Forschung im Ingenieurwesen 68 (2003) 101 – 104 Ó Springer-Verlag 2003 DOI 10.1007/s10010-003-0112-9
Chemical reaction effects on moving infinite vertical plate with uniform heat flux and variable mass diffusion R. Muthucumaraswamy, T. Kulandaivel
101
k thermal conductivity Kl chemical reaction parameter K dimensionless chemical reaction parameter N buoyancy ratio parameter Pr Prandtl number Sc Schmidt number T0 temperature of the fluid near the plate t0 time t dimensionless time u0 velocity of the fluid in the x0 -direction u0 velocity of the plate u dimensionless velocity Einfluß chemischer Reaktionen auf eine bewegte, coordinate axis normal to the plate vertikale Platte mit konstanter Wa¨rme-und variabler y0 y dimensionless coordinate axis normal to the plate Massenstromdichte Zusammenfassung In dem vorliegenden Artikel wird eine theoretische Lo¨sung fu¨r die Stro¨mung an einer impulsartig Greek symbols bewegten vertikalen Platte mit konstanter Wa¨rme- und a thermal diffusivity variabler Massentromdichte unter Beru¨cksichtigung einer b volumetric coefficient of thermal expansion homogenen, chemischen Reaktion 1. Ordnung vorgestellt. b volumetric coefficient of expansion with Das beschreibende, dimensionslose Gleichungssystem concentration wird mit Hilfe der Laplace-Transformation gelo¨st. Die l coefficient of viscosity erhaltenen Geschwindigkeits-, Temperatur- und Konzen- m kinematic viscosity trationsprofile werden graphisch dargestellt. Es wird die q density of the fluid Abnahme der o¨rtlichen Geschwindigkeit und der o¨rtlichen s dimensionless skin-friction Konzentration mit zunehmendem chemischen Reaktions- h dimensionless temperature parameter aufgezeigt. g similarity parameter erfc complementary error function List of symbols A constant Subscripts C0 w conditions at the wall species concentration in the fluid 1 conditions in the free stream C dimensionless concentration Cp specific heat at constant pressure D mass diffusion coefficient 1 Gm mass Grashof number Introduction Gr thermal Grashof number Chemical reactions can be codified as either heterogeg acceleration due to gravity neous or homogeneous processes. This depends on wheAbstract Theoretical solution to the problem of flow past an impulsively started infinite vertical plate in the presence of uniform heat flux and variable mass diffusion are presented here, taking into account the homogeneous chemical reaction of first-order. The dimensionless governing equations are solved using Laplace-transform technique. The velocity, temperature and concentration profiles are shown on graphs. It is observed that the velocity as well as concentration decreases with increasing the chemical reaction parameter.
Received: 18 March 2003 R. Muthucumaraswamy (&) Department of Information Technology, Sri Venkateswara College of Engineering, Sriperumbudur 602 105, India e-mail:
[email protected] T. Kulandaivel Department of Applied Mathematics, Sri Venkateswara College of Engineering, Sriperumbudur 602105, India e-mail:
[email protected]
ther they occur at an interface or as a single phase volume reaction. In well-mixed systems, the reaction is heterogeneous, if it takes place at an interface and homogeneous, if it takes place in solution. In most cases of chemical reactions, the reaction rate depends on the concentration of the species itself. A reaction is said to be of first order, if the rate of reaction is directly proportional to the concentration itself [1]. In many chemical engineering processes, there does occur the chemical reaction between a foreign mass and the fluid in which the plate is moving. These processes take place in numerous industrial applications like polymer production, manufacturing of ceramics or glassware and food processing.
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A polymer or metal sheet extruded continuously from a die, or a long fibre or filament travelling between a feed roller and a take-up roller is the typical example of moving continuous surface. Chambre and Young [2] have analyzed a first order chemical reaction in the neighbourhood of a horizontal plate. Das et al. [3] have studied the effect of homogeneous first order chemical reaction on the flow past an impulsively started infinite vertical plate with uniform heat flux and mass transfer. Again, mass transfer effects on moving isothermal vertical plate in the presence of chemical reaction studied by Das et al. [4]. The dimensionless governing equations were solved by the usual Laplace-transform technique. It is proposed to study the flow past an impulsively started infinite vertical plate with uniform heat flux and variable mass diffusion in the presence of a homogeneous chemical reaction of first-order. The dimensionless governing equations are solved using the Laplace-transform technique. The solutions are in terms of exponential and complementary error function.This study is found useful in chemical processing industries such as fibre drawing, crystal pulling from the melt and polymer production.
2 Analysis Here the flow of a viscous incompressible fluid past an impulsively started infinite vertical plate with uniform heat flux and variable mass diffusion is considered. It is assumed that the effect of viscous dissipation is negligible in the energy equation and there is a first order chemical reaction between the diffusing species and the fluid. The x0 -axis is taken along the plate in the vertically upward direction and the y0 -axis is taken normal to the plate. At time t 0 0, the plate and fluid are at the same temperature 0 0 T1 and concentration C1 . At time t 0 > 0, the plate is given an impulsive motion in the vertical direction against the gravitational field with uniform velocity u0 , the temperature from the plate to the fluid at a constant rate and the species concentration level near the plate is made to rise linearly with time. Then by usual Boussinesq’s and boundary layer approximation, the unsteady flow is governed by the following equations ou0 o2 u0 0 0 0 0 ¼ gbðT T Þ þ gb ðC C Þ þ m ; 1 1 ot 0 oy02 qCp
oT 0 o2 T 0 ¼ k ; ot 0 oy02
oC0 o2 C0 ¼ D Kl C0 ; ot 0 oy02
ð1Þ ð2Þ ð3Þ
with the following initial and boundary conditions
u0 ¼ 0;
0 0 T ¼ T1 ; C 0 ¼ C1 for all y0 ; t 0 0; oT 0 q 0 0 ¼ ; C 0 ¼ C1 þ ðCw0 C1 ÞAt 0 t0 > 0: u0 ¼ u0 ; oy k at y0 ¼ 0; 0 0 ; C0 ! C1 as y0 ! 1 ð4Þ u0 ¼ 0; T 0 ! T1
where A ¼ u20 =m.
On introducing the following non-dimensional quantities: 0 u0 t 0 u2 y0 u0 T 0 T1 ; t ¼ 0; y ¼ ; h¼ ; u0 Gr m m ðqm=ku0 Þ 0 0 gbqm2 C 0 C1 mgb ðCw0 C1 Þ Gr ¼ ; C ¼ ; Gm ¼ ; 3 4 0 0 ku0 Cw C1 u0 lCp m Gm mKl ; Sc ¼ ; N ¼ ; K¼ 2 ð5Þ Pr ¼ k u0 D Gr
u¼
in (1)–(4), leads to
ou o2 u ¼ h þ NC þ 2 ; ot oy
ð6Þ
oh 1 o2 h ; ¼ ot Pr oy2
ð7Þ
oC 1 o2 C KC : ¼ ot Sc oy2
ð8Þ
The initial and boundary conditions in non-dimensional quantities are
u ¼ 0;
h ¼ 0; C ¼ 0 for all y; t 0; 1 oh ¼ 1; C ¼ t at y ¼ 0; t > 0: u ¼ ; Gr oy u ¼ 0;
h ! 0;
ð9Þ
C ! 0 as y ! 1:
The Eqs. (6)–(8), subject to the boundary conditions (9), are solved by the usual Laplace-transform technique and the solutions are derived as follows:
pffiffiffiffiffi pffiffi expðg2 PrÞ p ffiffiffiffiffi h ¼ 2 t pffiffiffi g erfc ðg PrÞ ð10Þ p Pr 1 N erfcðgÞ þ u¼ Gr a2 ð1 ScÞ pffiffi t t 4 pffiffiffiffiffi pffiffiffi ð1 þ g2 Þ expðg2 Þ þ p 3ðPr 1Þ Pr 4 pffiffiffi ð1 þ g2 PrÞ expðg2 PrÞ p pffiffiffiffiffi pffiffiffiffiffi þ g Pr ð6 þ 4g2 PrÞerfcðg PrÞ 2 gð6 þ 4g ÞerfcðgÞ Nt 2g 2 2 ð1 þ 2g ÞerfcðgÞ pffiffiffi expðg Þ þ að1 ScÞ p pffiffiffiffiffi pffiffiffiffiffi N 2 expð2g atÞerfcðg atÞ 2a ð1 ScÞ pffiffiffiffiffi pffiffiffiffiffi þ expð2g atÞerfcðg þ atÞ h pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi N expð2g KtScÞerfcðg Sc 2 2a ð1 ScÞ pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffiffi i pffiffiffiffiffiffi þ KtÞ þ expð2g KtScÞerfcðg Sc KtÞ h pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffiffi Nt expð2g KtScÞerfcðg Sc þ Kt Þ 2að1 ScÞ pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffiffi i þ expð2g KtScÞerfcðg Sc KtÞ
R. Muthucumaraswamy, T. Kulandaivel: Chemical reaction effects on moving infinite vertical plate
pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffiffi Ng t Sc h þ pffiffiffiffi expð2g KtScÞerfcðg Sc KtÞ 2 K að1 ScÞ pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffiffi i expð2g KtScÞerfcðg Sc þ KtÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N expðatÞ h expð2g Scða þ KÞt Þ þ 2 2a ð1 ScÞ pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi erfcðg Sc ðK þ aÞtÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i þ expð2g Sc ða þ KÞt Þ erfc ðg Sc þ ðK þ aÞtÞ pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffiffi 1 C ¼ expð2g KtScÞerfcðg Sc þ KtÞ 2 pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffiffi þ expð2g KtScÞerfcðg Sc KtÞ
ð12Þ
pffiffi where, g ¼ y=2 t , and a ¼ KSc=ð1 ScÞ. The purpose of the calculations given here is to assess the effects of the parameters Sc; N and K upon the nature of the flow and transport. The numerical values of the velocity, temperature and concentration are computed for different parameters like Schmidt number, buoyancy ratio and chemical reaction parameter. The temperature profiles are calculated from the Eq. (10) and these are shown in Fig. 1 for different Prandtl number and time. The effect of Prandtl number is very important in temperature field. It is observed that the temperature increases with increasing the time. There is a fall in temperature due to increasing values of the Prandtl number. As expected heat transfer effect is more in air as compared to that of water. The effect of chemical reaction parameter and Schmidt number are very important in concentration field. The numerical values of the concentration profiles are computed from the Eq. (12) and these values are plotted in Fig. 2 for different values of the chemical reaction
Fig. 1. Temperature profiles
103
ð11Þ
Fig. 2. Concentration profiles for different K
parameter. It is observed that the concentration increases with decreasing chemical reaction parameter. The effect of Schmidt number and time in the presence of chemical reaction are studied in Fig. 3. It is clear that the concentration increases with decreasing Schmidt number. This trend is just reversed with respect to time. The effects of buoyancy ratio parameter for both aiding (N > 0) as well as opposing (N < 0) are shown in Fig. 4. It is observed that the velocity increases in the presence of aiding flows and decreases with opposing flows. The velocity profiles for different values of the Schmidt number, chemical reaction parameter and the time are shown in Fig. 5. It is clear that the velocity increases with increasing chemical reaction parameter. It is also observed that the velocity decreases with
Fig. 3. Concentration profiles for different Sc and t
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Fig. 5. Velocity profiles for different K, Sc and t
(i) The velocity as well as concentration increase with decreasing chemical reaction parameter. increasing values of the Schmidt number. Moreover, the (ii) The velocity increases in the presence of aiding flows (N > 0) and decreases with opposing flows (N < 0). velocity increases with increasing time. Fig. 4. Velocity profiles for different N
3 Conclusions The theoretical solution of flow past an impulsively started infinite vertical plate in the presence of uniform heat flux and variable mass diffusion is studied. A homogeneous firstorder chemical reaction between the fluid and the species concentration. The dimensionless governing equations are solved by the usual Laplace-transform technique. The effect of different parameters like Schmidt number, buoyancy ratio and chemical reaction parameter are studied. It is observed that the velocity increases due to the presence of the foreign mass. Conclusions of the study are as follows:
References 1. Cussler EL (1988) Diffusion Mass Transfer in Fluid Systems. Cambridge University Press, London 2. Chambre PL, Young JD (1958) On the diffusion of a chemically reactive species in a laminar boundary layer flow. Phys. Fluids 1, 48–54 3. Das UN, Deka RK, Soundalgekar VM (1984) Effects of mass transfer on flow past an impulsively started infinite vertical plate with constant heat flux and chemical reaction. Forschung im Ingenieurwesen 60, 284–287 4. Das UN, Deka RK, Soundalgekar VM (1999) Effects of mass transfer on flow past an impulsively started infinite vertical plate with chemical reaction. The Bulletin, GUMA, 5, 13–20