Dimensionless numbers, definitions

Dichotomous search, 407 Dimensionless numbers, definitions of 468-469 DIPPR, 117 Direct costs, 210 in capital investmenf 167 Direct production costs, 197-204,210 Discounf definition of, 226 Discount factors definition of 223/i 236-237, 303 tables of 219, 234-235, 240-246... 

Dimensionless number Definition Equation Scaling Value ... 

The value of tire heat transfer coefficient of die gas is dependent on die rate of flow of the gas, and on whether the gas is in streamline or turbulent flow. This factor depends on the flow rate of tire gas and on physical properties of the gas, namely the density and viscosity. In the application of models of chemical reactors in which gas-solid reactions are caiTied out, it is useful to define a dimensionless number criterion which can be used to determine the state of flow of the gas no matter what the physical dimensions of the reactor and its solid content. Such a criterion which is used is the Reynolds number of the gas. For example, the characteristic length in tire definition of this number when a gas is flowing along a mbe is the diameter of the tube. The value of the Reynolds number when the gas is in streamline, or linear flow, is less than about 2000, and above this number the gas is in mrbulent flow. For the flow... 

The above shows how the dimensionless numbers are used to provide the most accurate solution. Collecting these definitions together,... 

There are three dimensionless numbers used in these equations, and their definitions are ... 

The mass-to-charge ratio, m/z, (read m over z ) [29] is dimensionless by definition, because it calculates from the dimensionless mass number, m, of a given ion, and the number of its elementary charges, z. The number of elementary charges is often, but by far not necessarily, equal to one. As long as only singly charged ions are observed (z = 1) the m/z scale directly reflects the m scale. How-... 

Just as process translation or scaling-up is facilitated by defining similarity in terms of dimensionless ratios of measurements, forces, or velocities, the technique of dimensional analysis per se permits the definition of appropriate composite dimensionless numbers whose numeric values are process-specific. Dimensionless quantities can be pure numbers, ratios, or multiplicative combinations of variables with no net units. 

The definition of dimensionless numbers has been given in a simple way by Noble and Terry (2004) ... 

Definition 3.1. The recycle number of a material recycle loop in an integrated process is a process-wide dimensionless number, expressed as the ratio of the (steady-state) flow rates of the recycle stream and the process throughput, as captured by the (total) flow rate of the process feed stream(s) ... 

In Table 7.7 the most important dimensionless numbers introduced in this chapter are summarized (together with negligibility criteria). With these numbers, it is now possible to give generalized definitions for Aris numbers, for which two situations may be distinguished ... 

The basis of the solution of complex heat conduction problems, which go beyond the simple case of steady-state, one-dimensional conduction first mentioned in section 1.1.2, is the differential equation for the temperature field in a quiescent medium. It is known as the equation of conduction of heat or the heat conduction equation. In the following section we will explain how it is derived taking into account the temperature dependence of the material properties and the influence of heat sources. The assumption of constant material properties leads to linear partial differential equations, which will be obtained for different geometries. After an extensive discussion of the boundary conditions, which have to be set and fulfilled in order to solve the heat conduction equation, we will investigate the possibilities for solving the equation with material properties that change with temperature. In the last section we will turn our attention to dimensional analysis or similarity theory, which leads to the definition of the dimensionless numbers relevant for heat conduction. 

Heat and mass transfer coefficients are usually reported as correlations in terms of dimensionless numbers. The exact definition of these dimensionless numbers implies a specific physical system. These numbers are expressed in terms of the characteristic scales. 

For the single-reaction cases, we performed dimensional analysis and found a dimensionless number, the Thiele modulus, which measures the rate of production divided by the rate of diffusion of some component. A complete analysis of the first-order reaction in a sphere suggested a general approach to calculate the production rate in a pellet in terms of the rate evaluated at the pellet exterior surface conditions. This motivated the definition of the pellet effectiveness factor, which is a function of the Thiele modulus. 

If pressure and gravitational forces were not combined via the definition of dynamic pressure, then an additional dimensionless number appears explicitly... 

Dimensionless number Word definition Equation Range So what ... 

Here it is worth noting a few elementary notions of thermochemistry. The thermodynamic quantities depend on the specific state of the system. The thermodynamic state of a given system, i.e., usually a mixture of several components, is defined by the activity of each component. Let us recall the definition of the activity, which is a dimensionless number, in the case of ideal systems ... 

Dimensionless numbers are scale independent and contain a characteristic length term (L) that adjusts their magnitude to match the scale of the model. The velocity, v (m/sec), in these definitions is the superficial velocity of the fiuid. 

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