BENG 221 Mathematical Methods in Bioengineering Lecture 9 Heat and Diffusion Equation in Space and Time References Haberman APDE Sec 11 3 http en wikipedia org wiki Heat equation http en wikipedia org wiki Diffusion equation Cartesian Box Value Boundary Conditions h11 19 74 L2 h12 49 35 L2 h13 98 7 L2 h14 167 8 L2 h15 256 6 L2 h21 49 35 L2 h22 78 96 L2 h23 128 3 L2 h24 197 4 L2 h25 286 2 L2 h31 98 7 L2 h32 128 3 L2 h33 177 7 L2 h34 246 7 L2 h35 335 6 L2 h41 167 8 L2 h42 197 4 L2 h43 246 7 L2 h44 315 8 L2 h45 404 7 L2 h51 256 6 L2 h52 286 2 L2 h53 335 6 L2 h54 404 7 L2 h55 493 5 L2 Cartesian Box Flux Boundary Conditions h00 0 L2 h01 9 87 L2 h02 39 48 L2 h03 88 83 L2 h04 157 9 L2 h10 9 87 L2 h11 19 74 L2 h12 49 35 L2 h13 98 7 L2 h14 167 8 L2 h20 39 48 L2 h21 49 35 L2 h22 78 96 L2 h23 128 3 L2 h24 197 4 L2 h30 88 83 L2 h31 98 7 L2 h32 128 3 L2 h33 177 7 L2 h34 246 7 L2 h40 157 9 L2 h41 167 8 L2 h42 197 4 L2 h43 246 7 L2 h44 315 8 L2 BENG 221 M Intaglietta Lecture 9 Time dependent solution of the heat diffusion equation Derivation of the diffusion equation The diffusion process is describe empirically from observations and measurements showing that the flux of the diffusing material Fx in the x direction is proportional to the negative gradient of the concentration C in the same direction or Fx D dC dx 1 where D the diffusion constant is a coefficient that may be constant or a function of time location and concentration With reference to Figure 1 the flux of material through the face of the element of volume at x minus the flux through the face at x dx equals the rate at which the concentration changes in the volume assuming that fluxes occur only in the x direction or Fx Fx Fx F C dx x x t x Figure 1 Flux balance along the x direction in a region of space described in Cartesian coordinates This can be readily extended to effects in all directions yielding C Fx Fy Fz 0 t x y z And applying 1 we obtain C C C C D D D t x x y y z z 2 which in the nomenclature of vector analysis is expressed by C div D grad C D 2C t the Laplacian operator for D constant Solution for constant diffusion coefficient from a plane source Straight forward differentiation shows that x2 1 4 Dt 2 C At e 3 is a solution of C 2C D 2 t x 4 A 3 2 x A 3 2 4xDt Ax 5 2 Ax 5 2 D t e t t e 4 Dt t 2 2D 2 4 Dt 4 D t 2 2 This solution for C is symmetrical relative to x 0 tends to 0 as x tends to infinity and is everywhere zero for t 0 except for x 0 where it is infinite This solution shows the concentration of the diffusing material originating from a plane source with an amount of material M at zero time The diffusing material is not consumed and the amount of material is constant at all times To evaluate A we assume that material is diffusing in an infinite cylinder from a plane located at x 0 Mass balance requires that for all times M Cdx 5 Changing variables and substituting in 3 and 5 x2 4 Dt 2 M At 1 2 e 2x 2 d dx 4 Dt 2 1 2 2 Dt d 2 AD x 1 2 Dt 2 1 2 x d dx 4 Dt 1 e d 2 A D 2 2 1 2 dx 2 Dt d and substituting A in 3 we obtain M x2 C exp 4 Dt 4 Dt 5 In this solution half of the material diffuses in the positive x direction and the other half in the negative x This solution is also valid for a semi infinite cylinder where diffusion takes place in the positive x direction only from a plane located at x 0 Clearly the concentration will be double of that of the infinite cylinder In this case we indicate that the solution is reflected at the boundary and superposed Note that the gradient of concentration at x 0 is zero in both cases indicating that in either case no material crosses the plane source or boundary Diffusion from a finite region consisting of a volume source The solution for the diffusion of material occupying a volume in space can be obtained by assuming that the region is composed of an infinite number of plane sources and superposing the infinite number of related solutions This problem describes effects taking place in an infinite cylinder filled with water where the concentration of a solute is C Co for x 0 C 0 for x 0 t 0 Consider in the geometry of Figure 2 a plane of unit surface area containing diffusible material in a quantity Cod i located at i according to 5 will produce a distribution of concentration at any time t given by Ci x t Co d i x i 2 exp 4 Dt 4 Dt 6 Figure 2 Material at a concentration Co occupies the region along the negative x axis Therefore the effect due to the infinite number of planes at any given time t is obtained by adding the effect of each plane solution from 0 to or C x t 0 Ci i 0 x 2 exp d 4 Dt 4 Dt Co 7 and making the substitution of variables x 4 Dt d 4 Dt d and differentiating Changing variables and limits of integration in 7 we obtain for 0 x 4 Dt and for Substituting in 7 we obtain C x t Co x 4 Dt exp 2 d 0 Co Co exp 2 d 2 exp d 0 Co erf 2 2 Co x 4 Dt Co x 4 Dt exp 2 d 0 x 4 Dt exp 2 d 0 Co 1 erf 2 x 4 Dt Note that erfx 2 x e d 2 0 Diffusion from and in confined regions The methods described allow to describe diffusion from a substance confined between h x h along the x axis Solution of this problem gives the concentration in terms of 1 h x h x C x t C0 erf erf 2 4 Dt 4 Dt This solution is symmetrical about x 0 therefore the system can be cut in half providing the solution for the semi infinite system One dimensional diffusion from a finite system into a finite system that extends up to x l can be analyzed by the method of reflection …