Convective stationary state

Fig. 8.5. The product of the total rate of dissipation times temperature (solid line) in Js and the time derivative of excess work (dashed line) vs. time in the following processes for the Lorenz model (a) Gravity is initially set in the direction along which the temperature decreases, and the system is at a stable motionless conductive stationary state at t = 0, invert the direction of gravity the motionless conductive state becomes unstable and the system approaches the convective stationary state, (b) The reverse process. The temperature difference is AT = 4K for both cases...

There may be several reasons for a lack of quantitative agreement. First, the convective stationary state in the theory is a focus, not a node. (A node is approached with an eigenvalue that is real and negative and hence provides for a damped monotonic approach, whereas a focus is approached with a complex eigenvalue with the real part negative, that is a damped oscillatory approach.) In the experiments, however, the convective stationary state is a node due to the rigid boundaries. Second, because of the truncation to the first order in Fourier modes in the Lorenz model, this model can be a good approximation... 

Obviously, the heat transfer is the conjugating process here with respect to establishing the conjugate convection process. The controlling parame ter is obviously the differential temperature AT, while point AT r behaves as the bifurcation point of the potential stationary states of the system. 

For most liquids the product pc lies between 0.3 and 1.0 cal/cc-°C, while K varies between 0.5 and 1.5 X 10 cal/cm-sec-°C. If we choose mean values of O.G cal/cc- and 1.0 X 10 , we find that the stationary state is achieved in liquid systems in a time given by = GOro sec = min when n is in cm. Thus for a 500-cc vessel (r 5 cm) the mean time to reach the stationary state is about 25 min, so that it may be expected that convection will play a much more important role than conduction in a liquid system. 

We can use this equation by estimating the ratio between the convective and the diffusive flux. Taken in the z-direction only, the absolute value of this ratio becomes /D (0Cj /3z). If the process if fully diffusion-controlled, 3Cj /3z = cjA, where A is the diffusion layer thickness, so the concentration drops out and the ratio becomes v AID. Generally, A is 4(t) (the layer grows with time) in the stationary state A is fixed, but depends on and D. However, to get some feeling for the order of magnitude, if A 10 m, D 10 m s and 10" ms" the ratio is 10. In passing it may be noted that, however much convection may prevail over diffusion, the last part of transport towards a surface is always diffusion-controlled when the surface is inextendable. Note that ratio u A / D, is... 

The relation between the interfacial and bulk concentrations depends on mass transport, most often by diffusion (i.e., thermal motion) and/or convection (mechanical stirring). Often a stationary state is reached, in which the concentrations near the electrode can be described approximately by a diffusion layer of thickness 8. For a constant diffusion layer thickness the Nernst equation takes the form... 

Because the thermal lensing effect discussed here is very small, it is not practical to use the initial slope of the signal for a quantitative evaluation (cf. Section III). Instead, we determined the change in intensity at the detector after the stationary state had been reached. This was generally the case at time = 2 s after opening of the shutter. As the temperature change on irradiation is very small, convective currents in the liquid do not obfuscate the signal in these experiments. It should be noted that all experiments were done at a room temperature of 20°C. 

The system liquid - particles comes to a stationary state when the flux due to the translational diffusion is counterbalanced by the convective flux caused by the stationary hydrodynamic force acting on each particle. This force can be represented by the Stokes force... 

The most extensive study of thermal etplosion theory was carried out by Gray et al." on the decomposition of diethyl peroxide in the gas phase. The study was carried out under conditions where convective heat transfer within the reactant mass is negligible and where heat generation by the reactant and losses by conduction therefore determine the course of events. Their findings are in excellent qualitative agreement with the predictions of thermal theory and furthermore the quantitative agreement is remarkable, in view of the various assumptions of stationary-state conductive theory and the deviation in practice of the actual reaction system from these. The experimental results may be summarized as follows ... 

The equation above shows that the net amount of E exchanged through the boundary must be zero, and the divergence of the sum of the conduction and convection flows governed by a conservation law is equal to zero in the stationary state. For the values e = 1, J, = 0, and prod = 0, Eqn (3.22) becomes... 

In hydrocarbon reservoirs, the temperature distribution is often available from temperature measurements. Then for a ID or a 2D space with two components, there are two unknowns at each point, Xj and P (xg is not an independent variable since X2 — 1 i). For a ID space, because of the absence of convection, the problem becomes very simple. Let us derive an explicit expression for dx jdz at steady state (i.e., stationary state) for a two component system for a ID problem. 

We present a brief introduction to coupled transport processes described macroscopically by hydrodynamic equations, the Navier-Stokes equations [4]. These are difficult, highly non-linear coupled partial differential equations they are frequently approximated. One such approximation consists of the Lorenz equations [5,6], which are obtained from the Navier-Stokes equations by Fourier transform of the spatial variables in those equations, retention of first order Fourier modes and restriction to small deviations from a bifurcation of an homogeneous motionless stationary state (a conductive state) to an inhomogeneous convective state in Rayleigh-Benard convection (see the next paragraph). The Lorenz equations have been applied successfully in various fields ranging from meteorology to laser physics. 

Our theory based on the concept of exess work accounts for these experiments, at least qualitatively. According to our theory, when the system approaches a stable stationary state, either convective or conductive, there is a decrease in, the excess work, and a positive excess work is released, which... 

Forced convection can be used to achieve fast transport of reacting species toward and away from the electrode. If the geometry of the system is sufficiently simple, the rate of transport, and hence the surface concentrations cs of reacting species, can be calculated. Typically one works under steady-state conditions so that there is no need to record current or potential transients it suffices to apply a constant potential and measure a stationary current. If the reaction is simple, the rate constant and its dependence on the potential can be calculated directly from the experimental data. 

A CV voltammogram can be recorded under either a dynamic or a steady state depending on the electrode design and solution convection mode. In a stationary solution with a conventional disk electrode, if the scan rate is sufficiently high to ensure a non-steady state, the current will respond differently to the forward and backward potential scan. Figure 63 shows a typical CV for a reversible reduction.1... 

The complexities of turbulent flow are outside the province of this book. However, there are two further properties of laminar convective flow that are relevant to understanding the electrochemical situation. The first is easily understood by considering an excellent illustration of it—river flow. It is a matter of common observation that rivers (which flow convectively as a result of being pushed by gravity) move at maximum rale in the middle. At the river bank there is hardly any flow at all. This observation can be transferred to the flow of liquid through a pipe. The flow reaches a maximum velocity in the center. The liquid actually in contact with the walls of the pipe does not flow at all. The stationary layer is a few micrometers in thickness, about 1 % of the thickness of the diffusion layer set up by natural convection in an unstirred solution when an electrode reaction in steady state is occurring. 

All the electrode kinetic methodology described until now has assumed a steady state (or quasi-steady state in the case of the DME). Many techniques at stationary electrodes involve perturbation of the potential or current in combination with forced convection, this offers new possibilities in the evaluation of a wider range of kinetic parameters. Additionally, we have the possibility of modulating the material flux, the technique of hydrodynamic modulation which has been applied at rotating electrodes. Unfortunately, the mathematical solution of the convective-diffusion equation is considerably more complex and usually has to be performed numerically. 

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