Hea (iffusion) Equaion erivaion of iffusion Equaion The fundamenal mass balance equaion is I P O L A ( 1 ) where: I inpus P producion O oupus L losses A accumulaion Assume ha no chemical is produced or los wihin he conrol volume and hence ΣP ΣL 0. So ( 1 ) simplifies o O I A ( ) onsidering a conrol volume cell where here is flu in he direcion onl, we have z where indicaes he mass flu densi (ML - T -1 ) in he direcion a he poin.
In order o saisf ( ), we mus have ( ) z z ( 3 ) Tha is, he flu ino he lef wall imes he area over which i occurs, minus he flu ou of he righ wall imes is area in an inerval of ime mus equal he change in chemical mass in he conrol volume. iffusion occurs in response o concenraion gradiens (e.g., / 0.001) in accordance wih Fick s law: ( 4 ) where is he areal flu (ML - T -1 ) and is he diffusion coefficien (L T -1 ). Incorporaing Fick s law ino equaion ( 3 ), we have z z ( 5 ) ividing boh sides b z gives ( 6 ) Obviousl, we are jus aking he gradiens of he gradiens, and if we shrink o differenial size (and assume ha is consan in space) we have ( 7 ) or equivalenl, ( 8 ) This is he 1- version of he widel applicable Hea Equaion.
Finie ifference Epression of Hea Equaion Mos parial differenial equaions can no be solved analicall. Numerical soluions ha reduce he problem o algebraic equaions are ofen necessar. These equaions can hen be solved b direc or ieraive mehods. The essence of numerical mehods for PEs lies in convering he differenials o finie differences. onsider he following concenraion values on a linear domain from o. / / Esimae here The gradien of in he direcion a / (i.e., h/ / ) would be linearl approimaed b / ( 9 ) This should be a reasonable approimaion when is small enough. Wha abou second order derivaives like /? Remember from alculus ha / can be wrien as [/]/. So, basicall we need a gradien of gradiens for he second order derivaive. We have alread esimaed he gradien / /. We could look he oher direcion and compue a gradien a - /: / ( ) ( 10 ) - / -/ / / - Esimae here Esimae here / Esimae here Then, aking he gradien beween / / and / / gives
/ / ( 11 ) We need a finie difference esimae for / oo. This is a lile more complicaed because now we have o include ime. The simples approach is o use he known concenraion from he previous ime sep. A he sar of an numerical soluion, he Iniial ondiion will be he ime 0 concenraion. So, we can approimae he derivaive / b / or,, ( 1 ) Puing i all ogeher The hea equaion is: ( 13 ), / -/ Esimae here - -, -
Le s replace he derivaive erms wih heir finie difference approimaions.,,,,, ( 14 ) Wrien his wa, nearl all of he concenraions (all of hose wih he subscrip) are known from he previous ime sep or from he iniial condiion., is he onl ecepion and i is wha we are ring o deermine. Le s rearrange o solve for, : ( ),,,,, ( 15 ) This is he full eplici finie difference approimaion for he Hea equaion. I is eplici because i works wih he known concenraions from he previous ime sep. While his is relaivel simple, i is onl sable when he facor / is less or equal o ½. This can be a serious limiaion. Sa we wan o simulae a diffusion coefficien of 10 - cm s -1 (gas in soil). Tha means / has o be less han 0.0 s cm -. If we waned o simulae he process wih one second ime seps, we d need space seps of 0.0 cm. Simulaing a m soil profile would require calculaions a 10,000 space nodes! There are oher approaches for he ime derivaive ha avoid his sabili resricion. Iniial and Boundar ondiions Iniial and boundar condiions mus be specified when differenial equaions like he hea equaion are o be solved. Iniial condiions simpl specif he concenraions hroughou he problem domain a ime zero. Boundar condiions specif eiher 1. The concenraions a he boundaries, or. The chemical flu a he boundaries (usuall zero) The fied concenraion boundar concep is simple. The chemical flu boundar is slighl more difficul. We go back o Fick s law ( 16 ) and noice ha if / 0, hen here is no flu. The finie difference epression we developed for / is / ( 17 )
Seing his o 0 is equivalen o ( 18 ) So, if is he las poin inside he problem domain, we could pu some ghos poins ouside he domain a. Then, if we make he concenraion a he ghos poins equal o he concenraion inside he domain, here will be no flu. Ofen he boundar condiions are consan in ime, bu he need no be. Analical Soluion A number of analical soluions for he hea equaion are available. Here we presen one for he following condiions: Iniial condiion: (,0) 0 Lef boundar condiion: (0,) 1 Righ boundar condiion: (,) 0 Noe ha he righ boundar is a infinie disance. The soluion is erfc 4 ( 19 ) Analical soluions can be used o check he resuls of finie difference compuer programs when he boundar condiions are equivalen. Summar Hopefull, ou have he idea ha we ve come full circle in our derivaion of he Hea equaion and is finie difference approimaion. We began wih a macroscopic consideraion of flues and chemical mass changes in ime and incorporaed concenraion gradiens via Fick s law. Then we showed how his led o a second order parial differenial equaion, he Hea equaion. Our developmen of he finie difference epression for he Hea equaion was esseniall he same process in reverse; we considered how o epress he derivaives in erms of he concenraion and evenuall derived a simple algebraic equaion ha can be used o solve he Hea equaion numericall.