Unsteady State Heat Transfer from Cylinders to Air in Normal and

Mar 16, 2009 - Transient temperature measurements were made to determine the Nusselt number for both normal and parallel air flow over cylinders. The ...
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Ind. Eng. Chem. Res. 2009, 48, 4119–4126

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Unsteady State Heat Transfer from Cylinders to Air in Normal and Parallel Flow Vishnu T. Marla, Robert L. Shambaugh,* and Dimitrios V. Papavassiliou School of Chemical, Biological, and Materials Engineering, The UniVersity of Oklahoma, 100 East Boyd Street, SEC T335, Norman, Oklahoma 73019

Transient temperature measurements were made to determine the Nusselt number for both normal and parallel air flow over cylinders. The cylinders were fine polymer fibers, and the temperatures were determined with an infrared camera. The experimental results for normal flow agree with the well-established correlation of McAdams. For parallel flow, the results lie between the predictions of previous researchers. Unlike the work described herein, previous investigators used equilibrium conditions as a basis for determining Nusselt numbers. Our transient measurements showed that the Nusselt number is not strongly dependent on the transient character of temperature difference. Introduction Heat transfer from a fluid to a cylinder is commonly encountered in numerous industrial and domestic processes. For example, consider the commercial fiber-making processes such as melt spinning, spunbonding, and melt blowing. Forced convection of a fluid (usually air) upon a cylinder is the dominant mode of heat transfer in these fiber processes. Melt spinning is a process in which molten polymer is forced through a tiny orifice to produce a fiber. The fiber is then attenuated by a mechanical roll placed some distance away from the orifice. Spunbonding is similar to melt spinning, except that a venturi replaces the mechanical roll to provide the drawing force upon the fiber. In melt blowing, high velocity streams of hot air impact the molten polymer stream as it emerges from the spinning orifice.1 In melt spinning and spunbonding, the hot fiber is quenched by air into which the fiber is spun, and the quench air also exerts a drag force upon the fiber. In melt spinning, spunbonding, and melt blowing, the fiber exhibits lateral vibrations that complicate the prediction of the heat transfer coefficient. Kase and Matsuo2 introduced a multiplying factor (1 + K) in their correlation for Nusselt number. This factor accounts for the transverse velocity of cooling air and for fiber vibrations in the spinning process. Kase and Matsuo reported that the value of K was solely dependent on the ratio of transverse air velocity to the relative air velocity parallel to the fiber. However, Han and Lamonte3 argued that K was an adjustable constant in the heat transfer correlation, and that K depended on the type of polymer used in the process. The correlation of Kase and Matsuo2 is commonly used when modeling the melt spinning and the melt blowing process. In some investigations, the use of this correlation (in a mathematical model) overpredicts the temperature when compared with experimental data for melt spinning4 and melt blowing.5 Additional, independent experimental determination of the relationship between the Nusselt number and the Reynolds number might help explain this discrepancy, since these relationships are key inputs in the model solutions. The contributions of the present work include (a) measurements of the heat transfer coefficient for flow around filaments using modern, accurate infrared techniques, (b) development of heat transfer correlations for flow parallel and perpendicular to the axis of the filament, (c) investigation of the transient heat transfer * To whom correspondence should be addressed. E-mail: [email protected]. Tel.: (405) 325-6070. Fax: (405) 325-5813.

behavior around a fine cylinder, and (d) development of heat transfer correlations while using experimental conditions that are tailored to approximate process conditions during melt spinning, spunbonding, and melt blowing. (In terms of fiber diameter, our experimental work is at the high end for melt blown fibers.) Past Work. The general problem of heat transfer in flow parallel to cylinders and for flow normal to cylinders has been described by previous researchers including McAdams6 and Morgan.7 For parallel flow, some of the earliest work in heat transfer from cylinders to air was done by Mueller.8 However, these measurements were made on a fine wire inside a vertical tube, and the Nusselt number showed dependence on the diameter of the vertical tube. Sano and Nishikawa9,10 also measured the heat transfer coefficient of a fine wire with air flowing parallel to the wire. They used wires of different diameters placed in tubes of different diameters and found no effect of the tube diameter on the heat transfer correlation. They attributed the scatter in Mueller’s data to the effect of additional turbulence of air flow at the inlet of the tube. Sano and Nishikawa also observed that the Nusselt number was independent of wire lengths and air velocities. Kase and Matsuo2 placed a heated wire in a stream of air and measured the wire’s heat loss rate. Their results (Nusselt number versus Reynolds number) compared well with the results of Sano and Nishikawa.9 Kase and Matsuo used their correlation in their mathematical analysis of the melt spinning process. For all of the above-cited work, the range of Reynolds numbers used was 3-120. The effect of vibration on heat transfer from wires to air in both forced and free convection has been reported by a number of researchers.2,11-13 The study of Lemlich11 on free convection showed that the heat transfer coefficient increased both with the amplitude and frequency of vibration. Lemlich observed a quadrupling of the rate of heat transfer when vibrations were introduced. Martinelli et al.14 observed a 5-fold increase in heat transfer rate over free convection for a 3/4 in. cylinder vibrating in water at a frequency of 40 Hz and an amplitude of 2.54 mm. Anantanarayanan and Ramachandran12 studied the effect of vibrations during forced convection over wires. For their work, the wire was stationary and the range of Reynolds number was 198-585. They observed an increase in heat transfer coefficient with both amplitude and frequency, and the maximum increase observed was 130% above the value for the stationary wire. Gupta and Agrawal13 used an experimental setup that was similar to the one used by Anantanarayanan and Ramachandran.

10.1021/ie800946a CCC: $40.75  2009 American Chemical Society Published on Web 03/16/2009

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Figure 1. Effect of Reynolds number on the Nusselt number for the cases of air flowing parallel to and normal to a cylinder. Only a portion of the full McAdams correlation (that covers Re from 2 to 200 000) is shown on this figure. Table 1. Comparison of Correlations from Different Studies for Heat Transfer from Small Cylinders (Wires or Fibers) to Air in Parallel Flow. The Range of Reynolds Numbers over Which the Correlations Are Valid Is Also Shown authors Anantanarayanan and Ramachandran12 Sano and Nishikawa9 Kase and Matsuo2 Gupta and Agrawal13 present study

correlation Nu ) 0.0522Re0.729 Nu Nu Nu Nu Nu Nu

) ) ) ) ) )

0.32 + 0.155Re0.5 0.42Re0.334 1.2Re0.249 0.068Re0.762 1.4 + 0.0141Re 0.3575Re0.46

Figure 2. Test unit used for the experiments. In the configuration shown, a Pitot tube allowed the measurement of air velocity below the stainless steel pipe (see the data in Figure 4). In another configuration, the Pitot tube was replaced with a thermocouple (to produce the data in Figure 5). In a third configuration, the Pitot tube was replaced with a horizontal fiber holder (Figure 3), and in a fourth configuration a vertical fiber holder was used. The r and z directions (for the cylindrical coordinate system) are shown in the figure. The z axis coincides with the longitudinal axis of the stainless steel discharge pipe.

Re range 198-585 1-100 0.5-50 59 e Re e 200 200 < Re < 9600 24 e Re e 150 24 e Re e 923

The range of Reynolds numbers that they investigated was 59-9600, a range that included half of the range of Mueller (Re ) 3-120) and all of the range of Anantanarayanan and Ramachandran (Re ) 198-585). In comparison with these two previous studies, Gupta and Agrawal reported higher Nusselt numbers for the stationary wire in forced convection (with no vibrations). For the case of the wire under vibration, Gupta and Agrawal observed an increase of 300% in the heat transfer coefficient. Figure 1 shows a Nu-Re plot for heat transfer from a cylinder to air flowing parallel to the cylinder axis and from a cylinder to air flowing normal (perpendicular) to the cylinder axis. For parallel flow, the correlations of four different research teams are shown. The figure also shows a correlation for normal flow that is taken from McAdams.6 This McAdams’ correlation for normal flow is based on the work of 13 different groups of researchers. (See Table 1 for the analytical forms of the Nu-Re correlations suggested by different authors). The correlation of Kase and Matsuo overlaps the correlation suggested by Sano and Nishikawa. For higher Reynolds numbers, the correlation by Anantanarayanan and Ramachandran appears to be the extrapolation of the work of Kase and Matsuo and Sano and Nishikawa. However, the correlation by Gupta and Agrawal shows Nusselt numbers that are double those predicted by the other investigators for Nu in parallel air flow. Even though the experimental setup of Anantanarayanan and Ramachandran12 was similar to the setup of Gupta and Agrawal,13 these two research groups obtained different results. All of the aforementioned heat transfer studies were performed using an electrically heated wire. The present study makes use of an unsteady state heat transfer approach to

compute the heat transfer coefficient for flow over polymer fibers. The present study considers both normal flow and parallel flow. Also, unlike some studies that used a thermocouple wrapped around the test wire13san arrangement that could alter the flow fieldstemperature measurements in the present study were made using an infrared camera (which is a noncontact technique of temperature measurement). The infrared camera used in the experiments and the test stand that was built to determine the heat transfer coefficient are described in the next section. Experimental Details The infrared (IR) camera used in the present study was a ThermaCAM S60 manufactured by FLIR Systems (Portland, OR).15 The spectral range for this camera is 7.5-13 µm (i.e., it is a longwave camera). The thermal sensitivity of this camera is 0.06 at 30 °C. [Thermal imaging radiometers with about 3-5 µm (shortwave) or about 3-14 µm (broadband) spectral response are also commercially available.] The S60 camera’s IR detector is 320 elements × 240 elements (76 800 elements). The camera comes equipped with a “primary” IR lens with a field of view (FOV) of 24° × 18° and with a minimum focusing distance of 0.3 m (11.81 in.). Unless otherwise mentioned, all temperature measurements in the present study were made with a “100 µm” close-up lens attachment for which the focusing range was 80-110 mm. With this lens attachment, the field of view is the same as with the primary lens. Hence, with this lens attachment the IFOV (instantaneous field of view) values for the minimum and maximum focusing distances are approximately 80 and 108 µm, respectively. The IR camera has the capability of recording 60 frames (temperature scans) per second, and a real-time plot of these measurements can be seen on the monitor of a PC that is connected to the IR camera by means of an IEEE 1394 (firewire) connection. Figure 2 shows the schematic of the test unit that was built to determine the heat transfer coefficient. A Pitot tube is shown in this schematic; the Pitot tube was used to measure air velocity. In another configuration used in our experiments, the Pitot tube

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Figure 3. Aluminum holder used to position the fibers for normal (perpendicular) flow experiments.

was replaced with a thermocouple so that air temperature could be measured. In other configurations, the apparatus was modified to hold a fiber to take measurements for flow normal to fibers (Figure 3) or parallel to fibers. Champagne et al.16 used a similar setup for measuring the heat transfer coefficient from cylinders at angles of 30° to 90° to the flow and for Reynolds numbers in the range of 5-30. In our equipment, compressed air regulated at 70 psig was passed through a thermal mass flow meter, and the flowrate was adjusted to produce the desired flow rate at standard conditions of 21 °C and 1 atm pressure. The air then passed through a reservoir heated by two 750 W cartridge heaters. The heated air was then directed through a 0.704 cm (0.277 in.) and 0.952 cm (0.375 in.) o.d. stainless steel tube; this tube had three heated zones, each zone had a 420 W tape heater, and each of these zones had a thermocouple that was used to monitor and control the temperature. To further maintain uniform temperature, this tube was wrapped in two layers of insulation, an inner layer of fiberglass foam insulation and an outer layer of fiberglass cloth that was tightly wrapped around the inner layer. The air from the 0.952 cm (0.375 in.) tube was fed to the top of a vertically mounted, 304 stainless steel pipe with an i.d. of 1.25 cm (0.493 in.) and an o.d. of 1.71 cm (0.675 in.). The pipe length was 76.2 cm (30 in.). Hence, the pipe L/D was 60, which was long enough to establish a well-developed velocity profile in the pipe.17 A cylindrical coordinate system was chosen for the analysis of the air flow pattern below the pipe; see Figure 2. Mean air velocity measurements below the pipe discharge were taken using a conventional Pitot tube (see Figure 2) that had an outer diameter of 0.7 mm and an inner diameter of 0.4 mm. A twoway valve enabled the measurement of the stagnation pressure with either of two instruments: (a) an Ashcroft digital industrial pressure gauge (model 2074; range 0-30 psi with 0.25% accuracy) supplied by Cole Parmer and (b) a Dwyer model 25 liquid manometer. The model 25 manometer had a range of 1.27-76.2 mm of water. The Pitot tube was attached to a threedimensional Velmex manual traverse that could position the Pitot tube with 0.001 in. (25.4 µm) resolution. The air temperature measurements below the pipe discharge were made using a J-type, 0.01 in. diameter thermocouple connected to an Omega microprocessor thermometer (model HH21) that digitally displayed the measured temperature. For the temperature measurements, the thermocouple replaced the Pitot tube on the traverse unit. Polymer fibers of different diameters (0.1-2 mm) were prepared in our laboratory using melt spinning equipment. The

Figure 4. Vertical velocity profiles of air at different downstream (z) locations as a function of radial distance (r) for a centerline discharge velocity of 10.1 m/s and a temperature of 141 °C.

Figure 5. Temperature profiles of air at different downstream (z) locations as a function of radial distance (r) for a centerline discharge velocity of 10.1 m/s and a temperature of 141 °C.

details of this spinning equipment can be found in Marla and Shambaugh.18 The diameters of the circular fibers were measured using an optical microscope. The polymer used in most of the experiments was 88 MFR Fina Dypro isotactic polypropylene. This polymer had an Mw of 165 000 g/mol and an Mn of 41 500 g/mol. In addition to the polypropylene fibers, polybutylene fibers were also spun and were used in our heat transfer experiments. This resin was grade 0400 manufactured by Basell Polyolefins; the resin had an MFR of 20. For the case with air cross-flow over the fiber, the fiber was held horizontally below the pipe discharge using a fiber holder as shown in Figure 3. For the case of air flow parallel to the fiber, the fiber was oriented vertically and attached to the traverse in a manner similar to that used for the mounting of the Pitot tube (see Figure 2). See Marla et al.19 for a description of the techniques that were used for the determination of the temperature of fine polymer filaments. Results and Discussion Normal (Cross) Flow. For a discharge temperature of 141 °C and an air flow rate of 50 slpm, the velocity and temperature

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profiles of air at different z locations are shown in Figures 4 and 5, respectively. It can be seen that the air flow field at z ) 8 mm displays a flat velocity and flat temperature profile in the region around the z-axis. Thus, as described in Marla et al.,19 one can make the assumption that a fine polymer filament centered at r ) 0, the center of the flow field, would (at steady state) assume the same temperature as the air. Velocity and temperature profiles for several other downstream locations are also shown for comparison. The flat-topped velocity profile gradually changes into a bell-shaped profile as z increases; this type of behavior is typical for round jets. Further evidence of this flat-topped behavior can be seen in IR images of fibers positioned in the flow field. For example, a 0.855 mm diameter polypropylene fiber was positioned in cross-flow 8 mm below the pipe discharge (for 141 °C and an air flow rate of 50 slpm). The IR temperature profile along the axial length of the fiber clearly showed that the temperature gradient over a small length in the central portion of the fiber was zero. Additional experiments were run wherein the air flow rate was varied to obtain centerline discharge velocities in the range of 3-15 m/s and pipe discharge temperatures were varied in the range of 90-141 °C. These were the velocity and temperature ranges for our experiments. A region of zero temperature gradient was observed (for the center section of the fiber) over all these temperature and velocity ranges. For our studies with cross-flow, the determination of the heat transfer coefficient is based on measuring the heating rate of the polymer fiber as it is suddenly introduced in the flow field at z ) 8 mm. (Keep in mind that, as shown in Figure 2, the r and z refer to the stainless steel discharge tube, and not the fiber.) An energy balance can be written for a short segment of the fiber. For cross-flow, this segment straddles the center of the air flow field over which the air velocity and air temperature are essentially constant. The energy balance reduces to VFCp

dTf(t) ) h(t)A[Tair - Tf(t)] dt

(1)

where V is the volume of the fiber element of length ∆z′, F is the polymer density, Cp is the specific heat of the polymer, A is the outside area of the fiber element, h(t) is the instantaneous heat transfer coefficient, Tair is the temperature of the air as determined from the thermocouple, and Tf is the instantaneous fiber temperature as determined from the IR camera. Both F and Cp are assumed constant over the temperature interval of interest, which is reasonable when no dramatic changes in the values of these properties occur over the temperature range of interest. This constancy is common practice in the modeling of heat transfer from fine fibers.20,21 It is assumed that the fiber surface and the fiber core are at essentially the same temperature. Also, thermal conduction in the axial direction is assumed insignificant. This assumption is common for fine filaments.1 In addition, our infrared pictures show that, in the center region of the fiber length (directly below the air discharge tube), the temperature of the filament is axially constant over a length of several millimeters (the same region where the surrounding air temperature is essentially constant). Simplifying eq 1, one gets

h(t) )

dTf(t) dt - Tf(t)]

FCpdf 4[Tair

(2)

where df is the fiber diameter. With the use of eq. 2, the procedure for determining the heat transfer coefficient in cross-flow is as follows:

Figure 6. Heating of a polypropylene fiber under a crossflow stream of hot air. The air velocity and temperature at the pipe discharge are shown on the figure. The time t′ corresponds to the point at which the difference between the air and fiber temperature becomes less than 20 °C.

1. The traverse is used to center the fiber such that the z axis (see Figure 3) intersects the axis of the fiber at 8 mm below the pipe discharge. This position is defined as the initial position of the traverse unit and a steel bolt is placed to mark this position (the bolt is fastened to the base plate that supports the traverse). A second steel plate is placed adjacent to the traverse stand and acts as a guide so that the entire traverse unit can be moved in a straight line (such that the center of the fiber can be quickly moved into or out of the air discharge). 2. The IR camera with the 100 µm close-up lens is focused so that the center region of the fiber with the uniform temperature is in sharp focus. The camera lens is placed 81 mm from the fiber. 3. The entire traverse stand is moved out of the flow field by sliding along a straight line using the steel plate as the guide. 4. After a couple of minutes (to allow the fiber to cool down) the traverse unit (along with the fiber) is brought back to its initial position (using the steel plate and bolt as a guide) and the IR camera then records the heating of the fiber at 60 frames/ s. The raw fiber temperature readings (from the IR camera) are converted into true fiber temperatures by using the emissivity values and calibration procedure recently described in Marla et al.19 5. The corrected temperature readings are fitted to a functional form for the temperature (a fifth order polynomial in this case) and the instantaneous heating rate dTf(t)/dt is computed using this functional form to avoid noise due to numerical differentiation. The instantaneous rate is calculated for each frame (i.e., the rate is calculated at 1/60 s intervals). 6. Using eq 2, the instantaneous heat transfer coefficient, h(t), is calculated. 7. The time-averaged heat transfer coefficient is calculated from the equation hj ) 1/t′ ∫0t′h(t) dt, where t′ is defined as the time when the temperature difference between the air and the fiber becomes less than 20 °C, and t ) 0 corresponds to the time when the fiber is first brought into the flow field and into the focus of the IR camera. A representative plot of the heating of a fiber is shown in Figure 6. This figure shows the center temperature of a 0.49 mm diameter polypropylene fiber as a function of time. The fiber took about 5 s to reach the air temperature (133 °C), and the fiber temperature remained constant thereafter. Figure 6 also

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Figure 7. Instantaneous heat transfer coefficient calculated for a polypropylene fiber held normal to the air flow from the pipe. The flow conditions were the same as in Figure 6. The time-averaged heat transfer coefficient (hj) and t′ are shown on the figure.

shows the location of t′ (as defined in step 7 above) for the conditions shown in the figure. With eq 2, the data shown on Figure 6 were used to determine the instantaneous heat transfer coefficient, h(t). The results of these calculations are shown in Figure 7. The dotted horizontal line on Figure 7 corresponds to the average hj determined as described in step 7. For the 1.6 s time range of the calculations, the h(t) is slightly below average for low times, h(t) is slightly above average for intermediate times, and h(t) is again slightly below average for large times. However, the extent of these excursions is at most 10%, which indicates a relative constancy of h(t) in spite of the rapid change in thermal driving force (∆T) from 113 to 20 °C. Keep in mind that, although h is often used as a constant in engineering calculations, h is considered to be a function of parameters such as the mechanism of fluid flow, system geometry, and fluid properties (Welty et al., 2001; Bird et al., 2006). Usually, h is defined as q A∆T where q is the rate of convective heat transfer, A is the characteristic area, and ∆T is the temperature difference between the surface and the fluid. The ∆T is a separate factor in the definition of h, but h might in fact be additionally dependent on ∆T. Many previous investigators determined, through the use of a range of ∆T in their steady state experiments, that h did not appear to be additionally dependent on ∆T. (Most previous investigators did not even consider the possibility of additional dependence on ∆T. When using correlations for h, practicing engineers do not assume any additional dependence of h on ∆T, even for situations where ∆T is changing significantly. The classic Gurney-Lurie Charts17 are good examples of this assumption of constancy.) In our unsteady state experiments, we also had a range of ∆T, and Figure 7 shows that h(t) did not change greatly over this range. However, unlike what occurs in steady state experiments, our ∆T changed rapidly during each experimental run. Some recent work has addressed the issue of changes in ∆T with respect to time [d(∆T)/dt]. Crocker and Parang22 and Su and Davidson23 suggested that Nusselt number (and h) varied over the course of experiments with free convection and changing ∆T. However, the time scales in their free convection h≡

Figure 8. Instantaneous temperature profiles of five polypropylene fibers of different diameters for the conditions shown in the figure. The fiber was held normal to the air stream (cross-flow).

experiments were orders of magnitude longer than what we encountered in our forced convection experiments. The transient forced convection study by Lee24 is more useful for comparison to our experiments. Lee performed a numerical analysis of transient forced convective heat transfer to water near the critical region in developing flow through a vertical tube. Dimensionless time from 2 to 14 were used in the simulations. In a typical simulation, h varied from about 7600 to 7800 W/m2 K for dimensionless times of 2 to 10. This is not a big change. No experimental data were given by Lee. Bloem et al.25 presented both experimental and numerical results on transient heat transfer to supercritical helium at low temperatures. The temperatures of the test tube were measured with multipurpose cryogenic fast response thermometers. Nusselt number calculations were made, and the Nu decreased from about 5 to 1 kW/(m2 K) when time changed from 0 to 20 ms. In Figure 7, perhaps the increase in h in the first 200 ms is partially due to a real dependence of h on the rate of change of ∆T with respect to time. From the figure, one can see that the amount of this change is at most 10%. For a similar system, Lee’s simulations24 predicted a 2% rise in h. However, as mentioned above, in our experiments the fiber was quickly inserted into the airstream at the start of each experimental run. So, part of the variability in h in the first 200 ms is probably caused by this insertion process. (In other words, we cannot distinguish between the effect of d∆T/dt and the effect of the process of introducing the fiber into the flow field.) The runs for cross-flow were done with both polypropylene (PP) and polybutylene (PB) fibers. While the highest pipe discharge temperature used with the polypropylene fibers was 141 °C, the highest discharge temperature used for polybutylene fibers was 105 °C. This was because the melting point of polypropylene is 165 °C, while that for polybutylene is 125 °C. Discharge temperatures near or above the melting point of a polymer would distort or melt the fiber. Figure 8 shows the transient temperatures of five polypropylene fibers of different diameters. These fibers were run under the same flow conditions with air cross-flow. The rate of heating increased as the fiber diameter decreased. This is expected, since smaller fibers (which have less thermal mass) require less energy to heat them to a desired temperature and, consequently, these fibers will more rapidly rise to the final temperature. Similar results were obtained with polybutylene fibers.

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Figure 9. The Nusselt number as a function of Reynolds number for air flow normal to fibers. Data obtained from both polypropylene and polybutylene fibers are shown. For comparison, the correlation of McAdams6 is also included.

For all of the polypropylene and polyethylene fibers, the calculation of h(t) and hj gave results that were similar to those shown in Figure 7. In particular, results for all fibers showed that the instantaneous h typically varied from the time-averaged heat transfer coefficient by not more than (10% between t ) 0 and t ) t′. Also, for both the polypropylene and polybutylene fibers, the time-averaged heat transfer coefficient, hj, increased as the fiber diameter decreased. All of the hj values were used to produce an Nu-Re plot for cross-flow; see Figure 9. The data for both PP and PB fibers agree very well with the crossflow correlation of McAdams.6 The error bars on the data of Figure 9 are two standard deviations in height. Since our data indicate that the McAdams correlation works well for fine filaments in crossflow, the McAdams correlation is quite suitable for use in the modeling of fiber spinning processes. It can also be inferred that our method of measuring fiber temperature (with the IR techniques discussed in Marla et al.19) gives accurate results. Parallel Flow. For the parallel air flow studies, the fiber was oriented along the z axis (see Figure 2) and attached to the traverse such that the tip of the fiber was 3 mm away from the pipe discharge and the exposed length of the fiber was roughly 25 mm. The IR camera was used with the close-up lens attachment, and the fiber to lens distance was 81 mm. The key heat transfer equations (equations 1 and 2) also apply to a vertical fiber. However, one must be careful as to where along the fiber the equations are used (i.e., how far from the fiber tip is appropriate). As was done for the cross-flow experiments, the fiber was suddenly introduced into the hot air stream and the fiber temperature was followed as a function of time. The seven step procedure described for the cross-flow studies also applies to the parallel flow studies. Figure 10 shows the effect of z location on the fiber temperature profile for the heating of a 0.68 mm PP fiber. The profiles are given for z ) 7, 11.5, 16.5, and 22 mm. These z values correspond to distances to fiber tip of 4, 8.5, 13.5, and 19 mm. Now, as shown in Figures 4 and 5, the centerline air velocity and air temperature are essentially constant over this range of z values. Thus, the fiber is exposed to near constant conditions over this z range. Figure 10 shows that, though the temperature profiles have a slightly higher slope at lower z values, the profiles are fairly well established by z ) 7 mm. A

Figure 10. Temperature profiles at four different z locations for a polypropylene fiber in parallel flow. The profiles correspond to z values of 7, 11.5, 16.5, and 22 mm (the arrow on the figure indicates how the profiles shift with increasing z). The fiber diameter was 0.68 mm. The distance from the tip of the fiber to the pipe discharge was 3 mm. Hence, z ) 22 mm corresponds to 19 mm from the tip of the fiber. The air velocity and air temperature remain almost constant along the fiber length.

Figure 11. Temperature profiles for the heating of polypropylene fibers in parallel flow. Data for three different diameters are given.

calculation of hj for the z ) 11.5 mm profile gives a value that is only few percent different than the hj calculated with the z ) 22 mm profile. Figure 11 shows the temperature profiles taken during the heating of polypropylene fibers of three different diameters. As was the situation in cross-flow, fibers with smaller diameters heat more quickly than fibers with larger diameters. However, the temperature gradients are not as steep as the gradients for cross-flow (when comparing fibers of similar diameters). The hj values were determined for the heating of the four fibers shown in Figures 10 and 11. These hj values were then used to produce Figure 12, which is a plot of Nusselt number versus Reynolds number for parallel flow. Also shown on the graph are the correlations developed by previous investigators. The present data lie in between the results of Gupta and Agrawal13 and the results of the other three research groups. As discussed in the Background section of this paper, Gupta and Agrawal surmised that vibration caused the Nusselt number (or h) to increase by 300%. Some vibration was also probably the cause of the higher Nusselt numbers determined from our work. In the course of our experiments, our fibers were quickly introduced

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were designed to explore the Reynolds number functional form of the Nusselt number, rather than the Pr functionality. Conclusions

Figure 12. The Nusselt number versus the Reynolds number for the case of air flowing parallel to the fiber axis. The correlations from other studies are also shown on the figure.

into the air stream directly below the air discharge tube. Some fiber vibration undoubtedly occurred during this movement of the fiber. (Small fiber vibrations are present in the melt spinning process, and vibrations are even more pronounced in the melt blowing process.) The polymers used in the present study are the same as those used in our laboratory for the melt spinning and melt blowing processes. Since the present experimental study was designed to simulate these processes closely, the heat transfer correlation from the present study should yield an accurate temperature profile when used in a melt spinning/melt blowing model. For melt spinning, where the Reynolds numbers are typically less than 150, a correlation based on our data for 24 e Re e 150 (see Figure 12) is Nu ) 1.4 + 0.0141Re

(3)

A correlation based on the whole range of Reynolds numbers in the present study (24 e Re e 923) is Nu ) 0.3575Re0.46

(4)

Table 1 summarizes various correlations for the Nusselt number for parallel flow of air on small cylinders. In the table, our new correlations are compared with the work of previous investigators. For cross-flow, our data are very close to the correlation of McAdams (see Figure 9). Hence, no new correlation needs to be suggested: McAdams correlation can be used to represent the new data. For 40 < Re < 4000, the McAdams correlation is (for Pr ) 1) Nu ) 0.683Re0.466

(5)

The effects of the changes of the air viscosity between the free-stream air temperature and the air temperature at the cylinder surface have not been explicitly included in eqs 4 and 5. It is usually accepted that the effects of the viscosity can be included as a term of the form (µ∞/µw)1/4 in the right-hand side of the Nusselt correlation.26 In the cases examined herein, the value of this term is about 1.05. The effects of changes on the Pr have also not been considered, since the experiments

An unsteady state approach was used to determine the heat transfer coefficient for fine cylinders normal and parallel to an air stream. For the case of a cylinder in normal flow with a hot air stream, our time-averaged heat transfer coefficient compared well with the established McAdams correlation. For heat transfer from fine cylinders to air in parallel flow, our correlation predicts Nusselt numbers that are (a) lower than that predicted by Gupta and Agrawal13 and (b) higher than that predicted by three other groups of investigators. Unlike the work of previous investigators, our experimental technique gave h(t), wherein the temperature difference varied with time (as the fiber quickly heated up). Thus, Figure 7 gives information on how h depends on unsteady temperature difference. Figure 7 shows that h(t) varies by no more than (10% from the average hj. Hence, using hj as an approximation of h(t) appears to be quite reasonable. Since our cross-flow results match the McAdams correlation, then a rapidly changing ∆T does not measurably affect the Nusselt number. (At least for the conditions of our experiments.) The same can be said for parallel flow, since our results fit nicely between the results of previous research teams. The heat transfer correlations developed in the present study can be used in mathematical models for processes that involve heat transfer from and to fine cylinders. Examples of such processes include the production of fibers by melt spinning or melt blowing.27 Model predictions of fiber temperature can be (a) compared with the temperature profiles from online measurements of fiber temperature and (b) used to suggest optimum operating conditions for fiber processes. Acknowledgment This work was supported by an NSF GOALI grant (DMII0245324). The support of 3M and Procter & Gamble is also gratefully acknowledged. Nomenclature A ) area of heat transfer over a length ∆z′ (m2) F ) density of the polymer (kg/m3) Cp ) specific heat of the polymer (J/(kg · K)) df ) fiber diameter (m) h(t) ) instantaneous heat transfer coefficient ((W/m2 · K)) hj ) the time-averaged heat transfer coefficient ((W/m2 · K)) HFOV ) horizontal field of view (cm) IFOV ) instantaneous field of view (cm or mm) K ) Kase and Matsuo multiplicative factor kair ) thermal conductivity of air (W m-1 K-1) Mn ) number average molecular weight (g/mol) Mw ) weight average molecular weight (g/mol) MFR ) melt flow rate of polymer Nu ) Nusselt number; Nu ) hdf/kair q ) rate of convective heat transfer (W) r ) radial coordinate position defined relative to the air discharge tube; see Figures 2 and 3 (mm) Re ) Reynolds number; Re ) dfVFair/µair t ) time (s) t′ ) the time when the temperature difference between the air and the fiber becomes less than 20 °C (s) Tair ) temperature of the air as determined from the thermocouple (°C)

4126 Ind. Eng. Chem. Res., Vol. 48, No. 8, 2009 Tf(t) ) instantaneous fiber temperature as determined from the IR camera (°C) z ) axial coordinate position defined relative to the air discharge tube; see Figures 2 and 3 (mm) z′ ) axial coordinate position defined relative to the fiber axis (mm) V ) air velocity (m/s) V ) volume of the fiber element of length ∆z′ (m3) VFOV ) vertical field of view (cm) Greek Symbols µair ) air viscosity (Pa s) F ) polymer density (kg/m3) Fair ) air density (kg/m3) ∆T ) temperature difference between the surface and the fluid (°C) ∆z′ ) the length of a fiber element (m)

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ReceiVed for reView June 16, 2008 ReVised manuscript receiVed February 11, 2009 Accepted February 18, 2009 IE800946A