Basic solution to Heat Diffusion In general, one-dimensional heat diffusion in a material is defined by the linear, parabolic PDE

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1 New Meio e yd 5 ydrology Program Quaniaie Meods in ydrology Basi soluion o ea iffusion In general one-dimensional ea diffusion in a maerial is defined by e linear paraboli PE or were we assume a is defined on e domain of ineres. Subsiuion your omework will sow a a A ep a / is a general soluion of is equaion. See p. - of Crank 975. Wa is e oal ea onen in e domain a any ime? e ea onen a a poin is gien by e produ of speifi ea [J kg - C - ] e maerial densiy [kg m -3 ] and e emperaure [ C] or. e oal ea onen depends on e size of e domain. If we assume a e domain is infinie is is easier an dealing wi a finie domain for reasons a will beome learer laer en e oal ea onen will be gien by e inegral 6 d d A ep ζ dζ A A 3 were we e assumed a speifi ea and e densiy are onsans and were we e used e ange of ariables ζ /. We an sole for A wi is a onsan sine ea is onsered in is model no soure or sink in e ea diffusion equaion. A Consequenly we an now wrie ou e general soluion as ep b Bu wa are ere boundary and iniial ondiions a orrespond o is soluion? Wa appens a? a is wa is e iniial ondiion for e soluion in applied oer an infinie domain? Alim { ep lim { ep δ / 5 were we e inrodued a new ariable and onep e ira dela funion δ. e ira dela funion is no e ypial funion you sudy in fresman alulus. I as srange properies. For eample i is no differeniable using e rules wi wi you are familiar. ow is i defined? One way of defining i is simply e seond limi in 5. We an augmen is wi e following addiional properies. δ for all 6a 6 One would inegrae oer finie or semi-infinie domains similarly by anging e limis of inegraion. -93-

2 New Meio e yd 5 ydrology Program Quaniaie Meods in ydrology δ 6b a is e dela funion inegraes o one and is zero eep a e loaion were e argumen is zero. I s a spike or pulse. δ as unis of [L - ] a is one oer e unis of e argumen 7. e dela funion is a ery useful represenaion of a poin narrow inpu pulse in is ase of ea. Sould e iniial emperaure no be zero oer an a e dela funion inpu superposiion an be used o represen a effe. For uniform iniial emperaure one would simply add o a iniial emperaure. δ Wa are e boundary ondiions for is problem? Basially you an see from e soluion a ± A lim ± ep ζ { lim { { } / e ζ ± 7 a is e boundaries are a ± and ey are omogeneous irile boundaries. Sould e irile boundaries be non-omogeneous en superposiion of soluions an be used o represen e sum of e dela inpu and non-zero irile boundaries. Oer eample appliaions Anoer eample appliaion of e dela funion would be a narrow pulse of solue mass in a fluid were e solue redisribues by diffusion. e PE is C C wi sae ariable onenraion C and solue diffusion oeffiien. en AM/ M is e oal solue mass e iniial ondiion is C M δ e irile boundaries are C± and e soluion is C M/ ep- /. A ird eample appliaion of e dela funion would be a narrow pulse of waer injeed rearged ino a erially inegraed preai aquifer were e waer redisribues by so-alled ydrauli diffusion following ary s Law. e PE is were e sae ariable is e rise of waer able eleaion and is e ydrauli diffusiiy /S y wi ransmissiiy and S y speifi yield. en A V/ S y V is e oal waer olume injeed e iniial ondiion is V/ S y δ e irile boundaries are ± and e soluion is V/ S y ep- /. 7 δ as unis of one oer ime wile e riple produ δδyδ for a ira dela funion in wo spae dimensions and one ime dimension as SI dimensions s - m

3 New Meio e yd 5 ydrology Program Quaniaie Meods in ydrology -95- Uni response Reurning o e ea problem. Suppose a uni [[J m - ] 8 en in represens e uni impulse response of e domain o a uni dela inpu ep 8a Uni responses for e solue pulse and groundwaer rearge problems would ake be e quaniy in { } s muliplied by one and S y - respeiely. Wa would a uni response look like for an inpu a anoer ime say or anoer loaion? ese are respeiely ep 8b and ep 8 Superposiion of ime arying ea inpu. In our ea problem suppose a e ere is a ime arying rae of ea inpu a denoed as &. ow does e sysem respond? Sine our PE is linear we an use superposiion of e response o is ime arying inpu. Firs assume a e iniial ondiion is if is no we an superpose is effe laer. en onoluion of e inpu and e uni response desribes e emperaure a any laer ime d & 9a You an ink of e onoluion as e sum of e emperaure responses a a loaion o a sequene of ea inpus a differen imes. o make is easier o see e onoluion inegral an also be wrien as d & 9b e onoluions of solue onenraion or groundwaer ead for ime arying inpus work in e same way. Superposiion of a spaially disribued ea inpus. In our ea problem suppose a e ere are wo ira dela ea inpus a one is a and e oer is a ow does e sysem respond? One again sine our PE is linear we an superpose is ime in spae ep ep 5 8 Wy per meer squared? is is a one-dimensional problem bu e world is ree-dimensional. us is e oal ea per ross-seional area per m aken normal o e -ais. e usual ea diffusion eample is ea ransfer in a onsan-diameer rod. e oordinae is disane along e rod and e area is e area of a ross-seion of e rod.

4 New Meio e yd 5 ydrology Program Quaniaie Meods in ydrology is superposiion onep an be eended o any number of spaially and ime arying inpus. Semi-infinie domain Suppose e injeion of ea were o our a e origin of a semi-infinie domain su a e ea as o diffuse only oward posiie alues of. en we an assume e soluion aboe for negaie is simply refleed abou e origin and superposed doubling e emperaure for posiie from a gie aboe see Crank 975 p. 3. en e general soluion beomes A ep 5 / or for ea diffusion ep 5 e boundary ondiion a remains a omogeneous irile ondiion wile due o e refleion a e origin e boundary ondiion a is a Neumann zero gradien ondiion a 53 As / a in e original soluion i is learly sill zero afer refleion and superposiion. Inegral of basi soluion: sep ange in iniial emperaure Suppose a e iniial ondiion were a sep funion of e form for < and for > 5 An eample would be bring wo meal bars of similar omposiion ea a a differen emperaure ino ona a eir ends. We an use superposiion of e infinie domain general soluion o represen is proess. Consider e eended disribuion o be omposed o an infinie number of line soures and superpose e orresponding infinie number of elemenary soluions Crank 975 p. 3-. e ea soure in some infiniesimal inremen d a loaion en as sreng d. en e emperaure a some poin a disane - away due o is inremen of ea is d ep 55 e omplee soluion due o e iniial ondiion in 5 is en deermined by summing oer e inremens d or -96-

5 New Meio e yd 5 ydrology Program Quaniaie Meods in ydrology ep d erf.5 erf / e ζ dζ 56 were ζ / and we e used e definiion of e error funion and is omplemen ζ erf e dζ erf z e d erf z z ζ ζ erf -z -erf z erf erf Wa boundary ondiions does is saisfy? From e properies of e error funion we find a as - and as 57 us 56 is a soluion of e ea diffusion equaion wi boundary and iniial ondiions a lead o is ommon appliaion in ydrology and e geosienes. Anoer eample appliaions of e sep ange. Anoer eample appliaion would be a sep ange in iniial solue onenraion were e solue redisribues by diffusion. e PE is C C wi sae ariable onenraion C and solue diffusion oeffiien. e iniial ondiion is C< C and C> e irile boundaries are C- C and C and e soluion is C.5C erf/. Influene of mean flow. Suppose is ea or solue diffusion is aking plae in e presene of a mean flow wi eloiy. e PE for ea adeion-diffusion beomes or 58 In an infinie domain wi boundary and iniial ondiions as desribed aboe for e pulse inpu or sep ange we an sole is equaion using a ransformed ersion of e soluions already produed. e basi idea is o ransform 58 so a i looks like e diffusion equaion. We use e araerisi of e adeion equaion i.e. e seondary of 58 and a ain rule operaion. If is onsan en e araerisi is or onsan suggesing a ange of ariables o a moing oordinae sysem a follows a araerisi or 59 en we apply ain rules o implemen is new oordinae sysem as follows -97-

6 New Meio e yd 5 ydrology Program Quaniaie Meods in ydrology -98-6ab en subsiue ino e adeion-dispersion equaion and simplify 6 o yield a simply diffusion equaion in e moing oordinae. We almos always drop e erial lines and subsrips and wrie is as 6 were e subsrips are undersood. We know general soluions for 6 for bo pulse and sep inpus. Afer ransforming ese bak o e saionary oordinae sysem ese soluions and eir orresponding iniial and boundary ondiions are: Pulse inpu a a ime : ep 6 IC: δ BCs: ± 65ab Sep ange a a ime : erf.5 66 IC: for < and for > 67a BCs: as - and as 67b

7 New Meio e yd 5 ydrology Program Quaniaie Meods in ydrology Noe a for e sep ange problem.5 for >. e sep smears oer ime and unlike e diffusion problem e onenraion a e origin anges. I is no a boundary ondiion. ransform Soluions o ea iffusion see Crank. I an be sown by dimensional analysis a soluions o 3 are ofen of e form A Φ 68 / were Φ is a funion o be deermined 9. In pariular we e already found Gaussian and error funion soluions. Noie a e funion is self-similar. a is e funion is sreed one way or anoer bu e sape remains unanged oer ime-spae. You e seen su similariy soluions before in e eis Well funion wi as similariy ariable ur S/ were e parameers ake on e normal well-ydraulis definiions and ydrauli diffusiiy is /S. I is no unusual o assume a soluion of e form of 68 and en o seek e oeffiien A a maes e boundary and iniial ondiions. 9 ypially is some ombinaion of eponenial or error funions. -99-