Tube flow stream function

The Poisson process represents only one possible way of assigning joint distribution functions to the increments of counting functions however, in many problems, one can argue that the Poisson process is the most reasonable choice that can be made. For example, let us consider the stream of electrons flowing from cathode to plate in a vacuum tube, and let us further assume that the plate current is low enough so that the electrons do not interact with one another in the... 

Fig. 32. Streamlines in abrupt contraction flow computed at a midtravel distance of the piston at x = — 40 mm (the origin of the abscisse is taken at the orifice entrance). Due to axial symmetry, only half of the flow tube is shown. The dimensionless stream function

Consider one-dimensional laminar viscous flow through a straight mbe with circular cross section [i.e., u (r)] and obtain an expression for the axisym-metric stream function at the tube wall, where r = R. The average velocity through the tube is (v ). 

If one end of a vector is pinned on the symmetry axis at r = 0 and the other end lies somewhere on the lateral surface of the tube at r = R, then this vector maps out a circular cross section of n when it is rotated by Itt radians around the symmetry axis. The volumetric flow rate through this circle is jzR v ), where (u ) is the average fluid velocity through the tube. The axisymmetric stream function at r = is defined by... 

Calculate the stream function for axisymmetric fully developed creeping viscous flow of an incompressible Newtonian fluid in the annular region between two concentric tubes. This problem is analogous to axial flow on the shell side of a double-pipe heat exchanger. It is not necessary to solve algebraically for all the integration constants. However, you must include all the boundary conditions that allow one to determine a unique solution for i/f. Express your answer for the stream function in terms of ... 

S.2.3 Path Lines. Since the stream function is defined only for 2D flows, an alternative method is needed to visualize 3D flows in the same manner. Path lines can be nsed for this pnrpose. Path lines follow the trajectories that wonld be followed by massless particles seeded at any location within the domain. These particles move with the flow field and leave behind tracks in one form or another that allow the flow field to be visnalized. In Figure 5-17, path lines are used to illustrate the flow through a static mixer. Path lines can be drawn as simple lines or as tubes, ribbons, or a series of dots. They can usually be colored by problem variables, such as temperature. When colored by time, they give information on residence time if inflow and outflow of fluid are involved. 

Good heat transfer on the outside of the reactor tube is essential but not sufficient because the heat transfer is limited at low flow rates at the inside film coefficient in the reacting stream. The same holds between catalyst particles and the streaming fluid, as in the case between the fluid and inside tube wall. This is why these reactors frequently exhibit ignition-extinction phenomena and non-reproducibility of results. Laboratory research workers untrained in the field of reactor thermal stability usually observe that the rate is not a continuous function of the temperature, as the Arrhenius relationship predicts, but that a definite minimum temperature is required to start the reaction. This is not a property of the reaction but a characteristic of the given system consisting of a reaction and a particular reactor. 

This factor corrects for the main bypass stream, the flow between the tube bundle and the shell wall, and is a function of the shell to bundle clearance, and whether sealing strips are used ... 

The mobile phase reservoir is made of an inert material, usually glass. There is usually a cap on the reservoir that is vented to allow air to enter as the fluid level drops. The purpose of the cap is to prevent particulate matter from falling into the reservoir. It is very important to prevent particulates from entering the flow stream. The tip of the tube immersed in the reservoir is fitted with a coarse metal filter. It functions as a filter in the event that particulates do find their way into the reservoir. It also serves as a sinker to keep the tip well under the surface of the liquid. In addition, in specially designed mobile phase reservoirs, this sinker/filter is placed into a well on the bottom of the reservoir so that it is completely immersed in solvent, even when the reservoir is running low. This avoids drawing air into the line under those conditions. These details are shown in Figure 13.3. 

Propelling unit(s), aimed to move the samples. They are generally peristaltic pumps, although their function can also be served by piston pumps and the pressure exerted by a gas or gravitational force. They are intended to set and keep several streams in motion— the flow-rate of such streams should be regulatable and maintained as constant as possible (normally accomplished by using flexible tubes that withstand the mechanical pressure to which they are subjected). 

Fig. 12.14 Cell in which a 30% solution of HC1 in H2O evaporates into a column of air. Mole fractions at the liquid-vapor interface are taken to be their equilibrium values. A stream of dry air flows past the open top of the cylinder, dropping the mole fractions of HC1 and H2O to zero. Calculated mole fractions as a function of height along the tube are shown.

More recently, Brauer (B18) carried out a detailed analysis of the flow of smooth films and gas streams inside vertical tubes this work was subsequently extended by Feind (F2). In this treatment, all the possible cases of film/gas flow (countercurrent, upward cocurrent, and downward cocurrent) are dealt with in a unified manner by plotting the calculated results in the form of / as a function of NRe, . Here / is the absolute value of the dimensionless pressure drop in the gas stream ... 

Strain Distribution Function in Poiseuille Flow (a) Derive the SDF F(y) for fully developed isothermal laminar flow of a Newtonian fluid in a tube, (b) Calculate the mean strain, (c) If the length of the tube is 1 m and its radius 0.01 m, what fraction of the exiting stream experiences a total strain of less than 100 ... 

The time constant of any process is the result of its capacitance and resistance. Usually, the heat exchanger outlet temperature is the controlled variable, and the flow rate of the heat transfer fluid is the manipulated variable. The time constant of an exchanger is a function of the mass and the specific heat of the tube material, the mass flow, and the specific heat of the process and utility streams and their heat transfer coefficients. 

Accuracy is improved if the flowing gas stream is directed at the probe by a venturi nozzle, or by placing the sensor in the throat of the venturi (Figure 3.74). The venturi ensures a smooth velocity profile and eliminates boundary layer effects while concentrating the flow onto the sensor. These units are available for both liquid and gas services. Other designs are of the insertion probe type. Their flow ranges are a function only of the size of the pipe into which they are inserted, and their performance is a function of the correctness of the insertion depth (as are all Pitot tubes). 

The unknown surface and the corresponding swell ratio can be determined by considering only the peripheral stream tube involving the wall and the free surface. Fig 16 illustrates the peripheral stream tube and its corresponding stream band in the mapped computational domain. In relation to the particular shapes of the free smface and stream lines in the physical flow domain, we use, for approximating the mapping streamline function / in the peripheral stream tube, analytical forms derived from the equation proposed by Batchelor and Horsfall [66] and already used in previous papers [54,58] ... 

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