ABSTRACT
This article is concerned with a
numerical study of mixed convection flow and heat transfer in vertical annular
duct, numerical results are presented for air-water vapour systems. The
developing flow and heat transfer performance were examined by the fully
implicit finite difference scheme. A marching procedure is employed for
solution of the equation of mass momentum, energy and concentration in the
flow. The effects of outer wall temperature 
, Reynolds number 
and ratio of radius 
on the heat and mass transfer performance were investigate
in great detail. Results reveal that the convection of heat in the flow is
dominated by the transfer of latent heat is conjunction with water film
evaporation or condensation. In addition, better heat and mass transfer is
noticed in the case with lower radius ratio 
near the inner wall and viseversa near the outer wall.
1. INTRODUCTION
Latent heat transfer associated with liquid film evaporation in laminar mixed convection heat transfer in vertical ducts is important in many engineering applications. Noticeable examples include the process of the evaporative cooling for waste heat disposal, channel type solar energy collectors and protection of system components from a high-temperature gas stream. The purpose of this article is the investigation of the extent of the energy transport through mass diffusion, a latent energy exchange process in comparison with that through thermal diffusion, a sensible energy transport process, in vertical annular duct. A vast amount of work, both theoretical and experimental exists in the literature to study the effects of aiding and opposing buoyancy forces on laminar mixed convection. Naserddine et al. [1,2] numerically examined the effect of variable properties in laminar aiding and opposing mixed convection in vertical tube flows without mass diffusion. Chang et al. [3] studied the effects of combined buoyancy forces of heat and mass diffusion on the natural convection flows in a vertical open tube. Analysis of combined heat and mass transfer on laminar forced convection along a vertical open-ended vertical pipe has been reported by Lin et al. [4]. They found that the heat transfer along the wetted wall is dominated by the transfer of latent heat in association with film evaporation. Similar study was conducted by Feddaoui et al. [5] by analysing the combined buoyancy effects of thermal and mass diffusion in laminar mixed convection tube flows. This study was presented for both air-water vapour and air-ethanol systems. In addition Al-Arabi et al. [6] and Joshi [7] investigated the natural convection heat transfer in vertical open annular duct flows induced by the thermal buoyancy forces. Recently Yan and Lin [8] examined a natural convection flows in vertical annulus ducts induced by the combined buoyancy effects of thermal and mass diffusion. Their results show that the tremendous enhancement in heat transfer due to the exchange of latent heat in association with film evaporation and condensation was found. In spite of its importance in engineering applications, the laminar mixed convection heat and mass transfer in wetted annular duct has received less attention. In the present work attention is focused on the study of the effects of the film evaporation and condensation along the wetted walls on
the heat and mass transfer in vertical annular duct.
2. ANALYSIS
2.1. Physical model and assumptions
The physical
model under consideration and the co-ordinates are depicted in figure1.
The inner and outer radius are 
and 
, respectively. The inner tube is maintained at a uniform
temperature 
and the outer tube at , the flow of the mixture enters the tube from the bottom at
the ambient temperature 
with relative humidity 
. The flow is influenced by the combined buoyancy forces due
to the differences in temperature and concentration of vapour mixture between
the wetted walls and the ambient. The flow is considered laminar and steady.
Figure 1. Schematic diagram of the physical system.
The following simplification assumptions are made in the present study :
1. The moist mixture is an ideal gas with variables thermophysical properties with temperature and concentration. The thermophysical properties are available in [9].
2. Viscous heat dissipation, thermal radiation and other secondary effects are negligible.
3. Thermodynamic equilibrium exists at the air-water interface.
4. The liquid film on the wetted walls is assumed to be extremely thin. With this assumption, the liquid film can be regarded as the boundary condition for heat and mass transfer, the film is stationary and at the same uniform temperature as the tube walls temperature.
2.2. Dimensionless governing equations and boundary conditions
In order to nondimensionalise the governing equations derived form the basic conservation laws of mass, momentum, energy and concentration, the following variables are defined :
, 
, 
, 
, 
, 
,
, 
, 
, 
(1)
Where 
is a saturated mass fraction of vapour at 
and 
. Under the above assumptions, the nondimensional governing
equations in cylindrical co-ordinates are expressed as:
continuity equation
(2)
axial-momentum equation
(3)
energy equation
(4)
concentration equation of water vapour
(5)
Equations (2)-(5) are completed by the following set of boundary conditions :
, 
, 
, 
, 
, 
,
, 
, ,
, 
(6)
One constraint to be satisfied in the analysis of a steady channel flow is the overall mass balance at every axial location:
(7)
2.3. Heat and mass-transfer parameters
The local heat exchange between the air stream and the water film depends on two related factors: the interfacial temperature gradient on the air side results in sensible convective heat transfer, and the evaporative mass transfer rate on the water film side results in latent heat transfer (Manganaro and Hana [10] ; Fedorov et al. [11]). The total convective heat transfer rate from the film interface to the air stream can be expressed as follows:
(8)
The local Nusselt number can be defined as
(9)
Where 
and 
are, respectively, the local Nusselt numbers for sensible
and latent heat transfer, and are defined as :
,
(10)
,
(11)
where
,
(12)
Here 
and 
signify the importance of energy transport through species
diffusion relative to that through thermal diffusion.
Similarly, the local Sherwood number then becomes:
,
(13)
3. SOLUTION METHOD
The governing equations (2)-(5) with boundary conditions equations (6), are solved numerically using a fully implicit numerical scheme in which the axial convective term is approximated by the upstream difference and the radial convection and diffusion terms by the central difference. This scheme is employed to transform the governing equations into finite-difference equations. Each system of finite difference equations forms a tridiagonal set, which can be solved iteratively using the line by line TDMA algorithm [12].
After specifying the flow and thermal conditions, the numerical solution is advanced forward and step by step as follows:
1- For any axial
location, guess 
and solve the finite-difference forms of equations (2)-(5)
for 
, 
and 
.
2- Integrate numerically the continuity equation to find :
(14)
3- Check the satisfaction of the overall conservation of mass at any axial location :
(15)
4- Check the satisfaction of the convergence of velocity, temperature and mass fraction. If the relative error between two consecutive iterations is small enough, i.e. :
: , and
(16)
If not repeat
procedures 1 to 4 until the conditions (16) is fulfilled. If equation (15) is
not satisfied, guess a new and repeat procedure 1 to 4 for the current axial location.
The correction of
the pressure gradient and axial velocity profile at each axial station in
order to satisfy the global mass flow constraint is achieved using a method
due to the Raithby and Schneider [13],
described by Anderson et al. [14]. To illustrate,
we will let 
. We make an initial guess for 
and calculate provisional velocities 
and a provisional mass flow rate 
. Due to the linearity of the momentum equation with frozen
coefficients, the correct velocity at each point would be:
(17)
Where 
is the change in the pressure gradient required to satisfy
the global mass flow constraint. We define 
. The difference equations are actually differentiated with
respect to the pressure gradient in form. The coefficients for the unknown in
these equation will be the same as for the original implicit difference
equations. The Thomas algorithm is used to solve the system of algebraic
equations for 
.
The boundary
conditions on must be consistent with the velocity boundary conditions.
On boundaries where the velocity is specified, 
, whereas on boundaries where the velocity gradient is
specified 
. The solution for is then used to compute by noting that 
is the correction in velocity at each point required to
satisfy the global mass flow constraint thus we can write:
(18)
Where the integral
is evaluated by numerical means. The 
in equation (18) is the known value specified by the
initial conditions. The required value of is determined from equation (18). The correct values of
velocity 
can then be determined from equation (17). The continuity
equation is then used to determine 
.
In
order to obtain enhanced accuracy in the numerical computation, grids are
chosen to be uniform in the radial direction but non uniform in the axial
direction to account for the drastic variations of velocity, temperature and
mass fraction in the near entrance region. Grid independence of the results is
established by employing different size meshes, ranging from 
to 
as showed in Table I. It is found from Table I
that the differences in the local Nusselt number for the computations are
always less than 2 percent. The results to be presented are computed by
using 
grid.
Table I.
Comparisons of local interfacial Nusselt number 
on the outer wall for various
grid arrangements
for 
, 
, 
, 
, 
|
|
|
|
|
|
|
|
|
|||||||
|
0.00172 0.01203 0.08435 0.28447 0.5000 |
57.38 36.16 30.05 29.58 28.87 |
57.24 36.00 29.97 29.49 28.78 |
53.74 36.06 30.07 29.62 28.91 |
54.39 36.22 30.17 29.72 29.00 |
55.53 36.50 30.36 29.90 29.18 |
54.54 36.70 30.66 30.29 29.62 |
|
|||||||
at 
and 
. The flow enters from the bottom end along a vertical
annular ducts. The ratio of radius 
, 
or 
.
).
increase and becomes constant at the exit of the annular
duct. A large 
is observed for higher Reynolds number and at higher wall
temperature 
decrease along the outer wall, and a slightly larger 
is found for the low Reynolds number and low outer wall
temperature. This implies that the direction of the sensible heat transfer is
from the outer wall to the inner wall. Shown in figures 6a and
6b are the development of latent Nusselt numbers near the inner and outer
wall, respectively. The latent heat transfer increase along the inner wall and
decrease near the outer wall. A larger latent Nusselt number is experienced for
a higher 
in figure 6b, i.e., the flow with 
shows a higher values of 
is the some of 
, and near the outer wall, 
and 
are depicted in figures 7a and 7b,
respectively. Basically, the trends of the distributions of Sherwood numbers are
in resemblance with those of Nusselt numbers. This is due to the fact that the
magnitudes of 
and 
are of order of one in this study (
and 
). 
for a given hydraulic diameter on the total Nusselt number
and Sherwood number along the inner and outer walls. Figure 8a
shows that near the entrance (
) the larger 
) the reverse trend is noted. For the distribution near the
outer wall an opposite results is assigned (figure 8b). The axial
variations of Sherwood numbers are presented in figure 8b for
various ratio of radius 
