Modeling—developing a set of governing equations—of systems of interest to chemical engineers often starts with defining a differential element of the system. This differential element is a subset of the larger system, but with infinitesimally small dimensions. All the processes and phenomena occurring in the larger system are represented in the differential element. The modeling approach involves writing conservation of mass and/or conservation of energy equations for the differential element. These equations yield ordinary differential equations when all the quantities are functions of a single independent variable. For example, equation 4.10 is a first-order differential equation relating the rate of change of concentration to time in a chemical reaction [6]. The equation indicates that the rate at which the concentration of species A, CA, changes with time t is linearly dependent on the concentration of A itself—an example of a first-order reaction. The parameter k is called the rate constant.
Solution of this equation yields the concentration-time profile for the reactant A in the reaction, which provides the basis for the design of the reactor.
Higher-order differential equations are very common in chemical engineering systems. Figure 4.4 shows the cross-sectional view of a pipe conducting steam, the ubiquitous heat transfer medium in chemical plants. The pipe will inevitably be covered with insulation to minimize heat loss to the surroundings. Note that the heat loss can be reduced but not completely eliminated. Obviously, choosing proper insulation and determining the resultant heat loss is extremely important for estimating the energy costs. Heat loss can be calculated from the temperature-distance profiles existing in the system [5].
Figure 4.4 Temperature profile for an insulated steam pipe.
The governing equation describing the heat transfer for a cylindrical pipe follows:
Equation 4.11 is a second-order ordinary differential equation that governs the relationship between temperature T and radial distance r from the center of the pipe. kT is the thermal conductivity of the material, which depends on the temperature. As the temperature varies with respect to the radial position, the thermal conductivity is also a function of the radial position. Solution of this equation yields the temperature profile within that object, which in turn allows us to determine the heat lost to the surroundings.
The solution of differential equations requires specifying values of dependent variable(s) at certain values of the independent variable. These specifications are termed boundary conditions (at a specific location, with respect to dimensional coordinate) or initial conditions (with respect to time). Complete solution requires as many boundary/initial conditions as the order of the differential equation [4].
Frequently, modeling of a system leads to a set of ordinary differential equations, consisting of two or more dependent variables that are functions of the same independent variable. These equations need to be solved simultaneously to obtain the quantitative description of the system.
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