Space time definition

Most of these questions are not important beyond the problem of comparing results from different laboratories, since many of the space time definitions vary from each other by just a calculable constant. However, in sane cases other issues may be involved, such as the assumed volume of vaporized liquid feed at inlet or at STP conditions, the definition of catalyst density, and so on, making comparison between results from various laboratories difficult and raising the potential for distorting the calculated activation energies and frequency factors of a reaction. There is no generally accepted convention for the definition of space time and the best we can expect is to know exactly how it has been defined in each case. There is an urgent need for this information and it must be demanded of authors by referees and editors if we are to remove some of the fuzz from kinetic data in the literature. 

However, these investigations also point out that we need a proper definition of space-time yields for micro reactors. This refers to defining what essentially the reaction volume of a micro reactor is. Here, different definitions lead to varying values of the respective space-time yields. Following another definition of this parameter for ethylene oxide formation, a value of only 0.13 t h m is obtained -still within the industrial window [159, 162, 163]. 

GP 2] [R 2] The definition of space-time yield in a micro reactor depends on the definition of the reactor volume . Owing to the large amoimt of construction material relative to the reaction channels and the neglect of some reactor parts ( abstraction to the real reaction zone ), several more or less useful definitions can be made. In the following, two definitions concerning the time yield divided by the pure reaction channel volume and the platelet volume were used. 

Following the first definition, a space-time yield of 0.781 h m using a OAOR-modified silver is obtained, which exceeds the industrial performance considerably (0.13-0.26 t h m ) [4]. Following the second definition and hence orienting more on outer than on inner dimensions, a space-time yield of 0.13 t h m is obtained, still within the industrial window. 

Eq. (2.1) is immediately obvious with the definition of space time x = V/Q. 

Vocabulary of Terms Used in Reactor Design. There are several terms that will be used extensively throughout the remainder of this text that deserve definition or comment. The concepts involved include steady-state and transient operation, heterogeneous and homogeneous reaction systems, adiabatic and isothermal operation, mean residence time, contacting and holding time, and space time and space velocity. Each of these concepts will be discussed in turn. 

Like the definition of the space time, the definition of the space velocity involves the volumetric flow rate of the reactant stream measured at some reference condition. A space velocity of 10 hr-1 implies that every hour, 10 reactor volumes of feed can be processed. 

From the definition of the space time and the inlet volumetric flow rate,... 

The product selectivity is strongly affected by the flow rate, reactor geometry (i.e., internal diameter and "heated zone) and weight of catalyst. On this account, the space time yield to HCHO - or HCHO productivity -( HCHO Scat h ) appears to be the more definite parameter to evaluate the reactivity of the partial oxi tion catalysts. 

The absence of an E(3) field does not affect Lorentz symmetry, because in free space, the field equations of both 0(3) electrodynamics are Lorentz-invariant, so their solutions are also Lorentz-invariant. This conclusion follows from the Jacobi identity (30), which is an identity for all group symmetries. The right-hand side is zero, and so the left-hand side is zero and invariant under the general Lorentz transformation [6], consisting of boosts, rotations, and space-time translations. It follows that the B<3) field in free space Lorentz-invariant, and also that the definition (38) is invariant. The E(3) field is zero and is also invariant thus, B(3) is the same for all observers and E(3) is zero for all observers. 

The situation is more complicated if expansion or contraction of a volume element does occur and the volumetric flowrate is not constant throughout the reactor. The ratio VJv, where v is the volume flow into the reactor, no longer gives the true residence time or contact time. However, the ratio VJv may still be quoted but is called the space time and its reciprocal v/V, the space velocity. The space velocity is not in fact a velocity at all it has dimensions of (time) 1 and is therefore really a reactor volume displacement frequency. When a space velocity is quoted in the literature, its definition needs to be examined carefully sometimes a ratio Vi/ V, is used, where V/ is a liquid volume rate of flow of a reactant which is metered as a liquid but subsequently vaporised before feeding to the reactor. 

The precise definition of a vacuum in a curved space-time is still subject to some ambiguities. We refer the interested reader to Fulling (1979) Fulling(1989) Birrell Davis (1982) Wald(1994) and to the discussion in Chung, Notari Riotto (2003) and references therein. 

The space-charge definition within the dielectric (PVC) membrane changes with time. Such changes are a direct consequence of membrane exposure to sample solution and are controlled by membrane formulation. The two factors of consequence to pragmatic design (not accounted for by equations 1-5) are the transient response of the membrane and signal drift due to membrane dielectric constant and interfacial potential changes. 

The symmetry between curvature and matter is the most important result of Einstein s gravitational field equations. Both of these tensors vanish in empty euclidean space and the symmetry implies that whereas the presence of matter causes space to curve, curvature of space generates matter. This reciprocity has the important consequence that, because the stress tensor never vanishes in the real world, a non-vanishing curvature tensor must exist everywhere. The simplifying assumption of effective euclidean space-time therefore is a delusion and the simplification it effects is outweighed by the contradiction with reality. Flat space, by definition, is void. 

The apparent residence time is the ratio of the downcomer volume to the clear liquid flow in the downcomer, The downcomer volume is the tray spacing times the average downcomer cross-section area. The true residence time is the ratio of froth volume in the downcomer to the frothy liquid flow in the downcomer. The true residence time can alternatively be expressed as the ratio of the clear liquid volume in the downcomer to the clear liquid flow. The definition adopted here is that of the apparent downcomer residence time. The author found this definition to be better and to be consistent with riublished criteria for downcomer residence times (1]. 

A catalytic process is commercially viable if the catalyst transformation is achieved within definite, practical limits of space and time. To quantify this aspect, one can determine the so-called space-time yield. This measure of activity is simply the amount of product obtained per unit time and per unit reaction space (where reaction space is usually the reactor volume). Weisz (79) pointed out that in industry the useful space-time yield is rarely less than 10"6 g/mol of reactant per cubic centimeter of volume of reactor space per second. This has been called the Weisz window on reality. Figure 9 (79) shows the Weisz window and other windows of chemical activity that apply to biochemistry and petroleum geochemistry (79). 

Unambiguous definitions of activity, selectivity, and conversion are needed to interpret the results correctly. It is also important to know the quality of the results. What is the reproducibility and the statistical validity of the conclusions Were heat and mass transfer disguises eliminated What tests were done to confirm ideal flow pattern, isobaricity, and isothermality Often it is necessary to know whether catalyst performance is adequate. For that reason the space time yield and catalyst life characterization need to be addressed. 

Chemical definition of the vacuum is a region of three-dimensional space devoid of matter. Chemical matter is the source of gravitational and electromagnetic fields and the removal of matter from some region of space does not prevent these fields from permeating the vacuum so created. There is only one way to obtain a field-free vacuum and that is by complete annihilation of chemical matter. It is not obvious what effect such annihilation would have on the vacuum. A partial answer to this question is provided by the theory of general relativity that outlines a reciprocal relationship between matter and the geometry of space-time. The implication is clear the physi-... 

The definition integrals of Eqns. (1.4.1) and (1.4.2), tell us that any arbitrary space-time dependent functions can be thought of as an ensemble of large numbers of waves with different combinations of wave numbers and circular frequencies. This assembly could be a result of countably infinite numbers of waves or it could represent a continuous spectrum. There is a definitive relationship between the wave numbers and the circular frequencies, as identified before, as the dispersion relation. 

The moment of concrescence may occupy the same position with respect to time that a magnetic pole of the earth occupies relative to the geomagnetic field of the planet. Like the poles, which are physically characterized by climatological extreme, the temporal pole would mark a temporal extreme, the most extreme moment of density of the ingression of novelty, but like the poles, it would not be apparently different from its space-time environment. Such an understanding of time would mean no definitive concrescence could take place. 

The throughput of the plant is definitely limited by the capacity of heat radiation of the commercial fire tubes. The prototype reactor allowed a space-time yield of 1.3 t/m fluid bed per hour. The heating value of the pyrolysis gas produced was sufficient to balance the heat demand of the process. 

H20/n heptane=0.i9-0.95 mol/mol. The definition of the equivalent space time T ,has been described elsewhere [13]. 

Fourier and Floquet space. The definitions of the different block-elements are given in a for the Fourier representation and in b for the Floquet representation. The Hamiltonians retain the same general form, but the time-dependent elements in a become time-independent in b with the addition of the number operator elements on the diagonal... 

To convert Eq. 7.4.2 to dimensionless form, we use the definition of the dimensionless space time, Eq. 7.1.3, and divide both sides by the reference thermal energy. 

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