Similar temperature fields

The advantage of introducing dimensionless variables has already been shown in section 1.1.4. The dimensionless numbers obtained in that section provide a clear and concise representation of the physical relationships, due to the significant reduction in the influencing variables. The dimensionless variables for thermal conduction are easy to find because the differential equations and boundary conditions are given in an explicit form. 

The starting point for the derivation of the dimensionless numbers in thermal conduction is the differential equation 

L0 is a characteristic length of the conductive body and t0 is a characteristic time (interval) which still has to be determined. We choose 

The heat conduction equation takes the dimensionless form of 

The characteristic time is chosen The dimensionless time is then 

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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... 

Optical detection of magnetic resonance (ODMR) was attempted for measurements of the pH effects on the triplet state of purine to investigate the protonation site of purine at low temperatures (78JA7131). The ODMR spectrum did not show the presence of more than one triplet state at liquid helium temperatures. Since the protonated tautomers 1H,9H (3a) and H,1H (3b) have similar bond structures, their triplets should have similar zero-field parameters and are thus not easy to distinguish by ODMR. 

Effectively, Eqs. (86) and (87) describe two interpenetrating continua which are thermally coupled. The value of the heat transfer coefficient a depends on the specific shape of the channels considered suitable correlations have been determined for circular or for rectangular channels [100]. In general, the temperature fields obtained from Eqs. (86) and (87) for the solid and the fluid phases are different, in contrast to the assumptions made in most other models for heat transfer in porous media [117]. Kim et al. [118] have used a model similar to that described here to compute the temperature distribution in a micro channel heat sink. They considered various values of the channel width (expressed in dimensionless form as the Darcy number) and various ratios of the solid and fluid thermal conductivity and determined the regimes where major deviations of the fluid temperature from the solid temperature are found. 

When we want to look at the connection between the flow behavior and the amount of heat that is transferred into the fixed bed, the 3D temperature field is not the ideal tool. We can look at a contour map of the heat flux through the wall of the reactor tube. Fig. 19 actually displays a contour map of the global wall heat transfer coefficient, h0, which is defined by qw — h0(Tw-T0) where T0 is a global reference temperature. So, for constant wall temperature, qw and h0 are proportional, and their contour maps are similar. The map in Fig. 19 shows the local heat transfer coefficient at the tube wall and displays a level of detail that would be hard to obtain from experiment. The features found in the map are the result of the flow features in the bed and the packing structure of the particles. 

Figure 12.5 Calculated mean temperature fields in combustors with a set of similar open-edge V-gutter flame holders of height H = 3 cm and apex angle of 60°. The isoterms divide the entire temperature interval from the initial temperature To to combustion temperature Tc into 10 uniform parts and correspond to t = 27.5 ms. The combustor is 1 m long and the distance between the planes of flame holders is 0.05 m. Flame holders are shifted in longitudinal direction by OH (no shift) (a), IH (6), 2H (c), 3H (d), and 5H (e). Combustion of stoichiometric methane-air mixture at the mean inlet velocity Ui = 20 m/s, po = 0.1 MPa, To = 293 K, ko = 0.24 J/kg, /o = 4 mm. The lower and upper boundaries of the computational domain are the symmetry planes...

Fig. 2.15 (a) In low temperature pulsed-laser field evaporation of silicon, ions formed are all doubly charged. The energy distributions are very narrow, similar to those found in low temperature field evaporation of metals. However, the onset flight times are always shorter than the calculated values, indicating a photo-excitation effect as will be discussed in Sec.2.2.6. 

Develop a system of equations to describe the flow and temperature field in the annular space between the two cylinders. Assume a perfect-gas equation of state. To achieve a tubular flow in the similar form, discuss the approximations that must be made and the limitations of the assumptions. 

The equation of heat transfer does not contribute anything new even though the temperature field is not similar to the velocity field due to the presence of the buoyancy force. (This situation holds identically in turbulent and laminar flows.)... 

In the absence of catalysis on the surface, similarity of the concentration and temperature fields is achieved precisely at the ignition limit if the coefficients of diffusion and thermal diffusivity are equal, since in this case both the diffusion gradient and the temperature gradient at the igniting surface are equal to zero, and the equations of diffusion and thermal conductivity with the chemical reaction may be reduced to the form of an identity (see our work on flame propagation [3]). 

The content of formula (1) may be formulated as the similarity of the concentration fields to the temperature field in the flame, which means that the temperature and concentrations are interrelated just as in an adiabatic reaction, although the reaction develops in time quite differently. 

In the case of a so-called zero-order reaction, i.e., one whose reaction rate is independent of the concentration of the reacting materials throughout the reaction, violation of the similarity of the concentration field and temperature field as a result of heat transfer to the outside, which does not have an analogue in diffusion, still does not over-complicate the problem. In this particular case we were able to follow in detail the effect of the heat transfer on the entire distribution of the temperature in the flame and on the propagation rate. In our approximation the activation heat is much larger than ETth, and the results of this more detailed investigation confirm all of the formulas derived above (unpublished work by the author, 1938). 

In the paper cited [2] the similarity of the concentration fields (relative concentrations or partial pressures) and the temperature field were also established for the combustion of a gas, from which also follows constancy of the enthalpy throughout the combustion zone. 

Similarly to the above derivation, we can also use the technique to predict transient temperature fields. Again, as with finite elements and boundary elements, the time stepping is done using finite difference techniques. For a Crank-Nicholson transient energy equation formulation given by... 

The heat transfer problem just discussed can be solved in a fashion similar to the one used in Section 5.3, to yield T(z, t). In principle, once the temperature field is known in the preform at any time before fr, the plunger force can be calculated. The preform can be taken as a solid that slips at the mold surface and has a temperature-dependent compressive modulus. At any time t < tf, each layer of the preform will deform by an amount such that (a) the force on every layer of thickness Az is the same (and equal to the unknown quantity), and (b) the sum of the compressive deformations of all the layers equals the deformation imposed on the preform by the plunger at the given time. The force... 

The horizontal structure of the field of the minimal water temperatures (the core of the CIL) of the Black Sea in the extreme months of the annual cycle (February and August) is presented in Figs. 6a and 6b. The February field of the minimal water temperatures (Fig. 6a) is similar to the surface temperature field (Fig. 5a). Meanwhile, a detailed analysis shows that they are not fully identical from the northwest to the southeast the excess of the surface temperature over the minimal value grows up to 1.0 °C. The depth of the temperature minimum location increases in the same direction down to 70-80 m. This points to the absence of CIL water renewal owing to the winter convective mixing over some part of the Black Sea area. 

The prime variables should be selected so that each dimensionless product characterizes some distinct feature of the flow. For example, the forced velocity, U, should be used as one prime variable since it determines whether or not the problem involves forced convection. Similarly, the buoyancy variable (3g(Tw - T/) will determine the importance of free convective effects and should also be used as a prime variable. Also, since h is the variable whose value is required, it should be used as a prime variable. The fourth prime variable will be taken as cp. This choice is not as obvious as the others but stems from the fact that cp will determine the thermal capacity of the fluid and will, therefore, influence the relation between the velocity and temperature fields. 

Attention will first be given to the determination of conditions under which the velocity fields are similar, the temperature field being dealt with later. The following dimensionless coordinate system is introduced ... 

Consideration will next be given to the determination of the conditions under which the temperature fields are similar. In order to define a dimensionless temper ature variable, a convenient reference fluid temperature, T, and some convenient measure of the wall temperature, 7wr, are introduced. In external flows, T is usu ally most conveniently taken as the freestream temperature while in internal flows it is usually taken as a convenient mean temperature. Using these the following dimensionless temperature is defined ... 

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