Diffusion Flux in a Natural Convection

Consider a vertical plate in a gravitational field [3], Assume that a fast chemical reaction takes place at the surface of this plate, so that the concentration of reacting substance at the plate is equal to zero. Far from the plate, this concentration is equal to Cq. Suppose that the density of the solution does not strongly depend on the concentration Co, so ve can write 

Choose the system of coordinates such that the jc-axis is vertical and the y-axis is perpendicular to the plate (and directed toward the solution). The bottom edge of the plate corresponds to x = 0. Assume that the change of concentration mostly occurs in the diffusion boundary layer. Since the flow of liquid is driven by the gradient of concentration, the flow occurs in this layer, too that is, the viscous boundary layer coincides with the diffusion layer. The gravity force /)g acts on a unit solution volume. Since p changes with height, so does the force. At p = p(Co) = const, this force does not cause the flow since the force is counterbalanced by the pressure gradient. The flow may be caused by density variations from / (Co). Since the difference Ap = y9(Co) —/ (C) is small, the equations of motion in the first approximation can be written in the form 

The concentration distribution is described by the equation of convective diffusion  

The system of equations (6.145), (6.147), and (6.148) with the boundary conditions (6.149) has a self-similar solution. Indeed, if we introduce the dimensionless variable 

As it has been indicated already, for extremely diluted solutions, Sc 1. Due to this inequality, the integrals converge rapidly. Therefore at small values of tj, the integrals are mostly determined by the value of f, while at large tj, the behavior of f does not noticeably influence the distribution of p. Because of this, without any detriment to the accuracy, we can replace the boundary condition at the infinity by the condition at a finite distance )/o from the wall, taking it to be equal to the thickness of the boundary layer 5d, that is, 

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Diffusion Flux in a Natural Convection 1143 Accordingly, the boundary conditions are transformed to... 

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