Department of Chemical Engineering

ChE 101 CHEMICAL ENGINEERING CONCEPTS 2- LABORATORY

Winter 2015

Instructor: Mingqian (John) Zhang

Table of Contents

Preface .......................................................................................................................................................... 1

Academic Offences and Discipline: UW Policy #71 .................................................................................. 2

General Lab Instructions ............................................................................................................................ 3

Lab Safety .................................................................................................................................................... 4

Guideline for Lab Report ............................................................................................................................................. 5

Experiment 1: Solid-Liquid Phase Diagram for Binary Naphthalene-Biphenyl System ...................................... 10

Experiment 2: Energy Balance and Heat Transfer Coefficient of Heat Exchangers ........................................... 20

Experiment 3: Fluid Flow through an Orifice ........................................................................................................... 28

Experiment 4: Reaction Kinetics for the Hydrolysis of Ethyl Acetate with Sodium Hydroxide ......................... 33

Appendix A: Sample Title Page ................................................................................................................................ 41

Appendix B: Correlation of Experimental Data – Least Squares Principle ............................................................. 42

Appendix C: Experimental Errors and Error Propagation and Analysis ............................................................... 47

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PREFACE

The laboratory investigations have played an indispensable role in the development of all the underlying chemical engineering principles for the design and operation of various chemical engineering processes. As such, chemical engineering laboratory has always been an essential part of chemical engineering curriculum. Not only does the laboratory course allow students to appreciate numerous experimental process configurations, equipment details, and measuring devices which are not otherwise covered adequately in a lecture courses, but also complement and reinforce the understanding of the basic principles of chemical engineering through actual observations and quantitative analysis of the behaviors of physicochemical systems. ChE 101 laboratory is a preliminary chemical engineering laboratory which introduces students to experiments on physical chemistry and basic chemical engineering processes, the main objectives of the course are:

• To assist in the understanding of basic principles of Chemical Engineering through actual observations of the behavior of physicochemical systems.

• To help develop the skills in experimentation, data analysis, and interpretation of the results.

• To practice in the art of writing effective engineering reports.

The lab manual covers most of materials required for each experiment including some practical backgrounds, detailed theories, suggested experimental procedure, and data analysis requirements. You are also expected to go through your course notes and relevant textbooks related to the experiments.

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ACADEMIC OFFENCES AND DISCIPLINE: UW POLICY #71

Inappropriate academic behaviors and misconducts such as plagiarism, cheating, copying and sharing will be strongly prohibited for this laboratory course, and can result in very serious consequences. For detailed information on the Academic Integrity for UW students, see https://uwaterloo.ca/secretariat-general-counsel/policies-procedures-guidelines/policy-71

The following is some excerpts from the Polycy#71:

Academic integrity is a commitment to five basic values: honesty, trust, fairness, respect and responsibility. It applies to all academic endeavours – teaching, learning and scholarship, and applies to a range of academic activities, from conduct in research to the writing of co-op work term reports.

Students are expected to know what constitutes academic integrity, to avoid committing offences, and to take responsibility for their actions.

Students are responsible for demonstrating behaviour that is honest and ethical in their academic work. Such behaviour includes:

Abiding by University policies and provincial and federal legislation. Following the expectations articulated by instructors for referencing sources of

information and for group work. Submitting original work, citing sources fully, and respecting the authorship of

others. Preventing their work from being used by others, e.g. not lending assignments to

others, protecting access to computer files. Asking for clarification of expectations as necessary. Students who are in any doubt

as to whether an action on their part may be viewed as a violation of the standards of academic integrity should ask their instructors, lab assistants and/or advisors.

Adhering to the principles of academic integrity when conducting and reporting research

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GENERAL LAB INSTRUCTIONS

PRELIMINARY LAB SRSSION: For each experiment, a pre-lab session is scheduled a week before the date of the actual experiment. In the pre-lab session, the Teaching Assistant will show the equipment and offer adequate instructions on All the theoretical principles involved. Equipment and experimental procedure. Data analysis and calculations.

At the end of the session, the TA will give you a set of sample data for you to do the sample calculations for your Preliminary Report. LABORATORY SESSION: There will be a laboratory session scheduled for each experiment in which The preliminary lab reports will need to be submitted before you start the experiment. The Teaching Assistant will mark the reports and talk to you individually (if there are

serious errors or deficiencies in your calculations and prelab report) to make sure you avoid the same mistakes in the memo report.

You perform the experiment and ensure that your data sheet containing the acquired experimental data is signed by the TA at the end of the experiment.

Every member of each lab group is expected to actively participate in the lab. The pre-assigning of duties to each member of the group is strongly advised in order for the experiment to be performed effectively and efficiently.

PROFESSIONAL CONDUCT: Although you are going to work as a group, individual unprofessional conduct in the lab session will result in a penalty. Marks will be deducted for such acts as horse-play, dangerous practices in the lab equipment, not adhering to safety protocol, and lack of participation during the experiment.

ATTENDANCE: Lab attendance is a prerequisite for receiving the marks of the rest of each experiment. Mark deduction will apply to latecomers. If you have a conflict with a co-op interview, you must inform the instructor in advance to make alternative arrangements to fulfill your lab requirements.

In the event of a lab missed due to an illness, contact the lab instructor asap to make alternative arrangement. You must also provide the instructor and the first year office with a Doctor’s certificate or a Verification of Illness form (can be downloaded from: http://www.healthservices.uwaterloo.ca/Health_Services/verification.html) completed by a doctor. Arrangements will be made for you to make up for the missed lab.

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LABORATORY SAFETY

Lab Safety is paramount for chemical engineering labs and can’t be overstated. Dealing with flammable or corrosive chemicals, spattering reagents, violent chemical reactions, or the escape of steam can be quite dangerous in the absence of sensible protective measures or devices. Failure to follow the lab rules regarding safety and health may result in the loss of marks and even lab privileges.

All those who work in the lab must follow these safety regulations:

All students are required to have completed a WHMIS course. If you have any doubts about a procedure, ask the lab instructor for assistance.

Make sure to know the location of the following:

Fire extinguisher

Safety shower

First aid kit

Fire alarm

Nearest telephone

Safety goggles must be worn at all times. Lab coats are highly recommended. Closed-toe shoes and long pants must be worn.

Read the Material Safety Data Sheets (MSDS) prior to experiment if necessary.

All hazardous wastes must be emptied into the appropriate waste containers. Read waste labels before disposing into any receptacle.

Dispose of any broken glass in the bucket labeled “Broken Glass”.

Keep chemical bottles closed tightly when not in use.

Immediately inform your Teaching Assistant of any injuries or spills. If a chemical comes in contacts with your skin, or eyes, or mouth, flush immediately with water at the sink or safety station. You must file an accident report with the department. In the event of a serious injury, inform the TA that you wish to go to Health Services. The TA will send someone to accompany you.

Clean up work area before leaving.

Food and Beverages are Strictly Prohibited in the Labs

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GUIDELINE FOR LAB REPORT Two types of laboratory report are required for this course: a preliminary report and memo report. The preliminary laboratory report includes the following:

1. Title page (sample title page in Appendix A or on course page on LEARN)

University and department. Laboratory course number Experiment number and title. Group number. Your name for individual reports and the names of all the group members for the group

reports. Date of experiment. Date of submission of the report.

2. Table of Contents List of the divisions of the report with page numbers on opposite sides. Lengthy divisions should be subdivided under appropriate headings.

3. Introduction The purpose of the Introduction is to introduce the reader to the topic. It should include

Brief statement of the objective. General background information relating to the experiment. Significance to Chemical Engineering of the data obtained from the experiment.

This must be in your own words. Do not copy from the manual

4. Theoretical Principles Short discussion of the theoretical foundation of the experiment. Include all the necessary equations to be used in the calculations. Do not include

derivations of equations unless it is your own or from an outside reference. Define newly introduced symbols with units below the equations. Conclude with a brief statement indicating how the objectives will be met using the

theoretical principles outlined.

Use your own words. Do not copy from the lab manual or other references

5. Experimental Summarize what you will be doing in the laboratory based on what you have read in

the laboratory manual and learned in the pre-lab.

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Include a labeled sketch or schematic of the equipment State any modifications made to the procedure. Include a blank data sheet (table with headings showing quantities to be measured) if

necessary:

6. Appendices Appendix A: Sample data obtained from the TA at the pre-lab session. Appendix B: Sample prelab calculations and answers to the pre-lab questions Appendix C: Safety data including summarized MSDS data, safety hazards, chemical

and waste handling, and first aid issues. Experiments 2 and 3 require individual prelab reports and experiments 1 and 4 require group reports.

Memo Report

The memo report must include all the components in the prelab report with necessary additions (e.g. to Table of Contents and Experimental) and corrections plus the following:

Corrected Preliminary Report should be attached to the Memo Report.

Abstract

The abstract should include a brief description of experiment and a summary of all important findings and conclusions from the experiment.

Results and Discussions

Since ChE101 experiments are not long and involved, it is convenient to combine Results and Discussion of results into one section. It is the most important section of the report, make sure to follow the following guidelines:

Present the results in the most effective way using appropriate plots or tables. Plots of data can be easier to understand than tables in many cases. If you use a plot, the data for the plot should be listed in the appendix.

Do not include long tables of experimental data. Put these in the appendix, and present only the derived results.

Each table, graph, or diagram must have An adequate title with table or figure number. See the graphs and tables in the lab manual of

Experiment 1 and 2 for detailed examples. Proper labels with appropriate units for each axis of the graph (also see the examples in Experiment

1 and 2). “Symbols” for measured values and “line” for predicted values when plotting data for comparison.

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Use headings, subheadings to break up the discussion into well defined, small packages which the reader can easily understand.

Guide the reader through the analysis of results by referring to specific tables and figures. Remember to state any assumptions made. In discussing your results, try and answer the following questions.

1. Are your results in agreement with the theory? If not, suggest reasons for the differences.

2. Are your results in agreement with published data? (Use a reference to indicate your source of information). If not, suggest reasons for the differences.

3. Why is one or more of your data out of line? Discuss experimental error. Quantitative results must be accompanied by an estimate of

their precision. Results should be presented with an estimated or given precision calculated from potential sources of errors in measurements made during the experiment.

Conclusions Extract conclusions from the previous section and present them in numbered form. Be as specific as possible and start with the most important ones. Refer to your objectives when you make your conclusions.

Recommendations

Recommendations to improve the experiment based on your experience should be included here.

Be as specific as possible starting with the most important ones. Include methods to improve the accuracy and reliability of the measurements. Make sure your recommendations are feasible from a practical and budgetary point of

view. Give estimates (if possible) of the cost involved when making such suggestions.

Nomenclature

Define all the symbols used in the report in alphabetical order, if it is necessary.

References

For journals, give complete information including names of all the authors, title of the article, name of the journal, volume, page number(s), and the year of publication. For example, Rabe, A. E., and Harris, J. F., “Vapor liquid Equilibrium Data for the Binary System”, Journal of Chemical Engineering Data, 8, 333 (1963).

For reference books, give author(s), title, edition, number, publisher, year of publication, and page number(s). For example,

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McCabe, W. L., Smith, J. C., Unit Operations of Chemical Engineering, Chap. 6, 3rd Ed., McGraw Hill, New York, 1992,

Appendices

Appendices are generally used for material which is useful for the report, but cannot conveniently be included in the body of the report.

Appendix A – Raw Data Appendix B – Sample calculations Appendix C – Safety Data Appendix D – Answers to questions Appendix E – Corrected pre-lab report.

Experiments 2 and 3 require individual memo reports and experiments 1 and 4 require group reports.

Style of Writing

P.B. Hughes (The Engineering Report in the Undergraduate Laboratory, Longmans Canada Ltd., 1963) has suggested several principles of effective report writing. Some of these, and others, which you may find to be of help in writing reports are:

1. To use the plainest words and grammatical constructions adequate for what has to be expressed is the ideal in engineering (and probably in all ) writing ... [if this goal is to be attained] the author must exert a distinct and controlled literary effort, an effort [which must] be considered a task in its own right.

Use the past tense for the description of the experiment except for experimental procedure in a Preliminary Report.

Avoid the use of personal pronouns. Avoid unnecessary adjectives. Don't use colloquialisms (e.g. ‘the value is off’).

2. The writer is a unique person, and elements of his or her individuality are bound to appear in

the style of writing, but in this highly-objective sort of writing, emotions, moods, prejudices, and the like, must, so far as is possible, be completely withdrawn.

3. The sequence in which the various parts of the experiment were done often is not the order most desirable for lucid reporting. The main aim in scientific reports is to be as clear and precise as possible and to make each sentence mean exactly what it is intended to while being incapable of other interpretation. Words or phrases that do not have an exact meaning are to be avoided because once one has given a name to something; one immediately has a feeling that the position has been clarified, whereas often the contrary is true". (W.I.B. Beveridge, The Art of Scientific Investigation, Vintage Books, New York, 1950).

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Proof reading

It is very important to proof-read your report carefully at least twice and by every group member. It is not unusual to have reports submitted with pages, diagrams, sections missing, sentences without verbs, completely meaningless sentences, and inconsistent statements -- sometimes on the same page, numerous spelling and grammatical errors, etc. You cannot proof-read your report too carefully.

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Experiment 1

Solid-Liquid Phase Diagram for Binary Naphthalene-Biphenyl System

Objectives

1. To determine the solid-liquid phase diagram of binary naphthalene-biphenyl system at atmospheric pressure using thermal analysis.

2. To determine the enthalpy of fusion for naphthalene and biphenyl using the linear regression analysis of measured freezing points.

Introduction

The plant for a chemical process to manufacture a particular product is usually an integrated system of different units. The main unit, the heart of the process, is the reactor where the actual reaction takes place. However, there are other units that do not involve reactions. These include separation units where components of a mixture are separated. The separation processes typically involve 2 phases, solid and liquid, gas and liquid, gas and solid, or two immiscible liquids.

The conditions under which different phases can exist, is a matter of considerable practical importance in separation processes. Experimental determinations of these conditions are usually presented in graphs called phase diagrams, which are graphical representations of the equilibrium relationships between equilibrium conditions (temperature, pressure) and phase compositions. The state of equilibrium depends on the main variables such as temperature, pressure, and concentration of the different substances in the phases. Time is not a variable for systems at equilibrium.

Solid-liquid phase diagrams have been of great value in solving practical problems such as the preparation of alloys or salts from complicated mixtures. Phase diagrams are also useful for the determination of proper conditions (temperature, pressure) to separate solid or liquid mixtures into the desired product.

Theoretical Principles

1. Phase Rule

The phase rule, first stated by J. Willard Gibbs is a useful tool in the consideration of phase equilibrium. It is derived by considering the number of variables in a system together with the number of equations relating them. The fundamental equation of the phase rule is,

2+−= PCF

Where F is the number of degrees of freedom, C is the number of components in the system, and P is the number of phases in the system.

Number of Degrees of Freedom: is the least number of intensive variables (temperature, pressure, concentrations of components) that must be fixed in order to describe the system completely.

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Number of Phases: A phase is a homogeneous, physically distinct, and mechanically separable portion of a system which is separated from other parts of the system by definite boundaries, for example:

• 1=P for a pure gas or a mixture of gases because they are miscible in all proportions. • 1=P for a mixture of liquids which are fully miscible, while 2=P for a mixture of

liquids which turns into two immiscible liquid phases. If there is a surface separation between two liquids, a two phase system results.

• Every solid constitutes a separate phase. Allotropes of an element and different crystalline modifications of a substance constitute different phases.

• Note that the surface separation between two phases needs not to be continuous. Bubbles in a liquid represent two phases, even though bubbles are separated from each other by liquid.

Number of Components is the smallest number of independent chemical constituents needed to fix the composition of every phase in the system. It is equal to the difference between the number of chemical species in a system and the number of equations relating the concentrations of these substances in an equilibrium system.

For example, the three phase system, CaCO3 => CaO + CO2. There are 3 distinct chemical species involved, but there is also one chemical reaction which relates the concentrations of these species, therefore there are only 2 independent components.

Applications of Phase Rule

Example 1: For pure liquid water, 1=C ; 1=P , so 2=F , meaning that it is possible to vary both the pressure and temperature of a single pure liquid phase.

Example 2: For an equilibrium mixture of water and steam, 1=C ; 2=P , so 1=F . Thus only one intensive property can be varied independently. If the temperature is specified, pressure is automatically set.

Example 3: For the vapor-liquid equilibrium mixture of ethanol water system, 2=C ; 2=P , so 2=F . Thus two variables may be set. The two may be any combination of the temperature,

pressure and phase concentration. For example, if the composition and pressure of a phase are specified, the temperature is automatically set.

2. Solid-Liquid Phase Diagram of Binary System

2.1. Phase Rule for Binary Solid-Liquid System

In the study of solid-liquid equilibrium, it is usual to carry out the investigation at constant pressure. Moreover, the effect of pressure on the solid-liquid system is negligible. A system in which only the solid and liquid phases are considered and the gas phase is ignored is called a condensed system. Since the pressure is kept constant, one degree of freedom is surrendered, this means the phase rule for the constant pressure system becomes,

1+−= PCF .

The phase diagrams are thus temperature-composition diagrams.

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2.2. Thermal Analysis and Solid-Liquid Phase Diagram of Binary System

One of the easiest methods for determining solid-liquid phase diagram is thermal analysis, which involves cooling a liquid mixture of known composition at constant cooling rate until the liquid freezes. The slow cooling normally yields two types of cooling curves depending on the composition.

Pure substance usually exhibits a cooling curve shown in Figure 1.1. First, a straight line of temperature versus cooling time indicates the liquid cooling at a constant cooling rate. When the temperature is lowered to the freezing point of the substance, the temperature doesn’t fall at all while the pure substance is freezing. This is because energy (enthalpy of fusion) is released when new bonds in the solid form with the phase change at exactly the same rate as the cooling rate, resulting in the temperature plateau as observed on the graph. At the end of the plateau, all the liquid is solidified, and further cooling leads to the lowering of solid temperature. Note that just before the liquid freezes, there is sometimes a slight dip in the cooling curve below the freezing point. This is called supercooling. As soon as some solid forms, the temperature recovers to the normal freezing point.

Figure 1.1. Typical cooling curve of pure substance A binary liquid mixture usually exhibits different cooling curve. Figure 1.2 shows a typical cooling curve of binary mixture (A and B) rich in substance A. First, the liquid cooling shows a straight line. As soon as A starts freezing at the mixture freezing point, the slope of the cooling curve changes due to the heat of fusion. However, the heat is not enough to compensate for the heat loss from the cooling, and so the temperature continues to drop. When the temperature is lowered to the point where substance B starts freezing, the temperature doesn’t fall any more until all the liquid mixture is solidified. Further cooling leads to the lowering of solid temperature. It is important to note that the addition of B lowers the melting point of A or the

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Tem

pera

ture

(°C)

Cooling time (min)

Liquid cooling

Liquid freezing

solid cooling

supercooling

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mixture, and almost pure A is in solid phase before B freezes. A binary liquid mixture rich in B will show similar cooling curve except that B start freezing first.

Figure 1.2. Cooling curve of binary liquid mixture rich in A

For a binary system, if the two substances A and B are completely miscible in the liquid state and without any appreciable solid-solid solubility, the phase diagram constructed using the freezing points at different compositions from the above thermal analysis will consist of the two solubility curves in the liquid state intersecting at point C as shown in Figure 1.3.

Figure 1.3. Solid-liquid Phase diagram of binary system (A and B)

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Cooling time (min)

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A freezing

A and B freezing

Solid A and solid B cooling

Mixture freezing point

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Percent B (%)

A

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A B

Liquid (A+B)

Solid A+Liquid (A+B) Solid B+Liquid (A+B)

Solid A+Solid B

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Some important characteristics of the typical phase diagram are summarized as follows:

• The lines AC and BC are obtained by determining the freezing points of liquid mixtures of various compositions and plotting the freezing point against the corresponding compositions.

• The curve AC on the left is the freezing point curve of A. The addition of pure B to A lowers the freezing point of A. Similarly, curve BC is the freezing point curve of B, the addition of A to B depresses the freezing point of B.

• The two curves intersect at point C, called the eutectic point. The term eutectic is derived from the Greek word eutectos, meaning easily melted. The eutectic point C is the lowest temperature at which a solution of A and B can coexist. At temperatures below this point all the remaining liquid freezes forming two solid phases. As such, at the eutectic point C three phases coexist and so temperature and composition are all fixed ( 0132 =+−=F ). The eutectic point is also the lowest melting point of a mixture of solid A and solid B.

• The theory of phase diagram is based on the thermodynamic phase equilibrium condition that the chemical potentials of any species in all the phases in equilibrium are equal. This condition, along with other assumptions such as insoluble solid phase and ideal liquid phase, leads to the following relations between the freezing point and liquid composition:

)ln(2

, AA

AAAm X

HRTTT∆

+= (1)

)ln(2

, BB

BBBm X

HRTTT∆

+= (2)

Where AmT , , BmT , : Freezing point of species A or B in a liquid mixture.

AT , BT : Freezing point of pure A or B. AX , BX : Mole fraction of species A and B in liquid phase.

AH∆ , BH∆ : Enthalpy of fusion for pure A and B.

AH∆ and BH∆ are thermodynamic properties which can, according to Equations 1 and 2, be determined by plotting freezing points of corresponding species against its logarithmic mole fraction. Equations 1 and 2 can also be used to calculate solid-liquid phase diagram from known thermodynamic properties of pure species.

2.3. Importance of Solid-Liquid Phase Diagram of Binary System

The importance of solid-liquid phase diagram lies not only in its reflection of physical interactions between two species in solid and liquid states, but also in its practical applications in separation processes. As shown in Figure 1.4, the cooling of a binary liquid mixture can leads to a liquid phase coexisting with a relatively pure solid phase, which can be easily removed. Depending on the mixture composition, there are three typical cases.

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Figure 1.4. Schematic representation of the cooling of binary liquid mixtures

Case 1: If a binary liquid mixture has the eutectic composition (composition 1 in Figure 1.4), cooling the mixture will eventually result in a mixture of solid A and solid B, separation is almost impossible.

Case 2: For a binary liquid mixture of composition below the eutectic composition (composition 2 in Figure 1.4), when cooling temperature reaches the freezing point, A starts freezing out of liquid. Further cooling will lead to more pure solid A coexisting with a liquid of composition moving along the phase curve (line AC). As an example, when the temperature is lowered to point M, the liquid composition can be found at the intercept of a horizontal line at the temperature (tie line) with the phase line. If we further cool down the mixture and when the temperature is close to the eutectic temperature, we end up with a lot pure solid A and a liquid mixture similar to the eutectic mixture.

Case 3: For a binary liquid mixture of composition above the eutectic composition (composition 3 in Figure 1.4), cooling the mixture will lead to pure solid B coexisting with a liquid of composition moving along the right-side phase curve (line BC). Similar to case 2, separation is possible.

So a tie line tells us the composition of phases present in an equilibrium mixture. Knowing the composition of the two phases, we often need to know the fraction of each phase exists in the mixture at the given temperature. This can be accomplished by applying the lever rule.

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3 1

M Tie line

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Figure 1.5. Schematic representation of the cooling of binary liquid mixtures

The lever rule states that the proportion of a phase in a two-phase mixture is the ratio of the segment length to the other phase to the entire length of the tie line. Take the two-phase mixture of initial concentration oC on another typical phase diagram of binary component A and B in Figure 1.5 as an example, the fraction of α phase ( αf ) and liquid phase ( Lf ) is, respectively,

α

α CCCCf

L

oL

−−

= and

α

αα CC

CCffL

oL −

−=−=1

The lever rule can be easily understood by a simple component mass balance. The composition of the mixture of oC must be made up of the fraction of α ( αf ) at composition αC and the fraction of the liquid ( αf−1 ) at composition LC , that is, Lo CfCfC )1( ααα −+=

We can easily obtain the expression of αf from this mass balance equation. Note that the length of the left side of the tie line gives the proportion of the liquid phase (phase to the right), while

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Solid α + solid β

Solid β + Liquid (A+B)

Solid α

Solid α + Liquid (A+B)

Solid β

Co Cα CL

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the length of the right side of the tie line gives the proportion of the α phase (the phase to the left). Also keep in mind that unlike the pure solid phase shown in Figure 1.4, the solid α phase in the phase diagram of Figure 1.5 contains both component A and B but is substantially rich in A, similarly the solid β phase also contains both components but is substantially rich in B.

Experimental 1. Apparatus/Reagents

• Water bath heated by a hot plate for melting mixtures. • Cooling bath with temperature controller. • Glass sample tubes. • Thermocouples. • Stirrers. • Reagent grade naphthalene and biphenyl (since these chemicals are noxious, avoid

spilling them into the water bath). 2. Experimental Procedure

The temperature-time data must be plotted to determine freezing point as you record it. Therefore, you must have graph paper or a laptop ready at the start of the experiment.

1. This experiment intends to cover the entire composition range of binary mixtures, so the sample mixtures will be divided for two lab groups. Group 1 will do mixtures 2~4 with test tube 1; and Group 2 will do mixtures 6~8 with test tube 2. First, prepare the sample mixture into the test tube according to Table 1 and ensure that the water bath is set to the appropriate program. Then place the test tube in the heating tube, fit the test tube with the thermocouple and the stirrer, and wait until the solid is completely melted. Occasional agitation may be necessary to ensure complete melting.

2. Use the cooling bath’s knob to start the cooling. This is done by turning the knob to the right until you reach the “Run program” screen and then selecting the program number at the top of the screen.

3. Set temperature recording and display, then start the cooling program by pressing the button “start” at the bottom of the “Run Program” screen. While the mixture is cooling, use the stirring rod to stir the mixture vigorously. Watch the temperature-time plot and observe the sample phase change.

4. When a temperature plateau is reached or there is a significant change in slope, accompanied by the formation of precipitated solid in the test tube, stop the temperature program and properly transfer and save your data.

5. Add required amount of naphthalene to the biphenyl into the test tube, and change the temperature program accordingly.

6. Re-melt the content of the tube and repeat Steps 2~4 for the other two samples.

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Table 1.1. Experimental samples and corresponding temperature programs.

Mixture

#

Naphthalene

(g)

Biphenyl

(g)

Total weight

(g)

Starting bath temp

(°C)

Temperature

program Tube # 1

1 8.5 0 8.5 83 1 2 8.5 1.5 10.0 76 2 3 8.5 3.5 12.0 70 3 4 8.5 7.0 15.5 60 4

Tube # 2

5 0 9.0 9.0 75 5 6 1.0 9.0 10.0 70 6 7 3.0 9.0 12.0 60 7 8 6.0 9.0 15.0 45 8

Data Analysis and Result Discussion

1. Plot your own four sets of data as temperature versus time. Comment on the similarity and difference among the plots and determine the freezing temperature for each sample.

2. Construct the phase diagram (Temperature vs. mass fraction) for the naphthalene-biphenyl system using the data from all eight samples. Discuss your experimental findings in relation to the theoretical principles.

3. Convert mass fraction into mole fraction, and plot the naphthalene portion of data against logarithmic mole fraction (Equation 1), then use the linear regression formula in Appendix B to determine the freezing temperature of pure naphthalene and the enthalpy of fusion. How do your calculated AT and the enthalpy of fusion compare with your measured values or values in literature?

4. Repeat (3) for the biphenyl portion of data, discuss your findings.

Questions

1. Define intensive and extensive properties and give 5 examples each. 2. Answer the following:

(a) Why is “time” not a “degree of freedom” when using the phase rule? (b) Using the phase rule to determine the number of degrees of freedom at the eutectic

point. 3. Calculate the number of degrees of freedom for a system of

(a) Ice – water – vapor (b) Mixture of hydrogen and ammonia

4. Answer the following: (a) How does super cooling affect the accuracy of the data obtained from the cooling

curve? (b) Why must the temperature difference between the contents of the sample and the

cooling water in the jacket be kept constant? (c) In the cooling curve, the temperature drops more rapidly for the solid (below freezing

point) than for the liquid. Why?

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5. Explain the cooling curves obtained with mixtures 6, 7 and 8 with reference to the phase diagram.

6. What is the composition and weights of the phases present at equilibrium when a mixture of 20% biphenyl (80% naphthalene) is at 80°C, 60°C and 30°C.? The total weight of the system is 12.7 kg. Illustrate how you used your phase diagram to answer this question by using a sketch of the phase diagram. Identify points or lines on the sketch by numbers or letters.

7. Make a sketch of the naphthalene-biphenyl phase diagram. Choose a temperature about halfway between the melting point of biphenyl and the eutectic temperature. Starting at 100% naphthalene and (at the above temperature) explain what happens to the system as you gradually add biphenyl until you reach the 90% biphenyl point.

8. The phase diagram of a binary A and B is similar to Figure 1.5. A liquid mixture of A and B with 23.7% B by weight is brought down to a two phase region where the solid phase contains 5.2% B by weight and the liquid phase contains 34.5% B by weight. Answer the following question:

1. What is the solid phase? 2. What is the fraction of the solid phase and liquid phase? 3. If the mass of the initial liquid mixture is 120 kg, what is the mass of the solid and

liquid at equilibrium?

20

Experiment 2

Energy Balance and Heat Transfer Coefficient of Heat Exchangers

Objectives

1. To perform energy balance on different heat exchangers. 2. To measure overall heat transfer coefficients of the heat exchangers, and 3. To examine the effect of flow velocity on overall heat transfer coefficient.

Introduction Many engineering processes involve heat being transferred from one body or fluid to another, chemical engineering processes do even more so, since most of chemical engineering process units, from central chemical reactors to upstream feedstock units and to downstream separation processes, all require the transfer of heat from energy sources or sinks (electrical, thermal, steam, coolant) to the process units to bring temperature to certain desired processing temperatures. The transfer of heat is often accomplished by using a mechanical device and another transfer fluid that can absorb or release heat without direct contact with process fluids. The specialized device that facilitates heat transfer from one body or fluid to another and avoids direct contact of the fluids is referred to as heat exchanger Heat exchangers are typically classified according to the type of construction and flow arrangement. The simplest heat exchanger is the one through which the hot and cold fluids flow in concentric tubes (commonly called double pipe) in either co-current flow arrangement (Figure 2.1.a) or countercurrent flow arrangement (Figure 2.1.b). The double pipe heat exchanger is widely used in chemical engineering processes because it is easy to be constructed and it is handy for fluid handling. The second type of heat exchanger, or more efficient and widely used heat exchanger in chemical engineering applications, is shell-and-tube heat exchanger (Figure 2.1.c), which consists of one shell and multiple tubes. One fluid makes a single pass through the shell side with baffles that are usually installed to increase the convective heat transfer, and the other fluid distributes evenly into multiple tubes. Similar to double pipe heat exchanger, shell and tube heat exchange can be operated in either co-current flow arrangement or countercurrent flow arrangement, depending on the nature of process fluids and heat transfer requirement. Another major type of heat exchanger is so called cross-flow heat exchanger (Figure 2.1.d) or plate and frame heat exchanger in which the plates mainly function as fins to enhance convective heat transfer and to ensure cross-flow over the tubes. The cross flow heat exchanger is usually used for heat exchange between a gas and a liquid, with the gas normally passing over the fin surfaces and the liquid flowing through the tubes. Such exchangers are used for air-conditioning and refrigeration heat rejection applications.

21

Figure 2.1. Types of heat exchangers: (a) double pipe heat changer with co-current flow; (b) double pipe heat exchanger with countercurrent flow; (c) shell-and-tube heat exchanger; and (d) cross flow heat exchanger.

In this experiment, you are about to be introduced to two types of the most commonly used heat exchangers in chemical engineering, double pipe heat exchanger and shell-and-tube heat exchanger. By measuring the amount of sensible heat gained and lost by two process fluids and their temperature changes, the overall heat transfer coefficient of the heat exchanger will be determined. Theoretical Principles 1. Energy Balance of Heat Exchanger

The main function of a heat exchanger is to transfer energy from one process fluid to another. Since heat exchangers are often operated above or below ambient temperature, heat loss to surroundings is inevitable. When the heat loss of a heat exchanger is significant (e.g. > 20%), the heat exchanger operation is neither effective nor energy efficient. From experimental point of view, determining the heat loss or performing an energy balance on a heat exchanger is essential to validate experimental data and to justify the measurement of heat transfer coefficient.

Energy balance can be performed by examining the heat loss of hot fluid and heat gain by cold fluid. For a process that involves no appreciable phase change, latent heat is negligible. As such, the heat exchange is mainly through the form of sensible heat. The sensible heat for cold stream and hot stream for any heat exchanger can be determined by,

22

)( cicopccc TTCmQ −⋅= (1) and

)( hihophhh TTCmQ −⋅−= (2) Where Q : Heat transfer rate (kJ/min).

m : Mass flow rate (kg/min).

pC : Specific heat capacity ( CkgkJ o⋅/ ).

T : Temperature (°C).

subscript c and h : Cold and hot fluid, respectively. subscript i and o : Inlet and outlet of cold and hot fluids, respectively. The percentage difference between cQ and hQ can be calculated by,

%100% ×−

=avg

avgc

QQQ

diff (3)

Where avgQ is the average of cQ and hQ . In general, if the %diff is less than 5% (or 10% in many other cases), we can confidently claim that (1) the energy loss of the particular heat exchanger is negligible, and (2) the experimental data are valid on the particular heat exchanger which will justify the use of the measured heat transfer rate and temperatures to further determine overall heat transfer coefficient as follows. 2. Overall Heat Transfer Coefficient

The energy balance indicates the heat transfer rate and temperature change for each fluid stream in a heat exchange system. In engineering practice, we need more often to determine the size of a heat exchanger to achieve certain desired heat transfer rate and temperatures, such design can be achieved by using overall heat transfer coefficient. Take double pipe heat exchanger as an example, the heat transferred in any differential cross-section area along the pipe between the two ends can be expressed in terms of heat transfer coefficient, heat transfer area, and temperature difference between the two fluids in that area. If we assume the heat transfer coefficient is constant from one end to the other of the heat exchanger, the heat transfer rate can be integrated as [2],

lmTUAQ ∆= (4)

Where U is referred to as overall heat transfer coefficient, A is the total heat transfer area based usually on the outside area of the tube, and lmT∆ is referred to as logarithmic mean temperature difference (LMTD) defined as,

1

2

12

11

22

1122

lnln

)()(

TT

TT

TTTT

TTTTT

ch

ch

chchlm

∆∆∆−∆

=

−−

−−−=∆ (5)

Where subscript 1 and 2 represent each end of the heat exchanger as shown in Figure 2.2.

23

It is important to understand that lmT∆ is the average temperature difference of two fluids at any cross-section along the pipe, and so is the driving force for heat transfer. Equation 4 indicates that with the calculated lmT∆ from measured temperatures at the two ends, along with the measured heat transfer rate and known pipe dimension, the overall heat transfer coefficient can be determined. It is also important to note that, as indicated in Figure 2.2., depending on the flow arrangement, the temperature changes or profiles can be different, so understanding of the temperature profiles is of great help in identifying each temperature reading on the heat exchangers for Equation 5.

Figure 2.2. Temperature profiles of countercurrent flow (left) and co-current flow (right), arrows indicate temperature trends along the length of heat exchanger.

3. Correlation between Overall Heat Transfer Coefficient and Flow Velocity

As shown above, the overall heat transfer coefficient, U , is a measure of heat transfer capability of a given heat exchange, and is determined by such factors as the properties (density, viscosity, heat conductivity) of fluids, pipe material, flow velocity of each fluid, and flow arrangement. The reciprocal of the product of U and A is called overall thermal or heat transfer resistance commonly denoted by ovR (exactly the same definition of R value used to rate insulation materials in construction industry). This can be easily understood by rearranging Equation 4 into,

QRQUA

T ovlm ==∆1 (6)

Equation 6 mimics the Ohm’s law ( RIE = ) in which lmT∆ is the thermal potential or driving force for heat flow and Q is the heat current or flow. Similar to electrical resistance of multiple resistance in series, we can envision the resistance to heat flow from higher temperature fluid stream to lower temperature stream as a series of resistances due to inner pipe flow (inside film resistance

iR ), pipe wall (conduction resistance wR ), and annulus or shell side flow (outside film resistance,

T

h2

Tc2

Th1

Tc1

1 2 Length

T

Th2

Tc2

Th1

Tc1

1 2 Length

T

24

oR , you will learn the details of heat transfer theory in your third year), the overall resistance to heat transfer is the sum of individual resistances, namely,

owiov RRRR ++= (7)

In general, the resistance from metallic pipe material is negligible in comparison to the resistances from the flow of either side, which is inversely proportional to its flow velocity to certain power, i.e.,

mi

i VCR 1

1= or no

o VCR 1

2= (8)

Where iV and oV are the flow velocity in the pipe and annulus (or shell side), respectively, which can be calculated by dividing volumetric flow rate by the cross-section area of the flow; 1C and 2Care constants which measures the resistance of the pipe flow and annular (or shell side) flow streams; and the value of the velocity exponent, m and n , should be around 0.8 in most cases [3,4]. Equation 7 and 8 allow us to experimentally measure the heat transfer resistance of the fluid flow. As an example, if we set annular or shell-side flow stream at constant flow rate and if we assume the change in fluid properties are negligible, the overall heat transfer resistance or coefficient can be expressed as,

omi

ov RV

CRUA

+==11

1 (9)

If we take 8.0=m and plot ovR versus 8.01

iV, a linear relationship can lead to at least two

conclusions: (1) the experimental data follow the film resistance theory, and (2) we can have a fairly reliable measurement of heat transfer resistance of annular (shell side) steam ( oR ) and

velocity coefficient of the other stream. The plot of ovR vs. mV1 is called the Wilson plot and is one

of the methods for heat exchange analysis and design. Experimental 1. Apparatus The following apparatus and heat exchangers will be used for the experiment:

• TQ TD360 heat exchanger module. • TQ TD360A Concentric Tube (double pipe) heat exchanger. • TQ TD360C shell and tube heat exchanger.

The dimensions for the double pipe heat exchanger and the shell-and-tube heat exchanger are listed in Table 2.1 and Table 2.2.

25

Table 2.1. Dimension of the double pipe heat exchanger (TQ TD360A)

Outside diameter Do (mm)

Inside diameter Di (mm)

Length of tube (mm)

Tube 12 10 2×270 Annulus 30 20 2×270

Table 2.2. Dimension of the shell and tube heat exchanger (TQ TD360C)

Outside diameter Do (mm)

Inside diameter Di (mm)

# of tubes*

Length of tube (mm)

Tube 6 4 6 180 Shell 60 50 1 180

*Note: One of the seven tubes is blocked.

Domestic water is used for this experiment. The heat capacity of water can be taken as a constant value ( CkgkJC o

p ⋅= /182.4 ) for the temperature range in this experiment. The

density of hot and cold water streams can be estimated using the average temperature of the inlet and outlet in degree Celsius:

23 0044.00208.030.1000)/( avgavg TTmkg −−=ρ (10)

2. Experimental Procedure

1. Open the two red water supply valves located behind the heat exchanger module. 2. Turn on the main switch located on the left side of the module. 3. Under the hot system section of the module, there are three lights (empty, half, full)

which indicate the water level inside the hot water supply unit. Before starting the experiment, ensure that the full light is on. If it is not, push the “Press to fill tank” button till the “full” light comes on.

4. Turn on the pump and heater, and set heater temperature to the specified temperature. 5. Adjust the hot and cold water flow rates using the valves located next to the hot and cold

supply. 6. Wait till the heater reaches the set temperature. The hot water temperature can be seen on

the water temperature controller. 7. It is very important to ensure that there are no air bubbles in water flow on annular or

shell side as this will affect the accuracy of the data. If bubbles are evident, tilt the system towards the hot water return pipe so that the air bubbles can escape out of the system.

8. Once the set temperature is achieved, start timing the experiment. It is possible for the flow rates to change while the system heated up. Therefore, adjust the flow rates accordingly before you time your experiment.

9. After approximately 5 minutes, record the temperature and flow rate values. For the shell and tube heat exchanger, record a total of four temperature reading (TH1,TH2, TC1, and

26

TC2) and two flow rate readings (FlowH and FlowC). For the double pipe heat exchanger, record six temperature readings (TH1,TH2,TH3, TC1,TC2, TC3) and two flow rate readings (FlowH and FlowC). To ensure that these values are in fact in steady state, wait another 5 minutes and if the values do not change, steady state has been achieved and the data obtained is accurate.

10. If the values change significantly in the 5 minute waiting period, record the new set of values and wait till steady state is achieved.

11. Repeat the experiment (Steps 8~10) for 3 more cold water flow rates. 12. Repeat Steps 1 to 11 with the other heat exchanger. To change the heat exchanger

module, first switch off the pump, close both water supply valves, and then switch to the other heat exchanger module and connect all the signal connections according to Step 9.

13. After completing all the required runs, turn off the heater first, then switch off the pump, close both water supply valves, and switch off the heat exchanger module.

Data Analysis and Result Discussion

1. Calculate cQ , hQ , and %diff for the two heat exchangers, comment on the energy balance of the experimental heat exchangers.

2. Calculate lmT∆ and A based on outside pipe, and then calculate the overall heat transfer coefficient U based on reasonable Q values for all the runs of each heat exchanger.

3. Calculate water velocity iV of all runs for each heat exchanger and plot ovR versus 8.01

iV for

both heat exchangers on the same plot (the Wilson plot), comment on the linearity of your plot and its implications.

4. Use the linear regression formula in Appendix B to calculate 1C (the thermal resistance coefficient of variable stream) and oR (the heat transfer resistance of constant stream) value for each heat exchanger. Comment on the difference in these values and heat transfer efficiency between the two heat exchangers.

5. Assuming that the absolute error for lmT∆ is 3.0± °C and the relative error for heat flow rate is the same as %diff, use the error propagation to calculate the absolute error for the measured overall heat transfer coefficient.

Questions

1. Why is the energy balance so important for the heat transfer experiment? What would you do if the difference between cQ and hQ is too large (e.g. >15%)?

2. Define latent heat and sensible heat, respectively. What type of heat is involved in this experiment?

3. According to Figure 2.2, the temperature difference between the hot and cold stream seems very constant (i.e. relatively constant driving force and lmT∆ ) from one end to the other for

27

the countercurrent flow arrangement, what would you say about the temperature difference and driving force for the co-current flow arrangement ?

4. A double pipe heat exchanger with the following pipe dimension,

Outside diameter Do (mm)

Inside diameter Di (mm)

Tube 80 70 Annulus 150 140

is to be constructed to cool down an aqueous process fluid at a flow rate of 69.0 L/min from 85 °C to 38 °C. The countercurrent flow arrangement will be chosen with cooling water at 12.5 °C being cooling fluid. The operation flow rate of the cooling water is 125.3 L/min. The overall heat transfer coefficient from this experiment ( min)/80 2 ⋅⋅= CmkJU o can be used for this design calculation. Assuming the process fluid has similar properties as pure water, calculate the required length of the double pipe heat exchanger.

References

1. Felder, R. M. and Rousseau, R. W., Elementary principles of Chemical Processes, 3rd edition., John Wiley & Sons, New York, 2000.

2. Perry, R. H. and Chilton, C. H., Chemical Engineers' Handbook, 5th Ed., McGraw-Hill, NY, 1973.

3. Fernandez-Seara, J., Uhia, F. J., Sieres, J., and Campo, A., “A General View of the Wilson Plot Method and its Modifications to Determine Convection Coefficients in Heat Exchanger Devices”, Applied Thermal Engineering, 27, 2745-2757 (2007).

4. Rose, J. W., “Heat Transfer Coefficients, Wilson Plots and Accuracy of Thermal Measurements”, Experimental Thermal and Fluid Science, 28, 77-86 (2004).

28

Experiment 3

Fluid Flow through an Orifice

Objectives

The objectives of this experiment are

1. To measure liquid head using differential pressure (DP) transducer and flow rate through an orifice.

2. To investigate the transient behavior of liquid flowing through orifices and to determine orifice discharge coefficient.

Introduction

Many of the materials involved in chemical engineering processes are in liquid form. These materials are normally pumped to flow from one processing unit to another in order to have them processed in a plant. Each processing unit is often designed to operate at certain flow rates for optimum performance. The flow rate of a liquid depends mainly on the head or pressure applied to the fluid, and can also be affected by the physical properties of the fluid such as density and viscosity as well as the size and geometry of process equipment in which it flows. The principles that govern the flow of fluids thus become an essential component of the chemical engineering curriculum, a core course (Fluid Mechanics) offered in the second year in the department.

Pressure losses due to frictional forces between viscous flowing liquid and the equipment it flows through result in a reduced flow rate. As a common practice, pressure boost by booster pumps installed in between processing units is often necessary to increase the flow rate to a desirable flow rate. As such, the accurate determination of flow rate and pressure loss of a fluid system is very important in process design and operations.

In general, pressure can be easily measured in terms of both absolute pressure and differential pressure. Flow rate is however more difficult to measure, and hence, many different types of devices are being used in various processes. The simplest are invariably those that directly measure the volumetric flow rate of fluid such as bubble flow meters or rotameters. Other more common and indirect devices for fluid metering include pitot tube, venturi, nozzle, and orifice meters, which are all based on pressure differential across a restriction in the flow path. For example, an orifice meter is based on the principle that when an orifice much smaller than flow pipe is placed within the pipe system through which the liquid is flowing, the pressure difference between the high pressure upstream and the low pressure downstream is closely related to the flow rate of the fluid across the restriction, thus the flow rate can be conveniently determined by simply measuring the pressure difference. The flow through an orifice in this experiment imitates the characteristics of these indirect flow meters and also simulates many practical applications (e.g. discharge of a liquid from storage tanks, water tower, etc.).

29

Theoretical Principle

1. Steady-State Flow and Orifice Discharge Coefficient

Recall from physics that any object, dropped from rest through a distance of h in a vacuum, will obtain the speed, 2/1)2( ghV = , the same is true as the liquid flowing through an orifice .

Consider a container of liquid having in its side a small hole or orifice located a distance h below the liquid level, as shown in Figure 3.1. In many practical applications, we are interested in establishing a relationship between the volumetric flow rate of liquid flowing through the orifice and liquid height or "head", h .

Figure 3.1. Schematic diagram of orifice flow due to a constant head The principle we can use to establish the relationship is the "mechanical energy balance" (an energy balance which excludes chemical form of energy is called a mechanical energy balance, see Felder and Rousseau, Chapter 7.7 for additional discussion). First, let’s assume a steady-state flow which means that both the head and flow rate are constant. If we consider a fluid element of mass m at point 1, moving to point 2 in Figure 3.1, a simple mechanical energy balance leads to:

2211 PEKEPEKE +=+ (1)

Where KE stands for kinetic energy and PE for potential energy, hence,

30

)0(21

21 2

221 mgmvmghmv +=+ (2)

Since the liquid level is constant, so 1v is very small, and can be set approximately to zero, Equation 2 becomes, ghv 22

2 = or ghv 22 = (3)

Thus, the volumetric flow rate of the orifice is,

hAgAvQ oo 22 == (4)

Where oA is the cross-sectional area of the orifice. The simplified energy balance equation indicates that the flow rate out of an orifice is proportional to the square root of the fluid head. In reality, when fluid flows through an orifice, there are friction loss and flow area contraction, both of which reduces the fluid velocity, so a coefficient is usually introduced to account for those effects, i.e., hCAgQ do2= (5)

Where dC is referred to as orifice discharge coefficient which takes a value of 0.6~1.0, depending on the design and size of the orifice.

2. Measuring Orifice Discharge Coefficient from Transient Flow

Equation 5 allows us to determine the flow rate at a specified or constant head. If we fill the tank to a certain level and close the inlet flow instantaneously, the fluid will continue to drain out through the orifice, the liquid level or the head will go down with time, and so the flow rate out of the orifice will change with time as well. We can use unsteady-state mass balance to examine this transient process. A general mass balance equation for unsteady-state process can be written as, [mass accumulation] = [mass in] – [mass out] + [mass generation] Appling to the draining tank through an orifice with constant liquid density yields,

)(0)(2)(0)()( generationouthCAginonaccumulatidt

hAddo

t +−= (6)

Where tA is the cross-sectional area of the tank. Separating variables yields,

dtCAAg

hdh

dt

o2−= (7)

31

Integrating from the initial height (namely, ohht == ;0 ), we have,

odt

o htCAAgh +−=

2 (8)

Equation 8 suggests that if we collect the data of liquid height, h , at different draining times, we can determine the value of dC from the slope of the linear plot of h versus time.

Experimental

1. Apparatus • A vertical cylindrical constant head water tank of 7.5 inches in inside diameter with a side-

mounted holder for orifice. Constant water head can be maintained by adjusting the level of the overflow pipe.

• Three orifice plates of different sizes. • Volumetric cylinders of various sizes, a stopwatch, and digital balance for flow rate

measurement. • A differential pressure (DP) transducer connected to the bottom of the water tank, and a data

acquisition board (DAQ) converting DP signals to digital readings for LABVIEW to display and record.

2. Experimental Procedure 1. Insert an orifice plate with the cone-shaped side facing the inside of the water column. 2. Start the LABVIEW program. 3. Set the overflow pipe to about 60 cm, open the water inlet valve to fill the tank. Adjust the

overflow to a low flow rate so that the water level is steady and DP reading should be leveled off.

4. Select a suitable size of graduated cylinder and collect the effluent from the orifice for 20-30 seconds with a stopwatch and record the mass of water collected and time. Also measure the head above the centre of the orifice from the attached scale.

5. Repeat Step 4 one more time. 6. Start data logging on LABVIEW and wait for about 1 min. to collect initial height, then

quickly close the inlet valve and wait until the tank is completely drained and the DP reading is leveled again. Stop data logging.

7. Repeat steps 3 through 6 for the same orifice but at initial water level of 70 and 80 cm, respectively.

8. Repeat steps 3 through 6 for another orifice at initial water level of 70 cm.

Data Analysis and Result Discussion

1. Plot h versus t for all similar runs on the same graph and discuss about how the flow behaviors change with orifice size and initial height.

2. Use Equation 8 and the least-square procedure outlined in Appendix B to calculate the values of dC for different orifice sizes and different initial heights, and compare with the literature value? Discuss about your results and findings.

32

3. Use your measured dC to calculate the flow rate at initial liquid height for each run and compare with your measured flow rate. Discuss about the difference.

4. Use Equation 5 to estimate the standard error and maximum error in flow rate when liquid height is measured using DP transducer.

Questions

1. Why is it better to take multiple measurements of the amount of water and water head at initial height and steady state?

2. In this experiment, the water flow velocity out of an orifice at a constant head can be

measured by tA

Mvoρ

= with M being the mass collected in the collection time t . If the

water collection time is doubled, how does it affect the error for the measured flow rate? (Hints: calculate and compare the error with a time 1t and mass 1M , and a time 12 2tt = and mass 12 2MM = ).

3. What is your suggestion of a better calibration between water height and DP reading when the measured data for multiple runs are available?

4. A cylindrical storage tank has a diameter of 1.25 m and a bottom drain valve of 2.0 inches in diameter. The tank contains a liquid of initial level up to 4.5 m. How long will it take for the liquid level to go down to half of the initial height? (You may assume that the drain valve is an orifice of the same diameter and its dC value is 0.60).

References

1. McCabe, W. L. and Smith, J. C., Unit Operations of Chemical Engineering, 2nd Ed., McGraw-Hill, NY, 1967.

2. Felder, R. M., and Rousseau, R. W., Elementary Principles of Chemical Processes, John Wiley & Sons, NY, 2000.

3. Perry, R. H. and Chilton, C. H., Chemical Engineers' Handbook, 5th Ed., McGraw-Hill, NY, 1973.

4. Prohaska, P. D., Khan, A.A., and Kaye, N. B., “Investigation of Flow through Orifices in Riser Pipes”, Journal of Irrigation and Drainage Engineering, May, 340 (2010).

5. Jan, C. D., ASCE, M., and Nguyen, Q.T., “Discharge Coefficient for a Water Flow through a Bottom Orifice of a Conical Hopper”, Journal of Irrigation and Drainage Engineering, August, 567 (2010).

33

Experiment 4

Reaction Kinetics for the Hydrolysis of Ethyl Acetate with Sodium Hydroxide

Objectives

1. To determine the reaction rate law and the reaction rate constant for the hydrolysis of ethyl acetate.

2. To perform experimental data analysis, data manipulation, and interpretation of kinetic results by using in situ measured data and linear regression analysis.

Introduction

One of the main functions of chemical engineering is to make useful products from available raw materials. A plant for manufacturing a particular product is actually an integrated system of many process units performing different functions. The most important one among these process units is the reactor where the reactants are converted into products. Reaction Engineering is a core course in the 3rd year of your curriculum, dealing with principles, practice, and design of different types of reactors used in chemical industries. The fundamental basis of the reactor design is the nature of chemical reactions in terms of reaction thermodynamics and kinetics. Hence, it is important to investigate the reaction properly before embarking on the detailed design of a suitable reactor.

In an attempt to investigate a reaction to make a useful and economically feasible product, the two most important questions to be answered are,

• How far the reaction can go • How fast the reaction can go

The answer to the first question involves the branch of science called chemical thermodynamics. Applying the appropriate chemical thermodynamics equations enables the determination of the maximum possible conversion (equilibrium conversion) of reactants into products. When the thermodynamic data indicates that the equilibrium conversion is too low for the process to be economically feasible, the reaction conditions (temperature, pressure etc) may be changed to achieve higher conversion

Thermodynamics, however, is concerned only with the initial and final states of the system and not the rate at which the reaction proceeds. Time and money are important to process industries. Industrialists are not satisfied with merely turning one substance into another; they want to obtain the products rapidly. This brings us to the second question – that is, how fast can the reaction go? The branch of science that deals with the rate at which a chemical reaction yields products is called chemical kinetics. The key factors affecting the rate of a chemical reaction are

• Nature and properties of the reactants and medium in which the reaction takes place. • Concentrations of reactants and • Temperature.

One of the goals in chemical kinetics study is to determine the kinetic equation that can be used to predict the relationship between the rate of the reaction and concentrations of reactants. This

34

relationship is based on experimental measurements. Such an experimentally determined equation is called a rate law or rate equation.

Consider a hypothetical reaction

hHgGbBaA +=+

Where A , B , G , and H are chemical species, and a , b , g , and h are corresponding stoichiometric coefficients in the balanced equation. The rate of such a reaction usually follows power-law rate equation,

βαBACkCRate = , Where AC and BC are molar concentrations (moles/L) of reactants A and B .

The proportionality constant k relates the rate of reaction to reactant concentrations and is called the rate constant of the reaction at a specified temperature. The exponents α and β are the order of reaction with respect to reactant A and B, respectively. Both α and β are usually positive integers of 1 or 2 (although fractions and negative numbers are also possible), and they are not necessarily related to the stoichiometric coefficients of the reaction. The reaction rate constant k and the reaction order α and β must be determined experimentally by observing changes in the rate of a reaction as the reactants are consumed. The rate of reaction also needs to be determined experimentally by measuring concentration changes over time.

The concentration changes can be determined by any of the number of conventional methods like titration, gravimetry, or spectrophotometry, in which one may be obliged to collect samples of the reaction mixture for analysis. However, it is always more practical to use an in-situ or online method, if available, which eliminates the need for the removal of samples.

In any homogeneous mixture the reactant and/or product concentrations can be determined by measuring a property that is dependent on the species concentration in the mixture such as electrical conductivity, refractive index, pH, or optical density. Knowledge of the physical phenomena that forms the basis of an analytical technique allows one to develop mathematical relationship to correlate an intangible property such as species concentration in a mixture to a property that is readily measurable.

In this experiment, the reactant concentrations are determined by measuring the conductivity of the reaction solution mixture. The conductivity probe immersed inside the reaction mixture sends a voltage signal to a meter which amplifies and converts the signal to the conductivity of the solution mixture, from which the concentration of each chemical species in the mixture can be determined.

Theoretical Principles

1. Reaction Rate and Kinetic Equation

Ethyl acetate undergoes hydrolysis with sodium hydroxide by the reaction,

EtOHNaAcNaOHEtAc +=+ (1)

Where NaOH stands for sodium hydroxide, EtAc for ethyl acetate, NaAc for sodium acetate, and

35

EtOH for ethanol.

The reaction rate of a chemical reaction is defined as the change in the concentration of a reactant or product per unit time. We can monitor the concentration change of any one of these components to determine the rate of the reaction. Suppose we monitor the concentration change of sodium hydroxide in the reaction mixture, the reaction rate becomes,

tNaOH

tttatNaOHofConctatNaOHofConc

Rate∆

∆−=

−−

−=][..

12

12

The negative sign accounts for the disappearance of NaOH as a reactant. In general, the differential form of reaction rate law for the reaction can be written as,

βα )()( EtAcNaOH

EtOHNaAcEtAcNaOH CCkdt

dCdt

dCdt

dCdt

dCRate ===−=−=

(2)

Where

t : Reaction time in s . k : Reaction rate constant in 1)1()/( −−+− ⋅ sLmol βα .

NaOHC : Concentration of NaOH in Lmol / .

EtAcC : Concentration of EtAc in Lmol / . α : Reaction order with respect to NaOH . β : Reaction order with respect to EtAc .

Equation 2 simply states that the rate of disappearance of reactants equals to the rate of appearance of products for this reaction with all the stoichiometric coefficients being unity. For the brevity of derivation, we use subscript A to represent NaOH and subscript B for EtAc . Experimentally, we start the reaction with known initial concentration of NaOH and EtAc as

AoC and BoC , respectively. If we can measure the concentration of NaOH ( AC ) with reaction time, the concentration of EtAc ( BC ) at any reaction time can be determined by using the reaction stoichiometry as,

OHHCCOONaCHHCOOCCHNaOH 523523 +=+

At 0=t : AoC BoC 0 0 At tt = : AC )( AAoBoB CCCC −−= AAo CC − AAo CC −

Therefore, the reaction rate law can be written as,

βα )]([)( AAoBoAA CCCCk

dtdC

−−=− (3)

36

At certain concentration and temperature range, the reaction order with respect to both reactants for the hydrolysis reaction of Equation 1 is close to the first order. Hence, we can assume

1== βa and use experimental data to verify the validity of this assumption. Equation 3 now becomes,

)( AAoBoAA CCCkC

dtdC

+−=− (4)

Separating variables and integrating the rate equation with initial condition: 0=t , AoA CC = ,

∫∫ −=+−

tC

C AAoBoA

A kdtCCCC

dCA

Ao 0)( (5)

Note that the left hand side function can be split using the partial fraction into,

+−

−−

=+− AAoBoAAoBoAAoBoA CCCCCCCCCC

111)(

1 (6)

Substituting into Equation 5, we have

ktCCC

dCCdC

CC

A

Ao

A

Ao

C

C AAoBo

AC

C A

A

AoBo

−=

+−−

− ∫∫ )(1

(7)

Integrating and rearranging yields,

Ao

BoAoBo

AAoBo

A

CC

tCCkCCC

C ln)(ln −−−=+−

(8)

Equation 8 implies that if the reaction is the first order with respect to each reactant, a plot of the left-handed side of Equation 8 versus reaction time should follow a linear relationship, and the reaction rate constant, k , can then be determined from the slope. For chemical reactions, it is also very common to use conversion to measure the degree of a reaction. The conversion of a reactant, x , is defined as the ratio of the number of moles of the reactant reacted per mole of the reactant fed to the reactor. The conversion of NaOH in this reaction is

Ao

AAoA C

CCx −= (9)

Substituting into Equation 8 gives,

Ao

BoAoBo

AoBo CCtCCk

xCCx ln)(

/1ln −−−=

−−

(10)

Equation 10 relates the conversion of sodium hydroxide to reaction time, and can also be used to determine the reaction rate constant, k .

37

2. Relationship between Concentration and Solution Conductivity

Definition of Conductivity

The electrical resistance R of a material between two electrodes of cross sectional area A and length L is proportional to L and inversely proportional to A , i.e.,

KALR ρρ == (11)

The proportionality constant ρ is known as the resistivity or specific resistance, and ALK = is

called cell constant with the unit of 1−cm .

The reciprocal of resistance is often referred to as conductance G , which is a measure of the capacity of the electrical conductivity of the material,

LA

RG

ρ11

== (12)

The quantity that is usually used to characterize a material in terms of its ability to conduct electricity is conductivity, which is the specific conductance in a cell of 1 cm long and 1 cm2 in cross sectional area. The conductivity, κ , is related to measurable conductance by, GK=κ (13) Molar Conductivity for Solutions

For liquid solutions, the measured conductivity depends on solution concentration. So, it is often necessary to use molar conductivity which is defined as the conductance of a solution containing 1 mole of an electrolyte in 1 L of solution. The molar conductivity is related to the measured conductivity of a solution by,

Cκ

=Λ

Where C is the molar concentration in Lmol / . Thus the measured conductivity of the solution

is directly proportional to the concentration, which is the basis for conductivity probe, CΛ=κ (14)

When a dilute solution contains multiple ions, the ions are free to move and behave independently. The conductivity of the electrolytic solution is equal to the sum of the conductivities of all the cations and anions in the solution, that is, for a solution of M ionic species, the solution conductivity is,

∑=

Λ=M

jjjt C

1κ (15)

38

Where tκ : solution conductivity in cmmS / .

jC : concentration of species j Lmol / . jΛ : molar ionic conductivity of species j in molcmLmS ⋅⋅ / .

Now in order to determine the reaction rate constant, we need to relate the reactant concentration to solution conductivity as a function of reaction time. In the hydrolysis reaction, the contribution of EtAc and EtOH to the reaction solution conductivity is negligible. Thus, based on the aforementioned reaction stoichiometry and Equation 15, the conductivity at reaction time 0=t

is, AoNaOHo C)( +− Λ+Λ=κ

(16)

And, at any other time tt = , the concentrations of all ions in solution are:

ACOH =− ][

AAo CCCOOCH −=− ][ 3

AoAAoA CCCCNa =−+=+ )(][

So, the solution conductivity becomes,

)(3

AAoCOOCHAoNaAOHt CCCC −Λ+Λ+Λ= −+−κ (17)

At the end of the experiment ftt = (assuming NaOH is in excess and the reaction goes to

completion), BoCCOOCH =− ][ 3 , BoAo CCOH −=− ][ , and AoCNa =+ ][ , BoCOOCHAoNaBoAoOHf CCCC −+− Λ+Λ+−Λ=

3)(κ (18)

Subtracting Equation 17 from Equation 16 yields, ))((

3AAoCOOCHOHto CC −Λ−Λ=− −−κκ

(19)

Similarly, subtracting Equation 18 from Equation 16 yields,

BoCOOCHOHfo C)(3

−− Λ−Λ=−κκ

(20)

Combining Equation 19 and 20 leads to,

fo

toBoAoA CCC

κκκκ

−−

−= (21)

Equation (21) indicates that the concentration of NaOH at any reaction time can be determined from the measured solution conductivity. Knowing the concentration, the linear relations of Equation 8 and Equation 10 in the form bmxy += can be calculated and used to determine the rate constant, k .

39

Experimental Procedure

1. Click the data acquisition software “Loggerpro” on computer screen. 2. Click the clock icon. Set the mode to time based, length to 70 min, and the sampling rate

to 4 sample/min, then click “Done”. 3. Pour 1000 mL of 0.05 M NaOH solution into the reaction vessel. 4. Insert the conductivity probe into the reaction vessel. Start stirring the solution and make

sure there is sufficient agitation. 5. Prepare 2.0 mL pure ethyl acetate using an auto pipette. 6. Click “collect” button to record the conductivity of the solution, and quickly inject the

pure ethyl acetate into the reactor vessel at 45 seconds, and continue to record solution conductivity.

7. When the conductivity reading is leveled off at about 30 min., inject another 2.0 mL pure ethyl acetate using the auto pipette.

8. Wait until the conductivity reading is leveled off, then stop data recording. Copy and paste all the data points into an Excel spreadsheet, and then save your data.

9. Dispose of the reaction mixture into the correct waste container, and use deionized water to rinse the reaction vessel two at least three times.

Data Analysis and Result Discussion

1. For all the data points, calculate the concentration of NaOH , EtAc , and NaAc , and plot all the concentrations as a function of reaction time on the same graph. Describe what you observe from the concentration profiles.

2. Use Equation 8 and Excel linear regression to determine the reaction rate constant k value (note that if certain data points are off the straight line, you may ignore those data points in your regression analysis, but justify this in your report) for each injection case. Also predict BoC from the intercept of the linear regression and compare to the BoC calculated from the injection volume for each case. Discuss about the effect of initial concentration of NaOH on the measurement of the reaction rate constant k .

3. Calculate the measured conversion of NaOH using Equation 9 and the predicted conversion of NaOH using your k value and Equation 10 for all the reaction times. Plot the measured conversion with a symbol and the predicted conversion with line on the same graph. Compare with and discuss about the conversion curves in relation to the reaction kinetics and the measured k value.

4. Discuss some possible sources of errors in this experiment based on your data analysis above.

Questions

1. Derive the integrated form of the rate equation, similar to Equation 8, when the initial reactant concentrations of NaOH and EtAc are the same.

2. Using the Equations 8 and 9 in the manual, derive Equation 10. 3. Derive an expression for the half-life for this experiment and calculate the half life.

40

4. What is the NaOH concentration in the reaction vessel, 3 half lives after the start of the experiment?

5. A first order reaction reaches a conversion of 55% at the end of 48 minutes. What is the value of the rate constant in sec-1? In how many minutes will the reaction be 80 % complete?

6. Is conductivity an intensive or extensive property? Why can solution conductivities be used to determine the reaction rate constant in Equation 21?

7. How does the equivalent ionic conductance of OH- and OAc- compare qualitatively. 8. When a certain conductance cell filled with 0.02M KCl, it had a resistance of 82.4 ohms

at 25°C and, when filled with 0.01M K2SO4, it had a resistance of 326 ohms. Calculate (a) The cell constant (b) The specific conductance (conductivity) of the potassium sulphate solution. (c) The molar conductance of potassium sulphate solution.

(Specific conductance (conductivity) of 0.02M KCl solution at 25°C is 0.002768[S cm-1])

References

1. Castellan, G. W., Physical Chemistry, Addison-Wesley, Reading, Mass.,1983. 2. Farrington, D., Williams, J. W., Bender, P., and Cornwall, C. D., Experimental Physical

Chemistry, 6th Ed., McGraw Hill, NY, 1962. 3. Kendall, H. B., Chem. Eng. Prog. Symp. Series, 63, 3, 1962.

41

Appendix A: Sample Title Page

ChE101 Memo Report on Experiment 4

Reaction Kinetics for the Hydrolysis of Ethyl Acetate with Sodium Hydroxide

Submitted by

Michael E. Smith for individual report

Michael E. Smith, John Hill, and Bing W. Chan for group report

Department of Chemical Engineering, University of Waterloo, Waterloo, ON, N2L 3G1

Date of submission: March 20, 2013

Lab Group #: 16

Date of Experiment: 6 March, 2013

Lab TA:

Lab Instructor:

Honor Pledge

By electronically submitting this report I/we pledge that I/we am/are the sole contributor to the written lab report and the work contained in the report is our own group’s work. I (we) fully understood the consequences of academic misconducts including plagiarism, cheating, copying, sharing etc. as stipulated in UW Policy #71.

Name ID Date

42

Appendix B: Correlation of Experimental Data – Least Squares Principle

A mathematical equation is the most compact and convenient form of quantitatively expressing the relationship between two (or more) variables. In general, any equation consists of the variables (represented by appropriate symbols) in combination with various coefficients. The values of these coefficients remain constant over a defined range of values of the variables (dependent and independent variables).

The simplest form of representation is, of course, the two-variable linear equation

bmxy += (1)

In an experiment in which there are one dependent variable y (the value of which is imprecisely known) and one independent variable x (the value of which is precisely known, e.g., by "exact" measurement) data are generally collected in the form of specific paired values of the variables; that is, we have a set of N data points ),( 11 yx , ),( 22 yx …… ),( ii yx …… ),( NN yx .

Note that many correlations that are not exactly linear can be converted into a linear equation as Equation 1 by proper mathematical manipulation. As an example, in Experiment 3 you are asked to develop such an equation relating fluid velocity ( 2v ) through an orifice, to the head ( h ) of water in the tank. The governing equation is,

nkhv =2 (2)

Where k and n are two coefficients.

Equation 2 is obviously not a linear equation. But if we take logarithm of both sides,

hnkv logloglog 2 += (3)

Now if we set 2logvy = and hx log= , we can have,

knxy log+= (4)

Which is in the form of linear equation as Equation 1 with the slope of n and intercept of klog .

Equation 1 contains two coefficients, m and b . Of course, the use of any two pairs of data ),( ii yxand ),( jj yx would determine specific values of m and b . However, to obtain the values of m and b which "best" represent the linear relationship over the entire range of values of x , we need to use the Principle of Least Squares.

The Principle of Least Squares leads to a method (set of rules) for the actual computation of the "best" values of m and b. Before we examine the method in detail, there is one point -- which is frequently overlooked -- which must be kept in mind: the method can be applied whether or not the data are worth correlating, but the results are only useful when the data are good in the first

43

place. No purely mathematical procedure can make a meaningful number out of any number of poor ones; the use of statistics does not lessen the necessity of using common sense in the interpretation of results. The principle of least square estimates the "best" values of m and b by minimizing the deviations (errors) of observed y from the predicted y value ( *y ) by the linear equation, bmxy ii +=* (5) Where *

iy is the value of y predicted by the linear equation for ix as shown in the following.

The sum of the squares of the errors ( SSE ) of all N observations from the predicted values is defined as,

∑∑==

−−=−=N

iii

N

iii bmxyyySSE

1

2

1

2* )()( (6)

Minimization of SSE in Equation 6 requires 0)(=

∂∂

mSSE

and 0)(

=∂

∂b

SSE . These two conditions

lead to the equations for evaluating the best values of m and b ,

44

2

11

2

1 11)(

−

−

=

∑∑

∑ ∑∑

==

= ==

N

ii

N

ii

N

i

N

ii

N

iiii

xxN

yxyxNm (7)

xmyxNmy

Nb

N

ii

N

ii −=

−

= ∑∑

== 11

1 (8)

Where

= ∑

=

N

iix

Nx

1

1 and

= ∑

=

N

iiy

Ny

1

1

The goodness of the observed y values fitting the linear equation is often characterized by,

SSTSSER −=12

(9)

Where SST is the total sum of the squares of the deviations of the observations from the average y value,

2

11

2

1

2 1)(

−=−= ∑∑∑

===

N

ii

N

ii

N

ii y

NyyySST (10)

The closer the 2R to unity, the greater the degree of association between y and x described by the linear relationship.

Example 1: The ),( ii yx data in the following table follow a linear relationship (Equation 1), find the coefficient values using the least square method.

ix iy ii yx 2ix

1 5 5 1 2 6 12 4 4 13 52 16

7=∑ ix 24=∑ iy 69=∑ iyxi

212 =∑ ix

78.27213

2476932 =

−⋅⋅−⋅

=m

50.17378.224

31

=⋅−⋅=b

45

Thus the equation of the least squares line for these observations is .5.178.2 += xy Example 2: Supposing the following ),( 2vh data were obtained in Experiment 3, find the values of k and n using the least square method and determine whether Equation 2 adequately represents the data.

)(mh

)/(2 smv

hx log=

2logvy =

2x

xy

0.0750 0.03582 -1.1249 -1.4459 1.2654 1.6264

0.1550 0.05064 -0.8097 -1.2955 0.6556 1.0490

0.2600 0.06249 -0.5850 -1.2042 0.3422 0.7045

0.5085 0.09620 -0.2937 -1.0168 0.0863 0.2986

0.7520 0.11946 -0.1238 -0.9228 0.0153 0.1142

9371.2−=∑ ix 8852.5−=∑ iy

3649.22 =∑ ix

9728.3=∑ iyxi

525.0)9371.2(3649.25

)8852.5()9371.2(9728.352 =

−−⋅−⋅−−⋅

=m

869.0)9371.2(5525.0)8852.5(

51

−=−⋅−−⋅=b

Thus, the straight line for this case is, 869.0525.0 −= xy or 869.0log525.0log 2 −= hv

525.0525.0869.0

2 135.010 hhv == −

That is, the coefficients are 135.0=k and 525.0=n .

1044.7

1

2 =∑=

N

iiy

1774.0)8852.5(511044.71 2

2

11

2 =−−=

−= ∑∑

==

N

ii

N

ii y

NySST

46

0011.0)8852.5)(9371.2(519728.3(525.01774.01

111=−−−−=

−−= ∑∑∑

===

N

ii

N

ii

N

ii yx

NyxmSSTSSE

i

9936.01774.0001.0112 =−=−=

SSTSSER

The value of 2R is very close to unity, meaning that Equation 2 adequately represents the data. Note that modern electronic hand calculators and spreadsheet software have a least square function for linear regression analysis. Microsoft Excel can, for example, be conveniently used to calculate not only the values of the coefficients and 2R but also the errors or confidence intervals of the coefficients.

References

1. Baird, D. C., Experimentation, p. 133 (Prentice-Hall, Englewood Cliffs, NJ, 1962).

2. Deming, W. E., Statistical Adjustment of Data (Dover Publ. Inc., New York, 1964).

47

Appendix C: Experimental Errors and Error Propagation and Analysis 1. Introduction to Experimental Error and Error Analysis Any measurement in physical experiments or chemical engineering labs, either by direct observation or by means of a measurement device, carries a certain degree of deviation from its unknown “true” value. We often use the synonymous terms uncertainty or error to represent the deviation of the measured quantity. In presenting experimental result, it is essential not only to establish the best possible value (accuracy) for the measured quantity, but also to give a range of possible true values (precision) based on the limited number of measurements. Furthermore, in most physical measurements or in engineering labs, the desired quantity is often calculated based on certain fundamental formula from multiple measurable quantities, so the uncertainty of the calculated quantity has to be determined through the principles of error analysis and propagation. 2. Experimental Error: Absolute and Relative Error Experimental errors are often expressed in the form of absolute error and relative error. The absolute error is the difference between the measured value and the “true” value; and the relative or percent error is percent difference. For example, when we use a balance to measure the mass of water in a beaker, if the “true” mass is 145.2 g, and the balance reads 147.5 g for a measurement, then the absolute error for the measurement is,

gmmm 3.22.1455.147ˆ +=−=−=∆ And the relative error for the measurement is,

%6.1%1002.145/3.2ˆ/ +=⋅+=∆ mm 3. Types of Experiment Errors: Systematic and Random Error Based on their sources, experimental errors are often classified into two major types: systematic error and random error. Systematic error is the result of a miscalibrated instrument, or an improperly zeroed device, or consistent operation error, or theoretical error due to simplified mathematical model. Systematic errors normally have the same “sign” and magnitude for identical conditions, which always makes the measured value larger or smaller than the "true" value. Random errors result from random and unpredictable variations in experimental measurements such as observational errors when reading the scale of a measuring device to the smallest division, unpredictable fluctuations in readings beyond the experimenter control, and instrument sensitivity and fluctuations. If an experiment has low systematic error, it is said to be accurate; if an experiment has low random error, it is said to be precise. Obviously an experiment can be precise but inaccurate or accurate but imprecise. The accuracy and precision are always limited by the degree of refinement of the instruments, by the skills of the observer, and by the theory underlying the experiment. Systematic errors can only be corrected or eliminated by carefully design or calibration of the measurement techniques. Random errors can, however, be estimated or

48

determined through the following methods. It is important to note that (1) blunders are not a source of uncertainty and should always be eliminated completely by careful work, and (2) exclusion of any “outliner” from the collected data for data analysis should always be based on experimental observation and/or a thorough and statistical analysis of the data and lab theories. Determining Random Error through Statistical Analysis of Replicated Measurements Suppose a series of N values are measured for a quantity x as Ni xxxx ⋅⋅⋅⋅⋅⋅⋅⋅ ,,, 21 ,

The mean of x : ∑=

=N

iix

Nx

1

1 (1)

The standard deviation of x : ∑=

−−

=N

ii xx

Ns

1

2)(1

1 (2)

The standard error of the mean x : Ns

=s (3)

So the standard deviation is a measure of the variability of individual data points, and the standard error reflects the variability of the mean value. Statistically, the true value of x lies somewhere between )(λs−x and )(λs+x with certain confidence limit of λ . Therefore, when we report a measured value, we often need to express it in terms of the mean value and its confidence limit,

)(λs±= xx (4) With the normal error distribution and 95% confidence interval, Equation 4 becomes,

)96.1( s±= xx (5) Determining Random Error Using Instrument Limit or Tolerance If the repeated measurements are not feasible, the random error of a measurement can also be determined by instrument limit or least count, which is smallest division marked on the instrument. For Example, a meter stick will have a least count of 1.0 mm, a balance scale might have a least count of 1 g. In general, the error is taken to be the least count or some fraction (1/2, 1/4) of the least count). For some electronic devices, the percent tolerance or range specified by manufacturers can also be used as random errors, as an example, a differential pressure transducer may be specified as having a tolerance of 5%, meaning that the random error is 5% of the measured value. 4. Error Propagation and Error Analysis Once we have some experimental measurements, we usually combine them according to certain fundamental formula to determine a desired quantity. Accordingly, to find the estimated error (uncertainty) for the calculated result, we must know how to combine the errors in the measured quantities. The simplest procedure would be to add the errors, but this would be a conservative assumption, and it generally overestimates the uncertainty in the result. A more accurate and

49

systematic estimation of errors is based on mathematical differentiation and statistical variance analysis. Consider a simple case of a single variable function )(xfw = where x is the independent (measured) variable and w is the dependent (calculated) variable. The differential of w with respect of x is, dxxfdw )('= (6)

Where dxdwxf =)(' is the derivative of w . If we take dx as a small and random change in x (not

necessarily infinitesimal), then dw is an estimate of the change in w due to the change in x ( dx ). In other words, Equation 6 offers the mathematical foundation to estimate the uncertainty in w

due

to the uncertainty in x . Example 1: Consider the equation for determining the vapour pressure of acetone from the table of Antoine Equation constants in Felder and Rousseau, Chapter 6 and Table B.4 in Appendix B,

CTBAP+

−=*10log

Where *P is vapor pressure in mm Hg, T is temperature in °C, and the constants assume values of 11714.7=A , 595.1210=B , and 664.229=C .

To determine the error in the vapor pressure due to the error in measuring temperature, we first need to differentiate the vapor pressure equation,

dTdP

PP

dTd *

**

10 10ln1)(log⋅

=

22 )()1(

)( TCB

TCB

TCBA

dTd

+=−

+−=

+−

Therefore,

dT

TCPBdP ) + ()10(ln = 2

**

If we intended to determine *P for temperature at 90 °C, and the measured T had an error of 3.0ºC

(i.e., CdT o3= ), the error in calculated *P would be,

Hg mm 18 = 03.

)90 + 664.229(2138)10ln(595.1210 = 2

* ⋅⋅⋅dP

That means that the actual *P

at 90 °C can be between 182138 ±

. Therefore, the differential

method provides an easy and reliable way to estimate possible error propagated through the calculation if the measurement was in error.

50

The differential method can be extended to functions of more than one variable. Consider a general function, ),,( zyxfw = (7) If the variables x , y and z are measured quantities and each has an error associated with it ( xδ ,

yδ , and zδ ) we may write three differentials that represent the individual contributions of the errors in x , y and z to the error in w as

xxfwx δδ∂∂

= , yxfwy δδ∂∂

= , and zxfwz δδ∂∂

= (8)

Where xf∂∂ ,

yf∂∂ , and

zf∂∂ are partial derivative, and xwδ , ywδ , and zwδ are the error in x , y and

z , respectively. As a first approximation, the total error in w can be estimated by simply adding all individual contributions as,

+ max zzfy

yfx

xfw δδδδ

∂∂

+∂∂

∂∂

= (9)

The estimated error is often referred to as the "maximum possible error". This is because the quantity calculated from Equation 9 tends to overestimate the error in w due to the fact that it does not account for the situation in which errors can compensate one another as far as sign is concerned. A more realistic estimation of uncertainty in w is to combine the individual error terms in Equation 8 using the concept of the additivity of variances. Let )(xVar , )(yVar , and )(zVar be the variance of x , y and z , which can be normally evaluated from repeated measurements. Alternatively, these variances can be approximated by, ( )2)( xxVar δ= , ( )2)( yyVar δ= , and ( )2)( zzVar δ= (10) From statistics, the overall variance of w is,

)( )( )( )(222

zVarzfyVar

yfxVar

xfwVar

∂∂

+

∂∂

+

∂∂

= (11)

Combining Equations 10 and 11 yields,

22

22

22

2 zzfy

yfx

xfw δδδδ

∂∂

+

∂∂

+

∂∂

=

(12)

or the estimated standard error for w is,

51

22

22

22

zzfy

yfx

xfw δδδδ

∂∂

+

∂∂

+

∂∂

±=

(13)

Note that both the partial derivative terms and individual error terms are squared, hence, the standard error of w can then be realistically estimated. Equations 12 and 13 are the general equations for error propagation analysis.

Example 2: In Experiment 3, we intend to determine the water velocity out of an orifice, 2v , by measuring M , t at a steady-state water level through the following relationships,

tA

Mvρ

=2 ; 2

4dA π

=

Combining the two equations yields,

tdMv 22

4πρ

=

What is the error for each calculated 2v value due to the errors in measured M , t ?

Let’s set 2vw = , Mx = , ty = and use 24d

aπρ

= to represent all the constant values, then we can

reduce the equation for friction factor into a more general formula,

yxaw =

According to Equation 12,

22

22

22

22

22 y

yaxx

yay

ywx

xww δδδδδ

−+

=

∂∂

+

∂∂

=

Dividing both sides by 2

2

=

yaxw yields,

2

2

2

2

2

2

yy

xx

ww δδδ

+=

or 2

2

2

2

yy

xx

ww δδδ

+±=

This equation is a more generalized formula for calculating the relative error of w due to the errors in measured x and y in this common function format. The same derivation should be followed for any other error formula, and Table 1 lists the error propagation formulas for some common functions.

52

Table 1. Error propagation equation for some common formula

Function

Error propagation formula

byaxw ±= 22222 ybxaw δδδ +=

axyw = or yxaw = 2

2

2

2

2

2

yy

xx

ww δδδ

+=

baxw = 2

22

2

2

xxb

ww δδ

= or xxb

ww δδ ±=

bxaew ±= xbww δδ ±=

)ln( bxaw ±= xx

baw δδ ±=

For a set of typical measured values and errors such as,

gM 1600= ; gM 50.0±=δ st 60= , st 0.1±=δ cmind 635.0250.0 ==

3/0.1 cmg=ρ

)/(2.84)(60)(635.0)/(0.1

)(16004422322 scm

scmcmgg

tdMv =

⋅⋅⋅⋅

==ππρ

From the formula above,

2

2

2

2

2

2

tt

MM

vv δδδ

+±=

Hence, the standard error for 2v is:

)/(40.1108.2108.92.8460

0.11600

5.02.84 482

2

2

2

2

2

2

2

22 scmtt

MMvv ±=×+×±=+±=+±= −−δδδ

Thus, the measured 2v with error should be )/(4.12.84 2 scmv ±=

At this point of the analysis, we are also able to comment on which error source contributes most to the uncertainty to the error of 2v . As indicated above, the error in the time seems to have the largest

53

effect on the velocity, while error in mass contributes to a much smaller extent. This kind of information is useful for suggesting ways in which the experiment could be improved. Similarly, we can estimated the maximum error for 2v by using Equation 9,

+±=

tt

MM

vv δδδ

2

max,2

)/(43.160

0.11600

5.02max,2 scm

tt

MMvv ±=

+=

+±=δδδ

This example demonstrates that the quantity referred to as the maximum possible error overestimates what is likely to occur. Therefore, the uncertainty in a quantity which is dependent upon several measured quantities should be evaluated using the concept of the additivity of variances. 5. Significant Figures in Measured Value and Uncertainty It is important that all measured results are reported with units, uncertainties, and appropriate number of significant figures. The general rule for significant digits is that the estimated error of a value determines the number of significant figures to be reported in the value. A large error may require the value be rounded to fewer figures, and a very small error may imply that the value should be known to a greater number of significant figures. Besides, a number that is used as an uncertainty itself is fundamentally an estimate, so it should not have more than 2 significant figures. As a general rule, (1) uncertainties are given with one significant figure if the most significant figure in the uncertainty is greater than 2, and two significant figures if the most significant figure in the uncertainty is less than or equal to 2. (2) A value and its uncertainty should be stated with the same precision, i.e., they should have the same number of digits past the decimal point. As well one should also specify the type of uncertainty, such as an estimated first standard deviation or confidence limits. Table 2. Examples of common significant figures Example Comment

scm /6.12.84 ± Ok sm /02.084.0 ± Ok

sm /016.0842.0 ± Ok, two digits in error are a maximum sm /016.084.0 ± Not ok, too many digits in error sm /02.0842.0 ± Not ok, too many digits in value or two few digits in error.

sm /016.0102.84 2 ±× − Ok, both numbers stated to the same precision

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6. References

1. Perry, R. H. and Green, D. W., “Perry’s Chemical Engineers’ Handbook”, 8th ed., McGraw-Hill, 2008. Electronic version of the book is available at:

http://www.accessengineeringlibrary.com/subject/chemical_engineering 2. Coleman, H. W. and Steele, W. G., “Engineering Application of Experimental

Uncertainty Analysis”, AIAA Journal, 33(10), 1888, 1995. 3. Moffat, R. J., “Using Uncertainty Analysis in the Planning of an Experiment”, J. Fluids

Eng., 107, 173, 1985.