Infinite Subcritical Reactor with a plane Source; Deriving analytical solution with boundary conditions












1














This might as well be a mathematics question since I am only looking for a way to solve a basic model of the neutrondistribution in an infinite plane subcritical reactor.



So the model for the neutrondistribution looks like this:



$frac{d^{2}phi}{{dx}^{2}}-gamma^{2}phi+frac{Q}{D}delta(x)=0$



The source of the neutrons is located at $x=0$, the net flow of neutrons is $Q$ per second and per unit area (perpendicular to the $x$-axis).



Now, we seperate this into two regions, namely $x<0$ and $x>0$. For the region $x>0$ we obtain the second order differential equation $frac{d^{2}phi}{{dx}^{2}}-gamma^{2}phi = 0$ which can be solved directly, yielding
$phi(x) =Aexp{(gamma x)}+Cexp{(-gamma x)}$. Naturally as $xrightarrow infty$ then $phi rightarrow 0$, so $A=0$. We can yield a similar result for the other result, namely $phi(x)=Dexp{(gamma x)}$, $x<0$. At the boundary we have that these two functions (let's name them $phi_1$ and $phi_2$ for $x<0$ and $x>0$, respectively), have the same value: $phi_1(0)=phi_2(0)$. This yields $D=C$.



Now, the last boundary condition is tricky. My teacher tells me that that the last final BC that sets the value of $D$ comes from integrating the first equation:



$lim_{epsilon to 0}int_{-epsilon}^{epsilon}[frac{d^{2}phi}{{dx}^{2}}-gamma^{2}phi+frac{Q}{D}delta(x)]$



I am not sure how this gives me anything, since the integral can be seperated into the different regions where we know that the differential equation equals zero. I got a (reasonable) result by using Fick's law instead, with



$lim_{x to 0}(-Dfrac{dphi}{dx})=Q/2$



How do I solve this problem going the way my teacher hinted about? This is supposedly the more mathematical way.



Best regards SimpleP.






differential-equations physics







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    1














    This might as well be a mathematics question since I am only looking for a way to solve a basic model of the neutrondistribution in an infinite plane subcritical reactor.



    So the model for the neutrondistribution looks like this:



    $frac{d^{2}phi}{{dx}^{2}}-gamma^{2}phi+frac{Q}{D}delta(x)=0$



    The source of the neutrons is located at $x=0$, the net flow of neutrons is $Q$ per second and per unit area (perpendicular to the $x$-axis).



    Now, we seperate this into two regions, namely $x<0$ and $x>0$. For the region $x>0$ we obtain the second order differential equation $frac{d^{2}phi}{{dx}^{2}}-gamma^{2}phi = 0$ which can be solved directly, yielding
    $phi(x) =Aexp{(gamma x)}+Cexp{(-gamma x)}$. Naturally as $xrightarrow infty$ then $phi rightarrow 0$, so $A=0$. We can yield a similar result for the other result, namely $phi(x)=Dexp{(gamma x)}$, $x<0$. At the boundary we have that these two functions (let's name them $phi_1$ and $phi_2$ for $x<0$ and $x>0$, respectively), have the same value: $phi_1(0)=phi_2(0)$. This yields $D=C$.



    Now, the last boundary condition is tricky. My teacher tells me that that the last final BC that sets the value of $D$ comes from integrating the first equation:



    $lim_{epsilon to 0}int_{-epsilon}^{epsilon}[frac{d^{2}phi}{{dx}^{2}}-gamma^{2}phi+frac{Q}{D}delta(x)]$



    I am not sure how this gives me anything, since the integral can be seperated into the different regions where we know that the differential equation equals zero. I got a (reasonable) result by using Fick's law instead, with



    $lim_{x to 0}(-Dfrac{dphi}{dx})=Q/2$



    How do I solve this problem going the way my teacher hinted about? This is supposedly the more mathematical way.



    Best regards SimpleP.






    differential-equations physics







    share|cite|improve this question















    migrated from physics.stackexchange.com Dec 7 at 15:16


    This question came from our site for active researchers, academics and students of physics.


















      1












      1








      1







      This might as well be a mathematics question since I am only looking for a way to solve a basic model of the neutrondistribution in an infinite plane subcritical reactor.



      So the model for the neutrondistribution looks like this:



      $frac{d^{2}phi}{{dx}^{2}}-gamma^{2}phi+frac{Q}{D}delta(x)=0$



      The source of the neutrons is located at $x=0$, the net flow of neutrons is $Q$ per second and per unit area (perpendicular to the $x$-axis).



      Now, we seperate this into two regions, namely $x<0$ and $x>0$. For the region $x>0$ we obtain the second order differential equation $frac{d^{2}phi}{{dx}^{2}}-gamma^{2}phi = 0$ which can be solved directly, yielding
      $phi(x) =Aexp{(gamma x)}+Cexp{(-gamma x)}$. Naturally as $xrightarrow infty$ then $phi rightarrow 0$, so $A=0$. We can yield a similar result for the other result, namely $phi(x)=Dexp{(gamma x)}$, $x<0$. At the boundary we have that these two functions (let's name them $phi_1$ and $phi_2$ for $x<0$ and $x>0$, respectively), have the same value: $phi_1(0)=phi_2(0)$. This yields $D=C$.



      Now, the last boundary condition is tricky. My teacher tells me that that the last final BC that sets the value of $D$ comes from integrating the first equation:



      $lim_{epsilon to 0}int_{-epsilon}^{epsilon}[frac{d^{2}phi}{{dx}^{2}}-gamma^{2}phi+frac{Q}{D}delta(x)]$



      I am not sure how this gives me anything, since the integral can be seperated into the different regions where we know that the differential equation equals zero. I got a (reasonable) result by using Fick's law instead, with



      $lim_{x to 0}(-Dfrac{dphi}{dx})=Q/2$



      How do I solve this problem going the way my teacher hinted about? This is supposedly the more mathematical way.



      Best regards SimpleP.






      differential-equations physics







      share|cite|improve this question















      This might as well be a mathematics question since I am only looking for a way to solve a basic model of the neutrondistribution in an infinite plane subcritical reactor.



      So the model for the neutrondistribution looks like this:



      $frac{d^{2}phi}{{dx}^{2}}-gamma^{2}phi+frac{Q}{D}delta(x)=0$



      The source of the neutrons is located at $x=0$, the net flow of neutrons is $Q$ per second and per unit area (perpendicular to the $x$-axis).



      Now, we seperate this into two regions, namely $x<0$ and $x>0$. For the region $x>0$ we obtain the second order differential equation $frac{d^{2}phi}{{dx}^{2}}-gamma^{2}phi = 0$ which can be solved directly, yielding
      $phi(x) =Aexp{(gamma x)}+Cexp{(-gamma x)}$. Naturally as $xrightarrow infty$ then $phi rightarrow 0$, so $A=0$. We can yield a similar result for the other result, namely $phi(x)=Dexp{(gamma x)}$, $x<0$. At the boundary we have that these two functions (let's name them $phi_1$ and $phi_2$ for $x<0$ and $x>0$, respectively), have the same value: $phi_1(0)=phi_2(0)$. This yields $D=C$.



      Now, the last boundary condition is tricky. My teacher tells me that that the last final BC that sets the value of $D$ comes from integrating the first equation:



      $lim_{epsilon to 0}int_{-epsilon}^{epsilon}[frac{d^{2}phi}{{dx}^{2}}-gamma^{2}phi+frac{Q}{D}delta(x)]$



      I am not sure how this gives me anything, since the integral can be seperated into the different regions where we know that the differential equation equals zero. I got a (reasonable) result by using Fick's law instead, with



      $lim_{x to 0}(-Dfrac{dphi}{dx})=Q/2$



      How do I solve this problem going the way my teacher hinted about? This is supposedly the more mathematical way.



      Best regards SimpleP.






      differential-equations physics




      differential-equations physics






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      edited Dec 8 at 8:11









      Martin Sleziak

      44.6k7115270




      44.6k7115270










      asked Nov 30 at 20:44









      SimpleProgrammer

      276




      276




      migrated from physics.stackexchange.com Dec 7 at 15:16


      This question came from our site for active researchers, academics and students of physics.






      migrated from physics.stackexchange.com Dec 7 at 15:16


      This question came from our site for active researchers, academics and students of physics.
























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          Okay, so I got an answer (finally) to this question. I am going to write the solution below.



          $lim_{epsilonrightarrow 0}int_{-epsilon}^{epsilon}[frac{d^2phi}{dx^2}-gamma^2phi+frac{Q}{D}delta(x)]dx=lim_{epsilonrightarrow 0}(int_{-epsilon}^{epsilon}frac{d^2phi}{dx^2}dx-int_{-epsilon}^{epsilon}gamma^2phi dx+int_{-epsilon}^{epsilon}frac{Q}{D}delta(x)dx)$.



          Now the first trem integrates to $lim_{epsilonrightarrow0}[phi'(epsilon)-phi'(-epsilon)]=phi'(0+)-phi'(0-)$



          The second term becomes $lim_{epsilonrightarrow0}(-int_{-epsilon}^{epsilon}gamma^2phi dx)={Continuous:and:finite:at:boundary}=0$



          The third term just becomes, per definition $lim_{epsilonrightarrow0}int_{-epsilon}^{epsilon}frac{Q}{D}delta(x)dx=frac{Q}{D}$



          You put this back into the equation and you obtain Fick's law through purely mathematical means. Best regards.






          share|cite|improve this answer





















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            Okay, so I got an answer (finally) to this question. I am going to write the solution below.



            $lim_{epsilonrightarrow 0}int_{-epsilon}^{epsilon}[frac{d^2phi}{dx^2}-gamma^2phi+frac{Q}{D}delta(x)]dx=lim_{epsilonrightarrow 0}(int_{-epsilon}^{epsilon}frac{d^2phi}{dx^2}dx-int_{-epsilon}^{epsilon}gamma^2phi dx+int_{-epsilon}^{epsilon}frac{Q}{D}delta(x)dx)$.



            Now the first trem integrates to $lim_{epsilonrightarrow0}[phi'(epsilon)-phi'(-epsilon)]=phi'(0+)-phi'(0-)$



            The second term becomes $lim_{epsilonrightarrow0}(-int_{-epsilon}^{epsilon}gamma^2phi dx)={Continuous:and:finite:at:boundary}=0$



            The third term just becomes, per definition $lim_{epsilonrightarrow0}int_{-epsilon}^{epsilon}frac{Q}{D}delta(x)dx=frac{Q}{D}$



            You put this back into the equation and you obtain Fick's law through purely mathematical means. Best regards.






            share|cite|improve this answer


























              1














              Okay, so I got an answer (finally) to this question. I am going to write the solution below.



              $lim_{epsilonrightarrow 0}int_{-epsilon}^{epsilon}[frac{d^2phi}{dx^2}-gamma^2phi+frac{Q}{D}delta(x)]dx=lim_{epsilonrightarrow 0}(int_{-epsilon}^{epsilon}frac{d^2phi}{dx^2}dx-int_{-epsilon}^{epsilon}gamma^2phi dx+int_{-epsilon}^{epsilon}frac{Q}{D}delta(x)dx)$.



              Now the first trem integrates to $lim_{epsilonrightarrow0}[phi'(epsilon)-phi'(-epsilon)]=phi'(0+)-phi'(0-)$



              The second term becomes $lim_{epsilonrightarrow0}(-int_{-epsilon}^{epsilon}gamma^2phi dx)={Continuous:and:finite:at:boundary}=0$



              The third term just becomes, per definition $lim_{epsilonrightarrow0}int_{-epsilon}^{epsilon}frac{Q}{D}delta(x)dx=frac{Q}{D}$



              You put this back into the equation and you obtain Fick's law through purely mathematical means. Best regards.






              share|cite|improve this answer
























                1












                1








                1






                Okay, so I got an answer (finally) to this question. I am going to write the solution below.



                $lim_{epsilonrightarrow 0}int_{-epsilon}^{epsilon}[frac{d^2phi}{dx^2}-gamma^2phi+frac{Q}{D}delta(x)]dx=lim_{epsilonrightarrow 0}(int_{-epsilon}^{epsilon}frac{d^2phi}{dx^2}dx-int_{-epsilon}^{epsilon}gamma^2phi dx+int_{-epsilon}^{epsilon}frac{Q}{D}delta(x)dx)$.



                Now the first trem integrates to $lim_{epsilonrightarrow0}[phi'(epsilon)-phi'(-epsilon)]=phi'(0+)-phi'(0-)$



                The second term becomes $lim_{epsilonrightarrow0}(-int_{-epsilon}^{epsilon}gamma^2phi dx)={Continuous:and:finite:at:boundary}=0$



                The third term just becomes, per definition $lim_{epsilonrightarrow0}int_{-epsilon}^{epsilon}frac{Q}{D}delta(x)dx=frac{Q}{D}$



                You put this back into the equation and you obtain Fick's law through purely mathematical means. Best regards.






                share|cite|improve this answer












                Okay, so I got an answer (finally) to this question. I am going to write the solution below.



                $lim_{epsilonrightarrow 0}int_{-epsilon}^{epsilon}[frac{d^2phi}{dx^2}-gamma^2phi+frac{Q}{D}delta(x)]dx=lim_{epsilonrightarrow 0}(int_{-epsilon}^{epsilon}frac{d^2phi}{dx^2}dx-int_{-epsilon}^{epsilon}gamma^2phi dx+int_{-epsilon}^{epsilon}frac{Q}{D}delta(x)dx)$.



                Now the first trem integrates to $lim_{epsilonrightarrow0}[phi'(epsilon)-phi'(-epsilon)]=phi'(0+)-phi'(0-)$



                The second term becomes $lim_{epsilonrightarrow0}(-int_{-epsilon}^{epsilon}gamma^2phi dx)={Continuous:and:finite:at:boundary}=0$



                The third term just becomes, per definition $lim_{epsilonrightarrow0}int_{-epsilon}^{epsilon}frac{Q}{D}delta(x)dx=frac{Q}{D}$



                You put this back into the equation and you obtain Fick's law through purely mathematical means. Best regards.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Dec 8 at 11:37









                SimpleProgrammer

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