The Fractal Calculus for Fractal Materials

The major problem in the process of mixing fluids (for instance liquid-liquid mixers) is turbulence, which is the outcome of the function of the equipment (engine). Fractal mixing is an alternative method that has symmetry and is predictable. Therefore, fractal structures and fractal reactors find importance. Using F α -fractal calculus, in this paper, we derive exact F α -differential forms of an ideal gas. Depending on the dimensionality of space, we should first obtain the integral staircase function and mass function of our geometry. When gases expand inside the fractal structure because of changes from the i + 1 iteration to the i iteration, in fact, we are faced with fluid mixing inside our fractal structure, which can be described by physical quantities P, V, and T. Finally, for the ideal gas equation, we calculate volume expansivity and isothermal compressibility.

Accurate and precise lattice parameters of H2O and D2O ice Ihbetween 1.6 and 270 K from high-resolution time-of-flight neutron powder diffraction data

Accurate and precise lattice parameters for D2O and H2O varieties of hexagonal ice (ice Ih, space groupP63/mmc) have been obtained in the range 1.6 to 270 K. Precision of the lattice parameters (∼0.0002% inaand 0.0004% incfor D2O, 0.0008% inaand 0.0015% incfor H2O) is ensured by use of the time-of-flight method on one of the longest primary neutron flight-path instruments in the world, the High-Resolution Powder Diffractometer at the ISIS neutron source. These data provide a more precise description of the negative thermal expansion of the material at low temperatures than the previous synchrotron `gold standard' [Röttgeret al.(1994).Acta Cryst.B50, 644–648], including the region below 10 K where the lattice parameters saturate. The volume expansivity of both isotopologues turns negative below 59–60 K, in excellent agreement with a recent dilatometry study. The axial expansivities are highly isotropic (differing by < 1% in D2O ice Ih). Furthermore, thec/aratio of different D2O ice samples exhibit a statistically significant dispersion of ∼0.015% below 150 K that appears to depend on the thermal history of the sample, which disappears on warming above 150 K. Similarly, H2O ice exhibits a `kink' in thec/aratio at ∼115 K. The most plausible explanation is a freezing-in of the molecular reorientation process on cooling and subsequent relaxation on warming.

One-dimensional refraction properties of compression shocks in non-ideal gases

Non-ideal gases refer to deformable substances in which the speed of sound can decrease following an isentropic compression. This may occur near a phase transition such as the liquid–vapour critical point due to long-range molecular interactions. Isentropes can then become locally concave in the pressure/specific-volume phase diagram (e.g. Bethe–Zel’dovich–Thompson (BZT) gases). Following the pioneering work of Bethe (Tech. Rep. 545, Office of Scientific Research and Development, 1942) on shocks in non-ideal gases, this paper explores the refraction properties of stable compression shocks in non-reacting but arbitrary substances featuring a positive isobaric volume expansivity. A small-perturbation analysis is carried out to obtain analytical expressions for the thermo-acoustic properties of the refracted field normal to the shock front. Three new regimes are discovered: (i) an extensive but selective (in upstream Mach numbers) amplification of the entropy mode (hundreds of times larger than those of a corresponding ideal gas); (ii) discontinuous (in upstream Mach numbers) refraction properties following the appearance of non-admissible portions of the shock adiabats; (iii) the emergence of a phase shift for the generated acoustic modes and therefore the existence of conditions for which the perturbed shock does not produce any acoustic field (i.e. ‘quiet’ shocks, to contrast with the spontaneous D’yakov–Kontorovich acoustic emission expected in 2D or 3D). In the context of multidimensional flows, and compressible turbulence in particular, these results demonstrate a variety of pathways by which a supplied amount of energy (in the form of an entropy, vortical or acoustic mode) can be redistributed in the form of other entropy, acoustic and vortical modes in a manner that is simply not achievable in ideal gases. These findings are relevant for turbines and compressors operating close to the liquid–vapour critical point (e.g. organic Rankine cycle expanders, supercritical $\text{CO}_{2}$ compressors), astrophysical flows modelled as continuum media with exotic equations of state (e.g. the early Universe) or Bose–Einstein condensates with small but finite temperature effects.

Specific heat and entropy

Before beginning the discussion of directional properties, we pause to consider specific heat, an important scalar property of solids which helps illustrate the important thermodynamic relationships between measured properties. Heat capacity, compressibility, and volume expansivity are interrelated through the laws of thermodynamics. Based on these ideas, similar relationships are established for other electrical, thermal, mechanical, and magnetic properties. Several atomistic concepts are introduced to help understand the structure–property relationships involved in specific heat measurements. The heat capacity or specific heat is the amount of heat required to raise the temperature of a solid by 1K. It is usually measured in units of J/kg K. Theorists prefer to work in J/mole K, and older scientists sometimes use calories rather than joules. One calorie is 4.186 J. For solids and liquids, the specific heat is normally measured at a constant pressure: where ΔQ is the heat added to increase the temperature by ΔT. Measurements on gases are usually carried out at constant volume: Electrical methods are generally employed in measuring specific heat. A heating coil is wrapped around the sample and the resulting change in temperature is measured with a thermocouple. If a current I flows through a wire of resistance R, the heat generated by the wire in a time Δt is given by . . . ΔQ = I2R

Anisotropy of Intermolecular Interactions from the Study of the Thermal-Expansion Tensor

The anisotropy of the intermolecular interactions in the low-temperature ordered phases of three chemically and structurally related compounds [neopentylglycol, (CH3)2C(CH2OH)2, pivalic acid, (CH3)3C(COOH), and neopentylalcohol, (CH3)3C(CH2OH)], all of which display an orientationally disordered high-temperature phase, has been shown by means of the isobaric thermal-expansion tensor. The variation of the directions of the principal components of the thermal-expansion tensor as a function of temperature, as well as the variation of its principal coefficients, is evidence of the large differences in the intermolecular interactions for each compound; or, more precisely, between the strong intermolecular hydrogen bonds and the weak van der Waals interactions. In addition, the differences in the hydrogen-bonding schemes expecteda priorifrom the molecular structures of the studied compounds have been enhanced. Finally, the volume expansivity as well as the packing coefficient have been analysed in the orientationally disordered high-temperature phase of each of the three compounds.