

(4)

)

∂[uD(FCp)gTg,e] ∂Tg,e ∂[εg,e(FCp)gTg,e] ∂ + ) Ke,eff + he ∂t ∂x ∂x ∂x (5) The external domain effective thermal conductivity of the gas phase, i.e., moist air in the interstitial space throughout the bed, is calculated by Ke,eff ) εg,e(Ka + WeKv)

(6)


Figure 9. Comparison of predicted and experimental data of the transient average moisture content of the bed during an adsorption process.

Figure 10. Comparison of predicted and experimental data of the transient average moisture content of the bed during a desorption process.

where the dry air thermal conductivityKa ) 0.0263 J/(kg · K), the water vapor thermal conductivity Kv ) 0.0196 J/(kg · K), and W is the humid ratio in the air. The heat source/sink term, h˙e, means the heat transfer rate through the boundary interfaces from the internal domain to the external domain or vice verse. In the particle phase, the heat conduction in axial direction is governed by eq 7:

(

)

∂Ti ∂{[εp(FCp)p + (εl,iεp + εl,e)(FCp)l,e]Ti} ∂ ) Ki,eff ∂t ∂x ∂x h˙e + h˙i(7) Using the same weighted average method as that for the external domain, the internal effective thermal conductivity for a particle,Ki,eff, is expressed as Ki,eff ) εpKp + (εl,iεp + εl,e)Kl

(8)

where the liquid water thermal conductivity Kl ) 0.613 W/(m · K), but Kp, the dry and porous urea particle thermal conductivity, is unknown because up to now there has been no literature that shows the thermal conductivity of pure urea or

urea particles. Since the Biot number is small for each particle, it was assumed that the temperature of each particle was uniform at any time but each particle could have a different temperature. This will induce interparticle conduction at interparticle contact points. That is, in the external domain the effective conductivity of the interstitial particle bed space must be modified by a contact conduction term for the bed. This equation is Kb,eff ) Ki,eff + Ke,eff ) Kb,contact + Ke,eff

(9)

The value of Kb,contact can be determined indirectly from eq 9 by measuring the heat flux in a particle bed with a known temperature gradient across the bed. Kb,eff was measured experimentally by a Fox 314 heat flow meter (HFM). For a typical smaller urea particle bed (average particle size dp ) 1.8 mm), Kb,eff changes from 0.1856 ( 0.0029 to 0.2407 ( 0.0165 W/(m · K) over a range of moisture content from 0 to 0.8% (w/ w), while for a larger urea particle bed (average particle sizedp ) 2.2 mm), Kb,eff changes from 0.2205 ( 0.0044 to 0.2359 ( 0.0067 W/(m · K) over the same range of moisture content. Therefore,Kb,contact at dry condition was determined to be, for


example, Kb,contact ) 0.1761 W/(m · K) for urea beds with smaller mean particle diameters, while Kb,contact ) 0.2110 W/(m · K) for urea beds with larger mean particle diameters since Ke,eff was determined from eq 6 to be 0.0095 W/(m · K). The heat transfer rate, h˙e, between the two domains is coupled by the convective heat flux through the area boundary interfaces, Av,e. h˙e ) -ha(Tg,e - Ti)Av,e

(10)

2.3. Effective Internal Diffusion Coefficient. The volume averaged effective internal mass diffusivity within each particle is used to determine both the water vapor diffusion through the pore space and surface diffusion of the liquid film on the pore surfaces. This water movement is mostly governed by the internal structure of the particle and the distribution of internal pores. The effective internal diffusivity is expected to be a function of moisture content.5 The experimental data show that this internal diffusion coefficient exponentially increases with moisture content during the adsorption process inside a particle, and this relationship, given for X < 0.004, is presented by Nie et al.18 Di,eff ) Di,eff,t0(0.905 + 0.085e

10000X ⁄ 5.3)

(11)

where the internal diffusion coefficient is Di,eff,t0 ) Dv,iεg,i,t0 ⁄ τi + Ds,t0

(12)

tainty. Typically this uncertainty was (20% of the value of ha. These new values were used in the results presented in the graphs. 2.5. Thermodynamics of Urea Dissolution. The heat source term h˙i in eq 4 includes two parts: (1) the enthalpy due to phase change m ˙ p_avHfg and the enthalpy change m ˙ p_avCpv(Tg - Ti) due to temperature change, and (2) the heat of solution, m ˙ p__avβHs. The average rate of water adsorbed by a urea particle at different bed position is defined by R

m ˙ p_av(x) )

Jm ) StmSc2⁄3 ) Jh ) StPr2⁄3 ) 0.023Redh- 0.3

(14)

where the Stanton number St ) ha/(FupCp)a, the Prandtl number Pr ) υ/R, the Reynolds number Redh ) updh/υ, the Stanton number for mass transfer Stm ) hm/up, and the Schmidt number Sc ) υ/Dav Because the uncertainty in these correlations is large, a number of transient temperature step change experiments for flow through the urea bed with a 9 cm depth were performed. The data from these tests allowed for the calculation the convective heat transfer coefficient for the bed and its uncer-

4πr2m ˙ i(x, r) dr

∫

R

4πr2 dr

(15)

It is assumed that urea dissolution takes place everywhere liquid water exists on the particle surface and a saturated solution is formed. Condensation causes the urea temperature to increase, while dissolution causes a temperature decrease. Therefore, the sensible energy release rate for each particle is ˙ p_av(Hfg + Cpv(Tg - Ti) + βHs) h˙i ) m

(16)

where, at room temperature (approximately 22-24 °C), Hfg ) 2446 kJ/kg,15 Hs ) -13.8 kJ/mol,16 and β ) 19 mol/kg.17 2.6. Initial and Boundary Conditions. It is assumed that water vapor density and temperature are initially constant and uniformly distributed both inside the particles and within the bed.

(13)

In contrast, this diffusion coefficient is essentially a constant during a desorption process. Nie et al. explained the reason why the internal diffusion coefficient increases with moisture content during the adsorption process. It is that the number of pathways increases radially toward the center of the particles inside the particles. Thus, as moisture penetrates into the particle, it experiences a greater number of pathways, leading to an increase in the effective diffusion coefficient. Conversely, as the particles dry out, mass transfer is limited due to the lower number of pores closer to the external surface of the particles, and then the effective diffusion coefficient remains constant.14 2.4. Convection Coefficient. Many studies have been done using the indirect volume average technique to find the internal heat transfer coefficient between the interstitial fluid and the particles in different types of porous media.5,15-17,19 Variations or differences in the predictions of these correlations suggest an uncertainty of more than (100% for the case of low Reynolds number (1 e Redh e 20). Using the mass-heat transfer analogy, a modified Chilton-Colburn correlation presented in eq 14 was tried in the simulations for 1 e Redh e 20.

0

0

The particle moisture content,X, is calculated by X ) εlFl ⁄ (εsFs) ) (εl,e + εpεl,i)Fl ⁄ (εpεs,iFs)

∫

Fv,e(x, t ) 0) ) Fv,e_t0

(17)

Tg,e(x, t ) 0) ) Tg,e_t0

(18)

Fv,i(x, r, t ) 0) ) Fv,i_t0

(19)

Ti(x, t ) 0) ) Ti_t0

(20)

In the exterior domain, i.e., within the bed along the direction of air flow outside of the particles, the inlet air water vapor density and temperature are assumed to be constant. Fv,e(x ) 0, t) ) Fv,e_in

(21)

Tg,e(x ) 0, t) ) Tg,e_in

(22)

The gradient of mass and heat transfer at the outlet of the bed is negligible so that convective mass transfer dominates. ∂Fv,e (x ) L, t) ) 0 ∂x

(23)

∂Tg,e (x ) L, t) ) 0 ∂x

(24)

It is assumed that there is zero mass flux at the center of each particle. ∂Fv,i (x, r ) 0, t) ) 0 ∂r

(25)

At the particle interface surface the water vapor convected to the surface of the particle is diffused into the particle during adsorption. That is to say, the source/sink term for the external domain is the boundary conditions for the individual particles. Di,eff

∂Fv,i (x, r ) R, t) ) hm[Fv,e(x, t) - Fv,i(x, r ) R, t)] (26) ∂r

For particle sensible energy transfer from layer to layer within the bed it is assumed that there is a zero energy gradient at both the inlet and outlet.


∂Ti (x ) 0, t) ) 0 ∂x ∂Ti (x ) L, t) ) 0 ∂x

(27) (28)

3. Simulation Results and Comparisons with Data The governing equation of the heat and mass transfer in both internal and external domains are discretized using the finite volume method20 in a fully implicit scheme. The time step for both domains was set as dt ) 1 s, and the size step dx for the external domain along the bed was the same as a urea particle diameter, i.e., dx ) 2.2 mm for large particles and dx ) 1.8 mm for small particles, while the size step for the internal domain inside a spherical urea particle was 2% of the particle radius. Numerical simulations were carried out over each domain, which is coupled at the air-particle/liquid boundary interfaces. An iterative technique was used to solve the coupled equations and obtain solutions for selected initial and boundary conditions. Any property changes with moisture content and temperature were updated to the new value after each time step. Because the internal diffusion process is much slower than the external diffusion and convection process, simulation for each time step starts at the external domain and then the new water vapor density and temperature are used as the convective boundary conditions for the internal domain or particle phase; updated water vapor density at the particle surface and the particle temperature are again embedded into the simulation for the next time step. The numerical results are presented for an actual experimental condition (i.e., actual experimental inlet temperature and relative humidity data are used as the inlet condition in the numerical simulation during both adsorption and desorption processes) with dp ) 2.2 mm and Redh ) 5. Figures 3 and 4 compare the numerical predictions of the relative humidity at the inlet and outlet of the bed. In Figure 3, which is for an adsorption process, there is some initial deviation in the outlet predictions that becomes close to the uncertainty in the data after approximately 10 h. For the desorption process presented in Figure 4 there is good agreement at all times. In these simulations a constant inlet condition was prescribed and some of the deviation between the predicted and experimental results would have been eliminated if the actual inlet condition was used. This is more pronounced in Figure 3 because during the adsorption experiment there are significant variations in the inlet relative humidity. This is because as an adsorption experiment progresses the water level of the bubbler tanks, which provide the moisture in the air stream, drops gradually, while the same amount of dry air passes through the humidifier. This gives rise to the changes in the supply air humidity and the need to periodically adjust the bubble tank valves. However the outlet humidity follows the inlet one, but it is delayed due to the slow sorption processes in the bed. It is seen that the relative humidity at the outlet starts to change after approximately 2 min. However it only takes 6 s for air to pass through the bed at this flow rate, but, due to the slow sorption process, it takes 24 h for the outlet relative humidity to approach the inlet relative humidity. Figures 5-8 show the simulated transient air and particle temperature versus time at different positions in the bed during adsorption/desorption processes. Because of the water vapor phase change due to adsorption and dissolution heat release, the temperature tends to increase quickly for the first hour or so and then drop to inlet temperature during the adsorption process. The particle temperature increase starts from the inlet.

Then with the heat conduction of particle layers along the bed direction and the subsequent water adsorption from the humid air flowing through, particles at any position x start to increase in temperature layer by layer. However, because the inlet air temperature is kept constant, the inlet particle layer is cooled convectively by the flowing air, and then the subsequent particle layers at position x > 0 are cooled while they are heated by the conduction and adsorption and dissolution. That is why particles further in the bed start cooling later and then a higher particle temperature can be reached at these conditions. During the desorption process, the temperature decreases quickly and then increases to the inlet temperature. The particle temperature decrease starts from the inlet. Then with the heat conduction of particle layers along the bed direction and subsequent water desorption due to the dry air flowing through, particles at any position, x, in the bed start to decrease in temperature layer by layer. However, because the inlet air temperature is kept constant, the inlet particle layer is immediately heated by the convection with flowing air, and then the subsequent particle layers at positions x > 0 are heated by convection while they are heated by the conduction and desorption. That is why particles further in the bed start heating later and then a lower particle temperature can be reached at these conditions. The simulation results are based on the assumption that the inlet humid/dry air is kept constant at room temperature. Fluctuations of inlet air properties, which are not accounted for, cause the data to show some deviations from the simulations. Figures 9 and 10 show comparisons of experimental data and simulation results for average moisture content versus time during adsorption/desorption processes. These simulations show agreement with data that are within the uncertainty limits of the data. The model therefore indicates that the internal effective water vapor diffusion coefficient inside each particle is much lower than that for water vapor in air (i.e., the ratio of between internal and external diffusion coefficient is around 10-4 to 10-3). Also, because the internal particle surface area is larger than the external one, internal moisture diffusion processes dominate the moisture adsorption process in the particle bed. Second, the model points to the fact that the rate of internal diffusion during the desorption process is more rapid than during the adsorption process. Finally, sensitivity studies were done on the spacial and temporal step sizes for both domains. The simulation results were shown to be insensitive to changes in dx from 0.5 to 2 times dp. However, the thermal conductivity due to contact points between particles is very sensitive to the particle size; therefore dx was taken equal to dp in this study. The spatial step size in the internal domain, dr, was chosen carefully and was prescribed when there was negligible change (e1%) in simulated water vapor distributions inside a particle, water vapor density gradient on the particle surface, and the volume average moisture content when dr was decreased. Time step sizes are taken to be the same, dte ) dti, because the coupled nonlinear heat and mass transport process between the internal diffusion process and the external one at the same time need the latest properties to update the coupled source/sink term or boundary condition for the current time step. The time step dt ) 1 s is selected after the simulation does not have significant change (1%) on some important properties. 4. Conclusions In this paper a new model is developed that considers mass and heat transfer in a bed of porous urea particles with air flow


through the bed. The model considers two domains: a domain inside each particle and a second domain outside of the particle in the interstitial air space in the bed. These domains are coupled at the external surface of the particles. The model agrees well with experimental results for moisture adsorption and desorption processes. The following conclusions can be made from this research. The urea particle bed behaves somewhat similarly to a silica gel desiccant bed except that urea has a very small adsorption capacity. Both silica gel desiccants and urea particles have internal pore structure but, compared with silica gel, urea particles have much smaller internal porosity and internal surface area available for internal moisture adsorption. The moisture adsorption process in a bed of urea particles occurs very slowly. This indicates that internal particle transport processes are the rate limiting process since the internal surface is larger than the exterior. The model indicates the effective internal water vapor diffusion coefficient is 10-4 to 10-3 times smaller than that of water vapor in air. The effective internal diffusion coefficient is also dependent on the moisture content during an adsorption process but is independent of moisture content during moisture desorption. Acknowledgment The authors are grateful for financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Potash Corp. of Saskatchewan through a collaborative research and development grant. Nomenclature Av ) specific surface area, m2/m3 Cp ) specific heat capacity at constant pressure, kJ/(kg · K) D ) diffusion coefficient, m2/s dh ) hydraulic diameter, dh ) 4εe/Av,e, m dr ) size step in domain i, m or mm dt ) time step, s dx ) size step in domain e, m or mm Hg ) water vapor phase change heat energy, kJ/kg Hs ) heat of solution, J/mol h˙˙ ) energy source term, J/(m3 · s) ha ) heat convection coefficient, W/(m · K) hm ) mass convection coefficient, m/s K ) thermal conductivity, W/(m · K) m ˙ ) mass source term, kg/(m3 · s) R ) radius of a urea particle, m r ) radial position inside a particle, m T ) temperature, K or °C t ) time, s uD ) Darcy velocity, m/s W ) humid ratio X ) moisture content, w/w or g/g x ) bed position, m Greek Letters β ) solubility of urea in water, mol/kg ε ) volume fraction φ ) relative humidity F ) density, kg/m τ ) tortuosity Subscripts a ) dry air fraction

b ) bed e ) external domain eff ) effective g ) gas phase i ) internal domain l ) liquid phase p ) particle phase s ) solid phase t0 ) initial state v ) vapor fraction ∞ )environment

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ReceiVed for reView December 21, 2007 ReVised manuscript receiVed July 17, 2008 Accepted July 23, 2008 IE701744G

Simulation of Moisture Uptake and Transport in a Bed of Urea

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