Two-reservoir system

Fig. 4-6 A coupled two-reservoir system with fluxes proportional to the content of the emitting reservoirs.

As an illustration of the concept introduced above, let us consider a coupled two-reservoir system with no external forcing (Fig. 4-6). The dynamic behavior of this system is governed by the two differential equations... 

Before turning to nonlinear situations, let us consider two specific examples of coupled linear systems. The first describes the dynamic behavior of a multireservoir system the second represents a steady-state situation of an open two-reservoir system. 

Figure 7.12 Exchange of Na in an open two-reservoir system the flux J0(t) of Na weathered from evaporites introduces a forcing term into conservation equations.

Equation 1.2 assumes that the concentration of C is constant throughout the ocean, i.e., that the rate of water mixing is much fester than the combined effects of any reaction rates. For chemicals that exhibit this behavior, the ocean can be treated as one well-mixed reservoir. This is generally only true for the six most abundant (major) ions in seawater. For the rest of the chemicals, the open ocean is better modeled as a two-reservoir system (surface and deep water) in which the rate of water exchange between these two boxes is explicitly accoimted for. 

In order to write the Master Equation for the two reservoir system we need the transition rate between the state (particle number N, total energy E) and the state (N+r, E+ ). We know that the rate at which particles reach the hole is proportional to their velocity, and that we have a Maxwell-Boltzmann velocity distribution. From this we may write that the transition rate, W(N,E,N+r,E+e ), is. 

In developing the simulation for the two reservoir system we tried to meet two important criteria. First, the simulation had to be fast. Since we were interested in measuring fluctuations in a large system we required very accurate statistics, at least millions of events. Secondly, we tried to keep an eye towards developing simulations for the next generation of thermal fluctuation problems, those with exothermic chemistry and spatial extent. 

In deciding what type of code to use, we had several frameworks to chose from. Almost immediately, we rejected using a molecular dynamics code [ 5 1 because of our first criterion also a molecular dynamics code would contain much more detail than we were really interested in. Our second choice was to use a collisionless Monte Carlo code [ 6 1. In this context, by a collisionless code we mean one in which the collisions in the system were not explicitly calculated but rather were assumed to always keep the system in a Maxwell-Boltzmann distribution. This would certainly meet our first criterion but we were not certain whether it would provide enough microscopic detail for more complex systems. For the two reservoir system though it was... 

The collisionless Monte Carlo code has been very successful for the two reservoir system. It is small and can do ten million events in about one hour on a VAX 11/780 computer. For systems of some 500 and 1000 particles, we have made runs at several temperatures. Figure 1 summarizes our results so far for the temperature fluctuations and we are pleased at the agreement with the values predicted by Eq. (8). Though statistics were taken for the other fluctuations, the number fluctuations and the number-temperature correlations will not differ substantially from the equilibrium values and so are not conclusive. 

It is also instructive to start from the expression for entropy S = log(g(A( m)) for a specific energy partition between the two-state system and the reservoir. Using the result for g N, m) in section A2.2.2. and noting that E = one gets (using the Stirling approximation A (2kN)2N e ). 

The mechanisms that control dmg deUvery from pumps may be classified as vapor-pressure, electromechanical, or elastomeric. The vapor-pressure controlled implantable system depends on the principle that at a given temperature, a Hquid ia equiUbrium with its vapor phase produces a constant pressure that is iadependent of the enclosing volume. The two-chamber system contains iafusate ia a flexible beUows-type reservoir and the Hquid power source ia a separate chamber (142). The vapor pressure compresses the dmg reservoir causiag dmg release at a constant rate. Dmg maybe added to the reservoir percutaneously via a septum, compressing the fluid vapor iato the Hquid state. 

So far only two groups have reported details of the use of ionic liquids with wholecell systems (Entries 3 and 4) [31, 32]. In both cases, [BMIM][PF(3] was used in a two-phase system as substrate reservoir and/or for in situ removal of the product formed, thereby increasing the catalyst productivity. Scheme 8.3-1 shows the reduction of ketones with bakers yeast in the [BMIM][PF(3]/water system. 

The mobile phase supply system consists of a series of reservoirs normally having a capacity ranging from 200 ml to 1,000 ml. Two reservoirs are the minimum required and are usually constructed of glass and fitted with an exit port open to air. Stainless steel is an alternative material for reservoir construction but is not considered satisfactory for mobile phases buffered to a low pH and containing... 

It should be noted that the steady-state solution of Equation (12) is not necessarily unique. This can easily be seen in the case of the four-reservoir system shown in Fig. 4-7. In the steady state all material will end up in the two accumulating reservoirs at the bottom. However, the distribution between these two reservoirs will... 

As a working example consider the thermoelectric system consisting of two reservoirs of energy and electrons, in which the temperatures are 7 and T2 and the electrochemical potentials are fj, 1 and fj,2-... 

Onsager found a proof of the general reciprocal relationship by consideration of natural fluctuations that occur in equilibrium systems. It is argued that an imbalance like that which initially exists when two reservoirs are connected as in figure 2, could also arise because of natural fluctuations. When the occasional large fluctuation occurs in an equilibrium system, the subsequent decay of the imbalance would be indistinguishable from the decay that follows deliberate connection of the reservoirs. 

Figure 7.13 Evolution of the fractions /A(t) and /B(t) of total Na held by the two reservoirs A and B in the system described in Figure 7.12./0(r) is the fraction remaining in evaporites at t.

As the same change in state occurs in the irreversible process, A5 for the cold reservoir still is given by Equation (6.95). In the irreversible process, the two reservoirs are the only substances that undergo any changes. As T2 > Ti, the entropy change for the system as a whole is positive ... 

Figure 18.1. System with its ends in diffusive contact with two reservoirs of different chemical potential.

---