Pressure Drop and the Rate Law

We now focus our attention on accounting for the pressure drop in the rate law. For an ideal gas, we recall Equation (,V46) to write the concentration of reacting species i as 

When P Pn one is being carried out in a packed-bed reactor, the differential form of the mol mu.st use the balance equation in terms of catalyst weight is 

Note from Equation (4-20) that the larger the pressure drop (i.e the smaller P from frictional losses, the smaller the reaction rate  

Combining Equation (4-20) with the mole balance (2-17) and assuminj isothermal operation (T = Tq) gives 

For isothermal operation (T = To), the right-hand side is a function of only conversion and pressure . 

Vb= -Wa). We now must determine the ratio Pressure (P/Po) as a function of the PFR reactor volume. V, or the PBR catalyst weight, IV, to account for pressure drop. We then can combine the concentration, rate law, and design equation. However, whenever accounting for the effects of pressure drop, the differeniial form of the mole balance (design equation) must be used. 

When P P one must use the diffcremial forms of the PFR/PBR design equations. 

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Pressure drop and flow rate are usually linearly related - expressed by Darcy s law (Chapter 3.1.4). If unknown from initial tests, the pressure drop is measured for different volume flows, once with column in the plant and once without the column in the plant, using a zero-volume connector. The difference between the two values yields the pressure drop characteristic Apc of the column alone. By plotting Ap, vs. Mjnt the unknown coefficient k0 is readily determined from the slope of the curve by rearranging Eq. 3.7. 

A third approach is to describe the foam flow in terms of foam mobility, X. 141 The quantities typically measured in a displacement or flow experiment in porous media are pressure drop and flow rate. These results can be used with Darcy s law to calculate a mobility. 

For non-circular shapes, the equations of motion may result in nonlinear partial differential equations, which are difficult to solve analytically. Therefore, approximate methods such as the variational method (Kantorovich and Krylov, 1958) are generally used for solving non-Newtonian flow problems. Schechter (1961) used the application of the variational method to solve the non-linear partial differential equations of pressure drop and flow rate of the polymer for non-circular shapes such as a rectangle or square. Moreover, Mitsuishi and Aoyagi (1969 1973) used similar methods for other non-circular shapes such as an isosceles triangle. The results were based on the Sutterby model (1966), which incorporates a viscosity function based on the rheological constants. Flow curves with pressure drop and flow rate for both circular and non-circular shapes were generated and the results were compared with the power law model. 

Polyethylene at 170°C passes through the annular die shown, at a rate of 10 x 10 m /s. Using the flow curves provided and assuming the power law index n = 0.33 over the working section of the curves, calculate the total pressure drop through the die. Also estimate the dimensions of the extruded tube. 

The filtration, or superficial face, velocities used in fabric filters are generally in the range of 1 to 10 feet per minute, depending on the type of fabric, fabric supports, and cleaning methods used. In this range, pressure drops conform to Darcy s law for streamline flow in porous media, which states that the pressure drop is directly proportional to the flow rate. The pressure drop across the... 

A coal slurry is found to behave as a power law fluid, with a flow index of 0.3, a specific gravity of 1.5. and an apparent viscosity of 70 cP at a shear rate of 100 s 1. What volumetric flow rate of this fluid would be required to reach turbulent flow in a 1/2 in. ID smooth pipe that is 15 ft long What is the pressure drop in the pipe (in psi) under these conditions ... 

The tower is operating under a pressure of 70 psig, and a slot liquid seal of 2 in. is maintained. At the point of maximum volumetric vapor flow, the molecular weight of the vapor is 100, the rate of vapor flow is 1500 lb mol/h, the liquid density is 55 lb/ft3, and the temperature is 175°F. The pressure drop through the tower is negligible, and the ideal gas law is applicable to the rising vapors. Approximately what percent of the maximum allowable flow rate is being used in the tower ... 

When using an ordinary differential equation (ODE) solver such as POLYMATH or MATLAB, it is usually easier to leave the mole balances, rate laws, and concentrations as separate equations rather than combining them into a single equation as we did to obtain an analytical solution. Writing She equations separately leaves it to the computer to combine them and produce a solution. The formulations for a packed-bed reactor with pressure drop and a semibatch reactor are given below for two elementary reactions. 

We now proceed (Example 4-6) to combine pressure drop with reaction in a packed for the case where we will assume that eX 1 in the Ergun equation bat not in the rate law in order to obtain an anal3rtical solution. Example 4-7 removes this assumption and solves Equations (4-21) and (4-31) numerically. 

In many industrial reactions, the overall rate of reaction is limited by the rate of mass transfer of reactants and products between the bulk fluid and the catalytic surface. In the rate laws and cztalytic reaction steps (i.e., dilfusion, adsorption, surface reaction, desorption, and diffusion) presented in Chapter 10, we neglected the effects of mass transfer on the overall rate of reaction. In this chapter and the next we discuss the effects of diffusion (mass transfer) resistance on the overall reaction rate in processes that include both chemical reaction and mass transfer. The two types of diffusion resistance on which we focus attention are (1) external resistance diffusion of the reactants or products between the bulk fluid and the external smface of the catalyst, and (2) internal resistance diffusion of the reactants or products from the external pellet sm-face (pore mouth) to the interior of the pellet. In this chapter we focus on external resistance and in Chapter 12 we describe models for internal diffusional resistance with chemical reaction. After a brief presentation of the fundamentals of diffusion, including Pick s first law, we discuss representative correlations of mass transfer rates in terms of mass transfer coefficients for catalyst beds in which the external resistance is limiting. Qualitative observations will bd made about the effects of fluid flow rate, pellet size, and pressure drop on reactor performance. 

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