Unsteady free convection flow between two vertical plates with variable temperature and mass diffusion

1.

Introduction

The Natural convection has been analyzed extensively by many  researchers. Sometimes, along with the free convection currents caused by difference in temperature, the flow is also affected by the differences in concentration or material constitution. There are many situations where the convection heat transfer phenomena are accompanied by mass transfer also. When mass transfer takes place in a fluid  ??, the mass is transferred purely by molecular diffusion resulting from concentration gradients. For low concentration of the mass in the fluid and low mass transfer rates, the convective heat and mass transfer processes are similar in nature. A number of investigations have already been carried out with combined heat and mass transfer under the assumption of different physical situations. The illustrative examples of mass transfer can be found in the book of Cussler [1]. The Free convection resulting from the combined thermal and mass buoyancy forces has received significant attention of engineers and scientists due to many applications in engineering and technological processes. The significant applications include the thermal processes in the vertical mounting board electronic components, channel-chimney systems, thermal comfort dynamics in building services, nuclear reactor thermal hydraulics and frost formation. The Theoretical investigation of unsteady free convection flow due to heat and mass transfer in vertical parallel plate channel received less attention than numerical investigations. It is well known that the effects of radiation on free convection flow problems  have  become more important industrially. Many engineering processes occur at high temperatures, the knowledge of radiative heat transfer plays significant role in the design of equipments. Nuclear power plants, gas turbines and various propulsion devices for aircraft, missiles, satellites and space vehicles are examples of such engineering processes. At high operating temperature, the radiation effect can be quite significant. The study of free convection flow with radiative heat and mass transfer also plays an important role in biological sciences. The Effects of various parameters on human body can be studied and appropriate suggestions can be given to the persons working in hazardous areas with noticeable effects of magnetism and heat variation.

The results of theoretical analysis can be served as a guide for both experimental and numerical investigations. The Natural convection heat and mass transfer in a vertical channel consisting of two vertical parallel plates, one porous and the other non-porous, with opposing buoyancy forces was studied numerically by using finite difference method and experimentally by Lee et al. [2]. They observed that the mass transfer from the porous wall resulted in a downward flow while the heat transfer from the non-porous wall resulted in an opposing upward flow. The effects of latent heat transfer associated with the thin liquid film vaporization on the heat transfer in natural convection flows driven by the combined buoyancy forces of thermal and mass diffusion between vertical parallel plates by using an implicit finite difference method has been studied by Yan et al.[3]. A numerical analysis for developing laminar flow between vertical parallel plates with natural convection heat and mass transfer for uniform wall temperature and the concentration  have been presented by Nelson and Wood [4]. Nelson and Wood [5] have also obtained an analytical solution for fully developed heat and mass transfer natural convection flow between vertical parallel plates with asymmetric boundary conditions. Yan and Lin [6] have carried out a numerical analysis to investigate the effects of latent heat transfer and finite liquid film evaporation on the channel wall and on the natural convection heat and mass transfer. They specifically presented the results for ethanol film evaporation and water film evaporation. Yan and Lin [7] have also carried out a numerical study to investigate the evaporative cooling of liquid falling film through interfacial heat and mass transfer in natural convection channel flows. The results for heat and mass transfer rates have specifically presented for ethanol film evaporation and the predicted results were also contrasted with the experimental results obtained by Yan and Lin [8]. Desrayaud and Lauriat [9] have made a numerical study to investigate buoyancy induced by heat and mass transfer analogy for condensation of  humid air flowing in a vertically heated channel based on thin-film assumptions. A finite-difference analysis has been presented by Salah El-Din [10] to investigate the effect of the thermal and mass buoyancy forces on the development of laminar mixed convection between two vertical parallel plates with uniform heat and mass fluxes. The closed-form analytic solutions have been derived by Cheng [11] to examine the effect of the vortex viscosity parameter and the buoyancy ratio on the fully developed natural convection heat and mass transfer of a micropolar fluid in a vertical channel with asymmetric wall temperatures and concentrations. Narahari [12] has presented an analytical study by using the Laplace transform technique for the transient free convection flow of a viscous incompressible fluid between two vertical parallel plates in the presence of constant temperature and mass diffusion. The natural convection due to heat and mass transfer between vertical parallel plates has not received much attention despite important applications in many engineering systems.

The main purpose of the present investigation is to study the unsteady free convection flow of an optically dense viscous incompressible fluid between two long vertical parallel plates with variable temperature and mass diffusion in the presence of thermal radiation. Assume that the flow is laminar and the fluid is gray absorbing-emitting radiation but no scattering medium. It is considered that the fluid to be optically thick instead of optically thin in this problem. Rosseland diffusion approximation is used to describe the radiative heat flux in the energy equation. The Closed form solutions of the initial and boundary value problems that govern the flow are obtained by means of the Laplace transform technique. The effects of pertinent flow parameters on the fluid velocity, temperature and mass concentration profiles are presented graphically and the physical aspects of the problem are discussed.

2. Formulation of the problem and its solutions

Consider the unsteady free convection flow of a viscous incompressible fluid between two long vertical parallel plates separated by a distance . Choose a Cartesian  coordinates  system with the - axis taken along one of the plates of the channel and the -axis normal to the plates (See Fig. 1). Initially, at time , both the plates and the fluid are assumed to be at the same temperature  and concentration . At time , the temperature and mass concentration at the plate at   have been raised to  and , respectively,  and  the constant temperature and mass concentration and  the characteristic time while that of the other plate at  continues to remain at the initial temperature  and mass concentration . The flow is considered to be laminar without any pressure gradient in the flow direction. It is also assumed that the radiative heat flux in the - direction is negligible as compared to that in the - direction. The density is assumed to be linearly dependent on mass concentration and temperature buoyancy forces in the equations of motion. This approximation is exact enough for both dropping liquid and gases at small values of the temperature and diffusion differences. As the plates are long along  direction infinitely, the velocity and temperature fields are functions of  and  only.

It should be noted that the temperature and concentration differences were small enough in the analyzed heat and mass transfer process. Therefore,  use of the Boussinesq approximation and analogy between heat and mass transfer processes  is pertinent. The unsteady natural convection flow of a radiating fluid, under usual Boussinesq approximation, is governed by the following equations:

where  is the fluid velocity in the -direction,  the temperature and  the mass concentration of the fluid,  the acceleration due to gravity,  

the coefficient of thermal expansion,  coefficient of volumetric expansion due to concentration change,  the kinematic viscosity,  the fluid density,  the thermal conductivity,  the specific heat at constant pressure,  mass diffusivity and  the radiative heat flux. The heat due to viscous dissipation is neglected for small velocities in the energy equation (2).

The study of combined heat and mass transfer problems are important in many processes such as drying, evaporation at the surface of water body, energy transfer in a wet cooling tower, solidification of binary alloy, dispersion of dissolved materials, drying and dehydration operations in chemical food processing plants and combustion of atomized liquid fuels.

The initial and boundary conditions are

In order to simplify the physical problem, the optically thick radiation limit is considered in the present analysis. The radiative heat flux for an optically thick fluid can be found from Rosseland approximation [13] and its formula is derived from the diffusion concept of radiative heat transfer in the following way:

where  is the Stefan-Boltzman constant and  the spectral mean absorption coefficient of the medium. It should be noted that by using the Rosseland approximation we limit our analysis to optically thick fluids. If the temperature differences within the flow are sufficiently small, then the equation (5) can be linearized by expanding  into the Taylor series about the temperature  and neglecting higher order terms to give

by using the equations (5) and (6), equation (2) becomes

By Introducing the dimensionless variables

equations (1), (7) and (3) become

where  is the radiation parameter,  the Prandtl number,  the Grashof number,  the mass Grashof number,  Schmidt number and .

The corresponding initial and the boundary conditions are

by using Laplace transformation, equations (9) - (11) become

where

The corresponding boundary conditions for ,  and  are

The solution of the equations (13)- (15) subject to the boundary conditions (17) are given by

where  and  .

The inverse Laplace transforms of the equations (18)- (19) give the solution for the velocity, temperature and concentration distributions as

where

and  is a dummy variable and  and  are the dummy functions and erfc  is the complementary error function.

2.1  Solution when Schmidt number Sc=1

The solution for the velocity by equation (23) is not valid for . Since the Schmidt number is a measure of the relative importance of the viscosity and mass diffusivity of the fluid, the case  corresponds to those fluids whose momentum and concentration boundary layer thicknesses are of the same order of magnitude. Therefore, the solution for the velocity field when  has to be obtained separately from equations (9)-(11) subject to the initial and boundary conditions (12) and given by

where

and  is the dummy function and  is given by (24).

3. Results and Discussions

In order to  obtain the clear vision of the physical problem, a parametric study is performed and the obtained numerical results are displayed graphically. In Figs. 2-11, We have presented the non-dimensional fluid velocity  and the fluid temperature for several values of radiation parameter , Grashof number , mass Grashof number , Prandtl number , Schmidt number  and time .The values of Schmidt number are chosen to represent the presence of species by hydrogen (0.24), water  vapor (0.62), ammonia (0.78) and carbon dioxide (0.96) at temperature C and the pressure 1atmosphere. The values of are chosen 0.72 and 7.1 for air and water, respectively, at temperature C and pressure 1 atm.. The values of radiation parameter and the chemical reaction parameter are chosen arbitrarily. It is revealed in Fig. 2 that the absolute value of the fluid velocity increases with an increase in the radiation parameter . It is clearly seen that the velocity decreases with increasing value of , i.e. the larger value of  is causedthe thicker the

momentum boundary layer size. This is because an increase in  is caused an increase in the Rosseland mean radiation absorption coefficient  for fixed  and . It is also clear that the radiation has significant effect on the fluid velocity profiles in the presence of variable temperature and mass diffusion. The influence of the thermal Grashof number  on the fluid velocity  is evident from Fig. 3. It can be observed that the the absolute value of fluid velocity  increases for the increasing values of . It is true physically as the thermal Grashof number  describes the ratio of bouyancy forces to viscous forces. Therefore, an increase in the values of  leads to increase in buoyancy forces, consequently the fluid velocity increases. Here the thermal Grashof number represents the effect of free convection currents. Physically,  means the heating of the cooling fluid  of the boundary surface,  means the cooling of the heating fluid  of the boundary surface and  corresponds the absence of free

convection current. It is also noticed from Fig. 3 that there is a reverse type of flow in the vertical channel for small values of . Fig. 4 shows that the absolute value of the fluid velocity  decreases for the increasing values of mass Grashof number .  This means that the mass buoyancy force opposes the fluid flow in the vertical channel in the presence of variable mass diffusion. Fig. 5 shows that the absolute value of the fluid velocity  decreases with an increase in the Prandtl number . Physically, it is true as the Prandtl number describes the ratio between momentum diffusivity and thermal diffusivity and hence controls the relative thickness of the momentum and the thermal boundary layers. As  increases the viscous forces (momentum diffusivity) dominate the thermal diffusivity  consequently decreases the fluid velocity. The values of Schmidt number  were chosen to be , ,  and , that represent the most common

diffusing chemical species . It is seen from Fig. 6 that the absolute value of the fluid velocity  decreases with an increase in Schimdt number . The physics of this observation is that the increased Schmidt number decreases the molecular diffusivityof the chemical species, which reduces the fluid velocity finally. Also, an increase of  (a predominance of the diffusive transport of momentum over that of mass) represents an increase in the momentum boundary layer thickness with a fixed species diffusivity and this causes the decrease in fluid velocity. Fig. 7 reveals that the absolute value of the fluid velocity  increases near plate and it oscillates away from the plate with an increase in time . It is observed from Fig. 7 that as time increases, the absolute value of the fluid velocity in the vertical channel accelerates the upward direction. This is due to increasing buoyancy effects in the vertical channel  with time increasing.

The temperature variations are presented in Figs. 8-10 for different values of the radiation parameter , Prandtl number  and time. It is seen from Fig. 8 that the fluid temperature decreases with an increase in radiation parameter . This is in agreement with  thisfact that radiation provides an additional means to diffuse energy. From the definition of , an increase in the value of  implies to decreasing the radiation effects. In the presence of radiation, the thermal boundary layer always found to thicken which implies that the radiation provides an additional means to diffuse energy. This means that the thermal boundary layer decreases and there is more uniform temperature distribution across the boundary layer. Fig. 9 shows that the fluid temperature  decreases with an increase in Prandtl number . The reason is that smaller values of  are equivalent to increase in the thermal conductivity of the fluid and therefore, the heat is able to diffuse away from the heated surface more rapidly for higher values . Hence in the case of smaller Prandtl numbers the

thermal boundary layer is thicker and the heat transfer is reduced. The temperature and mass concentration profiles for different values of time  are presented in Fig. 10. It is observed that these profiles increase in the vertical channel  as  time increases. It is seen from Fig. 11 that the concentration profiles increase with an increase in Schimdt number . As expected, if all other physical parameters are kept constant, the mass transfer decreases as  increases, i.e. an increase in the value of the Schmidt number  is associated with  a decrease in the concentration profiles. Further, it may be observed from this figure that the effect of Schmidt number  on concentration distribution slowly decreases in the concentration boundary layer for higher values of .

The  heat and mass transfer rates at the plates  are, respectively given by

where

and ,  and

The Numerical results of the heat transfer rate  and  at the plates  for several values of radiation parameter , Prandtl number  and time  are presented in Figs. 12-13. It is observed from Figs. 12-13 that the heat transfer rates  and  increase with an increase in either radiation parameter  or Prandtl number  or time . This may be explained by the fact that frictional forces become dominant by increasing the values of Prandtl number  and hence yield the greater heat transfer rate. An increase in Prandtl number reduces the thermal boundary layer thickness. The Prandtl number signifies the ratio of momentum diffusivity to thermal diffusivity. The Fluids with lower Prandtl number will have the higher thermal conductivities so that the heat can diffuse from the plate faster than  thefluids with higher  (thinner boundary layers). Hence, Prandtl number can be used to increase the cooling rate in conducting flows. It is clear that the heat transfer rate  is more in the presence of thermal radiation. The negative value

• of  explains physically that there is the heat flow from the hot plate  to the ambient fluid. It is observed from Fig. 14 that the mass transfer rate  decreases while  increases with an increase in either Schmidt number  or time .

From an engineering point of view, the most important characteristic of the flow is the shear stresses at the plate  that are, respectively given by

where

The Numerical values of the non-dimensional shear stress at the plates  are presented in Figs. 15-19 for several values of thermal Grashof number , mass Grashof number , radiation parameter , Prandtl number  and time . Figs. 15-16 show that the shear stresses  and  increase with an increase in either thermal Grashof number  or mass Grashof number . Fig. 17 reveals that the shear stresses  and  reduce for increasing values of Schmidt number . Physically, it is true since an increase in  serves to decrease momentum boundary layer thickness. It is seen from Fig. 18 that the shear stress  reduces while the shear stresses   increases with an increase in Prandtl number . It is observed that an increase in the Prandtl number results in a decrease of the thermal boundary layer thickness and in general lower average temperature within the boundary layer. The reason is that smaller values of  are equivalent to increase in the thermal conductivity of the fluid and therefore, the heat is able to diffuse away from the heated surface more rapidly for higher values of . Hence, in the case of smaller Prandtl number as the

thermal boundary layer is thicker and the rate of hear transfer is reduced. Fig. 17 shows that the shear stresses  and  increase as time   increases. It is also seen that the  shear stresses  and   reduce  with increasing values of radiation parameter .

1. 4.  Conclusion

The Exact solutions for the unsteady free convection flow of an optically dense incompressible viscous fluid between two long vertical parallel plates with variable temperature and mass diffusion in the presence of thermal radiation have been obtained by using the Laplace transform technique. The effects of the pertinent parameters such as the radiation parameter, buoyancy forces, Schmidt number and time on the fluid velocity and temperature fields and shear stresses at the plate have been studied in detail. The study shows that these parameters have significant impact on the velocity, temperature, mass concentration and the shear stresses at the plates. The following conclusions are extracted from this study:

• • An increase in the radiation parameter leads to a decrease in the fluid velocity as well as the temperature.
• • The fluid velocity and temperature decrease with an increase in the Prandtl number.
• • The fluid velocity increases with increases in the Grashof number and an increase in time leads to increases in the fluid velocity, temperature and concentration.
• • It is expected that the results of this study will be served as a mechanism to develop the design of cooling and energy systems and nuclear thermo-hydraulics.

References

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[13]. Rosseland, S. (1936). Theoretical Astrophysics, Oxford University, New York, NY, USA.