I n d . Eng. Chem. Res. 1990,29, 829-841
829
Tjoe, T. N.; Linnholf, B. Using Pinch Technology for Process Retrofits. Chem. Eng. 1986, April 28,47. Townsend, D. W.; Linnholf, B. Heat and Power Networks in Process. AIChE J . 1983a,29, 742. Townsend, D. W.; Linnholf, B. Part 11: Design Procedure in Equipment Selection and Process Matching. AIChE J . 1983b,29, 748.
Kocis, G. R.; Grossmann, I. E. A Modeling and Decomposition Strategy for the MINLP Optimization of Process Flowsheets. Comp. Chem. Eng. 1989, 13, 797. Nelson, D. A. Preliminary Retrofit Design of Chemical Process Plants. Ph.D. Dissertation, Department of Chemical Engineering, University of Massachusetts, Amherst, 1989. Nikolaides, I. A Retrofit Procedure for Distillation Systems. Ph.D. Dissertation, Department of Chemical Engineering, University of Massachusetts, Amherst, 1987. Simulation Sciences Inc. Process Simulation Program, I / O Graphics Manual; Simulation Sciences Inc.: 1987.
Received f o r review May 22, 1989 Revised manuscript received December 4, 1989 Accepted December 15, 1989
Heat-Transfer Characteristics of a Rotary Disk Processort Pradip S. Mehta* Hoechst Celanese Corporation, Corpus Christi, Texas 78469-9077
Gary S. Donoian Farrel Corporation, Ansonia, Connecticut 06401
Poly(oxymethy1ene) melt was cooled with oil in a novel scraped surface rotary disk heat exchanger. T h e polymer is repeatedly spread as thin films over the cooled surfaces of the rotating disks and intermittently collected for surface renewal, as i t moves from one chamber to the next within the processor. T h e total heat removal could be accurately estimated by computing the unsteady-state heat transfer during each revolution from a composite flow of the incoming melt and internally recycled films, frozen over the moving disks, to the cooling oil. Viscous heat dissipation, especially a t the close clearance areas within the processor, was found to significantly offset the cooling capacity. Even so, surprisingly, the oil-side transport was found to be rate limiting. In a polymer processing line, it is often necessary to externally heat or cool the polymer to either facilitate a downstream operation or compensate for harsher conditions prevailing during an upstream processing step. For example, polymer melt exiting a polymer reactor may need to be cooled rapidly if the reaction is highly exothermic. Similarly, in many processes, dilute polymer solutions are superheated in heat exchangers before being devolatilized in a flash chamber or a falling strand devolatilizer. Precooling of polymer prior to a final shaping operation, which requires quick solidification, is also not uncommon. In fact, because of high-temperature sensitivity of polymers, it is desirable to have a good temperature control in most postreactor processing steps. In spite of an overwhelming need, inherent properties of polymers make it difficult to effectively transfer heat to or from polymers by conventional means. High viscosities of polymers limit transport via forced convection severely. On the other hand, heat conduction is constrained by the low thermal conductivity of polymers, the attainable temperature gradients, and, in many cases, the available contact area. A variety of heat exchangers have been developed for heating and cooling of viscous fluids, of which scrapedsurface or “closed-clearance” devices have been more popular. Uhl (1970) and Penny and Bell (1967) described some of these equipment which range from agitated vessels to spring-loaded scraper arrangements. Uhl found that addition of scraping augments heat transfer several folds and correlated their performance in terms of penetration and boundary layer theories. In addition to heat-transfer enhancement, continual scraping and renewal of stagnant layers is also desirable for polymers that are sensitive to extended heat histories. On the other end of the spectrum, in-line motionless static mixing elements have been successful in improving transverse temperature distribution ‘This paper was presented at the Annual AIChE Meeting, New York, Nov 1987.
through multiple flow divisions (Collins, 1979). Postreactor polymer processing equipment such as single-screw or multiscrew extruders have been traditionally used to perform a wide range of tasks such as melting of polymers, mixing and compounding of additives, alloying and blending of polymers, polymerization, devolatilization, and pressurization for the final shaping operation. The polymer is conveyed within the equipment through relative movement of some parts which are in close clearance with the stationary parts. For example, flights of a single-screw extruder closely wipe the barrel as the screw rotates relative to the barrel. This feature makes the equipment, in principle, equivalent to a scraped-surface heat exchanger if either surface or both surfaces in relative motion are temperature controlled externally. Indeed, polymer processors today take advantage of this feature by circulating heat-transfer medium through both the barrel and the screw of the extruder in many applications where a close control of the internal polymer temperature is required. Davis (1986) and Hold et al. (1982) have described procedures for calculating heat-transfer coefficients in single-screw extruders. Despite an apparent gain in heat transfer due to surface renewal in the wiped processing equipment, however, Japson (1953) also pointed out the problem of heat generation within the clearances of the wiping elements. Because of the high viscosities of the molten polymers, significant power is dissipated and appears as viscous heat. In processing equipment, therefore, cooling of a viscous polymer is much more difficult than heating it. The rotary disk polymer processor is a novel single-shaft machine (Tadmor et al., 1979), which like extruders has the inherent capability of performing most postreactor processing functions. It differs from screw devices in that it utilizes a twin-drag mechanism for polymer movement, in contrast to drag induction through a single moving surface in screw extruders. It enjoys a distinctly high surface-to-volume ratio even when compared with conventional scraped-surface heat exchangers (Tadmor, 1985). 0 1990 American Chemical Society
830 Ind. Eng. Chem. Res., Vol. 29, No. 5, 1990 PROCESSING
COROTATMG
f-F:&cE&%F
TO SLCCEEDNG CHANNEL
rTR~F4~~~ BARREL
/
f
REUEMD AREA OVER DISKTOP
-COOLANT CIRCULATION CAVITY
A
\
‘INCOMING FLOW -MELT POOL MOVING DISKS RECYCLED FILMS
LDISKTOP
VISCOSEALS
Figure 1. Schematic representation of the corotating disk processor depicting coolant circulation cavities within the rotor.
RM COATED OVER DISKS
cIRcvu\TK)N PASSAGE MAIN PROCESSHG CHANNEL
Figure 2. Cross section of a processing chamber with inserts; coolant circulation cavities in barrel.
The exposed surfaces are cooled or heated with an external heat-transfer medium. The polymer is spread as thin films over the moving surfaces and intermittently collected for renewal of already cooled layers, as the surfaces are wiped, resulting in a very effective transport of heat. In this work, heat-transfer characteristics of the rotary disk processor were explored. Highly viscous poly(oxymethylene) was cooled within the processor. Various factors that enhanced or limited the transport of heat were identified. The main objective of the investigation was to explore the impact of viscous heat generated, especially in the close clearance areas, on the overall cooling efficiency of the processor.
Polymer Flow Configuration The rotary disk processor (Tadmor et al., 1983) is comprised of a number of parallel disks mounted on a rotating shaft, fitted closely within an encasing barrel, as shown schematically in Figures 1 and 2. The space formed between the two disks and the barrel constitutes a processing channel. A channel block, a stationary insert, mounted on the barrel and placed between the two disks separates
MAIN PROCESSHG CHANNEL
Figure 3. Developed view of a processing chamber a t r = R d .
the inlet and the outlet of the channel. A cooling medium is circulated via a rotary joint through cavities formed within the shaft and the protruding disks for controlling the disk surface temperature. The cooling medium is also circulated in passages formed within the barrel, to maintain it a t a desired temperature level. Figure 3 shows an unfolded plane of a processing chamber of Figure 2 taken along the disk circumference. Polymer melt entering a channel is dragged forward by the rotating disks and is coated on them by a spreading insert (Mehta et al., 1984) shaped to form a converging wedge with each disk surface. The coated films are cooled by the cooling medium as they freely translate with the disks. In the absence of spreading inserts, the polymer either travels down the channel as a “rope” or a layer wound around the shaft or travels in the form of many “chunks”, as it is dragged by the disks. The form usually depends upon the degree of fill and the viscoelastic properties of the material. The coated layers (or translating ropes or chunks) are scraped off the disks by the channel block into a circulating pool formed at the block. The pool generates pressure just enough for melt transfer into the succeeding chamber through a transfer groove formed in the stationary housing. The flow through the short transfer groove is similar to that in the helical channel of an extruder. A processing chamber (Figure 3) is thus comprised of a main processing channel, a transfer channel, and inserts placed within the main channel. The processor consists of many such processing chambers placed in a series configuration. In the absence of any back-pressure, the flow within a processing chamber is almost pluglike. However, as in the case of most processing equipment, back-mixing occurs within a processing chamber via recycle of the polymer through close clearance areas. Due to polymer leakage through the finite clearance between the channel block and the rotating disks, for example, very thin films are coated over the disk surface and are recycled toward the entrance of the chamber (Figure 3). The incoming material is then deposited a t the spreading insert over the top of the recycled films instead of being directly exposed to the disk surface. Similarly, material is deposited over the top of the rotating disks through the clearance between the disk and the barrel and is recycled.
Experimental Section The heat-transfer characteristics of a pilot-scale rotary processor (Farrel DISKPACK Model FDD/ 1-3V) was experimentally determined using poly(oxymethy1ene) (Celcon M90 grade). The polymer was premelted in a single-screw extruder (Eagan, Reciproscrew, 3.5 in.) and metered into
Ind. Eng. Chem. Res., Vol. 29, No. 5, 1990 831 /-EXTR PREMELTlNG UDER -HEATED
INLET PIPE
BARREL COOLANT OUTLET
r DATA AOUlSlTlON SYSTEM
POLYMER DISCHARGE
Figure 4. Experimental setup. Table I. Assumed Equipment Parameters Processor Geometry 14 no. of chambers in series chambers containing spreaders 2-4, 6-8 chambers with seals at disktops 1, 4, 5, 8, 9 Typical Chamber Geometry Rd
R, H channel block length spreader block length clearances a t channel block
c
3.94 in. 2.36 in. 1.0 in. 0.5 in. 0.8 in. 0.020 in. 0.080 in.
the processor through a small connecting oil-jacketed pipe. Figure 4 is a schematic diagram of the experimental setup. The processor was comprised of 12 1-in.-wide cooling chambers connected in series via transfer channels within the barrel, followed by 2 narrow pumping channels. Many of the disks had visco-seals machined over their top (Mehta and Valsamis, 1985). A polymer seal between adjacent chambers was formed when the visco-seals were pressurized with polymer melt. Visco-seals could thus be used to separate pressures or vacuums between adjacent chambers. The disks were cored from within for circulating a cooling medium. The cooling medium entered the rotor shaft via a rotary joint and traveled through successive disk cavities in series. At the end, the cooling medium returned to the rotary joint via a concentric tube within the shaft, as shown in Figure 1. The coolant flow arrangement could be configured either cocurrent or countercurrent to the polymer flow. The housing was also cored with multiple passages around the circumference, as shown Figure 2. The passages circumvented locations for the inserts through the housing. The clearance between the rotor and the housing was relieved for a substantial circumference over the flat disktops. Table I gives the relevant geometrical dimensions of a typical processing chamber (see also Figure 3). A typical heat-exchange chamber is comprised of a spreading insert near the entrance, a blocking insert far from the entrance, and a transfer passage for transport to the successive chamber. Only chambers 2-4 and 6-8 contained spreading inserts. Therminol-66, a heat-transfer oil, was used as the cooling medium. An external unit was used to circulate the oil in the rotor and the barrel at pre-set temperatures. Temperatures were measured at the inlet and outlet of the rotor as well as the housing. The oil flow rate through the rotor and the barrel was measured independently at the desired temperature settings. Temperatures within the chambers were measured by inserting thermocouples through the barrel into the melt
pools formed at the channel blocks. The thermocouples were inserted at least l/z in. deep into the pools. Melt temperatures were recorded within the first nine chambers of the processor. The exit melt temperature was measured with a hand-held pyrometer. The desired flow rate was set by controlling the speed of the feed screw. The processor speed could be varied independently of the flow via a separate control at the motor. The power input into the processor was estimated from the current drawn by the motor. Experiments were designed to evaluate the overall heat-transfer capability of the processor and investigate the underlying mechanism for cooling of the polymers. The spectrum of variations included disk speed, throughput, oil temperature at the rotor, and film formation via spreading inserts. The oil to the housing was kept at 450 OF through all the conditions explored. Rotor oil was initially set high (>350 OF) and then lowered to the desired setting after a steady operation was attained.
Theory The bulk of the cooling within the processing chambers is achieved by depositing thin films over the rotating disks. The films are then collected and mixed transversely at the channel block as the material exits the chamber. The process is repeated in successive chambers. In analogy with a corresponding mass-transfer process (Biesenberger, 1980),heat transfer within the processor can be viewed as a series of two elemental steps: (1) rapid cooling of translating films coated over disks and (2) renewal of the cooled layers at the end of the translation. The short-term thin film exposure takes advantage of large initial temperature gradients, and intermittent mixing at the channel block renews the already cooled layers with those still warm. A mathematical model based on this idealization (Tadmor, 1985) showed that this process was very effective for rapid cooling of thin films and yielded a polymer-side heat-transfer coefficient that depended exponentially on the number of surface renewals. In practice, however, three fundamental departures were observed from the model. Even though the bulk of the material traveled in a pluglike manner, as described by the idealized model, leakage flow through close clearance areas introduced back-mixing. Further, for the highly viscous poly(oxymethy1ene) melt, a significant amount of power was dissipated during processing, a majority of which also occurred in the close clearance areas within the processor. The dissipated energy appears as increased bulk temperature due to viscous heat generation and is therefore counterproductive. During some experiments, in order to compensate for the power dissipation, the coolant was kept well below the temperature at which the polymer begins to freeze. Consequently, a layer of the deposited films was frozen over the rotating disks. The model thus needed to be modified to account for these three observations and estimate the average rate of heat removal within the processor. Heat-transfer effects are prominent at three distinct locations within a processing chamber (see also Figures 2 and 3): (a) translating films over the disk surfaces, where most of the cooling occurs; (b) circulating melt pools at the channel block and the partially filled helical channel of the transfer passage, where surface renewal takes place; some power is expended here; (c) flow within the close clearances where polymer is locally heated due to power dissipation. Even though a rigorous analysis would require solution of the coupled momentum and energy balances within each of these flow elements, with some simplifying assumptions, the hydrodynamics could be separated from the energy
832 Ind. Eng. Chem. Res., Vol. 29, No. 5, 1990 BULK FLOW
CHANNEL BLOCK
CLOSE CLEARANCEAREA-
7 REENTERING RECYCLEDFILM
RELIEVED CLEARANCE
CLEARANCE AREA
DISKS
CLEARANCE AT CHANNEL ROOT
Figure 5. Clearances a t a channel block; recycled films originate from the pool upstream of it.
balance. The resulting model analyzes the heat transfer within the three flow elements described above and combines the results through an iterative procedure for estimating the heat removal within the entire chamber. The complete temperature profile can then be obtained by repeating the procedure to the successive chambers.
Translating Films Thin films of polymer deposited over the rotating disks travel at a uniform velocity equal to that of the disks. During the period of their travel, they lose heat to the cooler disks primarily through conduction. As they are not sheared, power is not dissipated during their translation. Two types of films are deposited within the processing chamber: (i) bulk flow deposited as a layer for effective cooling and (ii) leakage from close clearances which appear as recycle streams. The following describes both of these streams and the methods used to estimate the heat transport occurring within them. Recycled Films. Internal leakage flow occurs in the main channel at the clearances between the channel block and the moving disks and at the transfer channel between the disktop and the stationary barrel. The leakage flow is deposited as very thin films over the moving surface at the trailing end of the clearance. The thin films are recycled and meet the corresponding bulk streams within the chamber. For example, three recycled films emerge from the clearance at the channel block (Figure 5), two over the disks on each side (side films), and one over the rotating shaft (bottom film). Because of the availability of melt from the pool upstream of the block, the thin recycled films coat the surfaces entirely. Similarly, the material entering the clearance between the disktop and the barrel at the trailing edge of the transfer channel emerges as a recycled film on the disktop (disktop recycle) when the clearance is relieved, as shown in Figure 6. All of the disk surfaces are wiped either at the channel block in the main channel or at the disktop by the barrel. Thus, the recycled films coat all of the disk surfaces. Following Latinen’s (1962) assumption, the thickness of the recycled streams can be assumed to be equal to half the clearance from which they emerge. Thus, their flow rates can be calculated from the mass balance and are given by
= aR,HdCdN
(2)
for the bottom recycled film a t the channel block, and (3) for the recycled film on the disk top.
Bulk Layers. The bulk of the polymer flow is either deposited on the sides of the disks, if a spreader is inserted within a chamber, or lies a t the shaft in the absence of a spreader. Since all the disk surfaces are precoated with the recycled films, the bulk flow is deposited on the top of the recycle coat. The resulting composite layer then translates over the majority of the disk circumference until the bulk layer is scraped off a t the channel block. However, a spreader does not necessarily coat the entire disk surface. Mehta et al. (1984) showed that for a given flow rate, V, and exit clearance, C, the disks will be completely coated with the incoming bulk layer only a t speeds less than a critical speed given by
where 6o depends primarily upon the viscoelastic properties of the melt. It was taken to be 0.5 for poly(oxymethy1ene) melt. The thickness of the emerging - film is found from the mass balance b=
v 2r(Rd2 - R:)N
(5)
If the processor is rotated a t speeds greater than Ncr,it was shown that patches of films covering only a fraction of the disk surface given by
will be formed with a thickness half that of the spreader exit clearance. In eq 6, pi is the fraction of the disk circumference between the spreader and the melt pool at the channel block. The rest of the disk surface would only be coated with the underlying recycled films. In applying eqs 4-6, the value of the spreader exit clearance, C, to be used is the actual spreader clearance minus the thickness of the recycled coat. When a spreader is not utilized in a processing chamber, the melt translates as a layer wound around the root (shaft) of the channel. The layer lies on top of the bottom recycled film and forms a composite with it. Its thickness is obtained from the mass balance b=-
for each side recycled film a t the channel block, vrb
Figure 6. Clearance over the disktop; recycled films originate from the pool in the transfer channel.
v 2rR,NH
(7)
Conductive Heat Transfer. All the layers of polymer deposited over the disks lose heat by conduction during their time of travel. The total time of exposure of the translating film to the cooler disk surfaces is obtained from t, = Pi/N (8)
Ind. Eng. Chem. Res., Vol. 29, No. 5, 1990 833 where pi is the fraction of revolution over which the film under consideration was exposed. The latter is fixed by the design of the chamber. Each film within a chamber experiences one or more exposures. For example, the bulk flow is exposed to the disk and the recycled film above it from the time it is spread (if a spreader is used) until it meets the channel block, and pi in eq 8 can be calculated from 0: -, pi = -
2a where 8; is the angle between the spreader and the surface of the pool formed at the block. Recycled streams, in general, see many exposures during a revolution. Consider a side recycle film. It is exposed by itself to the disk from the time it emerges from the channel block clearance until a bulk layer is deposited over it at the spreader. It is then exposed as a sandwiched layer during its travel from the spreader to the pool. If the bulk layer is not deposited over it, then it continues to lose heat a t diminishing rates. A t the channel block, it is sandwiched again between the pool and the disks. Thus, each exposure of a recycled stream needs to be accounted for separately in calculating the total heat removal from it. Heat transferred from a film during its translation can be estimated through equations for unsteady-state conduction (Carslaw and Jaeger, 1959a-c). Boundary conditions differ depending upon its location. During the period a recycled stream translates by itself, the finite film model can be applied with one surface of the film in contact with the disk and the other free surface adiabatic. The resulting heat-transfer coefficient is given by (Tadmor, 1985)
where f is obtained from f=-
7r2ffte
4b2 The total amount of heat lost to the disk is given by and the bulk averaged temperature of the film at the end if the exposure is calculated from a heat balance. The calculation procedure for heat removed from the bulk stream differs because of an additional resistance to heat transfer offered by the recycled film underneath it. The problem may be treated in a manner similar to that of unsteady-state conduction in composite films (Carslaw and Jaeger, 1959a) or alternatively simplified to that of heat transfer in one dimension with a linear boundary condition. Since the thickness of the recycled stream is considerably smaller than that of the bulk layer over it, the thin recycled layer may be considered to be in a pseudosteady state with respect to the dynamics of the bulk layer. This assumption is equivalent to assuming a linear temperature profile over the thickness of the recycled film. The problem of heat transfer within the bulk film is now reduced to that of unsteady-state conduction in a finite film with a linear resistance at one boundary and adiabatic conditions at the other (Carslaw and Jaeger, 1985b). The average temperature within the bulk film is given by
Table 11. Assumed Material Propertiesa p , lb/ft3 83.2 80.62 - 0.02087' K , BTU/h/ft/OF 0.185 -0.2196 + 9.3 X 10-4T enthalpy, BTU/lb -46.76 + 0.495T 11.36 + 0.567' viscosity, lb.s/in.
7' < 330 T > 330 T < 330 T > 330 150 < 7' < 330 T > 330 where
+
m = 5.2 exp(-0.0098T) n =1 < 50 m = 71.9 exp(-0.014T) n = 0.186 + 0.00147' 53 < j , < 250 m = 291.5 exp(-0.0147') n = -0.075 + 0.00147' 250 < y < 2000
"T = temperature in
OF;
y = shear rate in s-l.
where t, is the time of exposure, r is the ratio of the film thicknesses (bulk to recycle), and X j are the eigenvalues calculated from cot X j = X j / r (13) The corresponding temperature at the interface of the recycled film and the bulk stream is given by
The temperature profile within the recycled film is a linear interpolation between Ti(t,) and Td. An overall energy balance over the exposed period yields the total heat transferred to the disk surface: Equations 12-15 are applicable to translation of side composite films if a spreader is utilized or that of bottom composite films if a spreader is not used in the chamber. Frozen Recycled Films. The situation is different if the disk surface temperature is lower than the freezing point of the polymer. Because of its high degree of crystallinity, poly(oxymethy1ene) has a sharp transition temperature where it freezes. Table I1 gives the approximate enthalpy-temperature relationship for poly(oxymethylene), where an abrupt freezing point, T,, of 330 O F was assumed. In the analysis, an assumption was made that the recycled films froze completely as they emerged from the clearance, if the disk surface temperature was below the freezing point. The magnitude of most clearances within the equipment was small enough (0.01-0.02 in.; see Table I) to justify this assumption. An independent calculation using the Stefan-Neumann equation (Tadmor and Gogos, 1979) confirmed that a film with thickness of less than 0.01 in. froze within a very short time, under the conditions of this investigation. The cooling of the frozen layer can then be estimated from eq 10 and 11 prior to deposition of the molten bulk layer. The cooling of the bulk layer, when deposited over the frozen recycled layer, is also calculated from eq 10 and 11 with the contact surface temperature replaced by T,, the temperature at the interface. In practice, when a molten layer is deposited over the frozen layer within the main channel, part of the frozen layer will remelt at the interface, and its thickness will reduce. As the composite of the frozen and the molten layers travels further, the thickness of the frozen layer will grow again due to heat removal through the disk. The process continues until the layers reach the channel block, where part of the frozen layer will again shrink as it meets the warm rotating pool. The frozen layer may melt further or even completely when the recycled layer passes under the clearance of the channel block. Upon emerging from the clearance, the recycled stream would quickly freeze. Figure 7 depicts this cyclic process of remelting and re-
834 Ind. Eng. Chem. Res., Vol. 29, No. 5, 1990 SPREADER W E D G E 7 CHANNEL BLOCK
I
COLD M O W DISK MELT ENTERNG CHANNEL BLOCK CLEARANCE
RECYCLED
Figure 7. Remelting and refreezing of recycled film during a revolution: (a) frozen layer enters spreader wedge; (b) frozen layer melts due to newly deposited warm bulk layer; (c) frozen layer, grown due to heat removal during translation, enters melt pool and begins melting; (d) frozen layer melts further in the clearance; (e) emerging recycled film starts freezing again.
freezing of the recycled layer during each revolution. The model used in this study, however, neglected the growth-shrinking process and assumed a constant thickness of the frozen layer throughout the revolution. This assumption limits the model's capability to include the heat-transfer enhancement that occurs through continual phase transitions occurring within each revolution. The additional heat removal can be estimated by calculating the amount of polymer remelted (and refrozen) during a revolution and multiplying it with AHf,the heat of fusion.
Surface Renewal at Circulating Melt Pools Two melt pools are formed within each processing chamber (Figure 3), one a t the channel block where the incoming bulk films are scraped and pressurized just enough for melt to exit the main channel and the other a t a translating pool within the partially filled transfer channel. A significant amount of agitation occurs in both of the pools due to drag from the nearby moving surfaces. Edelist and Tadmor (1983) have examined the circulatory pattern in the pool formed at the channel block, whereas the helical motion in the transfer channel is similar to that found within the channel of a partially filled screw extruder (Tadmor and Klein, 1970). The processor used in this investigation was designed so as to facilitate melt conveying to successive chambers with minimal pressures. Consequently, pools occupied only a small fraction of the disk circumference. Pool Exposure. Unlike the translating films, melt pools are exposed to both cooler disks as well as the barrel. However, layers close to the barrel walls are relatively stagnant, and heat transfer to the barrel wall through the stagnant layers of polymer is minimal. On the other hand, constant renewal of layers close to the moving disk surfaces increases the rate of heat removal through the disks significantly (Tadmor, 1985). Only heat transfer to the cooler disk surfaces need thus be considered. Within the main channel, as shown in Figure 8, the pool formed at the channel block is exposed to the disk on both sides and to the root (shaft) of the main channel. The pool in the transfer channel is exposed on one side to the disktop (Figure 6). The center-line length of the pool, l,, formed a t the channel block can be estimated from equations derived by Tadmor et al. (1985),
where the pressure rise from the center of the pool to the
DEVELOPED TEMPERANRE PROFILE
CLOSE
CLEARANCE AKA
BULK RECYCLED FILM
i
Figure 8. Double-circulatory flow in the pool upstream of channel block; entrance region and temperature development within its clearance.
block due to drag from the jointly moving disks was equated to the pressure drop required for melt to exit from the center to the transfer opening. The shape factor, Fp (Tadmor and Klein, 1970), depends upon H/lp, the ratio of the channel gap to the pool length, making eq 16 implicit in 1,. Equation 16 was derived assuming a Newtonian fluid. Since viscosity does not appear explicitly in the equation, it may also be used for non-Newtonian fluids a t low shear rates. A length, l,, a t the center-line subtends an angle, 6,, given by 21..
The exposed area of the pool to the disk surface on each side is obtained from the pool length, and the exposed area to the root (shaft) is given by
Similarly, the area of the pool within the transfer channel exposed to the disktop is obtained from st,= gWTLT (20) where g, the degree of fill within the transfer channel, may be estimated by using the methods described by Squires (1958) for partially filled channels. Heat Transfer within the Pool. In analogy with a corresponding mass-transfer operation (Biesenberger, 1980), agitated pools may be visualized as ideal mixers where already cooled layers are homogenized with still warmer layers. Temperature in the pool may then be considered uniform, if the pool is small enough and well agitated. Calculations using eq 16 indicated that pools occupied less than l/&h of the disk circumference under all the conditions investigated. Examination of velocity profiles within the pool (Edelist and Tadmor, 1983) has shown flow patterns that indicate good mixing. Thus, the pool may be considered to be an ideal mixer. The incoming bulk streams can be assumed to be immediately mixed (see also Figure 8) with the rest of the pool upon entering it, and the outgoing stream may be assumed to be a t the temperature of the pool. The pool transfers heat to the disks through its contact with the recycled layers on the disk surfaces. Consider the pool formed upstream of the channel block (Figure 8). The pool temperature is obtained from an energy balance written around the pool,
Ind. Eng. Chem. Res., Vol. 29, No. 5, 1990 835 where Po is the bulk averaged temperature of the main stream entering the pool, Tp is the temperature of the exiting bulk stream, Q, is the heat transferred to the recycled stream, and P, is the power dissipated within the pool. In the absence of any throttling at the chamber exit, the power dissipation is small and given by
P, = rNH(Rd2- R;)Pb
(22)
where the pressure, Pb,developed at the center line of the pool can be calculated from equations derived by Tadmor et al. (1985). The conductive heat, Q,, transferred from the pool to the recycled stream, can be estimated from the penetration theory. Flow patterns studied by Edelist and Tadmor (1983) indicate that polymer layers near the disk surface travel at a speed close to that of the disk. Figure 8 shows schematically the double-circulatory-flow pattern within the pool. The average exposure time of the hot circulating layer to the cooler recycled film is tp =
0, 2xN
where both layers are assumed to be traveling in near plug fashion at the disk velocity. As in the case of cooling of the bulk deposited films, the thin recycled stream may once again be considered to be in a pseudosteady state with respect to the layer above it. Further, the thickness of the circulating layer may now be considered infinite. The heat removal is thus equivalent to that from an infinitely thick film to a cooler wall through linear resistance. The temperature at their interface at the end of the exposure is then given by (Carslaw and Jaeger, 1959c)
Ti(tp)=
Td
+ ( T , - Td) eXp(&'t,/b2) erfC ( ( d , ) l i 2 / 6 ) (23)
where 6 is the thickness of the recycled film. The heat removed from the pool is calculated from
Q,
= h,S,(T,
-
Td)
(24)
where S, is the total area exposed to the recycled films and h is the effective heat-transfer coefficient over period t,, ottained from
The temperature profile within the recycled stream is linear, and the bulk average temperature of the film entering the clearance is thus Tr =
(Ti(tp) + Td)/2
with a gap dimension of the order of 1-30 mils. The close clearance areas of significance within the rotary processor (Figures 2 and 3) are those between the rotating disk surfaces and (i) the barrel over the disktops, (ii) the channel block, sides and bottom, and (iii) the spreader block. With very small gaps between the two surfaces, the flow within the clearance may be viewed as that between two parallel plates, one in a relative motion with respect to the other. Figure 8 also shows a schematic diagram of the flow in the close clearance area on one side of a channel block. The disk is the moving surface that drags melt from the pool formed upstream of the block into and through the clearance. If pressure developed at the pool is small, then flow within the clearance is by drag alone and the thickness of the emerging recycled coat is half that of the clearance gap. Similarly, melt enters the clearance over the disktop from the pool formed in the transfer channel (Figure 6). After undergoing shear over the disktop, it exits the clearance as a recycle stream, where the barrel is relieved, or in case of disks with seals, it continues to be dragged around the circumference until it reenters the transfer channel and meets back a t the pool. On the other hand, the clearance at the spreader is larger so that both the bulk and the recycled streams are dragged through it and deposited as a composite layer on the disk wall. As in other processing equipment, a significant amount of power is dissipated in the close clearance areas. The local heat generated, if not removed efficiently, can result in local temperature hot spots. Under steady-state conditions, the temperature rise occurring within the parallel plate configuration was obtained by Gavis and Laurence (1968) and Martin (1967) for Newtonian and power law fluids. They solved the momentum and energy balance simultaneously and showed that the temperature rise depended critically on a dimensionless group,
(26)
In the foregoing analysis, a tacit assumption is made that, even though some power is dissipated within the pool, a negligible amount of it is dissipated within the layers close to the recycled streams. Heat removal from the pool within the transfer channel can be obtained by following a similar procedure.
Energy Dissipation and Heat Conduction in Close Clearances Processing equipment have close clearance areas between their stationary and moving parts due to mechanical limitations. In a screw extruder, for example, the gap between the flights and the barrel is a close clearance area
where m and n are the power law parameters, Vois the velocity of the disk, C is the clearance gap, K is the thermal conductivity of the polymer, and Tband Td are the barrel and the disk temperatures, respectively. The Brinkmann number defined by eq 27 can be viewed as the ratio of heat generated by power dissipation to the heat removed through conduction across the clearance. The model presented by Martin (1967) accounted for temperature variations across the gap and showed that, for large values of Brinkmann numbers, the velocity profile across the gap is no longer linear and the net flow rate through the gap can be reduced. In the present work, a simplified approach is taken. The energy equation can be decoupled from the momentum balance (Bird et al., 1960), if the temperature variation across the gap does not influence the velocity profile significantly. The steady-state solution for the temperature within the gap is T , = T , + (Tb- T d ) { l + (Br/2)9
-
Bra2/2) (28)
where 9 = y / C is the dimensionless distance from the moving plate. The profile exhibits a maximum at v* = 1 / 2 + 1/Br (29) with the peak temperature given by
836 Ind. Eng. Chem. Res., Vol. 29, No. 5, 1990
The average temperature of the fluid is obtained by integrating eq 28 over the gap Ts = T d + ( T b - Td)(1/3 + &/12) (31) and the heat fluxes to the disk and the barrel surfaces are given by
BARREL
/
-
COOLANT CIRCULATION CAVITY
(33)
BOTTOM RECYCLED FILM
The power dissipated over the length, L, of the clearance under steady-state conditions is given by
Figure 9. Temperature gradients across a cross section due to various resistances.
Pw= mCbL(Vo/C)R+l
Recently Rauwandaal and Ingen-Housz (198813) have suggested that significant errors could arise in using eqs 28-36, as viscosity variation due to temperature across the gap is not accounted for. The simplified approach taken here overestimates the peak temperature, given by eq 30, and the heat fluxes, given by eq 32 and 33. The viscosity parameters, m and n, to be used for determining the power dissipation and the Brinkmann number were, however, evaluated a t the bulk average temperature over the clearance. A three-segment fit (which accounts for an exponential temperature dependence) for the viscosity of Celcon over the shear rate range for poly(oxymethy1ene) is given in Table 11. Calculation of the average temperature within the clearance was first made by assuming the average temperature, estimating m and n using Table I1 a t this temperature, and then using eq 31. The process was iterated until the average temperature converged. If the disk surface temperature is lower than the transition temperature of the polymer, then the layer immediately close to it would be frozen. Because of a large heat flux resulting from power dissipation, the thickness of the frozen layer was estimated to be vanishingly small. With this assumption, eqs 27-37 could be used, with the disk surface temperature replaced with T,, the temperature at the interface of the frozen layer. The heat-transfer resistance through the frozen layer may be neglected.
(34)
The steady-state solutions, however, do not represent the entire temperature distribution over the length of the clearance. As in most boundary value problems, the temperature profile across the gap develops over a certain entrance length as melt enters the clearance. Meyer et al. (1978) investigated the temperature development in the leakage flow over screw flights and found that the entrance length is of the order of magnitude of the gap size. However, the initial heat fluxes to the disk and the barrel over the entrance length can differ significantly from those given by eqs 32 and 33. A rigorous solution of the unsteady-state energy equation, coupled with the momentum balance, would yield the correct temperature distribution. An analytical solution for heat transfer within the entrance region has been developed recently by Rauwandaal and Ingen-Housz (1988a) using a model that also neglects viscosity variations due to temperature rise across the gap. A good estimate of the heat fluxes can be obtained, however, if an energy balance is written over the entire clearance length, pCpV(Ts - To) = (Pw)t- ( L - Le)(Qds Qbs) - Le(Qdu + QbJ (35) where the left side represents the accumulation terms and the heat fluxes to the two boundaries on the right side are divided into two Parts-Qd, i-&bu over the entrance length and Qds i- Qbs over the steady-state portion. If entrance length is small enough, then the infinite film model for unsteady-state conduction may be used to estimate the heat fluxes on each side, to the disks and to the barrel. Rewriting eq 35 in terms of the heat-transfer coefficients,
where the fluxes for the steady-state portion are obtained from eqs 32 and 33, and h is the heat-transfer coefficient for the unsteady-state portion, estimated by using the Penetration theory. If the average fluid velocity in the clearance is taken as half the disk velocity, h can be written in terms of the length of exposure (i.e., the entrance length, Le)
)
2KpCpVo ' I 2 h=(
TL,
(37)
The bulk average temperature, Ts, in eq 36 is the steady-state temperature given by eq 31 and the total power dissipated, (Pw)t, can be obtained from eq 34, written for the entire clearance length. Thus, 36 becomes an algebraic equation in only one unknown, Le, the entrance length.
An Iterative Procedure The foregoing analysis describes procedures for estimating local heat-transfer coefficients on the polymer side a t various locations within a processing chamber. Figure 9 schematically depicts a typical temperature gradient across a cross section of the disk. The temperature of the disk surface in contact with the polymer may thus differ significantly from that of the oil in the cavity. Because of high conductivity of the disk metal, the entire disk surface will attain a temperature that does not vary significantly with angular positions. Determination of the disk temperatures requires an estimation of the heattransfer resistances within the cavity of the disk (oil side) and the metal. In the model used in this investigation, the heat-transfer coefficient within the cooling cavities in the rotor was determined from their geometries and independently measured oil flow rates. Either the Sieder-Tate relationship or Hausen's equation (Perry and Chilton, 1973) was used depending upon the calculated oil-side coefficients. Table I11 summarizes some properties of the oil used and the calculated oil-side coefficients. It also lists the combined oil and metal resistances obtained from 1
- - --Uod
1
Ad
boil Ai
+--A X Ad
(38)
Km A L
which is based on the exposed disk area per chamber.
Ind. Eng. Chem. Res., Vol. 29, No. 5, 1990 837 Table 111. Assumed Coolant Parameters parameter at 200 OF at 240 OF 7.99 7.86 P , lb/gal 0.44 0.46 C , BTU/lb/OF 0.0666 0.0657 (BTU/h/ft/OF 11 6.7 viscosity, (lb/h/ft) 1675 2527 Reynolds no. heat-transfer coeff, BTU/ h/ft2/OF oil sidea combined (coolant + metal) (based on disk exposed area)
59 34
-7"
Legend
:$
EXPERIMENTAL
\.
0 COMPUTED CHANNEL A
TEMP
COMPUTED DISK TEMP
42 24.7
OAt a disk surface temperature of 330 O F .
An iterative procedure may now be adopted for determining the disk surface temperature. For cocurrent oil flow configuration, the disk surface temperature is assumed for the first chamber, and the heat transferred from various elements to the disk (or barrel) within the chamber is calculated by the procedures outlined in the previous sections. The total heat transferred to the disk in the entire chamber, Q, is now known. The disk surface temperature for the next iteration is then found from (39) 8 = U o d A d ( T o i l - Td) where Ad is the exposed disk area, Ud is defined by eq 38, and To,is the oil temperature within the disk cavity. The procedure is repeated until the surface temperature converges to a value. A similar iteration loop is simultaneously set up for the barrel surface temperature. The temperature of the oil exiting the first chamber is found from a heat balance over the oil. This is also the temperature of the oil entering the second chamber. Similarly, the bulk average temperature of the melt exiting the first chamber becomes the entrance temperature to the successive chambers. For countercurrent oil flow, an additional chamber-to-chamber iteration loop would be required for estimating interchamber oil temperatures. In the current investigation, the convergence was attained quickly when a variant based on the Secant method to the proposed iterative procedure was used.
Results and Discussion Very high rates of heat transfer attained during an earlier investigation of the rotary disk processor (Duran and Valsamis, 1987) were attributed to its exceptionally high surface-to-volume ratio and large number of internal surface renewals. The temperature of a low-viscosity polymer was lowered by as much as 250 O F within only five processing chambers. The temperature profile within the processor followed very closely to that calculated by the surface renewal model (Tadmor, 1985). When a similar but higher viscosity polymer was used, however, the temperature drop over five chambers was limited to 150 O F . The theory deviated considerably from experimental findings in the latter case. One of the goals of the current investigation was to assess the effect of the internal heat generated during processing of the much more viscous poly(oxymethy1ene) on the overall cooling efficiency of the processor. Further, for poly(oxymethylene), the lowest attainable temperature was limited to T,, the temperature at which the polymer begins to freeze. A target was set for reducing the temperature from a typical processing temperature of 400 O F to less than 350 O F within the same processor used by Duran and Valsamis (1987). The number of chambers required to attain the desired reduction gave an indication of the overall heat-transfer capability of the processor. Figures 10-15 show temperature profiles taken under various conditions. The depicted experimental points
3201
' \
.-
I
300
0
1
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I
I
1
2
3
4
\
-_, ' 5
\
1
1
I
-,
8
7
8
9
0
CHANNEL NUMBER
Figure 10. Down channel temperature profiles-experimental and computed. Conditions: G = 130 lb/h; N = 37 rpm; oil flow cocurrent entering at 237 OF and exiting at 254 O F . 460
1
Legend EXPERIMENTAL
4401
-
420
-
400
-
360
.
360
.
340
.
$
z
i k
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0 COMPUTED CHANNEL TEMP
0,
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CHANNELNUMBER
Figure 11. Down channel temperature profile. Conditions: G = 100 lb/h; N = 35 rpm; transfer passage in chamber 4 throttled; oil flow countercurrent entering at 180 O F and exiting at 208 OF. 440
I
Legend EXPERIMENTAL
420
-
0
COMPUTED CHANNEL TEMP
0
400
-
Y
f
2 380 360
-
340tl 340
320 320
.
-
0
I
I
I
1
2
3
I 4
5
6
7
I 8
0
CHANNELNUMBER
Figure 12. Down channel temperature profile. Conditions: G = 130 lb/h; N = 37 rpm; oil flow cocurrent entering at 188 O F and exiting at 217 OF.
represent an average of three measurements recorded by thermocouples placed in the melt pools formed upstream
D
838 Ind. Eng. Chem, Res.. Vol. 29, No. 5 , 1990 440
0 EXPERIMENTAL 420
0 COMPUTED CHANNEL TEMP
I b
320
0
1
I
I
I
I
1
I
I
I
I
1
2
3
4
5
8
7
8
9
440
42C
40(
U
;
38(
k
361
10 EXPERIMENTAL
I 0 COMPUTE0 CHANNEL TEMP L
321
I
I
1
1
2
3
0 COMPUTED CHANNEL TEMP
1
I
3
Figure 13. Down channel temperature profile. Conditions: G = 130 lb/h; N = 50 rpm; oil flow cocurrent entering a t 184 O F and exiting at 216 O F .
341
1
Legend 0 EXPERIMENTAL
CHANNELNUMBER
CHANNEL NUMBER
-
1
1
J
I
I
I
I
I
I
4
5
6
7
8
9
I 10
CHANNEL NUMBER
Figure 14. Down channel temperature profile. Conditions: G = 100 lb/h; N = 50 rpm; transfer passage in chamber 4 throttled; oil flow countercurrent entering at 176 O F and exiting a t 215 OF.
of the channel block in the respective chamber, after a steady state was attained. Profiles calculated by using the theoretical model described in the previous section are also plotted in Figures 10-15. The temperature recorded in the short feed chamber was taken to be the entrance temperature. The temperatures in the successive chambers were then calculated. Nine chambers were found to be sufficient for cooling the polymer to the target value under all the conditions explored. Further, temperatures were not measured beyond the first nine chambers. Figure 10 also shows the calculated surface temperature of the rotating disks. Consider the temperature profiles shown in Figure 10. The initial rapid decline in the first three chambers (not counting the feed chamber) reflects the large initial temperature gradient. Little additional cooling and even a slight increase in temperature is seen between chambers 4 and 6. The trend is repeated between chambers 6-9. The calculated temperature profile is seen to predict heat removal in the initial chambers adequately but underestimates slightly the temperature increase between chambers 4 and 6 and between chambers 8 and 9. The
Figure 15. Down channel temperature profile. Conditions: G = 125 Ib/h; N = 37 rpm; oil flow cocurrent entering a t 183 "F and exiting at 209 O F ; spreaders from chambers 2-4 and 6-8 removed.
cooling in the seventh chamber was also underestimated. The deviation between the two profiles can be explained as follows. Even though the oil temperature was maintained around 250 O F in the oil cavities, the disk surface temperature remained above T , in chambers 1-4 which had high cooling loads. Therefore, all the polymer within these chambers was molten. Further, because of spreading inserts within chambers 2-4, the flow patterns could be estimated adequately. Consequently, temperatures at various locations could be accurately estimated. On the other hand, the disk surface temperature was calculated below T , for the rest of the chambers. Even though chambers 6-8 also contained spreading inserts, because of a frozen layer over the disk surfaces, the dynamics of the chambers differed. The entering bulk film in the model saw a smaller temperature gradient, as the interface of the frozen layer remained at T,. Thus, lower heat removal was predicted for this chamber. In practice, however, the total heat removal could have been enhanced by the continual remelting and refreezing of the frozen layer (Figure 7) during each revolution, a feature not included in the model. It might be intuitively expected that a frozen layer, if formed, would insulate the bulk stream from the disks. Unlike other polymers, however, the thermal conductivity of frozen poly(oxymethy1ene) does not differ significantly from its molten counterpart. In other words, a frozen layer is not different from a molten recycled layer in its thermal characteristics. Recycled layers, whether molten or frozen, effectively carried away heat generated in the clearances from which they emerged. Further, they covered the entire disk surface and utilized it well. The slight temperature increase between chambers 4 and 6 (and between chambers 8 and 9) came from the viscous heat generated during processing. The majority of power dissipation occurred over disktops. As seen in Table I, the disktops following channels 4, 5, 8, and 9 contained dynamic visco-seals. A small clearance between the barrel and the rotor was maintained throughout the disk circumference. Thus, a large amount of power was dissipated over the visco-seals. Application of eq 31 to the recycle stream over the disktop in chamber 5 indicates an average temperature of 399 O F over the seals at the end of a revolution. The overheated recycle stream would raise the bulk stream temperature when it mixes with it at the transfer channel. In contrast, the clearance over the
Ind. Eng. Chem. Res., Vol. 29, No. 5, 1990 839 disktops in other chambers was relieved after a short land at the trailing edge of the transfer channel (Figure 6), and the recycle streams over the disktops were not overheated. The visco-seals were designed (Mehta and Valsamis, 1985) to have self-pressurization capability. The model, however, accounted only for drag flow within the grooves of the seals and underestimated the flow rate of the recycled stream over the disktop. The calculated profile does not, therefore, show the temperature increase experimentally observed in chamber 6. A large temperature increase seen at the fourth chamber in Figure 11illustrates the role of viscous dissipation more dramatically. The transfer channel in this chamber was artificially pressurized by lowering a gate valve inserted from the barrel. A larger melt pool was formed within the main channel than that predicted by eq 16. In addition to the increased viscous dissipation within the larger pool, the recycle flow over the disktops on both sides of the main channel would have increased substantially due to increased pressure flow. Upon being heated within the respective clearances, these two recycle streams would indeed contribute to the bulk stream temperatures when they are recombined with it at the respective transfer channels. While the model was modified to recalculate the pool size, it failed to account for the increased pressure flow within the clearances and therefore shows a much smaller temperature increase. A large portion of the total power dissipation within the first nine chambers occurred over the disktops. For the run conditions of Figure 10, a total cooling duty of 4340 BTU/h was calculated for cooling the polymer from 410 O F in the first chamber to the target value of 350 "C. A total power dissipation of 9572 BTU/h was calculated for the first nine chambers, of which 5213 BTU/h was dissipated over disktops. Thus, the net effect of power dissipation was to raise the effective cooling duty from 4340 BTU/h to a total of 13912 BTU/h, and the polymer was cooled to only 355 O F . We note here that the majority of the power dissipation can be eliminated by either optimizing or relieving the close clearance areas. Lower oil temperatures can result in even greater heat-transfer rates because of the increased temperature gradient. As seen in Figure 12, the target temperature of 350 OF was attained in only four chambers when the oil temperature was lowered to 200 OF. By contrast, it took seven chambers to reach the target when oil temperature was set at 250 OF, under otherwise similar conditions (Figure 10). Interchamber temperature rise within the cooling oil was minimal because of relatively large flow of the coolant (20 lb of oil/lb of polymer). Thus, the direction of oil flow (cocurrent or countercurrent) did not affect the temperature profile significantly. Higher operating speeds at the same throughput required a larger number of chambers to reach the target because of the increased power dissipation per pound of the polymer. For non-Newtonian power law fluids, the energy dissipated within a clearance depends upon speed to the power (n + 11, where n is the power law exponent. As seen in Figure 13, in the absence of a throttling gate, it took seven chambers to reach the target temperature at a speed of 50 rpm, as compared to only four chambers at 37 rpm (Figure 12). If the throttling gate is lowered, the cooling gained in the first few chambers is nearly offset by the increased power dissipation in close clearances of chambers 4-6, as shown in Figure 14. Despite the lower coolant temperature, an average temperature of 434 O F was calculated within the seal area of the fourth chamber at a speed of 50 rpm, indicating a severe hot spot. The se-
verity of the hot spot could have indeed been reduced if the barrel coolant temperature was lowered during the experiments. Lower bulk temperature rise would occur if higher speed operation is accompanied by increased throughput, as more heat is convected away by the polymer. However, since power dissipation and the flow through the clearances are not significantly affected by the bulk flow rate but are speed dependent, local temperature within the clearances would be higher. One factor affecting the local temperature rise within the clearances is the size of the clearance gap. Both power dissipation and the conductive heat transfer increase with narrower gaps. As seen from eq 27, the net effect is a reduction in the average temperature for narrow gaps. It can be shown, however, that, if finite resistance at the two bounding surfaces are incorporated in the model, then an optimum gap size emerges, for which the temperature rise is minimal. The model described in the previous section did account for resistances within the disk and the barrel through an iterative procedure described therein. Figure 15 depicts the temperature profile within the processor when spreader blocks were removed from chambers 2-4, 6, and 8. It was expected that, with the contact area of the bulk stream substantially reduced, the heat-transfer rate would diminish substantially in these chambers. As in Figure 15, surprisingly, the heat removal rate was not significantly affected, and the target temperature was reached in six chambers. We emphasize here that, even though the spreaders were removed, the entire disk surfaces were coated by the recycled streams and therefore utilized. Indeed the calculated temperature profile shows a gradual decline. The model could predict well the total temperature reduction attainable within the processor. It failed, however, in predicting the initial rapid decline that was experimentally observed. Past observations have revealed that, in the absence of spreaders, fluid with low viscoelasticity could translate as chunks rather than as a uniform layer at the root of the channel. In the initial chambers, where the fluid is less viscous, this is more likely to happen. The increased contact area of the bulk stream could then explain the steeper initial decline in the temperature. As the temperature is lowered in subsequent chambers, the fluid would have a greater tendency to wind around the channel root and form a layer there, as assumed in the model. The most severe limitation to the overall heat transfer, surprisingly, was found to be within the oil circulation cavities. The cooling oil became viscous at low temperatures. The calculated heat-transfer coefficient (see Table 111) on the oil side was 42 BTU/h/ft2/OF at an oil temperature of 250 OF. When the metal-side resistance and the reduced internal cavity area were incorporated, the combined disk-side coefficient, U , using eq 38, was calculated to be 24.7 BTU/h/ft2/"F. Figure 16 illustrates the gain in the overall heat-transfer rate that could be attained if the oil-side coefficient could be improved. The parameter H,, represents the oil-side heat-transfer coefficient with its dependence on wall temperature factored out. Temperature profiles were calculated using different values of H,,, at the conditions of the experimental run of Figure 10. As seen in Figure 16, the target temperature of 350 OF can be attained in just two processing chambers, if the oil-side resistance is removed (Hrc= 250 BTU/h/ft2/OF). For example, by changing the cooling medium to pressurized water, H,, is expected to he in the range of 500 BTU/h/ft2/OF. Since metal-side resistance is also negligible, the overall heat-transfer
840 Ind. Eng. Chem. Res., Vol. 29, No. 5 , 1990
2
380
340
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9
10
Figure 16. Computed down channel temperature profiles with various oil-side heat-transfer coefficients; comparison with experimental profile at conditions of Figure 10.
coefficient is now approximately equal to the polymer-side heat-transfer coefficient, which can now be estimated from I7
=
Qtota~
(40)
A(T,, - Td) A value of 36 BTU/h/ft2/'F, based on total disk area of the two chambers following the feed chamber, was calculated at the conditions of Figure 16 (Hrc = 250 BTU/h/ft2/OF). The heat generated within the two chambers was accounted for in the definition (40) for calculating the overall coefficient. The actual polymer-side coefficient was back-calculated to 51 BTU/h/ft2/'F when the metal-side resistance and the difference between the external area and the internal cavity were taken into account. Such a high value for viscous polymers, inspite of significant power dissipation, would not be surprising if the processor is compared to a scraped surface heat exchanger in which close tolerance scraping and surface renewals a t many internal locations utilize the available contact area very efficiently.
Concluding Remarks Highly viscous poly(oxymethy1ene) melt was effectively cooled to the target temperature within a very short section of the processor. Indeed, an estimated polymer-side heat-transfer coefficient of 51 BTU/h/ft2/"F could be attained and the target temperature reached within just two processing chambers, if the heat transfer was not limited on the coolant side. In practice, while the polymer is being cooled, additional functions such as a reaction, devolatilization or compounding can be simultaneously performed within the processor. In this study, the benefits derived from the processor's high surface-to-volume ratio and frequent internal closeclearance scraping were offset by inadequate coolant-side heat transfer and the increased cooling load due to the dissipated power. The limitations imposed by the coolant can be surmounted with an appropriate coolant choice and by increasing its circulation rate. On the other hand, for very viscous materials, dissipated power, especially in close-clearance areas, can increase the net cooling duty of the equipment several folds and affect its operation. The majority of the power dissipated in the processor occurred over the tips of the rotating disks. Optimizing the clearance gap over the rotating disks or possibly relieving it completely after a short !and is essential for improved performance.
Despite intuitive expectations, the transport of heat was not significantly affected when layers close to the rotating disks froze at low coolant temperatures. It is believed, instead, that the frozen layers, which had thermal conductivity equivalent to their molten counterpart, could have augmented the heat transfer while continually melting and refreezing over the disk surface during each revolution. Another surprise was that it was not necessary to coat the bulk layers uniformly over the recycled layers. The disk surfaces were well covered with thin layers of the recycled streams. The recycled streams, which emerged from close clearances, carried with them the majority of the heat generated within them and transferred it to the disks effectively. A mathematical model based upon equations for unsteady-state conduction adequately described the heat transport within the chambers of the processor. Despite many simplifying assumptions used in the model, a detailed accounting of the heat transfer in all the flow elements gave satisfactory results for most runs. The total power dissipation and the total heat removed through the disk, calculated for all the 14 chambers, matched closely with the overall power drawn and the total heat gained by the coolant. The model, however, lacked the capability of accurately estimating the local temperatures over the close-clearance areas and the temperature increase caused by internal throttling within the processor.
Acknowledgment We are grateful to Prof. Z. Tadmor from Technion, Israel, and D. Steele and Dr. P. Hoffmann from Hoechst Celanese Corporation for helpful discussions.
Nomenclature Ad = total disk area available for exposure, in.' A, = heat-transfer area of the internal cavity, in.' A , = log-mean average of A, and Ad, in.' b = thickness of a film or a layer, in. Rr = Brinkmann number defined by eq 27 C, = heat capacity, BTU/lb/"F C = clearance gap size, in. F , = shape factor g = degree of fill of the transfer channel h = heat-transfer coefficient, BTU/h/ft2/OF H = gap between two disks, in. Hd = width of disktop, in. K = thermal conductivity of poly(oxymethy1ene). BTU/ft/ h/"F K , = thermal conductivity of disk metal, BTU/ft/h/OF L = length of a clearance, in. Le = entrance length for temperature development, in. LT = length of the transfer channel. in. 1, = length of melt pool, in. N = disk speed, rps ~V,, = critical disk speed, rps Ph = pressure developed at the channel block, psi P, = power dissipated, in.-lbf/s Q = amount of heat transferred, BTU/h r = ratio of bulk to recycled film thicknesses Rd = disk outer radius, in. R , = shaft radius, in. SF = exposed film area t , = exposure time for the film, s tp = exposure time of circulating layer in pool, s T = bulk average temperature, O F 7' = local or point temperature, O F L' = combined or overall heat-transfer coefficient, BTU/h/ ft?/"F V ( ,= velocity of disk surface, in./s
Ind. Eng. Chem. Res., Vol. 29, No. 5, 1990 841 V = volumetric throughput, inS3/s W , = width of the transfer channel, in. Ax = average distance of cavity from disk outer surface, in. a = pC,/K, thermal diffusivity, ft/h fi = fraction of the disk surface exposed 6 = thickness of the recycled film, in. 6,, = dimensionless film thickness Xj = eigenvalues defined by eq 13 7 = dimensionless distance within the gap p = density, lb/ft3 0 = angle occupied by bulk film or melt pool Subscripts
d = disk b = bulk film/layer i = interface m = freezing point for poly(oxymethy1ene) 0 = inlet or entering (temperature) p = pertaining to melt pool r = recycle stream rb = bottom recycle rd = disktop recycle rs = side recycle Registry No. Polyoxymethylene, 9002-81-7. Literature Cited Biesenberger, J. A. Polymer Devolatilization: Theory of Equipment. Polym. Eng. Sci. 1980,20 (15), 1015. Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Heat Conduction with Viscous Heat Source. Transport Phenomena; Wiley & Sons: New York, 1960. Carslaw, H. S.; Jaeger, J. C. Problems in Linear Flow: Composite solids. Conduction o f Heat in Solids; Oxford University Press: Oxford, U.K., 1959a. Carslaw, H. S.; Jaeger, J. C. Linear Flow of Heat in the Solid Bounded by two parallel Planes. Conduction of Heat in Solids; Oxford University Press: Oxford, U.K., 1959b. Carslaw, H. S.; Jaeger, J. C. Linear Flow of Heat: Semi-Infinite Solid. Conduction of Heat in Solids; Oxford University Press: Oxford, U.K., 1959c. Collins, S.H. Plast. Mach. Equip. 1979 (Feb). Davis, W. Extrusion Heat Transfer. Reactive Extrusion-Principles and Practice; Polymer Processing Institute: Hilton Head, SC, 1986. Duran, 0.; Valsamis, L. N. The Corotating Disk Processor as a Dynamic Heat Exchanger, Presented a t the Regional Technical Conference, Polyolefins V of the SPE, Houston, TX; Society of Plastics Engineers: Greeenwich, CT, 1987. Edelist, Y.; Tadmor, Z. Velocity Profiles in Corotating Disk Processors. Polym. Process. Eng. 1983,1, 1-36. Gavis, J.; Laurence, R. L. Viscous Heating in Plane and Circular Flow between Moving Surfaces. Ind. Eng. Chem. Fundam. 1968, 7 (2), 232.
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Received for review May 25, 1988 Revised manuscript received April 26, 1989 Accepted May 11, 1989
