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The interaction of transient fully developed natural convection flow with thermal radiation inside a vertical annulus is analyzed both analytically and numerically. The Rosseland approximation is used to describe the radiative heat flux in the energy equation. The mathematical model capturing the present physical situation is highly non-linear due to the presence of radiation effect. The solution of transient model is obtained by implicit finite difference method. To check accuracy of the numerical solution, steady state solution for energy and momentum equations are derived analytically using perturbation series method. Skin-friction and Nusselt number at the outer surface of inner cylinder as well as inner surface of the outer cylinder are obtained. Selected sets of graphical results illustrating the effects of various controlling parameters involved in the problem on flow formation are discussed.

Free convective heat transfers inside concentric and eccentric annulus has received an intensive attention of researchers owing to its scientific, technological and engineering applications such as in the construction of electrical motors and generators, cooling of electronic components, nuclear reactors, thermal storage systems, heating and cooling of underground electric cables, completion of oil source and aircraft fuselage insulation. [

The physical problem under consideration consists of a transient natural convection flow in an infinite vertical annulus formed by two infinite concentric vertical cylinders in presence of thermal radiation. The transient flow formation is due to sudden heating of outer surface of inner cylinder in presence of thermal radiation. The physical properties of the working fluid are assumed to be constant. A schematic diagram of the present problem is shown in

∂ u ′ ∂ t ′ = v r ′ ∂ ∂ r ′ ( r ′ ∂ u ′ ∂ r ′ ) + g β ( T ′ − T 0 ) (1)

∂ T ′ ∂ t ′ = α [ 1 r ′ ∂ ∂ r ′ ( ∂ T ′ ∂ r ′ ) − 1 K ∂ q r ∂ r ′ ] (2)

The radiation heat flux term in the problem is simplified by using the Rosseland approximation:

q r = − 4 σ ∂ T ′ 4 3 κ * ∂ y ′ (3)

The initial and boundary conditions for the present problem assumed the form:

t ′ ≤ 0 : u ′ = 0 , T ′ = T 0 , for a * ≤ r ′ ≤ b * t ′ > 0 : { u ′ = 0 , T ′ = T w at r ′ = a * u ′ = 0 , T = T 0 at r ′ = b * (4)

To obtain the solutions of Equations ((1) and (2)) subject to the conditions (4) in dimensionless form, the following appropriate dimensionless quantities are introduce to the problem:

u = u ′ [ g β a 2 ( T w − T 0 ) ] − 1 , t = t ′ ν a 2 , θ = T ′ − T 0 T w − T 0 , R = 4 σ ( T w − T 0 ) 3 κ * K C T = T 0 T w − T 0 , P r = ν α , r = r ′ a , λ = b * a * (5)

Using the dimensionless quantities introduced in Equation (5), the dimensionless form of equation in (1) and (2) are:

∂ u ∂ t = 1 r ∂ ∂ r ( r ∂ u ∂ r ) + θ (6)

P r ∂ θ ∂ t = [ 1 + 4 3 R ( C T + θ ) 3 ] 1 r ∂ ∂ r ( r ∂ θ ∂ r ) + 4 R [ C T + θ ] 2 ( ∂ θ ∂ r ) 2 (7)

while the dimensionless initial and boundary conditions are:

t ≤ 0 : u = θ = 0 for 1 ≤ r ≤ λ t > 0 : { u = 0 , θ = 1 at r = 1 u = 0 , θ = 0 at r = λ (8)

Analytical solution is often an opportunity to validate computer routines of complicated problems and comparison with data as well as inspecting the internal consistency of mathematical models. It is of interest to reduce the nonlinear governing equations presented in the previous section to a form that can be solved analytically. A special case of the present problem that can exhibits analytical solution is the problem of steady state free-convection flow in vertical annulus in presence of thermal radiation. The resulting steady state equations and the boundary conditions for the special case can be written as:

1 r d d r ( r d u d r ) + θ = 0 (9)

1 r d d r ( r d θ d r ) [ 1 + 4 3 R ( C T + θ ) 3 ] + 4 R [ C T + θ ] 2 ( d θ d r ) 2 = 0 (10)

The relevant boundary conditions to be satisfied are:

{ u = 0 ; θ = 1 ; at r = 1 u = 0 ; θ = 0 ; at r = λ (11)

We approximate solution to Equations ((9) and (10)) subject to (11) using regular perturbation method by taking power series expansion in the radiation parameter R.

θ ( r ) = θ 0 ( r ) + R θ 1 ( r ) + 0 ( R 2 ) u ( r ) = u 0 ( r ) + R u 1 ( r ) + 0 ( R 2 ) } (12)

Substituting Equation (12) into Equations ((9) and (10)) and equating the like power of R, the required analytical expressions for the steady state velocity and temperature fields subject to boundary conditions (11) are as follows:

u ( r ) = 1 4 [ 1 + 1 ln ( λ ) ] − 1 4 [ r 2 + r 2 ln ( λ ) ] + 1 4 [ r 2 ln ( r ) ln ( λ ) + λ 2 ( ln ( λ ) ) 2 ] − 1 4 [ ln ( r ) + r 2 ln ( λ ) ] + R { − C 1 2 [ r 2 [ ln ( r ) ] 2 4 − r 2 [ ln ( r ) ] 2 + 3 r 2 8 − r 2 ln ( λ ) ( 5 16 − ln ( r ) 2 ) ] − C 2 6 [ r 2 [ ln ( r ) ] 3 4 − 3 r 2 [ ln ( r ) ] 2 8 + 9 r 2 [ ln ( r ) ] 8 − 3 r 2 4 − r 2 [ ln ( λ ) ] 2 ( 5 16 − ln ( r ) 2 ) ] − C 3 12 [ r 2 [ ln ( r ) ] 4 4 − r 2 [ ln ( r ) ] 3 + 9 r 2 [ ln ( r ) ] 2 4 − 3 r 2 ln ( r ) + 15 r 2 8 − r 2 [ ln ( λ ) ] 3 ( 5 16 − ln ( r ) 2 ) ] + C 4 ln ( r ) + C 5 } (13)

θ ( r ) = 1 − ln ( r ) ln ( λ ) + R { C 1 [ ln ( r ) ] 2 [ ln ( r ) − ln ( λ ) ] + C 2 [ ln ( r ) ] 6 × [ [ ln ( r ) ] 2 − [ ln ( λ ) ] 2 ] + C 3 [ ln ( r ) ] 12 [ [ ln ( r ) ] 3 − [ ln ( λ ) ] 3 ] } (14)

From (13), the steady-state skin frictions at the outer surface of inner cylinder and inner surface of the outer cylinder of the annulus are:

τ 1 = d u d r | r = 1 = − 3 4 [ 1 + ln ( λ ) ] + R { C 1 2 ( 1 4 + 21 40 [ ln ( λ ) ] ) + C 2 6 ( 3 8 + 21 40 [ ln ( λ ) ] 2 ) + C 3 12 ( 3 + 21 40 [ ln ( λ ) ] 3 ) + C 4 } (15)

τ λ = d u d r | r = λ = − 1 4 ( 2 λ 2 + 1 λ ) − 3 4 ( λ ln ( λ ) ) + 1 2 ( λ ) + R { − C 1 2 [ λ ( ln ( λ ) ) 2 4 − 2 λ ln ( λ ) ( 5 16 − ln ( λ ) 2 ) − λ 4 ]

− C 2 6 [ λ ( ln ( λ ) ) 3 2 + λ ( ln ( λ ) ) 2 2 + 3 λ ln ( λ ) 2 − 3 λ 8 − 2 λ ln ( λ ) ( 5 16 − ln ( λ ) 2 ) ] − C 3 12 [ λ ( ln ( λ ) ) 4 2 − λ ( ln ( λ ) ) 3 − 15 λ ( ln ( λ ) ) 2 8 − 3 λ ln ( λ ) 2 − 3 λ − 2 λ ( ln ( λ ) ) 3 ( 5 16 − ln ( λ ) 2 ) ] + C 4 λ } (16)

Equally from (14), the steady state of heat transfers at the outer surface of inner cylinder and inner surface of outer cylinder of the annulus are:

N u 1 = − d θ d r | r = 1 = 1 ln ( λ ) + R { C 1 ln ( λ ) 2 + C 2 ( ln ( λ ) ) 2 6 + C 3 ( ln ( λ ) ) 3 12 } (17)

N u λ = d θ d r | r = λ = − 1 λ ln ( λ ) + R { C 1 2 ln ( λ ) λ + C 2 3 ( ln ( λ ) ) 2 λ + C 3 4 ( ln ( λ ) ) 3 λ } (18)

The momentum and energy equations given in Equations ((6) and (7)) are solved numerically using implicit finite difference method. The time derivatives in both equations are approximated using backward difference formula as:

∂ u ∂ t ( r i , t j ) ≈ u ( r i , t j ) − u ( r i , t j − 1 ) Δ t + O ( ( Δ t ) 2 ) (19)

∂ θ ∂ t ( r i , t j ) ≈ θ ( r i , t j ) − θ ( r i , t j − 1 ) Δ t + O ( ( Δ t ) 2 ) (20)

while the first and second order space derivatives are approximated by the central difference formula.

∂ θ ∂ r ( r i , t j ) ≈ θ ( r i + 1 , t j ) − θ ( r i − 1 , t j ) 2 ( Δ r ) + o ( ( Δ r ) 2 ) (21)

∂ 2 u ∂ r 2 ( r i , t j ) ≈ u ( r i + 1 , t j ) − 2 u ( r i , t j ) + u ( r i − 1 , t j ) ( Δ r ) 2 + O ( ( Δ r ) 2 ) (22)

∂ 2 θ ∂ r 2 ( r i , t j ) ≈ θ ( r i + 1 , t j ) − 2 θ ( r i , t j ) + θ ( r i − 1 , t j ) ( Δ y ) 2 + O ( ( Δ r ) 2 ) (23)

Replacing j by j + 1 in (19), (20), (22) and (23) gives an iterative system, which does not restrict the time step. Thus the transport Equations ((6) and (7)) at the grid point ( i , j ) are linearized. The momentum Equation reads:

u i j + 1 − u i j Δ t = u i − 1 j + 1 − 2 u i j + 1 + u i + 1 j + 1 ( Δ r ) 2 + 1 r ( i ) u i + 1 j − u i − 1 j 2 ( Δ r ) + θ i j (24)

θ i j + 1 − θ i j Δ t = 1 P r ( 1 + 4 3 R [ C T + θ i j ] 3 ) [ θ i + 1 j + 1 − 2 θ i j + 1 + θ i + 1 j + 1 ( Δ r ) 2 + 1 r ( i ) θ i + 1 j − θ i − 1 j 2 ( Δ r ) ] + 4 P r R 2 ( C T + θ i j ) 2 [ θ i + 1 j − θ i − 1 j 2 ( Δ r ) ] 2 (25)

The dimensionless mathematical model representing the transient natural convection flow in an annulus with thermal radiation is solved numerically using finite difference method. The problem examines two fluids air ( P r = 0.71 ) and water ( P r = 7.0 ) taking into account the influence of thermal radiation parameter R and temperature difference parameter C T . The numerical scheme is validated using the steady state analytical solution derived by perturbation method.Choosing small perturbation parameter R = 0.1 , the results are found in good agreement between numerical and analytical solution at large value of time as depicted in

higher value of Pr which create weak convection current inside the annulus.

From

The variation of skin friction for different values of time and temperature difference parameter ( C T ) is presented in

( r = 1 ) and inner surface of the outer cylinder ( r = 2 ) respectively. However, the values of skin friction are higher at the outer surface of the inner cylinder ( r = 1 ) in comparison to the skin friction at the inner surface of the outer cylinder ( r = 2 ) . In addition, as time increases skin friction increases and finally reached its steady state value.

From

The effect of temperature difference, thermal radiation and Prandtl number on

transient-natural convection flow inside vertical annulus is analysed analytically as well as numerically. The expression for velocity, temperature, skin-friction and Nusselt number for the nonlinear partial differential equation are obtained analytically by the well-known perturbation series method under steady state operating condition. The numerical solution is obtained by implicit finite difference method for transient situation. The outcome of the result shows that:

1) Velocity and temperature increase with increase of thermal radiation and temperature difference parameter.

2) During transient state, the maximum value of velocity occurs at smaller radial distance from the outer surface of inner cylinder and then decreases to zero asymptotically.

3) Thermal radiation and temperature difference parameter enhances the skin friction at the outer surface of inner cylinder as well as inner surface of the outer cylinder.

4) Nusselt number is higher at small values of C T and R while decreases with increase in dimensionless time (t) to zero with increase in C T and R at the outer surface of inner cylinder ( r = 1 ) .

5) Nusselt number increases with increase in C T and R and dimensionless time (t) at the inner surface of outer cylinder ( r = 2 ) .

6) During course of this investigation, our results are found in good agreement between steady state and transient solution after some sufficiently large time.

The authors wish to thank the referees for their valuable comments to improve the quality of the paper.

Jha, B.K., Yabo, I.B. and Lin, J.-E. (2017) Transient Natural Convection in an Annulus with Thermal Radiation. Applied Mathematics, 8, 1351- 1366. https://doi.org/10.4236/am.2017.89100

C 1 = − 4 [ C T 2 + 2 C T + 1 ] [ ln ( λ ) ] 2 ; C 2 = 8 [ C T + 1 ] [ ln ( λ ) ] 3 ; C 3 = − 4 [ ln ( λ ) ] 4

C 4 = C 1 2 [ 2 λ 2 ( ln ( λ ) ) 2 + 11 λ 2 ln ( λ ) 16 − ln ( λ ) 16 + 3 λ 2 8 ln ( λ ) − λ 2 2 − 3 8 ] + C 2 6 [ 5 λ 2 ( ln ( λ ) ) 3 12 − 5 λ 2 ( ln ( λ ) ) 2 48 − 3 λ 2 ln ( λ ) 8 − 3 λ 2 4 ln ( λ ) − ( ln ( λ ) ) 2 16 + 9 λ 2 8 + 3 4 ] + C 3 12 [ λ 2 ( ln ( λ ) ) 4 4 − 33 λ 2 ( ln ( λ ) ) 3 172 − ( ln ( λ ) ) 3 16 − λ 2 ( ln ( λ ) ) 2 + 9 λ 2 ln ( λ ) 4 + 15 λ 2 8 ln ( λ ) − 3 λ 2 ]

C 5 = C 1 2 ( 3 8 + 1 16 ln ( λ ) ) − C 2 2 ( 3 4 − 1 16 [ ln ( λ ) ] 2 ) + C 3 2 ( 5 16 + 1 96 [ ln ( λ ) ] 3 )

a radius of the inner cylinder

b radius of the outer cylinder

g acceleration due to gravity

C T temperature difference parameter

P r Prandtl number

R thermal radiation parameter

t ′ time

r ′ radial coordinate

T 0 initial temperature

t dimensionless time

T w temperature of the outer surface of inner cylinder

u vertical component of velocity

z ′ vertical co-ordinate, direction of flow

α thermal diffusivity

β coefficient of thermal expansion

κ mean absorption coefficient

K thermal conductivity

ρ density of the fluid

δ Stefan-Boltzmann constant