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MOJ
eISSN: 2573-2919

Ecology & Environmental Sciences

Mini Review Volume 3 Issue 3

Precluding oil leakage from the oil ship tankers under mechanical damage

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, Tel 8916-9263-952

Received: April 16, 2018 | Published: June 27, 2018

Citation: Boris M, Maria U. Precluding oil leakage from the oil ship tankers under mechanical damage. MOJ Eco Environ Sci. 2018;3(3):212-214 . DOI: 10.15406/mojes.2018.03.00089

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Abstract

The problem of Protection of sea medium with the help of low temperature engineering is discussed. The special item is the way how to preclude oil leakage which is shipped with sea tankers in case of mechanical damage of it. It is suggested to set up vertical flexible tubs which are fed by cryogenic fluids when the mechanical damage of the ship hull occurred. Water ices are forming on the tube surface and preclude oil leakage from the tanks.

The heat transfer task under formation of water ice is considered. Fourier equation for a layer of water ice forming on a cylindrical tube wall being at low temperature surrounded by a water medium was solved with a temperature dependence ice heat conductivity et it.

Keywords: water ice, fourier unstionale differential equation, temperature dependence of heat conductivity of water ice, heat transfer task

Introduction

Preservation and transportation of bulk quantities of oil products in tanks is strictly connected with a creation of safety conditions under its operation. Oil leakage in to environment medium may cause a detriment to it. This is equally attributed to the case when the tank is located on the dry land or at the transportation ships. It is known, that cost of water purification from the oil is reached more than 106 $ if the sea surface is one mile sq. The damage osf tanks wall is possible when the tankers are collided with underwater obstacles, icebergs or other ships. Now in case of such an emergency situation with the tankers, an auxiliary vessel with empty reservoir is called and oil product is pump out from the damaged tank. The span of arrayal time depends on geographical point of the incident and takes from one to a few days. During this time pollution of sea may spared to great scales. There is no garantec of complete protection from oil leakage if the tanker has double hull. Low temperature engineering technology gives a method which may solve the problem of precluding of oil leakage from the damaged wall of ship tanks with en a short span of time .It is necessary to equip each tank-bearer with a cryogenic reservoir containing liquid nitrogen. The outer side of each tank is supplied by flexible vertical tubs which are set up in parallel with interval of 300 mm. The free hanging of the tubs is preventing them from mechanical damage in case of sea incident. So let it consider the process of water freezing around a tube which is cooled down inside by a liquid nitrogen. To simplify the analysis the temperature of wall tube is accepted as constant at the level of 77 K (boiling temperature of liquid nitrogen at atmospheric pressure).

The heat transfer coefficient from the water side is constant αw = 180 Wt/ (m2 K).

The heat transfer process is described by Fourier unstationary differential equation for a tube being cooled inside and immersed in an aqueous medium.

In cylindrical coordinates it looks like this

c p ρ dT dτ = d dr [ λ( T ) dT dr ]+ λ r dT dr MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGJb WcdaWgaaqaaKqzadGaamiCaaWcbeaajugibiabeg8aYLqbaoaalaaa keaajugibiaadsgacaWGubaakeaajugibiaadsgacqaHepaDaaGaey ypa0tcfa4aaSaaaOqaaKqzGeGaamizaaGcbaqcLbsacaWGKbGaamOC aaaajuaGdaWadaGcbaqcLbsacqaH7oaBjuaGdaqadaGcbaqcLbsaca WGubaakiaawIcacaGLPaaajuaGdaWcaaGcbaqcLbsacaWGKbGaamiv aaGcbaqcLbsacaWGKbGaamOCaaaaaOGaay5waiaaw2faaKqzGeGaey 4kaSscfa4aaSaaaOqaaKqzGeGaeq4UdWgakeaajugibiaadkhaaaqc fa4aaSaaaOqaaKqzGeGaamizaiaadsfaaOqaaKqzGeGaamizaiaadk haaaaaaa@612F@      (1)

Equation (1) is attributed to unlinear unstationary type with variable transfer coefficient, which can’t be solved in quadrature by methods. More fertile results may be obtained on the bases of approximate analytical methods which give result with an desirable accuracy where

c p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGJbWcdaWgaaadbaGaamiCaaqabaaaaa@38AF@ is average specific heat accumulation of ice at a temperature T ¯ = T ph T wall 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aa0aaaO qaaKqzGeGaamivaaaacqGH9aqpjuaGdaWcaaGcbaqcLbsacaWGubWc daWgaaqaaKqzadGaamiCaiaadIgaaSqabaqcLbsacqGHsislcaWGub qcfa4aaSbaaSqaaKqzadGaam4DaiaadggacaWGSbGaamiBaaWcbeaa aOqaaKqzGeGaaGOmaaaaaaa@47C5@

T ph MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub WcdaWgaaqaaKqzadGaamiCaiaadIgaaSqabaaaaa@3AA5@ Temperature of phase change of water in to ice, K;

T wall MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub qcfa4aaSbaaSqaaKqzadGaam4DaiaadggacaWGSbGaamiBaaWcbeaa aaa@3D15@ Temperature of wall of tube, K;

The law of heat conductivity change of ice with temperature is correlated by K.D. Timmerhaus et al.1

λ= K T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH7o aBcqGH9aqpjuaGdaWcaaGcbaqcLbsacaWGlbaakeaajugibiaadsfa aaaaaa@3CB8@      (2)

Where K = 615.34 W/m – dimensional constant;

T–Temperature of ice, K

λ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeU7aSTWdamaaBaaabaqcLbmapeGaaGimaaWcpaqabaaa aa@3AA6@ heat conductivity of water ice at phase change temperature, W/(m K);

ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbsacq WFbpGCaaa@384A@ Is density, kg/m3;

r– Cylindrical coordinate, m;

τ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbsacq WFepaDaaa@384F@ Time of process, s.

An initial and bordering conditions is expressed as

T( 0, τ ) = T wall MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadsfajuaGpaWaaeWaaOqaaKqzGeWdbiaaicdacaGGSaGa aeiiaiabes8a0bGcpaGaayjkaiaawMcaaKqzGeWdbiaabccacqGH9a qpcaWGubqcfa4damaaBaaaleaajugWa8qacaWG3bGaamyyaiaadYga caWGSbaal8aabeaaaaa@473E@     (3)

T( r, 0 ) = T w MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadsfajuaGpaWaaeWaaOqaaKqzGeWdbiaadkhacaGGSaGa aeiiaiaaicdaaOWdaiaawIcacaGLPaaajugib8qacaqGGaGaeyypa0 JaamivaSWdamaaBaaabaqcLbmapeGaam4DaaWcpaqabaaaaa@431A@       (4)

T( η, τ ) = T ph MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadsfajuaGpaWaaeWaaOqaaKqzGeWdbiabeE7aOjaacYca caqGGaGaeqiXdqhak8aacaGLOaGaayzkaaqcLbsapeGaaeiiaiabg2 da9iaadsfajuaGpaWaaSbaaSqaaKqzadWdbiaadchacaWGObaal8aa beaaaaa@464E@        (5)

Where T wall MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadsfajuaGpaWaaSbaaSqaaKqzadWdbiaadEhacaWGHbGa amiBaiaadYgaaSWdaeqaaaaa@3D63@ temperature of tube wall, K;

η–Movable coordinates of ice water border, m;

T w MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadsfal8aadaWgaaqaaKqzadWdbiaadEhaaSWdaeqaaaaa @3A0D@ Water temperature, K.

Heat balance on the movable ice water border

λ 0 dT dr | r=η = α w ( T w T ph )+ρL dη dr MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaqGaaO qaaKqzGeGaeq4UdWwcfa4aaSbaaSqaaKqzadGaaGimaaWcbeaajuaG daWcaaGcbaqcLbsacaWGKbGaamivaaGcbaqcLbsacaWGKbGaamOCaa aaaOGaayjcSdqcfa4aaSbaaSqaaKqzGeGaamOCaiabg2da9iabeE7a ObWcbeaajugibiabg2da9iabeg7aHTWaaSbaaeaajugWaiaadEhaaS qabaqcfa4aaeWaaOqaaKqzGeGaamivaSWaaSbaaeaajugWaiaadEha aSqabaqcLbsacqGHsislcaWGubWcdaWgaaqaaKqzadGaamiCaiaadI gaaSqabaaakiaawIcacaGLPaaajugibiabgUcaRiabeg8aYjaadYea juaGdaWcaaGcbaqcLbsacaWGKbGaeq4TdGgakeaajugibiaadsgaca WGYbaaaaaa@62F9@     (6)

Where L–heat of phase changes (freezing of water into ice), Dj/kg.

The final result of the equation (1) decision without deriving of it is presented at.2 The procedure of deriving of the result provides an introduction of new variable ”V”

V= r r 0 τ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGwb Gaeyypa0tcfa4aaSaaaOqaaKqzGeGaamOCaiabgkHiTiaadkhajuaG daWgaaWcbaqcLbmacaaIWaaaleqaaaGcbaqcfa4aaOaaaOqaaKqzGe GaeqiXdqhaleqaaaaaaaa@4236@          (7)

T( r, τ ) =T( V ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadsfajuaGpaWaaeWaaOqaaKqzGeWdbiaadkhacaGGSaGa aeiiaiabes8a0bGcpaGaayjkaiaawMcaaKqzGeWdbiaabccacqGH9a qpcaWGubqcfa4damaabmaakeaajugib8qacaWGwbaak8aacaGLOaGa ayzkaaaaaa@4559@     (8)

( η r 0 )=( τ )=β τ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9aGeGaeq4TdGMaeyOeI0IaamOCaKqbaoaaBaaaleaajugWaiaa icdaadbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzSqabaaakiaawIcacaGLPaaajugibiabg2da9Kqbaoaabmaake aajugibiabes8a0bGccaGLOaGaayzkaaqcLbsacqGH9aqpcqaHYoGy juaGdaGcaaGcbaqcLbsacqaHepaDaSqabaaaaa@4B3E@     (9)

Where r 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGYb qcfa4aaSbaaSqaaKqzadGaaGimaaWcbeaaaaa@3A29@ outside of the tube, m;

ξ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbsacq WF+oaEaaa@384D@ Thickness of ice layer forming on the tube surface, m;

β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaqcLbsacq WFYoGyaaa@382B@ Variable factor office growth rate, m/c2.

This turning the equation (1) from partial derivation to a full of them

cρ dT dV = d dV [ K T dT dV ]+ 1 V τ + r 0 K T dT dV MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGJb GaeqyWdixcfa4aaSaaaOqaaKqzGeGaamizaiaadsfaaOqaaKqzGeGa amizaiaadAfaaaGaeyypa0tcfa4aaSaaaOqaaKqzGeGaamizaaGcba qcLbsacaWGKbGaamOvaaaajuaGdaWadaGcbaqcfa4aaSaaaOqaaKqz GeGaam4saaGcbaqcLbsacaWGubaaaKqbaoaalaaakeaajugibiaads gacaWGubaakeaajugibiaadsgacaWGwbaaaaGccaGLBbGaayzxaaqc LbsacqGHRaWkjuaGdaWcaaGcbaqcLbsacaaIXaaakeaajugibiaadA fajuaGdaGcaaGcbaqcLbsacqaHepaDaSqabaqcLbsacqGHRaWkcaWG Ybqcfa4aaSbaaSqaaKqzadGaaGimaaWcbeaaaaqcfa4aaSaaaOqaaK qzGeGaam4saaGcbaqcLbsacaWGubaaaKqbaoaalaaakeaajugibiaa dsgacaWGubaakeaajugibiaadsgacaWGwbaaaaaa@652B@       (10)

Equation (6) is taking the form

λ τ dT dV | V=β =ρL β 2 τ + α w ( T w T ph ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaqGaaO qaaKqbaoaalaaakeaajugibiabeU7aSbGcbaqcfa4aaOaaaOqaaKqz GeGaeqiXdqhaleqaaaaajuaGdaWcaaGcbaqcLbsacaWGKbGaamivaa GcbaqcLbsacaWGKbGaamOvaaaaaOGaayjcSdqcfa4aaSbaaSqaaKqz GeGaamOvaiabg2da9iabek7aIbWcbeaajugibiabg2da9iabeg8aYj aadYeajuaGdaWcaaGcbaqcLbsacqaHYoGyaOqaaKqzGeGaaGOmaKqb aoaakaaakeaajugibiabes8a0bWcbeaaaaqcLbsacqGHRaWkcqaHXo qyjuaGdaWgaaWcbaqcLbmacaWG3baaleqaaKqbaoaabmaakeaajugi biaadsfalmaaBaaabaqcLbmacaWG3baaleqaaKqzGeGaeyOeI0Iaam ivaSWaaSbaaeaajugWaiaadchacaWGObaaleqaaaGccaGLOaGaayzk aaaaaa@653F@     (11)

Initial and bordering conditions will be

T( 0 ) = T wall MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadsfajuaGpaWaaeWaaOqaaKqzGeWdbiaaicdaaOWdaiaa wIcacaGLPaaajugib8qacaqGGaGaeyypa0JaamivaKqba+aadaWgaa WcbaqcLbmapeGaam4DaiaadggacaWGSbGaamiBaaWcpaqabaaaaa@4426@      (12)

T( ) = T w MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadsfajuaGpaWaaeWaaOqaaKqzGeWdbiabg6HiLcGcpaGa ayjkaiaawMcaaKqzGeWdbiaabccacqGH9aqpcaWGubWcpaWaaSbaae aajugWa8qacaWG3baal8aabeaaaaa@4187@       (13)

T( β ) = T ph MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadsfajuaGpaWaaeWaaOqaaKqzGeWdbiabek7aIbGcpaGa ayjkaiaawMcaaKqzGeWdbiaabccacqGH9aqpcaWGubqcfa4damaaBa aaleaajugWa8qacaWGWbGaamiAaaWcpaqabaaaaa@432B@          (14)

Suppose the temperature profiles within the ice lager 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 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub qcfa4aaeWaaOqaaKqzGeGaamOvaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpcaWGubqcfa4aaeWaaOqaaKqzGeGaeqOSdigakiaawIcacaGLPa aajugibiabgUcaRKqbaoaalaaakeaajugibiaadsfajugWaiaacEca juaGdaqadaGcbaqcLbsacqaHYoGyaOGaayjkaiaawMcaaaqaaKqzGe GaaGymaiaacgcaaaqcfa4aaeWaaOqaaKqzGeGaamOvaiabgkHiTiab ek7aIbGccaGLOaGaayzkaaqcLbsacqGHRaWkjuaGdaWcaaGcbaqcLb sacaWGubqcLbmacaGGNaGaai4jaKqbaoaabmaakeaajugibiabek7a IbGccaGLOaGaayzkaaaabaqcLbsacaaIYaGaaiyiaaaajuaGdaqada GcbaqcLbsacaWGwbGaeyOeI0IaeqOSdigakiaawIcacaGLPaaalmaa CaaabeqaaKqzadGaaGOmaaaajugibiabgUcaRiaac6cacaGGUaGaai OlaiabgUcaRKqbaoaalaaakeaajugibiaadsfalmaaCaaabeqaaKqz adGaamOBaaaajuaGdaqadaGcbaqcLbsacqaHYoGyaOGaayjkaiaawM caaaqaaKqzGeGaamOBaiaacgcaaaqcfa4aaeWaaOqaaKqzGeGaamOv aiabgkHiTiabek7aIbGccaGLOaGaayzkaaqcfa4aaWbaaSqabeaaju gWaiaad6gaaaaaaa@8055@      (15)

The second derivative of T(V)

T''= (T') 2 273 T' τ β τ + r 0 + 273cρT' 2K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub qcLbmacaGGNaGaai4jaKqzGeGaeyypa0tcfa4aaSaaaOqaaKqzGeGa aiikaiaadsfacaGGNaGaaiykaSWaaWbaaeqabaqcLbmacaaIYaaaaa GcbaqcLbsacaaIYaGaaG4naiaaiodaaaGaeyOeI0scfa4aaSaaaOqa aKqzGeGaamivaiaacEcajuaGdaGcaaGcbaqcLbsacqaHepaDaSqaba aakeaajugibiabek7aILqbaoaakaaakeaajugibiabes8a0bWcbeaa jugibiabgUcaRiaadkhajuaGdaWgaaWcbaqcLbmacaaIWaaaleqaaa aajugibiabgUcaRKqbaoaalaaakeaajugibiaaikdacaaI3aGaaG4m aiaadogacqaHbpGCcaWGubGaai4jaaGcbaqcLbsacaaIYaGaam4saa aaaaa@60AD@      (16)

Taking into account expressions (11, 12, 13, and 14) the final result of the task is expressed.

T wall = T ph [ ρLβ 2λ + α w τ λ ( T w T ph ) ]β++[ ( ρLβ 2λ + α w τ λ ( T w T ph ) ) 2 T ph ( ρLβ 2λ + α w τ λ ( T w T ph ) ) τ β τ + r 0 + + ρ c p β T ph 2K [ ρLβ 2λ + α w τ λ ( T w T ph ) ] β 2 2 ] β 2 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub WcdaWgaaqaaKqzadGaam4DaiaadggacaWGSbGaamiBaaWcbeaajugi biabg2da9iaadsfalmaaBaaabaqcLbmacaWGWbGaamiAaaWcbeaaju gibiabgkHiTKqbaoaadmaakeaajuaGdaWcaaGcbaqcLbsacqaHbpGC caWGmbGaeqOSdigakeaajugibiaaikdacqaH7oaBaaGaey4kaSscfa 4aaSaaaOqaaKqzGeGaeqySde2cdaWgaaqaaKqzadGaam4DaaWcbeaa juaGdaGcaaGcbaqcLbsacqaHepaDaSqabaaakeaajugibiabeU7aSb aajuaGdaqadaGcbaqcLbsacaWGubWcdaWgaaqaaKqzadGaam4DaaWc beaajugibiabgkHiTiaadsfalmaaBaaabaqcLbmacaWGWbGaamiAaa WcbeaaaOGaayjkaiaawMcaaaGaay5waiaaw2faaKqzGeGaeqOSdiMa ey4kaSIaey4kaSscfa4aamWaaKqzGeabaeqakeaajuaGdaWcaaGcba qcfa4aaeWaaOqaaKqbaoaalaaakeaajugibiabeg8aYjaadYeacqaH YoGyaOqaaKqzGeGaaGOmaiabeU7aSbaacqGHRaWkjuaGdaWcaaGcba qcLbsacqaHXoqylmaaBaaabaqcLbmacaWG3baaleqaaKqbaoaakaaa keaajugibiabes8a0bWcbeaaaOqaaKqzGeGaeq4UdWgaaKqbaoaabm aakeaajugibiaadsfalmaaBaaabaqcLbmacaWG3baaleqaaKqzGeGa eyOeI0IaamivaKqbaoaaBaaaleaajugWaiaadchacaWGObaaleqaaa GccaGLOaGaayzkaaaacaGLOaGaayzkaaWcdaahaaqabeaajugWaiaa ikdaaaaakeaajugibiaadsfajuaGdaWgaaWcbaqcLbmacaWGWbGaam iAaaWcbeaaaaqcLbsacqGHsisljuaGdaWcaaGcbaqcfa4aaeWaaOqa aKqbaoaalaaakeaajugibiabeg8aYjaadYeacqaHYoGyaOqaaKqzGe GaaGOmaiabeU7aSbaacqGHRaWkjuaGdaWcaaGcbaqcLbsacqaHXoqy lmaaBaaabaqcLbmacaWG3baaleqaaKqbaoaakaaakeaajugibiabes 8a0bWcbeaaaOqaaKqzGeGaeq4UdWgaaKqbaoaabmaakeaajugibiaa dsfajuaGdaWgaaWcbaqcLbmacaWG3baaleqaaKqzGeGaeyOeI0Iaam ivaKqbaoaaBaaaleaajugWaiaadchacaWGObaaleqaaaGccaGLOaGa ayzkaaaacaGLOaGaayzkaaqcfa4aaOaaaOqaaKqzGeGaeqiXdqhale qaaaGcbaqcLbsacqaHYoGyjuaGdaGcaaGcbaqcLbsacqaHepaDaSqa baqcLbsacqGHRaWkcaWGYbWcdaWgaaqaaKqzadGaaGimaaWcbeaaaa qcLbsacqGHRaWkaOqaaKqzGeGaey4kaSscfa4aaSaaaOqaaKqzGeGa eqyWdiNaam4yaKqbaoaaBaaaleaajugWaiaadchaaSqabaqcLbsacq aHYoGycaWGubqcfa4aaSbaaSqaaKqzadGaamiCaiaadIgaaSqabaaa keaajugibiaaikdacaWGlbaaaKqbaoaadmaakeaajuaGdaWcaaGcba qcLbsacqaHbpGCcaWGmbGaeqOSdigakeaajugibiaaikdacqaH7oaB aaGaey4kaSscfa4aaSaaaOqaaKqzGeGaeqySde2cdaWgaaqaaKqzad Gaam4DaaWcbeaajuaGdaGcaaGcbaqcLbsacqaHepaDaSqabaaakeaa jugibiabeU7aSbaajuaGdaqadaGcbaqcLbsacaWGubWcdaWgaaqaaK qzadGaam4DaaWcbeaajugibiabgkHiTiaadsfalmaaBaaabaqcLbma caWGWbGaamiAaaWcbeaaaOGaayjkaiaawMcaaaGaay5waiaaw2faaK qbaoaalaaakeaajugibiabek7aILqbaoaaCaaaleqabaqcLbmacaaI YaaaaaGcbaqcLbsacaaIYaaaaaaakiaawUfacaGLDbaajuaGdaWcaa GcbaqcLbsacqaHYoGyjuaGdaahaaWcbeqaaKqzadGaaGOmaaaaaOqa aKqzGeGaaGOmaaaaaaa@0AE5@      (17)

Variable factor of growth rate β is found for a accepted time by solving (17). The ice thickness is detcrmed by equation (9). The calculation made according equation tube (17) and (9) revealed that cryogenic temperatures on the wall needs taking into account the temperature dependence of water ice layer (Figure 1).

Figure 1 Growth of ice on a flat wall being at cryogenic temperatures and immersed in aqua media (Tw = 278 К, αw = 180 W/(m2 K), Twall = 77 K, radius of the tube r = 0.04 m): 1 – with constant characteristics of ice; 2 – with variable characteristics of ice.

Conclusion

  1. The problem of Protection of sea medium with the help of low temperature engineering is discussed.
  2. It is suggested to set up vertical flexible tubs with cryogenic liquid nitrogen on the surface of the tanks.
  3. Fourier equation for a layer of water ice cylinder being cooled from inside and surrounded by a water medium was solved with a temperature dependence of ice heat conductivity.

Acknowledgements

None

Conflict of interest

The author declares there is no conflict of interest.

References

  1. Dean JW, Timmerhaus KD. Thermal Conductivity Of solid O and O at Low Temperatures. Adv in Cryogenic Engineering. 1963;8:63‒67.
  2. Marinyuk BT, Ugolnikova MA. The Dynamics of ice Formation on a tubs elements ice generators. J of Refrigeration. 2016;12:44–47.
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