Introduction to Chemical Engineering Processes/Converting Information into Mass Flows

Converting Information into Mass Flows - Introduction

In any system there will be certain parameters that are easier (often considerably) to measure and/or control than others. When you are solving any problem and trying to use a mass balance or any other equation, it is important to recognize what pieces of information can be interconverted. The purpose of this section is to show some of the more common alternative ways that mass flow rates are expressed, mostly because it is easier to, for example, measure a velocity than it is to measure a mass flow rate directly.

Volumetric Flow rates

A volumetric flow rate is a relation of how much volume of a gas or liquid solution passes through a fixed point in a system (typically the entrance or exit point of a process) in a given amount of time. It is denoted as:

 ${\displaystyle {\dot {V}}_{n}{\dot {=}}{\frac {Volume}{time}}}$ in stream n

Volume in the metric system is typically expressed either in L (dm^3), mL (cm^3), or m^3. Note that a cubic meter is very large; a cubic meter of water weighs about 1000kg (2200 pounds) at room temperature!

Why they're useful

Volumetric flow rates can be measured directly using flow meters. They are especially useful for gases since the volume of a gas is one of the four properties that are needed in order to use an equation of state (discussed later in the book) to calculate the molar flow rate. Of the other three, two (pressure, and temperature) can be specified by the reactor design and control systems, while one (compressibility) is strictly a function of temperature and pressure for any gas or gaseous mixture.

Limitations

Volumetric Flowrates are Not Conserved. We can write a balance on volume like anything else, but the "volume generation" term would be a complex function of system properties. Therefore if we are given a volumetric flow rate we should change it into a mass (or mole) flow rate before applying the balance equations.

Volumetric flowrates also do not lend themselves to splitting into components, since when we speak of volumes in practical terms we generally think of the total solution volume, not the partial volume of each component (the latter is a useful tool for thermodynamics, but that's another course entirely). There are some things that are measured in volume fractions, but this is relatively uncommon.

How to convert volumetric flow rates to mass flow rates

Volumetric flowrates are related to mass flow rates by a relatively easy-to-measure physical property. Since ${\displaystyle {\dot {m}}{\dot {=}}mass/time}$ and ${\displaystyle {\dot {V}}{\dot {=}}volume/time}$, we need a property with units of ${\displaystyle mass/volume}$ in order to convert them. The density serves this purpose nicely!

 ${\displaystyle {\dot {V}}_{n}*{\rho }_{n}={\dot {m}}_{n}}$ in stream n

The "i" indicates that we're talking about one particular flow stream here, since each flow may have a different density, mass flow rate, or volumetric flow rate.

Velocities

The velocity of a bulk fluid is how much lateral distance along the system (usually a pipe) it passes per unit time. The velocity of a bulk fluid, like any other, has units of:

 ${\displaystyle v_{n}={\frac {distance}{time}}}$ in stream n

By definition, the bulk velocity of a fluid is related to the volumetric flow rate by:

 ${\displaystyle {v}_{n}={\frac {{\dot {V}}_{n}}{A_{n}}}}$ in stream n

This distinguishes it from the velocity of the fluid at a certain point (since fluids flow faster in the center of a pipe). The bulk velocity is about the same as the instantaneous velocity for relatively fast flow, or especially for flow of gasses.

For purposes of this class, all velocities given will be bulk velocities, not instantaneous velocities.

Why they're useful

(Bulk) Velocities are useful because, like volumetric flow rates, they are relatively easy to measure. They are especially useful for liquids since they have constant density (and therefore a constant pressure drop at steady state) as they pass through the orifice or other similar instruments. This is a necessary prerequisite to use the design equations for these instruments.

Limitations

Like volumetric flowrates, velocity is not conserved. Like volumetric flowrate, velocity changes with temperature and pressure of a gas, though for a liquid velocity is generally constant along the length of a pipe.

Also, velocities can't be split into the flows of individual components, since all of the components will generally flow at the same speed. They need to be converted into something that can be split (mass flow rate, molar flow rate, or pressure for a gas) before concentrations can be applied.

How to convert velocity into mass flow rate

In order to convert the velocity of a fluid stream into a mass flow rate, you need two pieces of information:

1. The cross sectional area of the pipe.
2. The density of the fluid.

In order to convert, first use the definition of bulk velocity to convert it into a volumetric flow rate:

${\displaystyle {\dot {V}}_{n}=v_{n}*A_{n}}$

Then use the density to convert the volumetric flow rate into a mass flow rate.

${\displaystyle {\dot {m}}_{n}={\dot {V}}_{n}*{\rho }_{n}}$

The combination of these two equations is useful:

 ${\displaystyle {\dot {m}}_{n}=v_{n}*{\rho }_{n}*A_{n}}$ in stream n

Molar Flow Rates

The concept of a molar flow rate is similar to that of a mass flow rate, it is the number of moles of a solution (or mixture) that pass a fixed point per unit time:

 ${\displaystyle {\dot {n}}_{n}{\dot {=}}{\frac {moles}{time}}}$ in stream n

Why they're useful

Molar flow rates are mostly useful because using moles instead of mass allows you to write material balances in terms of reaction conversion and stoichiometry. In other words, there are a lot fewer unknowns when you use a mole balance, since the stoichiometry allows you to consolidate all of the changes in the reactant and product concentrations in terms of one variable.

Limitations

Unlike mass, total moles are not conserved. Total mass flow rate is conserved whether there is a reaction or not, but the same is not true for the number of moles. For example, consider the reaction between hydrogen and oxygen gasses to form water:

${\displaystyle H_{2}+{\frac {1}{2}}O_{2}\rightarrow H_{2}O}$

This reaction consumes 1.5 moles of reactants for every mole of products produced, and therefore the total number of moles entering the reactor will be more than the number leaving it.

However, since neither mass nor moles of individual components is conserved in a reacting system, it's better to use moles so that the stoichiometry can be exploited, as described later.

The molar flows are also somewhat less practical than mass flow rates, since you can't measure moles directly but you can measure the mass of something, and then convert it to moles using the molar flow rate.

How to Change from Molar Flow Rate to Mass Flow Rate

Molar flow rates and mass flow rates are related by the molecular weight (also known as the molar mass) of the solution. In order to convert the mass and molar flow rates of the entire solution, we need to know the average molecular weight of the solution. This can be calculated from the molecular weights and mole fractions of the components using the formula:

${\displaystyle {\bar {MW}}_{n}=[\Sigma ({MW}_{i}*y_{i})]_{n}}$

where i is an index of components and n is the stream number. ${\displaystyle y_{i}}$ signifies mole fraction of each component (this will all be defined and derived later).

Once this is known it can be used as you would use a molar mass for a single component to find the total molar flow rate.

 ${\displaystyle {\dot {m}}_{n}={\dot {n}}_{n}*{\bar {MW}}_{n}}$ in stream n