In an annulus

Example 2 Calculation of j Factors in an Annulus Calculate the heat-transfer / factors for both walls of an annulus for the following condi-... 

Barnett confirmed that dh does not fully describe the cross-sectional geometry for burn-out in an annulus, and he decided to introduce as an extra term—the wetted equivalent diameter, i.e., the hydraulic diameter dw = d0 — di. The final expressions found suitable for A, B, and C are... 

The connection that has been shown in Section VIII to exist between burn-out in a rod bundle and in an annulus leads to the question of whether or not a link may also exist between, for example, a round tube and an annulus. Now, a round tube has its cross section defined uniquely by one dimension—its diameter therefore if a link exists between a round tube and an annulus section, it must be by way of some suitably defined equivalent diameter. Two possibilities that immediately appear are the hydraulic diameter, dw = d0 — dt, and the heated equivalent diameter, dh = (da2 — rf,2)/ however, there are other possible definitions. To resolve the issue, Barnett (B4) devised a simple test, which is illustrated by Figs. 38 and 39. These show a plot of reliable burn-out data for annulus test sections using water at 1000 psia. Superimposed are the corresponding burn-out lines for round tubes of different diameters based on the correlation given in Section VIII. It is clearly evident that the hydraulic and the heated equivalent diameters are unsuitable, as the discrepancies are far larger than can be explained by any inaccuracies in the data or in the correlation used. 

Including capillary condensation with the Hertz approximation, as considered by Fogden and White [20], introduces pressure outside the contact area i.e., adhesion enters the problem nonenergetically through the tensile normal stress exerted by the condensate in an annulus around the contact circle. The resulting equations cannot be solved analytically however, their asymptotic analysis may be summarized as follows. 

Liu, Z. Sun, X., and Ma, C., 2005, Experimental study of the characteristics of solidification of satiric acid in an annulus and its thermal conductivity enhancement, Energy Comers. Mgmt. 46 971—984. 

Annular flow. In annular flow there is a continuous liquid in an annulus along the wall and a continuous gas/vapor phase in the core. The gas core may contain entrained droplets—dispersed mist—while the discontinuous gas phase appears as bubbles in the annulus. This flow pattern occurs at high void fractions and high flow velocities. A special case of annular flow is that where there is a gas/vapor film along the wall and a liquid core in the center. This type is called inverse annular flow and appears only in subcooled stable film boiling (see Sec. 3.4.6.3)... 

Correlations for CHF in polyphenyl Two empirical correlations were developed by Core and Sato (1958) for predicting the CHF of polyphenyl flow in an annulus. The correlations are as follows. 

Downward flow effects. The CHF correlation of a downflow at high or medium mass flux in an annulus is given by Mirshak and Towell (1961) for V > 10 ft/sec (3 m/s) in a steady flow ... 

Bennett, A. W., J. G. Collier, and P. M. C. Lacey, 1963, Heat Transfer to Mixtures of High Pressure Steam and Water in an Annulus, Part 111. The Effect of System Pressure on the Burnout Heat Flux for an Internally Heated Unit, UK Rep. AERE-R-3934, Harwell, England. (5)... 

Collier, J. G., and D. J. Pulling, 1962, Heat Transfer to Two-Phase Gas-Liquid System, Part II, Further Data on Steam-Water Mixtures in the Liquid Dispersed Region in an Annulus, UK Rep. AERE-R-3809, Harwell, England. (4)... 

The material balance for a reaction of order q in an annulus of length dL is... 

Determine the shear stress distribution and velocity profile for steady, fully developed, laminar flow of an incompressible Newtonian fluid in a horizontal pipe. Use a cylindrical shell element and consider both sign conventions. How should the analysis be modified for flow in an annulus ... 

In determining the flow in a whole pipe, as above, it is unnecessary to use the infinitesimal shell element the method used in Example 1.8, with an element extending from the centre to a general position r, is preferable because it is simpler. Where the infinitesimal cylindrical shell element is required is for flow in an annulus, for example between r = rx and r = r2. This is necessary because the flow region does not extend to the centre-line so a whole cylindrical element cannot be fitted in. In the case of flow in an annulus, equation 1.56 is valid but the constants of integration must be determined using the boundary conditions that the velocity is zero at both walls. (Note that this specifies a value of vx at two different values of r and therefore provides two boundary conditions as required.)... 

When the inlet length is expressed in terms of number of gap widths , the difference between the flow in a tube and the flow in an annulus of narrow gap differs only by 25% [(0.05 - 0.04)/0.05]. This situation is an indication that the growth of the laminar boundary layers from the wall to the center of the channel is similar in both cases. Because duct friction coefficients, a measure of momentum transfer, do not vary by more than a factor of 2 for ducts of regular cross sections when expressed in terms of hydraulic diameters, the use of the inlet length for tubes or parallel plates can be expected to be a reasonable approximation for the inlet lengths of other cross sections under laminar flow conditions. In the annular denuder, the dimensionless inlet length for laminar flow development, L, can be expressed as... 

Discuss the modifications to the program for developing flow in a pipe that are necessary to allow it to calculate developing flow in an annulus when the inner and outer surfaces of the annulus have diameters of D, and Z>0, respectively, and when both the inner and outer surfaces are kept at the same uniform temperature. 

Use the Reynolds analogy to derive an expression for the Nusselt number for fully developed turbulent flow in an annulus in which the inner wall is heated to a uniform temperature and the outer wall is adiabatic. Assume that the friction factor can be derived by introducing the hydraulic diameter concept. 

Example 4.5 Entropy production in a flow through an annular packed bed The introduction of suitable packing into a fluid flow passage considerably enhances wall-to-fluid heat transfer, and hence reduces the entropy production due to heat transfer but increases the entropy production due to fluid-flow friction. Heat transfer to a fluid flowing in an annulus has a technical importance because we can heat or cool either or both of the surfaces independently. Entropy production provides a new criterion in analyzing such processes. In terms of the velocity and temperature profiles, the local rate of entropy production per unit volume of an incompressible Newtonian fluid for a two-dimensional annular flow is... 

The simple hydrides of carbon include CH, CH2, (CH)2 and (CH2)2. Of these, CH(s3p3) has a non-classical barrel-shaped structure, as shown in figure 7, with hydrogen cylindrically delocalized around C. The bonding electrons, a non-bonding pair and one unpaired electron are distributed in an annulus... 

Figure 1-15. Schematic sections of a hypothetical cylindrical cell resembling the intemodal cells of Nitella or Chora, illustrating various dimensions, the hydrostatic pressure, and the stresses existing in the cell wall (a) section perpendicular to cylinder axis, and (b) section through cylinder axis. The colored region indicates an aqueous solution where the hydrostatic pressure P leads to the longitudinal stress trL, which acts in an annulus of area approximately equal to 2nr x f , and the tangential stress crT, which acts along the two sides each of area l x tcw.

Nusselt number fer fully developed laminar flow in an annulus v/ith one surface isothermal and the other adiabatic (Kays and Perkins, 1972)... 

The velocity profile was assumed to be fully-developed. The velocity distribution in a circular microchannel including the slip boundary condition was taken from the literature. However, for the other geometries, they derived the fully-developed velocity profiles from the momentum equation. It is straightforward for flow between parallel plates and flow in an annulus. They applied the integral transform technique to obtain the velocity in a rectangular channel. The problem was simplified by assuming the same amount of slip at all the boundaries. 

The details of convection in a microarmulus subject to the uniform wall temperature boundary condition will not be given in full due to the sizable resulting equations. The eomplete details of the derivation for convection in an annulus subject to slip-flow can be found in [14]. Here the solution to the velocity profile and constant temperature boundary condition will be given. The flow is assumed to be fully developed and therefore the momentum equation is given as ... 

Determine the RTD for an isothermal tubular-flow reactor in which the liquid is in laminar flow in an annulus of inner radius and outer radius r2 rjr2 = a). Neglect molecular diffusion. The velocity in the axial direction at any radius r (between i and r2) is given by... 

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