Aalborg Universitet. Melting of snow on a roof Nielsen, Anker; Claesson, Johan. Publication date: 2011

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1 Aalbog Univesie Meling of snow on a oof Nielsen, Anke; Claesson, Jan Publicaion ae: 211 Docuen Vesion Ealy vesion, also known as pe-pin Link o publicaion fo Aalbog Univesiy Ciaion fo publishe vesion (APA): Nielsen, A., & Claesson, J. (211). Meling of snow on a oof: Maheaical epo. Göebog: Chales ekniska högskola. Geneal ighs Copyigh an oal ighs fo he publicaions ae accessible in he public poal ae eaine by he auhos an/o ohe copyigh ownes an i is a coniion of accessing publicaions ha uses ecognise an abie by he legal equieens associae wih hese ighs.? Uses ay ownloa an pin one copy of any publicaion fo he public poal fo he pupose of pivae suy o eseach.? You ay no fuhe isibue he aeial o use i fo any pofi-aking aciviy o coecial gain? You ay feely isibue he URL ienifying he publicaion in he public poal? Take own policy If you believe ha his ocuen beaches copyigh please conac us a vbn@aub.aau.k poviing eails, an we will eove access o he wok ieiaely an invesigae you clai. Downloae fo vbn.aau.k on: augus 31, 218

2 MELTING OF SNOW ON A ROOF. MATHEMATICAL REPORT JOHAN CLAESSON, ANKER NIELSEN Depaen of Civil an Envionenal Engineeing Repo 211:3 Division of Builing Technology/Builing Physics CHALMERS UNIVERSITY OF TECHNOLOGY Göebog 211

3 MELTING OF SNOW ON A ROOF. MATHEMATICAL REPORT JOHAN CLAESSON, ANKER NIELSEN JOHAN CLAESSON, 211 REPORT NO. 211:3, ISSN Depaen of Civil an Envionenal Engineeing Division of Builing Technology/Builing Physics CHALMERS UNIVERSITY OF TECHNOLOGY SE Göebog Sween Telephone +46 ()

4 Conens 1. Inoucion Poble Meling of snow on a oof Hea flows an cieion fo snow eling Diffeenial equaion fo snow eph D( ) Soluion fo he invese elaion = ( D) Mele snow Feezing in he ovehang an ipping fo i Hea balance in ovehang. Dipping lii Dipping wae Feezing in ovehang Toal aouns of eling, feezing an ipping Oveview an suay Meling of snow on he oof Feezing an ipping a he ovehang A few exaples Winow on he oof Meling of snow on he oof an on he winow Cieia fo ipping Feezing in ovehang an ipping a he oue en Suay of foulas An exaple Concluing eaks Noenclaue

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6 1. Inoucion Snow on oofs gives any pacical pobles as exa loa on he oof, sliing of he snow, icing on he oof in gues, an geneaion of icicles. Sliing of snow an ice fo oofs can in wos case kill people an aage popey. A bee unesaning of he physics of snow an ice on oof can help in eucing he isk of aages. This eseach is suppoe by he eseach founaion a Länsfösäkinga, which is a Sweish banking an insuance alliance copany. A ypical wine poble is snow an ice on oofs. This inclues a nube of pobles ha ae elae o builing physics an heaing of he house. Slope oofs wih exenal gues can give pobles. An exaple is eling of he snow on he oof an he feezing of he wae on he ovehang. The esul is geneaion of an ice laye along he eaves. The ice laye can esul in ice as, so ha eling wae is collece behin. As he wae oes no feeze on he oof i will give a wae pessue on he lowe pa of he oof. This gives a isk fo wae leakage ino he builing if he oof is no waeigh. Anohe exaple is icing an geneaion of icicles on he oof eges. Icicles hanging fo he eaves ae a seious poble as hey can fall own an hi people walking beneah. The ipac of a falling icicle o ice fo ice as can in he wos case kill people. Such inciens have happene in Sween an Noway. Accoing o Sweish law, i is he owne of he builing who is esponsible fo pevenion of sliing of snow an ice fo he builing. The Sweish Associaion of Builings Ownes (Fasighebanchens Uviklingsfou) has ae a epo (Snö och is på ak 24) abou he pobles of snow an ice on oofs. I escibes soe law cases an exaples of conacs wih a fi o eove he ice an icicles, when hey fo in he wine. I is vey helpful fo he builing owne as a basis fo eucing he isk of snow an ice pobles bu i only skeches he physics behin he poble. A bee soluion is o peven o a leas euce he isk by a bee knowlege of snow eling, feezing an icicles geneaion on oofs. The poble wih icing an icicles on oof is a coplex poble involving achiecue, eeoology, glaciology an builing physics. We can ivie oofs in wo ypes: col (venilae) oofs an wa (non-venilae) oofs. In wa oofs, i is noal o have inenal ainage wih ownpipes in he builing. This soluion has no o vey lile isk fo icicles. Feezing of he eling wae on he oof can sill be a poble. Venilae oofs inouce a venilae gap o oof space o peven oisue pobles an o keep he suface of he oof col. These oofs ae in os cases slope. The ainage is exenal o gues along he eaves an o ownpipes. The esul is a high isk of ice foaion on he ovehang a he eave an icicles foaion if he eling wae feezes fo insance in he gue. In his epo we pesen calculaions fo venilae oofs wih a known insie epeaue. The insie epeaue can be efine in 2 cases: 1. The insie epeaue is he sae as he inoo epeaue. We use he inoo epeaue of he builing an he U-value fo he ineio of he builing o ousie oof suface. This is use if we have no venilae aispaces in he consucion o he venilaion wih ouoo ai is ahe low. 2. The insie epeaue is he sae as he aic epeaue. We use he aic epeaue of he builing an he U-value fo he aic o he ousie suface of he oof. This us be cases, whee he aic epeaue is influence by ai flows o hea souces. If we have hea souces as hea pipes, venilaion uc o venilaion syses, hen his will incease he epeaue in he aic an give a highe isk of icicles. If he consucion beween he builing an he aic is no aiigh hen we will have an ai flow fo he house o he aic ha will incease he aic epeaue. If he aic is venilae wih 5

7 ouoo ai, he aic epeaue an he isk of icicles will ecease. If he aic epeaue is aoun -2 o C o lowe in feezing peios, hee is no eling an no icicles. If we use he calculaion on exising builings, is i ipoan o ecie which case is os elevan fo he builing. As enione, he aic epeaue can in secon case be highe o lowe han in he fis case. 2 Poble Figue 1 shows he consiee oof. The insie epeaue below he oof is T (aoun 2 o C o any lowe aic epeaue) an he exeio o ouoo epeaue T e. The wih of he oof is L () fo oof-ige o ovehang. The U-value o heal conucance of he oof (beween T an he uppe, oue sie of he oof) is U (W/ 2,K). The hickness of he snow laye, D( ), eceases wih ie, if he snow els ue o sufficien heaing fo he inoo epeaue T. The wih of he ovehang is L an he U-value U (W/ 2,K). The heal conuciviy of he snow on he oof an ovehang is λ s (W/K,), an he ensiy of he snow is ρ s (kg/ 3 ). Changes ove ie of hese snow paaees ae neglece in his suy. Figue 1. Snow on a oof wih ovehang. The ask is o calculae of he eling of snow on he oof, an he ensuing ice an icicle foaion a he ovehang. The ouoo epeaue is below zeo an, in his analysis, consan. By assupion hee is no eling of snow fo above. The U-value of he snow on he oof, U, s is vaying wih he snow eph D( ). The iniial snow eph is D. The snow on he ovehang oes no el, which eans he U-value of he snow on he ovehang is equal o he iniial U-value of he snow, λ s / D. We have: 6

8 λs T >, Te <, Us ( ) =, D() = D. (2.1) D( ) The ai of his suy is o calculae of he eling of snow on he oof. The wae will flow o he ovehang an feeze o ice une he snow on he ovehang. Pa of he wae ay ip o ooze fo he lowe en of he ovehang o fo ice an icicles hee o leave he ovehang as wae ops. All ele snow ens up as ice again. Ou ai is o quanify as funcion of ie he ele snow, he foaion of ice une he snow on he ovehang an he aoun of ipping wae, which gives an uppe lii fo he ice foaion a he oue en of he ovehang. 3 Meling of snow on a oof The snow on he oof will el fo below if he heaing fo T is lage han he cooling o T. Le g (kg/s,) enoe he ae of snow eling on he oof (pe ee oof wih), e an (kg/) he accuulae aoun. The ele wae fo he oof enes he ovehang, whee i will feeze again ue o he col ouoo epeaue ha suouns he ovehang. Soe of he wae ay ip fo he ovehang an fo ice an icicles a he oue en of he ovehang. Le g ( ) (kg/s,) enoe he ae of ipping a he oue en of he ovehang, an (kg/) he accuulae aoun. We have: [ ] ( ) = ρ L D D( ) = g ( ), ( ) = g ( ). s The ie eivaive of ( ) becoes (3.1) D = g ( ) = ρsl. (3.2) 3.1 Hea flows an cieion fo snow eling The epeaue a he oof below he snow laye is equal o ( U T + U ( ) T ) / ( U + U ( ) ) s e s povie ha his epeaue lies below zeo. Thee will be eling when he value is posiive. We suy he case when his epeaue is posiive a he sa = wih he snow hickness D() = D : ( T ) λ λ U T + U T = U T + T > D > (3.3) s s e s () e e o. D U T The snow hickness lii D, above which eling occus, becoes: ( T ) λ D =, D = D() > D. (3.4) s e U T 7

9 The epeaue is zeo a he oof ajacen o he snow laye, when snow is eling. Le q (W/) enoe he hea flux hough he oof, an q e he hea flux hough he snow laye (when he epeaue is zeo a he bounay beween oof an snow): The eling lii ( ) Lλs Te D q = LU ( T ), qe ( ) = = q. (3.5) D( ) D( ) D an he ne hea flux o el snow ae: Lλs Te D D =, q qe ( ) = q 1. q D( ) (3.6) The snow els as long as his hea flux is posiive: D q qe( ) > 1 >, D( ) > D. (3.7) D( ) The lii fo snow eling, D, us lie below he iniial snow eph D, if eling is o occu. The cieia fo snow eling ae hen: q > q () D < D, D < D( ) D. (3.8). e 3.2 Diffeenial equaion fo snow eph D( ) The eling hea flux is equal o he ae of snow eling uliplie by he laen hea of eling fo snow h (334 kj/kg): D q qe ( ) = q 1 = h g ( ), D < D( ) D. D( ) (3.9) Cobining (3.2) an (3.9), we ge he iffeenial equaion fo he snow hickness D( ) : D h ρ L D = < = (3.1) s 1, D D( ) D D(). D( ) q o, inoucing a ie, (3.12): D D = 1, D() = D > D, <. (3.11) D D Hee, is he ie equie o el he snow laye wih he iniial hickness D fo e T =, i.e. fo zeo hea flux hough he snow: 8

10 h ρ LD h ρ D = =, U T = h ρ D. (3.12) s s s q U T 3.3 Soluion fo he invese elaion = ( D) The above iffeenial equaion (3.11) ay be solve by consieing he invese elaion = ( D). The equaion ay be wien: D D D D D D D D D = = 1 +, D < D D (3.13) The equaion is inegae fo any D, D < D < D, o D : D ( D ) ( D) = D + D ln ( D D ) (3.14) D Using ( D ) =, we ge he basic foula: D D D D D ( D) = 1 + ln, D < D D D D D D (3.15) The ie inceases o infiniy when D ens o he lowe lii D, whee he eling sops. The snow hickness D( ) is obaine by a nueical invesion of (3.15) fo any consiee ie. Foula (3.15) ay be wien in a iensionless fo using iensionless ie τ, snow eph, an eling lii : D D τ = = = (3.16) D D,,. The iensionless fo of elaion (3.15) beween ie an snow eph becoes: 1 τ = f (, ) = 1 + ln, < 1. (3.17) This funcion The funcion f (, ) is shown in Figue 2. f (, ) eceases fo infiniy o zeo in he ineval < 1: f ( +, ) =, f (1, ) = ; [ f (, ) ] =. (3.18) In he lii T e =, D an ae zeo, an he snow laye eceases linealy wih : 9

11 f D ( 1 / ) D < (,) = 1, < 1; =. (3.19) 2.5 f (,.9) 2.25 f (,.8) f (,.7) 2 f 1.75 (,.6) f (,.5) 1.5 f (,.4) 1.25 f (,.3) f (,.2) 1 f.75 (,.1) f (, ) Figue 2. The funcion τ = f (, ), τ = /, = D / D, = D / D, which gives = ( D) fo = (he lowes saigh line),.1,.2,....9 (he ighos cuve). Equaion (3.17) efines he invese elaion, i.e. he elaive snow hickness = D / D as funcion of τ = / wih = D / D as paaee: = f ( τ, ). This funcion is shown in Figue 3. The se of cuves is he sae as in Figue 2, bu he axes ae inechange. Fo any consiee τ an, we have o calculae he oo o he equaion f (, ) τ = o eeine : τ = f (, ) = f ( τ, ); τ = f ( f ( τ, ), ). (3.2) The oo ay be soewha ifficul o eeine nueically fo close o. The following appoxiaion ay hen be use: 1

12 ( 1 τ )/ f ( τ, ) = + (1 ) e, τ > 3, < < 1. (3.21) This elaion is obaine fo (3.17) by puing eo is salle han. 6 fo τ > 3. = in he secon igh-han e. The f ( τ,.9).9 f ( τ,.8).8 f ( τ,.7) f.7 ( τ,.6) f ( τ,.5).6 f ( τ,.4).5 f ( τ,.3) 1.4 f ( τ,.2) f.3 ( τ,.1) f ( τ, ) τ Figue 3. The funcion = f ( τ, ), he snow hickness = D / D as funcion of τ = / wih = D / D as paaee; = (he lowes saigh line),.1,.2,....9 (op cuve). We will nee he eivaive of = f ( τ, ) wih espec o τ. We have fo (3.18): 1 [ f τ ] τ τ (, ) = = = 1. (, ) [ f (, ) f τ ] (3.22) 3.4 Mele snow ( ) The accuulae aoun of ele snow a ie is fo (3.1): ( ) = ρ L D D( ) = 1 ( ), = ρ LD. (3.23) s s 11

13 Hee, (kg/) he iniial aoun of snow on he oof. The oal aoun of ele snow M (kg/) is obaine fo vey lage : M = ρ L D D = 1. (3.24) s Equaion (3.23) ay be wien in iensionless fo: ( ) = 1 ( ) = ( τ, ), ( τ, ) = 1 f ( τ, ). (3.25) The iensionless snow eph ( ) = D( ) / D is shown in Figue 3. We ge iecly he accuulae aoun of ele snow by changing o 1 ( ) = ( τ, ) on he veical axis. See Figue 4. ' ( τ, ) ' ( τ,.1) ' ( τ,.2) '.7 ( τ,.3) ' ( τ,.4).6 ' ( τ,.5).5 ' ( τ,.6).4 ' ( τ,.7) '.3 ( τ,.8) ' ( τ,.9) Figue 4. Accuulae aoun of ele snow, ( ) / = ( τ, ), as funcion of τ wih as paaee; = (uppe saigh line),.1,.2,....9 (boo cuve). τ Cobining (3.23) an (3.11), we have: 12

14 1 D D D = = 1 ρsl D( ) (3.26) Inegaion ove gives D D D D = o, inseing = ( D) fo (3.15):, (3.27) D( ) D D D D = ln. D( ) D D D (3.28) This elaion will be use below. 4 Feezing in he ovehang an ipping fo i The ele snow g will flow own he slope oof ino he ovehang, whee he suouning epeaue T e is below zeo. See Figue 1. All wae will feeze below he snow in he ovehang as long as he wae influx is sall. Thee is an uppe lii above which pa of he wae feezes an he es g ips fo he ovehang. This lae pa, he ipping flow, ay leave he ovehang as wae ops, o ceae icicles an ice a he lowe en of he ovehang. 4.1 Hea balance in ovehang. Dipping lii In he case of ipping, he hea balance fo he ovehang ( g ( ) > ) is: ( g g ) h q q K ( T ) =, =. (4.1) e Hee, q (W/) is he hea flux fo he ice/wae laye of zeo epeaue une he snow on he ovehang hough he snow upwas an hough he ovehang oof ownwas. The faco K (W/(K)) is he heal conucance beween he ice/wae laye in he ovehang an he suouning ai wih he epeaue T e. This hea flux gives he feezing capaciy of he ovehang. Assuing ha he hickness of snow on he ovehang is equal o he iniial value D wih he U-value λs / D, we have: λ K = L + L U. (4.2) s D Fo (3.2) an (3.9) we have: 13

15 h g( ) = h = q qe( ) (4.3) This expession is insee in (4.1) an we ge h g( ) = h = q qe ( ) q. (4.4) Dipping occus when his expession is posiive, an he expession becoes zeo a he ipping lii = : q q ( ) q > ; q q ( ) q =. (4.5) e e Below, we will analyze hese coniions fo feezing in he ovehang an ipping fo he ovehang in wo ways. In he fis analysis, we use (3.5), igh: q q q q q q D e =. q D( ) (4.6) The above hea flux is neve posiive fo q becoe: q. The coniions a he ipping lii q q q D > q an =. (4.7) q D( ) The snow hickness D( ) = D a he ipping lii is now: q D D q q =, > ; = 1. q q q D q D (4.8) We noe ha he ipping lii is lage han he eling lii: D > D. Thee ae now hee cases o consie: no eling, eling, an eling an ipping. The possibiliies ae illusae in Figue 5. The snow hickness D = D( ) ivie by he eling lii D is given by he hoizonal axis, an he veical axis gives he hea flux aio q / q. In he eling egion, snow els on he oof an feezes again o ice in he ovehang. The cuve fo he ipping lii is fo (4.8) given by: q D D/ D 1 = 1 = = f,li ( D/ D ), f,li (1) =, f,li ( ) = 1. (4.9) q D D / D This cuve inceases fo zeo o one as We have now hee possibiliies: D / D inceases fo one o infiniy. 1. No eling: D / D 1 14

16 2. Meling wihou ipping fo D / D > 1 an q / q > 1, an fo 1 < D / D < D / D an q / q < Meling an ipping: D / D < D / D < D / D an q / q < 1. The iniial snow eph is D() = D. Thee is no eling if D. < D The lines A, B an C show wha happens fo a ceain D > D. A an B: eling fo D o D. C: eling an ipping fo D, < D < D an eling only wih ice accuulaion a he ovehang fo D < D < D. Figue 5. Regions of no eling, eling, eling an ipping. In he secon analysis of feezing in he ovehang an ipping fo ovehang, we use he iniial hea flux hough he snow q = q () an ewie e e q e in he following way, (3.5): ( T ) e s e e e e ( ) D q Lλ q ( ) =, q = q () = = q. (4.1) The ipping cieia (4.5) ae hen: q q q q > ; q q =, = ( ). (4.11) e e ( ) The coniion fo ipping a he iniial ie is: q q q = o 1 = q + q. (4.12) e e 1 q q Figue 6 shows a cooinae syse wih he axes qe / q. an qe / q. Each poin epesens a se of hea fluxes q, qe an q. In he iangula egion below he line (4.12), igh, hee 15

17 is eling an ipping, an in he egion above he iangle hee is eling wihou ipping. To he igh of qe / q = = 1 no eling akes place. Figue 6. Meling an ipping epening of he hea fluxes q, qe an q. Figue 7. Dipping lii, (4.14), as lines in a plane wih he axis qe/ q = an q/ q. The ipping lii (4.11), igh, ay be wien in he following way: q q q = = (4.13) q q q e 1 o 1. 16

18 The elaion beween = qe / q an q / q is, fo any consan, a saigh line. I goes hough he poin 1 (,1) poin P ( ) : P = an has he slope 1/. The line cus he hoizonal axis in he =,. See Figue 7, lef. All poins along he line have he sae ipping lii q = =, < < < 1. e 1 q/ q q q (4.14) I is seen fo Figue 7, lef, ha he lines fall insie he iangle,( 1, ),(,) P fo 1 Fo ousie his ineval, he line lies wholly ousie he iangle. Then hee is no ipping. Figue 7, igh, shows he ipping lii as saigh lines hough P 1 = (,1) fo P o =.2,.4,.6,.8, an 1. Fo = 1 along he line fo 1 coincies wih he iniial snow eph: D = D. < < 1. 1,, he ipping lii 4.2 Dipping wae Fo (4.4)-(4.7), an fo (3.12) an (3.23), igh, we have: q D D q ρsld =, = =. h D D( ) h (4.15) We see ha ipping occus fo D D D. Inegaion of (4.15) wih () = gives D D ( ) = D (4.16) D( ) Using (3.28) we ge: D D D D ( ) = ln, D < D D D. D D D( ) D (4.17) We ay eliinae, (3.15), o ge as funcion of D = D( ) : D D( ) D D D D = 1 1 ln D D D D D( ) D (4.18) The ipping ay be expesse in iensionless fo. We use iensionless quaniies fo snow eph: 17

19 D( ) D D = = = (4.19),,. D D D Then we have: ( ) = ( τ,, ), ( ) = f ( τ, ), < f ( τ, ) 1. (4.2) 1 ( τ,, ) = 1 f ( τ, ) ( ) ln, τ τ. f ( τ, ) (4.21) The ipping sops when ( ) = f ( τ, ) =. The coesponing ie τ = / is fo (3.16)-(3.17): τ 1 = f (, ) = 1 + ln. (4.22) The incease of he accuulae ele snow sops a his ie: ( ) = ( ), ; ( τ,, ) = ( τ,, ), τ τ. (4.23) The axiu value ( τ,, ) = M (, ), (4.31), is iscusse fuhe in Secion 3.4. (,,.55 ) (,,.6) (,,.7) (,,.8) (,,.9) ' τ,.5 ' τ,,.51.4 ' τ ' τ ' τ ' τ ' τ.3.2 M' Figue 8. Aoun of eling ( τ,.5) an ipping ( τ,.5, ) fo =.5 fo a few. The os show he poins ( τ, M ). τ 18

20 Figue 8 shows a few cuves ( τ,, ) fo =.5. The op cuve shows he ele snow ( τ,.5), which inceases fo zeo o 1 =.5. The ohe cuves show ( τ,.5, ) fo (4.21) fo =.51 (op cuve),.55,.6,.7,.8,.9 (boo cuve). The os show he oal ipping M, (4.31), ha occus a he ie τ, (4.22). The cuves fo ipping ae hoizonal afe ha ie. 4.3 Feezing in ovehang The iffeence beween ele snow an ipping wae is accuulae as ice in he ovehang: ( ) = ( ) ( ). (4.24) We have fo (4.3) an (4.4): q h = q q ( ) q q ( ) q =, (4.25) e e h The accuulae ice a he ovehang inceases linealy as long as wae is ipping: q q D ( ) = = = 1,. h q D (4.26) Hee, (4.15), igh, an(4.8), igh, is use. This linea incease eans ha he full hea flux q is use o feeze eling snow. Afe he ie when ipping has soppe only a facion of his hea flux is neee o he feeze he ele wae in an uppe pa of he ovehang. The epeaue une he snow in he oue pa of he ovehang will fall below zeo. In iensionless fo (4.26) becoes: = = ( τ ) 1,. τ τ τ (4.27) Figue 9 shows as an exaple he case =.4 an =.6. The op cuve shows he ele snow, (3.25), which inceases fo zeo o 1.4 =.6. The ile cuve shows he ice in he ovehang, (4.24), an he boo cuve he ipping fo he ovehang, (4.21). The ipping inceases o he axiu M =.12 given by (4.3)-(4.31). The axiu is aaine a τ =.84 given by (4.22). 19

21 .6 (,, ) (,, ) ' τ, ' τ ' τ P Figue 9. Aoun of eling, ( τ, ), feezing in he ovehang, ( τ ), an ipping, ( τ ), fo he case =.4 an =.6. Dos (P): τ = τ an = M, = M. τ 4.4 Toal aouns of eling, feezing an ipping The oal aoun of ele snow (kg/) is fo (3.24): M = M, M = 1. (4.28) In he case wihou ipping, we have M =, M = M fo q > q, an fo D < D, q < q. (4.29) The oal aoun of ipping wae is given by: M = ( ) = M (, ), < < < 1. (4.3) The funcion M (, ), which gives he iensionless oal aoun of ipping wae, is obaine fo (4.21) fo τ = τ an f ( τ, ) = : 1 M (, ) = 1 ( ) ln, < < < 1. (4.31) The oal aoun of ice in he ovehang is: M = M M = M (, ),. (4.32) 2

22 1 M (, ) = 1 M (, ) = ln. (4.33) The funcions M (, ) an M (, ) ae efine in a iangula egion, an we have: M (, ) + M (, ) = M = 1, < < < 1. (4.34) Figues 1 an 11 show hese wo funcions fo =.1 (lefos cuve),.2,,.9,.95 (ighos cuve). The ashe line shows he lii 1. On he bounaies of he iangula egion we have in accoance wih he su (4.34): M (, ) =, M (, ) = 1, < < 1; M (,1) =, M (,1) = 1, < < 1; M (, ) = 1, M (, ) =, < < 1. (4.35) 1 M',.1 M',.2 M',.3 M',.4 M',.5 M',.6 M',.7 M',.8 M',.9 M', Figue 1. The funcion M (, ) fo he oal aoun of ipping fo he ovehang. 21

23 1 M',.1 M',.2 M',.3 M',.4 M',.5 M',.6 M',.7 M',.8 M',.9 M', Figue 11. The funcion M (, ) fo he oal aoun of ice in he ovehang. 5 Oveview an suay The above analysis an he os ipoan foulas ae suaize in his secion. 5.1 Meling of snow on he oof The piay paaees ae shown in Figue 1. The hea flux fo he insie of he oof, q, he iniial snow eph, D, he iniial aoun of snow on he oof, (kg/), an he ie o fully el he iniial snow laye wih he hea flux q ae: h q = LU T ( T > ), D() = D, = ρsld, =. (5.1) q Thee is a ceain eling lii above which he hea flux fo he insie is lage han he hea flux hough he snow. The snow on he oof els when he snow eph lies above he eling lii D : 22

24 ( T ) Lλ D > D = T < (5.2) s e, e. q Thee is no eling if he iniial snow eph lies below he eling lii. The snow eph D( ) eceases wih ie as he snow on he oof els. We have eive an explici foula fo he ie as a funcion of he snow eph, (3.15): D D D D ( D) = 1 + ln, D < D D. D D D D (5.3) The ie inceases fo zeo o infiniy as he snow hickness eceases fo he iniial value D o he eling lii D (fo D < D ). This elaion an ohe elaions below ay be foulae wih a few iensionless vaiables. We will use iensionless ie τ, snow eph, eling lii, an ipping lii :,,,. D D D τ = = = = (5.4) D D D Eq. (5.3) becoes in iensionless fo 1 τ = f (, ) = 1 + ln, < 1. (5.5) This elaion is shown in Figue 2. The invese elaion = f ( τ, ) is eaily ploe by inechanging he axes, Figue 3. In he copue pogas, i is obaine by a nueical soluion of (5.5) o ge = f ( τ, ) fo any τ an. Equaion (3.21) is use fo τ > 3. The ele snow,, is iecly obaine fo he snow eph D ( ) : D( ) ( ) = 1, D < D D, D (5.6) o, using iensionless vaiables, ( ) = ( τ, ), ( τ, ) = 1 f ( τ, ), τ <, < 1. (5.7) The iensionless elaion ( ) / = ( τ, ) fo he ele snow is shown in Figue 4. The oal aoun of ele snow is obaine fo D = D : M = M, M = 1. (5.8) 23

25 5.2 Feezing an ipping a he ovehang The wae fo he ele snow feezes again below he snow on he ovehang, which is expose o he col ouoo epeaue T e <. All ele wae feezes if q is salle han he hea flux q fo he feezing wae in he ovehang. Fo q, > q soe of he ele wae ay ip fo he ovehang o fo ice an icicles when he snow eph lies above he ipping lii D. We have fo (3.6), igh, (4.8) an (4.1): ( ) Lλ T q D =, D = D q > q, q = K T. s e e q q q (5.9) The heal conucance of he ovehang K is given by (4.2). Dipping wih he ensuing ice an icicle foaion a he oue en of he ovehang will occu if wo coniions ae fulfille: q > q an D. < D Ohewise hee is no ipping. We consie in his secion he case when ipping occus. Then we have D < D < D, < < 1. (5.1) Pa of he ele snow, ( ), feezes in he ovehang an he es,, ips o en up as ice an icicles: ( ) = ( ) + ( ). (5.11) The ipping sops a he ie when he snow hickness is equal o he ipping lii D( ) = D : D( ) = D = f (, ), < 1. (5.12) The accuulae aoun of ipping is fo (4.19)-(4.21) D D( ) D D D D ( ) = 1 ln,. D D D D( ) D (5.13) The oal aoun of ipping becoes M = ( ) = M (, ), < < < 1. (5.14) The iensionless funcion M (, ) fo he oal ipping becoes: 1 M (, ) = 1 ( ) ln. (5.15) The accuulae ipping oes no change afe he ie : 24

26 ( ) = ( ) = M, <. (5.16) The accuulae ice in he ovehang is obaine fo (5.11), (5.13) an (5.7). I inceases linealy uing he peio of ipping: q ( ) = ( ) ( ); ( ) = =,. (5.17) h The oal aoun of ice in he ovehang is: M = M M = M (, ). (5.18) The iensionless funcion M (, ) fo he oal aoun of ice une he snow on he ovehang becoes: M (, ) = 1 M (, ). (5.19) The funcions M (, ) an M (, ) ae shown in Figues 1 an A few exaples Le us consie a few exaples. We use he following inpu aa: T = 2 C, T = 1 C, L = 8, L =.4, D =.2, o o e λ =. W/ (,K), ρ = 2 kg/, U = 2. W/ (, K). 3 2 s s (5.2) In he fis exaple we consie a oof wih poo heal insulaion o lage U-value: U 2 = 1. W/ (, K) D =.3, = 32 kg/, q = 16 W/, K =.92 W/ (,K), q = 9.2 W/ (, K), = 7.7 ays, D =.32, = 11.8 ays. (5.21) Figue 12 shows he eceasing snow eph fo.2 o he eling lii D =.3. Figue 13 illusaes he eling an ipping in he consiee exaple. We ge fo ou foulas: M = 272 kg/, M = 31 kg/, M = 241 kg/, = 11.8 ays ( ) = 3 kg/. (5.22) Thee ae in Figue 13 wo hoizonal lines, hee cuves an wo poins (a cicle an a squae) in he figue. (The five on he hoizonal axis ae a he ie in ays fo he op five funcions on he veical axis, while he wo give he ipping lii in hous fo he wo poins.) The op hoizonal line shows he oal aoun of eling snow M =272 kg/. The fis cuve (op) shows he accuulae aoun of ele snow () afe ays. The secon cuve fo op gives he accuulae aoun of ipping wae, (), an he lowes cuve he accuulae aoun of ice in he ovehang, (). The cuve fo ipping wae inceases 25

27 up o he ipping ie lii =11.8 ays. Afe ha, he value is he consan an equal o M =241 kg/. The lowe hoizonal line shows he oal aoun of ice in he ovehang, M =31 kg/. The secon poin (a squae) shows he accuulae aoun of ice in he ovehang a he ie when ipping sops, ( )=3 kg/. The cuve () is a saigh line unil he ie..2.2 D (.ay ) D Figue 12. Snow eph as funcion of ie (in ays) own o he eling lii D =.3 3 M ay ay M ay M ( ) ,,,,,, Figue 13. Accuulae eling of snow, ice in ovehang an ipping as funcions of ie (in ays). 26

28 In he secon exaple we consie a oof wih fai heal insulaion o ineeiae U- value: U 2 =.3 W/ (, K) D =.1, = 32 kg/, q = 48 W/, K =.92 W/ (, K), q = 9.2 W/ (, K), = 26 ays, D =.124, = 28 ays. (5.23) Figue 14 shows he eceasing snow eph fo.2 o he eling lii D =.1..2 D( ay ) D Figue 14. Snow eph as funcion of ie (in ays) own o he eling lii D =.1 Figue 15 illusaes he eling an ipping in his exaple. We ge fo ou foulas: M = 16 kg/, M = 15 kg/, M = 55 kg/, = 28 ays ( ) = 68 kg/. (5.24) Thee ae in Figue 15 wo hoizonal lines, hee cuves an wo poins (a cicle an a squae) in he figue. The op hoizonal line shows he oal aoun of eling snow M =16 kg/. The fis cuve (op) shows he accuulae aoun of ele snow () afe ays. The ohe wo cuves give he accuulae aoun of ipping wae, (), an he accuulae aoun of ice in he ovehang, (). The cuve fo ipping wae inceases up o he ipping ie lii =28 ays. Afe ha, he value is he consan an equal o M =55 kg/. The lowe hoizonal line shows he oal aoun of ice in he ovehang, M =15 kg/. The secon poin (a squae) shows he accuulae aoun of ice in he ovehang a he ie when ipping sops, ( )=68 kg/. The cuve () is a saigh line unil he ie. 27

29 175 M ay M ay ay M ( ) ,,,,,, Figue 15. Accuulae eling of snow, ice in ovehang an ipping as funcions of ie (in ays). In he hi exaple we consie a oen oof wih goo heal insulaion o low U- value: U = = (5.25) 2.15 W/ (, K) D.2 This eans ha he eling lii an he snow eph ae equal. Thee is no eling. 6 Winow on he oof Thee ay be a winow on he oof. This is an ineesing coplicaion ha is suie in his secion. The lengh of he winow is L, w an he lengh of he eaining oof below an above he winow is L an L, L especively. The oal oof lengh fo ige o ovehang is L + L, an he oof lengh excluing he winow w L. See Figue 16. The noaions of Figue 1 fo a oof wihou a winow ae use. The hickness of he snow on he oof is D ( ) an on he winow D. w The U-value o heal conucance of he winow (beween T an he uppe, oue sie of he winow) is U w (W/ 2,K). We assue ha U > U, so ha he eling of snow is fase on he winow: w Dw ( ) < D ( ). The snow eling on he oof above an below he winow is ienical, an he posiion of winow efine by L oes no ae. The analyses an foulas o no epen on L. The winow occupies a ceain wih of he oof. Ousie he winow aea (pepenicula o he coss-secion of Figue 12) he pevious analyses fo a oof is vali. Hee we consie a uni wih of oof an winow. 28

30 Figue 16. Meling of snow on a oof wih a wiow. 6.1 Meling of snow on he oof an on he winow The accuulae aouns of ele snow on oof an winow ae iecly obaine fo he snow eph. We have as in (3.23): ρ ( ) = L ρ D D ( ), ( ) = L D D ( ). (6.1) s w w s w The iniial snow eph on oof an winow is D. We have fo iensionless snow eph: D ( ) D ( ) ( ) =, ( ) = ; () = 1, () = 1. (6.2) w w w D D ρ ( ) = L ρ D 1 ( ), ( ) = L D 1 ( ). (6.3) s w w s w The oal aoun of ele snow becoes: [ ] ( ) = ( ) + ( ) = 1 L ( ) L ( ). (6.4) w w w Hee, (kg/) is oal iniial ass of snow on oof an winow, an L w he elaive lengh of he winow: L L = L + L D, L =, L =. w ρ w s w L + Lw L + Lw (6.5) 29

31 The su of L an L w is 1, so L is iecly obaine fo L : w L + L = 1, L = 1 L. (6.6) w w The elaive snow eph fo oof an winow is obaine fo he foulas in Secion 2.3 applie fo he aa of oof an winow. We have fo (3.2), (3.17), (3.16), (3.12) an (3.4) ( ) = f ( /, ) ( ) = f ( /, ). (6.7) w w w h ρ D λ ( T ) h ρ D λ ( T ) =, =, =, =. (6.8) s s e s s e w w UT UT D U wt U wt D The explici foula fo he accuulae aoun of ele snow is now: [ ] ( ) = 1 L f ( /, ) L f ( /, ). (6.9) w w w Using he iensionless ie τ = /, we have: ( ) = ( / ), ( τ ) = 1 L f ( τ, ) L f ( τ /, ). (6.1) w w w Hee, we use he elaions: U, = = = = =. (6.11) U w w w w w w w w The iensionless aoun of ele wae, ( ) / = ( τ ), becoes a funcion of he iensionless ie τ wih hee paaees:, w an L w. The oal aoun of ele snow M (kg/) is obaine fo vey lage : M = M, M,, L = 1 1 L L. (6.12) w w w w w 6.2 Cieia fo ipping The hea flux fo he ineio inus he hea flux ove he snow laye o he exeio els he snow on oof an winow. The hea balance fo eling of snow on he oof is fo(3.5), (3.9) an (3.2) : L λ ( T ) h = LU T, q = LU T. (6.13) s e D ( ) Hee, q is he hea flux hough he oof fo he ineio. The coesponing elaions fo he winow becoe: 3

32 L λ ( T ) h = L U T, q = L U T. (6.14) w w s e w w w w w Dw ( ) We inouce special noaions fo he hea fluxes a he iniial ie = hough he snow on he oof an he winow, an hei su: L λ ( T ) L λ ( T ) q =, q =, q = q + q. (6.15) s e w s e e ew e e ew D D We ge he elaions: L + L λ ( T ) q q q =, = L, = L. (6.16) w s e e ew e w D qe qe We noe he fuhe elaions: q = q + q, q + q = h. (6.17) e w w w w The hea balances fo eling of snow on oof an winow ay now be wien: q e qe h = q = q 1, =. ( ) ( ) q (6.18) w q ew w qew h = qw = qw 1, w =. w ( ) w ( ) qw (6.19) The oal hea balance fo snow eling is now: o q q h = h + h = q + q, (6.2) w e ew w w L L w h = q + qw qe +. (6.21) ( ) w ( ) Dipping occus when he ne hea flux o el snow excees he cooling hea flux fo he ovehang: h q q + q q L + L q w w e ( ) w ( ). (6.22) The faco afe q e is equal o 1 fo =. The cieion fo ipping a he iniial ie =, an he cieion fo no ipping ae hen Dipping a = : q + q q > q ; No ipping : q + q < q. (6.23) w e w e 31

33 All ele wae fo oof an winow feezes on he ovehang wihou ipping in he case of no ipping. We inouce a iensionless ipping paaee: q q = q + q q w e. (6.24) The iensionless hea loss fo he ovehang as efine above is salle han one (an lage han zeo) when ipping occus: q + q q > q < q < 1. (6.25) w e The ipping sops a he ie : q + q q L L = h = q = + (6.26) w w : o. = qe ( ) w ( ) The lef-han aio of hea fluxes ay be wien in he following way: q q + q q ( q + q q )( 1 q ) w w e = = 1 +. (6.27) qe qe We have fo (6.18), (6.19) an (6.16): q q L q q L q q q q e w w w = =, = =. (6.28) e e e w e w So we have: L L w q = q + + ( 1 q ), L = 1 L w. w (6.29) Le τ enoe he iensionless ie when ipping sops: / = τ, / = τ /. (6.3) w w The equaion o eeine τ is now fo (6.26), igh: L L w q = + τ = τ ( q,, w, L w ). f ( τ, ) f ( τ / w, w ) h( τ ) (6.31) Hee, q is given by (6.29). The ie when ipping sops was given explicily by (4.22) in he case wihou winow. Hee, we have o solve he above equaion. The iensionless 32

34 ipping ie τ epens on he fou paaees q,, w an L w. Dipping occus in he ineval < q < 1, while τ ay vay fo zeo o infiniy. A he ineval ens we have: L L w q = 1: q = 1, h() = + = 1 h() = q ; 1 1 L L L L q = : q = +, h( ) = + h( ) = q. w w w w (6.32) This eans ha τ ( q,, w, L w ) vaies fo zeo o infiniy when q goes fo 1 o : τ (1,,, L ) =, τ (,,, L ) =. (6.33) w w w w 6.3 Feezing in ovehang an ipping a he oue en The accuulae ele snow is equal o he feezing in he ovehang an he ipping: ( ) = ( ) + ( ). (6.34) The coesponing iensionless quaniies ae enoe by pie: ( ) = ( τ ), ( ) = ( τ ), ( ) = ( τ ). (6.35) Duing he ipping peio, hee is a consan feezing of ele wae in he ovehang eeine by he ovehang hea loss q : 1 : h = h = q τ q q ( ) =, = q τ, q =. h τ h (6.36) The accuulae aoun of ice in he ovehang inceases linealy in ie. The slope fo he iensionless incease becoes: q ρ q q q + q q h ρ D q q + q q w e s w e = = = = h L + Lw sd h UT w q + qw q q + qw qe q q w = q + 1. qe + qew qe qe (6.37) In he las expession (6.28) is use. So we have he foula: q q L L w = + w 1. (6.38) 33

35 The iensionless aoun of ice in he ovehang uing he ipping peio is now: ( τ ) = q τ, τ τ. (6.39) The iensionless aoun of ipping is equal o he iffeence beween snow eling an he feezing in he ovehang. The ipping sops a he ie τ = τ. So we have: ( τ ) q τ τ τ τ = < <. ( τ ) τ τ (6.4) Hee, he consan value fo he ie τ an onwas becoes: ( τ ) = ( τ ) q τ = M. (6.41) The ele snow is given by (6.9)-(6.1) an he ipping by (6.4) an (6.35), igh. The iffeence gives he feezing in he ovehang fo all ies: ( τ ) = ( τ ) ( τ ), τ <. (6.42) Fo he oal aouns we have: M = M + M, M = M, M = M, M = M. (6.43) Fo (6.41) an (6.1) we ge: M q,,, L = 1 L f ( τ, ) L f ( τ /, ) q τ. (6.44) w w w w w Hee, τ is he soluion o (6.31), an q is efine by (6.38). Finally we have fo (6.43) Hee, M q,,, L = M,, L M q,,, L. (6.45) w w w w w w M is given by (6.12). 6.4 Suay of foulas The following iensionless quaniies ae use: h ρ D L L τ =, =, L =, L =, U T L L L L s w w + w + w λ ( T ) λ ( T ) =, =. U T D s e s e w UwT D (6.46) The eling of snow on oof an winow is: 34

36 ( ) = ( / ), = L + L ρ D, w s ( τ ) = 1 L f ( τ, ) L f ( τ /, ). w w w (6.47) Hee, f ( τ, ) is given by he invese o (3.17) in accoance wih (3.2). The cieion fo ipping is: q + q q > q ; q = q q + q q, < q < 1. (6.48) w e w e The iensionless ipping lii τ is he soluion o he equaion: L L w L L w + = q = q + + ( 1 q ). f ( τ, ) f ( τ / w, w ) w (6.49) I becoes a funcion of fou paaees: τ = τ ( q,,, L ), L = 1 L. (6.5) w w w Dipping occus uing he ie τ < τ. The iensionless accuulae asses ae given by: ( ) = ( τ ), ( ) = ( τ ), ( ) = ( τ ). (6.51) The accuulae ipping is given by: ( τ ) q τ τ τ L L w ( τ ) =, q = q + 1. M τ < τ < w (6.52) Hee, M is efine in (6.54). The accuulae ice in he ovehang is given by: ( τ ) = ( τ ) ( τ ), τ <. (6.53) The oal aouns of ele snow, ice in he ovehang an ipping ae given by: M = M, M = M, M = M, M = M + M, M = 1 L L, M = M M, w w M = 1 L f ( τ, ) L f ( τ /, ) q τ. w w w (6.54) 6.5 An exaple We consie an exaple wih a winow on he oof wih he following inpu aa: o o T = 2 C, Te = 1 C, L = 6.5, Lw = 1.5, L =.6, D =.2, (6.55) λ =. W/ (, K), ρ = 2 kg/, U =.2, U = 3., U = 2. W/ (, K). 3 2 s s w 35

37 Figue 17 shows he eceasing snow eph fo.2 o he eling liis D =.12 fo he oof an D w =.1 fo he winow. The ie scales ae in ays an ays ies 3, especively. We see ha he eling on he oof has a uch longe ie scale ue o he bee insulaion D ay 3 D w ay Figue 17. Snow eph as funcion of ie own o he eling lii D =.12 fo he oof in ays an own o he eling lii D w =.1 fo he winow in ays ies 3. Figue 18 illusaes he eling an ipping in he consiee exaple. We ge fo ou foulas: M = 161 kg/, M = 15 kg/, M = 56 kg/, = 3.3 ays. (6.56) The uppe gaph shows he accuulae aoun of ele snow () (op cuve), he accuulae aoun of ipping wae an he accuulae aoun of ice in he ovehang, (). (lowes cuve) uing he fis 5 ays. The incease of ipping wae () is consan up o =. The lowe gaph shows hese cuves uing he fis 1 ays. The cuves fo () an () coss each ohe a =2 ays. The hoizonal ashe lines show he oal aouns M =161an M =15. 36

38 8 ( ay ) ( ay ) ay M ( ay ) ( ay ) ( ay ) M Figue 18. Accuulae eling of snow, ice in ovehang an ipping as funcions of ie (in ays). 7 Concluing eaks The aheaical oels in his pape povie a calculaion eho fo eling of snow, feezing on he ovehang an ipping fo he oof. The iagas ake i possible o copae iffeen oof soluions an see he effec of changing soe of he paaees. This is useful in evaluaion of isk fo icicles on oofs. I is seen ha heal insulaion of he oof an venilaion of an open aic is vey ipoan o avoi pobles wih icicles. Roofs wih a winow (skyligh) in he oof will always give oe eling wae han oofs wihou an give a highe isk of icicles. Exaples on he use of he eho will be pesene a he 9h Noic Syposiu of Builing Physics in Tapee

39 Noenclaue iensionless hickness of he snow laye, = D / D - iensionless lii fo ipping, = D / D - iensionless snow hickness lii, = D / D - D( ) hickness of snow on he oof a ie D iniial hickness of snow on he oof a ie = D snow hickness lii above which eling occus, (3.4) D snow hickness lii above which ipping occus, (4.8), lef f (, ) iensionless funcion fo ie as funcion of snow eph, (3.16)-(3.17) - f ( τ, ) iensionless snow hickness, (3.2); invese o τ = f (, ) - g ae of snow eling kg/(s,) g ae of wae ipping fo ovehang o fo ice, icicles o ops kg/(s,) h laen hea of eling he snow, h = 334 J/kg K heal conucance in he ovehang wih is snow cove fo he ice une he snow o he ousie ai, (4.2) W/(K) L oof lengh (fo ige o ovehang) L lengh of ovehang L lengh of he oof (above an below he winow) L lengh of winow on he oof w accuulae ipping a ie kg/ iensionless accuulae ipping, (4.21) - accuulae ele snow a ie kg/ iensionless aoun of ele snow, (3.25) - accuulae ice a he ovehang fo ele snow a ie kg/ iensionless accuulae ice a he ovehang - iniial ass of snow on he oof, (3.23), igh kg/ M oal aoun of ele snow ha ips fo ovehang kg/ M iensionless oal aoun of ipping fo ovehang - M oal aoun of ele snow on he oof kg/ M iensionless oal aoun of ele snow on he oof - M oal aoun of ice on he ovehang kg/ M iensionless oal aoun of ice on he ovehang, - q he hea flux aio (6.27) an (6.29) fo he ipping lii (6.31) - q e hea flux fo he eling zone he hough he snow, (3.5), igh W/ q e hea flux he hough he snow wih he iniial hickness D W/ q hea flux fo he ineio o el he snow on he oof, (3.5), lef W/ i 38

40 q hea flux o feeze wae une he snow on he ovehang W/ q elaive hea flux o feeze wae on he ovehang, (6.24) W/ q slope fo iensionless feezing on he ovehang, (6.38) W/ ie s ie o el all snow wih he hickness D fo T e =, (3.12) s ie when ipping sops, (4.22) s T T e ineio epeaue (below oof insulaion) exeio epeaue U U-value of he oof W/( 2,K) U U-value of he ovehang W/( 2,K) U ( ) U-value of he snow on he oof a ie s W/(2,K) o C o C λ s heal conuciviy of snow on oof W/(,K) ρ ensiy of snow on oof kg/ 3 s τ iensionless ie, / - τ iensionless ipping lii, (4.22) - The subscips, e,,,, s, w an efe o ipping, exeio epeaue, eling, ovehang, oof, snow, winow an iniial ie, especively. The subscip (in ialics) in f ( τ, ) efes iensionless snow eph, an no o ipping. A pie is ofen use o enoe he iensionless fo of a quaniy. In he case wih a winow on he oof, a secon o w is ae in he subscip wheneve appopiae. 39