Derive Equation of Continuity Spherical Coordinates

Continuity and Navier-Stokes Equations in Different Coordinate Systems

Bastian E. Rapp , in Microfluidics: Modelling, Mechanics and Mathematics, 2017

13.4.1 Continuity Equation: Compressible

For spherical coordinates, ¹ we derived the divergence in Eq. 7.122, which allows us to express the continuity equation for compressible fluids as

$\begin{array}{ll}\frac{\partial \rho }{\partial t}+\overrightarrow{\nabla }\cdot \left(\rho \overrightarrow{\upsilon }\right)\hfill & =0\hfill \\ \frac{\partial \rho }{\partial t}+\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\rho F_r\right)+\frac{1}{r\text{sin}\theta }\left(\frac{\partial }{\partial \theta }\left(p{F}_0\text{sin}\theta +\frac{\partial \rho F_{\phi }}{\partial \phi }\right)\right)\hfill & =0\hfill \end{array}$

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Radiation and scattering of sound by the boundary value method

Leo L. Beranek , Tim J. Mellow , in Acoustics: Sound Fields and Transducers, 2012

Incident field

In spherical coordinates, the incident plane wave pressure is

(12.7) ${\tilde{p}}_I(r\text{,}\theta )={\tilde{p}}_0{e}^{-jkr\text{cos}\theta }\text{.}$

Fortunately, this expression can be expanded in terms of spherical Bessel functions j_n and Legendre functions P_n as follows:

(12.8) ${\tilde{p}}_I(r\text{,}\theta )={\tilde{p}}_0\sum _{n=0}^{\mathrm{\infty }}{(-j)}^n(2n+1)j_n(kr)P_n(\text{cos}\theta )\text{.}$

This expression, which is similar in form to Eq. (2.164) for the solution to the wave equation in spherical coordinates, is treated more rigorously later in the derivation of Eq. (13.63).

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The Elastic Solid

W. Michael Lai , ... Erhard Krempl , in Introduction to Continuum Mechanics (Fourth Edition), 2010

5.42 HOLLOW SPHERE SUBJECTED TO UNIFORM INTERNAL AND EXTERNAL PRESSURE

In spherical coordinates ( R, β, θ), where β is the angle between e _z and e _R and θ is the longitude angle, consider the following potential functions for spherical symmetric problems:

(5.42.1) $\begin{array}{lll}\psi =B{Re}_R,\hfill & \phi =\frac{A}{R},\hfill & A\text{and}B\text{are}\text{constants}.\hfill \end{array}$

From Example 5.38.5, we have the following nonzero stress components:

(5.42.2) $T_{RR}=-2\{(2-v)\frac{d\psi }{dR}+(2v-1)\frac{\psi }{R}-\frac{d^2\phi }{2d{R}^2}\}=-2(1+v)B+2\frac{A}{R_3},$

(5.42.3) $T_{\beta \beta }=T_{\theta \theta }=-\{(2v-1)\frac{d\psi }{dR}+\frac{3\psi }{R}-\frac{1d\phi }{dR}\}=-2(1+v)B+2\frac{A}{R^3},$

and the following displacement components:

(5.42.4) $2\mu u_R=R\frac{d\psi }{dR}+(-3+4v)\psi +\frac{d\phi }{dR}=2(2v-1)BR-\frac{A}{R^2},u_{\beta }=u_{\theta }=0.$

Let the internal and external uniform pressure be denoted by p _i and p _o, respectively, then the boundary conditions are

(5.42.5) $T_{RR}=-pi\text{at}\text{internal}\text{radious}R=R_{i,}$

and

(5.42.6) $T_{RR}=-p_{\text{o}}\text{at}\text{external}\text{radious}R=R_{\text{o}}.$

Thus,

(5.42.7) $\begin{array}{ll}-2\left(1+v\right)B+\frac{2A}{R_i^3}=-p_i,\hfill & -2\left(1+v\right)B+\frac{2A}{R_{\text{o}}^3}=-p_{\text{o}}.\hfill \end{array}$

from which we have

(5.42.8) $\begin{array}{ll}2A=(-p_i+p_{\text{o}})\frac{R_i^3{R}_{\text{o}}^3}{R_{\text{o}}^3-R_i^3},\hfill & 2(1+v)B=\frac{p_{\text{o}}{R}_{\text{o}}^3-R_i^{3}pi}{R_{\text{o}}^3-R_i^3}\hfill \end{array}.$

If p _i = 0, then

(5.42.9) $T_{RR}=\frac{p_{\text{o}}{R}_{\text{o}}^3}{R_i^3-R_{\text{o}}^3}-\left(\frac{p_{\text{o}}{R}_i^3{R}_{\text{o}}^{\text{3}}}{R_i^3-R_{\text{o}}^3}\right)\frac{1}{R^3},$

(5.42.10) $T_{\theta \theta }=T_{\beta \beta }=\frac{p_{\text{o}}{R}_{\text{o}}^3}{R_i^3-R_o^3}+\left(\frac{p_{\text{o}}{R}_i^{\text{o}}{R}_{\text{o}}^3}{R_i^3-R_{\text{o}}^3}\right)\frac{1}{2R^3}.$

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Potential Fields of Interacting Spherical Inclusions

Volodymyr I. Kushch , in Micromechanics of Composites, 2013

2.1.1 Scalar Spherical Harmonics

The spherical coordinates $(r\text{,}\theta \text{,}\phi )$ (Figure 2.1) relate to the Cartesian coordinates $(x_1\text{,}{x}_2\text{,}{x}_3)$ by

Figure 2.1. Spherical coordinate frame.

$\begin{array}{cc}\hfill x_1+\mathrm{i}{x}_2=& r\sin \theta \exp (\mathrm{i}\phi )\text{,}{x}_3=r\cos \theta \text{,}\hfill \\ \hfill & (r⩾0\text{,}0⩽\theta ⩽\pi \text{,}0⩽\phi <2\pi )\text{.}\hfill \end{array}$

Separation of variables in Laplace equation are written in spherical coordinates as

(2.1) ${▿}^{2}T(\mathbf{r})=\frac{\partial }{\partial r}\left(r^2\frac{\partial T}{\partial r}\right)+\frac{1}{\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta \frac{\partial T}{\partial \theta }\right)+\frac{1}{{\sin }^2\theta }\frac{{\partial }^{2}T}{\partial {\phi }^2}=0$

which gives us a complete set of partial ("normal," in Hobson [74] terminology) solutions of the form

(2.2) $r^t{P}_t^s(\cos \theta )\exp (\mathrm{i}s\phi )=r^t{\chi }_t^s(\theta \text{,}\phi )(-\infty <t<\infty \text{,}|s|⩽t)\text{,}$

referred [138] to as scalar solid spherical harmonics of degree t and order s. Here and below, the radius-vector (position-vector field, in Gurtin [58] terminology) $\mathbf{r}=x_k{\mathbf{i}}_k\text{,}{\mathbf{i}}_k$ being the unit base (reference) vectors of the Cartesian coordinate system. In Eq. (2.2), $P_t^s$ are the associated Legendre functions of the first kind [74]:

(2.3) $P_t^s(\eta )={\overline{\eta }}^s\frac{d^s{P}_t(\eta )}{d{\eta }^s}=\frac{(-1{)}^t{\overline{\eta }}^s}{2^{t}t!}\frac{d^{t+s}{\overline{\eta }}^{2t}}{d{\eta }^{t+s}}$

for $|\eta |⩽1$ . The relationships

(2.4) $P_t^{-s}(\eta )=(-1{)}^s\frac{(t-s)!}{(t+s)!}{P}_t^s(\eta )\text{;}$

$\begin{array}{cc}\hfill P_{-(t+1)}^s(\eta )=& P_t^s(\eta )(0⩽t<\infty \text{,}|s|⩽t)\text{;}\hfill \\ \hfill P_t^s(\eta )\equiv & 0(|s|>t)\text{,}\hfill \end{array}$

redefine the functions $P_t^s$ Eq. (2.3) to the arbitrary integer indices t and s. It also follows from Eq. (2.3) that

$P_t^s(-\eta )=(-1{)}^{t+s}{P}_t^s(\eta )\text{.}$

The following recurrent formulas [74] are valid for all the indices t and s:

(2.5) $\begin{array}{cc}\hfill (2t+1)\sqrt{1-{\eta }^2}{P}_t^s(\eta )=& P_{t+1}^{s+1}(\eta )-P_{t-1}^{s+1}(\eta )\hfill \\ \hfill =& (t+s)(t+s-1)P_{t-1}^{s-1}(\eta )\hfill \\ \hfill & -(t-s+1)(t-s+2)P_{t+1}^{s-1}(\eta )\text{;}\hfill \\ \hfill (2t+1)\eta P_t^s(\eta )=& (t-s+1)P_{t+1}^s(\eta )+(t+s)P_{t-1}^s(\eta )\text{;}\hfill \\ \hfill (1-{\eta }^2)\frac{\partial }{\partial \eta }{P}_t^s(\eta )=& (t+1)\eta P_t^s(\eta )-(t-s+1)P_{t+1}^s(\eta )\hfill \\ \hfill =& t\eta P_t^s(\eta )+(t+s)P_{t-1}^s(\eta )\text{.}\hfill \end{array}$

In Eq. (2.2), ${\chi }_t^s$ are the scalar surface spherical harmonics

(2.6) ${\chi }_t^s(\theta \text{,}\phi )=P_t^s(\cos \theta )\exp (\mathrm{i}s\phi )\text{.}$

They obey the differential equation [156]

$\frac{{\partial }^2}{\partial {\theta }^2}{\chi }_t^s+\cot \theta \frac{\partial }{\partial \theta }{\chi }_t^s+\frac{1}{{\sin }^2\theta }\frac{{\partial }^2}{\partial {\phi }^2}{\chi }_t^s+t(t+1){\chi }_t^s=0$

and possess the orthogonality property

(2.7) $\frac{1}{S}{\int }_S{\chi }_t^s\overline{{\chi }_k^l}\mathit{dS}={\alpha }_{\mathit{ts}}{\delta }_{\mathit{tk}}{\delta }_{\mathit{sl}}\text{,}{\alpha }_{\mathit{ts}}=\frac{1}{2t+1}\frac{(t+s)!}{(t-s)!}\text{,}$

where integral is taken over the spherical surface S; over bar means complex conjugate and ${\delta }_{\mathit{ij}}$ is the Kronecker delta. Also, in view of Eq. (2.4),

(2.8) ${\chi }_t^{-s}=(-1{)}^s\frac{(t-s)!}{(t+s)!}\overline{{\chi }_t^s}\text{;}$

$\begin{array}{cc}\hfill {\chi }_{-(t+1)}^s=& {\chi }_t^s(0⩽t<\infty \text{,}|s|⩽t)\text{;}\hfill \\ \hfill {\chi }_t^s\equiv & 0(|s|>t)\text{.}\hfill \end{array}$

The recurrent formulas for ${\chi }_t^s$ follow from Eq. (2.5): they are

$\begin{array}{cc}\hfill (2t+1)\cos \theta {\chi }_t^s=(t-s+1){\chi }_{t+1}^s+(t+s){\chi }_{t-1}^s\text{;}& \hfill \\ \hfill \sin \theta \frac{\partial }{\partial \theta }{\chi }_t^s=(t+1)\cos \theta {\chi }_t^s-(t-s+1){\chi }_{t+1}^s\text{;}& \hfill \\ \hfill \exp (\mathrm{i}\phi )\left[\cos \theta \frac{\partial }{\partial \theta }{\chi }_t^s-(t+1)\sin \theta {\chi }_t^s-\frac{s}{\sin \theta }{\chi }_t^s\right]=-{\chi }_{t+1}^{s+1}\text{;}& \hfill \\ \hfill \exp (-\mathrm{i}\phi )\left[\cos \theta \frac{\partial }{\partial \theta }{\chi }_t^s-(t+1)\sin \theta {\chi }_t^s+\frac{s}{\sin \theta }{\chi }_t^s\right]& \hfill \\ \hfill =(t-s+1)(t-s+2){\chi }_{t+1}^{s-1}\text{;}& \hfill \\ \hfill \exp (\mathrm{i}\phi )\left(s\frac{\cos \theta }{\sin \theta }{\chi }_t^s-\frac{\partial }{\partial \theta }{\chi }_t^s\right)={\chi }_t^{s+1}\text{;}& \hfill \\ \hfill \exp (-\mathrm{i}\phi )\left(s\frac{\cos \theta }{\sin \theta }{\chi }_t^s+\frac{\partial }{\partial \theta }{\chi }_t^s\right)=(t+s)(t+s-1){\chi }_t^{s-1}.& \hfill \end{array}$

An expansion of the any piecewise continuous at the sphere function $f(\theta \text{,}\phi )$ over a set of surface spherical harmonics Eq. (2.6) is given by the formula [74]:

$f(\theta \text{,}\phi )={\displaystyle \sum _{t=0}^{\infty }}{\displaystyle \sum _{s=-t}^t}{c}_{\mathit{ts}}{\chi }_t^s(\theta \text{,}\phi )\text{,}$

where,

$c_{\mathit{ts}}=\frac{1}{4\pi {\alpha }_{\mathit{ts}}}{\int }_0^{2\pi }d\phi {\int }_0^{\pi }f(\theta \text{,}\phi )\overline{{\chi }_t^s(\theta \text{,}\phi )}\sin \theta d\theta \text{.}$

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Polymer Characterization

Geoffrey R. Mitchell , in Comprehensive Polymer Science and Supplements, 1989

31.3.3 Structure and Scattering Intensity

In spherical coordinates, any cylindrically symmetric function with an inversion centre can be expanded in a series of Legendre polynomials of even order (as for the orientation distribution function). Thus the cylindrical distribution function W( r, α), the anisotropic version of the radial distribution function may be expanded

(27)

Similarly, the reduced intensity function i(s, α) may be expanded

(28)

where

(29)

The amplitudes of the spherical harmonics of the cylindrical distribution function may be related to those of the scattering function by ^55,56

(30)

where, j_2n are spherical Bessel functions; the first few terms are shown in Table 3. For small departures from isotropy, only a few terms will be needed to construct the complete cylindrical distribution function, for example a deformed glassy sample of poly(methyl methacrylate) required four terms. ⁵⁶ In such cases the first term W₀(r) will dominate and this is identical with the radial distribution function of Section 31.2.2. For such systems, the 'sampled transform' method may be applied in the calculation of equation (30). ⁵⁶ For a highly aligned material, such as a liquid crystal polymer system, many terms will be required ⁴⁹ and it may be more useful to employ an alternative relationship ⁵⁷

(31)

where J₀ is the Bessel function of zero order, and α′ and α the azimuthual angle in real and reciprocal space respectively. The use of a deconvolution route to give atomic cylindrical distribution functions ⁵⁶ is identical to that described for the radial functions (Section 31.2.2).

The expressions detailed above provide a route to the calculation of the cylindrical distribution function from the observed scattering. However, we can utilize the expansions of the various functions into series of spherical harmonics in a much more powerful manner, and one which provides a direct route to both the evaluation of the orientation distribution function and the chain conformation from arcing of the measured scattering patterns. ^58–60

The scattering for an aligned polymer system is given by the convolution of the orientation distribution function D(α) and the scattering for a perfectly aligned system I′(s, α) ^{55, 58}

(32)

Normally the convolution would involve terms coupling spatial (i.e. dependent upon s) and orientational (i.e. dependent upon α) order. Attempts have been made to deconvolute I(s, α) to obtain I′(s, α) for various glassy polymers using arbitrary orientation distribution functions. ^61–63 However, for independent scattering units, we may use the properties of Legendre polynomials ⁵² in terms of the general Legendre addition theorem to write the convolution as ^{55, 58, 59}

(33)

where I_2n (s), I′_2n (s) and D_2n are the spherical harmonics of the appropriate functions in equation (32). Obviously the relationship given in equation (33) is only valid for a system in which D(α) and I′(s, α) have cylindrical symmetry. Furthermore this equation is only valid over an s range in which the scattering arises solely from a single structural unit uncomplicated by interunit correlations. (A simple example of such a unit would be a crystallite, in semi-crystalline polymer. For a disordered system we would have to consider chain segments, unless some 'domain' structure existed.) Similar relationships may be established for the spherical harmonic expansion of the real space cylindrical distribution function W(r, α) (equation 27). ⁵⁶ We may utilize equation (33) in a number of different but interrelated ways. Of course if we had a numerical knowledge of the orientation distribution function D(α) or its harmonics D_2n , then equation (33) could be inverted to deconvolute the observed scattering in order to obtain the scattering function for a perfectly aligned system. This approach would only be valid if we postulate some 'domain' type model. Such a model assumes that the spread of orientation is due to the misorientation of the domains. This type of analysis has been used to prepare a cylindrical distribution function for a liquid crystal polymer system. ⁴⁹

However, in general the above procedures are unnecessary because of a particular feature of equation (33). The 'observed' spherical harmonics, whether they are reciprocal space or real space functions, are those of a perfectly aligned system, weighted with the appropriate harmonic of the distribution function. Since the values of D_2n are independent of s, the only difference between I_2n (s) and I′_2n (s) in equation (33) is a constant. In other words, we could compare the observed I_2n (s) functions with those calculated from perfectly aligned models, without requiring a knowledge of the orientation distribution function, as long as the comparison is restricted to peak positions, widths and shape, and relative intensities. When the two sets of curves match, not only is the structures known, but the ratio of the two gives the coefficients of the orientation distribution function! This type of analysis has been applied, in reciprocal space and real space, ⁵⁷ to both non-crystalline polymers ⁵⁹ and liquid crystal polymers ^{60, 64} using individual chain segments as the structural units.

In those cases described above the structural analysis has concentrated on the intrachain molecular scattering. In principle we could utilize the interchain scattering to provide a measure of the molecular orientation. To do so requires a model of the interchain scattering, or cylindrical distribution function. Remarkably, the model chosen by many is invariant and is a set of infinitely long parallel rods, a model almost appropriate to the liquid crystal state but certainly not for a non-crystalline polymer. Such a model exhibits scattering which is confined to the equatorial plane (α   =   90°). It is more convenient, before proceeding further, to rewrite equation (33) in terms of the normalized amplitudes of the spherical harmonics

(34)

where the subscripts D indicates the orientation distribution function, and for example 〈P_2n (cos α)〉 is given by

(35)

This is in essence the normalized version of equation (29). In order to operate this procedure we need values for the model harmonics 〈P_2n (cos α)〉 and for the simple model of a set of parallel rods the first few terms are ⁵⁸

(36)

A more complete description of the application of these procedures in measuring the molecular orientation of liquid crystal polymers is given in ref. 50.

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Attitude Representation

Enrico Canuto , ... Carlos Perez Montenegro , in Spacecraft Dynamics and Control, 2018

Exercise 13

Given the spherical coordinates of the unit vector r/r in Eq. (2.4), prove the following identity:

(2.43) $\left[\begin{array}{c}\cos \alpha \sin \beta \\ \sin \alpha \sin \beta \\ \cos \beta \end{array}\right]=Z\left(\alpha \right)Y\left(\beta \right)\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]\text{,}$

and provide a geometrical interpretation. □

Since the three angles of rotation are to each other different, the Euler rotations are denoted by

(2.44) $X\left(\phi \right),Y\left(\theta \right),Z\left(\psi \right)\text{,}$

or simply by the integers 1, 2, and 3. The three angles in {φ,θ,ψ} are known as Euler angles.

The factorization in Eq. (2.39) reveals another important property of 3D proper orthogonal matrices. They may be just represented by three appropriate Euler angles as the next theorem proves.

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Case Studies

Hebertt Sira-Ramírez , ... Eric William Zurita-Bustamante , in Active Disturbance Rejection Control of Dynamic Systems, 2017

5.2.1 Derivation of the Model via Euler–Lagrange Equations

We use spherical coordinates to define the position of the end effector, idealized to be a point on the surface of the sphere acting as the configuration space. The kinetic energy of the system is given by

(5.1) $K=\frac{1}{2}{m}_2{l}_2^2({\dot{\varphi }}^2+{\sin }^2{\dot{\varphi }}^2)$

The potential energy is

(5.2) $V=m_{2}g(l_1+l_2\cos \varphi )-{\tau }_1\theta -{\tau }_2\varphi$

The Lagrangian of the system is then obtained as

(5.3) $\mathcal{L}=\frac{1}{2}{m}_2{l}_2^2({\dot{\varphi }}^2+{\sin }^2{\dot{\varphi }}^2)-m_{2}g(l_1+l_2\cos \varphi )+{\tau }_1\theta +{\tau }_2\varphi$

The mathematical model of the system is then derived from the Euler–Lagrange equations, obtaining the following differential equations in matrix form:

(5.4) $\begin{array}{cc}\hfill & \left[\begin{array}{cc}\hfill m_2{l}_2^2{\sin }^2\varphi \hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill m_2{l}_2^2\hfill \end{array}\right]\left[\begin{array}{c}\hfill \ddot{\theta }\hfill \\ \hfill \ddot{\varphi }\hfill \end{array}\right]+\left[\begin{array}{cc}\hfill m_2{l}_2^2\dot{\varphi }\sin \varphi \cos \varphi \hfill & \hfill m_2{l}_2^2\dot{\theta }\sin \varphi \cos \varphi \hfill \\ \hfill -m_2{l}_2^2\dot{\theta }\sin \varphi \cos \varphi \hfill & \hfill 0\hfill \end{array}\right]\left[\begin{array}{c}\hfill \dot{\theta }\hfill \\ \hfill \dot{\varphi }\hfill \end{array}\right]\hfill \\ \hfill & +\left[\begin{array}{c}\hfill 0\hfill \\ \hfill -m_{2}g{l}_2\sin \varphi \hfill \end{array}\right]=\left[\begin{array}{c}\hfill {\tau }_1\hfill \\ \hfill {\tau }_2\hfill \end{array}\right]\hfill \end{array}$

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SURFACE AND INTERFACE ANALYSIS AND PROPERTIES

Kathleen J. Stebe , Shi-Yow Lin , in Handbook of Surfaces and Interfaces of Materials, 2001

APPENDIX A DERIVATION FOR EQUATION (3.5)

Fick's law in spherical coordinate is

(A.1) $\frac{D}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial C}{\partial r}\right)=\frac{\partial C}{\partial t}(r>b,t>0)$

with initial conditions

(A.2a) $C(r,t)=C_{\infty }(r>b,t=0)$

(A.2b) $\Gamma (t)={\Gamma }_b(t=0)$

and boundary conditions

(A.3a) $C(r,t)=C_{\infty }(r\to \alpha ,t>0)$

(A.3b) $d\Gamma /dt=D\left(\partial C/\partial r\right)\left(r=b,t>0\right)$

(A.3c) $C(r,t)=C_s(t)(r=b,t)$

Define dimensionless variables:

$\begin{array}{c}\theta =\frac{C_{\infty }-C}{C_{\infty }},\tau =\frac{t}{b^2/D},z=r/b,\\ \text{and}{\Gamma }^{*}=\frac{\Gamma -{\Gamma }_b}{{\Gamma }_{\infty }}\end{array}$

Here, Γ_∞ is the maximum surface concentration. Equation (A.1) becomes

(A.4) $\frac{\partial \theta }{\partial \tau }=\frac{1}{z^2}\frac{\partial }{\partial z}\left(z^2\frac{\partial \theta }{\partial z}\right)$

Initial and boundary conditions become

(A.5a) $\theta (z,\tau )=0(z>1,\tau =0)$

(A.5b) ${\Gamma }^{*}(z,\tau )=0(z=1,\tau =0)$

(A.6a) $\theta (z,t)=0(z\to \infty ,\tau >0)$

(A.6b) $\frac{{\Gamma }_{\infty }}{C_{\infty }b}\frac{d{\Gamma }^{*}}{d\tau }=-\frac{\partial \theta }{\partial z}(z=1,\tau >0)$

(A.6c) $\theta (z,t)=\frac{C_{\infty }-C_s}{C_{\infty }}={\theta }_1(\tau )(z=1,\tau )$

Taking Laplace transform with respect to time, Eqs. (A.4), (A.6b) and (A.6c) become

(A.7) $s\overline{\theta }(z,s)-\theta (z,\tau =0)=\frac{1}{z^2}\frac{d}{d_z}\left(z^2\frac{d\overline{\theta }}{dz}\right)$

(A.8a) ${\frac{{\Gamma }_{\infty }}{C_{\infty }b}\left[s{\overline{\Gamma }}^{*}(s)-{\Gamma }^{*}(0)\right]=-\frac{d\overline{\theta }(z,s)}{dz}|}_{z=1}$

(A.8b) $\overline{\theta }(z,s)={\overline{\theta }}_1(s)(z=1,s)$

According to Eq. (A.5b), Γ*(0) = 0; therefore, Eq. (A.8a) becomes

(A.8c) ${\frac{{\Gamma }_{\infty }s}{C_{\infty }b}{\overline{\Gamma }}^{*}(s)=-\frac{d\overline{\theta }(z,s)}{dz}|}_{z=1}$

Apply initial condition equation (A.5a), θ(z, τ = 0) = 0 and Eq. (A.7) then becomes

(A.9) $s\overline{\theta }(z,s)\frac{1}{z^2}\frac{d}{dz}\left(z^2\frac{d\overline{\theta }}{dz}\right)$

Equation (A.9) is a second-order ordinary differential equation of $\overline{\theta }\left(z\right)$ with variable coefficient. We can solve Eq. (A.9) by assigning a new variable f(z, s).

$\overline{\theta }(z)=\frac{f(z)}{z}$

The left- and right-hand sides of Eq. (A.9) become sf(z)/z and 1/z(d ² f/dz ²), respectively. Therefore, Eq. (A.9) becomes

(A. 10) $\frac{d^{2}f}{d{z}^2}=sf$

Equation (A. 10) can be solved easily by assuming

(A.11) $f=e^{\lambda z}$

Substitute Eq. (A.11) into Eq. (A.10) and solve for eigenvalue λ

$\lambda =\pm s^{1/2}$

Therefore,

(A.12) $f=A{e}^{-\sqrt{sz}+Be\sqrt{sz}}$

(A.13) $\overline{\theta }(z,s)=A\frac{e^{-\sqrt{sz}}}{z}+B\frac{e^{-\sqrt{sz}}}{z}$

By applying boundary condition equation (A.6a), coefficient B must be equal to zero.

Therefore, Eq. (A.13) becomes

(A.14) $\overline{\theta }(z,s)=A\frac{e^{-\sqrt{sz}}}{z}$

Apply boundary condition equation (A.8b) $\overline{\theta }$ (z = l, s) = ${\overline{\theta }}_1$ (s), and Eq. (A.14) becomes

${\overline{\theta }}_1(s)=A{e}^{-\sqrt{s}}$

$\therefore A={\overline{\theta }}_1(s)e^{\sqrt{s}}$

Therefore,

(A.15) $\overline{\theta }\left(z,s\right)=\frac{{\overline{\theta }}_1(s)}{z}{e}^{\sqrt{s}(1-z)}$

(a)Obtain the bulk concentration distribution C(z, τ):

By taking an inverse Laplace transform on Eq. (A. 15),

(A.16) $L^{-1}\left[F(s)G\left(s\right)\right]=f(t)*g(t)={\int }_0^{t}f(t-\omega )g(\omega )$

we get

(A.17) $\begin{array}{c}\theta (z,\tau )=L^{-1}\left[\overline{\theta }\left(z,s\right)\right]=L^{-1}\left[\frac{{\overline{\theta }}_1(s)}{z}{e}^{\sqrt{s}(1-z)}\right]\\ =L^{-1}\left[\frac{{\theta }_1(s)}{z}\right]*L^{-1}\left[e^{-\sqrt{s}(z-1)}\right]\\ ={\int }_0^1\frac{1}{z}{\theta }_1(t-\omega )\frac{(z-1)}{2\sqrt{\pi {\omega }^3}}{e}^{-\frac{{\left(z-1\right)}^2}{4\omega }d\omega }\end{array}$

Note that

(A.18) $L^{-1}\left[e^{-a\sqrt{s}}\right]=\frac{a}{2\sqrt{\pi {\omega }^3}}{e}^{-a^2/4t}$

Therefore,

(A.19) $\frac{C_{\infty }-C(z,\tau )}{C_{\infty }}={\int }_0^1\frac{1}{z}{\theta }_1\left(t-w\right)\frac{\left(z-1\right)}{2\sqrt{\pi {\omega }^3}}{e}^{-{(z-1)}^2/4wdw}$

(b)To obtain the surface concentration distribution Γ*(τ), recall Eq. (A.8c)

(A.20a) $\begin{array}{c}{\frac{{\Gamma }_{\infty }s}{C_{\infty }b}{\overline{\Gamma }}^{*}(s)=-\frac{d\overline{\theta }(z,s)}{dz}|}_{z=1}\\ =\frac{d}{dz}{\left[\frac{{\overline{\theta }}_1(s)}{z}{e}^{\sqrt{s}(1-z)}\right]}_{z=1}\end{array}$

(A.20b) $={\overline{\theta }}_1(s){\left[\frac{e^{\sqrt{s}(1-z)}}{z^2}+\frac{\sqrt{s}}{z}{e}^{\sqrt{s}(1-z)}\right]}_{z=1}$

(A.20c) $={\overline{\theta }}_1(s)\left[1+\sqrt{s}\right]$

Therefore,

(A.21) ${\overline{\Gamma }}^{*}(s)=\frac{C_{\infty }b}{{\Gamma }_{\infty }}\frac{{\overline{\theta }}_1(s)\left(1+\sqrt{s}\right)}{s}$

Now we obtain

(A.22) $\Gamma (\tau )=L^{-1}\left[{\overline{\Gamma }}^{*}(s)\right]=\frac{C_{\infty }b}{{\Gamma }_{\infty }}{L}^{-1}\left[\frac{{\overline{\theta }}_1(s)}{s}+\frac{{\overline{\theta }}_1(s)}{\sqrt{s}}\right]$

Note that

(A.23) $L^{-1}\left[\frac{F\left(s\right)}{s^n}\right]={\int }_0^{t}f(w)\frac{{(t-\omega )}^{n-1}}{\left(n-1\right)!}d\omega$

(A.24) $\therefore L^{-1}\left[\frac{{\overline{\theta }}_1\left(s\right)}{s^n}\right]={\int }_0^t{\theta }_1(\omega )d\omega$

and

(A.25) $L^{-1}\left[\frac{{\overline{\theta }}_1\left(s\right)}{\sqrt{s}}\right]=L^{-1}\left[\frac{1}{\sqrt{s}}\right]*L^{-1}\left[{\overline{\theta }}_1(s)\right]$

Note that

(A.26) $L^{-1}\left[\frac{1}{\sqrt{s}}\right]=\frac{1}{\sqrt{\pi t}}$

Therefore

(B.27) $\Gamma (\tau )=\frac{C_{\infty }b}{{\Gamma }_{\infty }}\left[{\int }_0^{\tau }{\theta }_1(\omega )d\omega +{\int }_0^{\tau }\frac{1}{\sqrt{\pi t}}{\theta }_1(\tau -\omega )d\omega \right]$

According to Eq. (A.6a),

(B.28) $\begin{array}{ccc}{\theta }_1(\tau )& =& \frac{C_{\infty }-C_s(\tau )}{C_{\infty }}\hfill \\ \Gamma (\tau )& =& \frac{b}{{\Gamma }_{\infty }}{\int }_0^{\tau }\left[C_{\infty }-C_s(\omega )\right]d\omega \hfill \\ & & +\frac{b}{{\Gamma }_{\infty }}{\int }_0^{\tau }\frac{1}{\sqrt{\pi \tau }}\left[C_{\infty }(\omega )-C_s(\tau -\omega )\right]d\omega \end{array}$

Integrate Eq. (A.28) and apply the following equation to it:

(B.29) ${\int }_0^{\tau }\frac{C_s\left(\tau -\omega \right)}{\sqrt{\tau }}d\omega =2{\int }_0^{\sqrt{\tau }}{C}_s\left(\tau -\omega \right)d\sqrt{\omega }$

Then change the dimensionless variables back to the dimensional variables. An equation relating the surface concentration, subsurface concentration and time is obtained:

(B.30) $\begin{array}{c}\Gamma (t)={\Gamma }_b+\frac{D}{b}\left[C_{\infty }t-{\int }_0^t{C}_s(\omega )d\omega \right]\\ +2\sqrt{\frac{D}{\pi }}\left[C_{\infty }{t}^{1/2}-{\int }_0^{\sqrt{t}}{C}_s(t-\omega )d\sqrt{\omega }\right]\end{array}$

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Propagation phenomena in boiling water reactors

Alfonso Prieto-Guerrero , Gilberto Espinosa-Paredes , in Linear and Non-Linear Stability Analysis in Boiling Water Reactors, 2019

4.1.6.1 Spatial deviations of the velocity potential

The averaging velocity potential in spherical coordinates is given by

(4.89) ${〈\varphi 〉}^{\ell }=\frac{1}{V_{\ell }}\underset{0}{\overset{2\pi }{\int }}\underset{0}{\overset{\pi }{\int }}\underset{R_a}{\overset{r\left(\theta ,\phi \right)}{\int }}\varphi \left(r,\theta \right)r^{2}drsin\theta \mathit{d\theta d\phi }$

where r ² dr  sin θdθdφ is the differential element of volume dV, V _ℓ   =   4/3π(b ³  − R_b ³) and r(θ, φ) is the eccentric radius given by Eq. (4.85), which depends on the bubble position within the cell (Fig. 4.4).

Substituting Eq. (4.88) into Eq. (4.89) and integrating, we obtain

(4.90) ${〈\varphi 〉}^{\ell }=\frac{{\mathit{UR}}_b^3}{2V_{\ell }}\underset{0}{\overset{2\pi }{\int }}\underset{0}{\overset{\pi }{\int }}r\left(\theta ,\phi \right)cos\theta sin\theta \mathit{d\theta }\mathit{d\phi }$

Substituting Eqs. (4.89), Eq. (4.90) into Eq. (4.62), the spatial deviation of the velocity potential is obtained

(4.91) $\tilde{\varphi }=\frac{1}{2}\frac{{\mathit{UR}}_b^3}{r^2\left(\theta ,\phi \right)}-\frac{{\mathit{UR}}_b^3}{2V_{\ell }}\underset{0}{\overset{2\pi }{\int }}\underset{0}{\overset{\pi }{\int }}r\left(\theta ,\phi \right)cos\theta sin\theta \mathit{d\theta }\mathit{d\phi }$

The integral on the interfacial area of $\tilde{\varphi }$ gives as a result

(4.92) $\frac{{\rho }_{\ell }}{V}\underset{A}{\int }\tilde{\varphi }{\mathbf{n}}_{\mathit{\ell g}}\mathit{dA}=\frac{1}{2}{\rho }_{\ell }{\varepsilon }_{g}U{\mathbf{e}}_z-\frac{1}{2}{\rho }_{\ell }\left(\frac{R_b^3}{V_{\ell }}\underset{0}{\overset{2\pi }{\int }}\underset{0}{\overset{\pi }{\int }}r\left(\theta ,\phi \right)cos\theta sin\theta d\theta \mathit{d\phi }\right)U\nabla {\varepsilon }_g$

where

(4.93) ${\mathbf{n}}_{\mathit{\ell g}}=-{\mathbf{e}}_r={\mathbf{e}}_{x}sin\theta cos\phi +{\mathbf{e}}_{y}sin\theta sin\phi +{\mathbf{e}}_{z}cos\theta$

It can be observed in Eq. (4.92) that the density of the liquid is included to analyze the physical interpretation. The first term is related to the virtual mass effect (Wallis, 1969) and the second term is governed by the gradient of the void fraction.

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26th European Symposium on Computer Aided Process Engineering

Shuey Z. Saw , Jobrun Nandong , in Computer Aided Chemical Engineering, 2016

3.1.2 Catalyst phase species balance

The mass balance of a species i inside the catalyst is given as:

(4) $\frac{d{C}_{c,i,j}}{dt}=\frac{1}{r^2}\frac{\partial }{\partial r}\left[D_{i,m}{r}^2\frac{\partial C_{c,i,j}}{\partial r}\right]+r_i$

(5) $k_{c,i}\left(C_{c,i}\left(z,r=0.5D_{\mathit{cat}}\right)-C_i\left(z\right)\right)=-D_{i,m}\frac{\partial C_{c,i}}{\partial r}\left(z,r=0.5D_{\mathit{cat}}\right)$

where r (m ) denotes the spherical coordinates of the catalyst pellet radius, D_cat (m) is the diameter of catalyst and D _i,m (m/h) is the effective gas diffusivity of species i in the mixture. Note that Eq. (5) is the boundary conditions for a catalyst pellet.

In order to determine the need of involving temporal and spatial concentration profiles into the reactor model, a simulation study is carried out to test the dynamic inside the catalyst pellet using the partial differential equation (pde) solver in Matlab 2014 software. The result obtained suggested that the changes of concentration against time inside a catalyst pellet is almost in pseudo-steady state due to fast reaction in the pellet. Hence, the dC _{c,i, j} /dt term in Eq. (4) is assumed as zero.

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Source: https://www.sciencedirect.com/topics/engineering/spherical-coordinate

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