Research Note pubs.acs.org/IECR
Short Communication: Comments on the Role of Gas-Phase Axial Thermal Dispersion and Solid-Phase Thermal Conduction for Heat Transfer in a Packed Bed of Solid Particles Michael G. Beaver and Shivaji Sircar* Department of Chemical Engineering, Lehigh University, Bethlehem, Pennsylvania 18015-4791, United States ABSTRACT: The effects of gas-phase axial thermal dispersion and solid-phase thermal conduction on gas−solid heat transfer in a packed bed of solid particles was numerically studied using a simplified conduction−dispersion model for heat transfer in conjunction with a realistic set of boundary conditions. It is reconfirmed that the exclusion of gas-phase thermal axial dispersion in the model analysis of gas−solid heat transfer coefficients can result in misleading interpretation of heat transfer in packed beds, particularly in the low Reynolds number gas flow region.
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BACKGROUND Gas−solid heat transfer coefficient in a packed bed of solid particles is an important variable for design of nonisothermal adiabatic processes such as gas adsorption, catalytic reaction, etc., as well as for heating and cooling of packed beds by direct flow of a hot or cold gas through the bed which is often used in such processes. It has been reported by many authors that the gas-phase axial thermal dispersion and solid-phase thermal conduction should be included in models for correct depiction of the gas−solid heat transfer in packed beds.1−4 Failure to account for gasphase thermal dispersion can lead to moderate errors in estimation of the gas−solid heat transfer coefficient at high particle Reynolds number (Re) values and very large errors (orders of magnitude) at low Re values.1 Model analysis of experimental heat transfer data in packed beds reported by many workers for measurement of gas−solid heat transfer coefficients without accounting for gas thermal axial dispersion verify this apparent anomalous behavior.5 Different interpretations of this behavior based on gas channeling in packed bed, irregular distribution of voids, particle agglomeration, unusual limiting behavior of heat transfer at the limit of zero flow rate, lack of central thermal symmetry in particles at very low Re, etc., have also been proposed.5−7 A review article by Lemcoff et al (1990) covers different experimental protocols and analysis of heat transfer in packed beds.8
Figure 1. Schematic drawing of conduction−dispersion model.
using a conduction−dispersion model. The model separately accounts for gas-phase and solid-phase heat balances inside the packed column. The gas-phase [Tg(z,t)] and solid-phase [TP(z,t)] temperatures at a distance z (m) from the inlet end of the packed bed at time t (s) are intrinsic average transient temperatures (K) of the two phases (no radial gradient) at z and t. Thus this model explicitly allows the investigation of the effect of a finite gas−solid-phase heat transfer coefficient in the packed column, which is the primary goal of this work. A model
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which defines the column temperature at z and t by a single
DISCUSSION AND RESULTS Conduction−Dispersion Model. We re-examined the influences of gas-phase thermal axial dispersion and solidphase thermal conduction on an estimation of the gas−solid heat transfer coefficient by a simplified “conduction−dispersion (CD) model” for step-heating of a packed bed of particles using a hot gas. Figure 1 is a schematic drawing of the model. The packed column is initially at temperature To. An inert gas is passed through it at To. The inlet gas temperature is changed to T* as a step function at time t = 0, and the effluent gas temperature [Tg(L,t)] is monitored as a function of time. The objective is to estimate the effective gas−solid heat transfer coefficient from the effluent gas temperature profile by © 2012 American Chemical Society
temperature [T(z,t) = Tg(z,t) = TP(z,t)] model preassumes that instantaneous thermal equilibrium exists between the gas and the solid phases (infinite heat transfer coefficient), which defeats the purpose of this work. Equations 1 and 2 describe the transient heat balances in gas and solid phases of the packed bed, respectively: Received: Revised: Accepted: Published: 5826
June 17, 2011 March 21, 2012 April 2, 2012 April 2, 2012 dx.doi.org/10.1021/ie2012888 | Ind. Eng. Chem. Res. 2012, 51, 5826−5829
Industrial & Engineering Chemistry Research
Research Note
Gas-phase heat balance:
dθg
⎛ δ2θ ⎞ ⎛ δθg ⎞ a(Pe) ⎛ δθg ⎞ g ⎜ ⎟ = a 2⎜⎜ 2 ⎟⎟ − ⎜ ⎟ ε ⎝ δx ⎠ ⎝ δτ ⎠x ⎝ δx ⎠τ τ
dx
⎡ α ⎤⎛ δ2θ ⎞ ⎡ k g ⎤⎡ α p ⎤ ⎛ δθg ⎞ p p ⎜ ⎟ = ca 2⎢ ⎥⎜⎜ 2 ⎟⎟ + 6(1 − c)⎢ ⎥⎢ ⎥(Nu) ⎝ δτ ⎠x ⎣⎢ k p ⎥⎦⎢⎣ αg ⎥⎦ ⎣⎢ αg ⎥⎦⎝ δx ⎠ τ
θp = [Tp(z , t ) − T o]
(2)
To (K) is the initial column temperature [time t ≤ 0, seconds] for all distance z within the packed column of length L; kg and kp are, respectively, the effective thermal conductivities (W m−1 K−1) of the gas and the solid particles; ρg and ρp are, respectively, the gas and the particle density (kg m−3) at z and t. Cg and Cp (J kg−1 K−1) are, respectively, the specific heat capacities of the gas and the particle; αg (= kg/(ρgCg)) and αp (= kp/(ρpCp)) are, respectively, the thermal diffusivities (m2 s−1) of the gas and the particle phases; dp is the particle diameter (m). The variable x is dimensionless distance (= z/L) inside the column. The dimensionless time is τ (= αgt/dp2). The variable a = (dp/L), ε is the external void fraction of the packed bed, and c is the fraction of particle external surface area for heat conduction in the particle phase due to particle to particle contact. (1 − c) is the fraction of external surface area of the particles in contact with the gas phase for gas−solid heat transfer. The variable he is the effective gas−solid heat transfer coefficient (J m−2 s−1 K−1) and Nu (= hedp/kg) is the corresponding Nusselt number. Pe (= QdpCg/kg) is the Peclet number, where Q (kg m−2 s−1 ) is the mass flow rate of the gas per unit cross-sectional area (A, m2) of the packed column of diameter D (m). Pe is equal to [(Re) × (Pr)], where Re (= Qdp/ μg) is the particle Reynolds number and Pr (= Cgμg/kg) is the Prandl number for the gas. The molar viscosity of the gas is μg (kg m−1 s−1 ). Initial and Boundary Conditions. The initial and boundary conditions for the governing equations are given as follows: Initial conditions at τ = 0 for both gas and solid phases: θg(x , 0) = θp(x , 0) = 0
dθg dτ
(7)
x=1
a(Pe) dθg ε dx at x = 1
− x=1
6(1 − ε)(1 − c)) N ε (8)
Simulation Protocol. Equations 1 and 2 were simultaneously solved numerically with initial and boundary conditions given by eqs 3−5 and using either eq 6 or 8 as an additional boundary condition. A MATLAB ordinary differential equation solver (ode15s), in conjunction with a numerical method of lines scheme were used to solve the spatial derivatives.12 The number of stages for calculating the spatial derivatives was generally large (∼400) and no improvements in the calculated results could be observed by further increasing the number of grid points. A typical CPU time for a solution was ∼21 s. The use of eq 6 as a boundary condition often resulted in numerical instability and oscillating solutions. In contrast, the use of eq 8 as a boundary condition did not create any such instability in the solution. Consequently, we carried out the following numerical simulations of the effects of including or excluding gas- and solid-phase axial thermal dispersions in a conduction− dispersion model used for estimation of the effective gas−solid heat transfer coefficient by analysis of the transient gas-phase temperature profiles at the column exit end or at an inside location within the packed bed being heated by a hot gas. The simulation protocol consisted of (a) generating the gasphase temperature−time profiles at column exit end (x = 1) and various other locations within the column (x = 0.5, 0.7) using different values of dp (0.02, 0.2 cm), Re (∼0.01 to 10) and gas temperature step sizes [(T* − T0) = 1, 50 K] by the full conduction−dispersion model with the assumption that the classic correlation by Ranz13 can be used to estimate the effective gas−solid heat transfer coefficient for a given value of Re [Nu = 2 + Pr0.33{Re/ε}0.5] followed by (b) curve fitting the temperature profiles with the conduction−dispersion model without the axial thermal dispersion in the gas phase and/or thermal conduction in the solid phase in order to estimate the “apparent effective heat transfer coefficients”.
(3)
for all τ (4)
Boundary condition for the gas phase at the inlet end: θg(0, τ) = T * − To
=−
u[θg − θs]
Boundary conditions for the solid phase at both ends of the packed bed which realistically implies that no heat enters or leaves the packed bed through the solid-phase end: ⎛ δθp ⎞ =0 ⎜ ⎟ ⎝ δx ⎠x = 0;1
at x = 1
Equation 6 is a frequently used boundary condition to account for gas axial dispersion of heat in packed columns.1,4 This classic boundary condition is attributed to Danckwerts et al.9 However, it implies that the gas-phase temperature gradient may be finite when the gas flow approaches the outlet end of the bed, but then it suddenly becomes equal to zero at the column exit. That appears to be unrealistic. The dubious nature of this boundary condition and the numerical problems associated with its use are discussed by Scheisser.10 Equation 7 is a more realistic boundary condition that has been proposed by several researchers.10,11 It implies that the gas-phase temperature gradient remains unchanged (frozen) at the outlet end as the gas flows out from the bed. It also implies that the rate at which heat enters the gas phase at the end of the packed bed is always equal to that at which it leaves the column with the out-flowing gas. Consequently it follows from eqs 1 and 7 that
(1)
Particle (solid)-phase heat balance:
[θg − θp];
(6)
⎛ δ2θ ⎞ ⎜ g⎟ = 0 ⎜ δx 2 ⎟ ⎝ ⎠τ
6(1 − ε)(1 − c) Nu[θg − θp]; − ε θg = [Tg(z , t ) − T o]
=0 x=1
(5)
Boundary conditions for the gas phase at the exit end: 5827
dx.doi.org/10.1021/ie2012888 | Ind. Eng. Chem. Res. 2012, 51, 5826−5829
Industrial & Engineering Chemistry Research
Research Note
The heating of a packed bed of porous alumina particles by nitrogen was used as an example for the simulation. The physical properties of the gas and solids are summarized in Table 1.
Table 2. Legends for Figure 2 Profiles
Table 1. Physical Properties Used in Numerical Solutions property density (kg m−3) heat capacity (KJ kg−1 K−1) thermal conductivity (W m−1 K−1 ) viscocity (kg m−1 s−1) thermal diffusivity (m2 s−1) void fraction length (m) c
gas phase (nitrogen at 1 atm., 300 K)
solid phase (alumina, particle diameter dp = 2 × 10‑3 m)
ρg = 1.138 Cg = 1.046
ρp = 670 Cp = 0.878
kg = 0.258
kp = 0.167
column
μg = 1.786 × 10−5 αg = 2.175 × 10−5
αp= 2.84 × 10−7 ε = 0.4 L = 0.2 c= 0.03
Simulation Results. Figure 2 shows several selected examples of temperature−time profiles at various column
Figure 3. Plots of Nu vs Re.
coefficient) can occur if axial gas-dispersion is not included in the gas-phase heat balance eq 1 of the conduction−dispersion model, even though the column temperature−time profiles during the heating process (Figure 2) can be described very well by the model. This is true of any particle size and temperature step size. The decrease in the apparent heat transfer coefficient is relatively small when Re > 10, but orders of magnitude decrease may be seen at low Re values. This study reconfirms that the exclusion of gas thermal dispersion in data analysis is the primary cause of observing very low Nu at low Re in many earlier studies.5 It may also be seen from Figure 3 that the estimated value of Nu does not change much if the solid-phase thermal conduction is ignored. The reason is that the particle to particle contact area for heat conduction is rather small (c ≈ 0.03). Figure 3 also shows an empirically correlated Nu versus Re plot which qualitatively describes scattered experimental data by various authors collected by Kunii and Suzuki.5 According to this reference plot, the value of Nu for any Re is much lower than those obtained by the present simulation. Figure 4 compares the temperature−time profiles at x = 0.5 [dp = 0.2 cm, (T* − T0) = 1 K, Re = 0.138 and 1.38] simulated by using heat transfer coefficients (he) given by the Ranz model with axial dispersion and the corresponding apparent value of he shown by Figure 3, which was obtained by excluding axial dispersion from the simulation model. The figure shows that the sizes of the heat transfer zones (HTZ) are close when Re is 1.38 because the he values are similar. On the other hand, the size of the HTZ is substantially longer for the low Re case where he is lowered by a factor of ∼435 due to exclusion of gas thermal axial dispersion in simulation.
Figure 2. Midpoint and 0.7 L/Lc temperature−time profiles.
distances generated using the full conduction−dispersion model and the heat transfer coefficient by Ranz model for different particle sizes and Re values. The figure also shows the corresponding fit of the profiles using the model without the axial conduction and dispersion terms. The ordinates in Figure 2 plot are dimensionless gas temperatures [θ̅ = (Tg − T 0)/(T * − T 0)]
Table 2 provides the legends for the simulation plots. It may be seen that the conduction−dispersion model can describe the profiles generated using the Ranz model of heat transfer extremely well even when the axial dispersion and conduction are ignored albeit with a very different effective heat transfer coefficient than that of the Ranz correlation. Figure 3 compares these values of heat transfer coefficients in the form of Nu versus Re plots. It may be seen that a drastic lowering of Nu values (hence, apparent heat transfer 5828
dx.doi.org/10.1021/ie2012888 | Ind. Eng. Chem. Res. 2012, 51, 5826−5829
Industrial & Engineering Chemistry Research
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Research Note
ACKNOWLEDGMENTS The authors thank Professor William E. Schiesser of Lehigh University for valuable discussions on boundary conditions and “method of lines” protocol.
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Figure 4. Comparative temperature−time profile at x = 0.5.
Two key implications of these results are (a) gas-phase axial thermal dispersion must be included in modeling gas−solid heat transfer in a packed bed particularly for the low Re flow region and (b) the gas-phase boundary condition described by eq 7 should be used in the solution of heat balance equations. It should be noted that Wakao et al. proposed an empirical correlation for the calculation of the gas−solid heat transfer coefficient [Nu = 2 + 1.1Pr0.33 × Re0.6] which was derived by the analysis of many published steady-state heat transfer data using a model that included axial thermal dispersion in conjunction with the Danckwarts boundary condition [eq 6].2
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SUMMARY The thermal profiles inside a packed bed of solid particles during stepwise, adiabatic heating using a hot gas was numerically simulated by a conduction−dispersion model of heat transfer. The model included gas- and solid-phase axial thermal dispersions. A transient boundary condition for the gasphase heat balance equation which assumes that the gas-phase temperature gradient remains unchanged at the exit end of that phase was found to give stable and physically consistent simulation. This was contrary to the conventional Danckwerts boundary condition which assumes that the gas-phase temperature gradient suddenly vanishes at the gas exit end. It was reconfirmed that the exclusion of gas-phase axial thermal dispersion in the model for estimation of gas−solid heat transfer coefficient from such thermal profiles can lead to a severely lower value of the gas−solid heat transfer coefficient as previously reported. The error was moderate at high values of Reynolds number (>10), but the error could be very large (orders of magnitude) when the Reynolds number is low (
