computation of optimal structural and technical … 4/issue 1/ijeit1412201407_47.pdf · computation...

ISSN: 2277-3754

ISO 9001:2008 Certified International Journal of Engineering and Innovative Technology (IJEIT)

Volume 4, Issue 1, July 2014

Computation of Optimal Structural and Technical Parameters of Solar Dryer

K. M. Khazimov; G. C. Bora; B. A. Urmashev; M. Z. Khazimov; Z. M. Khazimov

ABSTRACT: -determination of the geometrical measurement and technological parameters of a heliodryer are very important. This study engaged computational modeling using known mathematical relationship and new boundary statements of the problem in relation to the object. The movement of air was acting as drying agent in the dryer and the common heat conductivity equation was considered. According to the presented algorithm in this study, the differential equation was solved with taking into account dimensionless magnitudes (Grasgof, Reynolds and Prandtl).The results were identified as is olines, the function of an isotherm and an isobar current were found at various values of Grasgof, Reynolds and Prandtl numbers. Most efficient heat exchanges zones were recognized by various Nusselt's various values. The effective zone of drying was described graphically ina vicinity of the camera of the dryer, where the passive zone near site of the dryer camera was identified and it was presented also as a graph.

KEYWORDS: Computational Modeling, Grashof

Numbers, Nusselt numbers, Optimization, Solar Drying.

I. INTRODUCTION Computational modeling for justification of

technological and structural parameters of solar dryer allows reducing costs of experimental studies during its design. The laws of transfer of energy and mass in wet materials in the process of dehydration are very complex and have not been studied at sufficient level. Main provisions of the theories of drying were developed by A.V.Lykov, P.A.Rebinder, A.S.Ginzburg, V.V Krasnikov and other researchers [1- 4]. Wet material subjects to drying represents multiphase and multi component environment. During the passage of drying agent the complex process takes place which is accompanies by heat and mass transfer between different phases and components of the system. Depending on the set task, the system is studied at different levels of complexity. In accordance with the technology, regularity of heat and mass exchange in the system of solar dryer - wet material - the environment must be described in the appropriate equations, qualitatively and quantitatively satisfying the real process. There are many mathematical models of free convection. The case of thermo-gravitational convection has been most studied, when the equations of fluid motion are solved together with the heat equation. Changing of density is taking into consideration in the equations of motion through the approximation of Boussinesk.During last three decades a considerable amount of publications were devoted to the research of free (natural) convection. Such interest to a phenomenon

of thermal convection is mainly explained by its important role in various processes, such as air flow in the street canyons, a cooling tower, spread of smoke and fire in building, movement of blood in vessels, operation of solar collectors etc.The objective of this study is to find out the optimum technological parameters and structural measurements of a heliodryer.

II. MATERIALS AND METHODS The proposed construction of solar dryer relates to

mine types, which includes air heating element from solar energy, vertical drying chamber, heat accumulator, air passage (Figure 1a&b). The device is made with possibility of rotat able by means of supporting wheels 12, which are arranged at the basis of the device. On the outer surface of the drying chamber 6 there is coating which is made of heat isolating material. The proposed system works like this. In the daytime, sun rays pass through the screen 8, through a layer of greenhouse protection, heat the air in the heater of drying chamber 10 and heat accumulator 11. Warm air enters the drying chamber 6, which is used for drying of raw material, and the exhaust air enters the air ductwork 3, and is expelled out of the umbrella 1.

There are many mathematical models of free convection. The most studied is the case of thermo-gravitational convection, when the equations of fluid motion are solved together with thermal conductivity equation. Changing of density is taken into consideration in the equations of the motion through the approximation of Boussinesk. The system of equations of free convection in the Boussinesk approximation is as follows [5].

,])([ ρθβµρ gupuutu

−∆=∇+∇+∂∂

(1)

,0 =udiv (2)

,])([ θχθθρ ∆=∇+∂∂ u

tс (3)

The system of equations (1) - (3) takes into account the assumption of Boussinesk, which applies to the density, assuming that the change of density() can be taken into account only for. Practice shows that this system is a good description of a wide range of free convection. The solution of (1) - (3) depends on initial and boundary conditions.

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Fig 1 (a) – Schematic of the drying device

1 – Umbrella; 2 – ejector; 3 – dryer chamber air transfer; 4 – shelves for raw material; 5 – Ejector air transfer; 6 – dryer chamber; 7 – perforated damper; 8 – screen; 9 –ejector helio air Heater; 10 –drying chamber helio air heater; 11 – thermal accumulator; 12 – bearing wheels

Fig 1 (b) – General view of drying device

1 2 3

4 5

7

8 9

12 11

6

10

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We note that most often the literature highlights the problems in closed areas which contain solid boundaries with conditions of viscous adhesion [6,7] and flow in channels, in the input and output are given the values of the velocity component [8]. But in mathematical modeling of the flows of viscous fluid often there are problems in which normal velocity components in border sections of considered areas are not specified, i.e., it is impossible to determine the flow rate through the input and output section. Typical examples of such problems are given, for example, in works [8-12]. In [8] viscous fluid flow in a circular cylindrical tube of variable cross section is considered. The liquid moves under the action of pressure drop which is applied to the ends of the pipe. As an example there are solutions for stationary and harmonic pressure drop. [9] Considers the hydrodynamics of periodic fluid flow with an inverted symmetry in finned channel. (As the coefficients of heat transfer of smooth channels are small, to increase heat irradiation of contact thermal exchangers, usually edges on hard surfaces of channel are used which increase the heat transfer area and improve mixing). In the cited paper [9] the flow is considered as laminar, the pressure for the periodic fully developed flow is presented in the form:

p(x,y)=-βx+p(x,y), Where - a constant component of total pressure gradient. In numerical solution the different values can be iteratively picked so that solution will match the desired flow rate or Reynolds number. The Reynolds number is defined by the expression

,Reµ

ρ HDu=

Where - ,2 ,1 0

HHDudyH

u H

H

== ∫ рсchannelwidth.

The solution for the desired value Re is sought as follows. For a given valuethe velocity field is calculated and then u . On this value u the viscosity coefficient is selected which ensures constant Re. In work [10] numerically studied a viscous incompressible fluid in variables "current function-vorticity" in the area shown in Figure 1. Formulation of the boundary values on the input (AB) is defined in such a way that motion of fluid is determined only by heat convection. It is assumed that in input:

,0 ),( ,0 v 000 =∂∂

===yTor(y)TTyuuu

whereu0 (y), T0(y) - given functions, u0 = const which must be defined. That is at the input section of the considered area only the shape of profile of the normal component of velocity vector is given. To determine the constants u0 it is additionally required that a difference of pressure in two points must be given:

pBPAP ∆=− )()(

where p∆ - given value. In the future, consider the convective flow of a viscous incompressible fluid in the region shown in Figure 1 with given constant inlet pressure (AB) and output (DО′) and present a discussion of numerical calculations performed by algorithms proposed in this paper to solve the auxiliary grid of equations of Navier-Stokes. Further we will consider convective flow of a viscous incompressible fluid in the area shown in Figure 2 with given constant inlet values in input (AB) and output (DО ) and present a discussion of numerical calculations performed on algorithms proposed in this work for solution of supplement net equations Navie-Stocks.

Fig 2- Calculation area

III. SETTING OF THE PROBLEM AND BASIC

DIFFERENTIAL CORRELATIONS In two-dimensional domain , shown in Figure 2, we

consider the system of equations (1) - (3) and consider the following edge conditions: on top of solid wall (BC, CD)

u=v=0, θ =Т1=const; (4)

on lower wall (АО): u=v=0, θ =Т2=const; (5)

on input edge (АВ):

0 , ,0 v 2 =∂∂

==x

Pp θ; (6)

On output edge (DO ):

0 , ,0 1 =∂∂

==y

Ppu θ , (7)

In the future, we are interested in stationary flow regimes corresponding to the equations (7) and (9), existence and uniqueness of which are assumed. For doing numerical calculations of differential equations (5) and (7) can be written the following dimensionless form

,ReRe

1)( 2 Tg

gGrupuutu

−∆=∇+∇+∂∂ (8)

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,0 =udiv (9)

,RePr1)( TTu

tT

∆=∇+∂∂ (10)

Where ),,0( gg −=

−∆

= 2

3

υθβ LgGr Grash of number,

μpL ∆

=ρ

Re Raynolds number

λυ

=Pr -Prandtle number,

- typical difference of temperatures, v – cinematic coefficient of viscosity, - coefficient of temperature conductivity. Edge conditions corresponding to the expressions (7) - (9) in this case are as follows: on top of solid wall (BC, CD)

u=v=0, T=0; (11)

on low wall (АА ): u=v=0, T=1; (12)

on input edge (АВ):

;0 ,1 ,0 v =∂∂

==xTp , (13)

On output edge (DO):

0 ,0 ,0 =∂∂

==yTpu . (14)

IV. SOLUTION ALGORITHMS

To solve this problem, consider the following iterative process, which in differential level can be written as follows (Figure 3).

),( ,2/1

′==−+

QGRRuu nn

τ (15)

,)( 2/111 +++ =−+ nnnh

n uppgradu τ (16)

,01 =+nhudiv (17)

,)( 111

PrRe

1 +++

∆=+− n

hnn

h

nn

TTuLTT

τ(18)

which for internal nodes of final- difference grids in index form is as follows

,,2/1,2/1

2/1,2/1

mk

nmk

nmk G

uu+

+++ =

−

τ (19)

,2/1,2/1,

2/12/1,

++

++ =

−mk

nmk

nmk Q

vvτ

(20)

),,,1(1

2/1,2/1)1

,1,1(

11

,2/1n

mkpnmkp

hn

mkunmkpn

mkph

nmku −+++

+=+−++++

+ττ

(21)

),,1,(2

2/12/1,)1

,1

1,(2

12/1,

nmkpn

mkph

nmkvn

mkpnmkp

hn

mkv −++++=+−+

++++

ττ(22)

,2

12/1,

12/1,

1

1,2/1

1,2/1

h

nmkvn

mkv

h

nmkun

mku +−−+

+=

+−−+

+(23)

,1

,PrRe11

,)()(,

1, +

∆=+

+−+

n

mkTh

n

mkTn

uT

hL

nmkTn

mkT

τ (24) where

,,2/1Re1

1

,,1,2/1)()(

,2/1n

mkuhh

nmkpn

mkpn

mkunuuhLn

mkG +∆+−+−+−=+

,),1,(2Re22/1,Re1

2

,1,2/1,)(

)(

2/1,

n

mkTn

mkTGrn

mkvh

h

nmkpn

mkpn

mkvn

uv

hLn

mkQ

−+++∆+

+−+

−+−=+

( ) ,,2/1)()(2,)()(

1,,2/1)()( n

mkunuuhLnuu

hLn

mkun

uu

hL ++=+

( )( ),1,2/1

)(2/1,1,2/1

)(2/1,22

1,2/1)(

)(

2,

,,2/1)(

,2/1,2/3)(

,2/1121

,2/1)()(

1,

nmkuu

mkbnmkuu

mkbhn

mkun

uu

hL

nmkuu

mkanmkuu

mkahn

mkun

uu

hL

−+−−+++=+

−−−++=+

(25)

( )( ),2/1,2/1,15.0

)(

2/1,

,,2/1,2/35.0)(

,2/1

nmkvn

mkvu

mkb

nmkun

mkuu

mka

++++=+

+++=+

( ) ,2/1,)()(2,)()(

1,2/1,)()( nmkvnuv

hLnuvhLn

mkvnuvhL ++=+

( )( ),2/1,

)(2/1,2/3,

)(2/1,22

12/1,)(

)(

2,

,2/1,1)(

,2/12/1,1)(

,2/1121

2/1,)()(1,

nmkvv

mkbnmkvv

mkbhn

mkvn

uv

hL

nmkvv

mkanmkvv

mkah

nmkvnuv

hL

−−−++=+

+−−−+++=+

(26)

( )( ),2/1,2/3,5.0

)(

2/1,

,,2/11,2/15.0)(

,2/1

nmkvn

mkvv

mkb

nmkun

mkuv

mka

+++=+

++++=+

( ) ,,)()(2,)()(

1,,)()( nmkTnuT

hLnuThLn

mkTnuThL

+=

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( )( ).1,,(2/1,),1,(2/1,22

1,)(

)(

2,

,),1,(,2/1),,1(,2/1121

,)()(

1,

nmkTn

mkTnmkvn

mkTnmkTn

mkvhn

mkTn

uT

hL

nmkTn

mkTnmkun

mkTnmkTn

mkuhn

mkTn

uT

hL

−−−+−++=

−−−+−++=

(27)

on the wall (ОО′):

,1T,0u ,0

,

,2/12/1,

=

== −+

mN

mNmNv(28)-(30)

to find the values of flow parameters on the symmetry axis (PA ) we have corresponding Formula as that will be presented. Edge conditions (11) - (14) for the grid values of unknown functions u, p,T are as follows:

on hard wall (ВС): 1,1 ,2 kkkm ==

;02/12, ,02, ,02,2/1 =−==+ kkvkkTkku (31) on hard wall (СD): 2,2m ,1 Nkkk ==

;0 ,0 ,0 2/1,1,1,2/11 === ++ mkmkmk vTu (32) on low wall (АО): 1,0 ,0 Nkm ==

;0 ,1 ,0 2/1,0,0,2/1 ===+ kkk vTu (33) On input edge (АВ): 2,0m ,0 kk ==

;0 , ,1 , 2/1,0,1,0,0,2/3,2/1 ==== +mmmmmm vTTpuu (34)

;2/12,2/12, ,12,2, ,02, ,02,2/1 +=−−===+ NkvNkvNkTNkTNkpNku (35) Calculations by algorithm (15) - (30) taking into account edge (31) - (35) is as follows. 1. First by explicit formulas (19), (20) and (30) we determine the values of the component of the vector

2/1+nu in nodes corresponding grids. 2. In second step to find the values 11 , ++ nn pu

satisfying (21), (22) and (23) we use an iterative algorithm (36) from work / 9 /, which can be written as follows:

),2(

))

2(

(

1,1,2/1

1,1,2/1

1,1,2/32

1

0

21

,12/1,

,12/1,

,12/1,1

,12/1,1

21

,1,2/1

,1,2/1

,1,2/3

0

1

,1,

,1,11,1

,2/1

++−

+++

+++

+−

++

+−+

+++

+−

++

++

+++++

+

+−=

=+−−

+

++−

−

−+

snmk

snmk

snmk

snmk

snmk

snmk

snmk

snmk

snmk

snmk

snmk

snmksn

mk

h

hhvvvv

huuu

hpp

u

ξξξδττ

τ

τ

(36)

,,1,2/1

1,1,2/1

1,1,2/1

snmk

snmk

snmk uu +

+++

+++

+ −=ξ

With edge conditions, in 12,1m ,1,0 −== kNk ,

;1 ,0 , ,0,2/11,2/11,2/1,2/3 ==+= +− mmNmNmm puuuu in 2,2m ,1,11 NkNkk =−= ,

.0 ,0 ,1,2/11 ==+ mNmk uu

Fig 3 - Study area grid

For determination1

2/1,++

nmkv we have correlation

),2(

))2

(

(

1,12/1,

1,12/1,

1,12/3,2

1

0

22

,12/1,

,12/1,1

,12/3,

21

,1,2/1

,1,2/1

,11,2/1

,11,2/1

0

2

,1,

,11,1,1

2/1,

++−

+++

+++

+−

++

++

+−

++

++−

+++

+++++

+

+−=

=+−

+

++−−

−

−−

+

snmk

snmk

snmk

snmk

snmk

snmk

snmk

snmk

snmk

snmk

snmk

snmksn

mk

h

h

vvv

hh

uuuu

h

ppv

ηηηδττ

τ

τ

(37)

snmk

snmk

snmk vv ,1

2/1,1,12/1,

1,12/1,

++

+++

+++ +=η

With edge conditions: in 2,0m ,1,1 kkk == ,

;02/12,2/1, == −kkk vv in 2,0m ,1,11 NNkk =−= ,

.0 , ,0 2,2/12,2/12,2/1, === +− NkNkNkk pvvv

j=1

j=k2

j=n2+1/2 j=n2

i= 0

i=k1

ik

i=n1

-1/

2

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Values 1,11,1 , ++++ snsn vu from correlation (36), (37) we find with help of scalar sweep. Then onalgorithm /9/

pressure value1,1 ++ snp we determine by formula

1,1

0,11,1 +++++ −= sn

hsnsn udivpp τ (38)

3. At the third stage by solving equations (18) and (29) with appropriate edge conditions determine the temperature distribution in calculation area. The main feature of the proposed method of calculation of [14-16] is effective finding of solution of algebraic equations system as (21) - (23).Numerous calculations of thermal convection problems with various parameters Gr, Re, Pr, and isothermal flow (Gr = 0) have shown the reliability of algorithm (36) - (38) for solution of auxiliary grid equations using calculation of the Navier-Stokes equations. Table 1 shows the dependence of number of steps on s for different values 0 , needed for exit from iteration formulas (36) - (38) in observing particular criterion:

,10v 41,1 −++ ≤snhudi

(39)

In caseRe=100, Gr=0,,1.2 ,2 ,001.0 === δτ N

in iteration stepn = 600 As shows in table 1, the number of iterations doesn’t dependent on number of nodes of final -difference grid, which corresponds to uniform convergence on h. Process of successive approximations (15) - (27), (29), (30) with appropriate edge conditions(28) and (31) - (35) is considered steady if the following convergence criterion

,ε≤+ nh

n udivR (40)

In which - is ahead given small value.

Table 1 – Dependence of number of steps M x M 20 =τ

00 =τ

τ 0 =

τ 0 =

τ 0 =

101 · 201

8 9 10 20 48

251 ·501

9 10 11 22 50

501 · 1001

9 11 12 24 51

V. THE NUMERICAL CALCULATION RESULTS

AND THEIR DISCUSSION Calculations were done for H/L = 1, L*/ L = 0.3, H*/L

= 0.3 and Pr = 0.72 for different values of the Grash of

number (Gr = 10 ·10⁵ 105)."Bearing" value Re * supposed to be 100 700.Calculations were done on grids (101, 201), (01, 1001). Test calculations have shown that differences in worst case (Gr = 5 ·10 ⁵), for full Nusselt numbers in different grids 2%. For calculations were used Core i7 3400/S1155/4Gb/1Tb, working on the program Microsoft Visual Studio 2010. In all variants of calculations stationary regime is reached. For various Grash of numbers and different grids, typical number of internal iterations required for achieving convergence criterion is from 8 to 27. (The number of external (global) iterations ranged from 600 to 850). The characteristic calculation time for grid (101, 201), - 3 min. First, we provide detailed description of the results of calculations. Figures 4a, 5a, 6a, 7a, 8ashows the contours of current stream function for different numbers Gr and Re. In all cases, the main part of calculation area is occupied by "cross-cutting" flow. It was found that above the point C (y 0.4), liquid flow first moves away from side wall, and then it covers it. Like incase of the flow behind the projection in large values of Gr (Gr = 5 105 at Re = 300) above the point C near the wall there forms a vortex zone of low intensity. Dependence of the Reynolds number corresponding to the steady regimes of parameter Gr is shown in Figures 12, 14. As expected, with increasing of Gr, number Re increases, which also corresponds to the increase of fluid flow through inlet section. Thus, in increasing of Gr, the flow in calculated area becomes more intense. From practical point of view, one of the major objectives of the study of free convection is to determine the dependence of heat transfer on problem allows technologists to purposefully solve all sorts of problems on determining heat losses, heat transfer intensification, reduction of temperature difference, choice of material, etc. Figures 13 and 15 shows the dependence of Reynolds number corresponding to the steady regime, on parameter Re. As expected, with increasing of Re the number Re increases, the dependence is almost linear and with increase of Gr dependence becomes weaker. Figure 11shows that, dependence of Reynolds numbers on Re , for small Grashof numbers (Gr = 104 5 104) are almost identical. Figures 4b, 5b, 6b, 7b, 8b show the most characteristic temperature distributions for Gr = 104 5 * 105; corresponding isotherm lines are done in intervals T = 0.05. Tables 2, 3 and 4(respectively, Re = 300, Re = 500, Re = 700) include values of total Nusselt numbers for different sections of a solid edge of calculated area where through Nu(1),Nu(2), Nu(3) the total Nusselt numbers for lower (AO), top horizontal (BC) and upper vertical (CD) walls, are indicated respectively.

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Table 2- Dependence of total Nusselt numbers on Grash off numbers (Re=300)

Gr Nu(1) Nu(2) Nu(3)

104 -1.20711935 1.21368607 2.71421552

25·103 -1.14501685 1.25032469 2.80149099

5·104 -1.06250990 1.30866717 2.94254227

105 -.95039644 1.41649719 3.17271196

2·105 -.82435129 1.60518169 3.45053971

3·105 -.74823105 1.76227810 3.54576866

4·105 -.67409142 1.80170216 2.99786548

5·105 -.62978933 1.93340066 3.00173457


Gr Nu(1) Nu(2) Nu(3) 104 -1.19822014 1.20415896 2.48389472

25·103 -1.16663345 1.21699929 2.52143465 5·104 -1.11961953 1.23843673 2.58275753 105 -1.04304669 1.28149494 2.70138683

2·105 -.93610120 1.36702174 2.90152287 3·105 -.86360107 1.44891971 3.03714750 4·105 -.80805648 1.52368884 3.07299943 5·05 -.74173832 1.56032695 2.49497321


Gr Nu(1) Nu(2) Nu(3)

104 -1.18376544 1.19702134 2.18610035

25·104 -1.16282018 1.20318555 2.20873636

5·104 -1.13057284 1.21351415 2.24664922

105 -1.07475286 1.23435200 2.32009021

2·105 -.98901285 1.27659414 2.45177677

3·105 -.92553245 1.31885430 2.55051401

4·105 -.87478441 1.35976946 2.58983134

5·105 -.81685899 1.39002582 2.22452353

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As can be seen from Tables 2 to 4, for Re = 300, Re = 500, Re = 700, the relative deviation of values Nu (2), Nu (3) (with difference of values of numbers Gr almost in two orders) is less than14-35%, 10-17%, respectively. At the same time the values Nu(1) differ from each other by 30-50%, indicating an increase in the intensity of heat transfer process on low wall in increasing of Grassh of values. Isotherm shows that the temperature distribution in the fields in all of given variants in calculation area is homogeneous, i.e. in homogeneities, e.g. in the form of temperature boundary layers, even for relatively large values Grash of (Gr = 5 · 10 ⁴) are absent.

A deeper understanding of interaction of fluid flow and heat transfer is achieved when considering changes in local heat transfer coefficients for solid walls of the channel. Figures 16- 20 illustrate behavior of local Nusselt numbers on the lower and upper walls of AVSDOO 'area. Let’s consider the upper horizontal wall (BC) (Figures 17, 19, 23). Initially a "through" current deviate from it, and then gradually goes at the wall in the direction of angle top C. Therefore, in the left half of this boundary, the Nusselt number reduces, and in right half is greatly increased. Position of local minimum corresponds to the point of least interaction of flow and the wall, and the relative position of maximum corresponds to the point C, where the most Intensive interaction between the wall and the flow takes place.

Fig 4- Isolines of flow function, isotherm in Gr=3104, Re=300

Fig 5 - Isolines of flow function, isotherm in Gr=5104, Re=300



Fig 8 - Isolines of flow function, isotherm in Gr=5104, Re=600

Fig 9 - Graph of flow in Re=300, isobars а-Gr=3·104, b- Gr=5·104

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Fig 10 - Graph of flow Re=600, isobars а-Gr=104 ⁴, б- Gr=3·104 ⁴

Fig 11 - Graph of flow in Re=600, isobars

Figures 18, 21, 24 show a graph of changing of local Nusselt numbers (Nu(3)) for the top vertical wall ( CD ). As can be seen, the maximum value is also in point C. The most noticeable feature of the behavior of Nusselt numbers in this part of the border region is that there are local minima behind the corner (у 0.4). Point of intensity fall of thermal interaction of flow with the wall complies with the zone of formation of secondary motion of large Gr, i.e. the stream deviates from the wall and breaking off, serves as a thermal insulator.

Fig 12 - Dependence of Reynolds number on Grash of number

Fig 13- Dependence of Raynolds number on «Baring»

number Re

Fig 14 - Dependence of Raynolds number on Grash of

number

Fig 15 - Dependence of Raynolds number on «Baring»

number Re

Fig 16- Local Nusselt numbers on lower edge in Re=500

10000 20000 30000 40000 50000

50

100

150

200

250

300

350

Re*=100 Re*=200 Re*=300 Re*=400 Re*=500

Re

Gr

100 200 300 400 500

50

100

150

200

250

300

350

400

Gr=10 000 Gr=20 000 Gr=30 000 Gr=40 000 Gr=50 000

Re

Re*(δp)

0 100000 200000 300000 400000 500000

100

150

200

250

300

350

400

450

500

550

600

Re*=200 Re*=300 Re*=400 Re*=500 Re*=600 Re*=700

ReGr

200 300 400 500 600 700

100

200

300

400

500

600

T=const Gr=10000 Gr=25000 Gr=50000 Gr=100000 Gr=200000 Gr=300000 Gr=400000 Gr=500000

Re

Re*(δp)

-0.018

-0.016

-0.014

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0

00.

035

0.07

0.10

50.

140.

175

0.21

0.24

50.

280.

315

0.35

0.38

50.

420.

455

0.49

Nu(

1)

A O

Gr=5E+5Gr=4E+5Gr=3E+5Gr=2E+5

266

ISSN: 2277-3754



Fig 17 - Nusselt local numbers on upper Horizontal edge in Re=500

Fig 18 - Nusselt local numbers on upper horizontal edge in

Re=500

Fig 19 - Nusselt local numbers on lower edge in Re=300

Fig 20 - Nusselt local numbers on upper horizontal edge in

Re=300

Fig 21 - Nusselt local numbers on upper vertical edge in Re=300

Fig 22 - Nusselt local numbers on lower edge in Re=700

Fig 23- Nusselt local numbers on upper horizontal edge in Re=700

Fig 24 - Nusselt local numbers on upper vertical edge in Re=700

-0.018

-0.016

-0.014

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0

00.

025

0.05

0.07

50.

10.

125

0.15

0.17

50.

20.

225

0.25

0.27

50.

30.

325

0.35

0.37

50.

40.

425

0.45

0.47

50.

5

Nu(

1)

A O

Gr=5E+5

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.3

0.34

50.

390.

435

0.48

0.52

50.

570.

615

0.66

0.70

50.

750.

795

0.84

0.88

50.

930.

975

Nu(

3)

C D

Gr=5E+5Gr=4E+5Gr=3E+5Gr=2E+5Gr=1E+5

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

00.

015

0.03

0.04

50.

060.

075

0.09

0.10

50.

120.

135

0.15

0.16

50.

180.

195

0.21

0.22

50.

240.

255

0.27

0.28

50.

3

Nu(

2)

B C

Gr=5E+5Gr=4E+5Gr=3E+5Gr=2E+5

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.3

0.35 0.4

0.45 0.5

0.55 0.6

0.65 0.7

0.75 0.8

0.85 0.9

0.95 1

Nu(

3)

C D

Gr=5E+5

00.0050.01

0.0150.02

0.0250.03

0.0350.04

0.0450.05

0.3

0.35

50.

410.

465

0.52

0.57

50.

630.

685

0.74

0.79

50.

850.

905

0.96

Nu(

3)

C D

Gr=5E+5Gr=4E+5Gr=3E+5Gr=2E+5Gr=1E+5

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0 0… 0… 0… 0… 0… 0… 0… 0 … 0… 0… 0… 0…

Nu(

2)

B C

Gr=5E+5Gr=4E+5Gr=3E+5

0.0150.02

0.0250.03

0.0350.04

0.0450.05

0.055

00.

020.

040.

060.

08 0.1

0.12

0.14

0.16

0.18 0.2

0.22

0.24

0.26

0.28 0.3

Nu(

2)

B C

Gr=5E+5

Gr=4E+5

Gr=3E+5

Gr=2E+5

-0.018

-0.016

-0.014

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0

00.

025

0.05

0.07

50.

10.

125

0.15

0.17

50.

20.

225

0.25

0.27

50.

30.

325

0.35

0.37

50.

40.

425

0.45

0.47

50.

5

Nu(

1)

A O

Gr=5E+5

Gr=4E+5

267

ISSN: 2277-3754



Figures 16, 19, 22 shows a graph of changes Nu(1) on bottom wall. The negative value Nu(1)shows heat transfer. Indeed, approaching to the axis of symmetry, near the wall, liquid is heated more intensively; this corresponds to decrease in heat transfer. In the lower right area the convective heat transfer from the wall becomes substantial, and isotherms are pushed away from the wall, respectively, heat losses are reduced in this area. Also it should be noted that at sufficiently high Reynolds numbers (that is, insufficiently strong convective flows, Figure 19 - Nusselt local numbers on lower edge in Re=300increasing of the differences of the walls temperatures (for example, numbers Gr=5∗105characteristic temperature difference for the air is approximately 100Сdoes not lead to a noticeable increase in fluid flow.

VI. CONCLUSION

1. Calculation algorithm for solving stationary Navier-Stokes equation in simply connected and multiply connected areas has been proposed. 2. A new model of the boundary problem with a pressure drop for equations of free Convection has been formulated. An algorithm for the numerical solution of Hydrody namic problems with unknown flow rate of fluid through the cross section of the calculation area has been formulated. 3. Calculations of free convection problem with a given pressure drop in the channels of complex shape were done and flow patterns for different Grashof (Gr) numbers were obtained. The results are presented in graphs and tables and were analyzed. 4. It was found that in the increase of walls temperature difference, Reynolds number changes insignificantly

ACKNOWLEDGEMENT We would like to acknowledge and grateful to the

Ministry of education and science republic of Kazakhstan for providing funding for this research.

REFERENCES [1] Rebinder P.A. On the form of links of moisture with

materials in drying. All-Union Scientific and Technical Meeting. M: EC SPS 1958.

[2] Rebinder P.A. Physical and chemical mechanics and technical progress. - WCS: The Future of Science. - M.: Knowledge, 1973. - P.174 -189 p .

[3] A.V. Lykov. Theory of drying. -M.: Energy, 1968. - 470p.

[4] Lykov A.V., B.I. Leonchik. Spray dryers. -M.: Engineering, 1973. - 295p.

[5] Polezhaev V.I., V.L. Grjaznov. Method of calculating the boundary conditions for the Navier-Stokes equations in vorticity, stream function. / / DAN. USSR. -1974. -T. 219, № 2 .-p. 301-304.

[6] Tarunin E.L. On the choice of approximation formula for the vorticity at the solid boundary for solving problems of

viscous fluid dynamics. Numerical methods of continuum mechanics. Novosibirsk, 1979. T. 9, № 7.-p. 97-113.

[7] Tarunin E.L. Computer experiment in problems of free convection. - Irkutsk: Izd Irkutsk University, 1990. -p. 228.

[8] Rouch P. Calculational Fluid Dynamics. -M.: Mir, 1980. -p. 616.

[9] Kelkar K.M., Patankar S. Numerical calculation of flow and heat transfer in the channel between parallel plates with alternately opposed plates. / Heat. -1981, № 1. -p. 24-30.

[10] Urmashev B., Danaev N. The convergence of iterative processes for solving Navier –Stokes equations / / Search. Almaty, 2000, № 3. , p.174 -181

[11] Kaltayev A.A., B.A. Urmashev. Numerical solution of a problem of thermal convection //Bulletin of KazGU.Ser. Mat. , mechan. Inf. Number 1 (20), Ed. KazGU, Almaty, 2000, p.162 -170.

[12] Urmashev B.A., Danaev N.T. Numerical solution of thermal convection with unspecified flow / / Search. Almaty, 2000, № 5.

[13] Danaev N.T., B.A. Urmashev. Iterative schemes for solving auxiliary grid Navier- Stokes equations. Bulletin of the Kazakh State University .Ser. math. , mech. , inf. Number 4 (23), Ed. Kazakh State University, Almaty, 2000, p.74 -78 .

[14] B.A.Urmashev, N.T. Danaev, B.S. Darybaev On Problem of Thermal Convection with Unset Flow Rate / IECMSA- 2012. 1st International Eurasian Conference on Mathematical Sciences and Applications, Prishtine, Kosovo, September 03-07, 2012 , pp.309

[15] Bolekbaeva A.B., N.B. Zakariyanova. Numerical solution of the Navier -Stokes equations for incompressible viscous fluid variables “velocity-pressure” in two-dimensional space / /International conference of students and young scientists “World of Science”, 14th Scientific Student Conference “Elementary questions of Mathematics and Informatics” 17 -19 April2013.

[16] Urmashev B.A. Danaev N.T., Alimjanov E.S. Numerical solution of the Navier –Stokes equations for incompressible viscous fluid variables “velocity-pressure “in three-dimensional space / International Conference Computational and Informational Technologies for Science, Engineering and Education in 2013, Ust-Kamenogorsk, Kazakhstan, 18-22 September 2013.

AUTHOR’S PROFILE The authors are Kanat M Khazimov, PhD Student, Faculty of Engineering, Kazakh National Agrarian University, Almaty, Kazakhstan;Ganesh C. Bora, Ph D,Assistant Professor, Department of Agricultural and BiosystemsEngineering, North Dakota State University, Fargo, ND, USA;Baidaulet A Urmashev Assistant Professor, Faculty of Mechanics and Mathematics,Al-FarabiKazakh National University, Almaty, Kazakhstan; Marat Z Khazimov Ph D, Faculty of Engineering, Kazakh National Agrarian University, Almaty, Kazakhstan; Zhanat M Khazimov, MS, Faculty of Engineering, Kazakh National Agrarian University, Almaty, Kazakhstan; Corresponding author: Ganesh C. Bora, Department of Agricultural and Biosystems Engineering, North Dakota State University, NDSU Dept.7620, P.O. Box 6050, Fargo ND 58108 ‐6050;phone: +1-701 ‐231 ‐7271; fax:701 ‐231‐ 1008;e‐ mail:ga

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