Orders of magnitude

The order of magnitude of a real number is how many tens the number has. Thus, the order of magnitude of x is the integer part of the logiga . 

An n orders of magnitude difference between two real numbers is a difference by a factor of 10 . 

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Providing film coefficients vary by less than one order of magnitude, then Eq. (7.6) has been found to predict network area to within 10 percent of the actual minimum. ... 

Calculate the weighted network area Anetwork from Eq. (7.22). When the weighted h values i4>h) vary appreciably, say, by more than one order of magnitude, an improved estimate of Anetwork can be evaluated by linear programming. ... 

These small positive and negative errors partially cancel each other. The result is that capital cost targets predicted by the methods described in this chapter are usually within 5 percent of the final design, providing heat transfer coefficients vary by less than one order of magnitude. If heat transfer coefficients vary by more than one order of magnitude, then a more sophisticated approach can sometimes be justified. ... 

Table 5.10 gives octane number examples for some conventional refinery stocks. These are given as orders of magnitude because the properties can vary according to process severity and the specified distillation range. 

Octane numbers (RON and MON) of some conventional refinery streams (orders of magnitude). 

The safety triangle shows that there are many orders of magnitude more unsafe acts than LTIs and fatalities. A combination of unsafe acts often results in a fatality. Addressing safety in industry should begin with the base of the triangle trying to eliminate the unsafe acts. This is simple to do, in theory, since most of the unsafe acts arise from carelessness or failure to follow procedures. In practice, reducing the number of unsafe acts requires personal commitment and safety awareness. 

Permeability (k) is a rock property, while viscosity (fi) is a fluid property. A typical oil viscosity is 0.5 cP, while a typical gas viscosity is 0.01 cP, water being around 0.3 cP. For a given reservoir, gas is therefore around two orders of magnitude more mobile than oil or water. In a gas reservoir underlain by an aquifer, the gas is highly mobile compared to the water and flows readily to the producers, provided that the permeability in the reservoir is continuous. For this reason, production of gas with zero water cut is common, at least in the early stages of development when the perforations are distant from the gas-water contact. 

Using Equ. (3.1), we can now compute the optimum frequency for cracks in various depths (see Fig. 3.2). For comparison, the optimum excitation frequency for a planar wave or a sheet inducer (300 x 160 mm) is also displayed. One finds that for a planar excitation source, a much lower excitation frequency is required, which causes a reducfion in the response signal of the crack of up to an order of magnitude in case of a small circular coil. 

The geometric compensation by means of a gradiometric coil is realised by placing the SQUID exactly between the two halfs of the coil, in order to detect only the response of the sample. In both cases we could achieve a reduction of the excitation field at the location of the SQUID of up to 1000. Electronic and geometric compensation together leads to an improvement of six orders of magnitude in the dynamic range, compared to a system without excitation field compensation. 

T/cm s at 1 MHz. Further SQUID development will allow to improve these data by one order of magnitude. 

It should be noted that these results are only preliminary and have to be considered as a proof of concept. As is clear from eq. (2) the phase contrast can be improved drastically by improving the global resolution and sensitivity of the instrument. Currently, a high resolution desktop system is under construction [5] in which the resolution is much better than that of the instrument used in this work, and in which the phase contrast is expected to be stronger by one order of magnitude. 

At first we tried to explain the phenomenon on the base of the existence of the difference between the saturated vapor pressures above two menisci in dead-end capillary [12]. It results in the evaporation of a liquid from the meniscus of smaller curvature ( classical capillary imbibition) and the condensation of its vapor upon the meniscus of larger curvature originally existed due to capillary condensation. We worked out the mathematical description of both gas-vapor diffusion and evaporation-condensation processes in cone s channel. Solving the system of differential equations for evaporation-condensation processes, we ve derived the formula for the dependence of top s (or inner) liquid column growth on time. But the calculated curves for the kinetics of inner column s length are 1-2 orders of magnitude smaller than the experimental ones [12]. 

Fig. 4 illustrates the time-dependence of the length of top s water column in conical capillary of the dimensions R = 15 pm and lo =310 pm at temperature T = 22°C. Experimental data for the top s column are approximated by the formula (11). The value of A is selected under the requirement to ensure optimum correlation between experimental and theoretical data. It gives Ae =3,810 J. One can see that there is satisfactory correlation between experimental and theoretical dependencies. Moreover, the value Ae has the same order of magnitude as Hamaker constant Ah. But just Ah describes one of the main components of disjoining pressure IT [13]. It confirms the rightness of our physical arguments, described above, to explain the mechanism of two-side liquid penetration into dead-end capillaries. 

The next point of interest has to do with the question of how deep the surface region or region of appreciably unbalanced forces is. This depends primarily on the range of intermolecular forces and, except where ions are involved, the principal force between molecules is of the so-called van der Waals type (see Section VI-1). This type of force decreases with about the seventh power of the intermolecular distance and, consequently, it is only the first shell or two of nearest neighbors whose interaction with a given molecule is of importance. In other words, a molecule experiences essentially symmetrical forces once it is a few molecular diameters away from the surface, and the thickness of the surface region is of this order of magnitude (see Ref. 23, for example). (Certain aspects of this conclusion need modification and are discussed in Sections X-6C and XVII-5.)... 

When, for a one-component system, one of the two phases in equilibrium is a sufficiently dilute gas, i.e. is at a pressure well below 1 atm, one can obtain a very usefiil approximate equation from equation (A2.1.52). The molar volume of the gas is at least two orders of magnitude larger than that of the liquid or solid, and is very nearly an ideal gas. Then one can write... 

Oyy/Ais of the order of hT, as is Since a macroscopic system described by themiodynamics probably has at least about 10 molecules, the uncertainty, i.e. the typical fluctuation, of a measured thennodynamic quantity must be of the order of 10 times that quantity, orders of magnitude below the precision of any current experimental measurement. Consequently we may describe thennodynamic laws and equations as exact . 

Ionic conductors arise whenever there are mobile ions present. In electrolyte solutions, such ions are nonually fonued by the dissolution of an ionic solid. Provided the dissolution leads to the complete separation of the ionic components to fonu essentially independent anions and cations, the electrolyte is tenued strong. By contrast, weak electrolytes, such as organic carboxylic acids, are present mainly in the undissociated fonu in solution, with the total ionic concentration orders of magnitude lower than the fonual concentration of the solute. Ionic conductivity will be treated in some detail below, but we initially concentrate on the equilibrium stmcture of liquids and ionic solutions. 

For analytic theories, y is simply 1, and we have seen that for the van der Waals fluid F / F equals 2. Divergences with exponents of the order of magnitude of unity are called strong . 

The central quantity of interest in homogeneous nucleation is the nucleation rate J, which gives the number of droplets nucleated per unit volume per unit time for a given supersaturation. The free energy barrier is the dommant factor in detenuining J J depends on it exponentially. Thus, a small difference in the different model predictions for the barrier can lead to orders of magnitude differences in J. Similarly, experimental measurements of J are sensitive to the purity of the sample and to experimental conditions such as temperature. In modem field theories, J has a general fonu... 

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