Flow on a vertical wall

In his early paper, Yih (Yl) carried out numerical calculations for stability in the case of flow on a vertical wall, from which it appears that... 

Experimental determination of wavelengths near point of onset of rippling on films of various liquids flowing on a vertical wall. 

Heat transfer in laminar flow on a vertical wall... 

Fig. 3.45 Velocities in free flow on a vertical wall, according to [3.47]...

Helden W, Geld C, Boot P (1995) Forces on bubbles growing and detaching in flow along a vertical wall. Int J Heat Mass Transfer 38 2075-2088... 

M. Turbulent Mass Transfer near a Liquid-Fluid Interface Based on the Turbulent Diffusivity Concept Turbulent Flow of a Liquid Film on a Vertical Wall... 

The stability problem in the presence of a bounding gas stream (cocurrent and countercurrent) has been considered for a number of special cases (F3, G6, M10, Zl). The countercurrent solutions tend to confirm Benjamin s result that the flow is always inherently unstable on a vertical wall. 

The research at MIT has been done in the cold-wall vertical tube reactor shown in Figure 14. The wafer is aligned almost parallel to the flow on a vertical silicon carbide-coated susceptor. The wafer is heated by optical radiation from high-intensity lamps to a temperature of 775°C. Silane was introduced... 

Stuhltrager. E., Naridomi, Y., Miyara. A., and Uehara, H.. "Flow Dynamics and Heat Transfer of a Condensate Film on a Vertical Wall. I. Numerical Analysis and Flow Dynamics/ Int. J. Heat Mass Transfer, Vol. 36. pp. 1677-1686, 1993. 

We will now deal with free flow on a vertical, flat wall whose temperature d0 is constant and larger than the temperature in the semi-infinite space. The coordinate origin lies, in accordance with Fig. 3.43, on the lower edge, the coordinate x runs along the wall, with y normal to it. Steady flow will be presumed. All material properties are constant. The density will only be assumed as temperature dependent in the buoyancy term, responsible for the free flow, in the momentum equation, in all other terms it is assumed to be constant. These assumptions from Oberbeck (1879) and Boussinesq (1903), [3.45], [3.46] are also known as the Boussinesq approximation although it would be more correct to speak of the Oberbeck-Boussinesq approximation. It takes into account that the locally variable density is a prerequisite for free flow. The momentum equation (3.294) in... 

As Fig. 4.6 shows, saturated steam at a temperature s is condensing on a vertical wall whose temperature 0 is constant and lower than the saturation temperature. A continuous condensate film develops which flows downwards under the influence of gravity, and has a thickness 5 x) that constantly increases. The velocity profile w(y), with w for wx, is obtained from a force balance. Under the assumption of steady flow, the force exerted by the shear stress are in equilibrium with the force of gravity, corresponding to the sketch on the right hand side of Fig. 4.6... 

Example 4.1 Saturated steam at a pressure 9.8 10 3 MPa, condenses on a vertical wall. The wall temperature is 5 K below the saturation temperature. Calculate the following quantities at a distance of H = 0.08 m from the upper edge of the wall the film thickness 5(H), the mean velocity wm of the downward flowing condensate, its mass flow rate M/b per m plate width, the local and the mean heat transfer coefficients. 

Suppose that on a vertical wall whose temperature is constant and equal to Ts, stagnant dry saturated vapor is condensing. Let us consider the steady-state problem under the assumption that we have laminar waveless flow in the condensate film. According to [200], we make the following assumptions the film motion is determined by gravity and viscosity forces the heat transfer is only across the film due to heat conduction there is no dynamic interaction between the liquid and vapor phases the temperature on the outer surface of the condensate film is constant and equal to the saturation temperature Tg the physical parameters of the condensate are independent of temperature and the vapor density is small compared with the condensate density the surface tension on the free surface of the film does not affect the flow. 

For specificity let us first consider the soluble wall problem sketched in Fig. 4.2.3. Now in a channel (or pipe) flow u = 0, and near the surface the streamwise velocity component is given by u (lylh). This behavior of the velocity profile is the same as for a fully developed thin liquid film on a vertical wall, falling under gravity with a free surface at atmospheric pressure. The velocity profile is parabolic with the fall velocity and has a maximum at the free surface equal to... 

Substances intermediate between the two are materials which can permanently resist a small shear force but cannot permanently resist a large one. For example, if we put a blob of any obvious liquid on a vertical wall, gravity will make it run down the wall. If we attach a piece of steel or diamond securely to a wall, it will remain there, no matter how long we wait. If we attach some peanut butter to a wall, it will probably stay but if we increase the shear stress on the peanut butter by spreading it with a knife, the peanut butter will flow just as a fluid would. We cannot, of course, spread steel with a knife. 

In a variety of apparatus and chemical reactors, liquid films or rivulets are flowing around pipes, walls, or packing elements (packed colunuis, cooling towers, liquid film coolers). Things are very easy in the case of a laminar film ruiuiing on a vertical wall, see Fig. 3.1-10. 

Film-condensation coefficients for vertical surfaces. Film-type condensation on a vertical wall or tube can be analyzed analytically by assuming laminar flow of the condensate film down the wall. The film thickness is zero at the top of the wall or tube and increases in thickness as it flows downward because of condensation. Nusselt (HI, Wl) assumed that the heat transfer from the condensing vapor at 7, K, through this liquid film, and to the wall at 7 K was by conduction. Equating this heat transfer by conduction to that from condensation of the vapor, a final expression can be obtained for the average heat-transfer coefficient over the whole surface. 

Dukler and Taitel, 1991b). The simplest configuration of annular flow is a vertical falling film with concurrent downward flow. Given information on interfacial shear and considering the film to be smooth, the film thickness and heat transfer coefficient between the wall and the liquid film can be predicted from the following basic equations (Dukler, 1960) ... 

Sensor A is mounted onto an orifice plate inserted in the main supply pipeline for liquid urea. The orifice has a smaller hole diameter than the pipeline, which induces turbulence in the flowing urea downstream of the orifice. The vibrations produced by this turbulence will be detected by sensor A. Sensors B, C and D are mounted on the vertical wall on the granulator, about 30 cm above the perforated bottom plate to detect vibrations produced by the granules when they impact the reactor wall. Thus sensors B, C and D are used to monitor the process conditions inside the granulator, while sensor A is used to monitor the liquid supply of urea. The sensors used in this trial are all high temperature accelerometers. 

The other limiting solution is that in which the flow essentially consists of boundary layers on each wall of the duct, these boundary layers being so thin compared to W that there is no interaction between the flows in the two boundary layers, i.e., the boundary layer on each wall of the duct behaves as a boundary layer on a vertical plate in a large environment. Now, for free convective boundary layer flow over a vertical plate of height l, it was shown earlier in this chapter that ... 

An approximate model of the flow in a vertical porous medium-filled enclosure assumes that the flow consists of boundary layers on the hot and cold w alls with a stagnant layer between the two boundary layers, this layer being at a temperature that is the average of the hot and cold wall temperatures. Use this model to find an expression for the heat transfer rate across the enclosure and discuss the conditions under which this model is likely to be applicable. 

T3q3ical patterns of rivulet spreading on a vertical copper surface which was observed in [22] are shown in Fig. 13 for different overheating of the wall relative to saturation temperature AT=Tw-Ts. The Reynolds number was defined as Re=(Q/dv), where Q is the volumetric flow rate, and d is the width of the open cross-section of the nozzle. One can see that the higher the wall superheat, the narrower the spreading zone. At overheating of 0.39K the rivulet starts to contract on hot wall. The... 

When a vapour condenses on a vertical or inclined surface a liquid film develops, which flows downwards under the influence of gravity. When the vapour velocity is low and the liquid film is very thin, a laminar flow is created in the condensate. The heat will mainly be transferred by conduction from the surface of the condensate to the wall. The heat transferred by convection in the liquid film is negligibly small. 

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