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Report, Transport Processes, 2019

The Reynolds Analogy

Mojtaba Afzali Student number 230769

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Preface In this report the focus is on the Reynolds Analogy as assignment of course Transport Process by Professor Lars-Andre Tokheim in faculty of technology of the South-Eastern of Norway university (USN). The course is part of Master of Science Energy and Environmental Technology (EET). I try to explain and describe everything simple for deeper understanding. “ If you can’t explain it simply, you don’t understand it well enough” (Albert Einstein)

Porsgrunn, 01.11.2019 Mojtaba Afzali

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Contents Preface.....................................................................................................................2 Contents...................................................................................................................3 1 Introduction.........................................................................................................4 1.1 two use of the Reynolds Analogy 1.2 The Reynolds Analogy's base 4

4

2 Reynolds Analogy's Restrictions......................................................................7 3 Example...............................................................................................................9 4 Conclusion.........................................................................................................10 References.............................................................................................................11 Appendices............................................................................................................12

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1 Introduction The assignment is about the Reynolds Analogy, where it is popular as a relation between turbulent momentum and heat transfer (Wikipedia Nov 2019).An equation with basic and vital parameters of engineering for velocity, thermal, and concentration boundary layers. The equation describes as below:

Cf 2

St Stm

C Where f is friction coefficient, St and Stm are Stanton number and mass transfer Stanton number respectively.

1.1 Two use of Reynolds Analogy In a turbulent boundary layer for many aerospace vehicle utilizations have turbulent boundary layers therefor we describe the physical mechanism for heat transfer coefficient. The fluid movement close enough to wall is laminar and gentle and shear and molecular conduction are dc   important. The shear stress is equal to dy where  is dynamic viscosity and the heat q  k

flux is

dT dy .

The hydrodynamic theory of heat exchange’s foundation is Reynold’s Analogy in turbulent flow of exitance of common mechanism of transfer of heat and momentum. If this mechanism exists so we can describe a relation between the heat transfer and hydraulic resistance.

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1.2 Reynolds Analogy’s base From table 1.1 we can procure a secondary boundary layer analogy that, the boundary layer equations (1),(2),(3) are exactly identical if we have Pr = Sc = 1 and dp*/dx* = 0 .For the inbound flow in a flat parallel we have no change in free stream velocity outside the boundary layer and also dp*/dx* = 0. With u∞ = V, Equations from (4) to (6) are also identical. So solutions for u*, T*, and CA* in functional forms, must be in same form. From eqations below : Cf 

s 2 u *  V 2 / 2 ReL  y * │y*=0

(10)

2 C f  f ( x*, Re L ) Re L

(11)

hm L C *  A y* │ D AB

(12)

Sh 

y*=0

It follows that Cf

Re L Nu sh 2

(13)

By substituting Nu and with Stanton number (St) and mass transfer (Stm), we have St 

h Nu  Vc P RePr

(14)

Stm 

hm Sh  V Re Sc (15)

Equation (13) can also write as

Cf 2

St St m

(16)

Equation (16) is named the Reynold Analogy that key engineering parameters of the boundary layers such as velocity, thermal and concentration boundary layers can generate from this equation.

The Reynolds Analogy Table 1.1: The boundary layer equations and their y-direction boundary conditions in nondimensional form1 Boundary Conditions Boundary Layer

Conservation Equation

u * * u * dp * 1 2u * v  *  * * x y dx Re L y *2

Velocity

u*

Thermal

u*

T * * T * 1 2T * v  * * x y ReL Pr y *2

Concentration

u*

C A* C * 1 2C A*  v * A*  * x y Re L Sc y *2

Wall

(1)

(2)

(3)

,

Free Stream

u * (x * ) u* ( x* , )  u ( x ,0) 0 , V T * ( x* ,0) 0 ,

*

*

T * ( x *, ) 1

C A* ( x* ,0) 0 , C A* ( x * , ) 1

Similarity Parameters (s)

(4)

VL ReL  v

(7)

(5)

v Re L, Pr  

(8)

v Re L, Sc  D AB

(9)

(6)

1 6

The Reynolds Analogy

2 Reynolds Analogy’s Restrictions If we have velocity parameters, we can use Reynolds Analogy to procure other parameters also if we have other parameters, we can obtain velocity parameters. There are some restrictions in contact to using this result. Besides that, although counting on the validity of boundary conditions estimations the precision of equation (16.1) depends on if Pr = 1 and Sc = 1 and dp*/dx* = 0, modified Reynolds Analogy with addition precise correction can be used. The modification of Reynolds or Chilton -Colburn Analogies have the form like below:

C f St Pr 2/3  jH

(17)

C f  Stm Sc 2/3  jm

(18)

Here, j means Colburn j factors and jH is for heat transfer and jm for mass transfer. If dp*/dx*...