Elastic Curve Dealing With Continuity Condition

Subloading-multiplicative hyperelastic-based plastic and viscoplastic constitutive equations

Koichi Hashiguchi , in Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity, 2020

9.4 Continuity and smoothness conditions

The continuity condition expressed in the current configuration in Eq. (7.2) is extended for the intermediate configuration as follows:

(9.55) $\underset{\delta \overline{\mathbf{D}}\to \mathbf{O}}{\lim }\dot{\overline{\mathbf{M}}}(\overline{\mathbf{M}},\overline{\mathbf{H}},H;\overline{\mathbf{D}}+\delta \overline{\mathbf{D}})\to \dot{\overline{\mathbf{M}}}(\overline{\mathbf{M}},\overline{\mathbf{H}},\mathit{H};\overline{\mathbf{D}})$

where $\overline{\mathbf{H}}$ designates collectively the tensor-valued internal variables based in the intermediate configuration.

The smoothness condition expressed in the current configuration in Eq. (7.4) is extended for the intermediate configuration as follows:

(9.56) $\underset{\delta \overline{\mathbf{M}}\to \mathbf{O}}{\lim }\dot{\overline{\mathbf{M}}}(\overline{\mathbf{M}}+\delta \overline{\mathbf{M}},\overline{\mathbf{H}},H;\overline{\mathbf{D}})\to \dot{\overline{\mathbf{M}}}(\overline{\mathbf{M}},\overline{\mathbf{H}},H;\overline{\mathbf{D}})$

The elastoplastic modulus tensor

(9.57) ${\overline{\mathbb{M}}}^{\mathit{ep}}(\overline{\mathbf{M}},\overline{\mathbf{H}},H)=\frac{\partial \overline{\mathbf{M}}}{\partial \int \overline{\mathbf{D}}dt}$

in the constitutive equation fulfilling the smoothness condition satisfies

(9.58) $\underset{\delta \overline{\mathbf{M}}\to \mathbf{O}}{\lim }{\mathbb{M}}^{\mathit{ep}}(\overline{\mathbf{M}}+\delta \overline{\mathbf{M}},\overline{\mathbf{H}},H)\to {\mathbf{M}}^{\mathit{ep}}(\overline{\mathbf{M}},\overline{\mathbf{H}},H)$

Both of the continuity and the smoothness conditions are satisfied only in the subloading surface model, while the smoothness condition is violated in the other elastoplasticity models as described in Section 7.1.2.

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The Method of Cells Micromechanics

Jacob Aboudi , ... Brett A. Bednarcyk , in Micromechanics of Composite Materials, 2013

4.2.2 Thermal Conductivity

The continuity conditions of the temperature at the interfaces on an average basis provide:

(4.178) $\begin{array}{c}{d}_1{\xi }_1^{(1\beta \gamma )}+d_2{\xi }_1^{(2\beta \gamma )}=(d_1+d_2)\frac{\partial T}{\partial x_1}\\ h_1{\xi }_2^{(\alpha 1\gamma )}+h_2{\xi }_2^{(\alpha 2\gamma )}=(h_1+h_2)\frac{\partial T}{\partial x_2}\\ l_1{\xi }_3^{(\alpha \beta 1)}+l_2{\xi }_3^{(\alpha \beta 2)}=(l_1+l_2)\frac{\partial T}{\partial x_3}\end{array}$

The average heat flux in the subcell is given by:

(4.179) ${\overline{q}}_i^{(\alpha \beta \gamma )}=-k_i^{(\alpha \beta \gamma )}{\xi }_i^{(\alpha \beta \gamma )},i=1,2,3$

where i is not summed, and $k_i^{(\alpha \beta \gamma )}$ is the thermal conductivity of the material in the subcell in the i-direction.

The average heat flux $\overline{q_i}$ in the composite is determined from the local subcell heat flux ${\overline{q}}_i^{(\alpha \beta \gamma )}$ :

(4.180) ${\overline{q}}_i=\frac{1}{V}\sum _{\alpha ,\beta ,\gamma =1}^2{d}_{\alpha }{h}_{\beta }{l}_{\gamma }{\overline{q}}_i^{(\alpha \beta \gamma )}$

and continuity conditions of the local heat flux at the interfaces are:

(4.181) $\begin{array}{l}{\overline{q}}_1^{(1\beta \gamma )}={\overline{q}}_1^{(2\beta \gamma )}\\ {\overline{q}}_2^{(\alpha 1\gamma )}={\overline{q}}_2^{(\alpha 2\gamma )}\\ {\overline{q}}_3^{(\alpha \beta 1)}={\overline{q}}_3^{(\alpha \beta 2)}\end{array}$

Elimination of the microvariables ${\xi }_i^{(\alpha \beta \gamma )}$ from Eqs. (4.178) to (4.181) gives:

(4.182) ${\overline{q}}_i=-k_i^{\ast }\frac{\partial T}{\partial x_i},i=1,2,3$

where $k_i^{\ast }$ are the effective thermal conductivities and i is not summed. For example, Eq. (4.179) is substituted into the first of Eq. (4.181), and the resulting equation is solved for ${\xi }_1^{(\alpha \beta \gamma )}$ . This is then substituted into Eq. (4.178), which is solved for ${\xi }_2^{(\alpha \beta \gamma )}$ in terms of $\partial T/\partial x_1$ . Substituting this expression into Eq. (4.180) with i  =   1 and again using Eq. (4.179) results in an equation of the form of Eq. (4.182), from which $k_1^{\ast }$ can be readily identified. A similar procedure can be used to determine $k_2^{\ast }$ and $k_3^{\ast }$ . The effective thermal conductivities in the three directions are given by:

(4.183) $\begin{array}{l}{k}_1^{\ast }=\frac{d}{hl}\sum _{\beta ,\gamma =1}^2{h}_{\beta }{l}_{\gamma }{\left(\frac{d_1}{k_1^{(1\beta \gamma )}}+\frac{d_2}{k_1^{(2\beta \gamma )}}\right)}^{-1}\\ k_2^{\ast }=\frac{h}{dl}\sum _{\alpha ,\gamma =1}^2{d}_{\alpha }{l}_{\gamma }{\left(\frac{h_1}{k_2^{(\alpha 1\gamma )}}+\frac{h_2}{k_2^{(\alpha 2\gamma )}}\right)}^{-1}\\ k_3^{\ast }=\frac{l}{dh}\sum _{\alpha ,\beta =1}^2{d}_{\alpha }{h}_{\beta }{\left(\frac{l_1}{k_3^{(\alpha \beta 1)}}+\frac{l_2}{k_3^{(\alpha \beta 2)}}\right)}^{-1}\end{array}$

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2D thermo-elastic solutions for laminates and sandwiches with interlayer delaminations and imperfect thermal contact

H. Darban , R. Massabò , in Dynamic Response and Failure of Composite Materials and Structures, 2017

Interfacial continuity conditions and global transfer matrix

The continuity conditions between the layer k and $k-1$ , Eq. (1.11), are written in matrix form:

(1.39) ${}^{\left(k\right)}M\left(x_3=x_3^{k-1}\right)=\left(B^{k-1}\right){}^{\left(k-1\right)}M\left(x_3=x_3^{k-1}\right)$

with

(1.40) $B^{k-1}=\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1/K_S^{k-1}\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 1/K_N^{k-1}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\text{.}$

The matrix $B^{k-1}$ depends on the interfacial stiffnesses and for the case of perfect bonding, with $1/K_S^{k-1}=1/K_N^{k-1}=0$ , it becomes the identity matrix.

The procedure used for the solution of the heat conduction problem (Fig. 1.2) is used to derive a relationship between ${}^{\left(k\right)}M\left(x_3=x_3^k\right)$ and ${}^{\left(1\right)}M\left(x_3=x_3^0\right)$ :

(1.41) $\begin{array}{ll}{}^{\left(k\right)}M\left(x_3=x_3^k\right)=& {\left(B^k\right)}^{-1}\left\{{\displaystyle \prod _{i=k}^1\left\{\left(B^i\right){}^{\left(i\right)}E\left(x_3=x_3^i\right){}^{\left(i\right)}E^{-1}\left(x_3=x_3^{i-1}\right)\right\}}{}^{\left(1\right)}M\left(x_3=x_3^0\right)\right.\hfill \\ & -{\displaystyle \sum _{i=1}^k\left\{{\displaystyle \prod _{j=k}^i\left\{\left(B^j\right){}^{\left(j\right)}E\left(x_3=x_3^j\right){}^{\left(j\right)}E^{-1}\left(x_3=x_3^{j-1}\right)\right\}}{}^{\left(i\right)}Q\left(x_3=x_3^{i-1}\right)\right\}}\left.+{\displaystyle \sum _{i=2}^k\left\{{\displaystyle \prod _{j=k}^i\left\{\left(B^j\right){}^{\left(j\right)}E\left(x_3=x_3^j\right){}^{\left(j\right)}E^{-1}\left(x_3=x_3^{j-1}\right)\right\}\left(B^{i-1}\right){}^{\left(i-1\right)}Q\left(x_3=x_3^{i-1}\right)}\right\}}\right\}\hfill \\ & +{}^{\left(k\right)}Q\left(x_3=x_3^k\right)\text{.}\hfill \end{array}$

The explicit expressions relating the four unknown constants, ^(k) a ₁₁, ^(k) a ₂₁, ^(k) a _12, and ^(k) a _22, in Eq. (1.32) to those of the first layer are then derived substituting ${}^{\left(k\right)}M\left(x_3=x_3^k\right)$ and ${}^{\left(1\right)}M\left(x_3=x_3^0\right)$ defined by Eq. (1.35) into Eq. (1.41):

(1.42) $\begin{array}{c}{}^{\left(k\right)}\left[\begin{array}{c}\hfill a_{11}\hfill \\ \hfill a_{21}\hfill \\ \hfill a_{12}\hfill \\ \hfill a_{22}\hfill \end{array}\right]={}^{\left(k\right)}E^{-1}\left(x_3^k\right){\left(B^k\right)}^{-1}\left\{{\displaystyle \prod _{i=k}^1\left\{\left(B^i\right){}^{\left(i\right)}E\left(x_3^i\right){}^{\left(i\right)}E^{-1}\left(x_3^{i-1}\right)\right\}}{}^{\left(1\right)}E\left(x_3^0\right){}^{\left(1\right)}\left[\begin{array}{cccc}\hfill a_{11}\hfill & \hfill a_{21}\hfill & \hfill a_{12}\hfill & \hfill a_{22}\hfill \end{array}\right]^T\right.\\ \left.+{\displaystyle \sum _{i=2}^k\left({\displaystyle \prod _{j=k}^i\left\{\left(B^j\right){}^{\left(j\right)}E\left(x_3^j\right){}^{\left(j\right)}E^{-1}\left(x_3^{j-1}\right)\right\}\left\{\left(B^{i-1}\right){}^{\left(i-1\right)}Q\left(x_3^{i-1}\right)-{}^{\left(i\right)}Q\left(x_3^{i-1}\right)\right\}}\right)}\right\}\text{.}\end{array}$

The equation can be used for $k=2,\dots ,n$ to define the integration constants of the layers as function of those of the first layer. Eq. (1.42) reduces the thermo-elasticity problem to finding only four unknown constants in the solution of the first layer, which requires four equations. Two equations are the boundary conditions at the bottom surface of the plate, which directly depend on the unknowns. The other two boundary conditions, at the top surface of the plate, are restated in terms of the unknown constants of the first layer using Eq. (1.42) with $k=n$ . The stress components of each layer are then obtained using the constitutive and compatibility equations (1.1) and (1.9). Explicit expressions for stresses and displacements are given in Ref. [34].

The expressions of the integration constants in Eq. (1.42) are valid also for the other solution cases (different values of the discriminant), provided the matrix ^(k) E is changed according to the forms given in Appendix.

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Flow Mechanism of Fractured Low-Permeability Reservoirs

Wanjing Luo , ... Bo Ning , in Petrophysical Characterization and Fluids Transport in Unconventional Reservoirs, 2019

2.3 Coupling

According to the continuity condition that the pressure and the flux must be continuous along the fracture surface, the following conditions must hold along the fracture plane:

(17) ${\overrightarrow{\overline{p}}}_{\mathit{fD}}={\overrightarrow{\overline{p}}}_D,{\overrightarrow{\overline{q}}}_{\mathit{fD}}={\overrightarrow{\overline{q}}}_D$

and the flow rate in the wellbore is

(18) $\sum _{i=1}^N{\overline{q}}_{\mathit{fDi}}={\overline{q}}_{\mathit{fwD}}$

Combining Eqs. (6), (13), (17), and (18), we can obtain

(19) $\left(\begin{array}{cc}A& -I\\ I^T& 0\end{array}\right)\cdot \left(\begin{array}{c}{\overrightarrow{\overline{q}}}_{\mathit{fD}}\\ {\overline{p}}_{\mathit{wD}}\end{array}\right)=\left(\begin{array}{c}0\\ {\overline{q}}_{\mathit{fwD}}\end{array}\right)$

where

(20) $A=\mathit{AR}+\mathit{AF}$

and unit vector I

(21) $I={\left(1,1,\cdots ,1\right)}^T$

While the flow q _fwD is specified, we can obtain wellbore pressure p _wD and flow rate in Laplace domain by solving Eq. (19).

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Validation and Verification

Zhuming Bi , in Finite Element Analysis Applications, 2018

12.4.4.3 Convergence of energy

For an FEA model with the satisfaction of completeness and continuity conditions, the convergence of energy warrantees that the solution to the model is converged. It is the user's responsibility to ensure the convergence of energy.

Mesh sizes relate closely to both computation and accuracy. The finer a mesh is, the better result an FEA model can obtain; however, it demands more computation. A trial and error method can be applied to determine mesh sizes. Once an FEA solution is found, a mesh with a fine size is applied to find a new solution, two solutions are compared to see if the solution is improved by refining the mesh. The iteration will be continued until the difference of two solutions is within the specified percentage of errors.

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30th European Symposium on Computer Aided Process Engineering

Morgan T. Kelley , ... Michael Baldea , in Computer Aided Chemical Engineering, 2020

3 Case Study

The previously derived methods are applied to a batch reactor, whose model is an example supplied by Process Systems Enterprise with gPROMS Process Builder v1.3.1.

3.1 Process Description

Two exothermic reactions Eq. (7), with four total components, ℓ = {A, B, C, D}, take place in the batch reactor over a time horizon of t _f   = 1000 units, with desired product being C.

(7) $\begin{array}{l}A+B\to C\\ B+C\to D\end{array}$

The model equations for the batch reactor are given in Eq. (8), with the control input as the flow rate of the cooling water, F _cw . N _ℓ , C _ℓ , and h _ℓ are the number of moles, concentration, and enthalpy of each component, respectively. The volumetric holdup of the reactor is represented by V, the rate of reaction, r, is based on the temperature-dependent (T) rate constant, k, and C _A and C _B . C _C is the rate of cooling, U is the cumulative amount of cooling water used in the process, and H is the total enthalpy of the process.

The discretized model has additional constraints in the form of state continuity conditions:

Table 1. Model parameters for the batch reactor case study.

Parameter	Value
a	{150,175,200,175}
b	{0,0,0,0}
ρ	{10000,8000,11000,11000}
ν	–1,–1,1,1
ΔH	–60000
E	19000
R	8.314
L	7.5 * 10–4
$\overline{T}$	296
M	168000

Initial conditions are imposed on all differential state variables:

The cooling water flowrate, F _cw is bounded according to:

(11) $0.1\le F_{\mathit{cw}}\le 3\left|0.1\le F_{{\mathit{cw}}_i}\le 3\right.$

and the reactor temperature, T, is bounded by a path Eq. (12) and an end-point Eq. (13) constraint.

(12) $T\left(t\right)\le 400\left|T_{i,j}\le 400\right.$

(13) $310\le T\left(t_f\right)\le 321\left|310\le T_{N_i,N_j}\le 321\right.$

The sequential optimization problem (referred to as P1) utilizes Eq. (8), (10)-(13), and the simultaneous optimization problem (referred to as P2) uses Eq. (8)-(13). The objectives, which seek to maximize the moles of the desired product, N _C while reducing the cumulative moles of cooling water used, U are given in Eq. (14).

In order to apply LR to P2, continuity constraints Eq. (9) were removed and used to calculate the constraint violation, γ_i = {γ_N, γ_U, γ_H}, (following Eq. (4) with ε = 10^–8). Applying the LR scheme to Eq. (14) yields P3, which follows the format of PIII with Eq. (3), (8), and (10-13) and objective function:

(15) $\underset{F_{{\mathit{cw}}_i}}{\text{max}}\mathrm{L}=\mathrm{J}-\sum _{\mathrm{i}}\left({\mathrm{\gamma }}_{\mathrm{i}}^{\mathrm{U}}{\mathrm{\lambda }}_{\mathrm{i}}^{\mathrm{U}}+{\mathrm{\gamma }}_{\mathrm{i}}^{\mathrm{H}}{\mathrm{\lambda }}_{\mathrm{i}}^H+\sum _{\mathrm{\ell }}{\mathrm{\gamma }}_{\mathrm{\ell },\mathrm{i}}^{\mathrm{N}}{\mathrm{\lambda }}_{\mathrm{\ell },\mathrm{i}}\right)$

3.2 Results and Discussion

P1-P3 were solved on a 64 bit PC running Windows 10® with a 3.6 GHz Intel Core i7 processor with 32 GB of RAM and 8 threads available. P1 was solved using gPROMS Process Builder v1.3.1 and P2-P3 were solved using CONOPT3 version 3.17I in GAMS 25.0.3. In all cases, wall clock time is reported rather than CPU time since multiple threads were used to solve the decomposed problem, P3.

Figure 2 presents the solution times for P2 and P3 for N _j varied from 150 to 500 and N _i   = 4 control intervals. The solution times of P2 and P3 are nearly equal for small subproblem sizes, demonstrating that the LR-based decomposition strategy in P3 does not lead to added overhead solution time in the generation of independent subproblems. Furthermore, the solution time plot of P3 stays relatively flat until the number of subproblem discretization points reaches 500. Conversely, solution time for P2 increases almost exponentially with the number of discretization points used in each control interval. Figure 2 also demonstrates that the LR decomposition-based problem, P3, reaches nearly the same optimal solution as the non-decomposed problem, P2.

Figure 2. Solution time and optimal objective value versus problem size for P2 and P3.

The optimization problems P1-P3 were also solved considering different control intervals, N _i = [2,10] with the total number of time points kept constant (N_iN_j = 1000). Figure 3 demonstrates that the solution of the sequential optimization problem (P1) is consistently slower than simultaneous methods, P2 and P3. The simultaneous method using LR (P3) consistently solves fastest up until 10 intervals are used. This is expected with a constant number of time points, given that the size of the subproblems decreases as the number of intervals, N _i , grows, thereby increasing the number of penalty terms in the objective and the number of γ _i variables to calculate for each state variable. The sequential problem P1 achieves a higher objective function value than the two simultaneous problems (P2 and P3). This can possibly be attributed to the variable time step utilized by the single-shooting solver for P1, whereas the discretization of the time variable in P2 and P3 is based on a constant time step. However, the original problem (P2) and the relaxed problem (P3) achieve nearly the same result, demonstrating that it is possible to significantly reduce the computation time required to get the same solution by employing a LR-based decomposition strategy.

Figure 3. Solution time and optimal objective function value versus problem structure for P1-P3.

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Buckling/Plastic Collapse Behavior and Strength of Rectangular Plate Subjected to Uni-Axial Thrust

Tetsuya Yao , Masahiko Fujikubo , in Buckling and Ultimate Strength of Ship and Ship-Like Floating Structures, 2016

4.3.2 Derivation of Interactive Buckling Strength

In the following, a method of analysis is explained to evaluate the local panel buckling strength considering the interaction between the plate and the stiffener web as well as that between stiffener web and flange. In the case of a continuous stiffened plating, symmetry condition or periodically continuous condition can be imposed along the centerlines in both longitudinal and transverse directions. So, it is enough if the shaded region partitioned by four centerlines is analyzed instead of analyzing the whole stiffened plating; see Fig. 4.12.

The following boundary conditions are considered in the formulation.

(1)

Continuity condition for rotation angle along panel/web intersection:

(4.23) $\begin{array}{l}\hfill {\left.\frac{\partial w}{\partial y}\right|}_{y=0}={\left.\frac{\partial v_{\mathrm{web}}}{\partial z}\right|}_{z=0}\end{array}$

(2)

Equilibrium condition for bending/torsional moment along panel/web intersection (see Fig. 4.14):

(4.24) $\begin{array}{l}\hfill D_w{\left.\left(\frac{{\partial }^2{v}_{\mathrm{web}}}{\partial z^2}+\nu \frac{{\partial }^2{v}_{\mathrm{web}}}{\partial x^2}\right)\right|}_{z=0}+2D_p{\left.\left(\frac{{\partial }^{2}w}{\partial y^2}+\nu \frac{{\partial }^{2}w}{\partial x^2}\right)\right|}_{y=0}=0\end{array}$

where,

$D_w=\frac{E{t}_w^3}{12(1-{\nu }^2)},D_p=\frac{E{t}_p^3}{12(1-{\nu }^2)}$

and t _p and t _w are the thicknesses of panel and stiffener web.

Fig. 4.14. Equilibrium condition and deflection mode.

(3)

Equilibrium condition for bending/torsional moment along web/flange intersection of stiffener considering continuity of rotation (see Fig. 4.14):

(4.25) $\begin{array}{l}\hfill G{J}_f{\left.\frac{{\partial }^3{v}_{\mathrm{web}}}{\partial x^2\partial z}\right|}_{z=h}-D_w{\left.\left(\frac{{\partial }^2{v}_{\mathrm{web}}}{\partial z^2}+\nu \frac{{\partial }^2{v}_{\mathrm{web}}}{\partial x^2}\right)\right|}_{z=h}=0\end{array}$

where GJ _f is a torsional stiffness of the flange of a stiffener

Applying the Principle of Minimum Potential Energy, the elastic buckling interaction equation is finally derived in the following form:

(4.26) $\begin{array}{l}\hfill {\kappa }_1{\sigma }_x^2+{\kappa }_2{\sigma }_x{\sigma }_y+{\kappa }_3{\sigma }_y^2-{\kappa }_4{\sigma }_x-{\kappa }_5{\sigma }_y+{\kappa }_6=0\end{array}$

Detail of the derivation and the coefficients in Eq. (4.26) are given in Fujikubo and Yao [6].

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BEM in Biomechanics

M. Perrella , ... R. Citarella , in Numerical Methods and Advanced Simulation in Biomechanics and Biological Processes, 2018

8.3.3 Discontinuous Elements

To apply DBEM, discontinuous and semidiscontinuous elements must be used to cope with continuity conditions inherent the use of the traction equation. For two-dimensional problems, several types of elements are used: the most important are the linear and quadratic elements, with respectively two or three nodes; all these types of element can be discontinuous or semidiscontinuous as shown in Fig. 8.7.

Figure 8.7. Bidimensional discontinuous or semidiscontinuous elements.

For three-dimensional problems, quadrilateral and triangular elements are used, with a varying number of nodes depending on the adopted interpolation order. All these types of element can be discontinuous or semidiscontinuous. Fig. 8.8 displays the discontinuous and semidiscontinuous quadrilateral eight-node elements are shown.

Figure 8.8. Three-dimensional discontinuous or semidiscontinuous elements.

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Physics coupling phenomena and formulations

In Multiphysics Modeling, 2016

2.8 Conjugate heat transfer problem

When the thermal fluid flow is coupled with a thermal stress problem, the continuity conditions of temperature and conservative conditions of heat flow need to be satisfied across the coupling interface. This kind of coupling is called the conjugate heat transfer problem. We need to solve this conjugate heat transfer problem with temperature being transferred from one field to another and also the heat flux. Similar to the fluid–structure-coupling problem, this is also a fully coupled (essential coupling, i.e., ${\Gamma }_c\ne 0$ ) problem.

The fully coupled matrix is shown as Equation (2.37).

(2.37) $\left[\begin{array}{ccc}\hfill C_{ii}^f\hfill & \hfill C_{ic}^f\hfill & \hfill 0\hfill \\ \hfill C_{ci}^f\hfill & \hfill C_{cc}^f+C_{cc}^s\hfill & \hfill C_{ci}^s\hfill \\ \hfill 0\hfill & \hfill C_{2c}^s\hfill & \hfill C_{ii}^s\hfill \end{array}\right]\left\{\begin{array}{c}\hfill {\dot{T}}_i^f\hfill \\ \hfill {\dot{T}}_c\hfill \\ \hfill {\dot{T}}_i^s\hfill \end{array}\right\}\text{+}\left[\begin{array}{ccc}\hfill K_{ii}^f\hfill & \hfill K_{ic}^f\hfill & \hfill 0\hfill \\ \hfill K_{ci}^f\hfill & \hfill C_{cc}^f+C_{cc}^s\hfill & \hfill K_{ci}^s\hfill \\ \hfill 0\hfill & \hfill K_{ic}^s\hfill & \hfill K_{ii}^s\hfill \end{array}\right]\left\{\begin{array}{c}\hfill T_i^f\hfill \\ \hfill T_c\hfill \\ \hfill T_i^s\hfill \end{array}\right\}=\left\{\begin{array}{c}\hfill Q_i^f\hfill \\ \hfill Q_c\hfill \\ \hfill Q_i^s\hfill \end{array}\right\}\text{.}$

Although the coupled Equation (2.37) can be solved directly by the SC method, the weak coupling method is also suitable and commonly used for the thermal coupling of the conjugate heat transfer problems.

As mentioned earlier, to solve the conjugate heat transfer problems, we need to exchange heat flux and temperature across the fluid–structure interface (Figure 2.9). Transferring temperature from the higher conductivity side to the lower conductivity side, rather than transferring the heat flux or flow from the lower conductivity side to the higher conductivity side, provides a better convergence in interface load transfers. A detailed explanation about the reason of this is presented in Chapter 3.

Figure 2.9. The data transfers for conjugate heat transfer problems.

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Membranes for advanced biofuels production

S. Curcio , in Advanced Membrane Science and Technology for Sustainable Energy and Environmental Applications, 2011

Dense and spongy layers

The velocity profiles in both the dense and the spongy layer are obtained after utilization of the continuity conditions referred to the different regions of the membrane:

[12.41] $v_{r,i}\left(r\right)=\frac{1}{{\mathrm{\epsilon }}_{\mathrm{i}}}{L}_{\mathrm{p}}\mathit{·TMP}\frac{R_1}{r}i=2,3\mathrm{and}{R}_1<r<R_3$

where the term L _p is the membrane hydraulic permeability and the product between L _p and TMP (Equation [12.37]) represents the permeation velocity, v _p, in the radial direction; ε _i is the porosity of each of the two regions.

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