Bulk mean temperatures

The author s work has included the development of the Sandia Bear and Bertha explosive recovery fixtures, that provide a standardized set of fixtures in which recovery experiments can be routinely carried out at peak shock pressures from 4 to 500 GPa. Shock-induced, mean-bulk temperatures from 50 to 1200°C are achieved with variation in the density of the powder compacts under study. 

Use of the term mean-bulk temperature is to define the model from which temperatures are computed. In shock-compression modeling, especially in porous solids, temperatures computed are model dependent and are without definition unless specification of assumptions used in the calculations is given. The term mean-bulk temperature describes a model calculation in which the compressional energy is uniformly distributed throughout the sample without an attempt to specify local effects. In the energy localization case, it is well known that the computed temperatures can vary by an order of magnitude depending on the assumptions used in the calculation. 

It should be observed that every element except the powder system in the recovery system is chosen for favorable shock properties which can be confidently simulated numerically. The precise sample assembly procedures assure that the conditions calculated in the numerical simulations are actually achieved in the experiments. The influence of various powder compacts in influencing the shock pressure and mean-bulk temperature must be determined in computer experiments in which various material descriptions are used. Fortunately, the large porosity (densities from 35% to 75% of solid density) leads to a great simplification in that the various porous samples respond in the same manner due to the radial loading introduced from the porous inclusion in the copper capsule. 

Fig. 6.7. The predicted, one-dimensional, mean-bulk temperatures versus location at various times are shown for a typical powder compact subjected to the same loading as in Fig. 6.5. It should be observed that the early, low pressure causes the largest increase in temperature due to the crush-up of the powder to densities approaching solid density. The "spike in the temperature shown on the profiles at the interfaces of the powder and copper is an artifact due to numerical instabilities (after Graham [87G03]).

As shown in Fig. 6.6, the peak pressures are little affected by the density of the powder. This behavior is characteristic of a reverberation process to achieve peak pressure. On the other hand, the mean bulk temperature is strongly affected by the powder density, representing the volume compression to achieve solid density. 

The influence of crush strength on mean-bulk temperature achieved in one-dimensional compression of powders has been extensively investigated... 

Fig. 6.8. The peak mean-bulk temperatures predicted in one-dimensional numerical simulation are investigated for powder compacts of different crush strengths. For the explosive loadings of the Bear fixtures, no difference in temperature is predicted for crush strengths up to about 2 GPa. This value is about that of the initial loading wave into the samples. Above that pressure the crush strength has a strong effect on temperature. The predicted behavior can be understood in terms of the various loading paths.

Fig. 6.10. Two-dimensional simulations of the same arrangement of Fig. 6.9 show the predicted mean-bulk temperature contours at various times. The principal unusual feature is the hot region developed in the outside area due to simultaneous longitudinal and radial loading (after Graham [87G03]).

Mean-bulk temperatures are representative of a powder compact at a density of 55% of solid density. 

A summary of peak pressure and mean bulk temperatures in the various fixtures is shown in Table 6.3. Included in the characterization is the peak pressure along the axial few millimeter region along the axis of the samples (called focus) for which the radial focusing produces a high pressure region for a period of about 100 ns. 

Shock-synthesis experiments were carried out over a range of peak shock pressures and a range of mean-bulk temperatures. The shock conditions are summarized in Fig. 8.1, in which a marker is indicated at each pressure-temperature pair at which an experiment has been conducted with the Sandia shock-recovery system. In each case the driving explosive is indicated, as the initial incident pressure depends upon explosive. It should be observed that pressures were varied from 7.5 to 27 GPa with the use of different fixtures and different driving explosives. Mean-bulk temperatures were varied from 50 to 700 °C with the use of powder compact densities of from 35% to 65% of solid density. In furnace-synthesis experiments, reaction is incipient at about 550 °C. The melt temperatures of zinc oxide and hematite are >1800 and 1.565 °C, respectively. Under high pressure conditions, it is expected that the melt temperatures will substantially Increase. Thus, the shock conditions are not expected to result in reactant melting phenomena, but overlap the furnace synthesis conditions. 

Four different material probes were used to characterize the shock-treated and shock-synthesized products. Of these, magnetization provided the most sensitive measure of yield, while x-ray diffraction provided the most explicit structural data. Mossbauer spectroscopy provided direct critical atomic level data, whereas transmission electron microscopy provided key information on shock-modified, but unreacted reactant mixtures. The results of determinations of product yield and identification of product are summarized in Fig. 8.2. What is shown in the figure is the location of pressure, mean-bulk temperature locations at which synthesis experiments were carried out. Beside each point are the measures of product yield as determined from the three probes. The yields vary from 1% to 75 % depending on the shock conditions. From a structural point of view a surprising result is that the product composition is apparently not changed with various shock conditions. The same product is apparently obtained under all conditions only the yield is changed. 

In these equations all of the physical properties ate taken at the mean bulk temperature of the fluid (T, + T0)/2, where 7) and T0 are the inlet and outlet temperatures. The difference in the value of the index for heating and cooling occurs because in the former case the film temperature will be greater than the bulk temperature and in the latter case less. Conditions in the film, particularly the viscosity of the fluid, exert an important effect on the heat transfer process. 

For the individual (film) coefficient h for heating or cooling of fluids, without phase change, in turbulent flow through circular tubes, the following dimensionless equation [2] is well established. In the following equations all fluid properties are evaluated at the arithmetic-mean bulk temperature. 

Viscosity Correction for the Dirty Exchanger. These overall coefficients are based upon the uncorrected film heat transfer coefficients, i.e. those without the viscosity correction parameter (gfgw)014. As the temperature difference across a layer is proportional to the layer s thermal resistance (see Equations (2) and (10)), the relative resistances, above, allow estimation of the mean tube wall temperature, and hence evaluation of the viscosity corrections (jib/hw)° 14 to the film coefficients. Mean bulk temperature difference ... 

If Tm is the mean bulk temperature at any point in a flow channel then the rate of change of Tm with respect to distance z along the duct is given by applying an energy... 

Water at a mean bulk temperature of 35°C flows through a pipe with a diameter of 3 mm whose wall is maintained at a uniform temperature of 55°C. The tube length is 2 m. If the mean velocity of the water in the pipe is 0.05 m/s, find the heat-transfer rate to the water assuming fully developed flow. 

Water flows in a rectangular 5 mm x 10 mm duct with a mean bulk temperature of 20°C. If the duct wall is kept at a uniform temperature of 40°C and if fully developed laminar flow is assumed to exist, find the heat transfer rate per unit length of the duct. 

In this formula the average heat-transfer coefficient is based on the arithmetic average of the inlet and outlet temperature differences, and all fluid properties are evaluated at the mean bulk temperature of the fluid, except /j., which is evaluated at the wall temperature. Equation (6-10) obviously cannot be used for extremely long tubes since it would yield a zero heat-transfer coefficient. A comparison by Knudsen and Katz [9, p. 377] of Eq. (6-10) with other relationships indicates that it is valid for... 

We base the heat transfer on mean bulk temperature of S2°C, so that... 

Conversion Mean bulk temperature Catalyst-bed depth, ft... 

Conversion Mean bulk temperature, °C Catalyst-bed depths cm... 

Assuming the temperature dependence of the reaction rate follows the Arrhenius form, the ratio of the maximum rates at the maximum radial mean (bulk) temperature and the wall temperature must satisfy the following constraint defined earlier ... 

The Reynolds number is calculated on the assumption that the fluid temperature equals the mean bulk temperature, which is defined as the arithmetic average of the inlet and outlet temperatures. 

The friction factor corresponding to the mean bulk temperature is divided by a factor j/, which in turn is calculated from the following equations ... 

Calculate the mean bulk temperature as the average of the inlet and outlet bulk fluid temperatures. 

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