## REq

where p is the density in kg/m3, v is the average flow rate in m/s, and d is the diameter of the tube in m. When Re is small (< 100), the flow regime is laminar.

Consider the case of parallel flow over a flat plate. The velocity distribution in the fluid phase is shown in Fig. 6. The flow is uniform, with velocity V0, before the plate is inserted into the system. After the plate is introduced, a contiguous, low-velocity boundary layer develops. The width of the resulting boundary a layer for the condition z > S0 is given by:

where z is the distance measured from the leading edge of the susceptor, D is the diffusion coefficient, and V0 is the entry velocity (Ref 62). The boundary layer width is proportional to the square root of z and inversely proportional to the square root of V0.

Effect of Substrate Heating. The introduction of heat adds another dimension of complexity to the gas flow, because convection is important in controlling gas flow along length of pipe. In a horizontal reactor chamber, the temperature difference from the chamber wall to the heated susceptor is enough to influence the flow of gas. When the thermal diffusivity is large and the temperature difference is small, the exchange of momentum is so fast, on a microscopic scale, that no real macroscopic gas-density gradients can develop. Hence, no buoyancy is observed, and the gas is said to be metastable. For higher temperature gradients and smaller heat diffusivities, the heavy, cold gas moves downward and the light, hot gas near the susceptor moves upward. The buoyancy forces are so large that free convection occurs. The onset of free convection is characterized by the Rayleigh number, Ra:

a gCp p 2h3AT h k

where a is the coefficient of thermal expansion in 1/K, g is the gravitational constant in m/s2, Cp is the specific heat in J/kg • K, p is the gas density in kg/m3, h is the free height above the susceptor in m, AT is the difference in temperature between the susceptor and the reactor wall, n is the dynamic viscosity in kg/ms, and k is the thermal conductivity in J/m • s • K). When Ra is less than or equal to 1700, the gas is stable. When Ra is greater than 1700, free convection occurs.

The flux, J-, of a chemical species, p,, toward the susceptor surface now depends on the local temperature, the local temperature gradient, and the creation or annihilation of molecules during the reaction (Ref 63). Flux can be expressed as:

Dirt

P dPi dT

where D,(T) = D0(T/T0)2 is the binary diffusion coefficient of species i as a function of T, R is a gas constant equal to 8.31 J/mol • K, Pt is the total pressure, P, is the partial pressure of component i, n is the ratio of the number of molecules after and before the reaction, z is the direction normal to the susceptor surface, and ai is the thermal diffusion factor for species i.

Modeling Gas Flow Patterns to Optimize Reactor Design. Two approaches are used to study gas flow patterns and the effect of reactor geometry on those patterns. The first approach is based on experimental observations, whereas the second is based on numerical calculation.

Flow patterns can be visualized by using smoke particles (Ref 64, 65, 66). Generally, titanium dioxide smoke is used. These experiments provide information about the existence of a boundary layer adjacent to the susceptor, the presence of temperature gradients perpendicular to the flow, and additional evidence of a convection-induced gas motion. Flow visualization experiments also provide a qualitative insight into the momentum transfer in the gas phase. The problem with a smoke test is that it does not give a clear boundary layer thickness, because smoke is heavier than the process gases.

Giling used holographic interference techniques to visualize temperature gradients (Ref 63). In this method, a light beam that passes through the reactor is compared with a reference beam that does not pass through the reactor. Small differences in the light paths become visible as interference patterns. If the flow is turbulent, then all fringe lines will move around, giving an incoherent image. When the flow is convective, the isotherms will be curved such that an upward or downward flow can be recognized. When the flow is laminar and stable against convection, fringes parallel to the hot surface will be formed. Giling analyzed flow images for hydrogen, nitrogen, argon, and helium carrier gases.

Berkman et al. (Ref 67) developed a detailed model of the flow dynamics, mass, and thermal gradients in a horizontal cold-walled reactor heated from below. The engineering formulas derived from hydrodynamic flow theories are easy to apply and correlate well with experimental observations.

There are two distinct zones above the susceptor in a reactor. The gas near the susceptor flows in a nearly laminar manner. Above that, the gas flow is turbulent, with oscillations of 3 to 5 Hz. The turbulence is caused by both thermal entry effects and buoyancy forces originating at the leading edge of the hot susceptor. The Berkman model treats the upper turbulent zone as a cold finger of well-mixed gas that acts as a reservoir of reactants. The lower zone acts like a laminar boundary layer, with a velocity profile that has average characteristics of both laminar and turbulent flow, through which the reactant is transported by diffusion. For a flat susceptor, the growth rate decreases exponentially along the susceptor as:

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