Circular ducts

The convective heat-transfer coefficient and friction factor for laminar flow in noncircular ducts can be calculated from empirically or analytically determined Nusselt numbers, as given in Table 5. For turbulent flow, the circular duct data with the use of the hydrauhc diameter, defined in equation 10, may be used. 

Cascade coolers are a series of standard pipes, usually manifolded in parallel, and connected in series by vertically or horizontally oriented U-bends. Process fluid flows inside the pipe entering at the bottom and water trickles from the top downward over the external pipe surface. The water is collected from a trough under the pipe sections, cooled, and recirculated over the pipe sections. The pipe material can be any of the metallic and also glass, impeiMous graphite, and ceramics. The tubeside coefficient and pressure drop is as in any circular duct. The water coefficient (with Re number less than 2100) is calculated from the following equation by W.H. McAdams, TB. Drew, and G.S. Bays Jr., from the ASME trans. 62, 627-631 (1940). 

For rectangular ducts, the area is evenly divided into the necessary number of measurement points. For circular ducts. Table 32-3 can be used to... 

Fig. 32-5. Circular duct divided into three equal areas, as described in Table 29-3. Numbers refer to sampling points.

For circular ducts, at least 10% of the total surface shall be tested, and for rectangular ducts, at least 20% shall be tested. In either case the area to be tested shall normally be at least 10 ra. Note that there is no specific information regarding the determination of the duct surface area. 

The graphical integration method is based on graphical presentation of the average flow profile. For a circular duct, the cross-section is virtually divided into several concentric ring elements. The spatial mean velocity of such an element is determined as an arithmetical mean of local velocities along the circumference of the corresponding radius. For a circular cross-section the flow rate can be expressed as... 

Cross-sectional area of a duct The area of a duct perpendicular to airflow, A. . In the case of a circular duct,... 

For flow in an open channel, only turbulent flow is considered because streamline flow occurs in practice only when the liquid is flowing as a thin layer, as discussed in the previous section. The transition from streamline to turbulent flow occurs over the range of Reynolds numbers, updm/p = 4000 — 11,000, where dm is the hydraulic mean diameter discussed earlier under Flow in non-circular ducts. 

HARTNETT and KOSTIC 26 have recently examined the published correlations for turbulent flow of shear-thinning power-law fluids in pipes and in non-circular ducts, and have concluded that, for smooth pipes, Dodge and Metzner S(27) modification of equation 3.11 (to which it reduces for Newtonian fluids) is the most satisfactory. 

The shear stress Ri at the pipe wall in the upper portion of the pipe may be calculated on the assumption that the liquid above the bed is flowing through a non-circular duct, bounded at the top by the wall of the pipe and at the bottom by the upper surface of the bed. The hydraulic mean diameter may then be used in the calculation of wall shear stress. However, this does not take account of the fact that the bottom boundary, the top surface of the bed, is not stationary, and will have a greater effective roughness than the pipe... 

For the heat transfer for fluids flowing in non-circular ducts, such as rectangular ventilating ducts, the equations developed for turbulent flow inside a circular pipe may be used if an equivalent diameter, such as the hydraulic mean diameter de discussed previously, is used in place of d. 

Derive a relationship between the pressure difference recorded between the two orifices of a pitot tube and the velocity of flow of an incompressible fluid. A pitot tube is to be situated in a large circular duct in which fluid is in turbulent flow so that it gives a direct reading of the mean velocity in the duct. At what radius in the duct should it be located, if the radius of the duct is r l... 

Kostic M (1994) On turbulent drag and heat transfer reduction phenomena and laminar heat transfer enhancement in non-circular duct flow of certain non-Newtonian fluid. Int J Heat Mass Transfer 37 133-147... 

The human cochlea, which derives its name from the Greek word koch-lias, consists of approximately three turns of a circular duct, wound in a manner similar to that of a snail shell (Figure 9.4). The ahihty to resolve different tones is determined hy the pattern of vibration of the flexible cochlear partition. This partition has three main components (Figure 9.5). When... 

The equivalent diameter is used in place of the pipe diameter for non-circular ducts or partially full pipes. For example, it is used to calculate Re as a means of obtaining f. In determining ... 

An AIG designed for circular ducts is shown in Figure 17.21. It is made from carbon steel pipe and there are multiple planes for injection. This design offers a high degree of tuning as each injection lance can be controlled by a valve. 

FIGURE 17.21 Ammonia injection grid for circular duct in the FCCU SCR. (With permission from Haldor-Topsoe, Inc.)... 

Fig. 9.4 Drag coefficient and fractional drag increase (Kj. — 1) for rigid spheres on the axis of circular ducts.

For fully developed laminar flow, the shear stress at the wall of a circular duct is... 

This well-known relationship is valid for laminar flow in circular ducts, but it also sets the stage for more general scaling relationships in noncircular cross sections and turbulent flows. 

Fig. 4.8 Illustration of the axial-velocity profile in the entry region of a circular duct.

We have just discussed several variations of the flow in ducts, assuming that there are no axial variations. In fact there well may be axial variations, especially in the entry regions of a duct. Consider the situation illustrated in Fig. 4.8, where a square velocity profile enters a circular duct. After a certain hydrodynamic entry length, the flow must eventually come to the parabolic velocity profile specified by the Hagen-Poiseuille solution. 

Consider a long circular duct in which an incompressible, constant-property fluid is initially at rest. Suddenly a constant pressure gradient is imposed. The axial momentum equation that describes the transient response of the velocity profile for this situation is... 

Fig. 4.9 Transient nondimensional axial velocity profiles in a long circular duct, responding to a suddenly imposed pressure gradient. The fluid is initially at rest.

Consider the transient flow in a circular duct where the pressure gradient can vary periodically in time, but at any instant in time is uniform axially. The axial momentum equation, for a constant-viscosity fluid, can be written as... 

Fig. 4.10 Instantaneous nondimensional velocity profiles in a circular duct with an oscillating pressure gradient. The mean velocity, averaged over one full period, shows that the parabolic velocity profile or the Hagen-Poiseuille flow. These solutions were computed in a spreadsheet with an explicit finite-volume method using 16 equally spaced radial nodes and 200 time steps per period. The plotted solution is that obtained after 10 periods of oscillation.

Fig. 4.16 Illustration of the Graetz problem. A fully developed parabolic velocity profile is established in a circular duct and remains unchanged over the length of the duct. There is a sudden jump in the wall temperature, and the fluid temperature is initially uniform at the upstream wall temperature. The thermal-entry problem is to determine the behavior of the temperature profile as it changes to be uniform at the downstream wall temperature. Because the flow is incompressible, the velocity distribution does not depend on the varying temperatures.

Fig. 4.17 Nondimensional temperature profiles (left-hand panel) in the thermal entry of a circular duct with a fully developed velocity profile. The profiles are shown at various nondimensional downstream locations z. Also shown is the nondimensional heat-transfer coefficient, Nu as a function of the nondimensional downstream position.

Consider the flow of an incompressible fluid in the entry region of a circular duct, assuming that the inlet velocity profile is flat. As is often the case, the problem can be generalized by casting into a nondimensional form. A set of nondimensional variables may be chosen as... 

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