Mathematical Model Predictions of a Plugging Phenomenon in an

Digester plugging leads to a significant downtime with possible structural damage. In this work, we make use of a rigorous fundamental thermal−hydra...
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Ind. Eng. Chem. Res. 2004, 43, 5225-5232

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PROCESS DESIGN AND CONTROL Mathematical Model Predictions of a Plugging Phenomenon in an Industrial Single-Vessel Pulp Digester Sharad Bhartiya†,‡ and Francis J. Doyle, III*,§ Department of Chemical Engineering, University of California-Santa Barbara, Santa Barbara, California 93101, and Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716

Strong interactions between the mass, energy, and momentum variables make operation of a continuous pulp digester a challenging task. Steady-state operation is difficult to attain owing to stochastic variations associated with the naturally occurring feedstock and frequent disturbances due to the integrated nature of modern pulping mills. Further, it is imperative to operate the digester safely while maximizing pulp quality and yield, as the digester represents one of the most capital-intensive pieces of equipment in the pulping mill. Among the various scenarios that can lead to loss of production, the plugging of the digester is the most serious. Digester plugging leads to a significant downtime with possible structural damage. In this work, we make use of a rigorous fundamental thermal-hydraulic model of the digester to investigate a digester plugging incident in a large, commercial digester. The model accurately predicts digester plugging as well as the location of plug formation. It is demonstrated that digester hydraulics and their interaction with mass and energy variables is primarily responsible for operating conditions that can lead to plugging. Case studies are presented that study the sensitivity of digester hydraulics to operating conditions. Thus, the model can be utilized to build safe standard operating procedures to aid in digester operation. 1. Introduction Pulping mills convert wood chips to pulp suitable for paper production by displacing lignin from cellulose fibers. The conversion is achieved through a combination of strategies involving thermal, chemical, and mechanical degradation of the wood chips. Continuous kraft processes use large, vertical, tubular reactors called digesters in which the chips react with an aqueous solution of sodium hydroxide and sodium sulfide, known as white liquor, at elevated temperature. Most continuous digesters consist of three basic zones: an impregnation zone, a cooking zone, and a wash zone. A schematic of a single-vessel digester is shown in Figure 1. White liquor and presteamed chips are introduced at the top of the digester into the impregnation zone where the liquor penetrates the wet chips. However, the majority of the delignification reaction occurs only after the two streams flow downward into the subsequent cooking zone, where the mixture is heated to reaction temperatures achieved by liquor circulation through external heaters. The spent liquor is withdrawn from the digester at extraction screens located at the end of the cook zone, whereas the cooked * To whom correspondence should be addressed. Tel.: (805) 893-8133. Fax: (805) 893-4731. E-mail: doyle@ engineering.ucsb.edu. † University of Delaware. ‡ Current address: Department of Chemical Engineering. IIT Bombay, Powai 400 076, India. E-mail: bhartiya@ che.iitb.ac.in. § University of California-Santa Barbara.

Figure 1. Schematic of a single-vessel continuous digester.

chips continue the downward flow to the wash zone. Here, the chips are washed by the countercurrent flow

10.1021/ie034314h CCC: $27.50 © 2004 American Chemical Society Published on Web 07/20/2004

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presented later assume control volumes (CVs) with fixed cross-sectional areas A. The inventory of each phase at a fixed vertical location, z, in the digester is measured using the following volume fraction definitions

η)

Figure 2. Cross-sectional slice of the digester showing solid, entrapped liquor and free liquor phases.

(1)

The remainder of the control volume is occupied by the chip phase and is a measure of chip compaction in the digester. Thus

1-η) of cold, dilute liquor. This effectively quenches the delignification reaction. The quality of the resulting pulp is described by Kappa number, which is a measure of the residual lignin content. A typical control objective of digester operation is to minimize variation in the kappa number from a prescribed value. Continuous digesters present challenging problems in modeling and control. From a modeling perspective, the interplay between heat, mass, and momentum transport during the thermal-hydraulic degradation of the wood chips creates rich dynamic behavior. For example, softening of the chips, as cooking proceeds, causes them to compact more densely, which, in turn, affects the chip velocity profiles. Digester operation represents a complex interaction between the slower kinetics and heat-transfer phenomena and the fast-occurring digester hydraulics. A significant effort has been directed toward the development of first-principles models that describe the thermalhydraulic degradation of wood chips in the continuous pulp digester. Most fundamental continuous digester models in the literature can be classified into two broad categories depending on the attributes they emphasize (1) the pulping chemistry1,2 and (2) the hydraulic description of the chip and liquor streams.3,4 Recently, a thermal-hydraulic model of the Kraft pulping process in a single-vessel continuous digester has been reported.5 This model integrates the extended Purdue model6 with a rigorous description of momentum transport. The model illustrated the effect of hydraulics influenced by production rate, white liquor flow rate, chip level, and compaction (distribution of chips and liquor along the length of the digester) on cooking conditions. Similarly, the impact of cooking conditions (through changes in cooking temperatures, white liquor concentrations, etc.) on the digester hydraulics is illustrated through simulation examples. As in the Purdue model, each control volume is assumed to contain three phases: solid phase, entrapped liquor phase, and free liquor phase. The entrapped liquor phase resides within the pores of the wood chips where it reacts with the solid substance. However, dynamic and thermal equilibria are assumed between these two phases. The combined solid and entrapped liquor phases are referred to as the chip phase. A schematic of an infinitesimal slice of the digester is show in Figure 2. Components of the three phases are also shown. External flows from heaters and extraction screens are modeled as flows entering and/or leaving the control volume. The temporal variations within each infinitesimal control volume are described by conservation statements, resulting in a set of coupled nonlinear partial differential equations (PDEs). The interested reader is referred to the original reference5 for additional details. The results

[ ]

volume of free liquor phase in ∆V ∆Vf(z) volume of ∆V ∆V(z)

[ ]

volume of chip phase in ∆V ∆Vc(z) volume of ∆V ∆V(z)

(2)

The compaction profile along the length of the digester is influenced by the volumetric flow rates of the chip and free liquor phases as well as the degree of cooking. Thus, a compaction, 1 - η, of 1 indicates the absence of free liquor, as the digester is completely filled with chips at that location. Absence of free liquor adversely affects the drainage property of the chip plug through the digester. This situation is referred to in the industry as “plugging” of the digester and can result in significant downtime for the digester. The extended Purdue model assumes the compaction profile to be a fixed parameter of the model. Thus, changes in the hydraulic properties of the digester cannot be predicted using the extended Purdue model. On the other hand, the thermal-hydraulic Purdue model explicitly models chip compaction. In this work, we use the thermal-hydraulic model to investigate a plugging scenario in a real, large commercial digester. Because compaction is not a measurable quantity, the approach used here is to simulate the model with inputs gathered from the mill historian and observe the predicted compaction to infer information about the drainage property of the chips. Section 2 provides a brief introduction to the thermal-hydraulic model and discusses tuning of the fundamental model to match the real mill conditions. Section 3 describes the plugging incident that occurred in the mill. Finally, comparison of the model simulation results with mill data during this period is provided in section 4. 2. Tuning of the Thermal-Hydraulic Purdue Model to Pulp Mill Operation From a modeling perspective, the interplay between heat, mass and momentum transport during the thermal-hydraulic degradation of the wood chips leads to a system of tightly coupled and ill-conditioned equations. For example, softening of the chips, as the cooking proceeds, causes them to compact more densely, which, in turn, affects the chip velocity profiles. On the other hand, large transport delays, complex dynamics, biological feedstock variability and process integration make control applications difficult. As a first step, details of the vessel dimensions were obtained from the vessel drawings and mapped to the model. Also, data pertaining to normal operation of the digester were gathered and used to tune the model to mill conditions. The model inputs comprised mill measurements of the chip meter rpm (volumetric flow rate of raw chips), pulp production rate, and the flow rates and temperatures of the various liquor streams, sampled at a 10-min interval. The aim of model tuning was to achieve a predicted end-point kappa number and a

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Figure 3. Steady-state profiles along the length (dimensionless) of the impregnation vessel (IV) and digester vessel (cook and wash zones): (a) kappa number, (b) EA in free liquor, (c) compaction profile, and (d) chip temperature (solid) and free liquor temperature (dashed).

predicted residual effective alkali (EA) at the extraction screens similar to the mill measurements of these quantities. However, no attempt was made to validate the various constitutive rules (for example, kinetic rate constants and the coefficient of friction) because of the lack of appropriate data. Rather, the values documented in literature4,5 were adopted. The reaction rate effectiveness factor, ef, for the delignification kinetics served as the primary tuning knob. The model slightly overpredicted the digester end-point kappa number, as well as the residual EA in the spent liquor. It was conjectured that this is due to the difference in the chip heating mechanisms at the mill and in the model. Chips are heated at the mill through direct steam condensation in the vapor phase of the digester vessel. Because direct latent heat transfer is more effective than convective heat transfer, it is expected that the mill measurements indicate a higher conversion (lower end-point kappa number) than the model predictions for same wood-toliquor ratio. However, because the main goal of the current work is the study of hydraulic characteristics during digestion, no further efforts were expended toward matching of the end-point kappa number. For purposes of simulation, the digester was divided into 128 axial control volumes, and the PDEs were discretized. The resulting 2953 ordinary differential equations (ODEs) were implemented in MATLAB. Figure 3 shows the steady-state characteristics of the digester along its length. The simulation model inputs were based on actual mill measurements corresponding to nominal digester operation. The model calculated the compaction profile using momentum conservation. The kappa number (Figure 3a) shows an initial increase in the impregnation zone due to rapid consumption of araboxylan relative to lignin. Subsequently, the kappa number decreases in the cooking zone, beyond which the reaction is quenched by the cooler and dilute wash liquor. Figure 3c shows the compaction profile along the length of the digester. With progression of cooking in the cook zone, it is observed that the fraction of chip phase (solid fiber + entrapped liquor in pores) continues to increase until it reaches the extraction screens at the end of the cook zone. The increase in compaction is

Figure 4. Validation of the tuned thermal-hydraulic model using dynamic data representing digester operation for 24 h. Comparison of (a) model predictions of the digester end-point kappa number (dashed line) with mill measurements of the kappa number (thick line) and (b) model predictions of the residual EA of the liquor at the extraction screens (dash-dotted line, upper extract; dashed line, lower extract) with mill measurements (thick line). Mill measurements of the residual EA represent the EA of total extract.

explained as follows: as the chips travel downstream in the digester vessel, they soften as a result of cooking, which helps the chips pack more tightly, thereby reducing the room for free liquor. In the wash zone, the chip phase encounters the upflow of wash liquor, which aids in dispersing the chip phase, and the compaction profile levels off or shows a slight decrease. The gradual increase of chip temperature in the cook zone in Figure 3d is attributed to the exothermic nature of the delignification reaction. To validate the predictive capability of the model, the tuned model was simulated with actual mill measurements (chip meter rpm, pulp production rate, temperatures and flow rates of various streams) representing 24 h of mill operation. During this period, the production rate was maintained at its nominal value; however, there was an increase in the wash liquor flow rate after 15 h of operation. Comparisons of model predictions for the digester end-point kappa number and residual EA with real mill measurements are shown in Figure 4. As stated previously, the model end-point kappa number (dashed line in Figure 4a) was higher than the mill measurement (thick line in Figure 4a) during tuning. This trend was consistently observed during the dynamic simulation (see Figure 4a). The mill measurement for the residual EA (thick line in Figure 4b) refers to the residual EA of the combined upper and lower extracts. The model predictions for the residual EA in Figure 4b are documented separately for the upper extract (dash-dotted line) and the lower extract (dashed line). Whereas no significant end-point kappa number movement is exhibited in the mill measurements, the predicted kappa number shows wider variation. Despite this variation, the predicted kappa number remains bounded and does not exhibit a definite shift to a different operating region. The immediate divergence of the residual EA between the mill measurement and model prediction might be attributable to difference in initial conditions. Although, the absolute values do not match, the trends in the residual EA between the measurements and predictions are similar. Improve-

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Figure 5. Chip level control in the thermal-hydraulic Purdue model using PI controllers. The PI controllers were designed for (a) tight level control in the digester vessel and (b) loose averaging control in the impregnation vessel. Mills typically use customized strategies for chip level management (mill data not available).

Figure 7. Timeline of the plugging incident based on mill data for production rate (scaled). The tuned thermal-hydraulic Purdue model was validated with mill data corresponding to 0-24 h (see Figures 4-6). At about t ) 30 h, the digester was short-stopped and then restarted at about t ) 48 h and maintained at a lower production rate until about 60 h when the chip feed was ramped to its nominal rate. Plugging was observed in the form of a drop in the load of the outlet device at about 64 h.

between the initial and final time of the 24-h period, the compaction profile displays a markedly different behavior in the wash zone. The larger amount of free liquor available in the wash zone at the end of the validation (dashed) is due to the larger upflow rate (data not shown) after 15 h of operation. Previous models that assumed fixed hydraulic properties are unable to track the internal rearrangement of the chips and liquor due to changes in flow conditions. The next section describes the plugging incident and the corresponding analysis using the thermal-hydraulic Purdue model. 3. Plugging Incident Figure 6. Model predictions for kappa number and compaction profiles along the length (dimensionless) of the impregnation vessel (IV) and digester vessel (cook and wash zones) at the start of the dynamic simulation (solid) and at the end of 24 h (dashed). It is observed that that the kappa number profiles are similar. However, the compaction profile in the wash zone at the end of the 24-h period is markedly different from that at the start.

ments in model tuning will typically require extra effort and data, in addition to the need for closed-loop observers for matching initial conditions. Parts a and b of Figure 5 show the model predictions of the chip levels in the digester and impregnation vessels, respectively, during the dynamic validation exercise. The thermalhydraulic model uses PI controllers to accomplish chip level control using the upstream flow rate as the manipulated variable, whereas mills typically use customized (proprietary) strategies for chip level management. It is noted that end-point kappa number is usually strongly correlated with the chip level in the digester. Thus, if a different level controller is employed, it is expected that the model predictions of the end-point kappa number will be different from the ones shown in Figure 5. Figure 6 shows the initial (solid) and final (dashed) steady state profiles of the kappa number and compaction for the 24-h dynamic model validation period discussed above. It is observed that, whereas the kappa number profile shows only a slight variation

The tuned model was used to validate the model for “normal” operation of the mill immediately prior to the digester plugging incident. After the normal operation, the digester was short-stopped as a result of a recovery outage, after which the digester was restarted. However the chip meter was maintained at a lower production rate because of recovery limitations. Eventually, the digester rate began ramping up from the lower production rate toward its nominal aim rate. These rate manipulations are evident from the production rate trajectory shown in Figure 7, which is based on mill data and provides the timeline for the operating loss incident. Immediately prior to the plugging incident, the mill was operating at a lower production rate. Steady-state results for operation at the lower production rate are depicted in Figure 8. The solid lines in Figure 8a-c show the steady-state digester profiles when operating at the lower fixed production rate between 48 and 60 h (prior to plugging) as shown in Figure 7. The dashed lines represent steady-state profiles when operating the digester at the higher nominal production rate and have been reproduced from Figure 3. Figure 8d shows the temperature profiles for the two cases. It is clear that the internal digester profiles change significantly with the change in the production rate. However, the current study focuses on the hydraulic character of the digester described by the compaction in Figure 8c. It is observed that the compaction profile corresponding to the lower

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Figure 8. Steady-state profiles during the lower production rate operation (solid line) at t ) 59 h along the length (dimensionless) of the impregnation vessel (IV) and digester vessel (cook and wash zones) obtained by model simulation. Also, profiles corresponding to nominal rate operation (dashed line) at t ) 0 h are shown for comparison. The chip compaction profile shows a pronounced kink in the vicinity of the extraction screens during operation at the lower production rate.

rate (solid line) has a significantly pronounced kink at the extraction screens. Also, the amount of free liquor in the vicinity of the extraction screens reduce from about 18% for the higher nominal rate to about 12% for the current lower production rate. The implication is that the digester exhibits a poorer chip drainage capability at the lower production rate than the higher production rate. A compaction of 1 implies that the chips have softened and compacted tightly with no possibility of free liquor passage. Simulations show that, for the same extent of cooking (end-point kappa number), operation at lower production rates usually moves the digester toward plugging-like conditions. However, the countercurrent flow of dilute liquor aids in dispersing the fibers as the pulp enters the wash zone. Next, dynamic simulation of the model using mill measurements leading up to the plugging of the digester at t ) 64.5 h (see Figure 7) was performed. Figure 9 is a reproduction of the mill data during the period 6064.5 h and shows the ramping up of production rate from the lower production rate toward the nominal aim rate. The mill data also reflect the corresponding changes that were initiated in the various process variables to accomplish this rate transition (data not shown). However, approximately 9 min before the digester plugging (or sudden drop in the load of the outlet device), there was a sudden decrease in the total extract flow rate (data not shown). Because no drop in the white liquor flow rate was noted in the data, it was assumed that the upflow of wash liquor had suddenly dropped. It appears that this event is related to the plugging of the digester. Comparisons of the model predictions with mill measurements for the end-point kappa number and residual EA are shown in Figure 10. It is observed that predictions of the digester end-point kappa number (solid line in Figure 10a) and residual EA (upper extract EA, solid line; lower extract EA, dashed line in Figure 10b) closely follow the trends of the mill measurements (denoted by × in Figure 10). The model generates a diagnostic message if the compaction falls below 0.06

Figure 9. Mill data showing the production rate ramp-up from the initial lower production rate toward the nominal aim rate. Time t ) 0 indicates the beginning of the ramp-up and corresponds to t ) 60 h in Figure 7. The digester showed symptoms of plugging at 4.5 h after the ramp-up of the production rate was initiated.

Figure 10. Comparison of predicted (a) end-point kappa number (solid) and (b) residual EA at the extraction screens (upper extract, solid line; lower extract, dashed line) with mill measurements (denoted by ×) during the ramp-up period leading to digester plugging.

at the extraction screens and 0.1 at any other point along the length of the digester. The values were selected on an arbitrary basis. Model predictions for compaction profiles in the digester at the start of the simulation (start of the chip meter ramp-up at t ) 60 h in Figure 7) and end of the simulation (approximately 4.5 h after the ramp-up, when the model raised the flag to denote plugging) are shown in Figure 11. It is evident from this figure that, after 4.5 h of ramping up the production rate, the compaction near the exit of the digester increases to about 0.9, denoting plugging-like conditions. Note that, at this time, the compaction in the vicinity of the extraction screens actually decreases, indicating better drainage of the chips at that location. Simulations reveal that plugging of the digester near the extraction screens generally occurs while ramping down to lower production rates rather than while ramping up to higher rates (unlike the current incident). The current model does not include the outlet device. The exit point of the digester is assumed to be located above the outlet device where the only flow of liquor is

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Figure 11. Steady state compaction profiles along the length (dimensionless) of the impregnation vessel (IV) and digester vessel (cook and wash zones) based on model simulation. The solid line indicates the chip compaction profile at the beginning of the production rate ramp-up period. The model predicts a sudden decrease in the fraction of liquor near the exit of the digester, denoting plugging (dashed line). Loss of liquor corresponds to formation of a solid plug, which adversely affects drainage of pulp from the digester.

the countercurrent flow (that is, the sluicing of liquor to help pulp move out is not modeled). The time history of compaction (fraction of free liquor) at this exit point is shown in Figure 12. The sudden increase in the proportion of chips near the exit of the digester from about 79% to greater than 90% occurs instantaneously and correlates with the sudden drop in total extract flow, resulting in almost no upflow of wash liquor. Unlike quality variables such as the end-point kappa number, hydraulic variables react to changes very quickly. Because of the incompressible nature of the liquor, any changes in flow rates are transmitted downstream almost instantaneously. Judging from this preliminary study, it is inferred that the plugging occurred because of sudden loss in upflow of dilute wash liquor. 4. Sensitivity of Digester Compaction (Plugging) to Operating Conditions Safe operation of the pulp digester represents a careful orchestration of mass, energy, and momentum phenomena. The thermal-hydraulic Purdue model considers digester hydraulics in addition to mass and energy transport. The previous section demonstrated that the model adequately captured the occurrence of the digester plugging phenomena. Because of the instantaneous nature of hydraulics, there will not be sufficient time to implement corrective action if the digester operation represents unsafe conditions. Thus, the model can be utilized to study the sensitivity of the compaction profile (indicator of digester plugging) to the operating conditions. This will also enable the formulation of safe operating procedures as well as the incorporation of safety constraints directly into model-based control strategies. This section explores causes of digester plugging with respect to changes in operating conditions. It is assumed that the digester is operating at the lower production rate. In each case study, a given input is perturbed from its nominal value, and the

Figure 12. Time history of compaction (based on model simulations) near the exit of the digester where the thermal-hydraulic Purdue model predicts plugging after approximately 4.5 h of the ramping up of the production rate toward its aim rate.

Figure 13. Sensitivity of digester compaction (fraction of free liquor) to changes in the upflow rate: variation of the compaction profile along the length (dimensionless) of the impregnation vessel (IV) and digester vessel (cook and wash zones) with changes in upflow rates of the wash liquor.

compaction (fraction of free liquor) is noted. Simulation studies show that a different behavior emerges when a sudden change in input is made versus a gradual change. Case I: Change in Upflow Rate. The model assumes that the dilute wash liquor is water devoid of any chemicals needed for delignification. Thus, changes in the upflow rate do not significantly affect end-point kappa number. However, depending on the magnitude of the step change, the hydraulic behavior might be greatly impacted. Figure 13 shows a step reduction in upflow by 8% from its nominal value. Implementation of the step change (instantaneous) leads to near plugging of the digester at the extraction screens (dotted line). Also shown in the figure is the compaction profile when the upflow is gradually decreased by 8% from its nominal value (dash-dotted line). The compaction profiles for the two cases of reduced upflows (step and gradual reductions) show similar behaviors in the cook zone, but a very different situation emerges in the vicinity of the extraction screens and the wash zone. At

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Figure 14. Compaction, i.e., fraction of free liquor, at the exit of the digester when an instantaneous 95% loss of upflow rate is simulated. The sudden drop in the amount of free liquor leads to poorer drainage of the chips at the digester exit.

the extraction screens, a sudden decrease in upflow leads to a greater compaction of chips. Clearly, a large step change in the upflow can lead to plugging at the extraction screens, whereas if the same magnitude of change is accomplished in a gradual manner, the digester hydraulics is stable. The case of a sudden loss in upflow, simulated as an instantaneous step decrease by 95%, is depicted by the dashed line. This situation is similar to the production loss incident described in section 3. As observed before, this sudden drop in upflow leads to plugging near the exit of the digester (unlike the smaller instantaneous decrease, which led to movement toward plugging at the extraction screens). The plugging occurs within 42 s after implementation of the sudden loss in upflow. Figure 14 shows the time history of compaction at the exit of the digester. The lower fraction of free liquor at the exit of the digester indicates poorer drainage of chips and hence plugging-like conditions. It is noteworthy that the internal hydraulic profiles in the digester change from stable operation to plugging-like conditions almost instantaneously. The following observations are made: (1) The wash liquor performs not only the important function of washing spent liquor from the pulp in the wash zone, but also the hydraulic function of keeping the column watered. (2) The upflow helps break the tightly packed pulp in the wash zone. (3) Both the magnitude of the change in upflow and the period of time over which the change occurs affect the distribution of chips and liquor in the column. (4) For large (8%) instantaneous step decrease in upflow, simulations indicate that the plugging is likely to occur in the vicinity of the extraction screens. For near-complete loss of upflow, the plugging occurs near the exit of the digester. (5) Changes in the upflow rate have a negligible effect on the end-point kappa number. Case II: Change in Cook Temperature. Effects of temperature changes are relatively well understood. An increased cook temperature results in increased cooking or a lower kappa number. From a hydraulic standpoint, higher cooking leads to increased softening of the chips and tighter packing. As discussed earlier, tight packing of wood causes the digester to move toward plugging-like conditions. Figure 15 shows compaction profiles when the cook temperature was instan-

Figure 15. Sensitivity of digester compaction (fraction of free liquor) to changes in the cook temperature: Steady-state compaction profiles along the length (dimensionless) of the impregnation vessel (IV) and digester vessel (cook and wash zones) for various changes in cook temperature. Higher cook temperatures cause the digester to move toward plugging-like conditions.

Figure 16. Sensitivity of digester compaction (fraction of free liquor) to changes in the wood-to-liquor ratio: Steady-state compaction profiles along the length (dimensionless) of the impregnation vessel (IV) and digester vessel (cook and wash zones) for various changes in the wood-to-liquor ratio. The white liquor flow rate is maintained constant. The wood-to-liquor ratio is varied by changing the chip flow rate from its nominal value. When the chip flow is decreased by 10%, digester plugging is observed.

taneously increased from its nominal value by 2 °C (dashed) and 6 °C (dotted), while the other model inputs were maintained constant. For the 2 °C increase, the chips compact more tightly at the extraction screens. However, for the 6 °C increase in the cook temperature, the simulations predict plugging of the digester at the extraction screens. The softening of the chips due to excessive cooking leads to the formation of a chip plug. It is emphasized that a step change of 6 °C is unrealistic in practice but has been used here merely to illustrate that large temperature changes can lead to plugginglike conditions in the digester. Case III: Change in Wood-to-Liquor Ratio (Assuming Constant White Liquor EA). Here, the white liquor flow rate is maintained constant. The wood-toliquor ratio is varied by changing the flow rate of uncooked chips (that is, changing the chip meter rate).

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The increase in the chip flow rate results in a lower residence time and, hence, a lower conversion or higher kappa number. Conversely, a reduction in the chip flow rate results in a lower kappa number. As noted earlier, chip softening is directly related to the kappa number. When chips are cooked to lower kappa number, they tend to soften, which enables tighter packing of the chips or higher chip compaction. Figure 16 illustrates the effect on the compaction profile of changing woodto-liquor ratio from its nominal value. The ratio is varied by keeping the white liquor flow rate constant and varying the chip feed rate by +10% (dotted) and -10% (dash-dotted). The lower chip-to-liquor ratio leads to excessive cooking of the chips, causing plugging-like conditions at the extraction screens.

the chips. This reduces the fraction of free liquor and chip velocity, thereby disrupting the flow of chips out of the digester. Although changes in upflow have little effect on kappa number, a large and sudden decrease in upflow can also cause plugging of the digester. The thermal-hydraulic Purdue model employed in the current work ignores radial variations in the transport of mass, energy, and momentum. Further opportunities exist to study the behavior of chip movement and plugging in the vicinity of the extraction screens where the exiting extract liquor exerts a radial force of the chips, as this might also contribute to plugging-like conditions.

5. Conclusions

The authors gratefully acknowledge funding from the Department of Energy (Grant DE-FC07-00ID-13882).

The modeling of interactions between mass/energy conservation and hydraulic features enables the simulation of a fault scenario in the digester with disastrous economic consequences, namely, plugging of the digester. The thermal-hydraulic Purdue model was used to analyze plugging in a large commercial digester. Mill measurements constituted model inputs. The model was tuned for normal operation and validated using a data record representing 24 h of normal operation. The tuned model was then used to investigate the plugging incident. The model predicted plugging at the exit of the digester approximately 4.5 h after a rate transition was initiated. The location of the plug formation was confirmed through discussions with the mill personnel. Also, the time of the plugging incident was predicted accurately. Unlike quality variables that vary slowly because of the large time constants of the digester, plugging is a relatively fast process. Thus, the thermal-hydraulic Purdue model can be used to gain insights and design standard operating policies at the mill to avoid plugging. Changes in operating conditions that increase cooking, for example, increased cooking temperatures and lower wood-to-liquor ratios, tend to cause tighter packing of

Acknowledgment

Literature Cited (1) Smith, C. C.; Williams, T. J. Mathematical Modeling, Simulation and Control of the Operation of Kamyr Continuous Digester for Kraft Process; Technical Report 64; PLAIC, Purdue University: West Lafayette, IN, 1974. (2) Christensen, T.; Albright, L. F.; Williams, T. J. A Mathematical Model of the Kraft Pulping Process; Technical Report 129; PLAIC, Purdue University: West Lafayette, IN, 1982. (3) Ha¨rko¨nen, E. J. A Mathematical Model for Two-Phase Flow in a Continuous Digester. Tappi J. 1987, 70, 122. (4) Michelsen, F. A. A Dynamic Mechanistic Model and ModelBased Analysis of a Continuous Kamyr Digester. Ph.D. Thesis. University of Trondheim, Trondheim, Norway, 1995. (5) Bhartiya, S.; Dufour, P.; Doyle, F. J., III Fundamental Thermal-Hydraulic Pulp Digester Model with Grade Transition. AIChE J. 2003, 49, 411. (6) Wisnewski, P. A.; Doyle, F. J., III; Kayihan, F. Fundamental Continuous Pulp Digester Model for Simulation and Control. AIChE J. 1997, 43, 3175-3192.

Received for review December 16, 2003 Revised manuscript received May 20, 2004 Accepted May 26, 2004 IE034314H