Variable volumetric flow rate

Batch Reactors with Variable Volume Although variable volume batch reactors are seldom encountered because they are usually solid steel containers, we will develop the concentrations as a function of conversion because (1) they have been u.sed to collect reaction data for gas-phase reactions, and (2) the development of the equations that express volume as a function of conversion will facilitate analyzing flow systems with variable volumetric flow rates. 

Flow Reactors with Variable Volumetric Flow Rate. An expression similar to Equation (3-38) for a variable-volume batch reactor exists for a variable-volume flow system. To derive the concentrations of each species in terms of conversion for a variable-volume flow system, we shall use the relationships for the total concentration. The total concentration. Cj-. at any point in the reactor is the total molar flow rate. Fj, divided by volumetric flow rate v [cf. Equation (3-27). In the gas phase, the total concentration is also found from the gas law. Cr = P/ZRT. Equating these two relationships gives... 

Expected value of a given random variable Volumetric flow rate (L /t)... 

Vacuum flow is usually described with flow variables different from those used for normal pressures, which often leads to confusion. Pumping speed S is the actual volumetric flow rate of gas through a flow cross section. Throughput Q is the product of pumping speed and absolute pressure. In the SI system, Q has units of Pa m vs. 

Flow reactors usually operate at nearly constant pressure, and thus at variable density when there is a change of moles of gas or of temperature. An appai ent l e.sidence time is the ratio of reactor volume and the inlet volumetric flow rate. 

The variables involved in the performance of a centrifugal pump include the fluid properties (p, and p), the impeller diameter (cl), the casing diameter (/)), the impeller rotational speed (N), the volumetric flow rate of the fluid (0, the head... 

The pressure developed by a centrifugal pump depends on the fluid density, the diameter of the pump impeller, the rotational speed of the impeller, and the volumetric flow rate through the pump (centrifugal pumps are not recommended for highly viscous fluids, so viscosity is not commonly an important variable). Furthermore, the pressure developed by the pump is commonly expressed as the pump head, which is the height of a column of the fluid in the pump that exerts the same pressure as the pump pressure. 

We will use the Bernoulli equation in the form of Eq. (6-67) for analyzing pipe flows, and we will use the total volumetric flow rate (Q) as the flow variable instead of the velocity, because this is the usual measure of capacity in a pipeline. For Newtonian fluids, the problem thus reduces to a relation between the three dimensionless variables ... 

Following the steps for formulation of a CFD model introduced earlier, we begin by determining the set of state variables needed to describe the flow. Because the density is constant and we are only interested in the mixing properties of the flow, we can replace the chemical species and temperature by a single inert scalar field (x, t), known as the mixture fraction (Fox, 2003). If we take = 0 everywhere in the reactor at time t — 0 and set / = 1 in the first inlet stream, then the value of (x, t) tells us what fraction of the fluid located at point x at time t originated at the first inlet stream. If we denote the inlet volumetric flow rates by qi and q2, respectively, for the two inlets, at steady state the volume-average mixture fraction in the reactor will be... 

Since the volumetric flow rate is a function of X, T, and P, the residence time V/v depends on these variables. Instead of using the reactor residence time T to describe performance, an analogous quantity called the space time ST, defined as... 

Similar transient problems in the CSTR can be easily formulated. If the reactor volume is not constant (such as starting with an empty reactor and beginning to fill it at f = 0) or if the volumetric flow rate v into the reactor is changed), then the variable-densily version of this reactor involving Fa must be used. 

These equations are only vahd if there is no density change in the reactor, because otherwise W 7 v and Cj is not an appropriate variable. With mass transfer to and from a phase, one expects the volumetric flow rates in and out of the phase to not be identical. For... 

Variables Affecting Measurement Flow measurement methods may sense local fluid velocity, volumetric flow rate, total or cumulative volumetric flow (the integral of volumetric flow rate with respect to elapsed time), mass flow rate, and total mass flow. 

A variable-volume batch reactor is a constant-pressure (piston-like) closed tank. On the other hand, a variable-pressure tank is a constant-volume batch reactor (Fogler, 1999). Thus, in batch reactors, the expansion factor is used only in the case of a constant-pressure tank whereas and not in a constant-volume tank, even if the reaction is realized with a change in the total moles. However, in continuous-flow reactors, the expansion factor should be always considered. In the following section and for the continuous-flow reactors, the volume V can be replaced by the volumetric flow rate Q, and the moles N by the molar flow rate F in all equations. 

Consider liquid flowing through a tank of uniform area of cross-section A with variable volumetric inlet rate of flow Q0 (Fig. 7.15). Let the volumetric rate of flow... 

Time is still an important variable for continuous systems, but it is modified to relate to the steady-state conditions that exist in the reactor. This time variable is referred to as space time. Space time is the reactor volume divided by the inlet volumetric flow rate. In other words, it is the time required to process one reactor volume of feed material. Since concentration versus real time remains constant during the course of a CSTR reaction, rate-data acquisition requires dividing the difference in concentration from the inlet to the outlet by the space time for the particular reactor operating conditions. 

It is useful to have a measure of time for a flow reactor even though the major design variable is reactor or fluid volume. A commonly used quantity in industrial reactor design is space time. Space time is defined as the time required to process one reactor volume of feed, measured at some set of specified conditions. The normal conditions chosen are the inlet concentration of a reactant and inlet molar or volumetric flow rate. 

Variables whose symbols include a dot (-) are rales for example, m is mass flow rate and V is volumetric flow rate. 

Note on Notation Although any symbol may be used to represent any variable, having a consistent notation can aid understanding. In this text, we will generally use m for mass, m for mass flow rate, n for moles, h for molar flow rate, V for volume, and V for volumetric flow rate. Also, we will use x for component fractions (mass or mole) in liquid streams and y for fractions in gas streams. 

A volumetric flow rate is given for the feed stream, but mass flow rates and fractions will be needed for balances. The mass flow rate of the stream should therefore be considered an unknown process variable and labeled as such on the chart. Its value will be determined from the known volumetric flow rate and density of the feed stream. 

The flow of air to a gas-fired boiler furnace is regulated by a minicomputer controller. The fuel gases used in the furnace are mixtures of methane (A), ethane (B), propane (C), n-butane (D), and isobutane (E). At periodic intervals the temperature, pressure, and volumetric flow rate of the fuel gas are measured, and voltage signals proportional to the values of these variables are transmitted to the computer. Whenever a new feed gas is used, a sample of the gas is analyzed and the mole fractions of each of the five components are determined and read into the computer. The desired percent excess air is then specified, and the computer calculates the required volumetric flow rate of air and transmits the appropriate signal to a flow control valve in the air line. 

Crude oil pumped from a storage unit to a tanker is to be expressed in tons/hr, but the field variables of density and the volumetric flow rate are measured in Ib/ft and gal/min, respectively. Determine the units and the numerical value of the factor necessary to convert the field variables to the desired output. 

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