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MOJ
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Ecology & Environmental Sciences

Review Article Volume 2 Issue 2

The influence of water ice conductivity on its formation

Marinyuk Boris, Ugolnikova Maria

Department of Low Temperature Engineering, Moscow State University of Mechanical Engineering (MAMI), Russia

Correspondence: Marinyuk Boris, Department of Low Temperature Engineering named by PL Kapitsa, Moscow State University of Mechanical Engineering (MAMI), Moscow, Russia

Received: February 03, 2017 | Published: April 10, 2017

Citation: Boris M, Maria U. The influence of water ice conductivity on its formation. MOJ Eco Environ Sci. 2017;2(2):57-59. DOI: 10.15406/mojes.2017.02.00019

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Abstract

Protection of sea medium with the help of cryogenic engineering is discussed the special problem is considered how to preclude oil leakage at the sea platform digging out raw material on the shelf. Cryogenic engineering may be a great help of freez the water around the injury tube immersed in an aqueous medium. Heat transfer task of ice formation was solved with the help of information approach. Heat transfer under formation of water ice on a flat wall being at low temperature and immersed into aqua media was considered. The heat conductivity of water ice data, obtaind by different authors within a great range et temperature down to cryogenic level were correlated by a hyperbolic expression. The latter gave an opportunity to solve Fourier equation for a layer of water ice forming on a flat wall being at low temperature and immersed in to aqua media the heat transfer task was solved for a constant heat conductivity of ice and with a temperature dependence of it.

Keywords: water ice, heat transfer process, cryogenic temperature, movable border, ınitial and boudering conditions; temperature field, fourier equation, heat conductivity, ıce formation, water media, heat transfer characteristics, specific heat accumulation, ınformation approach

Introduction

Heat transfer under formation of water ice is a keen process which actual for operation of different types of ice generators, cold accumulators, gasification systems of cryogenics liquid products (LNG, nitrogen, argon) where water media is a source of heat. It is also actual for underwater works, connected with extraction of subjects from the sea shelf as well as liquidation of water oil pollution at the platforms in case if extremely situations arise. Water oil mixture is freeze down into ice slab, which cover the oil leakage location. Such a solid composition is water proved, has a great cohesion if it is kept at a low temperature.

Heat conductivity factor

Heat transfer process which is followed by a moving border between water and ice is a quite a hard task for analyses, the math complications is increased if the process is developed under super low temperatures (so called cryogenic level). The problem is concentrated around variability of ice heat transfer characteristics. The most sensitive to temperature is heat conductivity of water ice. Among those who studied temperature dependence of water ice conductivity are.1-4 Most publications on this problem attributed to the temperature range 0÷-100 °C. The information connected with heat conductivity of water ice at lower temperature (cryogenic level and near it) is not wide spread. We may refer to publications of KD Timmerhaus et al.1 Figure 1 suggests experimental data on heat conductivity of water ice at a wide range of temperatures down to cryogenics. As goes from the graph, heat conductivity of water ice is increased with reduction of temperature and at liquid nitrogen level (-196 K) its conductivity is comparative with stainless steel. To describe the heat transfer process with movable water ice border, an analytical correlation between ice conductivity and temperature ought to be known. The analysis of the experimental dater obtaind by different authors and presented on Figure 1 draw us to a conclusion that they may be correlated by KD Timmerhaus1 hyperbolic law.

λ= K T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeq4UdW Maeyypa0tcfa4aaSaaaOqaaKqzGeGaam4saaGcbaqcLbsacaWGubaa aaaa@3CAD@                                      1

where K=615,34 W/m – dimensional constant, T- temperature of ice K.

Figure 1 Experimental data of heat conductivity of water ice vs. temperature, obtained by different authors. 1-DS Dillard, KD Timmerhaus, 2–EHRatcliffe, 3–JWDean, KD Timmerhaus, 4–M Jakob, S Erk, 5–CH Lees.

Expression (1) gives quite a good correlation with experimental data of different authors within a great range of temperature including cryogenic level (Figure 2). Later on it will be used for analytic description of heat transfer under formation of ice on the low temperature. Specific heat accumulation of water ice Cp is also depends its temperature but it is less susceptible to it so this dependence can be approximated by a formula    

                         Cp=C.P MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaSkqcLbsaca WGdbqcLbmacaWGWbqcLbsacqGH9aqpcaWGdbGaaiOlaiaadcfaaaa@3DFA@                                            2       

Where C=7,970 Dj/(kg·K) – dimensional constant.

Figure 2 Experimental data on heat conductivity of ice correlated by formula (1).1-DS Dillard, KD Timmerhaus, 2-JW Dean, KD Timmerhaus, 3-E.H.Ratcliffe, 4-formula (1).

Heat transfer process under formation of water ice water ice on the low temperature wall
Fourier unstationary differential equation for a flat wall being at low temperature and immersed into aqua media has the following reduction

C.T. ρ ice . T τ = x ( K T . T x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaam4qai aac6cacaWGubGaaiOlaiabeg8aYTWaaSbaaWqaaiaadMgacaWGJbGa amyzaaqabaqcLbsacaGGUaqcfa4aaSaaaOqaaKqzGeGaeyOaIyRaam ivaaGcbaqcLbsacqGHciITcqaHepaDaaGaeyypa0tcfa4aaSaaaOqa aKqzGeGaeyOaIylakeaajugibiabgkGi2kaadIhaaaqcfa4aaeWaaO qaaKqbaoaalaaakeaajugibiaadUeaaOqaaKqzGeGaamivaaaacaGG Uaqcfa4aaSaaaOqaaKqzGeGaeyOaIyRaamivaaGcbaqcLbsacqGHci ITcaWG4baaaaGccaGLOaGaayzkaaaaaa@5A19@                            3

Temperature field within ice layer can be obtained by solving equation (3) taking into account initial and bordering conditions, which for that case of heat transfer is expresses as
Heat balance on the border ice-water.

T( Q τ )= T wall MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamivaK qbaoaabmaakeaajugibiaadgfalmaaBaaameaacqaHepaDaeqaaaGc caGLOaGaayzkaaqcLbsacqGH9aqpcaWGubqcfa4aaSbaaeaajugWai aadEhacaWGHbGaamiBaiaadYgaaKqbagqaaaaa@457D@                                4

T( xO )= T n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamivaK qbaoaabmaabaqcLbsacaWG4bGaam4taaqcfaOaayjkaiaawMcaaKqz GeGaeyypa0JaamivaSWaaSbaaWqaaiaad6gaaeqaaaaa@3FF1@                                 5

T( ξ,T )=2TBK MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamivaK qbaoaabmaaleaajugibiabe67a4jaacYcacaWGubaaliaawIcacaGL Paaajugibiabg2da9iaaikdacaWGubGaamOqaiaadUeaaaa@421C@                                6

λ. T x | x=ξ =α( T w T ph )+ρice.L. dξ dt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaSkqcLbsacq aH7oaBcaGGUaqcfa4aaSaaaOqaaKqzGeGaeyOaIyRaamivaaGcbaqc LbsacqGHciITjugWaiaadIhaaaqcLbsacaGG8bqcfa4aaSbaaeaaju gWaiaacIhacqGH9aqpcqaH+oaEaKqbagqaaKqzGeGaeyypa0JaeqyS dewcfa4aaeWaaOqaaKqzGeGaamivaSWaaSbaaWqaaiaadEhaaeqaaK qzGeGaeyOeI0IaamivaSWaaSbaaWqaaiaadchacaWGObaabeaaaOGa ayjkaiaawMcaaKqzGeGaey4kaSIaeqyWdiNaamyAaiaadogacaWGLb GaaiOlaiaacYeacaGGUaqcfa4aaSaaaOqaaKqzGeGaamizaiabe67a 4bGcbaqcLbsacaWGKbGaamiDaaaaaaa@63DD@                7

where λ - heat conductivity of water ice at 0 °C, W/(m·K); α – heat transfer coefficient from the side of water, W/(m2·K); ρice - ice density, kg/m3; L - heat of phase changes (freezing of water into ice), Dj/kg; ξ – thickness of ice layer forming on the flat wall, m; τ - process time, s.

Equation (3) is attributed to unlinear unstationary type with variable transfer coefficients, which can’t be solved in quadratures by a traditional. Methods more fertile results may be obtained on the bases of approximate analytical methods which give result with an acceptable accuracy. An introduction of new variable (information approach5)

v= x τ T( x,τ )=T( v ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabceqabaaRceaaca WG2bGaeyypa0ZaaSaaaeaacaWG4baabaWaaOaaaeaacqaHepaDaSqa baaaaaGcbaqcLbsacaWGubqcfa4aaeWaaOqaaKqzGeGaamiEaiaacY cacqaHepaDaOGaayjkaiaawMcaaKqzGeGaeyypa0JaamivaKqbaoaa bmaakeaajugibiaadAhaaOGaayjkaiaawMcaaaaaaa@4952@                              8

turning the equation (3) from partial derivation to a full of them.

d d .[ ( K T )( dT dv ) ]= v 2 .ρ . ice .C.T. dT dv =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaSkqcfa4aaS aaaOqaaKqzGeGaamizaaGcbaqcLbsacaWGKbaaaiaac6cajuaGdaWa daGcbaqcfa4aaeWaaOqaaKqbaoaalaaakeaajugibiaadUeaaOqaaK qzGeGaamivaaaaaOGaayjkaiaawMcaaKqbaoaabmaakeaajuaGdaWc aaGcbaqcLbsacaWGKbGaamivaaGcbaqcLbsacaWGKbGaamODaaaaaO GaayjkaiaawMcaaaGaay5waiaaw2faaKqzGeGaeyypa0tcfa4aaSaa aOqaaKqzGeGaamODaaGcbaqcLbsacaaIYaaaaiaac6cacqaHbpGCca GGUaWcdaWgaaadbaGaamyAaiaadogacaWGLbaabeaajugibiaac6ca caWGdbGaaiOlaiaadsfacaGGUaqcfa4aaSaaaOqaaKqzGeGaamizai aadsfaaOqaaKqzGeGaamizaiaadAhaaaGaeyypa0JaaGimaaaa@6173@                     9

An initial and boundaring conditions take the reduction

T( O )= T Wall MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamivaK qbaoaabmaakeaajugibiaad+eaaOGaayjkaiaawMcaaKqzGeGaeyyp a0JaamivaSWaaSbaaWqaaiaadEfacaWGHbGaamiBaiaadYgaaeqaaa aa@412B@                                           10

T( α )= T w T( α )= T w MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaSkqcLbsaca WGubqcfa4aaeWaaOqaaKqzGeGaeqySdegakiaawIcacaGLPaaajugi biabg2da9iaadsfalmaaBaaameaacaWG3baabeaajugibiaadsfaju aGdaqadaGcbaqcLbsacqaHXoqyaOGaayjkaiaawMcaaKqzGeGaeyyp a0JaamivaSWaaSbaaWqaaiaadEhaaeqaaaaa@4962@                                      11

T( β )=2E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaSkqcLbsaca WGubqcfa4aaeWaaOqaaKqzGeGaeqOSdigakiaawIcacaGLPaaajugi biabg2da9iaaikdacaWGfbaaaa@3F7A@                                           12

where β - is variable factor of ice growth rate

β= ξ τ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaSkqcLbsacq aHYoGycqGH9aqpjuaGdaWcaaqcaawaaKqzGeGaeqOVdGhajaaybaqc fa4aaOaaaKaaGfaajugibiabes8a0bqcbawabaaaaaaa@4159@                                               13

              λ. dT τ.dv | v=β = ρ ice .L. β 2 τ +α( T w T ph ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeq4UdW MaaiOlaKqbaoaalaaabaqcLbsacaWGKbGaamivaaqcfayaaKqzGeGa eqiXdqNaaiOlaiaadsgacaWG2baaaiaacYhalmaaBaaameaacaWG2b Gaeyypa0JaeqOSdigabeaajugibiabg2da9iabeg8aYTWaaSbaaWqa aiaadMgacaWGJbGaamyzaaqabaqcLbsacaGGUaGaamitaiaac6caju aGdaWcaaqaaKqzGeGaeqOSdigajuaGbaqcLbsacaaIYaqcfa4aaOaa aeaajugibiabes8a0bqcfayabaaaaKqzGeGaey4kaSIaeqySdewcfa 4aaeWaaeaajugibiaadsfalmaaBaaameaacaWG3baabeaajugibiab gkHiTiaadsfalmaaBaaameaacaWGWbGaamiAaaqabaaajuaGcaGLOa Gaayzkaaaaaa@6419@                                     14

Suppose the temperature field within the ice layer is expressed as a series

T( v )=T( β )+ ! T( β ) 1! ( .vβ )+ !! T( β ) 2! ( .vβ )2+... T n ( β ) n! ( .vβ ) n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamivaK qbaoaabmaakeaajugibiaadAhaaOGaayjkaiaawMcaaKqzGeGaeyyp a0JaamivaKqbaoaabmaakeaajugibiabek7aIbGccaGLOaGaayzkaa qcLbsacqGHRaWkjuaGdaWcaaGcbaqcfa4aaWbaaeqabaGaaiyiaaaa jugibiaadsfajuaGdaqadaGcbaqcLbsacqaHYoGyaOGaayjkaiaawM caaaqaaKqzGeGaaGymaiaacgcaaaqcfa4aaeWaaOqaaKqzGeGaaiOl aiaadAhacqGHsislcqaHYoGyaOGaayjkaiaawMcaaKqzGeGaey4kaS scfa4aaSaaaOqaaKqbaoaaCaaabeqaaiaacgcacaGGHaaaaiaadsfa daqadaGcbaqcLbsacqaHYoGyaOGaayjkaiaawMcaaaqaaKqzGeGaaG Omaiaacgcaaaqcfa4aaeWaaOqaaKqzGeGaaiOlaiaadAhacqGHsisl cqaHYoGyaOGaayjkaiaawMcaaKqzGeGaaGOmaiabgUcaRiaac6caca GGUaGaaiOlaKqbaoaalaaakeaajugibiaadsfalmaaCaaameqabaGa amOBaaaajuaGdaqadaGcbaqcLbsacqaHYoGyaOGaayjkaiaawMcaaa qaaKqzGeGaamOBaiaacgcaaaqcfa4aaeWaaOqaaKqzGeGaaiOlaiaa dAhacqGHsislcqaHYoGyaOGaayjkaiaawMcaaSWaaWbaaWqabeaaca WGUbaaaaaa@7BA6@                    15

Taking in to account expressions (10,11,12,13) the final result of the task is expressed

T wall =2Β( ρ ice .L.β 2λ + α( T w T ph ) τ λ )β+( ( T 2 ) 2B β 2K .C.2 B 2 .T ) β 2 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamivaS WaaSbaaWqaaiaadEhacaWGHbGaamiBaiaadYgaaeqaaKqzGeGaeyyp a0JaaGOmaiabfk5acjabgkHiTKqbaoaabmaakeaajuaGdaWcaaGcba qcLbsacqaHbpGClmaaBaaameaacaWGPbGaam4yaiaadwgaaeqaaKqz GeGaaiOlaiaadYeacaGGUaGaeqOSdigakeaajugibiaaikdacqaH7o aBaaGaey4kaSscfa4aaSaaaOqaaKqzGeGaeqySdewcfa4aaeWaaOqa aKqzGeGaamivaKqbaoaaBaaabaGaam4DaaqabaqcLbsacqGHsislca WGubWcdaWgaaadbaGaamiCaiaadIgaaeqaaaGccaGLOaGaayzkaaqc fa4aaOaaaOqaaKqzGeGaeqiXdqhaleqaaaGcbaqcLbsacqaH7oaBaa aakiaawIcacaGLPaaajugibiabek7aIjabgUcaRKqbaoaabmaakeaa juaGdaWcaaGcbaqcfa4aaeWaaOqaaKqzGeGaamivaSWaaWbaaWqabe aacaaIYaaaaaGccaGLOaGaayzkaaaabaqcLbsacaaIYaGaamOqaaaa cqGHsisljuaGdaWcaaGcbaqcLbsacqaHYoGyaOqaaKqzGeGaaGOmai aadUeaaaGaaiOlaiaadoeacaGGUaGaaGOmaiaadkealmaaCaaameqa baGaaGOmaaaajugibiaac6cacaWGubaakiaawIcacaGLPaaajuaGda WcaaGcbaqcLbsacqaHYoGylmaaCaaameqabaGaaGOmaaaaaOqaaKqz GeGaaGOmaaaaaaa@8079@                     16

Where T´- fist derivative of T(ν)

            T v=β =( ρ ice .L.β 2λ + α( T w T ph ) τ λ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamivaS WaaSbaaWqaaiaadAhacqGH9aqpcqaHYoGyaeqaaKqzGeGaeyypa0tc fa4aaeWaaOqaaKqbaoaalaaakeaajugibiabeg8aYTWaaSbaaWqaaK qzadGaamyAaiaadogacaWGLbaameqaaKqzGeGaaiOlaiaadYeacaGG UaGaeqOSdigakeaajugibiaaikdacqaH7oaBaaGaey4kaSscfa4aaS aaaOqaaKqzGeGaeqySdewcfa4aaeWaaOqaaKqzGeGaamivaSWaaSba aKqbagaajugWaiaadEhaaKqbagqaaKqzGeGaeyOeI0IaamivaKqbao aaBaaameaajugWaiaadchacaWGObaameqaaaGccaGLOaGaayzkaaqc fa4aaOaaaOqaaKqzGeGaeqiXdqhaleqaaaGcbaqcLbsacqaH7oaBaa aakiaawIcacaGLPaaaaaa@63C1@                                      17

T´´-is a second derivative of T(ν)

                        T ' v=β = ( T 2 ) 2B β 2B .C.2 B 2 T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamivai aacEcalmaaBaaameaacaGG2bGaeyypa0JaeqOSdigabeaajugibiab g2da9KqbaoaalaaakeaajuaGdaqadaGcbaqcLbsacaWGubqcfa4aaW baaeqabaqcLbmacaaIYaaaaaGccaGLOaGaayzkaaaabaqcLbsacaaI YaGaamOqaaaacqGHsisljuaGdaWcaaGcbaqcLbsacqaHYoGyaOqaaK qzGeGaaGOmaiaadkeaaaGaaiOlaiaadoeacaGGUaGaaGOmaiaadkea lmaaCaaameqabaGaaGOmaaaajugibiaadsfaaaa@5240@                                               18

Variable factor of growth rate β is found for a accepted time τ by solving (16) if the process is goes on at temperature of the wall Twall > 223 K.

The result is simplificated to

ξ= α.( T w T ph ) ρ ice .L + λ ρ ice .L [ α. ( TwTph ). τ λ ] 2 2. ρ ice .L.( Twd2B ).τ λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaSkqcLbsacq aH+oaEcqGH9aqpjuaGdaWcaaGcbaqcLbsacqGHsislcqaHXoqycaGG Uaqcfa4aaeWaaOqaaKqzGeGaamivaKqbaoaaBaaabaqcLbmacaWG3b aajuaGbeaajugibiabgkHiTiaadsfalmaaBaaameaacaWGWbGaamiA aaqabaaakiaawIcacaGLPaaaaeaajugibiabeg8aYLqbaoaaBaaaba qcLbmacaWGPbGaam4yaiaadwgaaKqbagqaaKqzGeGaaiOlaiaadYea aaGaey4kaSscfa4aaSaaaOqaaKqzGeGaeq4UdWgakeaajugibiabeg 8aYTWaaSbaaWqaaiaadMgacaWGJbGaamyzaaqabaqcLbsacaGGUaGa amitaaaajuaGdaGcaaGcbaqcfa4aamWaaOqaaKqzGeGaeqySdeMaai OlaKqbaoaalaaakeaajuaGdaqadaGcbaqcLbsacaWGubGaam4Daiab gkHiTiaadsfacaWGWbGaamiAaaGccaGLOaGaayzkaaqcLbsacaGGUa qcfa4aaOaaaOqaaKqzGeGaeqiXdqhaleqaaaGcbaqcLbsacqaH7oaB aaaakiaawUfacaGLDbaaaSqabaWaaWbaaWqabeaacaaIYaaaaKqbao aalaaakeaajugibiaaikdacaGGUaGaeqyWdi3cdaWgaaadbaGaamyA aiaadogacaWGLbaabeaajugibiaac6cacaWGmbGaaiOlaKqbaoaabm aakeaajugibiaadsfacaWG3bGaamizaiabgkHiTiaaikdacaWGcbaa kiaawIcacaGLPaaajugibiaac6cacqaHepaDaOqaaKqzGeGaeq4UdW gaaaaa@8C6F@                       19

                The calculations made according equations (16) and (17) revealed that cryogenic temperatures on the wall needs taking into account the temperature dependence of water ice layer (Figure 3).

Figure 3 Growth of ice on a flat wall being at cryogenic temperatures and immersed in aqua media (Tw=285 K, α=270 W/(m2·K), Twall=77 K).1 - with constant characteristics of ice, 2-with variable characteristics of ice.

Conclusion

  1. The heat conductivity of water ice data obtaind by different authors within a great range of temperature down to cryogenic level were correlated by a hyperbolic expression.
  2. Fourier equation for a layer a ice forming on a flat wall being at low temperature and immersed in to aqua media was solved taking in to account temperature dependence of water ice conductivity.
  3. In case of cryogenic level of temperatures on the wall, it is necessary to take in to account temperature dependence of water ice conductivity.

Acknowledgements

None.

Conflict of interest

The author declares no conflict of interest.

References

  1. Dillard DS, KD Timmerhaus. Low temperature thermal conductivity of solidified H20 and D20 Pure Appl. Cryogenics. 1966;4:35–44.
  2. Dean JW, Timmerhaus KD. Thermal conductivity of solid O And O At Low Temperatures. Adv. In Cryogen Engineering. 1963;8:263–267.
  3. Radcliffe EH. The thermal conductivity of ice new data on the temperature coefficient. Philosophical Magazine. 1962;7(79):1197–1203.
  4. Nazintsev JL, Panov VV. Phase composition and thermal characteristics of sea ice. SPb Gidrometioizdat; 2000.
  5. Marinyuk BT. The calculation of heat transfer in the apparatus and the low-temperature cooling systems. Russia: Mashinostroenie; 2015.
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