The Effect of Impinging Jet on the Flow and Heat Transfer from Isothermal Moving Plate

Abstract The effect of impinging uniform jet on flow and heat transfer over isothermal moving plate is investigated. The flow and temperature fields are studied numerically with different jet velocity ratio (jet to plate velocity ratio 0-5). The laminar flow field is analyzed numerically by solving the steady, two-dimensional incompressible NavierStokes and energy equations. A collocated (non-staggered) grid is used in the momentum equations, which discretized by finite volume method, SIMPLE algorithm is used to adjust the velocity field to satisfy the conservation of mass. The range of Reynolds number is (Re = 10 100). The results show that at high jet velocity ratio (V/U = 5) and Reynolds number (Re = 100), the rate of heat transfer from the plate is doubled.


Introduction
A continuously moving surface through an otherwise quiescent medium has many applications in manufacturing processes. Such processes are hot rolling, wire drawing, metal extrusion, crystal growing, continuous casting, glass fiber production, and paper production Altan et al., [1], Fisher, [2], Tadmor and Klein, [3]. The study of Flow Field and heat transfer is necessary for determining the quality of the Final products of such processes as explained by Karwe and Jaluria [4] and [5]. In almost all the papers cited earlier, the studies concentrated on the boundary layer far away from the extrusion slit where the boundary layer approximations and, hence, the similarity method are valid. However, in most practical applications, the flow and heat-transfer characteristics are very important close to the slit at which both the friction and heat transfer coefficients attain their largest values. On the other hand, the finite-difference method using the full governing equations was used by Karwe and Jaluria [4] and [5], for uniformly moving flat plate with a uniform temperature at the slit, to study the effects near the slit from which the plate emerges. Furthermore, Kang and Jaluria [6] included the buoyancy effects on moving plate in materials processing. In these papers, the effect of jet on the moving plate was not included. However, during the manufacture of some materials such as continuous casting it is often necessary to cool the moving material Jaluria [7]. The present study focuses on the effect of impinging jet on the moving surface using the full governing equations. Figure 1 shows a continuously moving plate emerges from a slit at a dimensionless velocity U=1 and a dimensionless temperature q =1. The motion of the fluid is assumed to be laminar, steady, and two dimensional with thermal active incompressible viscous fluid with constant properties. Subject to these assumptions the governing equations in dimensionless form can be written as:

Boundary Conditions
The boundary conditions used in numerical solution are illustrated in figure 1. A uniform impinging jet of thickness (D) is assumed.

Discretization of the Governing Equations
The governing equations are discretized by using the finite volume method based on nonstaggered (collocated) grid. Since all variables are stored in the center of the control volume, the interpolation method is used to avoid the decoupling between velocity and pressure; this interpolation method is presented in this paper. The continuity and the momentum equations are discretized over the collocated gird shown in figure 2. By integrating the xmomentum equation (2) over the control volume shown in figure (2), we have For numerical stability, the convection term (left side of equation 6) are approximated by the upwind differencing scheme, Patankar [8] as , are the mass flux at the control volume faces e and w respectively. By the same scheme the second convective term can be approximated as , are the mass flux at the control volume faces n and s respectively. The diffusion terms can be approximated by the center difference as Substituting of equs. (7),(8), (9), and (10) into equ. (6) gives:- Where the index (nb) runs over all neighboring points E, W, N, and S, and Where (a U ) is the under-relaxation parameter, necessary for stability, Patankar [8]. The quantity U P 0 in equation (11)

Pressure-Velocity Coupling
The U-and V-momentum equations are solved using guessed values for the pressure field and mass fluxes. The velocity components U * and V * , calculated with these guessed values will not satisfy the continuity equation, so that the velocity components must be corrected as  To enforce mass conservation, velocity and pressure correction are introduced by Rhie and Chow [9] ( ) ( )

The discretization of continuity equation is
To correct the velocities at the faces of the control volume: Where S P is the mass source and is given by

Solution Procedure
For the steady and non-staggered (collocated), the overall SIMPLE solution procedure takes the following steps (1) Guess the pressure field ( * P ).
(11) Correct the control volumes pressure by underrelaxed the pressure correction as (12) Calculate the temperature field (q ) by solving the descritized energy equation.
Return to step 1 and repeat step 1 to 12 until the convergence is attained. For each variable, the sum of the absolute value of the residuals over all the control volume is calculated, and normalized by an appropriate quantity F n , typically the inlet mass or momentum flux, this normalized sum of the absolute residuals should be satisfies

Grid Independence
The

Validation
The numerical solution is validated by comparing results of the axial velocity at the trailing edge of the plate with the numerical results of Sami and Mohamed [1] for Reynolds number (Re = 100) , (There is no jet V= 0). Figure 3 shows an excellent agreement between our results and the numerical results of Sami and Mohamed [10].
In figure (4a-d, left) the contours of the predicted streamlines are shown for jet velocity ratio (0, 1, 3, and 5) for Reynolds number (Re = 10). As the jet velocity ratio increases while the velocity of the plate remain constant, at this low Reynolds number there is no effect from the jet on the isothermal lines as shown in figure (4a-d, right), so that the average Nusselt number remains constant as shown in figure (6).
In figure (5a-d, left) the contours of the predicted streamlines are shown for Reynolds number (Re = 100) and for (0, 1, 3, and 5). As the jet velocity ratio increases, and this high Reynolds number, the rate of heat transfer (Nusselt number) is doubled due to the increase of the temperature gradient below the jet as shown in figure (5b-d, right), which causes this increasing in heat transfer rate.

Conclusions
The finite volume method with collocated grid is used to analyze the two dimensional steady flow and heat transfer over isothermal moving plate. The results show that at high jet velocity ratio (V/U = 5) and high Reynolds number (Re =100), the jet velocity causes high temperature gradient so that the heat transfer rate is doubled.