Steady temperature gradient

Fig. 12. Pressure and temperature stresses in a cylinder, k = 2 subjected to a steady temperature gradient of 100°C and an internal pressure of 138 MPa...

Dufour effect Establishment of steady temperature gradient due to fixed concentration gradient. There can be two types of time-invariant states. One such is equilibrium state. In the equilibrium state, thermodynamic variables such as temperature T, pressure P and chemical potentials p, are adjusted in a way so that there is no (i) flow of matter, (ii) flow of energy and current and (iii) occurring in the system. Typical examples are vapour-liquid, liquid-Uquid, solid-liquid and chemical equilibria. However, time-invariant non-equilibrium steady states are also possible when opposite flows are balanced and gradients are maintained constant. 

After a steady temperature gradient is established with the heat flux running from the heat somce to the heat sink, we can calculate the thermal conductivity K using the Fourier s law as ... 

Figure 7.5 Conversion of melt into spherulites measured during crystallization of a thin iPP film in the steady temperature gradient of 35°C/mm determined on isotherms at 128 C and 132°C (filled circles) and in isothermal conditions at corresponding temperature (empty circles) [44]. Dashed and solid lines result from calculations [10] based on Equation (7.7b) and Equation (7.44) for 2D crystallization in isothermal conditions and in a temperature gradient, respectively.

Thermal Stresses. When the wak of a cylindrical pressure vessel is subjected to a temperature gradient, every part expands in accordance with the thermal coefficient of linear expansion of the steel. Those parts of the cylinder at a lower temperature resist the expansion of those parts at a higher temperature, so setting up thermal stresses. To estimate the transient thermal stresses which arise during start-up or shutdown of continuous processes or as a result of process intermptions, it is necessary to know the temperature across the wak thickness as a function of radius and time. Techniques for evaluating transient thermal stresses are available (59) but here only steady-state thermal stresses are considered. The steady-state thermal stresses in the radial, tangential, and axial directions at a point sufficiently far away from the ends of the cylinder for there to be no end effects are as fokows ... 

Unsteady-State Direct Oxidation Process. Periodic iatermption of the feeds can be used to reduce the sharp temperature gradients associated with the conventional oxidation of ethylene over a silver catalyst (209). Steady and periodic operation of a packed-bed reactor has been iavestigated for the production of ethylene oxide (210). By periodically varyiag the inlet feed concentration of ethylene or oxygen, or both, considerable improvements ia the selectivity to ethylene oxide were claimed. 

FIG. 5-1 Temperature gradients for steady heat conduction in series through three solids. 

FIG. 5-6 Temperature gradients for a steady flow of heat by conduction and convection from a warmer to a colder fluid separated by a solid wall. 

Temperature gradient normal to flow. In exothermic reactions, the heat generation rate is q=(-AHr)r. This must be removed to maintain steady-state. For endothermic reactions this much heat must be added. Here the equations deal with exothermic reactions as examples. A criterion can be derived for the temperature difference needed for heat transfer from the catalyst particles to the reacting, flowing fluid. For this, inside heat balance can be measured (Berty 1974) directly, with Pt resistance thermometers. Since this is expensive and complicated, here again the heat generation rate is calculated from the rate of reaction that is derived from the outside material balance, and multiplied by the heat of reaction. 

In the simplest case of one-dimensional steady flow in the x direction, there is a parallel between Eourier s law for heat flowrate and Ohm s law for charge flowrate (i.e., electrical current). Eor three-dimensional steady-state, potential and temperature distributions are both governed by Laplace s equation. The right-hand terms in Poisson s equation are (.Qy/e) = (volumetric charge density/permittivity) and (Qp // ) = (volumetric heat generation rate/thermal conductivity). The respective units of these terms are (V m ) and (K m ). Representations of isopotential and isothermal surfaces are known respectively as potential or temperature fields. Lines of constant potential gradient ( electric field lines ) normal to isopotential surfaces are similar to lines of constant temperature gradient ( lines of flow ) normal to... 

Mathews and Kruh , in similar experiments, were unable to obtain steady-state pressures, but this was probably due to the larger size of vessel used, to temperature gradients, and to difficulties due to creeping of the fused sodium hydroxide up the vessel walls. 

Temperature gradients within the porous catalyst could not be very large, due to the low concentration of combustibles in the exhaust gas. Assuming a concentration of 5% CO, a diffusion coefficient in the porous structure of 0.01 cms/sec, and a thermal conductivity of 4 X 10-4 caI/sec°C cm, one can calculate a Prater temperature of 1.0°C—the maximum possible temperature gradient in the porous structure (107). The simultaneous heat and mass diffusion is not likely to lead to multiple steady states and instability, since the value of the 0 parameter in the Weisz and Hicks theory would be much less than 0.02 (108). 

A steady-state heat balance for a plug flow reactor with no radial temperature gradients is given by ... 

Topaz was used to calculate the time response of the model to step changes in the heater output values. One of the advantages of mathematical simulation over experimentation is the ease of starting the experiment from an initial steady state. The parameter estimation routines to follow require a value for the initial state of the system, and it is often difficult to hold the extruder conditions constant long enough to approach steady state and be assured that the temperature gradients within the barrel are known. The values from the Topaz simulation, were used as data for fitting a reduced order model of the dynamic system. 

Meiron (12) and Kessler et al. (13) have shown that numerical studies for small surface energy give indications of the loss-of-existence of the steady-state solutions. In these analyses numerical approximations to boundary integral forms of the freeboundary problem that are spliced to the parabolic shape far from the tip don t satisfy the symmetry condition at the cell tip when small values of the surface energy are introduced. The computed shapes near the tip show oscillations reminiscent of the eigensolution seen in the asymptotic analyses. Karma (14) has extended this analysis to a model for directional solidification in the absence of a temperature gradient. 

The presentation in this paper concentrates on the use of large-scale numerical simulation in unraveling these questions for models of two-dimensional directional solidification in an imposed temperature gradient. The simplest models for transport and interfacial physics in these processes are presented in Section 2 along with a summary of the analytical results for the onset of the cellular instability. The finite-element analyses used in the numerical calculations are described in Section 3. Steady-state and time-dependent results for shallow cell near the onset of the instability are presented in Section 4. The issue of the presence of a fundamental mechanism for wavelength selection for deep cells is discussed in Section 5 in the context of calculations with varying spatial wavelength. 

A steady-state method is disadvantageous in measurements on a mixture because for a long time the temperature gradient is likely to generate separation of the mixture due to thermal diffusion. Accurate measurement itself seems to be still one of the most pressing concerns for thermal diffusion of high-temperature melts. 

This result shows that the most likely rate of change of the moment due to internal processes is linearly proportional to the imposed temperature gradient. This is a particular form of the linear transport law, Eq. (54), with the imposed temperature gradient providing the thermodynamic driving force for the flux. Note that for driven transport x is taken to be positive because it is assumed that the system has been in a steady state for some time already (i.e., the system is not time reversible). 

The steady-state probability distribution for a system with an imposed temperature gradient, pss(r p0, pj), is now given. This is the microstate probability density for the phase space of the subsystem. Here the reservoirs enter by the zeroth, (10 = 1 /k To, and the first, (i, = /k T, temperatures. The zeroth energy moment is the ordinary Hamiltonian,... 

Figure 8 shows the r-dependent thermal conductivity for a Lennard-Jones fluid (p = 0.8, 7o = 2) [6]. The nonequilibrium Monte Carlo algorithm was used with a sufficiently small imposed temperature gradient to ensure that the simulations were in the linear regime, so that the steady-state averages were equivalent to fluctuation averages of an isolated system. 

We would be remiss in our obligations if we did not point out that the regions of multiple solutions are seldom encountered in industrial practice, because of the large values of / and y required to enter this regime. The conditions under which a unique steady state will occur have been described in a number of publications, and the interested student should consult the literature for additional details. It should also be stressed that it is possible to obtain effectiveness factors greatly exceeding unity at relatively low values of the Thiele modulus. An analysis that presumed isothermal operation would indicate that the effectiveness factor would be close to unity at the low moduli involved. Consequently, failure to allow for temperature gradients within the catalyst pellet could lead to major errors. 

The most common technique for the measurement of k(T) at low temperatures is the steady longitudinal flow method a steady thermal power flow along a sample of section A (i.e. a cylinder or a bar) is produced by a temperature gradient AT. If the power flows only in the x direction, eq. (11.1) becomes ... 

U-shaped curve, we have mixtures that can be ignited for a sufficiently high spark energy. From Equation (4.25) and the dependence of the kinetics on both temperatures and reactant concentrations, it is possible to see why the experimental curve may have this shape. The lowest spark energy occurs near the stoichiometric mixture of XCUi =9.5%. In principle, it should be possible to use Equation (4.25) and data from Table 4.1 to compute these ignitability limits, but the complexities of temperature gradients and induced flows due to buoyancy tend to make such analysis only qualitative. From the theory described, it is possible to illustrate the process as a quasi-steady state (dT/dt = 0). From Equation (4.21) the energy release term represented as... 

Figure 12.5 Simplified scheme of the temperature gradient along the sample cell of a DSC instrument, on heating and under steady state conditions. A heating element B heat conduction path ...

While a one-dimensional model for steady-state heat conduction might seem an oversimplifying assumption, values for a typical heat flux range from 0 up to 10 kW/m With a one-dimensional model, this translates to about a maximum 0.5 °C temperature gradient within the barrel metal in the radial direction. If a two-dimensional model is used, the temperature gradient will decrease even further, and thus have virtually no impact on the interfacial temperature calculations. 

Energy efficiency. Temperature gradients are steady state and take place in space rather than in time, reducing the need for the expensive heating and cooling cycles common in batch equipment. 

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