Flow in Uniform Ducts

When elevation head and work transfer are neglected, the mechanical energy balance equation (6.13) with the friction term of Eq. (6.18) become 

The average density may be found with the aid of an approximate evaluation of P2 based on the inlet density a second trial is never justified. Eqs. (6.76) and (6.77) and the approximation of Eq. (6.76) obtained by neglecting the logarithmic term are compared in Example 6.12. The restriction to ideal gases is removed in Section 6.7.4. 

This is integrated term-by-term between the inlet and outlet conditions, 

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Consider a series of rectangular channels of width W, and height H, all with the same cross-sectional area whose walls are kept at a uniform temperature. The flow in these ducts can be assumed to be fully developed and laminar. 

Isothermal Row in Uniform Ducts 110 Adiabatic Flow 110 Nonideal Gases 111... 

Heat Transfer on the Walls With Uniform Heat Flux. The solutions for simultaneously developing flow in circular ducts with uniform wall heat flux are reviewed by Shah and London [1], Recently, a new integral or boundary layer solution has been obtained by Al-Ali and Selim [33] for the same problem. However, the most accurate results for the local Nusselt numbers [1] are presented in Table 5.6. 

Thermally Developing Flow. The local Nusselt number for thermally developing flow in elliptic ducts with uniform wall temperature NutT was obtained by Dunwoody [183] in terms of a double infinite series. These results are considered the most accurate as x > 0.005. Dun-woody s formula for calculating the mean Nusselt number is as follows ... 

Figure 23-2 Dimensionless correlations between reactant molar density and channel length for viscous flow in square ducts, plug flow in square ducts, and viscous flow in tubes with the same effective diameter for first-order irreversible chemical kinetics and uniform catalyst activity when the Damkohler number is 1.

A flow is uniform when the velocity of flow is the same at any given instant at every point in the fluid. This state of affairs can exist only with an ideal fluid. However, steady flow (uniform flow) is assumed to take place in a duct with the velocity constant along a streamline. 

Flow may be steady but have a variation of velocity, pressure, etc., with position. If one optional coordinate is used to describe the flow it is onedimensional, a typical case being uniform flow in a constant-area duct. 

Consider the pressure loss in a duct with straight, uniform cross-sectional area. The pressure loss is caused by friction. When different air sheets move against each other, friction is generated. The velocity and thermodynamic properties of air influence the friction. The duct wall has an overall roughness, which causes vortices to be formed with resulting friction in gas. The velocity has a pronounced effect in flow with low velocity, the vortices are small. Eor a straight duct the pressure loss Ap can be determined from... 

As previously noted, if the Reynolds number in the tube is larger than about 2000, the flow will no longer be laminar. Because fluid elements in contact with a stationary solid boundary are also stationary (i.e., the fluid sticks to the wall), the velocity increases from zero at the boundary to a maximum value at some distance from the boundary. For uniform flow in a symmetrical duct, the maximum velocity occurs at the centerline of the duct. The region of flow over which the velocity varies with the distance from the boundary is called the boundary layer and is illustrated in Fig. 6-3. 

Low flow velocities prevented uniform dust concentrations in the ducts. (This was reflected in the dust buildup at duct cleanouts). 

The model results show the velocity profile of a FCCU flue gas traveling vertically upward then making a 90° turn leading to the SCR. Notice the uniformity of the stream velocity as it travels upward. As soon as the stream encounters the 90° turn, the velocities stratify with nearly stagnant flow at the comer, and very high velocities at the far wall. This occurs because the denser catalyst particles are carried out further in the duct by their momentum relative to the lighter gas molecules. 

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... 

Suspension in a Fluid Stream—While diffusional and eddy processes tend to keep a suspension more or less homogeneous, there are many occasions when in spite of these processes conveyed material tends to stratify or distribute itself in a non-uniform manner. This difficulty is not present when particles are conveyed vertically, and yet we must appreciate the fact that the movement of a fluid in any duct creates a velocity gradient which is a maximum along the axis and decreases to very low values next to the duct surface. Thus, all the particles do not move at a Uniform rate and in the case, of vertical motion the largest particles may move only along the duct axis. Those near the duct surface may, in fact, tend to fall unless projected by eddies and particle impacts toward the duct axis. In any case, it is generally easier to sample a suspension flowing in a vertical duct than in a horizontal duct. 

The thermal entrance region in a hydrodynamically fully developed flow in a rectangular duct may be studied by the use of the integral method. In this section, the uniform wall temperature and the uniform wall heat flux cases are discussed. The physical model is based on the following assumptions ... 

Now for a circular duct, i.e., a pipe with a uniform heat flux at the wall, the analysis discussed in the previous section gave Nup = 4.364. Therefore, for fully developed flow in a plane duct with a uniform heat flux at the wall, this would indicate using the hydraulic diameter concept, that ... 

Nusselt number and center line temperature variation in developing flow in a plane duct with a uniform wall temperature. 

In some situations it is possible to find the heat transfer rate with adequate accuracy by assuming that the velocity is constant across the duct. Le to assume that so-called slug flow exists. Find the temperature distribution and the Nusselt number in sllug flow in a plane duct when the thermal field is fully developed and when there is a uniform wall heat flux. 

Consider fully developed flow in a plane duct in which uniform heat fluxes qw and qwi are applied at the two walls. Derive expressions for the temperature distribution in the duct and the Nusselt number. 

Water flows in a rectangular 5 mm x 10 mm duct with a mean bulk temperature of 20°C. If the duct wall is kept at a uniform temperature of 40°C and if fully developed laminar flow is assumed to exist, find the heat transfer rate per unit length of the duct. 

Consider air flow in a plane channel when there is uniform heat flux at one wall and when the other wall is adiabatic. If the inlet air temperature is known, find how the temperature of the heated wall at the exit end of the duct varies with die distance, W, between the two walls. Assume that the mean air velocity is the same in all cases and that the flow is fully developed. 

Discuss the modifications required to the computer program given for flow in the thermal entrance region of a plane duct to deal with the case where there is a uniform heat flux at one wall of the duct and where there is a uniform specified temperature at the other wall of the duct. 

It will be assumed here that the flow enters the duct through a shaped unheated inlet section and that the velocity and temperature are therefore uniform across the inlet plane as illustrated in Fig. 7.12. 

In separation processes and chemical reactors, flow through cylindrical ducts filled with granular materials is important. In such systems conduction, convection, and radiation all contribute to the heat flow, and thermal conduction in axial ke x and radial ke r directions may be quite different, leading to highly anisotropic thermal conductivity. For a bed of uniform spheres, the axial and radial elements are approximated by... 

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