Temperature wave

In view of this, on the front of the temperature wave x = Dt the temperature and heat flow vanish for cr > 0 and the partial derivative... 

As a matter of fact, the front of the temperature wave becomes immovable, since Xj = const is independent of t and depends only on the parameters >Co) o problem concerned. Moreover, at the front the... 

A greater gain in accuracy in connection with the temperature wave depends significantly on how well we calculate the coefficients a (v). In the case where k = k u is a power function of temperature, numerical experiments showed that formula (38) is useless and formula (36) is much more flexible than (37), so there is some reason to be concerned about this. Further comparison of schemes (34) and (35) should cause some difficulties. Both schemes are absolutely stable and have the same error of approximation 0 r + h ). The scheme a) is linear with respect to the value of the function on the layer and so the value y7+i on every new layer... 

Calculations of the temperature waves. Of special interest is the case where the coefficient k u) is a function of temperature such that... 

Temperature, wave function properties, 214 Tetraatomic molecules ... 

It is easy to check that sometimes k can exceed the Cu conductivity. The very high thermal conductivity of 4He n allows for the propagation of temperature waves (second sound) [42-48]. 

As a matter of fact, the front of the temperature wave becomes immovable, since x1 — const is independent of t and depends only on the parameters x0, a, u0 of the problem concerned. Moreover, at the front the heat flow and temperature vanish for any 0, while the partial derivative becomes du/dx = oo for 2 (at the front of the travelling wave du/dx = oo for a > 1). 

No theoretical criterion for flammability limits is obtained from the steady-state equation of the combustion wave. On the basis of a model of the thermally propagating combustion wave it is shown that the limit is due to instability of the wave toward perturbation of the temperature profile. Such perturbation causes a transient increase of the volume of the medium reacting per unit wave area and decrease of the temperature levels throughout the wave. If the gain in over-all reaction rate due to this increase in volume exceeds the decrease in over-all reaction rate due to temperature decrease, the wave is stable otherwise, it degenerates to a temperature wave. Above some critical dilution of the mixture, the latter condition is always fulfilled. It is concluded that the existence of excess enthalpy in the wave is a prerequisite of all aspects of combustion wave propagation. 

There are no ISO standards at present for polymers. However, a series of methods are being developed in TC 61 for conductivity and diffusivity of plastics. At the time of writing there are drafts for general principles, laser flash method, temperature wave analysis method and the Gustafsson method. The general principles draft is a bit misleading as it appears to deal only with transient methods, and the specific procedures so far drafted appear to have been selected at random from the many transient methods available. 

The theory of kinematic waves, initiated by Lighthill Whitham, is taken up for the case when the concentration k and flow q are related by a series of linear equations. If the initial disturbance is hump-like it is shown that the resulting kinematic wave can be usually described by the growth of its mean and variance, the former moving with the kinematic wave velocity and the latter increasing proportionally to the distance travelled. Conditions for these moments to be calculated from the Laplace transform of the solution, without the need of inversion, are obtained and it is shown that for a large class of waves, the ultimate wave form is Gaussian. The power of the method is shown in the analysis of a kinematic temperature wave, where the Laplace transform of the solution cannot be inverted. 

We shall see that the mean of the temperature wave travels with just this velocity even in the presence of these dispersive effects and obtain an expression for the growth of the variance. 

This shows that the mean of the temperature wave moves with the kinematic wave velocity and that an apparent diffusion coefficient may be defined to describe the dispersion. This coefficient is the sum of the diffusion coefficients which would be obtained if each effect were considered independently. Such an additivity has been demonstrated by the author for the molecular and Taylor diffusion coefficients elsewhere (Aris 1956) and is assumed in a paper by Klinkenberg and others (van Deemter, Zuiderweg Klinkenberg 1956) in their analysis of the dispersion of a chromatogram. 

Let us consider the increase in recycle flow. The higher flow initially decreases temperatures in the reactor. Pressure begins to build. Then the temperatures in the reactor start to increase because of the reaction rate increase due to higher pressure. At about 7 min the exit temperature increases to a value slightly above its steady-state level. The higher exit temperature increases the reactor inlet temperature through the FEHE, and this starts a temperature wave that moves down the reactor. Temperature Tt2, which is located at about 40% of the way down the reactor, spikes first. Then... 

Decreases in recycle flow produce similar effects but in the reverse direction. The decrease in flow raises reactor temperatures, and a temperature wave starts to move down the reactor. Since the flowrate is lower, the temperature spike moves more slowly than when the recycle flowrate is increased. This explains the longer period of the cycles. These results demonstrate that the optimal design with the hot reaction is openloop unstable. 

At 316C inlet temperature, reaction rate controls, and the coke oxidizes slowly everywhere in the bed. There is no dis-cernable temperature wave and a significant concentration of oxygen leaves the bed throughout the bum. 

A theoretical and experimental study of multiplicity and transient axial profiles in adiabatic and non-adiabatic fixed bed tubular reactors has been performed. A classification of possible adiabatic operation is presented and is extended to the nonadiabatic case. The catalytic oxidation of CO occurring on a Pt/alumina catalyst has been used as a model reaction. Unlike the adiabatic operation the speed of the propagating temperature wave in a nonadiabatic bed depends on its axial position. For certain inlet CO concentration multiplicity of temperature fronts have been observed. For a downstream moving wave large fluctuation of the wave velocity, hot spot temperature and exit conversion have been measured. For certain operating conditions erratic behavior of temperature profiles in the reactor has been observed. 

Here T is the temperature x is the coordinate / is the thermal conductivity c and p are, respectively, the heat capacity and density of the solid mixture of reactants Q is the rate of reaction heat release, and V is the propagation velocity of the temperature wave front. 

The problem of the stability of the two modes of steady-state propagation of the temperature wave over the reaction sample needs a separate study. From qualitative considerations it follows that the faster process is less sensitive to disturbances (both mechanical and thermal) than the slower one. It is evident that in the case of slow motion any kind of inhomogeneities in the sample [e.g., a local reduction in the strength, leading to a decrease in (dT/dx) 1 may cause a displacement of the reaction-onset coordinate to the fore part of the front and thereby induce a spontaneous transformation of the slower wave into the faster one. 

In these parameters s designates some characteristic dimensions of the body for the plate it is the half-thickness, whereas for the cylinders and sphere it is the radius. The Biot number compares the relative magnitudes of surface-convection and internal-conduction resistances to heat transfer. The Fourier modulus compares a characteristic body dimension with an approximate temperature-wave penetration depth for a given time r. 

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