# Introduction to Modeling of Mass Transfer Processes

This chapter is from the book

## 1.10 Mesoscopic or Cross-Section Averaged Models

Mesoscopic models are useful when there is a principal direction over which the flow takes place, as in pipe flow, for instance. Control volume is taken as differential in this direction but assumed to span the entire cross-section in the cross-flow directions (Figure 1.12). The rationale for this assumption is as follows: The concentration variation in the flow direction is more significant, meaning it changes significantly from inlet to outlet, rather than in the radial direction. Changes in the radial direction at any fixed axial position may not be very large. Hence the mesoscopic control volume shown in Figure 1.12 is adequate in lieu of a complete differential model. The key point to note is that the information on the radial variation of concentration and its effect of the various terms in the conservation model is lost with this kind of model. Hence an appropriate closure model is needed the supplement the information lost due to area or radial averaging. More details on this approach are provided in Chapter 4. Here, we provide some simple examples of closure models used for this level of modeling.

### 1.10.1 Solid Dissolution from a Wall

Consider the mesocopic model applied to a simple system of solid dissolution from a wall in a tube shown in Figure 1.12. Here we have a pipe coated with a dissolving or subliming solute material; a fluid is flowing in the pipe. The concentration of the solute is to be calculated as a function of the length. Two specific examples are a naphthalene-coated pipe with air flow; a pipe coated with benzoic acid (which is a solid at room temperature) with water flow. We assume steady state so that the accumulation is zero.

The conservation law (inout = 0) is now represented as

in from flow at z + in by transported from the walls –out from flow at z + Δz = 0

To calculate the in and out terms due to flow, we must recognize that the velocity can vary across the cross-section. Concentration also varies along the pipe, with the maximum concentration found at the dissolving pipe wall. The flow can be laminar or turbulent. Laminar flow of a Newtonian fluid has a parabolic profile, whereas the profile is very steep in turbulent flow as shown in Figure 1.13. The velocity for the turbulent flow is the time-averaged value, a concept that is discussed more fully in Chapter 12.

A cross-sectionally averaged velocity is an useful quantity and is defined as

which is the integral average of the local velocity. The volumetric flow rate is then equal to 〈vA; v is the axial velocity.

The plug flow shown in the Figure 1.13 is an idealization that is often used to simplify the models. Here the velocity is assumed to be the same at each local radial position. Consequently, the cross-sectional average velocity is also equal to the local velocity.

The in and out terms are the integral average of the axial flux values. The flux component in the axial direction is primarily due to convection and is locally equal to vCA. For a small differential area, ΔA, the moles of A crossing is therefore equal to vCA ΔA. Hence the moles crossing over the entire tube area is the integral of the product of local velocity and the local concentration:

where the notation 〈...〉 is used as shorthand to indicate the averaged value of any quantity within the brackets. Thus we define

We need 〈vCA〉 to evaluate the in and out terms, which cannot be calculated because local concentration values are not available at the meso-level. Hence we define and use an average concentration. This average concentration is called the cup mixing average concentration, of the flow weighted average. It is denoted as CAb and defined as follows:

Hence

The in and out terms are then represented simply as AvCAb. Thus the cup mixing concentration is used as the representative concentration variable in the context of our mesoscopic model. The local concentration has no relevance because the local information is not available. Note: In the context of the meso-level model, the notation CA is used in many books as a simpler notation; by implication, this is the cup mixed average value and should be interpreted accordingly. The results in such models are the variation of the cup mixing concentration with the axial distance.

Example 1.2 shows a calculation of the cup mixing concentration to clarify the definition. Here we assume that the radial variation is known from other methods, such as from a differential model or by experimental measurements.

Having defined the in and out terms using the cup mixing concentration, we need an expression for the transport rate to the walls so as to close the model. This is done by defining and using a mass transfer coefficient. It is common practice to use the cup mixing concentration as the representative concentration away from the solid; the driving force for mass transfer is then defined using (CAbCAw) as the driving force. Hence the mass transfer rate is defined as

NAw = km(CAbCAw)

The mass transfer coefficients are available for a large number of flows (such as pipe flows) and, in turn, this approach to modeling is widely used in practice. The mesoscopic formulation can be cast into mathematical framework to account for variation of the concentration (cup mixing) as a function of axial position. Further details and the mathematical details follow in Chapter 4. At this stage, however, you should be able to appreciate the essential concepts that go into the construction of mesoscopic models.

### 1.10.2 Tubular Flow Reactor

Another common application of the meso-modeling concept is the tubular flow reactor. This is a pipe or channel in which a homogeneous chemical reaction is taking place. We will show that not only the cup mixing concentration is needed, but also another average, the cross-sectional average.

in – out + generation = 0

The generation is the average of the local rate:

generation = (A Δx) 〈RA

where

For a first-order reaction, for example, RA = –k1CA, where CA is the local concentration. Hence

To incorporate this into the conservation law, it is customary to define and use the the cross-sectional average concentration:

Hence the generation term (first order) to be used in the reactor model is –(AΔx)k1CA〉.

In and out terms are AvCAb as for the solid dissolution. Thus the conservation law in mathematical terms is

In the limit Δx → 0, this produces the following differential equation:

We find that both the cup mixing and the cross-sectional average appear in the reactor model. Some closure relationship is needed, however, and the commonly used closure models are the plug flow model and the axial dispersion model.

#### Plug Flow Model

The plug flow model is an idealized reactor model in which the velocity is assumed to be constant, as shown in the Figure 1.13. The cup mixing and the cross-sectional averages are the same since the velocity profile is uniform and, therefore, CAb = 〈CA〉. Only the cross-sectional concentration is needed; that is, no additional closure is needed. Equation 1.26 reduces to

#### Axial Dispersion Model

The actual reactor may not be a plug flow model, in which case the deviation from plug flow can be modeled by including an additional correction term. The resulting model is known as the axial dispersion model. The representation for the in or out term is

The first term on the right side is the mole flow crossing a cross-section if the plug flow prevails. The second term is a correction term. This system is modeled as though some diffusion type of mechanism is superimposed on the plug flow value. Hence

where DE is called the axial dispersion coefficient. This is only a model, and not a fundamental principle, but nevertheless it provides an important crutch to proceed further in modeling these systems. The axial dispersion coefficient is a commonly used parameter when modeling both reactors and separation processes. Details of the origin of this term and the calculation and application of this concept are discussed in Section 14.2. For now, recognize that the dispersion model closure is

The key points to appreciate from this section are the various definitions and closures that are needed in the context of meso-scale models.

Now we discuss models at an even higher level of simplicity—namely, the compartmental models.

### InformIT Promotional Mailings & Special Offers

I would like to receive exclusive offers and hear about products from InformIT and its family of brands. I can unsubscribe at any time.

## Overview

Pearson Education, Inc., 221 River Street, Hoboken, New Jersey 07030, (Pearson) presents this site to provide information about products and services that can be purchased through this site.

This privacy notice provides an overview of our commitment to privacy and describes how we collect, protect, use and share personal information collected through this site. Please note that other Pearson websites and online products and services have their own separate privacy policies.

## Collection and Use of Information

To conduct business and deliver products and services, Pearson collects and uses personal information in several ways in connection with this site, including:

### Questions and Inquiries

For inquiries and questions, we collect the inquiry or question, together with name, contact details (email address, phone number and mailing address) and any other additional information voluntarily submitted to us through a Contact Us form or an email. We use this information to address the inquiry and respond to the question.

### Online Store

For orders and purchases placed through our online store on this site, we collect order details, name, institution name and address (if applicable), email address, phone number, shipping and billing addresses, credit/debit card information, shipping options and any instructions. We use this information to complete transactions, fulfill orders, communicate with individuals placing orders or visiting the online store, and for related purposes.

### Surveys

Pearson may offer opportunities to provide feedback or participate in surveys, including surveys evaluating Pearson products, services or sites. Participation is voluntary. Pearson collects information requested in the survey questions and uses the information to evaluate, support, maintain and improve products, services or sites, develop new products and services, conduct educational research and for other purposes specified in the survey.

### Contests and Drawings

Occasionally, we may sponsor a contest or drawing. Participation is optional. Pearson collects name, contact information and other information specified on the entry form for the contest or drawing to conduct the contest or drawing. Pearson may collect additional personal information from the winners of a contest or drawing in order to award the prize and for tax reporting purposes, as required by law.

If you have elected to receive email newsletters or promotional mailings and special offers but want to unsubscribe, simply email information@informit.com.

### Service Announcements

On rare occasions it is necessary to send out a strictly service related announcement. For instance, if our service is temporarily suspended for maintenance we might send users an email. Generally, users may not opt-out of these communications, though they can deactivate their account information. However, these communications are not promotional in nature.

### Customer Service

We communicate with users on a regular basis to provide requested services and in regard to issues relating to their account we reply via email or phone in accordance with the users' wishes when a user submits their information through our Contact Us form.

## Other Collection and Use of Information

### Application and System Logs

Pearson automatically collects log data to help ensure the delivery, availability and security of this site. Log data may include technical information about how a user or visitor connected to this site, such as browser type, type of computer/device, operating system, internet service provider and IP address. We use this information for support purposes and to monitor the health of the site, identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents and appropriately scale computing resources.

### Web Analytics

Pearson may use third party web trend analytical services, including Google Analytics, to collect visitor information, such as IP addresses, browser types, referring pages, pages visited and time spent on a particular site. While these analytical services collect and report information on an anonymous basis, they may use cookies to gather web trend information. The information gathered may enable Pearson (but not the third party web trend services) to link information with application and system log data. Pearson uses this information for system administration and to identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents, appropriately scale computing resources and otherwise support and deliver this site and its services.

This site uses cookies and similar technologies to personalize content, measure traffic patterns, control security, track use and access of information on this site, and provide interest-based messages and advertising. Users can manage and block the use of cookies through their browser. Disabling or blocking certain cookies may limit the functionality of this site.

### Do Not Track

This site currently does not respond to Do Not Track signals.

## Security

Pearson uses appropriate physical, administrative and technical security measures to protect personal information from unauthorized access, use and disclosure.

## Children

This site is not directed to children under the age of 13.

## Marketing

Pearson may send or direct marketing communications to users, provided that

• Pearson will not use personal information collected or processed as a K-12 school service provider for the purpose of directed or targeted advertising.
• Such marketing is consistent with applicable law and Pearson's legal obligations.
• Pearson will not knowingly direct or send marketing communications to an individual who has expressed a preference not to receive marketing.
• Where required by applicable law, express or implied consent to marketing exists and has not been withdrawn.

Pearson may provide personal information to a third party service provider on a restricted basis to provide marketing solely on behalf of Pearson or an affiliate or customer for whom Pearson is a service provider. Marketing preferences may be changed at any time.

## Correcting/Updating Personal Information

If a user's personally identifiable information changes (such as your postal address or email address), we provide a way to correct or update that user's personal data provided to us. This can be done on the Account page. If a user no longer desires our service and desires to delete his or her account, please contact us at customer-service@informit.com and we will process the deletion of a user's account.

## Choice/Opt-out

Users can always make an informed choice as to whether they should proceed with certain services offered by InformIT. If you choose to remove yourself from our mailing list(s) simply visit the following page and uncheck any communication you no longer want to receive: www.informit.com/u.aspx.

## Sale of Personal Information

Pearson does not rent or sell personal information in exchange for any payment of money.

While Pearson does not sell personal information, as defined in Nevada law, Nevada residents may email a request for no sale of their personal information to NevadaDesignatedRequest@pearson.com.

## Supplemental Privacy Statement for California Residents

California residents should read our Supplemental privacy statement for California residents in conjunction with this Privacy Notice. The Supplemental privacy statement for California residents explains Pearson's commitment to comply with California law and applies to personal information of California residents collected in connection with this site and the Services.

## Sharing and Disclosure

Pearson may disclose personal information, as follows:

• As required by law.
• With the consent of the individual (or their parent, if the individual is a minor)
• In response to a subpoena, court order or legal process, to the extent permitted or required by law
• To protect the security and safety of individuals, data, assets and systems, consistent with applicable law
• In connection the sale, joint venture or other transfer of some or all of its company or assets, subject to the provisions of this Privacy Notice
• To investigate or address actual or suspected fraud or other illegal activities
• To exercise its legal rights, including enforcement of the Terms of Use for this site or another contract
• To affiliated Pearson companies and other companies and organizations who perform work for Pearson and are obligated to protect the privacy of personal information consistent with this Privacy Notice
• To a school, organization, company or government agency, where Pearson collects or processes the personal information in a school setting or on behalf of such organization, company or government agency.

This web site contains links to other sites. Please be aware that we are not responsible for the privacy practices of such other sites. We encourage our users to be aware when they leave our site and to read the privacy statements of each and every web site that collects Personal Information. This privacy statement applies solely to information collected by this web site.