Capua et al 2009b - Revision history

Scipediacontent: Scipediacontent moved page Draft Content 179546005 to Capua et al 2009b

2018-06-07T13:26:36Z

Scipediacontent moved page Draft Content 179546005 to Capua et al 2009b

Cinmemj at 13:15, 7 June 2018

2018-06-07T13:15:20Z

Cinmemj at 13:08, 7 June 2018

2018-06-07T13:08:19Z

Cinmemj at 12:57, 7 June 2018

2018-06-07T12:57:44Z

Cinmemj at 12:52, 7 June 2018

2018-06-07T12:52:26Z

Cinmemj at 12:37, 7 June 2018

2018-06-07T12:37:35Z

Cinmemj at 12:34, 7 June 2018

2018-06-07T12:34:41Z

Cinmemj at 12:02, 7 June 2018

2018-06-07T12:02:55Z

Cinmemj at 11:56, 7 June 2018

2018-06-07T11:56:14Z

Cinmemj at 11:52, 7 June 2018

2018-06-07T11:52:00Z

@@ Line 5,260: / Line 5,260: @@
 |-
 | <span id='ZEqnNum366648'></span>  <math>\begin{array}{c}
-\frac{d{\psi }_s}{dt}+{\phi }_0\frac{dU}{dt}+S_s\frac{dT}{dt}+T\frac{dS_s}{dt}+T\frac{d\left(\sum_{\alpha }m_{\alpha }s_{\alpha }\right)}{dt}=\\
+\displaystyle \frac{d{\psi }_s}{dt}+{\phi }_0\frac{dU}{dt}+S_s\frac{dT}{dt}+T\frac{dS_s}{dt}+T\frac{d\left(\sum_{\alpha }m_{\alpha }s_{\alpha }\right)}{dt}=\\
-\sigma :\frac{d\epsilon }{dt}+\sum_{\alpha }p_{\alpha }\frac{d{\phi }_{\alpha }}{dt}-T\sum_{\alpha }\nabla .\left(s_{\alpha }\boldsymbol{w}^{\alpha }\right)-\nabla .\boldsymbol{q}-g_l{\dot{m}}_{dehyd}+{\varphi }_f
+\displaystyle \sigma :\frac{d\epsilon }{dt}+\sum_{\alpha }p_{\alpha }\frac{d{\phi }_{\alpha }}{dt}-T\sum_{\alpha }\nabla .\left(s_{\alpha }\boldsymbol{w}^{\alpha }\right)-\nabla .\boldsymbol{q}-g_l{\dot{m}}_{dehyd}+{\varphi }_f
 \end{array}</math>
 |}
@@ Line 5,316: / Line 5,316: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <span id='ZEqnNum970466'></span>  <math>T\left(\frac{dS_s}{dt}+\frac{d\left(\sum_{\alpha }m_{\alpha }s_{\alpha }\right)}{dt}+\right. </math><math>\left. \sum_{\alpha }\nabla .\left(s_{\alpha }\boldsymbol{w}^{\alpha }\right)\right)=</math><math>-\nabla .\boldsymbol{q}+\sigma :\frac{d\epsilon }{dt}+</math><math>\pi \frac{d\phi }{dt}-{\overset{\mbox{ˆ}}{S}}_s\frac{dT}{dt}-</math><math>\frac{d{\psi }_s}{dt}+{\varphi }_f-g_l{\dot{m}}_{dehyd}</math>
+| <span id='ZEqnNum970466'></span>  <math>T\left(\frac{dS_s}{dt}+\frac{d\left(\sum_{\alpha }m_{\alpha }s_{\alpha }\right)}{dt}+ \sum_{\alpha }\nabla .\left(s_{\alpha }\boldsymbol{w}^{\alpha }\right)\right)=</math><math>-\nabla .\boldsymbol{q}+\sigma :\frac{d\epsilon }{dt}+\pi \frac{d\phi }{dt}-{\overset{\mbox{ˆ}}{S}}_s\frac{dT}{dt}-\frac{d{\psi }_s}{dt}+{\varphi }_f-g_l{\dot{m}}_{dehyd}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (5.90)
@@ Line 5,331: / Line 5,331: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <span id='ZEqnNum742147'></span>  <math>T\left(\frac{d{\overset{\mbox{ˆ}}{S}}_s}{dt}-\frac{d\left({\phi }_0\frac{\partial U}{\partial T}\right)}{dt}+\right. </math><math>\left. \frac{d\left(\sum_{\alpha }m_{\alpha }s_{\alpha }\right)}{dt}+\right. </math><math>\left. \sum_{\alpha }\nabla .\left(s_{\alpha }\boldsymbol{w}^{\alpha }\right)\right)=</math><math>-\nabla .\boldsymbol{q}+{\Phi }_M-g_l{\dot{m}}_{dehyd}</math>
+| <span id='ZEqnNum742147'></span>  <math>T\left(\frac{d{\overset{\mbox{ˆ}}{S}}_s}{dt}-\frac{d\left({\phi }_0\frac{\partial U}{\partial T}\right)}{dt}+ \frac{d\left(\sum_{\alpha }m_{\alpha }s_{\alpha }\right)}{dt}+ \sum_{\alpha }\nabla .\left(s_{\alpha }\boldsymbol{w}^{\alpha }\right)\right)=</math><math>-\nabla .\boldsymbol{q}+{\Phi }_M-g_l{\dot{m}}_{dehyd}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (5.91)
@@ Line 5,385: / Line 5,385: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <span id='ZEqnNum541437'></span>  <math>C_{eff}\frac{dT}{dt}-\left[\phi \left(1-S_l\right)+\right. </math><math>\left. \left(3{\alpha }_{\phi }+\phi 3{\alpha }_lS_l\right)T\right]\frac{dp_g}{dt}+</math><math>\left[3{\alpha }_{\phi }S_l-{\phi }_0\frac{{\partial }^2U}{\partial T\partial p_c}+\right. </math><math>\left. \phi 3{\alpha }_lS_l\right]T\frac{dp_c}{dt}+</math><math>C_p^l\boldsymbol{w}^l\cdot \nabla T+C_p^v\boldsymbol{w}^v\cdot \nabla T+</math><math>C_p^a\boldsymbol{w}^a\cdot \nabla T-\nabla .\left(\kappa \nabla T\right)=</math><math>+\left(A_{\omega }\frac{d\omega }{dt}-g_l{\dot{m}}_{dehyd}\right)-</math><math>L_{l\rightarrow v}{\overset{\circ}{m}}_{{}_{l\rightarrow v}}+</math><math>\left(-\mbox{ }l\frac{d\omega }{dt}-Ts_l{\dot{m}}_{dehyd}\right)+</math><math>\Theta </math>
+| <span id='ZEqnNum541437'></span>  <math>C_{eff}\frac{dT}{dt}-\left[\phi \left(1-S_l\right)+ \left(3{\alpha }_{\phi }+\phi 3{\alpha }_lS_l\right)T\right]\frac{dp_g}{dt}+</math><math>\left[3{\alpha }_{\phi }S_l-{\phi }_0\frac{{\partial }^2U}{\partial T\partial p_c}+ \phi 3{\alpha }_lS_l\right]T\frac{dp_c}{dt}+</math><math>C_p^l\boldsymbol{w}^l\cdot \nabla T+C_p^v\boldsymbol{w}^v\cdot \nabla T+C_p^a\boldsymbol{w}^a\cdot \nabla T-\nabla .\left(\kappa \nabla T\right)=</math><math>+\left(A_{\omega }\frac{d\omega }{dt}-g_l{\dot{m}}_{dehyd}\right)-</math><math>L_{l\rightarrow v}{\overset{\circ}{m}}_{{}_{l\rightarrow v}}+</math><math>\left(-\mbox{ }l\frac{d\omega }{dt}-Ts_l{\dot{m}}_{dehyd}\right)+</math><math>\Theta </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (5.95)
@@ Line 5,417: / Line 5,417: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>\Theta =\boldsymbol{\sigma }:\frac{d{\boldsymbol{\epsilon }}^i}{dt}+</math><math>A_{d^+}\frac{d\left(d^+\right)}{dt}+A_{d^{-}}\frac{d\left(d^{-}\right)}{dt}+</math><math>\left(b\frac{\partial U}{\partial T}{\phi }_0-TC_{ijkl}^s\alpha (\omega ){\delta }_{kl}\right)\left(\frac{d{\epsilon }_{ij}}{dt}-\right. </math><math>\left. \frac{d{\epsilon }_{ij}^i}{dt}\right)-\mbox{ }j^+\frac{d\left(d^+\right)}{dt}-</math><math>\mbox{ }j^{-}\frac{d\left(d^{-}\right)}{dt}+\frac{\left(3{\alpha }_l-1\right)}{{\rho }_l}\nabla p_l.\boldsymbol{w}^l+</math><math>\sum_{\alpha }\left(\boldsymbol{b}.\boldsymbol{w}^{\alpha }\right)</math>
+| <math>\Theta =\boldsymbol{\sigma }:\frac{d{\boldsymbol{\epsilon }}^i}{dt}+</math><math>A_{d^+}\frac{d\left(d^+\right)}{dt}+A_{d^{-}}\frac{d\left(d^{-}\right)}{dt}+</math><math>\left(b\frac{\partial U}{\partial T}{\phi }_0-TC_{ijkl}^s\alpha (\omega ){\delta }_{kl}\right) \left(\frac{d{\epsilon }_{ij}}{dt}- \frac{d{\epsilon }_{ij}^i}{dt}\right)-\mbox{ }j^+\frac{d\left(d^+\right)}{dt}-</math><math>\mbox{ }j^{-}\frac{d\left(d^{-}\right)}{dt}+\frac{\left(3{\alpha }_l-1\right)}{{\rho }_l}\nabla p_l.\boldsymbol{w}^l+</math><math>\sum_{\alpha }\left(\boldsymbol{b}.\boldsymbol{w}^{\alpha }\right)</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (5.97)
@@ Line 5,448: / Line 5,448: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>k_{pg}=-\left[\phi \left(1-S_l\right)+\left(3{\alpha }_{\phi }+\right. \right. </math><math>\left. \left. \phi 3{\alpha }_lS_l\right)T\right]</math>
+| <math>k_{pg}=-\left[\phi \left(1-S_l\right)+\left(3{\alpha }_{\phi }+  \phi 3{\alpha }_lS_l\right)T\right]</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (5.99)
@@ Line 5,458: / Line 5,458: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>k_{pc}=\left[3{\alpha }_{\phi }S_l-{\phi }_0\frac{{\partial }^2U}{\partial T\partial p_c}+\right. </math><math>\left. \phi 3{\alpha }_lS_l\right]T</math>
+| <math>k_{pc}=\left[3{\alpha }_{\phi }S_l-{\phi }_0\frac{{\partial }^2U}{\partial T\partial p_c}+\phi 3{\alpha }_lS_l\right]T</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (5.100)

@@ Line 5,178: / Line 5,178: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <span id='ZEqnNum615471'></span>  <math>\frac{dE}{dt}=\sigma :\frac{d\epsilon }{dt}-\nabla .\left(\sum_{\alpha }h_{\alpha }\boldsymbol{w}^{\alpha }+\right. </math><math>\left. \boldsymbol{q}\right)+\sum_{\alpha }\boldsymbol{b}.\boldsymbol{w}^{\alpha }</math>
+| <span id='ZEqnNum615471'></span>  <math>\frac{dE}{dt}=\sigma :\frac{d\epsilon }{dt}-\nabla .\left(\sum_{\alpha }h_{\alpha }\boldsymbol{w}^{\alpha }+ \boldsymbol{q}\right)+\sum_{\alpha }\boldsymbol{b}.\boldsymbol{w}^{\alpha }</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (5.80)
@@ Line 5,210: / Line 5,210: @@
-Teniendo en cuenta <span id='cite-ZEqnNum528944'></span>[[#ZEqnNum528944|(4.76)]], la entalpía específica del fluido  [[Image:Draft_Content_179546005-image947.png|12px]] se escribe como
+Teniendo en cuenta <span id='cite-ZEqnNum528944'></span>[[#ZEqnNum528944|(4.76)]], la entalpía específica del fluido <math>\alpha </math>  se escribe como
 {| class="formulaSCP" style="width: 100%; text-align: center;"
@@ Line 5,217: / Line 5,217: @@
 {| style="text-align: center; margin:auto;"
 |-
-| [[Image:Draft_Content_179546005-image948.png|90px]]
+|  <math>h_\alpha = g_\alpha +Ts_\alpha   </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (5.83)
@@ Line 5,223: / Line 5,223: @@
-A partir de la expresión anterior y teniendo en cuenta <span id='cite-ZEqnNum955364'></span>[[#ZEqnNum955364|(4.77)]], el termino  [[Image:Draft_Content_179546005-image949.png|96px]] que aparece en <span id='cite-ZEqnNum615471'></span>[[#ZEqnNum615471|(5.80)]] resulta
+A partir de la expresión anterior y teniendo en cuenta <span id='cite-ZEqnNum955364'></span>[[#ZEqnNum955364|(4.77)]], el termino  <math> \nabla . \left(\sum_{\alpha }h_{\alpha }\boldsymbol{w}^{\alpha }\right)</math> que aparece en <span id='cite-ZEqnNum615471'></span>[[#ZEqnNum615471|(5.80)]] resulta
 {| class="formulaSCP" style="width: 100%; text-align: center;"
@@ Line 5,230: / Line 5,230: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <span id='ZEqnNum526811'></span>  [[Image:Draft_Content_179546005-image950.png|414px]]
+| <span id='ZEqnNum526811'></span>  <math>\nabla .\left(\sum_{\alpha }h_{\alpha }\boldsymbol{w}^{\alpha }\right) = \sum_{\alpha } (\nabla g_\alpha )_T . \boldsymbol{w}^{\alpha }+</math><math>\sum_{\alpha } g_\alpha \nabla . \boldsymbol{w}^{\alpha } + T \sum_{\alpha } \nabla . (s_\alpha \boldsymbol{w}^{\alpha })
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (5.84)
@@ Line 5,236: / Line 5,237: @@
-donde  [[Image:Draft_Content_179546005-image951.png|120px]] .
+donde <math>(\nabla g_\alpha )_T = \frac{1}{\rho_\alpha}\nabla p_\alpha </math>.
 Considerando que  [[Image:Draft_Content_179546005-image952.png|96px]] y que  [[Image:Draft_Content_179546005-image953.png|72px]], teniendo en cuenta las ecuaciones de balance de masa de las fases fluidas <span id='cite-ZEqnNum866801'></span>[[#ZEqnNum866801|(5.50)]], <span id='cite-ZEqnNum366676'></span>[[#ZEqnNum366676|(4.53)]] y <span id='cite-ZEqnNum894139'></span>[[#ZEqnNum894139|(4.54)]], el termino  [[Image:Draft_Content_179546005-image954.png|78px]] resulta

@@ Line 5,090: / Line 5,090: @@
 {| style="text-align: center; margin:auto;"
 |-
-|<math>{\rho }_a\left(1-S_l\right)\left(b\frac{d\epsilon \left(\boldsymbol{u}\right)}{dt}+\right. </math><math>\left. \frac{1}{N}\frac{dp_g}{dt}-\frac{1}{N}\left(S_l+\right. \right. </math><math>\left. \left. p_c\frac{d\chi }{dS_l}\frac{\partial S_l}{\partial p_c}\right)\frac{dp_c}{dt}-\right. </math><math>\left. \left(3{\alpha }_{\phi }+\frac{p_c}{N}\frac{d\chi }{dS_l}\frac{\partial S_l}{\partial T}\right)\frac{dT}{dt}\right)-</math><math>\phi {\rho }_a\left(\frac{\partial S_l}{\partial p_c}\frac{dp_c}{dt}+\right. </math><math>\left. \frac{\partial S_l}{\partial T}\frac{dT}{dt}\right)+</math><math>\phi \left(1-S_l\right)\frac{M_a}{RT}\left[\left(1-\right. \right. </math><math>\left. \left. \frac{\partial p_v}{\partial p_g}\right)\frac{dp_g}{dt}-\right. </math><math>\left. \frac{\partial p_v}{\partial p_c}\frac{dp_c}{dt}-\right. </math><math>\left. \left(\frac{\partial p_v}{\partial T}+\frac{p_a}{T}\right)\frac{dT}{dt}\right]+</math><math>\nabla .\boldsymbol{w}^a={\rho }_a\left(1-S_l\right)\left(b\frac{d{\epsilon }^i}{dt}-\right. </math><math>\left. \frac{d{\phi }^{dehyd}}{dt}\right)</math>
+|<math>{\rho }_a\left(1-S_l\right)\left(b\frac{d\epsilon \left(\boldsymbol{u}\right)}{dt}+ \frac{1}{N}\frac{dp_g}{dt}-\frac{1}{N}\left(S_l+ p_c\frac{d\chi }{dS_l}\frac{\partial S_l}{\partial p_c}\right)\frac{dp_c}{dt}- \left(3{\alpha }_{\phi }+\frac{p_c}{N}\frac{d\chi }{dS_l}\frac{\partial S_l}{\partial T}\right)\frac{dT}{dt}\right)</math><math> -\phi {\rho }_a\left(\frac{\partial S_l}{\partial p_c}\frac{dp_c}{dt}+ \frac{\partial S_l}{\partial T}\frac{dT}{dt}\right)+</math><math>\phi \left(1-S_l\right)\frac{M_a}{RT}\left[\left(1- \frac{\partial p_v}{\partial p_g}\right)\frac{dp_g}{dt}- \frac{\partial p_v}{\partial p_c}\frac{dp_c}{dt}- \left(\frac{\partial p_v}{\partial T}+\frac{p_a}{T}\right)\frac{dT}{dt}\right]+</math><math>\nabla .\boldsymbol{w}^a={\rho }_a\left(1-S_l\right)\left(b\frac{d{\epsilon }^i}{dt}-\right. </math><math>\left. \frac{d{\phi }^{dehyd}}{dt}\right)</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (5.77)
@@ Line 5,103: / Line 5,103: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <span id='ZEqnNum937466'></span>  <math>C_{CG}\frac{dp_g}{dt}+C_{CC}\frac{dp_c}{dt}+C_{CT}\frac{dT}{dt}-</math><math>\nabla .\left(K_{CG}\nabla p_g+K_{CC}\nabla p_c+K_{CT}\nabla T\right)=</math><math>f_C</math>
+| <span id='ZEqnNum937466'></span>  <math>C_{CG}\frac{dp_g}{dt}+C_{CC}\frac{dp_c}{dt}+C_{CT}\frac{dT}{dt}-</math><math>\nabla .\left(K_{CG}\nabla p_g+K_{CC}\nabla p_c+K_{CT}\nabla T\right)=f_C</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (5.78)
@@ Line 5,113: / Line 5,113: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <span id='ZEqnNum899540'></span>  <math>C_{GG}\frac{dp_g}{dt}+C_{GC}\frac{dp_c}{dt}+C_{GT}\frac{dT}{dt}-</math><math>\nabla .\left(K_{GG}\nabla p_g+K_{GC}\nabla p_c+K_{GT}\nabla T\right)=</math><math>f_G</math>
+| <span id='ZEqnNum899540'></span>  <math>C_{GG}\frac{dp_g}{dt}+C_{GC}\frac{dp_c}{dt}+C_{GT}\frac{dT}{dt}-</math><math>\nabla .\left(K_{GG}\nabla p_g+K_{GC}\nabla p_c+K_{GT}\nabla T\right)=f_G</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (5.79)

@@ Line 4,959: / Line 4,959: @@
 Teniendo en cuenta <span id='cite-ZEqnNum681818'></span>[[#ZEqnNum681818|(4.107)]] se puede expresar la expresión de la variación temporal de la porosidad <span id='cite-ZEqnNum409216'></span>[[#ZEqnNum409216|(5.27)]] en función de las variables de estado
 {| class="formulaSCP" style="width: 100%; text-align: center;"
@@ Line 4,967: / Line 4,965: @@
 {| style="text-align: center; margin:auto;"
 |-
+| <math>\frac{d\phi }{dt}=b\frac{d\epsilon \left(\boldsymbol{u}\right)}{dt}+</math><math>\frac{1}{N}\frac{dp_g}{dt}-\frac{\chi \left(S_l\right)}{N}\frac{dp_c}{dt}-</math><math>\frac{p_c}{N}\frac{d\chi \left(S_l\right)}{dt}-\frac{dT}{dt}-</math><math>b\frac{d{\epsilon }^i}{dt}+\frac{d{\phi }^{dehyd}}{dt}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (5.67)
@@ Line 4,990: / Line 4,988: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>\frac{d\chi \left(S_l\right)}{dt}=\frac{d\chi }{dS_l}\left(\frac{\partial S_l}{\partial p_c}\frac{dp_c}{dt}+\right. </math><math>\left. \frac{\partial S_l}{\partial T}\frac{dT}{dt}\right)</math>
+| <math>\frac{d\chi \left(S_l\right)}{dt}=\frac{d\chi }{dS_l}\left(\frac{\partial S_l}{\partial p_c}\frac{dp_c}{dt}+ \frac{\partial S_l}{\partial T}\frac{dT}{dt}\right)</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (5.69)
@@ Line 5,003: / Line 5,001: @@
 {| style="text-align: center; margin:auto;"
 |-
-|<math>\boldsymbol{w}^l+\boldsymbol{w}^v=\left(-{\rho }_l\frac{\mbox{k}_{\mbox{int}}k_{rl}}{{\eta }_l}\nabla p_g+\right. </math><math>\left. {\rho }_l\frac{\mbox{k}_{\mbox{int}}k_{rl}}{{\eta }_l}\nabla p_c+\right. </math><math>\left. {\rho }_l\boldsymbol{b}\right)+\left(-{\rho }_v\frac{\mbox{k}_{\mbox{int}}k_{rg}}{{\eta }_g}\nabla p_g-\right. </math><math>\left. {\rho }_vD_{ev}\nabla \left(\frac{p_v}{p_g}\right)\right)</math>
+|<math>\boldsymbol{w}^l+\boldsymbol{w}^v=\left(-{\rho }_l\frac{\mbox{k}_{\mbox{int}}k_{rl}}{{\eta }_l}\nabla p_g+{\rho }_l\frac{\mbox{k}_{\mbox{int}}k_{rl}}{{\eta }_l}\nabla p_c+ {\rho }_l\boldsymbol{b}\right)+</math><math> \left(-{\rho }_v\frac{\mbox{k}_{\mbox{int}}k_{rg}}{{\eta }_g}\nabla p_g- {\rho }_vD_{ev}\nabla \left(\frac{p_v}{p_g}\right)\right)</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (5.70)
@@ Line 5,042: / Line 5,040: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <span id='ZEqnNum833758'></span>  <math>\nabla \left(\frac{p_v}{p_g}\right)=\left(\frac{1}{p_g}\frac{\partial p_v}{\partial p_g}-\right. </math><math>\left. \frac{p_v}{p_g^2}\right)\nabla p_g+\frac{1}{p_g}\frac{\partial p_v}{\partial p_c}\nabla p_c+</math><math>\frac{1}{p_g}\frac{\partial p_v}{\partial T}\nabla T</math>
+| <span id='ZEqnNum833758'></span>  <math>\nabla \left(\frac{p_v}{p_g}\right)=\left(\frac{1}{p_g}\frac{\partial p_v}{\partial p_g}- \frac{p_v}{p_g^2}\right)\nabla p_g+\frac{1}{p_g}\frac{\partial p_v}{\partial p_c}\nabla p_c+</math><math>\frac{1}{p_g}\frac{\partial p_v}{\partial T}\nabla T</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (5.73)
@@ Line 5,055: / Line 5,053: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>\boldsymbol{w}^l+\boldsymbol{w}^v=\left[-{\rho }_l\frac{\mbox{k}_{\mbox{int}}k_{rl}}{{\eta }_l}-\right. </math><math>\left. {\rho }_v\frac{\mbox{k}_{\mbox{int}}k_{rg}}{{\eta }_g}-\right. </math><math>\left. {\rho }_vD_{ev}\left(\frac{1}{p_g}\frac{\partial p_v}{\partial p_g}-\right. \right. </math><math>\left. \left. \frac{p_v}{p_g^2}\right)\right]\nabla p_g+</math><math>\left[{\rho }_l\frac{\mbox{k}_{\mbox{int}}k_{rl}}{{\eta }_l}-\right. </math><math>\left. \frac{{\rho }_vD_{ev}}{p_g}\frac{\partial p_v}{\partial p_c}\right]\nabla p_c+</math><math>\left[-\frac{{\rho }_vD_{ev}}{p_g}\frac{\partial p_v}{\partial T}\right]\nabla T+</math><math>{\rho }_l\boldsymbol{b}</math>
+| <math>\boldsymbol{w}^l+\boldsymbol{w}^v=\left[-{\rho }_l\frac{\mbox{k}_{\mbox{int}}k_{rl}}{{\eta }_l}- {\rho }_v\frac{\mbox{k}_{\mbox{int}}k_{rg}}{{\eta }_g}- {\rho }_vD_{ev}\left(\frac{1}{p_g}\frac{\partial p_v}{\partial p_g}- \frac{p_v}{p_g^2}\right)\right]\nabla p_g+</math><math>\left[{\rho }_l\frac{\mbox{k}_{\mbox{int}}k_{rl}}{{\eta }_l}-\frac{{\rho }_vD_{ev}}{p_g}\frac{\partial p_v}{\partial p_c}\right]\nabla p_c+</math><math>\left[-\frac{{\rho }_vD_{ev}}{p_g}\frac{\partial p_v}{\partial T}\right]\nabla T+{\rho }_l\boldsymbol{b}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (5.74)
@@ Line 5,068: / Line 5,066: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <span id='ZEqnNum962135'></span>  <math>\boldsymbol{w}^a=\left[-{\rho }_a\frac{\mbox{k}_{\mbox{int}}k_{rg}}{{\eta }_g}+\right. </math><math>\left. {\rho }_aD_{ea}\left(\frac{1}{p_g}\frac{\partial p_v}{\partial p_g}-\right. \right. </math><math>\left. \left. \frac{p_v}{p_g^2}\right)\right]\nabla p_g+</math><math>\left[\frac{{\rho }_aD_{ea}}{p_g}\frac{\partial p_v}{\partial p_c}\right]\nabla p_c+</math><math>\left[\frac{{\rho }_aD_{ea}}{p_g}\frac{\partial p_v}{\partial T}\right]\nabla T</math>
+| <span id='ZEqnNum962135'></span>  <math>\boldsymbol{w}^a=\left[-{\rho }_a\frac{\mbox{k}_{\mbox{int}}k_{rg}}{{\eta }_g}+ {\rho }_aD_{ea}\left(\frac{1}{p_g}\frac{\partial p_v}{\partial p_g}- \frac{p_v}{p_g^2}\right)\right]\nabla p_g+</math><math>\left[\frac{{\rho }_aD_{ea}}{p_g}\frac{\partial p_v}{\partial p_c}\right]\nabla p_c+</math><math>\left[\frac{{\rho }_aD_{ea}}{p_g}\frac{\partial p_v}{\partial T}\right]\nabla T</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (5.75)
@@ Line 5,081: / Line 5,079: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <span id='ZEqnNum951132'></span> <math>\left[{\rho }_lS_l+{\rho }_v\left(1-S_l\right)\right]\left(b\frac{d\epsilon \left(\boldsymbol{u}\right)}{dt}+\right. </math><math>\left. \frac{1}{N}\frac{dp_g}{dt}-\frac{1}{N}\left(S_l+\right. \right. </math><math>\left. \left. p_c\frac{d\chi }{dS_l}\frac{\partial S_l}{\partial p_c}\right)\frac{dp_c}{dt}-\right. </math><math>\left. \left(3{\alpha }_{\phi }+\frac{p_c}{N}\frac{d\chi }{dS_l}\frac{\partial S_l}{\partial T}\right)\frac{dT}{dt}\right)+</math><math>\phi \left({\rho }_l-{\rho }_v\right)\left(\frac{\partial S_l}{\partial p_c}\frac{dp_c}{dt}+\right. </math><math>\left. \frac{\partial S_l}{\partial T}\frac{dT}{dt}\right)-</math><math>\phi S_l3{\alpha }_l{\rho }_l\frac{dT}{dt}+\phi \left(1-\right. </math><math>\left. S_l\right)\frac{M_v}{RT}\left[\frac{\partial p_v}{\partial p_g}\frac{dp_g}{dt}+\right. </math><math>\left. \frac{\partial p_v}{\partial p_c}\frac{dp_c}{dt}+\right. </math><math>\left. \left(\frac{\partial p_v}{\partial T}-\frac{p_v}{T}\right)\frac{dT}{dt}\right]+</math><math>\nabla .\left(\boldsymbol{w}^l+\boldsymbol{w}^v\right)=</math><math>{\dot{m}}_{dehyd}+\left[{\rho }_lS_l+{\rho }_v\left(1-\right. \right. </math><math>\left. \left. S_l\right)\right]\left(b\frac{d{\epsilon }^i}{dt}-\right. </math><math>\left. \frac{d{\phi }^{dehyd}}{dt}\right)</math>
+| <span id='ZEqnNum951132'></span> <math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (5.76)

@@ Line 4,884: / Line 4,884: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>\frac{\partial p_v}{\partial p_g}=p_{vs}(T)\frac{M_v}{{\rho }_lRT}exp\left[\frac{M_v}{{\rho }_lRT}\left(p_g-\right. \right. </math><math>\left. \left. p_c-p_{atm}\right)\right]</math>
+| <math>\frac{\partial p_v}{\partial p_g}=p_{vs}(T)\frac{M_v}{{\rho }_lRT}exp\left[\frac{M_v}{{\rho }_lRT}\left(p_g- p_c-p_{atm}\right)\right]</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (5.60)
@@ Line 4,894: / Line 4,894: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>\frac{\partial p_v}{\partial p_c}=-p_{vs}(T)\frac{M_v}{{\rho }_lRT}exp\left[\frac{M_v}{{\rho }_lRT}\left(p_g-\right. \right. </math><math>\left. \left. p_c-p_{atm}\right)\right]</math>
+| <math>\frac{\partial p_v}{\partial p_c}=-p_{vs}(T)\frac{M_v}{{\rho }_lRT}exp\left[\frac{M_v}{{\rho }_lRT}\left(p_g- p_c-p_{atm}\right)\right]</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (5.61)
@@ Line 4,904: / Line 4,904: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>\frac{\partial p_v}{\partial T}=\frac{\partial p_{vs}(T)}{\partial T}exp\left[\frac{M_v}{{\rho }_lRT}\left(p_g-\right. \right. </math><math>\left. \left. p_c-p_{atm}\right)\right]-\frac{M_v}{{\rho }_lRT^2}\left(p_g-\right. </math><math>\left. p_c-p_{atm}\right)\mbox{ }p_v\left(p_g,\mbox{ }p_c,\mbox{ }T\right)</math>
+| <math>\frac{\partial p_v}{\partial T}=\frac{\partial p_{vs}(T)}{\partial T}exp\left[\frac{M_v}{{\rho }_lRT}\left(p_g- p_c-p_{atm}\right)\right]-\frac{M_v}{{\rho }_lRT^2} <math>T</math><math>\left(p_g-p_c-p_{atm}\right)\mbox{ }p_v\left(p_g,\mbox{ }p_c,\mbox{ }T\right)</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (5.62)

← Older revision	Revision as of 13:26, 7 June 2018
(No difference)

@@ Line 4,740: / Line 4,740: @@
 Se denominan variables de estado a un subconjunto de variables termodinámicas en función de las cuales se pueden definir todas las demás. También se las conoce como variables independientes o libres. Una correcta selección de las variables de estado es de vital importancia para conseguir una adecuada resolución numérica del modelo matemático resultante.
-Como variables de estado termo-mecánicas la temperatura  [[Image:Draft_Content_179546005-image887.png|12px]] y el vector de desplazamientos del esqueleto sólido  [[Image:Draft_Content_179546005-image888.png|12px]], son las opciones más naturales. No obstante la selección de una variable de estado relacionada con la humedad no resulta tan evidente. Existen varias posibilidades termodinámicamente equivalentes: masa o volumen del contenido de humedad, presión de vapor, humedad relativa y presión capilar.
+Como variables de estado termo-mecánicas la temperatura  <math>T</math> y el vector de desplazamientos del esqueleto sólido  <math>\bf u</math>, son las opciones más naturales. No obstante la selección de una variable de estado relacionada con la humedad no resulta tan evidente. Existen varias posibilidades termodinámicamente equivalentes: masa o volumen del contenido de humedad, presión de vapor, humedad relativa y presión capilar.
 Es importante recordar que a temperaturas por encima del punto crítico del agua (374.15 &#x00b0;C), solo existe agua en estado gaseoso (vapor) y que por tanto ya no se encuentra agua libre (capilar) en los poros del hormigón. Debido a esto el uso de la presión capilar como variable de estado relacionada con la humedad deja de tener sentido físico para temperaturas por encima del punto crítico del agua.
@@ Line 4,746: / Line 4,746: @@
 En Gawin et al. (2003) se realiza una extensa discusión de las posibles variables de estado relacionadas con la humedad. En dicha referencia se concluye que la elección más apropiada es la presión capilar pero con una interpretación física distinta para el rango de temperaturas superiores al punto crítico del agua (Gawin et al., 2002). En este rango de temperaturas se considera a la presión capilar como el potencial termodinámico del agua físicamente absorbida.
-En resumen las variables de estado consideradas en el presente modelo son las siguientes: la presión del gas  [[Image:Draft_Content_179546005-image889.png|18px]], la presión capilar  [[Image:Draft_Content_179546005-image890.png|18px]], la temperatura  [[Image:Draft_Content_179546005-image887.png|12px]] y el vector de desplazamientos del esqueleto sólido  [[Image:Draft_Content_179546005-image888.png|12px]].
+En resumen las variables de estado consideradas en el presente modelo son las siguientes: la presión del gas  <math>p_g</math>, la presión capilar  <math>p_c</math>, la temperatura  <math>T</math> y el vector de desplazamientos del esqueleto sólido  <math>\bf u</math>.
 <span id='_Toc205018949'></span><span id='_Toc209246133'></span>

@@ Line 4,507: / Line 4,507: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <span id='ZEqnNum903560'></span>  <math>K_{SEC}\mbox{=}\frac{\mbox{1}}{\mbox{9}}{\delta }_{ij}{\delta }_{kl}\mbox{ }\left[\left(1-\right. \right. </math><math>\left. \left. d^+\right)\frac{\partial {\overline{\sigma }}_{ij}^+}{\partial {\epsilon }_{kl}^e}+\right. </math><math>\left. \left(1-d^{-}\right)\frac{\partial {\overline{\sigma }}_{ij}^{-}}{\partial {\epsilon }_{kl}^e}\right]</math>
+| <span id='ZEqnNum903560'></span>  <math>K_{SEC}\mbox{=}\frac{\mbox{1}}{\mbox{9}}{\delta }_{ij}{\delta }_{kl}\mbox{ }\left[\left(1- d^+\right)\frac{\partial {\overline{\sigma }}_{ij}^+}{\partial {\epsilon }_{kl}^e}+ \left(1-d^{-}\right)\frac{\partial {\overline{\sigma }}_{ij}^{-}}{\partial {\epsilon }_{kl}^e}\right]</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (5.33)

@@ Line 4,099: / Line 4,099: @@
 | <math>\begin{array}{c}
 {\psi }_s={\psi }_s\left({\epsilon }_{ij}-{\epsilon }_{ij}^i,\mbox{ }\phi ,\mbox{ }T,\mbox{ }d^+\mbox{, }d^{-},\mbox{ }\omega \right)\Rightarrow \mbox{   }\\
-{\sigma }_{ij}=\frac{\partial {\psi }_s}{\partial \left({\epsilon }_{ij}-{\epsilon }_{ij}^i\right)};\mbox{   }\pi =\frac{\partial {\psi }_s}{\partial \phi };\mbox{   }{\overset{\frown}{S}}_s=-\frac{\partial {\psi }_s}{\partial T}
+\displaystyle {\sigma }_{ij}=\frac{\partial {\psi }_s}{\partial \left({\epsilon }_{ij}-{\epsilon }_{ij}^i\right)};\mbox{   }\pi =\frac{\partial {\psi }_s}{\partial \phi };\mbox{   }{\overset{\frown}{S}}_s=-\frac{\partial {\psi }_s}{\partial T}
 \end{array}</math>
 |}
@@ Line 4,141: / Line 4,141: @@
 | <span id='ZEqnNum650572'></span>  <math>\begin{array}{c}
 G_s=G_s\left({\epsilon }_{ij}-{\epsilon }_{ij}^i,\mbox{ }\pi ,\mbox{ }T,\mbox{ }d^+\mbox{, }d^{-},\mbox{ }\omega \right)\Rightarrow \mbox{   }\\
-{\sigma }_{ij}=\frac{\partial G_s}{\partial \left({\epsilon }_{ij}-{\epsilon }_{ij}^i\right)};\mbox{   }\phi =-\frac{\partial G_s}{\partial \pi };\mbox{   }{\overset{\frown}{S}}_s=-\frac{\partial G_s}{\partial T}
+\displaystyle{\sigma }_{ij}=\frac{\partial G_s}{\partial \left({\epsilon }_{ij}-{\epsilon }_{ij}^i\right)};\mbox{   }\phi =-\frac{\partial G_s}{\partial \pi };\mbox{   }{\overset{\frown}{S}}_s=-\frac{\partial G_s}{\partial T}
 \end{array}</math>
 |}
@@ Line 4,156: / Line 4,156: @@
 |-
 | <span id='ZEqnNum142101'></span>  <math>\begin{array}{c}
-\frac{\partial {\sigma }_{ij}}{\partial \left({\epsilon }_{kl}-{\epsilon }_{kl}^i\right)}=\frac{\partial {\sigma }_{kl}}{\partial \left({\epsilon }_{ij}-{\epsilon }_{ij}^i\right)};\mbox{  }\frac{\partial {\sigma }_{ij}}{\partial \pi }=-\frac{\partial \phi }{\partial \left({\epsilon }_{kl}-{\epsilon }_{kl}^i\right)};\mbox{ }\\
+\displaystyle\frac{\partial {\sigma }_{ij}}{\partial \left({\epsilon }_{kl}-{\epsilon }_{kl}^i\right)}=\frac{\partial {\sigma }_{kl}}{\partial \left({\epsilon }_{ij}-{\epsilon }_{ij}^i\right)};\mbox{  }\frac{\partial {\sigma }_{ij}}{\partial \pi }=-\frac{\partial \phi }{\partial \left({\epsilon }_{kl}-{\epsilon }_{kl}^i\right)};\mbox{ }\\
-\frac{\partial {\sigma }_{ij}}{\partial T}=-\frac{\partial {\overset{\frown}{S}}_s}{\partial \left({\epsilon }_{ij}-{\epsilon }_{ij}^i\right)};\mbox{    }\frac{\partial \phi }{\partial T}=\frac{\partial {\overset{\frown}{S}}_s}{\partial \pi }
+\displaystyle\frac{\partial {\sigma }_{ij}}{\partial T}=-\frac{\partial {\overset{\frown}{S}}_s}{\partial \left({\epsilon }_{ij}-{\epsilon }_{ij}^i\right)};\mbox{    }\frac{\partial \phi }{\partial T}=\frac{\partial {\overset{\frown}{S}}_s}{\partial \pi }
 \end{array}</math>
 |}
@@ Line 4,191: / Line 4,191: @@
 | <math>\begin{array}{c}
 G_s^e\left({\epsilon }_{ij}-{\epsilon }_{ij}^i,\mbox{ }T,\mbox{ }d^+\mbox{, }d^{-},\mbox{ }\omega \right)=\\
-{\sigma }_{ij}^0\left({\epsilon }_{ij}-{\epsilon }_{ij}^i\right)+\frac{1}{2}\left(1-d^+\right){\overline{\sigma }}_{ij}^+{\left[\left(1-\omega \right)C_{ijkl}^0\right]}^{-1}{\overline{\sigma }}_{kl}+\frac{1}{2}\left(1-d^{-}\right){\overline{\sigma }}_{ij}^{-}{\left[\left(1-\omega \right)C_{ijkl}^0\right]}^{-1}{\overline{\sigma }}_{kl}
+\displaystyle{\sigma }_{ij}^0\left({\epsilon }_{ij}-{\epsilon }_{ij}^i\right)+\frac{1}{2}\left(1-d^+\right){\overline{\sigma }}_{ij}^+{\left[\left(1-\omega \right)C_{ijkl}^0\right]}^{-1}{\overline{\sigma }}_{kl}+\frac{1}{2}\left(1-d^{-}\right){\overline{\sigma }}_{ij}^{-}{\left[\left(1-\omega \right)C_{ijkl}^0\right]}^{-1}{\overline{\sigma }}_{kl}
 \end{array}</math>
 |}
@@ Line 4,303: / Line 4,303: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <span id='ZEqnNum687161'></span> <math>\begin{array}{c}
+| <span id='ZEqnNum687161'></span> <math>
-{\sigma }_{ij}=\frac{\partial G_s}{\partial \left({\epsilon }_{ij}-{\epsilon }_{ij}^i\right)}=\frac{\partial \left(G_s^e+G_s^{HM}\right)}{\partial \left({\epsilon }_{ij}-{\epsilon }_{ij}^i\right)}=\\
+{\sigma }_{ij}=\frac{\partial G_s}{\partial \left({\epsilon }_{ij}-{\epsilon }_{ij}^i\right)}=\frac{\partial \left(G_s^e+G_s^{HM}\right)}{\partial \left({\epsilon }_{ij}-{\epsilon }_{ij}^i\right)}=
-={\sigma }_{ij}^0+\left(1-d^+\right){\overline{\sigma }}_{ij}^++\left(1-d^{-}\right){\overline{\sigma }}_{ij}^{-}-b{\delta }_{ij}\left(\pi -{\pi }_0\right)
+{\sigma }_{ij}^0+\left(1-d^+\right){\overline{\sigma }}_{ij}^++\left(1-d^{-}\right){\overline{\sigma }}_{ij}^{-}-b{\delta }_{ij}\left(\pi -{\pi }_0\right)</math>
-\end{array}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (5.19)
@@ Line 4,316: / Line 4,315: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <span id='ZEqnNum297009'></span> <math>\begin{array}{c}
+| <span id='ZEqnNum297009'></span> <math>
-\phi =-\frac{\partial G_s}{\partial \pi }=-\frac{\partial \left(G_s^{HM}+G_s^H+G_s^{HT}\right)}{\partial \pi }=\\
+\phi =-\frac{\partial G_s}{\partial \pi }=-\frac{\partial \left(G_s^{HM}+G_s^H+G_s^{HT}\right)}{\partial \pi }
 ={\phi }^0+b\left(\epsilon -{\epsilon }^i\right)+\frac{\left(\pi -{\pi }_0\right)}{N}-3{\alpha }_{\phi }\left(T-T_0\right)+{\phi }^{dehyd}
-\end{array}</math>
+</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (5.20)

@@ Line 3,582: / Line 3,582: @@
 | <span id='ZEqnNum180124'></span>  <math>\begin{array}{c}
 {\Psi }_s={\Psi }_s\left({\epsilon }_{ij},\phi ,S_l,T\right)\Rightarrow \mbox{   }\\
-{\sigma }_{ij}=\frac{\partial {\Psi }_s}{\partial {\epsilon }_{ij}};\mbox{   }\pi =\frac{\partial {\Psi }_s}{\partial \phi };\mbox{   }{\phi }_0p_c=-\frac{\partial {\Psi }_s}{\partial S_l};\mbox{   }S_s=-\frac{\partial {\Psi }_s}{\partial T}
+\displaystyle {\sigma }_{ij}=\frac{\partial {\Psi }_s}{\partial {\epsilon }_{ij}};\mbox{   }\pi =\frac{\partial {\Psi }_s}{\partial \phi };\mbox{   }{\phi }_0p_c=-\frac{\partial {\Psi }_s}{\partial S_l};\mbox{   }S_s=-\frac{\partial {\Psi }_s}{\partial T}
 \end{array}</math>
 |}
@@ Line 3,689: / Line 3,689: @@
 |-
 | <math>\begin{array}{c}
-\overline{M}\left(T\right)=\left(1-0.625a\right)\left(0.2358a^{1.256}\right)\mbox{    con }a=\frac{374.15-T}{647.3}\mbox{   para }T\leq 360\mbox{°}\mbox{C}\\
+\displaystyle \overline{M}\left(T\right)=\left(1-0.625a\right)\left(0.2358a^{1.256}\right)\mbox{    con }a=\frac{374.15-T}{647.3}\mbox{   para }T\leq 360\mbox{°}\mbox{C}\\
-\overline{M}\left(T\right)=\mbox{0}\mbox{.00191060 exp(-0}\mbox{.003136}T^2\mbox{+2}\mbox{.1702}T\mbox{-374}\mbox{.8383)   para }T>360\mbox{°}\mbox{C}
+\displaystyle \overline{M}\left(T\right)=\mbox{0}\mbox{.00191060 exp(-0}\mbox{.003136}T^2\mbox{+2}\mbox{.1702}T\mbox{-374}\mbox{.8383)   para }T>360\mbox{°}\mbox{C}
 \end{array}</math>
 |}