Effective tube diameter

The accumulation of CR-jumps results in the chain motion in the direction lateral to the tube axis over a distance well above the tube diameter a (oc M buik)/ as schematically shown at the bottom of part (b-3) of Figure 3.6. Thus, the CR mechanism d5mamically dilates an effective tube diameter defined in a coarse-grained time scale, and the chain is regarded to be constrained in the dilated tube (supertube) in that time scale, as first pointed out by Marrucci (1985). A model based on this molecular picture was first proposed by Marrucci (1985) and later refined by Ball and McLeish (1989) and by Milner and McLeish (1998). Because this coarse-graining molecular picture... 

An alternative picture called tube dilution was employed to describe CR during relaxation of branched polymers. If one end of the chain is fixed to a branch point, reptation is suppressed and the whole relaxation must proceed by CLFs. However, large fluauations are exponentially suppressed, resulting in a large separation between the characteristic relaxation times of the tube segments 5 and s-tl. Ball and McLeish su ested that because of this separation, one can assume that the effective tube diameter also depends on s since on the timescale r(s) constraints created by all segments closer to the chain end than s had disappeared many times and can be discarded. Thus, the... 

Where aiN ) is the effective tube diameter for a system of chains with... 

The case of star/linear blends is a challenging one, because the description of constraint release that works best for pure star polymers is dynamic dilution, while for pure linear polymers, double reptation , or some variant of it, seems to be the better description. However, Milner, McLeish and coworkers [23] have developed a rather successful theory for the case of star/ linear blends. In the Milner-McLeish theory, at early times after a step strain both the star branches and the ends of the linear chains relax by primitive-path fluctuations combined with dynamic dilution, the latter causing the effective tube diameter to slowly increase with time. Then, at a time corresponding to the reptation time of the linear chains, the tube surrounding the unrelaxed star arms increases rather quickly, because of the sudden reptation of the linear chains. The increase in the tube diameter would be very abrupt, if it were not slowed by inclusion of the constraint release-Rouse processes, which leads to a square-root-in-time decay in the modulus (see Section 7.3). With this formulation, the Milner-McLeish theory yields very favorable predictions of polybutadiene data for star/linear blends see Fig. 9.13, where the parameters have the same values as were used for pure linears and pure stars. 

Figure 46 shows the time dependence of the mean squared segment displacement as predicted by the harmonic radial potential theory [70]. The three anomalous diffusion limits, (I)de> (H)de> and (III)de> of the tube/repta-tion model are well reproduced. Note the extended width of the transition regimes between these limits, which should be kept in mind when discussing experimental data with respect to a crossover between different dynamic limits. Increasing the effective tube diameter is accompanied by the gradual transition to Rouse-like dynamics of an unconfined chain (where entanglement effects are not considered). 

Fig. 47. Spin-lattice relaxation dispersion for a chain of 1 =1,600 Kuhn segments (of length b) confined to a randomly coiled tube with a harmonic radial potential with varying effective diameters d. The data were calculated with the aid of the harmonic radial potential theory [70]. c is a constant. At low frequencies the curves visualize the crossover from Rouse dynamics depending on the effective tube diameter. The latter case is described by a Tj dispersion proportional to characteristic for limit (II)de of the tube/ reptation model...

Thus, the DTD molecular picture has been refined through incorporation of the maximum length scale flcR(t) for the CR-equilibration in a given timescale The effective tube diameter for this refined partial-DTD picture is given by fl ( ) = min[fle ( ) ", AcrCO] which includes the expression for fiill-DTD (for which fle 5 ( ) star-branched chains and the blends. The current versions of the tube model ° are conceptually consistent with the partial-DTD situation and may serve as a reliable base for the rheological characterization of homopolymer liquids. 

Effect of Uncertainties in Thermal Design Parameters. The parameters that are used ia the basic siting calculations of a heat exchanger iaclude heat-transfer coefficients tube dimensions, eg, tube diameter and wall thickness and physical properties, eg, thermal conductivity, density, viscosity, and specific heat. Nominal or mean values of these parameters are used ia the basic siting calculations. In reaUty, there are uncertainties ia these nominal values. For example, heat-transfer correlations from which one computes convective heat-transfer coefficients have data spreads around the mean values. Because heat-transfer tubes caimot be produced ia precise dimensions, tube wall thickness varies over a range of the mean value. In addition, the thermal conductivity of tube wall material cannot be measured exactiy, a dding to the uncertainty ia the design and performance calculations. 

The highly exothermic nature of the butane-to-maleic anhydride reaction and the principal by-product reactions require substantial heat removal from the reactor. Thus the reaction is carried out in what is effectively a large multitubular heat exchanger which circulates a mixture of 53% potassium nitrate [7757-79-1/, KNO 40% sodium nitrite [7632-00-0], NaN02 and 7% sodium nitrate [7631-99-4], NaNO. Reaction tube diameters are kept at a minimum 25—30 mm in outside diameter to faciUtate heat removal. Reactor tube lengths are between 3 and 6 meters. The exothermic heat of reaction is removed from the salt mixture by the production of steam in an external salt cooler. Reactor temperatures are in the range of 390 to 430°C. Despite the rapid circulation of salt on the shell side of the reactor, catalyst temperatures can be 40 to 60°C higher than the salt temperature. The butane to maleic anhydride reaction typically reaches its maximum efficiency (maximum yield) at about 85% butane conversion. Reported molar yields are typically 50 to 60%. 

It is assumed that process conditions and physical properties are known and the following are known or specified tube outside diameter D, tube geometrical arrangement (unit cell), shell inside diameter D shell outer tube limit baffle cut 4, baffle spacing and number of sealing strips N,. The effective tube length between tube sheets L may be either specified or calculated after the heat-transfer coefficient has been determined. If additional specific information (e.g., tube-baffle clearance) is available, the exact values (instead of estimates) of certain parameters may be used in the calculation with some improvement in accuracy. To complete the rating, it is necessary to know also the tube material and wall thickness or inside diameter. 

Fig. 29. Main characteristics of the effect of tube diameter on burn-out for uniformly heated round tubes using water.

Govier, G. W, and Short, W. L. Canad. J. Chem. Eng. 56 (1958) 195. The upward vertical flow of air-water mixtures II. Effect of tubing diameter on flow- pattern, hold-up and pressure drop. 

Pratt 921 used both circular and square tanks up to 0.6 m in size and a series of different arrangements of a simple paddle as shown in Figure 9.60. The effect of altering the arrangement of the coil was investigated and the tube diameter da, the gap between the turns dg. the diameter of the helix dc. the height of the coil dp, and the width of the stirrer 14 were all varied. The final equations tanks were ... 

Because most applications for micro-channel heat sinks deal with liquids, most of the former studies were focused on micro-channel laminar flows. Several investigators obtained friction factors that were greater than those predicted by the standard theory for conventional size channels, and, as the diameter of the channels decreased, the deviation of the friction factor measurements from theory increased. The early transition to turbulence was also reported. These observations may have been due to the fact that the entrance effects were not appropriately accounted for. Losses from change in tube diameter, bends and tees must be determined and must be considered for any piping between the channel plenums and the pressure transducers. It is necessary to account for the loss coefficients associated with singlephase flow in micro-channels, which are comparable to those for large channels with the same area ratio. 

On the other hand Bao et al. (2000) reported that the measured heat transfer coefficients for the air-water system are always higher than would be expected for the corresponding single-phase liquid flow, so that the addition of air can be considered to have an enhancing effect. This paper reports an experimental study of non-boiling air-water flows in a narrow horizontal tube (diameter 1.95 mm). Results are presented for pressure drop characteristics and for local heat transfer coefficients over a wide range of gas superficial velocity (0.1-50m/s), liquid superficial velocity (0.08-0.5 m/s) and wall heat flux (3-58 kW/m ). 

There are some experimental data available on the effects of tube diameter, initial pressure, and temperature on the run-up distance to detonation for smooth... 

In order to simplify the analysis, we will consider the capillary flow of a hquid L in a horizontal small tube (diameter much smaller than the capillary length), in order to avoid complications due to gravity effects (Fig. 14). 

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