Convection and Conduction Heat Transfer Part 4 pdf

Conclusion Two dimensional steady, mixed convection heat transfer in a two-dimensional trapezoidal cavity with constant heat flux from heated bottom wall while the isothermal moving top

Ri As a result, the maximum temperature decreases monotonously which can be recognized from the isothermal plots As the aspect ratio increases from 0.5 to 1 the Nu av increases for a

particular Ri

At higher Reynolds number i.e Re=600, with increasing aspect ratio some secondary eddy

at the bottom surface of the cavity has been observed This is of frictional losses and

stagnation pressure As the Ri increases, natural convection dominates more and the bottom

secondary eddies blends into the main primary flow For A>1.5 the variation is almost flat indicating that the aspect ratio does not play a dominant role on the heat transfer process at that range

4.5 Effect of Reynolds number, Re

This study has been done at two different Reynolds numbers They are Re=400 and Re=600

With a particular case keeping Ri and A constant, as the Reynolds number increases the

convective current becomes more and more stronger and the maximum value of the isotherms

reduces As we know Ri=Gr/Re 2 Gr is square proportional of Re for a fixed Ri So slight

change of Re and Ri causes huge change of Gr Gr increases the buoyancy force As buoyancy force is increased then heat transfer rate is tremendously high So changes are very visible to

the change of Re From figure 19-20, it can be observed that as the Re increases the average

Nusselt number also increases for all the aspect ratios

5 Conclusion

Two dimensional steady, mixed convection heat transfer in a two-dimensional trapezoidal cavity with constant heat flux from heated bottom wall while the isothermal moving top wall in the horizontal direction has been studied numerically for a range of Richardson number, Aspect ratio, the inclination angle of the side walls and the rotational angle of the cavity A number of conclusions can be drawn form the investigations:

• The optimum configuration of the trapezoidal enclosure has been obtained at γ=45º, as

at this configuration the Nuav was maximum at all Richardson number

• As the Richardson number increases the Nuav increases accordingly at all Aspect ratios, because at higher Richardson number natural convection dominates the forced convection

• As Aspect Ratio increases from 0.5 to 2.0, the heat transfer rate increases This is due to the fact that the cavity volume increases with aspect ratio and more volume of cooling air is involved in cooling the heat source leading to better cooling effect

• The direction of the motion of the lid also affects the heat transfer phenomena Aiding flow condition always gives better heat transfer rate than opposing flow condition Because at aiding flow condition, the shear driven flow aids the natural convective flow, resulting a much stronger convective current that leads to better heat transfer

• The Nu av is also sensitive to rotational angle Ф At Re=400 it can be seen that, Nusselt

number decreases as the rotational angle, Φ increases Nu av increases marginally at

Φ=30 from Φ=45º but at Φ=60º, Nu av drops significantly for all the aspect ratios

6 Further recommandations

The following recommendation can be put forward for the further work on this present research

1 Numerical investigation can be carried out by incorporating different physics like radiation effects, internal heat generation/ absorption, capillary effects

2 Double diffusive natural convection can be analyzed through including the governing equation of concentration conservation

3 Investigation can be performed by using magnetic fluid or electrically conducting fluid within the trapezoidal cavity and changing the boundary conditions of the cavity’s wall

4 Investigation can be performed by moving the other lids of the enclosure and see the heat transfer effect

5 Investigation can be carried out by changing the Prandtl number of the fluid inside the trapezoidal enclosure

6 Investigation can be carried out by using a porous media inside the trapezoidal cavity instead of air

7 References

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a stably stratified fluid, Appl Math Model 19 (1995) 465–472

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Convective Heat Transfer of Unsteady Pulsed Flow in Sinusoidal Constricted Tube

J Batina1, S Blancher1, C Amrouche2, M Batchi2 and R Creff1

1Laboratoire des Sciences de l’Ingénieur Appliquées à la Mécanique et l’Electricité Université de Pau et des Pays de l’Adour, Avenue de l’Université – 64000 Pau;

2Laboratoire de Mathématiques Appliquées- CNRS UMR 5142 Université de Pau et des Pays de l’Adour, Avenue de l’Université – 64000 Pau;

in order to obtain convective heat transfer enhancement, most of the studies are linked to:

- Firstly, the search for optimal geometries (undulated or grooved channels, tube with periodic sections, etc.) : among those geometrical studies, one can quote the investigations of Blancher, 1991; Ghaddar et al., 1986, for the wavy or grooved plane geometries, in order to highlight the influence of the forced or natural disturbances on heat transfer

- Secondly, the search for particular flow conditions (transient regime, pulsed flow, etc.): for example those linked to the periodicity of the pressure gradient (Batina, 1995; Batina

et al 2009; Chakravarty & Sannigrahi, 1999; Hemida et al., 2002), or those which impose

a periodic velocity condition (Lee et al., 1999; Young Kim et al., 1998) or those which carry on time periodic deformable walls

The main objective of this study is to analyse the special case of convective heat transfer of

an unsteady pulsed, laminar, incompressible flow in axisymmetric tubes with periodic sections The flow is supposed to be developing dynamically and thermally from the duct inlet The wall is heated at constant and uniform temperature

One of the originality of this study is the choice of Chebyshev polynomials basis in both axial and radial directions for spectral methods, the use of spectral collocation method and the introduction of a shift operator to satisfy non homogeneous boundary conditions for spectral Galerkin formulation A comparison of results obtained by the two spectral methods is given A Crank - Nicolson scheme permits the resolution in time

1.1 Nomenclature

a thermal diffusity ⎡⎣m s2 ⎤⎦ λ dimensionless total wavelength

h wall function θ=(T T− ∞) (T W−T∞)

H periodic sinusoidal radius [ ]m μ dynamic viscosity ⎡⎣Ns m2⎤⎦

L geometric half-length tube [ ]m ν μ ρ= kinematic viscosity: ⎡⎣m s2 ⎤⎦

R tube radius at the constriction [ ]m ρ fluid density ⎡⎣Kg m3⎤⎦

r radial co-ordinate [ ]m τ modulation flow rate

T fluid temperature [ ]K ω vorticity function [ ]1s

T∞ duct inlet temperature [ ]K ψ stream function ⎡⎣m s3 ⎤⎦

u axial velocity [ ]m s Dimensionless numbers

0

u mean bulk velocity [ ]m s Re Reynolds number:Re = Ru ν0

v radial velocity [ ]m s Pr Prandtl number: Pr=ν a

z axial co-ordinate [ ]m Nu Nusselt number

Greek symbols θ0m( )x averaged bulk temperature

is heated at constant and uniform temperature, and the fluid inlet temperature is equal to

the upstream ambient temperature Physical constants are supposed to be independent of

the temperature, which involves that the motion and energy equations are uncoupled

2.2 Governing equations

With the 2D hypothesis, we use the vorticity-stream function formulation (ω ψ, ) for the

Navier-Stokes equations in which the incompressibility condition is automatically satisfied

In fact, the essential advantage of this formulation compared to the primitive variables

(velocity-pressure formulation) is the reduction of the number of unknown functions and

the non-used of the pressure On the other hand, Navier-Stokes equations become a fourth

order Partial Differential Equations whose expressions in cylindrical coordinates are:

It is important to note that we have only one unknown function, i.e.: ψ The vorticity

function ω is linked to ψ by the relation:

∂

∂+

∂

∂+

∂

∂

r

T r z

T r

T a r

T z

T u t

2

2 2

2

3 Boundary conditions

The present problem is unsteady This unsteadiness is generated at the initial instant t=0,

and is sustained during all the time by a source of upstream pulsations For both steady and

unsteady flow, the following boundary conditions are available for any time t ≥ : 0

• Entry: for the thermal problem, the inlet fluid temperature is equal to the upstream

For dynamic conditions at the entry section, we impose:

- Steady flow (t=0 time step)

• Entry: for the dynamic problem, Poiseuille profile boundary condition is chosen

• Entry: the source of imposes a periodic pressure gradient modulation Then the velocity

axial component and the stream function ψ have a Fourier series expansion in time:

where f represents u or ψ At this section, to avoid reverse flow, we impose:τ< 1

4 New formulation and resolution of the dynamic and thermal problem

4.1 New formulation of the dynamic problem

4.1.1 Dimensionless quantities and variables transformations

One chooses for dimensionless variables:

0

L

=u

In order to obtain a computational square domain permitting the use of two dimensional

Chebyshev polynomials, we proceed to a space variables transformation This one is

inspired by Sobey, 1980, and modified by Blancher, 1991 It has been adapted to the

axisymmetric geometry used in this study Afterwards, we note by H z the duct periodic ( )

radius Then we define:

and (see equation 73)

Finally, the study domain is transformed into a rectangle 1− ≤ ≤ and 0x 1 ≤ ≤ ρ 1

representing the half - space of the square:[−1,1] [× −1,1]

4.1.2 New system of unsteady dynamic governing equations

Considering the transformation of variables defined before, the new stream – vorticity

formulation of this problem is:

4.1.3 The dynamic steady problem formulation

The dynamic steady problem corresponding to problem (16) is written as follows:

Important: for reason of convenience, the radius ρ will be noted r

4.2 New formulation of the thermal problem

For the thermal problem, the temperature θ is made dimensionless in a classic way:

4.2.1 The thermal unsteady problem formulation

Using (1) and (10)-(15), the dimensionless energy equation can be written as follows:

4.2.2 The thermal steady problem formulation

The dimensionless steady state energy problem related to the equation (24) is:

5 Numerical resolution using spectral methods

5.1 Trial functions and development orders

The spectral methods consist in projecting any unknown function f x r t on trial ( , , )

functions as follows:

( ) ( )

0 l 0( , , ) x r ( )

where N x and N r are the development orders according to the axis x and r respectively

The basis functions P r and l( ) Q x are generally trigonometric or polynomial functions k( )

(Chebyshev, Legendre, etc.) according to different boundary conditions situations The time

dependant coefficients f t are the unknowns of the problem For our problem, the kl( )

function f representsω, ψ orθ For a steady problem, the coefficients f t are time kl( )

independent

It is necessary to study the influence of the physical parameters such as the Reynolds

number to remain in 2D hypothesis From a numerical point of view we will show the

influence of the polynomials degrees particularly for the thermal problem

5.2 The choice of basis functions

Because no symmetry condition is imposed at the boundaries of our half-domain of study,

we choose basis functions constructed from Chebyshev polynomials (Bernardi & Maday,

1992; Canuto et al., 1988) instead of trigonometric trial functions Then, P r and l( ) Q x are k( )

written as linear combination of Chebyshev polynomials Their expressions depend on the

boundary conditions and the spectral method used (Galerkin or collocation method)

Generally, with Galerkin method, Dirichlet or Neuman boundary conditions imposed to

trial functions must be homogeneous, but it is not necessary for collocation method (see

Galerkin and collocation methods below)

The basisP r and l( ) Q x are written as a linear combination of Chebyshev polynomials k( )

such as (Gelfgat, 2004; Shen, 1994, 1995, 1997):

where n (respectively m) is the number of boundary conditions according to the radial

direction r (respectively the axial direction x), and T x is the Chebyshev polynomial of k( )

degree k

5.2.1 Advantages and limitations of spectral methods

Spectral methods are used successfully in many problems of physics, mainly those involving

periodic physical phenomena in space and / or in time Its main advantage is its high

degree of accuracy, compared with some methods such as finite differences, finite elements

or finite volumes (Bernardi & Maday, 1992; Canuto et al., 1988; Gelfgat, 2004; Shen, 1994,

1995, 1997) Spectral methods are particularly suitable to study instabilities phenomena,

self-maintained or forced, occuring in Computational Fluid Dynamics However, spectral

methods are limited to simple geometries For complicated study domains, an alternative

way may be using spectral finite elements The second disadvantage of these methods is

their cost of implementing and their high CPU calculations The matrices obtained are

usually full and strategies for solving linear or nonlinear systems remain limited

6 Numerical resolution of the dynamic and thermal problem using spectral

galerkin formulation

6.1 Numerical resolution of the dynamic steady problem

The steady dynamic problem is given by the equation (22) Generally, this problem is

written with classical homogeneous boundary conditions One of the originalities of this

study is the use of a relevment function allowing the introduction of non homogeneous

boundary conditions For this reason, the unknown stream function ( , )ψ x r is written by

mean of the Poiseuille stream function ϕ0( )r corresponding to the Poiseuille velocity

imposed at the duct entry as:

( , )x r ( , )x r ( )r

where the stream function ψ0( , )x r verifies homogeneous boundary conditions in both

The corresponding Galerkin method consists in projecting the discretized equations on a

Chebyshev polynomials basis, taking into account the whole boundary conditions (Canuto

et al., 1988) Then, according to the general formulation of spectral methods, the

stream-function ψ0 is projected on trial functions as follows:

P r will be an even function To construct the basis P r , we choose a linear 2l( )

combination of Chebyshev polynomials such as (Gelfgat, 2004; Shen, 1994, 1995, 1997):

where 3m = here (see bellow) The velocity boundary conditions imply that the stream

function must satisfy the corresponding homogeneous boundary conditions as:

( 1) 0

k

Q′ − = at x -1= (v = 0 at x = − ) (40) 1( 1) 0

k

Q − = at x -1= (Poiseuille profile x = − ) (41) 1(1) 0

where Δ is the square: Δ = −[ 1,1] [× −1,1]

Taking as test function:

( , )x r Q x P r k l , for 0 k N , 0 l N

the Galerkin spectral method consist to make scalar products between the non linear

equation (30) and each test function Q x P r , by writing: i( ) ( )2j

0

2

,Re

Finally, we obtain a system of N xr=(N x+1)(N r+1) non linear equations with N xr

unknowns, solved by Newton algorithm

6.2 Numerical resolution of the dynamic unsteady problem

From equation (16), introducing the unknown ψ function such as:

(x r t, , ) (x r t, , ) ( ) ( )r A t

and using the equations (46), we define the operator in which the unknown coefficients

depend now on time:

Then the previous problem (16) can take the following form:

The operator L x r tψ( , , ) is nonlinear Notice that ωΦ is the contribution coming from

Poiseuille extension The temporal discretization of (49) is made by using the ε –method,

reduced here to Crank - Nicolson method The advantage of this method is to be

unconditionally stable It leads to the equation below withε=1 / 2, which corresponds to a

two order scheme:

where the initial condition is given by the solution of the steady problem

The unknowns ψkl( )t are obtained by solving with Newton algorithm, at each time step, the

non linear system obtained with scalar products between relation (50) and test

functionsQ x P r i( ) ( )2j , as in equation (46)

6.3 Numerical resolution of the thermal unsteady problem

6.3.1 Choices of the basis functions

The dimensionless energy equation is given by (25) and (25) The choice of the temperature

basis functions is made in the same way as in the dynamic problem In order to apply the

Galerkin method, we consider the boundary conditions (heading 3) for the temperature θ

Let us set:

where θ is the solution satisfying the homogeneous boundary conditions and θR( )r is a

smoothed gap temperature imposed at the entry The homogeneous temperatureθ,

truncated at development orders M x according to the axis x and M r according to the

radius r, is projected on the trial functions as follows:

where p r and 2l( ) q x are built from Chebyshev polynomials as in heading 5 According k( )

to temperature boundary conditions (heading 3), we obtain, at last:

6.3.2 Resolution of the steady energy equation

With (24) and ( , )θx r =θ( , )x r +θR( )r , the steady thermal problem is written as follows:

This problem is discretized by Galerkin spectral method explained above The linear system

obtained is solved by a Gauss type classical method

6.3.3 Resolution of the unsteady energy equation

The unsteady problem is written as follows:

integrated in time by using the second order Crank-Nicolson scheme ( 1

2

ε= ) which is formulated as follows:

q x p r , one obtains at each time step a system

of linear equations solved by the classical Gauss method

One can notice that the use of Chebyshev polynomials in both axial and radial directions is

not obvious, and contribute to emphasize this numerical method

7 Numerical resolution of the dynamic and energy problem using spectral

collocation method

7.1 Numerical resolution of the dynamic problem

For reasons of simplicity, we describe explicitly only the resolution of the steady dynamic

problem For the unsteady problem, we use Crank-Nicolson method for time integration as

in (50); the unsteady problem resolution in space is identical to the steady case

The main interest of collocation method compared with Galerkin formulation is its

simplicity: it is not necessary to build a relevment function to take into account non

homogeneous boundary conditions We introduce these conditions directly in the matrix of

the system and/or in the basis trial functions For this reason, it is easy to compute

collocation procedure Let us explain this method for the steady dynamic problem