Length entrance region

Entrance and Exit Effects In the entrance region of a pipe, some distance is required for the flow to adjust from upstream conditions to the fuUy developed flow pattern. This distance depends on the Reynolds number and on the flow conditions upstream. For a uniform velocity profile at the pipe entrance, the computed length in laminar flow required for the centerline velocity to reach 99 percent of its fully developed value is (Dombrowski, Foumeny, Ookawara and Riza, Can. J. Chem. Engr, 71, 472 76 [1993])... 

Simulations of water in synthetic and biological membranes are often performed by modeling the pore as an approximately cylindrical tube of infinite length (thus employing periodic boundary conditions in one direction only). Such a system contains one (curved) interface between the aqueous phase and the pore surface. If the entrance region of the channel is important, or if the pore is to be simulated in equilibrium with a bulk-like phase, a scheme like the one in Fig. 2 can be used. In such a system there are two planar interfaces (with a hole representing the channel entrance) in addition to the curved interface of interest. Periodic boundary conditions can be applied again in all three directions of space. 

Water flows at a rate of 1 kg/s through a 3-cm diameter smooth pipe. The water enters the pipe a( a temperature of 15°C and must leave the the pipe at a mean temperature of 50°C. The pipe wall is heated in such a way that the wall temperature is 14°C higher than the local mean water temperature at all points along the pipe. What length of pipe is required Ignore entrance region effects. 

Water flows at a mean velocity of 5 m/s through a tube in a condenser that has a diameter of 3 cm and a length of 1.2 m. Because steam is condensing on the tube, its wall surface temperature is constant and equal to 90°C. If the water enters the tube at a temperature of 3S°C, what will be its mean temperature at the tube exit Assume that the tube is smooth and ignore entrance region effects. 

Th regioii of flow over which the thermal boundary layer develops and re.iches (he tube center i.s called the thermal entrance region, and the length of this region is called the thermal entry length L,. Flow in the thermal... 

Precise correlations for the friction and heat transfer coefficients for the entrance regions are available in Ihe literature. However, the tubes used in practice in forced convection are usually -several times the length of either entrance region, and thus the flow through the tubes is often assumed to be fully developed for the entire length of the tube. This simplistic approach gives reasonable results for the rate of heat transfer for long tubes and conservative results for short ones. 

For a circular tube of length L subjected to constant surface temperature, the average Nussell number for the thermal entrance region can be determined from (Ed vards et al., 1979)... 

The above boundary conditions are however insufficient for the unique determination of the two-dimensional flow field U(x,z), V(x,z), an additional information about p(x,z) is necessary. For the original problem with smooth walls, the problem closure was suggested by the transition to a steady-state flow solution at a certain sufficient distance Lx called the entrance region length [566, 605], To this end, the pressure gradient can be taken constant = -/ , and the steady-state solution gives the relation/ = A = ... 

Problem (3.29) with the formulated boundary conditions possesses a unique solution over the length of the entrance flow region 0 < x < Lx, but the value Lx, the unknown length of the entrance region, is to be chosen at a distance, where no further transformation of the flow field takes place. Lx can be easily adjusted in the course of the numerical performance. 

Figure 3.11 Entrance region length for a linear EPR near walls (see the insert) as a function of the EPR density A for different Reynolds numbers.

Transition length for laminar and turbulent flow. The length of the entrance region of the tube necessary for the boundary layer to reach the center of the tube and for fully developed flow to be established is called the transition length. Since the velocity varies not only with length of tube but with radial distance from the center of the tube, flow in the entrance region is two dimensional. 

Figure 3.33). There is a pressure drop down the channel or tube which means that the first part of the channel may be gel-polarized while the exit region may not be. (Usually, a restrictor is placed on the exit retentate stream to keep the exit pressure high so as to maximize flux throughout the channel length.) For laminar flow, the entrance region of the channel may not be gel-polarized either, because the boundary layer is not well developed at this point.

Prakash and Liu [266] have numerically analyzed laminar flow and heat transfer in the entrance region of an internally finned circular duct. In this study, the fully developed / Re is compared with those reported by Hu and Chang [265] and Masliyah and Nandakumar [267]. The incremental pressure drop K(°°) and hydrodynamic entrance length L+hy together with /Re are given in Table 5.48, in which the term n refers to the number of fins, while / denotes the relative length of the fins. 

---