Polar Spherical Coordinates

The elastic parameters of fruit and vegetables are normally monitored in quality control processes as there is a good correlation to the degrees of firmness, turgidity and humidity. These parameters have been traditionally measured by means of penetration tests, which are destructive. This has resulted in the increased attention recently given to ultrasonic techniques applied to the quality evaluation of horticultural commodities. Nevertheless, since most of the fruit and vegetables display a viscoelastic behavior, the penetration test should be considered to be quasi-static, especially when compared with the speeds associated with the ultrasonic tests. Both methods should provide different values for the elastic parameters. The aim of this work is to study this discrepancy in the values of the elastic parameters and interpret the elastodynamic behavior of the vegetable tissue under an ultrasound test. Thus, the paper presents an ultrasonic nondestructive method to evaluate the elas...

In this study, an exact Eulerian solution is derived for the bidirectional vortex in a conical chamber. Our model is applicable to idealized representations of cyclone separators and liquid rocket engines with slowly expanding chamber cross-sections. The bulk fluid motion is assumed to be nonreactive, steady, rotational, inviscid, and incompressible. Our approach is founded on the Bragg-Hawthorne equation and seeks to overcome some of the deficiencies encountered by Bloor and Ingham (Bloor, M. I. G., and Ingham, D. B., “The Flow in Industrial Cyclones,” Journal of Fluid Mechanics, Vol. 178, 1987, pp. 507-519). Despite inevitable similarities with Bloor and Ingham’s model, ours is constructed using a judicious framework that connects the solution to the swirl number and the cone divergence angle. In consequence, a self-similar formulation is produced that is independent of the cone’s finite body length. closed-form representations for the principal variables This enables us able to c...

In this study, an exact Eulerian solution is derived for the bidirectional vortex in a conical chamber. Our model is applicable to idealized representations of cyclone separators and liquid rocket engines with slowly expanding chamber cross-sections. The bulk fluid motion is assumed to be nonreactive, steady, rotational, inviscid, and incompressible. Our approach is founded on the Bragg-Hawthorne equation and seeks to overcome some of the deficiencies encountered by Bloor and Ingham (Bloor, M. I. G., and Ingham, D. B., “The Flow in Industrial Cyclones,” Journal of Fluid Mechanics, Vol. 178, 1987, pp. 507-519). Despite inevitable similarities with Bloor and Ingham’s model, ours is constructed using a judicious framework that connects the solution to the swirl number and the cone divergence angle. In consequence, a self-similar formulation is produced that is independent of the cone’s finite body length. closed-form representations for the principal variables This enables us able to characterize the problem by providing of flow motion. parameters of interest, the mantle divergence angle and the maximum latitudinal velocity are obtained explicitly. The mantle consists of a spinning cone that separates circumferential inflow region (outer vortex) from the central outflow (inner vortex). interface bisects the fluid domain at 60% of the cone’s divergence half-angle. Its precise determination helps to estimate the crossflow velocity responsible for mass transfer, or spillage as it were, from the outer vortex into the inner region. Finally, results are illustrated while varying the cone divergence angle and spatial location in both spherical and polar cylindrical coordinates. Nomenclature

The plant tissues, in particular the orange’s skin, are essentially aqueous-filled structures of varying density and elasticity composed by fat and acids substances, as well as areas of intercellular air. In recent years many techniques have been developed for ultrasonic characterization of fruit and vegetables in postharvest processes. All have found common macroscopic acoustic parameters: a slow propagation speed and a large absorption. It is intended therefore to obtain a prediction model of ultrasounds propagation in this viscoelastic heterogeneous media by FDTD (Finite Difference Time Domain) techniques. Thus, this paper presents a time domain numerical model for simulating acoustic propagation in plant tissue. This model correctly describes the special characteristics of wave propagation experimentally detected in these biological tissues, emphasizing the need for elastic frequency dependent characteristics: a frequency dependent elasticity modulus. These simulation results wi...

There are hundreds of PTC Mathcad functions. The following is a partial list of the different categories that have functions: Bessel Functions, Complex Numbers, Curve Fitting, Data Analysis, Differential Equations, Finance, Fourier Transforms, Graphing, Hyperbolic Functions, Image Processing, Interpolation, Logs, Number Theory, Probability, Solving, Sorting, Statistics, Trigonometry, Vectors, and Waves.

Lactoferrin (Lf) samples with ca. 25%, 50%, 75%, 85% and 100% iron saturation were prepared for the purpose of evaluating Chromametry, Differential Scanning Calorimetry (DSC) and Circular Dichroism (CD) spectropolarimetry for their suitability in determining the iron saturation level. Numerical values for colour from Chromametry, enthalpy change of denaturation (ΔHcal) from DSC and molar ellipticities from CD were statistically analysed to evaluate their correlation with the level of iron saturation in Lf. Linear regression analysis of colour coordinates Chroma (C(∗)) and hue (h°) angle on percentage iron saturation level of Lf showed that the values can be used to estimate the iron saturation level. The ΔHcal for the iron saturated peak and the CD ellipticities in the 310-340 nm region provided reliable data for the estimation of iron saturation level of Lf up to 75%. Mono- and di-saturated Lf displayed the same thermal stability and very similar tertiary structures.

In a numerical study of the electric field inside an Electro Capacitance Tomography (ECT) device and other applications, often a Poisson equation with a discontinuous coefficient needs to be solved in polar coordinates. This paper is devoted to the Immersed Interface Method (IIM) in polar coordinates and the application to the solution of the electric potential inside an ECT device.