Numerical and Experimental Investigation of Heat and MassTransfer in Unsaturated Porous Media with Low Convective

Drying Intensity

Tao Lu,1 Shengqiang Shen,2 and Xiaohua Liu2

1School of Mechanical and Electrical Engineering, Beijing University of Chemical Technology,Beijing 100029, China

2School of Energy and Power Engineering, Dalian University of Technology, Dalian 116024, China

The heat and mass transfer in an unsaturated wet cylindrical bed packed withquartz particles was investigated theoretically and experimentally for relatively lowconvective drying rates. The medium was dried by blowing dry air over the top of theporous bed which was insulated by impermeable, adiabatic material on the bottom andsides. Local thermodynamic equilibrium was assumed in the mathematical modeldescribing the multi-phase flow in the unsaturated porous medium using the energyand mass conservation equations for heat and mass transfer during the drying. Thedrying model included convection and capillary transport of the moisture, and con-vection and diffusion of the gas. The wet and dry regions were coupled with a dynamicboundary condition at the evaporation front. The numerical results indicated that thedrying process could be divided into three periods: the initial temperature rise period,the constant drying rate period, and the reduced drying rate period. The numericalresults agreed well with the experimental data, verifying that the mathematical modelcan evaluate the drying performance of porous media for low drying rates. © 2008Wiley Periodicals, Inc. Heat Trans Asian Res, 37(5): 290–312, 2008; Published onlinein Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/htj.20205

Key words: heat and mass transfer, porous medium, drying

1. Introduction

Convective drying of porous media is encountered in many industrial and agricultural fieldssuch as lumber production, building material processing, and farm products drying. Drying iscommonly done by blowing hot air or steam over the porous media surface. Experiences have shownthat if the drying rate is too slow the production efficiency decreases and considerable energy andtime are wasted. However, if the drying rate is too fast the product quality deteriorates. For example,lumber can experience serious cracking, warping, and fiber collapse during intensive drying. There-

© 2008 Wiley Periodicals, Inc.

Heat Transfer—Asian Research, 37 (5), 2008

Contract grant sponsor: Key Course of Chemical Process Machinery from Beijing Municipal Education Commission (No.XK100100541).

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fore, the drying process should be controlled to provide both better quality and higher efficiency.Lumber and paper drying rates are typically controlled through the drying parameters such astemperature, relative humidity, and velocity of the hot air.

The porous media drying model was given by Lewis [1], who suggested that the drying processconsisted of moisture diffusion within the solid and evaporation from the solid surface. Lewis’diffusion theory for the solid material was extended to the entire drying process by Sherwood [2].Examination of liquid migration within an unsaturated porous media lead Comings and Sherwood[3] to notice that the liquid motion induced by the capillary effect may play an important role in thedrying process. Gurr et al. [4] developed evaporation-condensation theory to explain the physicalphenomena of phase change between the liquid and vapor with a temperature gradient within theporous media. In the past several decades, Huang [5], Berger and Pei [6], and Harmathy [7] developedmore comprehensive theories to describe the simultaneous heat and mass transfer processes based ondiffusion theory, capillary flow theory, and evaporation-condensation theory. Increasing attention onthe heat and mass transfer in porous media has resulted in many experimental and theoreticalinvestigations for different applications. These models have either used Luikov’s approach [8] usingthermodynamics theory of irreversible processes to describe the temperature, moisture, and pressuredistributions within the porous media during drying in a phenomenological manner or Whitaker’smethod [9] which introduces a volume-averaging technique to deal with the fluid flow and heattransfer in the porous media as a continuum. Eckert and Faghri [10] and Liu [11] used Luikov’sequations to study moisture transport in porous media; however, the kinetic coefficients are difficultto obtain so they depend on experimental results for various kinds of porous media. Moreover, Moyneand Perre [12] pointed out that Luikov introduced a phase conversion factor within the phase changeterm which was not based on any sound physical principle, thereby leading to potential difficulties indrying analyses. The advantage of the Whitaker approach is that the heat and mass conservation arefor a continuous media and the porous media properties can be obtained by a volume-averagingmethod within a REV according as a function of the solid, liquid, and gas volume fractions inside theporous media. Whitaker [9] mathematically validated the use of macroscopic equations to describethe microscopic transport phenomena. The volume-averaging method has become a popular approachfor theoretically modeling the drying of porous media, for example, Spolek and Plumb [13], Plumbet al. [14], Ilic and Turner [15, 16], Nasrallah and Perre [17], Turner and Ilic [18], Quintard andWhitaker [19], Rogers and Kaviany [20], Turner [21], Perre and Turner [22], Lu and Shen [23, 24],Lu et al. [25, 26], and Shen et al. [27]. These studies typically used mass and energy conservationequations with phenomenological relationships to describe the liquid and vapor mass fluxes and theheat flux within the porous media. The drying model developed in this study is also based on thevolume-averaging approach of Whitaker.

Scheidegger [28] pointed out that the porous media structure is difficult to describe either inmacro-scale or in micro-scale because of the complexity of the porous media, not to mention thecombination of liquid and gas within the matrix. During drying of unsaturated porous media, thesystem essentially includes four fluids, free water, bound water, vapor, and air, and their transportphenomena must all be analyzed. For simplicity, the free water is defined as all liquid water existingin the media pores but not the bound water in the extremely small diameter capillaries. Liquid canalso be absorbed to the solid surfaces. This adsorption, which is due to the van der Waals’ dispersiveforces, is significant when the pores are very small (smaller than 10–6 m). In the present study the poresizes were assumed to be larger than 10–6 m, so surface adsorption/desorption will not be considered.

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Thus, the discussion will focus on the movement of the fluids including the convection of free waterand the air/vapor mixture due to the bulk pressure gradient, and the diffusion of air and water vaporin the gaseous phase. In addition, capillary transport is regarded as one of the most important transportmechanisms. Therefore, the convective and the diffusion that occur during the drying process mustbe analyzed as multi-phase flow in porous media where the volume fractions and force affect the flowprocesses.

Drying models differ depending on their emphases on different applications. Since thephysical principles of porous media drying processes with low convective drying intensity are thesame, this study is focused on the coupling between heat and mass transfer and on a general dryingmodel for convective drying of porous media, specially in a packed bed. To our best knowledge, fewexperimental investigations on convective drying of porous media have been reported. In order tovalidate the drying model, an experiment for the drying of a packed bed with unsaturated water wascompleted, which was the first motivation of the present work.

Nomenclature

c: specific heat, J/(kg⋅K)D: diffusivity, m2/sg: gravitational constant, m/s2

H: thickness of porous media, mhvap: latent heat of evaporation, J/kgJ: Leverett functionK: intrinsic permeability, m2

kg: relative permeability of gaskl: relative permeability of liquidkm: mass transfer coefficient, m/skT: heat transfer coefficient, W/(m2⋅K)L: characteristic length, mLe: Lewis numberM: molar mass, kg/molmi: mass concentration of the ith-component per unit volume of the porous system, kg/m3

n: direction vectorp: pressure, PaR: gas constant, J/(mol⋅K)S: saturationT: temperature, Ku: averaged velocity, m/sV: velocity of hot drying air, m/sx: x-coordinate

Greek symbols

β: surface tension constant, kg/(K⋅s2)δ: the rate of phase change, kg/(m3⋅s)φ: porosity, m3/m3

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φi: volume fraction of the ith-component, m3/m3

ϕ: relative humidityλi: thermal conductivity of the ith-component, W/(m⋅K)µ: dynamic viscosity, kg/(m⋅s)ρ: density, kg/m3

σ: surface tension of gas-liquid interface, kg/s2

σ0: surface tension at reference temperature, kg/s2

τ: time, s

Subscripts

a: airc: capillarycr: criticalg: gasi: ith-component of the mixtureini: initialir: irreduciblel: liquidp: porouss: solidsat: saturated statev: vapor0: atmosphere1: reference condition

3. Mechanism, Model, and Numerical Method

3.1 Drying mechanism

The drying process model is shown in Fig. 1 where the shaded region, 0 ≤ x ≤ H, representsthe unsaturated porous media to be dried at a relatively low convective drying rate by hot dry airflowing over the top surface of the porous media (x = H) whose temperature is not higher than thesaturation temperature at atmosphere pressure.

The bottom of the porous media (x = 0) was sealed with an impermeable, adiabatic surface.Rogers and Kaviany believed that the moisture removal from the porous media has two distinctregimes: the funicular regime and the evaporative moving front regime [20]. Ilic and Turner [16]divided the drying process into the wet and dry regions divided by the irreducible saturation Sir, wherethe funicular and wet regions are the same and the evaporative moving front regime and the dry regionare similar. In the wet region (Sir < S ≤ 1), the liquid phase is connected throughout the media and theliquid and its vapor are in thermodynamic equilibrium. The liquid flows in the porous media due togravity, capillary pressure gradients, temperature, and saturation. The vapor evaporating from theliquid moves out of the media due to the difference of vapor pressure between the top surface of theporous media and the ambient gas, so that the liquid volume fraction at the top surface decreases first.The flows of vapor and air inside the media are driven by the bulk pressure gradient and the

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concentration gradient. The wet region interface is defined as the surface where the saturation hasdecreased to the irreducible saturation, Sir, which remains constant. When S = Sir, the porous mediais defined as dry. As the drying process advances, the surface temperature increases rapidly and theheat transferred to the porous medium interior results in penetration of the evaporation front into theporous media. Therefore, two regions coexist in the porous media which are coupled by the movingevaporation front. The dry region has no phase change and vapor transport is the only mechanism formoisture transport to the atmosphere.

The motivation of the present study is to establish a drying model for porous media dryingthat can be used in practical applications. Therefore, the drying process is assumed to be one-dimen-sional, and the transient analysis in the x direction only can reasonably describe the process. Furtherassumptions are given below.

(1) The solid phase is rigid and homogeneous throughout the porous media during the wholedrying process.

(2) Both the liquid and air/vapor phases are continuous in the wet region and the air/vaporphases are continuous in the dry regions. The residual water as irreducible water, which cannot bemoved by heating method, is unable to flow due to the absence of a continuous wetting network indry region.

(3) Local thermodynamic equilibrium exists among the phases in the medium.

(4) Darcy’s law holds for the gas and liquid phases.

(5) The binary mixture of air and vapor behaves like an ideal gas and Fick’s law can be usedto describe the air and vapor diffusion.

Fig. 1. (a) Physical model and coordinate system, (b) Macroscopic averaging volume element.

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(6) Conduction occurs in each phase. All thermodynamic properties such as the thermalconductivity and specific heat of each phase are constant and the enthalpy in each phase is a linearfunction of temperature.

(7) In the wet region, the liquid and vapor are in equilibrium and the vapor pressure isdetermined by the Clausius–Clapeyron equation. The capillary pressure is a function of the saturationand the temperature is given by Leverett’s formula.

(8) The gas and liquid permeabilities can be expressed in terms of the relative permeability.

Since the vapor density in the gas phase is much less than the liquid density, it is reasonableto assume that

S = φl

φ(1)

so the saturation S shows the volume ratio of the liquid in the pores.

3.2 Governing equations

The mathematical model including conservations of mass, momentum, and energy for the wetzone is first established, then constitutive equations, boundary conditions, and initial conditions aregiven, and lastly the mathematical model for the dry region is established.

3.2.1 Conservation of mass

The one-dimensional mass continuity equations [16] for each phase are

∂(ρlφS)∂τ

+ ∂∂x

(ρlφSul) = −δ (2)

∂[ρvφ(1 − S)]∂τ

+ ∂∂x

[ρvφ(1 − S)uv] = δ (3)

∂[ρaφ(1 − S)]∂τ

+ ∂∂x

[ρaφ(1 − S)ua] = 0 (4)

∂[ρgφ(1 − S)]∂τ

+ ∂∂x

[ρgφ(1 − S)ug] = δ (5)

where δ denotes the mass source for the vapor phase due to evaporation. Subscripts l, v, a, and grepresent liquid, vapor, air, and gaseous mixture of air and vapor, respectively.

The moisture conservation equation including the liquid and vapor is obtained by adding Eqs.(2) and (3):

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∂∂τ

[ρlφS + ρvφ(1 − S)] + ∂∂x

[ρlφSul + ρvφ(1 − S)uv] = 0 (6)

The overall conservation equation including the liquid and the gaseous mixture is given by addingEqs. (2) and (5):

∂∂τ

[ρlφS + ρgφ(1 − S)] + ∂∂x

[ρlφSul + ρgφ(1 − S)ug] = 0 (7)

3.2.2 Conservation of momentum

In conventional drying processes, the drying process generally takes a long time and the heatand mass transfer are rather slow. Therefore, the fluid velocities are very slow and the Reynoldsnumbers are in the range of Re < 5. Therefore, the momentum equations can be written explicitlyusing Darcy’s law [5, 14]:

ul = − klK

µlφS

∂pl

∂x − ρlg

(8)

ug = − kgK

µgφ(1 − S)

∂pg

∂x − ρgg

(9)

Diffusion plays an important role in the drying process in porous media. Huang [5], Nasrallahand Perre [17], and Turner and Ilic [18] using Fick’s law to describe diffusion transport, the vaporand air momentum equations can be expressed as

uv = ug − ρgDMaMv

ρvM2

∂∂x

pv

pg

(10)

ua = ug + ρgDMaMv

ρaM2

∂∂x

pv

pg

(11)

The velocity of each fluid can then be directly used in Eqs. (8), (9), (10), and (11).

3.2.3 Conservation of energy

During drying, thermal conduction occurs in the solid, liquid, and gas phases with convectionin the liquid and gas phases. The conservation of energy in terms of the enthalpies is [22, 29]

∂∂τ

[ρshs(1 − φ) + ρlhlφS + ρvhvφ(1 − S) + ρahaφ(1 − S)]

+ ∂∂x

[ρlφSulhl + ρvφ(1 − S)uvhv + ρaφ(1 − S)uaha] = ∂∂x

λp

∂T

∂x

(12)

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Substitution of Eqs. (2)–(4) into Eq. (12) gives the energy equation in terms of the temperature andthe latent heat is

[ρscs(1 − φ) + ρlclφS + ρvcvφ(1 − S) + ρacaφ(1 − S)] ∂T

∂τ + δhvap

+ [ρlφSul + ρvφ(1 − S)uv + ρaφ(1 − S)ua] ∂T∂x

= ∂∂x

λp

∂T∂x

(13)

where hvap = hv − hl is the latent heat of evaporation and λp is the effective thermal conductivity ofthe porous media. Equations (6), (7), and (13) are the governing equations describing the heat transportin the wet region.

3.2.4 Constitutive equations

The Clausius–Clapeyron equation for the vapor pressure in the wet region and evaporationfront is [30]

pv = pv1exp −

Mvhvap

R

1T

− 1T1

(14)

Leverett’s formula for the capillary force is [28]

pc = pg − pl = √φK

σ(T)J(S) (15)

σ(T) = σ0 − βT (16)

J(S) = 0.364{1 − exp[−40(1 − S)]}

+ 0.221(1 − S) + 0.005 / (S − Sir) (17)

Thermodynamic relations:

ρv = Mvpv

RT(18)

M = Ma − (Ma − Mv) pv

pg(19)

h = cT + const (20)

Relative permeabilities [17]:

kl =

S − Sir

1 − Sir

3

, S > Sir(21)

Kg = 1 − 1.1S, S < Scr (22)

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Effective diffusivity [31]:

D = 2.17 × 10−5(101,325 / pg)(T / 273.16)1.88 (23)

3.2.5 Boundary conditions

At the adiabatic and impermeable surface (x = 0), the mass fluxes and heat flux are zero:

Mass flux of air,

ρaφ(1 − S)ua = 0 (24)

Mass flux of moisture including the liquid and vapor,

ρlφSul + ρvφ(1 − S)uv = 0 (25)

Heat flux,

λp ∂T

∂x + ρlφSulhvap = 0 (26)

On the exposed surface (x = H), the mass fluxes, heat flux, and pressure are continuous:

Mass flux of moisture,

ρlφSul + ρvφ(1 − S)uv = kmMv

R

pv

T −

ϕpv,sat

Tatm

(27)

Heat flux,

λp ∂T

∂x + ρlφSulhvap = kT(T0 − T) (28)

Pressure,

pg = p0 (29)

3.2.6 Initial conditions

Initially, the pressure of the gaseous mixture and the temperature are constant in the porousmedia.

T(x, 0) = Tini (30)

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pg = p0 + (ρl + ρg)g(H − x) (31)

The moisture content distribution is assumed to result in a hydrostatic capillary pressuredistribution.

∂pc

∂x = − ρlg (32)

3.2.7 Governing equations in the dry region

The dry region appears in the upper part of the porous media as the drying front advances.The main characteristics of the drying region are that the saturation remains constant, there is no phasechange, and no liquid flow. Therefore,

S = Sir (33)

δ = 0 (34)

ul = 0 (35)

The vapor and gaseous mixture conservation equations are

∂∂τ

[ρvφ(1 − Sir)] + ∂∂x

[ρvφ(1 − Sir)uv] = 0 (36)

∂∂τ

[ρgφ(1 − Sir)] + ∂∂x

[ρgφ(1 − Sir)ug] = 0 (37)

The energy conservation equation is

[ρscs(1 − φ) + ρlclφSir + ρvcvφ(1 − Sir) + ρacaφ(1 − Sir)] ∂T

∂τ

+ [ρvφ(1 − Sir)uv + ρaφ(1 − Sir)ua] ∂T

∂x =

∂∂x

λp

∂T∂x

(38)

The governing equations in the dry region are then Eqs. (36), (37), and (38).

3.2.8 Evaporation front boundary

When the wet and dry regions coexist in the porous media, they are coupled by an evaporationfront boundary. The evaporation front, the saturation is assumed to be Sir, and the liquid and vaporphases are assumed to be in equilibrium, so Eqs. (33) and (14) hold at the evaporation front. The air

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is assumed to be continuous so the mass function is given by Eq. (4) with S equal to Sir. The evaporationfront temperature is determined by the energy balance

λp

∂T∂x

f

+

= λp

∂T

∂x

f

−

+ ρvφ(1 − Sir)hvap[uv(f +) − uv(f −)] (39)

where f denotes the evaporation front.

3.2.9 Other parameters

The bed permeability was defined by the Carman–Kozeny equation [20]:

K = d2φ3

180(1 − φ)2(40)

The convection heat transfer coefficient was predicted using an average Nusselt number correlationfor turbulent flow over a flat plate [32]:

kT = 0.037Va0.8

µa

ρa

−0.8

Pr1 / 3λaL−0.2 (41)

The mass transfer coefficient at the convective surface was given by [33]

km = kT

(ρc)a Le−2 / 3 (42)

Expressions for the effective thermal conductivity of the porous media can be developed fromtwo extreme geometric models for the packed bed by modeling the structure as a bundle of conduitseither in parallel or in series. Using the definition of the phase volume fraction, the effective thermalconductivity predicted by a parallel model is

λpparallel = λgφg + λlφl + λsφs (43)

The effective thermal conductivity expressed by a series model is

λpseries =

1φg

λg +

φl

λl +

φs

λs

(44)

For real porous media, the solid, liquid, and gas structure is a combination of the parallel and seriesmodels. The effective thermal conductivity was calculated by using a weighing factor Λ

λp = Λλpparallel + (1 − Λ)λpseries

(45)

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In this paper, make the assumption of Λ = 0.5.

Since the vapor and air densities are less than that of the liquid, the moisture content wasdefined as

w = φSρl + φ(1 − S)ρv

φSρl + φ(1 − S)ρg + (1 − φ)ρs ≈

φSρl

φSρl + (1 − φ)ρs

(46)

3.3 Numerical method

Governing Eqs. (6), (7), and (13) in the wet region, and boundary conditions Eqs. (24)–(29),and (39) can be rewritten as the functions of the variables such as moisture content, w, temperature,T, and gas pressure, pg, through using other relations mentioned above. Governing Eqs. (36)–(38) inthe dry region can be rewritten as the functions of vapor pressure, pv, temperature, T, and gas pressure,pg. The numerical solution technique to solve the system of equations was a fully implicit finitedifference scheme. The convection terms in the energy equation were modeled using the first-orderupwind scheme with central differencing for other terms. A grid with 43 meshes in the x directionand time step 0.01 s, which can ensure the solution accuracy is independent of mesh size and timestep.

For a one-dimensional model in the wet region, the calculation domain contains J grid nodes.The discrete equations has the form:

Anψn = bnψn−1 + cn (47)

where the two-dimensional coefficient matrices, An, is 3J × 3J, bn and cn are arrays with dimensionof 3J. The variables are moisture content, w, temperature, T, and gas pressure, pg, written in vectorform as yn = (w0

n, T0n, pg0

n , . . . , wJ−1n , TJ−1

n , pgJ−1

n )T, where n is the number of time step. All coefficientmatrices, An, bn, and cn are the function of yn. Therefore, Eq. (47) is nonlinear. In order to solve thenonlinear problem, a predictor-corrector method is implemented. In the first step, the nonlinearcoefficients As–1, bs–1, and cs–1 at previous time step n – 1 is evaluated using the following equation:

As−1 ⋅ ψs = Bs−1 ⋅ ψn−1 + Cs−1 (48)

Let ψn−1 = ψs−1 as the initial value to compute the coefficient matrices, Eq. (48) become linear.The variable vector could be solved by Gauss elimination. Let s be the iteration step in the n – 1 timestep. Let ψs as new value compute the coefficient matrices again, ψs+1 could be obtained. Repeat theloop in an n – 1 time step, we can get the predictor value, ψn

∗ of the variable ψn as accuracy reaches

convergence criterion. The convergence criterion for each state variable in each node is

minψ j

n − ψ jn

∗

ψ jn

≤ 1.0 × 10−6 (49)

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In the second step, using the variable predictor value, ψn∗, to calculate the coefficient matrices

of Eq. (47), it becomes linear. Therefore, the corrector value, ψn, is solved. Following the loopmentioned above, variables of n + 1 time step, ψn+1, could be solved using a similar scheme:

As−1 ⋅ ψs = Bs−1 ⋅ ψn−1 + Cs−1 (50)

The flow diagram of numerical simulation is shown in Fig. 2.

4. Experiment

Figure 3 shows a schematic of the experimental system used to measure the weight and thetemperature in the porous packed bed directly during the steady drying rate tests. The bed was packedwith quartz particles with a diameter of 1 mm to 1.5 mm, having a bed porosity of about 46%. Quartz

Fig. 2. Flow diagram of numerical simulation.

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particles were selected as the solid material due to their inert chemistry and their thermal stability, sono gaps would form within the particles after high temperature baking. The packed bed was a squareof 45 mm side and 15 mm in depth, and its open surface was flush with the lower surface of a windtunnel with the cross section of 200 mm × 100 mm with insulation around the surfaces. Air blowninto the tunnel from upstream was heated by a 1.5 kW to 5 kW heater. A mixer at the upstream entranceensured temperature and velocity uniformity. The hot air velocity and temperature were measuredwith a hot wire anemometer. The relative humidity of the hot air was measured with a digital humidityrecorder. The packed bed was placed on an electronic balance every 40 min to measure the masschange during drying. The temperature field in the porous media was measured every 10 min bythermocouples located at different heights with the wires placed horizontally to avoid the influenceof heat transfer in the wires. The accuracy of the hot wire anemometer was 0.1 m⋅s–1 and 0.1 °C, theaccuracy of the humidity recorder was 0.1% and the accuracy of thermocouple was 0.1 °C.

The water was injected into the bed with a small injector from the bottom of the bed until thetop surface was completely flooded. The initial saturation distribution was determined by hydrostaticequilibrium equation (32) using the known saturation at the top surface, which was assumed as criticalsaturation calculated by Eq. (52). The irreducible saturation is the ratio of mass difference betweeninitial mass and final mass measured by mass weighting method to mass of water saturated in allporosity:

Sir = Wini − Wfnl

φρlV(51)

where Wini, Wfnl, and V are initial mass, final mass, and volume of porous media respectively. Thecritical saturation is given by

Scr = Wini − (1 − φ)ρsV

φρlV(52)

An initially nonuniform temperature distribution would occur due to evaporation prior to thebeginning of the convective drying. The heat for the evaporation both inside the porous media and at

Fig. 3. The experimental system.

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the top surface would be supplied mostly from the bed, resulting in a reduced bed temperature, mostlynear the top surface. However, nonuniformity of the initial temperature distribution was very smallso that the temperature distribution was assumed to be uniform. The initial temperature throughoutthe porous media was taken as 289.8 K. The temperature, relative humidity, and velocity of the hotair were 321 ± 1 K, 33 ± 2%, and 1.89 ± 0.1 m/s during the experiments.

5. Results and Discussion

The physical parameters used in calculation are summarized in Table 1. The numerical andexperimental results for the moisture content and temperature are presented in Figs. 4–6. The moisturecontent in the porous bed decreased from the initial value of 22.6% to about 1% after 360 min dryingin Fig. 4. The decrease in moisture content w for both the numerical and experimental results with anerror 0.05 was fast and linear before 300 min, then slowed dramatically. Note that the difference

Table 1. Values of Physical Parameters Used in Calculation

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Fig. 4. Variation of moisture content with drying time.

Fig. 5. Experimentally measured variation of temperature with drying time.

Fig. 6. Hot air stream and bed temperature variations with drying time.

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between experimental results and numerical results might result from the difference of dryingconditions such as temperature and relative humidity of hot air in experiment and simulation. Thedrying experiment was done from 8 AM to 2 PM when the temperature and the relative humidity ofambient increased and decreased respectively. Temperature of hot air was manually controlled byadjusting the power of heater and the response of temperature of hot air delayed. However, there wasno measure to stabilize the relative humidity of hot air. In general, temperature of hot air in theexperiment is higher than that in the numerical simulation. The relative humidity of hot air in theexperiment is lower than that in the numerical simulation. Both higher temperature and lower relativehumidity of hot air in the experiment caused higher drying rates in the experiment than those in thenumerical simulation. The drying process could be separated into two intervals: the constant dryingrate and the reduced drying rate. The total drying time in the numerical simulations was about 30 minlonger than those in the experiment, but the length of the constant drying rate period in both thenumerical simulations and the experiments was very nearly 300 min.

The experimentally measured bed temperature at heights of 0, 6, 8, 10, and 13 mm are shownin Fig. 5. Temperatures initially rose in the first 20 minutes, then remained constant at 303 ± 2 Kwhich is the ambient air wet bulb temperature from 40 to 280 min, and finally gradually rose as thedrying ended.

Figure 6 compares the numerical and experimental temperature distributions at 8 mm. Initially,the drying rate is slow and the temperature increases due to heat absorption from the hot air withoutmuch evaporation. Higher air temperatures resulted in higher drying rates. The temperatures wereessentially constant at the wet bulb temperature from 40 to 280 min since the heat transfer fromambient drying air was balanced by the latent heat absorbed by the evaporation, resulting in a constantdrying rate from 40 to 280 min. As the drying rate slowed, the heat transfer from the drying air resultedin the temperature increases since most of liquid water had evaporated. Some differences between thenumerical and experimental results occurred because the numerical simulations assumed steady-statedrying conditions while the actual drying conditions in the experiments were unstable since thetemperature, velocity, and relative humidity of the hot air were not effectively steady. The relativehumidity increased within the range of 2% during the experiment from 9:00 AM to 16:00 PM, whichresults in the increase of wet bulb temperature. The variations of hot air temperature during the dryingprocess shown are shown in Fig. 6. In addition, the bottom boundary was not a fully adiabatic boundarywhich would result in some heat loss.

Further numerical results that could not be measured experimentally are presented in Figs.7–13. The variation of the saturation distribution in the packed bed with the drying time is shown inFig. 7. The saturation uniformly decreases during the first 320 min with ∂S / ∂x < 0 showing that theliquid migrates toward the convective drying boundary by the drive of capillary force and liquid bulkpressure. The evaporation front does not advance into the porous media until the saturation atconvective drying boundary reaches the irreducible saturation, which means that the liquid easilymigrates to the convective boundary. The saturation profiles in Fig. 8 with ∂S / ∂x < 0 shows that theliquid diffuses towards the evaporation front when the wet and dry region coexist together as thedrying front moves towards the bottom. As soon as the dry region appears marked by the saturationS = Sir on the convection drying boundary at 340 min, temperatures begin to rise (Fig. 6).

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Fig. 7. Numerically calculated saturation distributions.

Fig. 8. Saturation distributions near the end of the drying process.

Fig. 9. Numerically predicted temperature distributions.

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Fig. 10. Variations of gas pressure distributions.

Fig. 11. Vapor pressure variations.

Fig. 12. Position of evaporation front.

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The variation of the temperature distribution with drying time shown in Fig. 9 reveals thattemperature change during the drying had three stages: the initial heating stage, the constanttemperature stage, and the final heating stage as also shown in Figs. 5 and 6. Temperature keepingconstant in the second constant temperature stage means thermal balance between heating from hotconvection drying air and cooling from water evaporation. The temperature became uniform atapproximately the hot air temperature throughout the porous media as the drying ended.

The gaseous pressure distribution is shown in Fig. 10. There was a small decrease in the gaspressure initially since in the initial heating stage, the packed bed pressure is initially reduced asmoisture is removed from the pores at a greater rate than it can be replaced by gas rapidly since thegas has a lower relative permeability in a highly saturated media. The lower pressure will initiallyretard moisture migration. Later, the packed pressure increases due to the thermal expansion of thevapor and air.

The variation of the vapor pressure distribution with drying time is shown in Fig. 11. Thevapor pressure depends on the porous media temperature during the initial heating and the constantdrying rate stage when the entire porous media is wet. So the vapor pressure is equal to the saturationpressure for the wet bulb temperature. After the porous media is dry, ∂pv / ∂x < 0, which means thatthe moisture motion is due to diffusive transport in dry region. The vapor pressure in the reduceddrying rate stage is less than that in the constant drying rate stage while the temperature is higher,which indicates that the drying rate is dropping. The vapor pressure distribution is related to the wetregion size during the reduced drying rate stage until the porous media is completely dry.

Figure 12 shows the position of the evaporation front during the drying process. Theevaporation front does not begin to advance until the saturation at the convective drying boundaryreaches the irreducible saturation. Figure 13 shows that the evaporation at τ = 100 min mainly takesplace near the convective drying boundary indicating that liquid transport in the interior is moreimportant than vapor transport in order to supply water to vaporize.

6. Conclusions

A one-dimensional model of multi-phase flow in a packed bed was developed using macro-scopic mass and energy conservation equations for convective drying. The model assumes Darcy flow

Fig. 13. Phase change rates at 100 min.

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for the momentum equations. The wet and dry regions were coupled by the evaporation front boundarycondition. The model includes convective and capillary transport of the liquid, diffusive transport ofvapor and air. The numerical results were compared with experimental results for convective dryingof a porous bed packed with quartz particles. The weight and temperature variations of the packedbed were measured during the drying. The heat and mass transfer processes were then analyzed usingthe numerical and experimental results.

The numerical results using one-dimensional drying model showing good agreement in anacceptable error range with the experimental results in terms of the moisture content, the temperature,and the total drying time proves the drying model valid. The change of state parameter such assaturation, temperature, or pressure could be predicted during drying by solving the drying model. Inthe wet region, the moisture transport is depended on the liquid movement derived by the capillaryforce and bulk pressure with ∂S / ∂x < 0. In the dry region, the moisture is mainly dominated by vapordiffusion with ∂pv / ∂x < 0 while vapor pressure in the wet region is uniform because temperature keepsa wet bulb temperature. As soon as the evaporation front begins to move from convective dryingboundary to inside, the drying rate also starts to fall from the constant drying rate, the temperaturealso commences to rise from constant temperature.

According to the results of both experimental and numerical simulation, the drying processin the packed bed can be divided into two periods based on the moisture content changes: the constantdrying rate and the reduced drying rate periods. In the constant drying rate period, temperaturesthroughout the whole porous media keep uniform with the wet bulb temperature since heat transferfrom the convective drying boundary was balanced by the latent heat absorbed by the evaporation,resulting in a constant drying rate. In the reduced drying rate period, temperatures increase since thedry region appears in which there is no free water to evaporate, vapor in the wet region transportingto ambient has to overcome the resistance to diffusion in the dry region resulting in the drying ratereducing. According to the analysis mentioned above, the temperature undergoes three stages: theinitial heating stage, the constant temperature stage, and the final heating stage. The changes oftemperature during drying in the porous media is close to heat and mass transfer in the porous mediaduring convective drying.

Acknowledgments

This work was partially supported by Key Course of Chemical Process Machinery fromBeijing Municipal Education Commission (No. XK100100541).

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Originally published in J Dalian University of Technol 45, 2005, 542–546.Translated by Tao Lu, School of Mechanical and Electrical Engineering, Beijing University of

Chemical Technology, Beijing 100029, China.

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