What needs to be found now is the length a reactor that will convert 65% of the ethane that is in the gas stream to ethylene. There are two equations that govern the reaction of ethane to ethylene. These equations are shown below.

The reactions that are dictated by the above equations tend to be irreversible at low to intermediate conversions (0% to 70%). At high conversions (>70%) the reactions become reversible and reactants start to form. That is why the conversion is limited to 65%.

The rate of consumptions of ethane based on equations 1 and 2 are shown below:

Equations for the rate of consumption of reactants and rate of formation of products can be written based on equation 5 shown below:

The R(i)'s are found by using equations 3 and 4 and the stoichiometry coefficients of equations 1 and 2. The final form of the R(i)'s are shown below:

The final equations for the rate of consumption and rate of formation (the set of ordinary differential equations, ODE's) are found by combining equation 5 with equations 6 to 10. These equations are displayed below:

Now the issue of temperature is considered. As can be seen below, the heat of reaction of equations 1 and 2 are positive. This means that the reactions are endothermic

The gas feed to the reactor needs to be heated in order for any product to form. That is why the pipe is enclosed inside a furnace. Due to this heating and the change in temperature of the gas stream a sixth ordinary differential equation is needed. This ODE will represent the change of temperature along the volume of the reactor. Another issue to consider is the set point for the furnace temperature. This temperature is set at a point where the heat transfer by radiation dominate all other forms of heat transfer. The picture below displays the schematic of how the furnace, the pipe's wall, and the gas stream temperature interact.

Equation 18 below displays how the temperature of the furnace, the pipe's wall, and the gas stream interact with one another.

Equation 19 below displays the ODE for the temperature of the gas stream along the volume of the reactor.

The final form of the temperature's ODE is found by combining equations 18 and 19. This ODE is displayed below as equation 20.

Now the temperature's ODE is in terms of the gas stream, and the wall temperature. But the wall temperature is not known. Only the furnace temperature is given. In fact, the wall temperature changes as the the gas stream temperature change. A form of a successive substitution method is used to find the temperature of the wall. This method method is called the Newton-Raphson method. The equations that are used in the Newton-Raphson method to calculate the wall temperature at a given gas stream temperature are displayed below.

In order for the Newton-Raphson method to work, an initial guess for the wall temperature is needed. For this guess the wall temperature is set equal to the gas stream temperature.

The modeling of the reaction process in now complete. To solve for the length of the reactor, equations 3, 4,11-15, 20 and the equations for the Newton-Raphson method are plugged into MATLAB along with the different constants that are involved in the equations (NOTE: units are very important. They need to be checked to make sure that they are correct.) The inlet flow rates of the different components of the gas stream as well as the inlet temperature must be specified. The initial and final volume of the reactor need to be specified to provide a rang over which MATLAB will integrate the differential equations. The command that is used to integrate the ordinary differential equations in MATLAB is the ode45 command. To avoid a stiff problem when integrating the temperature's ODE, the temperatures of the gas stream and the pipe's wall are divided by 100. However, the true value for the gas stream temperature is the value that is used to solve for the wall temperature in the Newton-Raphson method.

There are two checks that need to be done to confirm that the right problem is being solved. The first check is done to see if the equations are written correctly. This check is done by first changing the initial values so that they are not zero or one, and then executing a number of right hand side calculations by hand and comparing the answers obtained with what MATLAB register as values for the equations when using the initial values in them. The second check is done to see if the Newton-Raphson method provides the correct wall temperature. This check is done by plugging different values for the gas stream temperature and calculating the wall's temperature. Then the value found for the wall temperature is compared with the gas stream, and the furnace temperature. If the value for the wall temperature is in between the value for the gas stream temperature and the furnace temperature then the Newton-Raphson method calculates the correct wall temperature.

A typical case is solved using the procedure discussed above and the results are shown here. For this case the inlet flow rate of ethane and water are 571 and 190Ib-mole/hr, respectively. All the other inlet flows are equal to zero. The inlet temperature is 350K and the furnace temperature is set at 1311K. After solving the problem, the final reactor volume obtained that will result in a 65% conversion of ethane to ethylene turn out to be 160 cubic-feet. Using a pipe radius of two inches, the length of the reactor can be calculated. After all the calculations, the final length turn out to be 1833.5 feet.

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