4092
Ind. Eng. Chem. Res. 2005, 44, 4092-4098
Carbonaceous Material Deposition from Heavy Hydrocarbon Vapors. 2. Mathematical Modeling Wenxing Zhang and A. Paul Watkinson* Department of Chemical and Biological Engineering, University of British Columbia, 2216 Main Mall, Vancouver, Canada V6T 1Z4
A two-dimensional mathematical model was developed to predict the deposit formation from hydrocarbon vapor products in a straight tube with either constant and uniform wall heat flux or constant and uniform outside wall temperature, assuming physical condensation as the mechanism. A single condensable pseudo-component is modeled to transport, condense, and form the deposit layer on the wall. Two cases corresponding to different physical situations are simulated. In the first case, the model was used to simulate deposition results in a long tube downstream of a bitumen coking reactor. Experimental results of the effects of vapor temperature, addition of secondary steam, and vapor residence time or vapor velocity on deposition compared favorably with model calculations. The second case modeled a single transfer line exchanger tube, which is characterized by highly turbulent flow. The temperature profiles of quenched vapor and tube skin, vapor pressure drop, and the thickness of the deposit layer are predicted. 1. Introduction An experimental study of the deposition from hydrocarbon vapors in a tubular test section downstream of a bitumen coking reactor1 indicated that physical condensation is the dominant deposit formation mechanism in the bench-scale system. To obtain further insight of the fouling formation process, both on the bench scale and in larger equipment, a mathematical model is desirable to predict the deposition rate. In addition, the model can also be used for calculations in industrial transfer line exchangers (TLEs) downstream of heavy hydrocarbon cracking reactors, where deposit formation from the vapor phase seriously plagues the operation. Among the fouling mechanisms in TLEs, physical condensation is considered to be a primary contribution when cracking heavier hydrocarbons,2-4 such as gas oils. The fouling process is hypothesized to occur when high boiling hydrocarbons condense on the cooled tube wall and subsequently continue to react in the boundary layer and transform to a high-molecular-number carbonlike substance. In a detailed study by Kopinke et al.,4 the authors used 14C-labeled hydrocarbons and concluded that the condensation of heavy components of the fuel oil fraction and precipitation of tar droplets is the dominant mechanism in TLE fouling. In several published papers,5-7 the TLE fouling process is simulated, permitting calculation of the tube fouling profiles, variation of gas outlet temperature and pressure drop, etc. However, these approaches either do not take into account the contribution of physical condensation or do not treat the known physics of the condensation step. Horak and Beranek5 discussed the possible mechanisms of coke formation in coolers for the products of pyrolysis. They assumed that fouling causes are probably due to both chemical and mechanical (physical condensation) processes and simulated the deposit formation rate using kinetic expressions for * To whom correspondence should be addressed: Tel: 1-604822-2741. Fax: 1-604-822-6003. E-mail:
[email protected].
various mechanisms. More specifically, the physical condensation rate was modeled by assuming that the decisive step was the sticking of the fouling material to the surface. This sticking process is investigated as a coke layer growth rate with negative activation energy, such that higher surface temperatures yielded lower coke deposition rates. In their model, the transport effect of coke precursor was not taken into account and the handling of the physical condensation step is questionable. Huntrods et al.6 presented a one-dimensional model to simulate coke formation in gas quenchers of industrial ethane cracking furnaces. Their model incorporated existing molecular kinetic data for predicting coke formation rates and a semiempirical coke deposition model for the thermal and hydrodynamic behavior of the TLE. In comparison with the paper by Horak and Beranek,5 this model included the transport effects on the coke deposition; however, the authors did not account for the physical condensation contribution, although in their case (cracking light gaseous hydrocarbons), this type of contribution could be minor. Recently, Manafzadeh et al.7 proposed a modification of the approach of Huntrods et al.,6 to incorporate the coke deposition by physical condensation in TLE operation. The authors postulated that, in the first half of TLE, the deposit formation rate was determined by a chemical reaction mechanism, and in the downstream half of the tubes, the deposition was mainly formed because of physical condensation of polycyclic hydrocarbons, as a result of lower temperature (T < 500 °C). Based on the paper by Kopinke et al.,4 they ascribed the main condensation components to be acenaphene, anthracene, and chrysene and assumed the condensation rate to be equal to their formation rate in the gas stream. However, it is unclear how the authors determined the condensation temperature of individual components. The literature review revealed that, although some efforts have been made, a proper deposit formation model by physical condensation is still lacking. In this
10.1021/ie0490334 CCC: $30.25 © 2005 American Chemical Society Published on Web 05/03/2005
Ind. Eng. Chem. Res., Vol. 44, No. 11, 2005 4093
2.1. Case of Constant and Uniform Wall Heat Flux. 2.1.1. Temperature Profiles. In cylindrical coordinates, the simplified energy equation in the system is written as
FC ˆ pvz
∂T 1 ∂ )(rq ) ∂z r ∂r r
(1)
Integrating eq 1 across the radius, the axial temperature gradient can be obtained:
-2q0 dT ) dz RFC ˆ vj
(2)
p z
The radial temperature profile is readily obtained from ref 8. For laminar flow,
T(z ) 0) - Tin 7 r 2 1 r 4 + + )q0R/k R 4R 24
()
Figure 1. Physical situation analyzed.
paper, a two-dimensional, steady-state model is developed to describe the physical condensation process quantitatively and predict the deposition rate. A single condensable pseudo-component is assumed to transport, condense, and form the deposit layer on the wall. In contrast to the usual situations for the design of heat exchange condensers, where the condensable component is known and is a major constituent in the stream to be cooled, in fouling cases, the precursors are present at very low concentrations and more often than not the species are unknown. The present model can simulate the temperature profile and condensation rate in a straight tube with either constant and uniform wall heat flux or constant and uniform outside wall temperature. Two different physical situations are considered, namely, the bench-scale study of deposition from bitumen coking vapors by the present authors (in Part 1)1 and a typical industrial TLE tube with conditions taken from the literature.
(3)
where Tin is the gas stream inlet temperature and k is the gas mixture thermal conductivity. For turbulent flow, the dimensionless temperature profiles at the turbulent core and near the wall are expressed separately. In the turbulent core,
T+ - T1+ )
( )
1 s+ ln + κ1 s 1
(for s+ g 26)
(4)
s1+ ) 26 is taken as the boundary between the turbulent core and the near-wall region, and κ1 ) 0.36. Near the wall,
T+ )
∫0s
+
ds+ (1/Pr) + n v s [1 - exp(-n2v+s+)] (for s+ < 26) (5) 2 + +
where
2. Model Development The physical situation considered is depicted in Figure 1. A multiple component gas mixture with one condensable species (denoted as B) flows in a smooth circular tube with either constant and uniform wall heat flux (q0) or constant and uniform outside wall temperature (Two). The condensable species B coming into contact with the tube wall will condense and form as a layer of deposit on the wall, if the tube wall temperature is lower than its dew point at a given partial pressure. To obtain a mathematically manageable model, several assumptions are made, including (i) no entrance effects of the momentum, heat, and mass transfer; (ii) an ideal gas mixture with a low concentration of condensable species; and (iii) no bulk condensation. All condensed material is incorporated into the deposit, and no reentrainment of the deposit into the bulk stream is considered. Thus, the deposit is considered to be a static layer of pure B. The model does not account for aging reactions. Many of these restrictive assumptions could be relaxed, to give a less simple but more realistic model.
()
v/ )
x
s+ )
sv/F µ
τ0 F
v+ ) T+ )
vz v/
FC ˆ pv/(T - Twi) q0 n ) 0.124
2.1.2. Physical Condensation Rate-Mass Transfer. The simplified overall mass balance equation for the condensable species B is expressed as
1 ∂ ∂ (rNBr) + NBz ) 0 r ∂r ∂z
(6)
where NBr and NBz are the radial and axial molar fluxes of species B, respectively. Through integration of eq 6, the axial decrease of the fouling precursor due to radial physical condensation
4094
Ind. Eng. Chem. Res., Vol. 44, No. 11, 2005
can be obtained:
dNBz vj z dcB -2NBr ) ) dz dz R
transfer coefficient between the gas stream and the tube is expressed as
(7)
The radial mass-transfer rate of B from bulk to the wall is expressed as
(
NBr ) km
/ PB,wi
PB,bulk RTbulk RTwi
)
(8)
where the mass-transfer coefficient for established laminar flow is
0.023Re0.8Sc0.33DBm dt
ki ) 24.4 + 0.0041(Two - 977.6)
For a pure condensable species, P/B,wi can be calculated through eq 9:
B + C log10 Twi + DTwi + ET2wi Twi (9)
where the coefficients A, B, C, D, and E can be obtained from the chemical database for a specified component. In the case of unknown condensable species, an alternative method, as shown in eqs 10 and 11, could be applied to determine the relationship between P/B,wi and Twi, provided the boiling point of the pseudocondensable species TB is specified. The relationship between the vaporization enthalpy ∆HvB and TB is written as9
∆HvB ) KF(8.75 + R ln TB) TB
(10)
where KF is a coefficient with a value range of 0.991.01, which is determined by the carbon number and molecular structure. In the present case, a value of unity is taken for KF and R ) 1.92 cal mol-1 K-1. The Clausius-Clapeyron equation is then applied to determine the saturation pressure of B (P/B,wi) at the particular inner wall temperature:
ln
( )
/ ∆HvB PB,wi )(1/Twi - 1/TB) P1 R
(11)
where P1 is the absolute atmosphere pressure. For the purpose of reducing the number of variables, an empirical equation (eq 12) is applied to determine the relationship between the normal boiling point (TB) and the molecular weight (MB):10 0.1326
TB ) 79.23FB
0.023Re0.8Pr0.33k dt
where k is the thermal conductivity of the gas mixture. Tube wall thermal conductivity (ki) is given as
and, for established turbulent flow,
/ log10 PB,wi )A+
(13)
in which the terms on the right side of eq 13 represent the inside convective heat-transfer resistance (1/hi), the resistance of the coke layer (δc/kc), and the tube wall (δt/kt). The convective heat-transfer coefficient hi for established turbulent flow is written as
hi )
3.66DBm km ) dt
km )
1 1 δc δt ) + + Uo hi kc kt
MB
0.3709
(12)
where FB is the bulk density of component B (given in units of g/cm3) at 20 °C and 1 atm. Here, we used a value of 1.27 g/cm3. 2.2. Case of Constant and Uniform Outside Wall Temperature. First, the simplified overall heat-
and the deposit layer thermal conductivity (kc) used is 2.2 W m-1 K-1 (from Huntrods et al.6). The corresponding heat flux at a specified longitudinal position then is calculated as
qz ) Uo(Tbulk - Two)
(14)
where Two is the outside wall temperature. The subsequent procedures to calculate temperature profiles and condensation rate are the same as those described in Section 2.1, except that, at each specified longitudinal location, qz replaces q0 and its value varies along the tube length, because of the change in bulk temperature. 2.3. Numerical Simulation Steps. Step 1: Input system parameters, operating conditions, and physical properties of individual gas components to calculate the gas mixture properties and the Reynolds number Re, to determine the flow regime. In the case of a constant and uniform outside wall temperature, eqs 13 and 14 are used to obtain the wall heat flux qz. Step 2: The radial temperature profile is calculated using eq 3 for laminar flow and eqs 4 and 5 for turbulent flow; the variation in axial temperature is obtained using eq 2. In particular, the inner wall temperature Twi is essential for the next step in the simulation. Step 3: Calculate the saturation pressure of condensable species B (P/B,wi) at a given inner wall temperature, using eq 9 for pure component and eqs 10 and 11 for a pseudo-component with only the boiling point fixed. The radial mass-transfer rate is given in eq 8 at a specified longitudinal position (local condensation rate), and the variation in the bulk fouling precursor along the axis is given by eq 7; the global deposition rate is obtained by averaging the local condensation rate over the tube length, as in eq 15.
rc )
z)L NBz dz ∫z)0
1 L
(15)
Step 4: Use eq 16 to calculate the instantaneous growth rate of the deposit layer:
∆δc MBrc ) ∆t FB
(16)
Ind. Eng. Chem. Res., Vol. 44, No. 11, 2005 4095
Figure 2. Comparison of experimental data and simulation results of vapor temperature effects on the deposition rate. TR ) 535 °C.
where δc is the thickness of the deposit film. The pressure drop across the tube length due to friction is expressed as
dP -4f 1 2 ) Fvj dz dt 2 z
(
)
(17)
where f is the friction factor. Repeat steps 1, 2, 3, and 4 to calculate for the next time step with the updated tube diameter dt - 2δc. The time to reach blockage of the tube can be estimated; however, this requires the velocity to be corrected for blockage and also the temperature at the deposit/vapor interface to be calculated. In the above calculation, the physical and transport properties of the vapor mixture (C ˆ p, k, µ, F, DBm) vary with temperature, pressure, and composition of the gas stream. The corresponding properties are calculated using common mixing rules. For simplicity, these equations are omitted here. 3. Results and Discussion Two different physical situations are simulated. In the first case, the situation corresponding to the conditions in the long test tube of the pilot plant1 is modeled. This case is characterized by laminar flow with either a constant and uniform wall heat flux or a constant and uniform outside wall temperature. The fouling precursor molar fraction (XB) and boiling point (TB), which are essential for the calculation of condensation rate, are unknown. Thus, the experimental data in Part 11 are applied to calibrate the model results to obtain these values. With these values fixed, the effects of operating conditions on the direction of fouling trends are predicted. In the second case, the situation is extended to simulate the fouling trend of an industrial TLE tube, which is characterized by high turbulent flow and a constant and uniform outside wall temperature. The findings and implication of the simulation results are discussed. 3.1. Simulation of Long Tube Results of the Pilot Plant.1 First, the ability of the model to predict gas stream temperature effect trends on the deposition rate is examined, as shown in Figure 2. Experimental data were extracted from Figure 3 in Part 1,1 with the following conditions: dt ) 9 mm, Re ) 912, and a
Figure 3. (a) Comparison of experimental data and simulation results of secondary steam/nitrogen addition effects on the deposition rate. Experimental conditions: feedstock, ATB-B; TR ) 535 °C; Tin ) 535 °C; Two ) 500 °C; and main vapor flow rate, 9.5 L/min (STP). (b) Simulated results of secondary steam temperature effects on the deposition rate.
constant and uniform wall heat flux of q0 ) (1157 W/m2. The feedstock was atmosphere topped bitumen ATB-A. In the following figures, the units of the experimental deposition rate were converted from g/kg_bitumen_feed to g m-2 s-1, for compatibility with calculated results. It can be observed that adjusting the precursor molar fraction (XB) under otherwise identical conditions moves the curve up and down (see curves 1 and 2 in Figure 2), i.e., increasing concentration yields a higher condensation rate and results in curves of relatively sharper slope. The normal boiling point of the fouling precursor (TB) determines the temperature at which condensation occurs. Variation of TB results in horizontal translation of the predicted curves (see curves 2 and 3 in Figure 2). By regulating these two parameters (XB, TB), a relatively good fit between experimental data and predicted results was obtained. Although the data cannot be claimed as strict validation of simulation results, because of the unknown XB and TB values in the product stream, the agreement nevertheless confirmed that results were consistent with the postulated physical condensation mechanism in this system. Figure 3a and b shows the predicted influence of secondary steam or nitrogen addition on the deposition rate. Experimental data were extracted from Figure 6 in Part 1.1 The simulation is based on laminar flow and a constant and uniform outside wall temperature. First, the model calibrated the experimental data at the base
4096
Ind. Eng. Chem. Res., Vol. 44, No. 11, 2005 Table 1. Simulated Conditions of a Single Transfer Line Exchanger (TLE) Tube
Figure 4. Comparison of model prediction and experimental data of tube diameter/vapor residence time/vapor velocity effect on the deposition rate. Conditions were as follows: feedstock, ATB-B; TR ) 535 °C; Tin ) 535 °C; and Two ) 500 °C.
point (without secondary steam or nitrogen) and the obtained model parameters were XB ) 4.44 × 10-5 and TB ) 1074 °C. The model then was used to calculate the deposition rate with varying feeding rate of secondary steam or nitrogen at fixed TB; however, XB varied with the addition of secondary steam/nitrogen. It can be observed that the calculation follows the experimental trend well. The decrease of the calculated deposition rate is due to the lower concentration of fouling precursor in the stream, because of the addition of secondary steam or nitrogen; hence, the driving force for mass transfer is decreased. Although some literature11 claims that the steam might be chemically involved in alleviating coke formation, the good agreement between the physical model and the experimental data suggests that the steam might function simply as a dilution agent under these conditions. If steam dilution is used to reduce deposition, the steam temperature should be high. The addition of steam sufficiently below the main vapor temperature could increase the deposition, despite its dilution effect. In the calculation of Figure 3a, the temperature of secondary steam is assumed to be the same as that of the main vapor, i.e., 535 °C. Figure 3b shows a prediction of the secondary steam temperature effects on the deposition rate. The volumetric flow rates of the main vapor and the secondary steam are 9.5 and 5.5 L/min (at standard temperature and pressure, STP), respectively. With the main vapor temperature fixed at 535 °C, the addition of secondary steam at temperatures of
