ChE229 Homework 5

Laminar Flow Reactor Problem1. Derive equations 1 to 4 and confirm Figures 1 and 2Answer:

uou(r)

Figure 1. Schematic Diagram of the velocity profile (left) and fluid elements (right) in a laminar flow reactor.

In a laminar flow reactor we know that the axial velocity varies in the radial direction according to the Hagen-Poiseuille equation:(1)

RrL

dQCACA + d CAzz+dzu(r)Qo

Figure 2. Schematic diagram of a laminar flow reactor.Where u(r) is the velocity of an element of the fluid (a cylindrical shell of radius r and thickness dr in the tube of radius R as decribed by the figure above), and uo is the velocity at the center of the tube (r=0), the maximum velocity. The mean velocity of the elements of the fluid is given by the ratio of the total volumetric flow, Q, to the cross-sectional area of flow, Ac:(2)

If we make a material balance in with the volume element that we considered(3)

(4)

(5)

On cancellation of (6)

(7)

In the case of an isothermal irreversible reaction whose rate is (-R), the performance equation of the laminar flow tube reactor is(8)

or(9)

Since the material balance derived above refer to a particular radial position, we must integrate radially to obtain an average concentration CA, at any position z. At any position z, the rate of flow of A in terms of total and average quantities is(10)

On integrating over the entire cross-section(11)

Equating Equation 10 to 11, we get an expression for CA(12)

To obtain a concentration profile CA(z) as well as the outlet concentration CA(L), we express Equation 12 in terms of t rather than r. To achieve this we use the relationship(13)

(14)

Where is the mean residence time to the point at z. At the outlet these equations becomes(15)

(16)

Where is the mean residence time of the entire tube. Manipulating equations 13 and 14, eliminating z from the expression, yields(17)

We can obtain another relation from the expression above(18)

On differentiation at fixed z and letting t=t(z,r)(19)

Or(20)

Substituting Equation 17 and 20 to equation 12, and change the limits as folllows(21)

Consequently,(22)

The interpretation of CA(t) comes from the realization that each cylindrical shell passes through the vessel as an independent batch. Thats why we will be using the integration of the batch reactor to find the expression for CA(t). Dividing both sides by CAo(23)

From the design equation of an ideal batch reactor(24)

For second order kinetics with equimolar amounts of reactants (A+BC + D), the rate expression is(25)

(26)

(28)(27)

(29)

(30)

And knowing that , the expression in terms of conversion is(31)

Using the definition of which is equal to the ratio of volume and volumetric flow rate(32)

Integrating it from 0 to L, we get(33)

Reynolds number is(34)

(35)

(36)

By using the integral method in which the reaction rate expression is assumed and tested using experimental data, it was confirmed that the reaction rate constant was 0.102 m3 mol-1 s-1 at 26oC. It is also given that V=0.4dm3, ID=4mm, CAo=0.1 mol/m3 and T=25.26oC.

Using Equation 31 and 36 we generate the following data and plots.

xANRexA

1300.5175422000.847719

1500.5514693000.789162

2000.61824000.738832

2500.6674165000.69504

3000.7052816000.65654

3500.7353477000.62239

4000.7598118000.591866

4500.7801159000.564399

5000.79724110000.539535

Figure 3. The effect of Reynolds number on conversion of reactant .

Figure 4. The effect of mean residence time on conversion.References Burden, R. L. and Faires, J. D., Numerical Analysis, 8th ed.,Thompsom Brooks/Cole, 2005 Ernst, W. R., Second-Order Reaction in a Semibatch Reactor, AIChE Journal (1997) Vol 43 No.4. Fogler,H.S., Essentials of Chemical Reaction Engineering, 1st ed., Pearson Education, Inc., 2011 Levenspiel, O., Chemical Reaction Engineering, 3rd ed., John Wiley & Sons, Inc., 1999 Sales, J. et al., Study of Major Accidents Involving Chemical Reactive Substances Analysis and Lessons Learned, Institution of Chemical Engineers ( 2007) Vol 85(B2)117-124. Thoenes, D., Chemical Reactor Development: from Laboratory Synthesis to Industrial Production, Kluwer Academic Publishers, 1994.Mathcad symbolic evaluationsince