Nondimensional groups,

Reynolds Number. The Reynolds number, Ke, is named after Osborne Reynolds, who studied the flow of fluids, and in particular the transition from laminar to turbulent flow conditions. This transition was found to depend on flow velocity, viscosity, density, tube diameter, and tube length. Using a nondimensional group, defined as p NDJp, the transition from laminar to turbulent flow for any internal flow takes place at a value of approximately 2100. Hence, the dimensionless Reynolds number is commonly used to describe whether a flow is laminar or turbulent. Thus... 

Commonly used nondimensional groups. The commonly used nondimen-sional groups in boiling heat transfer and two-phase flow are summarized as follows. Some are used more frequently than others, but all represent the boiling mechanisms in some fashion. 

This nondimensional group describes the spheroidal state of film boiling. 

Applying these definitions to the governing equations and rearranging them so the parameters form nondimensional groups, the continuity equations become... 

It turns out to be a surprisingly difficult task to determine accurately the thermal conductivity of polyatomic gases from the viscosity. Many of the approaches are motivated by the ideas of Eucken. The so-called Eucken factor is a nondimensional group determined by dividing the kinetic-theory expression for a monatomic gas by that for viscosity, yielding... 

The nondimensional groups, Reynolds number, Prandtl number, and Mach number take... 

The nondimensional group on the right-hand side, defined as... 

Fig. 4.4 Velocity profiles in the annular gap between a rod moving with velocity U to the right and a stationary guide. The figure on the left is for a relatively thin gap, rj/Ar = 10, and the solution on the right is for a relatively wide gap, rj/Ar = 0.05. The solutions are both parameterized by the nondimensional group, P = These solutions were...

The nondimensional group of parameters Pr = cp xfX, which is usually called the Prandtl number, characterizes the particular problem. 

The Hagen-Poiseuille solution at steady state provides a specific relationship among the factors in the nondimensional group of constants involving the pressure gradient. From Eq. 4.59,... 

Using the nondimensional variables, transform the governing equation to a nondimensional form. What characteristic nondimensional groups emerge ... 

Transform the governing equations into a nondimensional form, collecting and identifying appropriate nondimensional groups. 

The following nondimensional groups (Reynolds, Schmidt, and Damkohler numbers) should be relevant ... 

In these relationships, Ya, is the mass fraction that enters the channel, Da is the diffusion coefficient of species A relative to the bulk fluid, and IV is the mean molecular weight. Based on the scaling and the nondimensional groups, discuss the circumstances under which the axial diffusion may be neglected. Show also how the nondimensional axial coordinate z may be written in terms of the Reynolds and Schmidt numbers. 

The nondimensional groups that appear in these equations are the Reynolds, Prandtl, and Schmidt numbers,... 

The distinguishing characteristics of the stagnation-flow subcases depend on the domain and on the rotation. The characteristic scales are different for the subcases, but the equations themselves are the same. The boundary conditions also differ among the subcases. Table 6.1 shows the applicable scales and nondimensional groups that apply to each of the four subcases. 

Assume reference values for the transport properties at the temperature of the inner tube Tj. As appropriate, base the nondimensional groups on reference properties at the inner tube. 

D8 - E8 Cell E8 contains the value of the nondimensional group /Re, which is essentially the desired general solution to this problem. For every value of the aspect ratio a, entered in cell Al, there will be a corresponding value of /Re. Given a value of a, the value of /Re in cell E8 will be correct when the pressure-gradient parameter in cell B8 is chosen such that mean velocity in cell E9 is 1. Referring to Eq. 4.87, we have... 

For nodes with Ax = Ay and no heat generation, the form of Eq. (3-32) has been listed as the second equation in segments of Table 3-2. The nondimensional group... 

Here ae is the effective thermal dififusivity. The nondimensional group y is called the Arrhenius group, and represents a nondimensional activation energy for the chemical reaction. 

Groups of symbols may be put together, either by theory or experiment, that have no net units. Such collections of variables or parameters are called dimension less or nondimensional groups. One example is the Reynolds number (group) arising in fluid mechanics. 

Accurate and reliable surface heat transfer and flow friction characteristics are a key input to the exchanger heat transfer and pressure drop analyses or to the rating and sizing problems (see Fig. 17.36). After presenting the associated nondimensional groups, we will present important experimental methods, analytical solutions, and empirical correlations for some important exchanger geometries. 

Our main interest is to know how various LJICF characteristics such as jet trajectory and droplet size depend on basic fluid and geometric variables. As is the norm in many physical problems, it is useful to express the dependence in the form of nondimensional groups. 

Although this list of variables is not inclusive by any means, it is adequate to help understanding the basic physics of the problem as we will show in the subsequent sections. For an angled injection, one could add another parameter to include the effect of the tilt of the nozzle. Using the Buckingham n theorem, we can form four nondimensional groups of parameters out of the mentioned seven parameters. So, a particular characteristic of the LJICF (such as the jet trajectory or droplet size distribution) can be written in the form... 

Since we are allowed to multiply any two of the above nondimensional groups, we can reorder the above relation and write... 

The scale-up problems arise from the fact that all STR gas-liquid mass transfer correlations are empirical. They are, for the most part, unable to account for hydrodynamic or liquid property changes with scale and time. Extensive attempts have been made in using nondimensional groups, especially toward solving gas-liquid processes involving non-Newtonian liquids. These correlations tend to be more complicated and require numerous static, but only few dynamic, inputs. One of the simplest correlations is presented by Ogut and Hatch (1988), which involves four dimensionless groups and requires six inputs. One of the more complicated forms. 

Consider a fluidic system with surface tension effects. The characteristic scales for length and velocity are L and U. The physical parameters are density p, viscosity v, gravity g, and surface tension y. By using the Buckingham n theorem, one can obtain three independent nondimensional groups from these six variables. An option for a set of three independent nondimensional numbers is... 

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