Space time defined

First of all, the space time defined in Eq. (11-5) or (11-6) depends on the volume of the reactor and the total volumetric feed rate. Thus, for a given reactor volume, space time is inversely proportional to the total feed rate. Itoh et al. [1993] studied the use of a dense yttria-stabilized zirconia membrane reactor for thermal decomposition of carbon dioxide. The reactor temperature was not kept constant everywhere in the reactor but varying with the reactor length instead. The resulting temperature profile is parabolic with the maximum temperature at the midpoint of the reactor length. This nonisothermal... 

This is not the end of the story. Space time is usually defined at room temperature and pressure, whereas the entrance condition to the reactor may be at a different temperature and a higher or lower pressure. Moreover, these conditions will change in subsequent runs as data is collected at several temperatures and pressures. The effect of using space time defined at room temperature and pressure for all such runs, without taking into account actual entry and reaction conditions, has an effect on the rate parameters of the fitted rate equations. It may even lead to differences of opinion as to the correct mechanistic rate expression that applies to a given reaction, if other workers define the space time differently. The same problems also lead to misconceptions as to the thermodynamic and kinetic values appropriate to the reaction, resulting in fundamental misunderstanding of the reaction itself. More obviously, this procedure can interfere with comparability between studies. 

The equation (8.51) can be integrated analytically or numerically for simple reactions in isothermal conditions to get the reactor volume or the space-time defined as r = F. ... 

The reciprocal of space velocity, the space time, is also sometimes employed. One s first thought is to associate this with the residence time. This can be misleading, however, since the space time (defined in terms of inlet conditions) is, in general, equal to the actual residence or contact time within the reactor only if the temperature, pressure, and reaction mixture density are constant throughout the reactor. A generalized expression for the residence time in a PFR can be derived as follows, in terms of molar flow rates ... 

The concentrations of reactant and products at the outlet of a packed bed reactor can be easily calculated with the mass balances for each compound supposing ideal plug flow behavior. For irreversible first-order consecutive reactions (Eq. (11.5)), the concentrations at the reactor outlet depend on the inlet concentration, Cj g, the rate constant, and the residence time, r. For reaction systems with constant fluid density, the residence time corresponds to the space-time defined as, r = V/Vg, with V the reactor volume and Vq the volumetric inlet flow. The space time... 

An analogous situation can be found with the time constant t for a relaxation process. It is also a variable pertaining to space-time, defined as an eigen-value of the time derivation (without imaginary number as here), which matches a system time constant Xq defined as a... 

Space time and space velocity are used to determine the performance of flow reactors. Space velocity, SV, is defined as F /VC Q dial is. 

Space time, ST, is defined as the time required to process one reactor volume of feed measure at specified conditions. The relationship between space velocity SV and ST is as follows ... 

Two systems of units are in common usage in mechanics. The first, the SI system, is an absolute system based on the fundamental quantities of space, time, and mass. All other quantities, including force, are derived. In the SI system the basic unit of mass is the kilogram (kg), the basic unit of length (space) is the meter (m), and the basic unit of time is tbe second (s). The derived unit of force is the Newton (N), which is defined as the force required to accelerate a mass of 1 kg at a rate of 1 m/s-. 

There is a fundamental relationship between d-dimensional PCA and d + 1)-dimensional Ising spin models. The simplest way to make the connection is to think of the successive temporal layers of the PCA as successive hyper-planes of the next higher-dimensional spatial lattice. Because the PCA rules (at least the set of PCA rules that we will be dealing with) are (1) Markovian (i.e. the probability of a state at time t + T depends only on a set of states at time t, and (2) local, one can always define a Hamiltonian on the higher-dimensioned spatial lattice such that the thermodynamic weight of a configuration 5j,( is equal to the probability of a corresponding space-time history Si t). ... 

As we have already observed in section 3.1.4.5,CA evolution may conserve locally defined quantities. Moreover, local conservation laws often cause walls to appear that prohibit sites sitting on opposite sides of those walls from exchanging any information. Figure 8.3, which shows the space-time plots of ERCA 18, 73r, and 129/j, provides three such examples. 

It-from-bit embodies the central notion that every it - that is, every aspect of reality electrons, protons, photons, fields of force, or even the what we call space-time itself - is in the deepest sense a derivative of experimentally deduced answers to yes/no questions that is, to bits. If we allow ourselves for a moment to go back to the roots of what it is that we by convention call reality, we see that it is something that is literally defined by a particular sequence of yes/no responses elicited from either a mechanical or (our own biological) sensory apparatus in other words, reality s origin is fundamentally information-theoretic. 

Forget for a moment that you know that figure 12.11 shows the space-time pattern due to a well defined local deterministic rule, and that the underlying universe really consists of nothing but bits. Suppose you are told only that this figure represents some sort of alien physics, and that you may see as many different samples of this alien world s behavior as you wish. How are you to make any sense of what is really going on ... 

An intriguing possibility thus presents itself. If some kind of a primordial information, and not higher-level constructs such as mass, energy, spin, and so forth, is indeed the real substance out of which all stuff is made - leaving aside for the moment, the question of form of that information - is it not natural to suppose that a discrete space-time structure, our heretofore pre-defined and static dynamical mediator, is itself built out of the same substance i.e. to suppose that space-time is not just a backdrop for information processing, there only to define what is local and what is not and where to and where from information is allowed to flow, but is itself a construct of primordial information This supposition is not entirely without precedent. 

In general, Greek indices will be used to denote the components (0,1,2,3) of a space-time tensor, whereas Latin indices will be used to denote spatial components (1,2,3). The raising and lowering of indices is defined by... 

Actually due to the coincidence of the space time point % in f and the operator iu(x) as given by (10-13) is ill-defined. This difficulty is remedied by defining the current in terms of the limit e— 0 applied to the operator... 

In a quantum mechanical framework, Postulate 1 remains as stated. It implies that there exists a well-defined connection and correspondence between the labels attributed to the space-time points by each observer, between the state vectors each observer attributes to a given physical system, and between observables of the system. Postulate 2 is usually formulated in terms of transition probabilities, and requires that the transition probability be independent of the frame of reference. It should be stated explicitly at this point that we shall formulate the notion of invariance in terms of the concept of bodily identity, wherein a single physical system is viewed by two observers who, in general, will have different relations to the system. 

The terms space time and space velocity are antiques of petroleum refining, but have some utility in this example. The space time is defined as F/2, , which is what t would be if the fluid remained at its inlet density. The space time in a tubular reactor with constant cross section is [L/m, ]. The space velocity is the inverse of the space time. The mean residence time, F, is VpjiQp) where p is the average density and pQ is a constant (because the mass flow is constant) that can be evaluated at any point in the reactor. The mean residence time ranges from the space time to two-thirds the space time in a gas-phase tubular reactor when the gas obeys the ideal gas law. 

Reaction products were analyzed by on-line gas chromatography with a Shimadzu GC-14A gas chromatograph equipped with a 50 m CP Sil-5 fused silica capfllary column and a flame ionization detector. Reaction intermediates were identified by GC-MS. Samples were taken after 50 h on stream when the activity of the catalyst was stable, with n-nonane and n-dodecane as internal standards. Space time was defined as t = e Voat/vgas, where e is the void fraction of the... 

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]. 

The above-mentioned space-time yields were referred solely to the reaction volume, i.e. the micro channel volume. When defining this quantity via an idealized reactor geometry, taking into accoimt the construction material as well, natarally the difference in space-time yield of the micro reactors from the laboratory bubble column becomes smaller. Still, the performance of the micro reactors is more than one order of magnitude better [38], The space-time yields for the micro reactors defined in this way ranged from about 200 to 1100 mol monofluorinated product... 

The design optimization of an electrolytic cell aims at a high throughput with a low energy consumption at the lowest feasible cost. The throughput of an electrochemical reactor is measured in terms of the space time yield, Yt, defined as the volumetric quantity of the metal produced per unit time per unit volume of the process reactor. This quantity is expressed as ... 

The missing steps are very similar to how we did Example 2.11 (p. 2-28). Divide the equations by the respective reactor volumes and define space times x, = V,/qs and x2 = V2/qs, we obtain... 

Analogous to time as a measure of batch process performance, space-time (r) can be defined for a continuous reactor ... 

---