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eISSN: 2576-4500

Aeronautics and Aerospace Open Access Journal

Research Article Volume 2 Issue 3

Mathematical modeling of limiting heat exchange for the account turbulent of a flow at turbulent current in flat channels with unilateral turbulizers

Lobanov IE

Moscow Air Institute, State Technical University, Russia

Correspondence: Lobanov IE, Moscow Air Institute, State Technical University, Russia

Received: May 14, 2018 | Published: June 13, 2018

Citation: Lobanov IE. Mathematical modeling of limiting heat exchange for the account turbulent of a flow at turbulent current in flat channels with unilateral turbulizers. Aeron Aero Open Access J. 2018;2(3):178-182. DOI: 10.15406/aaoaj.2018.02.00048

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Abstract

The theoretical model of calculation is generated and analytical dependences of the limiting heat transfer and hydraulic resistance for a flat channel with two-sided heating are obtained depending on the flow conditions of the coolant and also the boundary conditions of heat exchange, which allows to reveal the maximum reserves of heat exchange intensification in heat exchangers with flat channels for various branches of engineering production.

Keywords: limiting, intensification, heat exchange, turbulence, pipe, mathematical modeling

Introduction

Intensification of heat exchange by turbulence of a flow does not require essential increase of the external sizes of flat channels and consequently is applicable in any flat channels. The manufacturing turbulizers on an outside surface of pipes is not connected to significant technological difficulties. The circuit intensification of heat exchange for the flat channel by means of turbulizers is shown in a Figure 1.

The bottom surface of the flat channel (Figure 1) we shall be by analogy to the ring channel will name conditionally internal, and top-outside. Turbulizers are established on an internal pipe. The modeling of limiting isothermal heat exchange and resistance at turbulent current in flat channels for the account turbulent of a flow is made on a technique to a similar technique applied for round pipes1,2 and ring channels3,4 with turbulizers. At modeling limiting heat exchange for the flat channel, intensification by means of periodically located superficial turbulizers all assumptions used at account of limiting heat exchange for round pipes and ring channels with turbulizers (except for an assumption of a hypothesis about limiting radius of the maximal speed) will be fair. At intensification of heat exchange height of the maximal speed in the flat channel (Figure 1 ), intensive by means of periodically located superficial turbulizers, is displaced in the party of a surface, with smaller factor of resistance.5

The situation of height of the maximal speed is determined with an average error up to 0,5% (maximal divergence has made less than 2%) empirical formula received on the basis of processing of experimental data:5

Y max = y max Н = 1 2 ( ξ ξ sm ) 0,287 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGzb qcfa4aaSbaaSqaaKqzadGaciyBaiaacggacaGG4baaleqaaKqzGeGa eyypa0tcfa4aaSaaaOqaaKqzGeGaamyEaKqbaoaaBaaaleaajugWai Gac2gacaGGHbGaaiiEaaWcbeaaaOqaaKqzGeGaamyheaaacqGH9aqp juaGdaWcaaGcbaqcLbsacaaIXaaakeaajugibiaaikdaaaqcfa4aae WaaOqaaKqbaoaalaaakeaajugibiabe67a4LqbaoaaBaaaleaaaeqa aaGcbaqcLbsacqaH+oaEjuaGdaWgaaWcbaqcLbmacaWGZbGaamyBaa WcbeaaaaaakiaawIcacaGLPaaajuaGdaahaaWcbeqaaKqzadGaaGim aiaacYcacaaIYaGaaGioaiaaiEdaaaqcLbsacaGGSaaaaa@5D30@   (1)

where ( ξ ξ sm ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqbaoaalaaakeaajugibiabe67a4LqbaoaaBaaaleaaaeqaaaGc baqcLbsacqaH+oaEjuaGdaWgaaWcbaqcLbmacaWGZbGaamyBaaWcbe aaaaaakiaawIcacaGLPaaaaaa@420E@ -the relation of hydraulic resistance of surfaces of the flat channel with turbulizers to smooth accordingly.

Factor of limiting hydraulic resistance we shall determine, proceeding from expression for mean expense of speed turbulent of a flow w x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aa0aaaO qaaKqzGeGaam4DaKqbaoaaBaaaleaajugWaiaadIhaaSqabaaaaaaa @3B1A@ :

w x ¯ = 0 1 w x dY= 0 Y max ( w x ) SS dY + Y max 1 ( w x ) TS dY, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aa0aaaO qaaKqzGeGaam4DaKqbaoaaBaaaleaajugWaiaadIhaaSqabaaaaKqz GeGaeyypa0tcfa4aa8qCaOqaaKqzGeGaam4DaKqbaoaaBaaaleaaju gWaiaadIhaaSqabaqcLbsacaWGKbGaamywaiabg2da9aWcbaqcLbma caaIWaaaleaajugWaiaaigdaaKqzGeGaey4kIipajuaGdaWdXbGcba qcfa4aaeWaaOqaaKqzGeGaam4DaKqbaoaaBaaaleaajugWaiaadIha aSqabaaakiaawIcacaGLPaaajuaGdaWgaaWcbaqcLbmacaWGtbGaam 4uaaWcbeaajugibiaadsgacaWGzbaaleaajugWaiaaicdaaSqaaKqz adGaamywaSWaaSbaaWqaaKqzadGaciyBaiaacggacaGG4baameqaaa qcLbsacqGHRiI8aiabgUcaRKqbaoaapehakeaajuaGdaqadaGcbaqc LbsacaWG3bqcfa4aaSbaaSqaaKqzadGaamiEaaWcbeaaaOGaayjkai aawMcaaKqbaoaaBaaaleaajugWaiaadsfacaWGtbaaleqaaKqzGeGa amizaiaadMfacaGGSaaaleaajugWaiaadMfalmaaBaaameaajugWai Gac2gacaGGHbGaaiiEaaadbeaaaSqaaKqzadGaaGymaaqcLbsacqGH RiI8aaaa@7E47@    (2)

where ( w x ) SS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaam4DaKqbaoaaBaaaleaajugWaiaadIhaaSqabaaakiaa wIcacaGLPaaajuaGdaWgaaWcbaqcLbmacaWGtbGaam4uaaWcbeaaaa a@403F@ and ( w x ) ТS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaam4DaKqbaoaaBaaaleaajugWaiaadIhaaSqabaaakiaa wIcacaGLPaaajuaGdaWgaaWcbaqcLbmacaWGIqGaam4uaaWcbeaaaa a@4012@ -axial making speeds of a flow on the part of smooth and turbulent of surfaces accordingly.

Further we shall arrive the same as and for a case of modeling of limiting heat exchange in a round pipe with turbulizers;1,2 the appropriate structures of speeds hypothetically are accepted same, as well as for a case in a pipe. After the appropriate reductions, we shall receive the nonlinear equation for limiting factor of hydraulic resistance of the flat channel:

1= 2κ A 2 Re + 1 2 { ξ 2 B+ ξ 16σ ( 1 4A 2 ξ Re ) 2 }[ 1 8A 2 ξ Re ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaaIXa Gaeyypa0tcfa4aaSaaaOqaaKqzGeGaaGOmaiabeQ7aRjaadgeajuaG daahaaWcbeqaaKqzadGaaGOmaaaaaOqaaKqzGeGaciOuaiaacwgaaa Gaey4kaSscfa4aaSaaaOqaaKqzGeGaaGymaaGcbaqcLbsacaaIYaaa aKqbaoaacmaakeaajuaGdaWcaaGcbaqcfa4aaOaaaOqaaKqzGeGaeq OVdGhaleqaaaGcbaqcfa4aaOaaaOqaaKqzGeGaaGOmaaWcbeaaaaqc LbsacaWGcbGaey4kaSscfa4aaSaaaOqaaKqzGeGaeqOVdGhakeaaju gibiaaigdacaaI2aGaeyyXICTaeq4WdmhaaKqbaoaabmaakeaajugi biaaigdacqGHsisljuaGdaWcaaGcbaqcLbsacaaI0aGaamyqaKqbao aakaaakeaajugibiaaikdaaSqabaaakeaajuaGdaGcaaGcbaqcLbsa cqaH+oaEaSqabaqcLbsaciGGsbGaaiyzaaaaaOGaayjkaiaawMcaaK qbaoaaCaaaleqabaqcLbmacaaIYaaaaaGccaGL7bGaayzFaaqcfa4a amWaaOqaaKqzGeGaaGymaiabgkHiTKqbaoaalaaakeaajugibiaaiI dacaWGbbqcfa4aaOaaaOqaaKqzGeGaaGOmaaWcbeaaaOqaaKqbaoaa kaaakeaajugibiabe67a4bWcbeaajugibiGackfacaGGLbaaaaGcca GLBbGaayzxaaqcLbsacqGHsislaaa@7B36@

1= 2κ A 2 Re + 1 2 { ξ 2 B+ ξ 16σ ( 1 4A 2 ξ Re ) 2 }[ 1 8A 2 ξ Re ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaaIXa Gaeyypa0tcfa4aaSaaaOqaaKqzGeGaaGOmaiabeQ7aRjaadgeajuaG daahaaWcbeqaaKqzadGaaGOmaaaaaOqaaKqzGeGaciOuaiaacwgaaa Gaey4kaSscfa4aaSaaaOqaaKqzGeGaaGymaaGcbaqcLbsacaaIYaaa aKqbaoaacmaakeaajuaGdaWcaaGcbaqcfa4aaOaaaOqaaKqzGeGaeq OVdGhaleqaaaGcbaqcfa4aaOaaaOqaaKqzGeGaaGOmaaWcbeaaaaqc LbsacaWGcbGaey4kaSscfa4aaSaaaOqaaKqzGeGaeqOVdGhakeaaju gibiaaigdacaaI2aGaeyyXICTaeq4WdmhaaKqbaoaabmaakeaajugi biaaigdacqGHsisljuaGdaWcaaGcbaqcLbsacaaI0aGaamyqaKqbao aakaaakeaajugibiaaikdaaSqabaaakeaajuaGdaGcaaGcbaqcLbsa cqaH+oaEaSqabaqcLbsaciGGsbGaaiyzaaaaaOGaayjkaiaawMcaaK qbaoaaCaaaleqabaqcLbmacaaIYaaaaaGccaGL7bGaayzFaaqcfa4a amWaaOqaaKqzGeGaaGymaiabgkHiTKqbaoaalaaakeaajugibiaaiI dacaWGbbqcfa4aaOaaaOqaaKqzGeGaaGOmaaWcbeaaaOqaaKqbaoaa kaaakeaajugibiabe67a4bWcbeaajugibiGackfacaGGLbaaaaGcca GLBbGaayzxaaqcLbsacqGHsislaaa@7B36@     (3)

where A,B,κ,σ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGbb GaaGjbVlaacYcacaaMe8UaaGjbVlaadkeacaGGSaGaaGjbVlaaysW7 cqaH6oWAcaGGSaGaaGjbVlaaysW7cqaHdpWCaaa@4872@ -constants, determinate the same as in:1,2 А-characterizes average border of jet area; В-characterizes factor of proportionality between longitudinal speed and speed of friction (constants А=15¸114 and В=6,5¸8,5 at change Re=104¸105); k=0,443-constant for a universal structure of speed in turbulent a flow; s=0,013-constant for a jet nucleus of a flow describing initial turbulence; Re-number Reynolds on an equivalent diameter of the channel; x-factor of resistance to friction for the flat channel with turbulizers.

It is accepted, that the diameter of the smooth channel is equal to a diameter of the channel carrying ribbing, and the speed of a flow was determined on section of the channel, which would be at absence ribbing. The above-stated approach to the full is lawful, as at a considered type limiting turbulent the rather low ledges are used. In the further comparative analysis will be made on an equivalent diameter of the flat channel: deqv=2∙H.

In Figure 2 the settlement absolute meanings of limiting factor of hydraulic resistance for the flat channel are given depending on number Reynolds, equations, received as a result of the numerical decision, (3), from which it is visible, that to increase of number Reynolds there is a fall of factor of resistance. The important parameter is relative limiting factor of hydraulic resistance for the flat channel x/xsm, shown on a Figure 3; for comparison are given similar given for a round pipe taken from. 1,2 The hydraulic resistance for the smooth flat channel paid off, proceeding from empirical dependences given in.5 From a Figure 3 the maximum of the relation is visible, that x/xsm occurs at higher numbers Reynolds and his meaning a little bit above, than for a round pipe.

Last specifies that the limiting current for a flat pipe with turbulizers has higher relative resistance. The numerical decision of the nonlinear equation (3) for limiting factor of hydraulic resistance for the flat channel allows to determine limiting heat exchange for these conditions. Number Nusselt at the stabilized current for an internal wall Nu1∞ the flat channel with bilateral admission of heat, agrees,6,7 is equal:

Nu 1 = 1 2 0 1 ( 0 Y w x w x ¯ dY1 ) 2 ( 1+ Pr Pr T μ T μ ) dY+ 1 2 q с2 q с1 0 1 { ( 0 Y w x w x ¯ dY1 ) ( 1+ Pr Pr T μ T μ ) [ 0 Y w x w x ¯ dY ] } dY 1 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaqGob GaaeyDaKqbaoaaBaaaleaajugWaiaaigdacqGHEisPaSqabaqcLbsa cqGH9aqpjuaGdaaabaGcbaqcfa4aaSaaaOqaaKqzGeGaaGymaaGcba qcLbsacaaIYaaaaKqbaoaapehakeaajuaGdaWcaaGcbaqcfa4aaeWa aOqaaKqbaoaapehakeaajuaGdaWcaaGcbaqcLbsacaWG3bqcfa4aaS baaSqaaKqzadGaamiEaaWcbeaaaOqaaKqbaoaanaaakeaajugibiaa dEhajuaGdaWgaaWcbaqcLbmacaWG4baaleqaaaaaaaqcLbsacaWGKb GaamywaiabgkHiTiaaigdaaSqaaKqzadGaaGimaaWcbaqcLbmacaWG zbaajugibiabgUIiYdaakiaawIcacaGLPaaajuaGdaahaaWcbeqaaK qzadGaaGOmaaaaaOqaaKqbaoaabmaakeaajugibiaaigdacqGHRaWk juaGdaWcaaGcbaqcLbsaciGGqbGaaiOCaaGcbaqcLbsaciGGqbGaai OCaKqbaoaaBaaaleaajugWaiaadsfaaSqabaaaaKqbaoaalaaakeaa jugibiabeY7aTLqbaoaaBaaaleaajugWaiaadsfaaSqabaaakeaaju gibiabeY7aTbaaaOGaayjkaiaawMcaaaaaaSqaaKqzadGaaGimaaWc baqcLbmacaaIXaaajugibiabgUIiYdGaamizaiaadMfaaOGaayzkJa qcLbsacqGHRaWkjuaGdaaacaGcbaqcfa4aaSaaaOqaaKqzGeGaaGym aaGcbaqcLbsacaaIYaaaaKqbaoaalaaakeaajugibiaadghajuaGda WgaaWcbaqcLbmacaWGbrGaaGOmaaWcbeaaaOqaaKqzGeGaamyCaKqb aoaaBaaaleaajugWaiaadgebcaaIXaaaleqaaaaajuaGdaWdXbGcba qcfa4aaiWaaOqaaKqbaoaalaaakeaajuaGdaqadaGcbaqcfa4aa8qC aOqaaKqbaoaalaaakeaajugibiaadEhajuaGdaWgaaWcbaqcLbmaca WG4baaleqaaaGcbaqcfa4aa0aaaOqaaKqzGeGaam4DaKqbaoaaBaaa leaajugWaiaadIhaaSqabaaaaaaajugibiaadsgacaWGzbGaeyOeI0 IaaGymaaWcbaqcLbmacaaIWaaaleaajugWaiaadMfaaKqzGeGaey4k IipaaOGaayjkaiaawMcaaaqaaKqbaoaabmaakeaajugibiaaigdacq GHRaWkjuaGdaWcaaGcbaqcLbsaciGGqbGaaiOCaaGcbaqcLbsaciGG qbGaaiOCaKqbaoaaBaaaleaajugWaiaadsfaaSqabaaaaKqbaoaala aakeaajugibiabeY7aTLqbaoaaBaaaleaajugWaiaadsfaaSqabaaa keaajugibiabeY7aTbaaaOGaayjkaiaawMcaaaaajuaGdaWadaGcba qcfa4aa8qCaOqaaKqbaoaalaaakeaajugibiaadEhajuaGdaWgaaWc baqcLbmacaWG4baaleqaaaGcbaqcfa4aa0aaaOqaaKqzGeGaam4DaK qbaoaaBaaaleaajugWaiaadIhaaSqabaaaaaaajugibiaadsgacaWG zbaaleaajugWaiaaicdaaSqaaKqzadGaamywaaqcLbsacqGHRiI8aa GccaGLBbGaayzxaaaacaGL7bGaayzFaaaaleaajugWaiaaicdaaSqa aKqzadGaaGymaaqcLbsacqGHRiI8aiaadsgacaWGzbaakiaawQYiaK qbaoaaCaaaleqabaqcLbmacqGHsislcaaIXaaaaKqzGeGaaiilaaaa @E185@   (4)

where Pr MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaciGGqb GaaiOCaaaa@3851@ -number Prandtl; Pr Т MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaciGGqb GaaiOCaKqbaoaaBaaaleaajugWaiaadkcbaSqabaaaaa@3AEF@ -turbulent number Prandtl; μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBaaa@383B@ -Dynamic viscosity; μ Т MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaWcbaqcLbmacaWGIqaaleqaaaaa@3AD9@ -turbulent dynamic viscosity; ( q c2 q c1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqbaoaalaaakeaajugibiaadghajuaGdaWgaaWcbaqcLbmacaWG JbGaaGOmaaWcbeaaaOqaaKqzGeGaamyCaKqbaoaaBaaaleaajugWai aadogacaaIXaaaleqaaaaaaOGaayjkaiaawMcaaaaa@430A@ -the given relation of thermal flows at outside and internal heating accordingly.

Thus, for account of limiting heat exchange in the flat channel with turbulizers it is necessary to determine integrals contained in (4), for all channel. For this purpose the flat channel is broken into three sub-layer from each of the parties, i.e. is simulated hexa-layer by the circuit turbulent of a boundary layer: for smooth and intensification of the parties-viscous sub-layer, intermediate sub-layer, turbulent a nucleus. At account of limiting heat exchange for a round pipe with turbulizers in6,7 use of an assumption was shown, that w x w x ¯ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaam4DaKqbaoaaBaaaleaajugWaiaadIhaaSqabaaakeaa juaGdaqdaaGcbaqcLbsacaWG3bqcfa4aaSbaaSqaaKqzadGaamiEaa WcbeaaaaaaaKqzGeGaeyyrIaKaaGymaaaa@42C4@ , the precisely same conclusion rather insignificantly influences final result of account of heat exchange-it is possible to make and for limiting heat exchange for conditions of the flat channel with turbulizers. Hence, the opportunity of the analytical decision of a task about limiting heat exchange for the flat channel with turbulizers with bilateral admission of heat takes place; according to principle additions, expression for integrals which are included in the right part of expression (4), for each of appropriate sub-layer I i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGjb qcfa4aaSbaaSqaaKqzadGaamyAaaWcbeaaaaa@3A34@ will accept a kind:

Nu 1 = 1 i=1 6 I i ; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaqGob GaaeyDaKqbaoaaBaaaleaajugWaiaabgdacqGHEisPaSqabaqcLbsa cqGH9aqpjuaGdaWcaaGcbaqcLbsacaaIXaaakeaajuaGdaaeWbGcba qcLbsacaWGjbqcfa4aaSbaaSqaaKqzadGaamyAaaWcbeaaaeaajugW aiaadMgacqGH9aqpcaaIXaaaleaajugWaiaaiAdaaKqzGeGaeyyeIu oaaaGaai4oaaaa@4E1B@      (5)

where by an index "i" are designated: 1, 2, 3-viscous sub-layer, intermediate sub-layer, turbulent a nucleus accordingly for internal (which surface intensive) surface of the flat channel; 4, 5, 6-turbulent a nucleus, intermediate sub-layer, viscous sub-layer accordingly for outside (which surface smooth) surface of the flat channel.

After calculation of the given integrals, analytical dependences for I i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGjb qcfa4aaSbaaSqaaKqzadGaamyAaaWcbeaaaaa@3A34@ ( i=1÷6 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaeyiaIiIaaGjbVlaadMgacqGH9aqpcaaIXaGaey49aGRa aGOnaaGccaGLOaGaayzkaaaaaa@40B7@ in (5) after calculation of the given integrals, analytical dependences for:

I 1 = 2 2 Re ξ { 2304 Re 2 ξ ( q c2 q c1 +1 ) 72 2 Re ξ q c2 q c1 144 2 Re ξ +6 }; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGjb qcfa4aaSbaaSqaaKqzGeGaaGymaaWcbeaajugibiabg2da9Kqbaoaa laaakeaajugibiaaikdajuaGdaGcaaGcbaqcLbsacaaIYaaaleqaaa GcbaqcLbsaciGGsbGaaiyzaKqbaoaakaaakeaajugibiabe67a4bWc beaaaaqcfa4aaiWaaOqaaKqbaoaalaaakeaajugibiaaikdacaaIZa GaaGimaiaaisdaaOqaaKqzGeGaciOuaiaacwgajuaGdaahaaWcbeqa aKqzadGaaGOmaaaajugibiabe67a4baajuaGdaqadaGcbaqcfa4aaS aaaOqaaKqzGeGaamyCaKqbaoaaBaaaleaajugWaiaadogacaaIYaaa leqaaaGcbaqcLbsacaWGXbqcfa4aaSbaaSqaaKqzadGaam4yaiaaig daaSqabaaaaKqzGeGaey4kaSIaaGymaaGccaGLOaGaayzkaaqcLbsa cqGHsisljuaGdaWcaaGcbaqcLbsacaaI3aGaaGOmaKqbaoaakaaake aajugibiaaikdaaSqabaaakeaajugibiGackfacaGGLbqcfa4aaOaa aOqaaKqzGeGaeqOVdGhaleqaaaaajuaGdaWcaaGcbaqcLbsacaWGXb qcfa4aaSbaaSqaaKqzadGaam4yaiaaikdaaSqabaaakeaajugibiaa dghajuaGdaWgaaWcbaqcLbmacaWGJbGaaGymaaWcbeaaaaqcLbsacq GHsisljuaGdaWcaaGcbaqcLbsacaaIXaGaaGinaiaaisdajuaGdaGc aaGcbaqcLbsacaaIYaaaleqaaaGcbaqcLbsaciGGsbGaaiyzaKqbao aakaaakeaajugibiabe67a4bWcbeaaaaqcLbsacqGHRaWkcaaI2aaa kiaawUhacaGL9baajugibiaacUdaaaa@880F@    (6)

I 2 = 2 Pr T B 3Reξ( Pr T B+PrA ) [ 32 2 A 3 ξ ( q c2 q c1 +1 )24 A 2 Re12 A 2 Re q c2 q c1 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGjb qcfa4aaSbaaSqaaKqzadGaaGOmaaWcbeaajugibiabg2da9Kqbaoaa laaakeaajugibiaaikdaciGGqbGaaiOCaKqbaoaaBaaaleaajugWai aadsfaaSqabaqcLbsacaWGcbaakeaajugibiaaiodaciGGsbGaaiyz aiabe67a4LqbaoaabmaakeaajugibiGaccfacaGGYbqcfa4aaSbaaS qaaKqzadGaamivaaWcbeaajugibiaadkeacqGHRaWkciGGqbGaaiOC aiaadgeaaOGaayjkaiaawMcaaaaajuaGdaWabaGcbaqcfa4aaSaaaO qaaKqzGeGaaG4maiaaikdajuaGdaGcaaGcbaqcLbsacaaIYaaaleqa aKqzGeGaamyqaKqbaoaaCaaaleqabaqcLbmacaaIZaaaaaGcbaqcfa 4aaOaaaOqaaKqzGeGaeqOVdGhaleqaaaaaaOGaay5waaqcfa4aaeWa aOqaaKqbaoaalaaakeaajugibiaadghajuaGdaWgaaWcbaqcLbmaca WGJbGaaGOmaaWcbeaaaOqaaKqzGeGaamyCaKqbaoaaBaaaleaajugW aiaadogacaaIXaaaleqaaaaajugibiabgUcaRiaaigdaaOGaayjkai aawMcaaKqzGeGaeyOeI0IaaGOmaiaaisdacaWGbbqcfa4aaWbaaSqa beaajugWaiaaikdaaaqcLbsaciGGsbGaaiyzaiabgkHiTiaaigdaca aIYaGaamyqaKqbaoaaCaaaleqabaqcLbmacaaIYaaaaKqzGeGaciOu aiaacwgajuaGdaWcaaGcbaqcLbsacaWGXbqcfa4aaSbaaSqaaKqzad Gaam4yaiaaikdaaSqabaaakeaajugibiaadghajuaGdaWgaaWcbaqc LbmacaWGJbGaaGymaaWcbeaaaaqcLbsacqGHRaWkaaa@8E2D@    (7)

+3 2 A Re 2 ξ 18 2 Re 2 ξ +864Re+432Re q c2 q c1 6912 2 ξ q c2 q c1 6912 2 ξ ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aamGaaO qaaKqzGeGaey4kaSIaaG4maKqbaoaakaaakeaajugibiaaikdaaSqa baqcLbsacaWGbbGaciOuaiaacwgajuaGdaahaaWcbeqaaKqzadGaaG OmaaaajuaGdaGcaaGcbaqcLbsacqaH+oaEaSqabaqcLbsacqGHsisl caaIXaGaaGioaKqbaoaakaaakeaajugibiaaikdaaSqabaqcLbsaci GGsbGaaiyzaKqbaoaaCaaaleqabaqcLbmacaaIYaaaaKqbaoaakaaa keaajugibiabe67a4bWcbeaajugibiabgUcaRiaaiIdacaaI2aGaaG inaiGackfacaGGLbGaey4kaSIaaGinaiaaiodacaaIYaGaciOuaiaa cwgajuaGdaWcaaGcbaqcLbsacaWGXbqcfa4aaSbaaSqaaKqzadGaam 4yaiaaikdaaSqabaaakeaajugibiaadghajuaGdaWgaaWcbaqcLbma caWGJbGaaGymaaWcbeaaaaqcLbsacqGHsisljuaGdaWcaaGcbaqcLb sacaaI2aGaaGyoaiaaigdacaaIYaqcfa4aaOaaaOqaaKqzGeGaaGOm aaWcbeaaaOqaaKqbaoaakaaakeaajugibiabe67a4bWcbeaaaaqcfa 4aaSaaaOqaaKqzGeGaamyCaKqbaoaaBaaaleaajugWaiaadogacaaI YaaaleqaaaGcbaqcLbsacaWGXbqcfa4aaSbaaSqaaKqzadGaam4yai aaigdaaSqabaaaaKqzGeGaeyOeI0scfa4aaSaaaOqaaKqzGeGaaGOn aiaaiMdacaaIXaGaaGOmaKqbaoaakaaakeaajugibiaaikdaaSqaba aakeaajuaGdaGcaaGcbaqcLbsacqaH+oaEaSqabaaaaaGccaGLDbaa aaa@8705@

I 3 + I 4 = ( 1 12 10 Re ξ 2 1 3 A Re ξ 2 )/ ( 1+σRe Pr Pr T ) × MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGjb qcfa4aaSbaaSqaaKqzadGaaG4maaWcbeaajugibiabgUcaRiaadMea juaGdaWgaaWcbaqcLbmacaaI0aaaleqaaKqzGeGaeyypa0tcfa4aaS GbaOqaaKqbaoaabmaakeaajuaGdaWcaaGcbaqcLbsacaaIXaaakeaa jugibiaaigdacaaIYaaaaiabgkHiTKqbaoaalaaakeaajugibiaaig dacaaIWaaakeaajugibiGackfacaGGLbqcfa4aaOaaaOqaaKqzGeGa eqOVdGhaleqaaaaajuaGdaGcaaGcbaqcLbsacaaIYaaaleqaaKqzGe GaeyOeI0scfa4aaSaaaOqaaKqzGeGaaGymaaGcbaqcLbsacaaIZaaa aKqbaoaalaaakeaajugibiaadgeaaOqaaKqzGeGaciOuaiaacwgaju aGdaGcaaGcbaqcLbsacqaH+oaEaSqabaaaaKqbaoaakaaakeaajugi biaaikdaaSqabaaakiaawIcacaGLPaaaaeaajuaGdaqadaGcbaqcLb sacaaIXaGaey4kaSIaeq4WdmNaciOuaiaacwgajuaGdaWcaaGcbaqc LbsaciGGqbGaaiOCaaGcbaqcLbsaciGGqbGaaiOCaKqbaoaaBaaale aajugibiaadsfaaSqabaaaaaGccaGLOaGaayzkaaaaaKqzGeGaey41 aqlaaa@728D@  

× [ 64 A 2 Re 2 ξ ( q c2 q c1 +1 ) 1920A Re 2 ξ ( q c2 q c1 +1 )+ 57600 Re 2 ξ ( q c2 q c1 +1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGHxd aTjuaGdaWabaGcbaqcfa4aaSaaaOqaaKqzGeGaaGOnaiaaisdacaWG bbqcfa4aaWbaaSqabeaajugWaiaaikdaaaaakeaajugibiGackfaca GGLbqcfa4aaWbaaSqabeaajugWaiaaikdaaaqcLbsacqaH+oaEaaaa kiaawUfaaKqbaoaabmaakeaajuaGdaWcaaGcbaqcLbsacaWGXbqcfa 4aaSbaaSqaaKqzadGaam4yaiaaikdaaSqabaaakeaajugibiaadgha juaGdaWgaaWcbaqcLbmacaWGJbGaaGymaaWcbeaaaaqcLbsacqGHRa WkcaaIXaaakiaawIcacaGLPaaajugibiabgkHiTKqbaoaalaaakeaa jugibiaaigdacaaI5aGaaGOmaiaaicdacaWGbbaakeaajugibiGack facaGGLbqcfa4aaWbaaSqabeaajugWaiaaikdaaaqcLbsacqaH+oaE aaqcfa4aaeWaaOqaaKqbaoaalaaakeaajugibiaadghajuaGdaWgaa WcbaqcLbmacaWGJbGaaGOmaaWcbeaaaOqaaKqzGeGaamyCaKqbaoaa BaaaleaajugWaiaadogacaaIXaaaleqaaaaajugibiabgUcaRiaaig daaOGaayjkaiaawMcaaKqzGeGaey4kaSscfa4aaSaaaOqaaKqzGeGa aGynaiaaiEdacaaI2aGaaGimaiaaicdaaOqaaKqzGeGaciOuaiaacw gajuaGdaahaaWcbeqaaKqzadGaaGOmaaaajugibiabe67a4baajuaG daqadaGcbaqcfa4aaSaaaOqaaKqzGeGaamyCaKqbaoaaBaaaleaaju gWaiaadogacaaIYaaaleqaaaGcbaqcLbsacaWGXbqcfa4aaSbaaSqa aKqzadGaam4yaiaaigdaaSqabaaaaKqzGeGaey4kaSIaaGymaaGcca GLOaGaayzkaaqcLbsacqGHsislaaa@9251@  

16A 2 Re ξ 4A 2 Re ξ q c2 q c1 120 2 Re ξ q c2 q c1 + 240 2 Re ξ q c2 q c1 +2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGHsi sljuaGdaWcaaGcbaqcLbsacaaIXaGaaGOnaiaadgeajuaGdaGcaaGc baqcLbsacaaIYaaaleqaaaGcbaqcLbsaciGGsbGaaiyzaKqbaoaaka aakeaajugibiabe67a4bWcbeaaaaqcLbsacqGHsisljuaGdaWcaaGc baqcLbsacaaI0aGaamyqaKqbaoaakaaakeaajugibiaaikdaaSqaba aakeaajugibiGackfacaGGLbqcfa4aaOaaaOqaaKqzGeGaeqOVdGha leqaaaaajuaGdaWcaaGcbaqcLbsacaWGXbqcfa4aaSbaaSqaaKqzad Gaam4yaiaaikdaaSqabaaakeaajugibiaadghajuaGdaWgaaWcbaqc LbmacaWGJbGaaGymaaWcbeaaaaqcLbsacqGHsisljuaGdaWcaaGcba qcLbsacaaIXaGaaGOmaiaaicdajuaGdaGcaaGcbaqcLbsacaaIYaaa leqaaaGcbaqcLbsaciGGsbGaaiyzaKqbaoaakaaakeaajugibiabe6 7a4bWcbeaaaaqcfa4aaSaaaOqaaKqzGeGaamyCaKqbaoaaBaaaleaa jugWaiaadogacaaIYaaaleqaaaGcbaqcLbsacaWGXbqcfa4aaSbaaS qaaKqzadGaam4yaiaaigdaaSqabaaaaKqzGeGaey4kaSscfa4aaSaa aOqaaKqzGeGaaGOmaiaaisdacaaIWaqcfa4aaOaaaOqaaKqzGeGaaG OmaaWcbeaaaOqaaKqzGeGaciOuaiaacwgajuaGdaGcaaGcbaqcLbsa cqaH+oaEaSqabaaaaKqzGeGaeyOeI0scfa4aaSaaaOqaaKqzGeGaam yCaKqbaoaaBaaaleaajugWaiaadogacaaIYaaaleqaaaGcbaqcLbsa caWGXbqcfa4aaSbaaSqaaKqzadGaam4yaiaaigdaaSqabaaaaKqzGe Gaey4kaSIaaGOmaaaa@8AB9@    (8)

I 5 =( 2304/ ( 1+ Pr Pr T A B ) ) 1 Re 2 ξ [ 62 Re 32 ξ ( q c2 q c1 +1 )3 q c2 q c1 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGjb qcfa4aaSbaaSqaaKqzadGaaGynaaWcbeaajugibiabg2da9Kqbaoaa bmaakeaajuaGdaWcgaGcbaqcLbsacaaIYaGaaG4maiaaicdacaaI0a aakeaajuaGdaqadaGcbaqcLbsacaaIXaGaey4kaSscfa4aaSaaaOqa aKqzGeGaciiuaiaackhaaOqaaKqzGeGaciiuaiaackhajuaGdaWgaa WcbaqcLbmacaWGubaaleqaaaaajuaGdaWcaaGcbaqcLbsacaWGbbaa keaajugibiaadkeaaaaakiaawIcacaGLPaaaaaaacaGLOaGaayzkaa qcfa4aaSaaaOqaaKqzGeGaaGymaaGcbaqcLbsaciGGsbGaaiyzaKqb aoaaCaaaleqabaqcLbmacaaIYaaaaKqzGeGaeqOVdGhaaKqbaoaadm aakeaajuaGdaWcaaGcbaqcLbsacaaI2aGaaGOmaaGcbaqcLbsaciGG sbGaaiyzaaaajuaGdaGcaaGcbaqcfa4aaSaaaOqaaKqzGeGaaG4mai aaikdaaOqaaKqzGeGaeqOVdGhaaaWcbeaajuaGdaqadaGcbaqcfa4a aSaaaOqaaKqzGeGaamyCaKqbaoaaBaaaleaajugWaiaadogacaaIYa aaleqaaaGcbaqcLbsacaWGXbqcfa4aaSbaaSqaaKqzadGaam4yaiaa igdaaSqabaaaaKqzGeGaey4kaSIaaGymaaGccaGLOaGaayzkaaqcLb sacqGHsislcaaIZaqcfa4aaSaaaOqaaKqzGeGaamyCaKqbaoaaBaaa leaajugWaiaadogacaaIYaaaleqaaaGcbaqcLbsacaWGXbqcfa4aaS baaSqaaKqzadGaam4yaiaaigdaaSqabaaaaaGccaGLBbGaayzxaaaa aa@84E0@    (9)

I 6 = 4608 2 Re 3 ξ 3 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGjb qcfa4aaSbaaSqaaKqzadGaaGOnaaWcbeaajugibiabg2da9Kqbaoaa laaakeaajugibiaaisdacaaI2aGaaGimaiaaiIdajuaGdaGcaaGcba qcLbsacaaIYaaaleqaaaGcbaqcLbsaciGGsbGaaiyzaKqbaoaaCaaa leqabaqcLbmacaaIZaaaaKqzGeGaeqOVdGxcfa4aaWbaaSqabeaada WcaaqaaKqzadGaaG4maaWcbaqcLbmacaaIYaaaaaaaaaaaaa@4DC7@     (10)

The settlement importance of limiting numbers Nusselt for Pr=0,72 and Pr=10 depending on number Reynolds are given in a Figure 4.

Hence, from the point of view of limiting turbulence greatest intensification takes place for a round pipe with turbulizers, for the flat channel with turbulizers on one of surfaces intensification of heat exchange is lower, it is even lower for the ring channel with turbulizers on an internal wall. After comparison of the theoretical data on limiting heat exchange for the flat channel with the theoretical data for a round pipe,1,2 and also with the ring channel,3,4 it is necessary to proceed to comparison it with the appropriate existing experimental data. Most suitable for the above-stated comparison are the experimental data given in.5 In work,5 it is underlined, that the existing experimental data allow, at saturation intensification of heat exchange occurring in a case, when turbulizers fall outside the limits laminar and transitive layers, to reach meanings of growth heat irradiation in 2¸2,8 times at growth of hydraulic resistance in 3,35¸6 times,5 while the theoretical importance of limiting heat exchange received on a technique, developed within the framework of the given work, is equal 2,9. Hence, the theoretical data well correlating with experimental, specify that at the given method intensification of heat exchange his reserves are revealed almost all. In a Figure 6 relative limiting heat exchanges Nu  1 /Nu 1sm MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaqGob GaaeyDaiaabccajuaGdaWgaaWcbaqcLbmacaqGXaGaaGPaVlabg6Hi LcWcbeaajugibiaab+cacaqGobGaaeyDaKqbaoaaBaaaleaajugWai aaykW7caqGXaGaaCjaVlaaykW7cqGHEisPcaaMc8Uaae4Caiaab2ga aSqabaaaaa@4DC5@ at bilateral heating for flat channel and for ring channel with R1 with other things being equal (Re=104; Pr=0,72) is resulted depending on the attitude of thermal flows at outside and internal heating accordingly ( q c2 q c1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqbaoaalaaakeaajugibiaadghajuaGdaWgaaWcbaqcLbmacaWG JbGaaGOmaaWcbeaaaOqaaKqzGeGaamyCaKqbaoaaBaaaleaajugWai aadogacaaIXaaaleqaaaaaaOGaayjkaiaawMcaaaaa@430A@ . From a Figure 7 the heating of an outside wall is clearly visible, that ( q c2 q c1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqbaoaalaaakeaajugibiaadghajuaGdaWgaaWcbaqcLbmacaWG JbGaaGOmaaWcbeaaaOqaaKqzGeGaamyCaKqbaoaaBaaaleaajugWai aadogacaaIXaaaleqaaaaajugibiabggziTcGccaGLOaGaayzkaaaa aa@4584@ essentially reduces relative limiting heat exchange Nu  1 /Nu 1sm MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaqGob GaaeyDaiaabccajuaGdaWgaaWcbaqcLbmacaqGXaGaaGPaVlabg6Hi LcWcbeaajugibiaab+cacaqGobGaaeyDaKqbaoaaBaaaleaajugWai aaykW7caqGXaGaaCjaVlaaykW7cqGHEisPcaaMc8Uaae4Caiaab2ga aSqabaaaaa@4DC5@ on an internal surface, i.e. the surfaces, on which are located turbulizers, both for the flat channel, and for the ring channel. Decrease of limiting relative heat exchangefor flat channel atoccurs a little bit more poorly, than for the ring channel, therefore in this respect flat channel more preferably ring. Hence, maximal limiting heat exchange Nu  1 /Nu 1sm MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaqGob GaaeyDaiaabccajuaGdaWgaaWcbaqcLbmacaqGXaGaaGPaVlabg6Hi LcWcbeaajugibiaab+cacaqGobGaaeyDaKqbaoaaBaaaleaajugWai aaykW7caqGXaGaaCjaVlaaykW7cqGHEisPcaaMc8Uaae4Caiaab2ga aSqabaaaaa@4DC5@ for the flat channel ( q c2 q c1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqbaoaalaaakeaajugibiaadghajuaGdaWgaaWcbaqcLbmacaWG JbGaaGOmaaWcbeaaaOqaaKqzGeGaamyCaKqbaoaaBaaaleaajugWai aadogacaaIXaaaleqaaaaajugibiabggziTcGccaGLOaGaayzkaaaa aa@4584@ can be received at unilateral heating ( q c2 q c1 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqbaoaalaaakeaajugibiaadghajuaGdaWgaaWcbaqcLbmacaWG JbGaaGOmaaWcbeaaaOqaaKqzGeGaamyCaKqbaoaaBaaaleaajugWai aadogacaaIXaaaleqaaaaajugibiabgkziUkaaicdaaOGaayjkaiaa wMcaaaaa@4640@ . In the given research the cases with negative importance are not considered ( q c2 q c1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqbaoaalaaakeaajugibiaadghajuaGdaWgaaWcbaqcLbmacaWG JbGaaGOmaaWcbeaaaOqaaKqzGeGaamyCaKqbaoaaBaaaleaajugWai aadogacaaIXaaaleqaaaaaaOGaayjkaiaawMcaaaaa@430A@ .

After consideration of influence on limiting heat exchange in the flat channel with turbulizers on an internal surface of a mode of current of the heat-carrier (Figure 5), boundary conditions of heat exchange (Figure 6), is necessary to consider his(its) dependence on Prandtl number. In a Figure 77 the dependence of limiting heat exchange in the flat channel with turbulizers on an internal surface is given at heating only of internal wall Nu  11 /Nu 11 sm MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaqGob GaaeyDaiaabccajuaGdaWgaaWcbaqcLbmacaqGXaGaaeymaiabg6Hi LcWcbeaajugibiaab+cacaqGobGaaeyDaSWaaSbaaeaajugWaiaabg dacaqGXaGaeyOhIukaleqaamaaBaaabaqcLbmacaqGZbGaaeyBaaWc beaaaaa@4845@ depending on Prandtl number at the fixed number Reynolds Re=104; the settlement data for other numbers Reynolds have similar character.

For comparison the appropriate data on relative limiting heat exchange are given Nu  11 /Nu 11 sm MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaqGob GaaeyDaiaabccajuaGdaWgaaWcbaqcLbmacaqGXaGaaeymaiabg6Hi LcWcbeaajugibiaab+cacaqGobGaaeyDaSWaaSbaaeaajugWaiaabg dacaqGXaGaeyOhIukaleqaamaaBaaabaqcLbmacaqGZbGaaeyBaaWc beaaaaa@4845@ for ring channel with turbulizers on internal pipe with R1 and for a round pipe with turbulizers (Nu  /Nu sm ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaqGOa GaaeOtaiaabwhacaqGGaqcfa4aaSbaaSqaaKqzadGaeyOhIukaleqa aKqzGeGaae4laiaab6eacaqG1bqcfa4aaSbaaSqaaKqzadGaeyOhIu Qaae4Caiaab2gaaSqabaqcLbsacaqGPaaaaa@468F@ with other things being equal (Re=104). The analysis of the data on limiting heat exchange depending on Prandtl number, submitted on a Figure 7, shows, that relative limiting heat exchange Nu  11 /Nu 11 sm MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaqGob GaaeyDaiaabccajuaGdaWgaaWcbaqcLbmacaqGXaGaaeymaiabg6Hi LcWcbeaajugibiaab+cacaqGobGaaeyDaSWaaSbaaeaajugWaiaabg dacaqGXaGaeyOhIukaleqaamaaBaaabaqcLbmacaqGZbGaaeyBaaWc beaaaaa@4845@ the analysis of the data on limiting heat exchange depending on number Prandtl, submitted on a Figure 7, shows, that the relative limiting heat exchange for the flat channel with turbulizers is rather sharply reduced with increase of number Prandtl. The decrease of relative limiting heat exchange is obvious, that Nu  11 /Nu 11 sm MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaqGob GaaeyDaiaabccajuaGdaWgaaWcbaqcLbmacaqGXaGaaeymaiabg6Hi LcWcbeaajugibiaab+cacaqGobGaaeyDaSWaaSbaaeaajugWaiaabg dacaqGXaGaeyOhIukaleqaamaaBaaabaqcLbmacaqGZbGaaeyBaaWc beaaaaa@4845@ for the flat channel at increase of number Prandtl occurs much faster, than for a round pipe and practically the same as for the ring channel with turbulizers on an internal pipe. The greatest meanings of limiting relative heat exchange for the flat channel, as well as in case of limiting heat exchange for a round pipe, are in area gasiform of heat-carriers. The reduction of relative limiting heat exchange is obvious, that b Nu  11 /Nu 11 sm MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaqGob GaaeyDaiaabccajuaGdaWgaaWcbaqcLbmacaqGXaGaaeymaiabg6Hi LcWcbeaajugibiaab+cacaqGobGaaeyDaSWaaSbaaeaajugWaiaabg dacaqGXaGaeyOhIukaleqaamaaBaaabaqcLbmacaqGZbGaaeyBaaWc beaaaaa@4845@ for the flat channel at increase of number Prandtl occurs much faster, than for a round pipe and practically the same as for the ring channel with turbulizers on an internal pipe. Hence, the round pipe with turbulizers has advantage for intensification of heat exchange at higher numbers Prandtl not only above ring channels with turbulizers on an internal pipe,3,4 but also above flat channels with turbulizers on an internal surface. The important parameter of dependence limiting intensification of heat exchange for various channels from number Prandtl is such this number, at which limiting intensification becomes to equal unit.

For example, ( Nu /Nu sm )| max =1,00 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaqGaaO qaaKqbaoaabmaakeaajugibiaab6eacaqG1bqcfa4aaSbaaSqaaKqz adGaeyOhIukaleqaaKqzGeGaae4laiaab6eacaqG1bqcfa4aaSbaaS qaaKqzadGaeyOhIuQaae4Caiaab2gaaSqabaaakiaawIcacaGLPaaa aiaawIa7aKqbaoaaBaaaleaajugWaiGac2gacaGGHbGaaiiEaaWcbe aajugibiabg2da9iaaigdacaGGSaGaaGimaiaaicdaaaa@519A@ at Pr=85,6 (at use for account Nu sm MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaqGob GaaeyDaKqbaoaaBaaaleaajugWaiabg6HiLkaabohacaqGTbaaleqa aaaa@3D98@ the formulas Dittus–Bolter). Relevant it is necessary to note, that meanings ( Nu 11 /Nu 11sm )| max MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaqGaaO qaaKqbaoaabmaakeaajugibiaab6eacaqG1bqcfa4aaSbaaSqaaKqz adGaaeymaiaabgdacqGHEisPaSqabaqcLbsacaqGVaGaaeOtaiaabw hajuaGdaWgaaWcbaqcLbmacaqGXaGaaeymaiabg6HiLkaabohacaqG TbaaleqaaaGccaGLOaGaayzkaaaacaGLiWoajuaGdaWgaaWcbaqcLb maciGGTbGaaiyyaiaacIhaaSqabaaaaa@4FF6@ and ( Nu /Nu sm )| max MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaqGaaO qaaKqbaoaabmaakeaajugibiaab6eacaqG1bqcfa4aaSbaaSqaaKqz adGaeyOhIukaleqaaKqzGeGaae4laiaab6eacaqG1bqcfa4aaSbaaS qaaKqzadGaeyOhIuQaae4Caiaab2gaaSqabaaakiaawIcacaGLPaaa aiaawIa7aKqbaoaaBaaaleaajugWaiGac2gacaGGHbGaaiiEaaWcbe aaaaa@4D26@ are realized at various numbers Reynolds for different channels.

For the ring channel with turbulizers on an internal pipeis ( Nu 11 /Nu 11sm )| max =1,00 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaqGaaO qaaKqbaoaabmaakeaajugibiaab6eacaqG1bqcfa4aaSbaaSqaaKqz adGaaeymaiaabgdacqGHEisPaSqabaqcLbsacaqGVaGaaeOtaiaabw hajuaGdaWgaaWcbaqcLbmacaqGXaGaaeymaiabg6HiLkaabohacaqG TbaaleqaaaGccaGLOaGaayzkaaaacaGLiWoajuaGdaWgaaWcbaqcLb maciGGTbGaaiyyaiaacIhaaSqabaqcLbsacqGH9aqpcaaIXaGaaiil aiaaicdacaaIWaaaaa@546A@ reached at Pr=11,6 for R1; at Pr=7,9 for R1 and at Pr=7,5 for R1. For the flat channel with turbulizers on an internal surface at unilateral heating ( Nu /Nu sm )| max =1,00 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaqGaaO qaaKqbaoaabmaakeaajugibiaab6eacaqG1bqcfa4aaSbaaSqaaKqz adGaeyOhIukaleqaaKqzGeGaae4laiaab6eacaqG1bqcfa4aaSbaaS qaaKqzadGaeyOhIuQaae4Caiaab2gaaSqabaaakiaawIcacaGLPaaa aiaawIa7aKqbaoaaBaaaleaajugWaiGac2gacaGGHbGaaiiEaaWcbe aajugibiabg2da9iaaigdacaGGSaGaaGimaiaaicdaaaa@519A@ is achieved at Pr=17,3. Hence, for flat channels limiting intensification of heat exchange is possible at higher numbers Prandtl, than for ring channels, but also on this parameter they as concede to round pipes. In works1,2 was theoretically proved, that for round pipes with turbulizers in case of limiting heat exchange in the certain area of numbers Reynolds Re( 1÷2 ) 10 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaciGGsb GaaiyzaiabgIKi7MqbaoaabmaakeaajugibiaaigdacqGH3daUcaaI YaaakiaawIcacaGLPaaajugibiabgwSixlaaigdacaaIWaqcfa4aaW baaSqabeaajugWaiaaisdaaaaaaa@4758@ carry of heat prevails above carry of a pulse, and then, down to Re5 10 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaciGGsb GaaiyzaiabgIKi7kaaiwdacqGHflY1caaIXaGaaGimaKqbaoaaCaaa leqabaqcLbmacaaI0aaaaaaa@411C@ , the carry of heat becomes smaller, than carry of a pulse; then the distinction between them is reduced.1,2 The developed above technique of account of limiting heat exchange and hydraulic resistance for flat channels with turbulizers allows to construct for them similar law.

The specified law for air is reflected in a Figure 8, where the dependence is shown:

Ξ(Re)=: Nu 11  /Nu 11 sm ξ / ξ sm ( Re ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqqHEo awcaGGOaGaciOuaiaacwgacaGGPaGaeyypa0JaaiOoaKqbaoaalaaa keaajugibiaab6eacaqG1bqcfa4aaSbaaSqaaKqzadGaaeymaiaabg dacqGHEisPaSqabaqcLbsacaqGGaGaae4laiaab6eacaqG1bWcdaWg aaqaaKqzadGaaeymaiaabgdacqGHEisPaSqabaWaaSbaaeaajugWai aabohacaqGTbaaleqaaaGcbaqcLbsacqaH+oaEcaqGGaGaae4laiab e67a4LqbaoaaBaaaleaajugWaiaabohacaqGTbaaleqaaaaajuaGda qadaGcbaqcLbsaciGGsbGaaiyzaaGccaGLOaGaayzkaaaaaa@5DA6@     (11)

For comparison are given appropriate given for a round pipe with turbulizers.1,2 From a Figure 8 parameter is visible, that Ξ for the flat channel makes much smaller size, than for a round pipe.

Last specifies that the round pipe has advantage from the point of view of limiting intensification of heat exchange in comparison with the flat channel. Besides for flat channel already is not present areas with prevalence of carry of heat above carry of a pulse, and also is absent area of decreases of prevalence of carry of a pulse above carry of heat (at approximation to Re 10 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaciGGsb GaaiyzaiabgIKi7kaaigdacaaIWaqcfa4aaWbaaSqabeaajugWaiaa iwdaaaaaaa@3E14@ importance Ξ only are somewhat stabilized), that is characteristic for a round pipe. Last is an additional reason for a reduction of the flat channel in relation to a round pipe. Last specifies that the round pipe has advantage from the point of view of limiting intensification of heat exchange in comparison with the flat channel. Besides for the flat channel already there are no areas with prevalence of carry of heat over carry of a pulse, and also there is no area of reduction of prevalence of carry of a pulse above carry of heat (at approximation to Re 10 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaciGGsb GaaiyzaiabgIKi7kaaigdacaaIWaqcfa4aaWbaaSqabeaajugWaiaa iwdaaaaaaa@3E14@ importance Ξ only are somewhat stabilized), that is characteristic for a round pipe. Last is an additional reason for a reduction of the flat channel in relation to a round pipe. The urgency of a problem intensification of heat exchange in modern metallurgical manufacture, was most in detail shown in the monographies,8‒13 where it was specified on advantage of theoretical research intensification of heat exchange above experimental. The theoretical research intensification of heat exchange in heat exchangers for metallurgical manufacture is based on available development in this area,8‒13 which successfully proceed now.

Figure 1 Cross section of the flat channel with turbulizers.

Figure 2 Absolute meanings of limiting factor of hydraulic resistance for the flat channel depending on number Reynolds.

Figure 3 Meanings of the relations of limiting factor of hydraulic resistance for the flat channel to the appropriate smooth channel x/xsm depending on number Reynolds (for comparison the similar ratio for a round pipe) are given.

Figure 4 Absolute meanings of limiting numbers Nusselt for Pr=0,72 and Pr=10 for the flat channel depending on number Reynolds.

Figure 5 Settlement meanings of limiting relative heat exchange N u 11 /N u 11sm MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGob GaamyDaKqbaoaaBaaaleaajugWaiaaigdacaaIXaGaeyOhIukaleqa aKqzGeGaai4laiaad6eacaWG1bqcfa4aaSbaaeaajugWaiaabgdaca qGXaGaaGPaVlabg6HiLkaaykW7caqGZbGaaeyBaaqcfayabaaaaa@4A7B@ for air in the flat channel with turbulizers on an internal surface at heating only of internal wall depending on Re.

Figure 6 Relative limiting heat exchange N u 11 /N u 11sm MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGob GaamyDaKqbaoaaBaaaleaajugWaiaaigdacaaIXaGaeyOhIukaleqa aKqzGeGaai4laiaad6eacaWG1bqcfa4aaSbaaeaajugWaiaabgdaca qGXaGaaGPaVlabg6HiLkaaykW7caqGZbGaaeyBaaqcfayabaaaaa@4A7B@ at bilateral heating for flat channel and for ring channel with R1=½ with other things being equal (Re=104; Pr=0.72) depending on the attitude of thermal flows at outside and internal heating accordingly ( q c2 q c1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aadaWcaaqaaiaadghadaWgaaqaaKqzadGaam4yaiaaikdaaKqbagqa aaqaaiaadghadaWgaaqaaKqzadGaam4yaiaaigdaaKqbagqaaaaaai aawIcacaGLPaaaaaa@410A@ .

Figure 7 Dependence of limiting heat exchange in the flat channel with turbulizers on an internal surface at heating only of internal wall N u 11 /N u 11sm MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGob GaamyDaKqbaoaaBaaaleaajugWaiaaigdacaaIXaGaeyOhIukaleqa aKqzGeGaai4laiaad6eacaWG1bqcfa4aaSbaaeaajugWaiaabgdaca qGXaGaaGPaVlabg6HiLkaaykW7caqGZbGaaeyBaaqcfayabaaaaa@4A7B@ from Prandtl number (for comparison are given appropriate given for a round pipe and for the ring channel with R1=½).

Figure 8 Dependence of parameter X at Re=104 and Pr=0,72 for the flat channel with turbulizers on an internal surface at unilateral heating (for comparison are given similar given for a round pipe with turbulizers).

Conclusion

  1. The theoretical model of account is developed and the analytical dependences of limiting heat exchange and hydraulic resistance for the flat channel with bilateral heating are received depending on modes of current of the heat-carrier, and also boundary conditions of heat exchange. Intensification is received by means of periodically located superficial turbulizers on an internal surface.
  2. The importance of relative limiting heat exchange for flat channels are a little bit higher, than ring and is always lower, than for a round pipe.
  3. For flat channels the limiting carry of heat cannot prevail above carry of a pulse, while for round pipes in the certain area of numbers Reynolds it is possible.
  4. The generated theory allows revealing the maximal reserves intensification of heat exchange in ceramic heat exchangers with flat channels for modern metallurgical manufacture.

Acknowledgements

None.

Conflicts of Interest

The author declares that there is no conflict of interest.

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