Time-space equivalence

On the continuum level of gas flow, the Navier-Stokes equation forms the basic mathematical model, in which dependent variables are macroscopic properties such as the velocity, density, pressure, and temperature in spatial and time spaces instead of nf in the multi-dimensional phase space formed by the combination of physical space and velocity space in the microscopic model. As long as there are a sufficient number of gas molecules within the smallest significant volume of a flow, the macroscopic properties are equivalent to the average values of the appropriate molecular quantities at any location in a flow, and the Navier-Stokes equation is valid. However, when gradients of the macroscopic properties become so steep that their scale length is of the same order as the mean free path of gas molecules,, the Navier-Stokes model fails because conservation equations do not form a closed set in such situations. 

That is, we have recovered a Boltzmann distribution according to the Hamiltonian at time t, equivalent to (5.14). Jarzynski s identity (5.8) then follows simply by integration over phase space (p, q). 

The above simple example shows that in the special case of constant fluid density the space-time is equivalent to the holding time hence, these terms can be used interchangeably. This special case includes practically all liquid phase reactions. However, for fluids of changing density, e.g., nonisothermal gas reactions or gas reactions with changing number of moles, a distinction should be made between r and t and the correct measure should be used in each situation. 

Remark 3 Batch and semibatch reactors can also be studied by considering their space equivalent PFRs. The space equivalent of a single batch reactor can be regarded as a PFR with no side streams. The optimal holding time for the batch operation can be determined by the optimal PFR length and the assumed linear velocity of the fluid in the PFR. 

The physical mechanism described by this equation can be understood by starting at time zero with a velocity distribution sharply peaked at v = vo- As time passes, the maximum of this distribution is shifted toward smaller velocities, as a result of a systematic friction undergone by the particles (first term on the right-hand side of the equation). Furthermore, the peak broadens progressively as a result of diffusion in velocity space (second term on the right-hand side, which is the velocity space equivalent of the similar coordinate space term in Fick s law of diffusion). The final time-independent distribution reached by the Brownian particle is nothing more than the familiar Maxwell distribution ... 

For broad distributions this equation predicts that at least several hundred linearly-spaced channels are required. Nonlinear sample time spacing is superior. Both the fixed-angle instrument and the correlator used in this work operate with up to the equivalent of several thousand linearly-spaced data channels. 

The collagen fibrils occupy considerable space and thereby increase the path of diffusion. The net effect of impeding diffusion is to increase by several times the equivalent fluid layer thickness of the actual stroma. Nevertheless, the stroma is transparent to molecular species below approximately 500,000 Da. The stroma serves as the major ocular depot for topically applied hydrophilic drugs, and the keratocytes presumably provide a reservoir for lipophilic compounds as well. 

In the special case of a nondemolition interaction Hamiltonian, the master equation of the total system reduces to uncoupled systems of first order differential equations, whose dimensions are the same as the dimension Na of the ancilla Hilbert space. After having traced over the ancilla state, the master equation of the dynamic system can be expressed either as N,fh order differential equations in time or, equivalently, as Zwanzig equations with an explicit memory over the system evolution. 

Problem 1.4. Show that if the response is local in space but not in time the equivalent expressions for homogeneous stationary systems are... 

When we consider the same two cases as they appear in flow reactors the concepts of reaction time become less clear. In the case of a constant-volume homogeneous phase reaction taking place in a PFR, an increment of feed has a transit time, or space time t, during which it traverses the length of the constant temperature plug flow reactor. Both these times are equivalent in this case. Assuming that the reactants were preheated in a relatively small volume and are cooled in a similarly small volume, the main course of the reaction takes place in the reactive volume of the reactor proper. The time spent in this volume can be calculated as the volume of the reactor divided by the volumetric feed rate of the reactants. 

This expression for the reactor space time is equivalent to the mean residence time of the aqueous phase (growth medium) in the bioreactor if no significant volumetric expansion or contraction effects accompany the biochemical reaction. Moreover, solution of (13.3.8) for X2 gives... 

Here, Wr(C) is the equivalent rate vector in mass fraction space. Assuming that we are working in residence time space,... 

The volume of a Y -space-volume-element does not change in the course of time if each of its points traces out a trajectory in Y space determined by the equations of motion. Equivalently, the Jacobian... 

Average square barrier models witlr p=l. 4 A (equation (C3.2.6 )) would predict tire His 72 rate to be 1000 times faster. This equivalency of rates despite tire great difference in distairces is understood because tire strongest patlrways in tire His 72 derivative contain a tlrrough-space tumrelling gap. 

From a mathematical point of view, conformations are special subsets of phase space a) invariant sets of MD systems, which correspond to infinite durations of stay (or relaxation times) and contain all subsets associated with different conformations, b) almost invariant sets, which correspond to finite relaxation times and consist of conformational subsets. In order to characterize the dynamics of a system, these subsets are the interesting objects. As already mentioned above, invariant measures are fixed points of the Frobenius-Perron operator or, equivalently, eigenmodes of the Frobenius-Perron operator associated with eigenvalue exactly 1. In view of this property, almost invariant sets will be understood to be connected with eigenmodes associated with (real) eigenvalues close (but not equal) to 1 - an idea recently developed in [6]. 

This jacket is considered a special case of a helical coil if certain factors are incorporated into equations for calculating outside-film coefficients. In the equations at left and below, the equivalent heat transfer diameter D. for a rectangular cross-section IS equal to four times the width of the annular space, w and IS the mean or centerline diameter of the coil helix. Velocities are calculated from the actual cross-section of the flow area. pw. where p IS the pitch of the spiral baffle, and from the effective mass flowrate. W. through the passage. The leakage around spiral baffles is considerable, amounting to 35-50% of the total mass flowrate. The effective mass flowrate is about 60% of the total mass flowrate to the jacket W =... 

The European philosophy on area classification varies from that of the United. States and Canada. Specifically, in Europe and most other inter national areas, the Zone concept is utilized. An area in which an expio sive gas-air mixture is continuously present, or present for long perioiK of time, is referred to as Zone 0. The vapor space of a closed, but vented, process vessel or storage tank is an example. An area in which an explosive gas-air mixture is likely to occur in normal operations is designated Zone 1. An area in which an explosive gas-air mixture is less likely to occur, and if it does occur will exist only for a short time, is designated Zone 2. Zone 0 and Zone 1 correspond to Division 1 in the U.S. and Canada System. Zone 2 is equivalent to Division 2. 

Computer simulations of bulk liquids are usually performed by employing periodic boundary conditions in all three directions of space, in order to eliminate artificial surface effects due to the small number of molecules. Most simulations of interfaces employ parallel planar interfaces. In such simulations, periodic boundary conditions in three dimensions can still be used. The two phases of interest occupy different parts of the simulation cell and two equivalent interfaces are formed. The simulation cell consists of an infinite stack of alternating phases. Care needs to be taken that the two phases are thick enough to allow the neglect of interaction between an interface and its images. An alternative is to use periodic boundary conditions in two dimensions only. The first approach allows the use of readily available programs for three-dimensional lattice sums if, for typical systems, the distance between equivalent interfaces is at least equal to three to five times the width of the cell parallel to the interfaces. The second approach prevents possible interactions between interfaces and their periodic images. 

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