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Material Balances 1

Outline !

The Balance Equation

!

Mass Balance Analysis

!

Tie Component

!

Degree of Freedom Analysis

!

Mass Balance Calculations – Single Unit

!

Multiple Unit Balances

The Balance Equation The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

The Balance Equation The Balance Equation

!

Conservation Laws !

Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

!

!

Mass (except in nuclear reactions) !

Moles are NOT conserved when there are reactions!

!

Volume is generally NOT conserved

Energy (1st Law of Thermodynamics)

For any conserved property, In – Out + Generation = Accumulation !

IN: sum of all ﬂow rates into the system

!

OUT: OUT: sum of all ﬂow rates going out of the system

!

GEN: formation or consumption within the system

! ACC:

net rate of change of the property property in the system (zero at steady-state)

The Balance Equation The Balance Equation

!

!

Mass Balance Analysis Tie Component

!

!

Multiple-Unit Balances

! !

!

Terms are in rate form Usually for continuous processes

Integral Balance !

Degree of Freedom Analysis Mass Balance Calculations – Single Unit

Di"erential Balance

Describes two points in time Terms are in amounts Usually for batch processes

Basic Rules ! ! !

For total mass, GEN = 0 For non-reacting species, GEN = 0 For steady-state, ACC = 0

The Balance Equation The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

!

If there is no chemical reaction and the system is in steady-state:

Mass Balance Analysis The Balance Equation

1. Draw the ﬂowchart of the problem.

Mass Balance Analysis

2. Write the values and units of all known variables, preferably at the location of the stream.

Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

3. Assign algebraic symbols to unknown stream variables. 4. Select a convenient basis if needed. 5. Determine the appropriate set of equations. 6. Solve algebraically then numerically.

Mass Balance Analysis The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

General Material Balance Equation The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

!

Ex. Three students rent a room the night before an exam and pay the desk clerk P6000. A new clerk comes on duty and ﬁnds that the discount rate for students should have been P5500. The new clerk gives the bellhop P500 to return to the students, but the bellhop, not having change and being slightly dishonest, returns only P100 to each student and keeps the remaining P200. Now each student paid P2000 – P100 = P1900, and 3 # P1900 = P5700 paid in total. The bellhop kept P200 for a total of P5900. What happened to the other P100?

Material Balance Calculations - Mixing The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

!

Recall: Algebra Problems ! A

manufacturer of soft drinks advertises its orange soda as “naturally ﬂavored,” although it contains only 5% orange juice. A new federal regulation stipulates that to be called “natural,” a drink must contain at least 10% fruit juice. How much pure orange juice must this manufacturer add to 900 gal of orange soda to conform to the new regulation?

Material Balance Calculations - Mixing The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

!

Recall: Algebra Problems ! A

pharmacist, needs 100 L of a 50% alcohol solution. He has on hand a 30% alcohol solution and an 80% alcohol solution, which he can mix. How many liters of each will be required to make the 100 L of a 50% alcohol solution?

Material Balance Calculations - Mixing The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

Tie Component The Balance Equation

!

Enters in one input stream only

Mass Balance Analysis

!

Exits in one output stream only

!

Is easily traced, and can be used in relating other components

!

Ex. Find the mass M4

Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

Tie Component The Balance Equation

!

Ex. You buy 100 kg of cucumbers that contain 99% water. A few days later they are found to be 98% water. What is the remaining weight of the cucumbers?

!

Ex. Drying: A batch dryer can handle 5000 kg feed per hour. 5000 kg of a material containing 30% moisture is to be dried. How much moisture evaporates per hour if the dried material contains 5% moisture?

!

Ex. Evaporation: An evaporator has a capacity to evaporate 30,000 kg of water per hour. We want to concentrate a NaCl-water solution from 10% to 40%. If the evaporator is to operate at its rated capacity, what must the feed rate be? How much product is obtained per hour?

Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

Degree of Freedom Analysis The Balance Equation

!

Concept: DF = #Unknowns - #Indep.Eqn !

Mass Balance Analysis

!

Tie Component

!

Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

DF = 0 ! solvable DF > 0 ! underspeciﬁed, need more equations DF < 0 ! overspeciﬁed, possible redundant or inconsistent equations

!

Independent equation: Cannot be derived algebraically from other equations

!

Use to check if enough given information is available to solve the problem

!

Requires a complete ﬂowchart

Degree of Freedom Analysis The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

!

Example

Degree of Freedom Analysis The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

!

Sources of Equations !

Mass balance: total and component

!

Energy balance

!

Process speciﬁcations

!

Physical properties

!

Physical constraints

!

Stoichiometry

General Steps The Balance Equation Mass Balance Analysis

Read

Tie Component

Draw

Degree of Freedom Analysis Mass Balance Calculations – Single Unit

DFA

Basis

MB

Multiple-Unit Balances Check

• Identify the streams and equipment involved • Define the system. Sketch a BFD or PFD with all streams • Identify known and unknown parameters and relations • Choose and clearly state a convenient basis • Write overall and component balances • Consider the given conditions and common sense

Mass Balance Calculations The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

Mass Balance Calculations The Balance Equation

!

Mass Balance Analysis

Since the process is at steady-state condition and no chemical reaction is involved, the material balance equation becomes Input = Output

Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

!

This balance equation can be applied to: !

The total mass entering and leaving the process

!

Mass of individual component entering and leaving the process.

Mass Balance Calculations The Balance Equation

!

m1 = m2 + m3

Mass Balance Analysis Tie Component

!

Multiple-Unit Balances

Component A Balance m A1 = m A2 + m A3

Degree of Freedom Analysis Mass Balance Calculations – Single Unit

Total Mass (Overall Material Balance)

m1x A1 = m2x A2 + m3x A3 !

Component B Balance mB1 = mB2 + mB3 m1xB1 = m2xB2 + m3xB3

Mass Balance Calculations The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis

!

For the given process, 3 material balance equations can be written:

Total Balance:

m1 = m2 + m3

(E1)

A-Balance:

m A1 = m A2 + m A3

(E2)

B-Balance:

mB1 = mB2 + mB3

(E3)

Mass Balance Calculations – Single Unit Multiple-Unit Balances

Are these material balances independent equations?

Mass Balance Calculations The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis

!

For the given process, 3 material balance equations can be written:

Total Balance:

m1 = m2 + m3

(E1)

A-Balance:

m A1 = m A2 + m A3

(E2)

B-Balance:

mB1 = mB2 + mB3

(E3)

Mass Balance Calculations – Single Unit Multiple-Unit Balances

Are these material balances independent equations?

Mass Balance Calculations The Balance Equation

!

Mass Balance Analysis

Two methanol-water mixtures are contained in separate tanks.

Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

Example. Mixing of Methanol-Water Mixtures

The ﬁrst mixture contains 40.0 wt% methanol and the second contains 70.0 wt% methanol. If 200 kg of the ﬁrst mixture is combined with 150 kg of the second, what are the mass and composition of the product?

Mass Balance Calculations The Balance Equation

Step 1. Draw a ﬂowchart to visually organize the data.

Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

0.40 0.60

Mass Balance Calculations The Balance Equation

Step 2. Determine the degrees of freedom (DF)

Mass Balance Analysis

!

Number of unknowns (U): 3 unknowns !

m3, xM3, xW3

Tie Component Degree of Freedom Analysis

!

Number of independent equations (V):

!

DF = U – V = 3 – 3 = 0 , the problem is solvable

Mass Balance Calculations – Single Unit Multiple-Unit Balances

Mass Balance Calculations The Balance Equation

Step 3. Write down the equations

Mass Balance Analysis

!

Material Balances (Steady-State, Non-Reactive Process):

Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit

Total Balance:

m1 + m2 = m3

Methanol-Balance: m1xM1 + m2xM2 = m3xM3 Water-Balance:

m1xW1 + m2xW2 = m3xW3

!

choose only 2 equations since one of them is no longer independent

!

Physical Constraint (applied to mixture 3):

Multiple-Unit Balances

xM3 + xW3 = 1.00

Mass Balance Calculations The Balance Equation

Step 4. Solve the unknowns ( m 3 , x M3 , x W3 )

Mass Balance Analysis

! Always

Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

start with the equation with the least number of unknowns if possible and minimize solving equations simultaneously.

Mass Balance Calculations The Balance Equation Mass Balance Analysis

Step 4. Solve the unknowns ( m 3 , x M3 , x W3 ) !

m3 = (200 kg) + (150 kg) = 350 kg

Tie Component Degree of Freedom Analysis

!

CH3OH balance: (200 kg)(0.40) + (150 kg)(0.70) = (350 kg)x M3

Mass Balance Calculations – Single Unit Multiple-Unit Balances

Total balance:

xM3 = 0.529 !

Physical constraint: xW3 = 1.00 – xM3 = 1 – 0.529 xW3 = 0.471

Homework The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

Three hundred gallons of a mixture containing 75.0 wt% ethanol and 25% water (mixture speciﬁc gravity = 0.877) and a quantity of a 40.0 wt% ethanol-60% water mixture (SG=0.952) are blended to produce a mixture containing 60.0 wt % ethanol. Determine the required volume of the 40% mixture. Ans.: 207 gal

Questions?

Steady-State Material Balance The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

Input = Output !

In a mass balance set-up involving N components: ! ! ! !

N+1 relationships are available (one overall mass balance and N component balances) Maximum number of independent equations is equal to N Physical constraints (mass fractions) Process speciﬁcations

Steady-State Material Balance The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

! ! ! ! !

Total Mass (Overall Material Balance) M1 + M2 + M3 = M4 + M5 Component A Balance M1x A1 + M2x A2 = M4x A4 + M5x A5 Component B Balance M1xB1 + M2xB2 + M3xB3 = M4xB4 + M5xB5 Component C Balance M2xC2 + M3xC3 = M4xC4 + M5xC5 Component D Balance M = M

Steady-State Material Balance The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

!

Sum of the mass fractions of the components of each stream is one. x A1 + xB1 = 1 x A2 + xB2 + xC2 = 1 xB3 + xC3 + xD3 = 1 x A4 + xB4 + xC4 + xD4 = 1 x + x + x = 1

Steady-State Material Balance The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

!

Methods of Solution: ! Arithmetic ! !

Stream-to-stream analysis Tie component

! Algebraic ! Assign

! ! !

method

solution symbols to unknowns

Combination of arithmetic and algebraic methods Graphical technique Computer solutions using programs or packaged softwares

Single-Unit Balance: Drying The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

" Removal

of water or another liquid from a solid by vaporization

!

Ex. Drying:

A batch dryer can handle 5000 kg feed per hour. 5000 kg of a material containing 30% moisture is to be dried. How much moisture evaporates per hour if the dried material contains 5% moisture?

Single-Unit Balance: Evaporation The Balance Equation

!

Unit operation where part of the solvent in a solution is vaporized to concentrate the solution

!

Ex. Evaporation:

Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

An evaporator has a capacity to evaporate 30,000 kg of water per hour. We want to concentrate a NaCl-water solution from 10% to 40%. If the evaporator is to operate at its rated capacity, what must the feed rate be? How much product is obtained per hour?

Single-Unit Balance: Distillation The Balance Equation

!

Mass transfer due to di "erence in volatility !

Mass Balance Analysis

!

Tie Component

!

Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

!

!

Vapor is richer in the more volatile component as it ascends from plate to plate Product withdrawn from the top = distillate Bottom plate = bottoms Side products can be withdrawn from any other plate

Ex. Distillation:

We desire to produce two ethyl alcohol-water mixtures each containing 90% and 50% alcohol by mole from a dilute mixture containing 20% mole alcohol by distillation. 98% of the ethyl alcohol in the feed is to be recovered in these two products, that is, only 2% of the ethyl alcohol in the feed may go into the bottoms. If the bottoms contains 0.6% alcohol, calculate the amount of the di "erent streams per 1000 kmol of feed stream.

Single-Unit Balance: Crystallization The Balance Equation

!

Mass Balance Analysis

! Attained !

Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

Unit operation in which a solution is brought to a concentration where it cannot hold all the soluble component (solute) in solution

!

by cooling a hot solution and/or by evaporating some of the solvent Solute precipitates out as crystals

Ex. Crystallization:

One thousand kilograms of a 64% NaNO3 solution in water is at a temperature of 100°C. This is sent to a crystallizer where it is cooled to 30°C. At this temperature, the solubility of NaNO3 is 96 parts per 100 parts of water (the solubility of NaNO 3 is maximum amount of NaNO3 that can stay in solution; the rests crystallizes out). Calculate the amount of crystals that precipitates out if a. b.

No water evaporates during cooling 5% of the original water evaporates during cooling

Single-Unit Balance: Solid-Liquid Extraction The Balance Equation

!

Valuable component (solute) in a solid can be separated by soaking the solid in a solvent where the solute is soluble and allowing the solute to di"use out

!

Ex. Solid-Liquid Extraction:

Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

A copra batch contains 44% oil and is to be extracted with n-hexane. The extracted meal contains 0.07 kg oil/kg oil-free matter, and the mass fraction of n-hexane is 0.05. The extract contains 49% n-hexane. If 10,000 kg of copra are to be processed, how many kilograms of nhexane is required?

Single-Unit Balance: Practice The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

!

Ex. An aqueous solution of NaOH contains 20% NaOH by mass. It is desired to produce and 8% NaOH solution by diluting a stream of the 20% solution with a stream of pure water. Calculate the ratios [L H2O/kg feed] and [kg product/kg feed].

Single-Unit Balance: Practice The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

!

Ex. A liquid mixture containing 45% benzene (B) and 55% toluene (T) is fed to a distillation column. The overhead product contains 95% B, and the bottom product stream contains 8% of the benzene fed to the column. The volumetric ﬂow rate of the feed stream is 2000L/h and the speciﬁc gravity of the mixture is 0.872. Determine the mass ﬂow rate of overhead product and the mass ﬂow rate and mass fractions of the bottom product.

Multiple-Unit Balances The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

Overall balance Mixing point

Unit balance Splitting point

Multiple-Unit Balances The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

!

Ex. How will we solve for the ﬂow rates and compositions of (1), (2) and (3)?

Multiple-Unit Balances: Multiple-effect Evaporation The Balance Equation Mass Balance Analysis

!

Evaporator unit – “e"ect” ! !

electricity or steam Steam side: !

Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

Heat exchangers: steam enters tubes/coils, transfers heat to the solution, condenses, and exits as liquid water

Mass of steam in = mass of condensed water out !

Solution side: ! !

Liquid solution boils Water converted to vapor – can be used as source of heat for another evaporator (multiple-e "ect)

Mass feed = mass product + mass water evaporated

Multiple-Unit Balances: Multiple-effect Evaporation The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

!

Ex. A triple-e"ect evaporator concentrates a 10% caustic soda to 50% NaOH. Assuming equal amounts of evaporation in each e "ect, calculate the concentration of the solution leaving each e"ect.

Multiple-Unit Balances The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

!

Ex. NaCl crystals are to be produced from 1,000 kg of a solution containing 10% NaCl, 1% KOH, and 89% H 2O. The solution is ﬁrst sent to an evaporator where 790 kg of the water from the solution is removed. The hot concentrated solution is then sent to a crystallizer where the solution is cooled to 20°C and NaCl crystals precipitate. Each kg of crystals carries with it 0.1 kg of adhering mother liquor. The wet crystals are then treated in a drier where 95% of the water is removed. Calculate: a. b. c.

the composition of the concentrated solution from the evaporator. the composition of the ﬁnal crystal product. the percentage recovery of NaCl from the original solution.

Data: At 20°C, the solubility of NaCl in H 2O is 36 g/100g H 2O. KOH is non-volatile and very soluble in H2O. Assume that no water evaporates in cooling the solution. Assume that the presence of KOH does not a "ect the solubility of NaCl in H 2O.

Multiple-Unit Balances The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

!

Ex. Acetone is used in the manufacture of many chemicals and also as a solvent. In its latter role, many restrictions are placed on the release of acetone vapor to the environment. An acetone recovery system is illustrated below. All of the concentrations shown of both the gases and liquids are speciﬁed in weight percent in this special case to make the calculations simpler. Calculate A, F , W , B, and D in kilograms per hour. Assume that G = 1400 kg/hr.

! Answers:

A = 1336.7 F = 221.05 W = 157.7 D = 34.91 B = 186.1

Summary The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

!

By now you should have learned #

The Balance Concept

#

How to analyze and solve material balance equations

#

How to analyze solutions using Degree of Freedom Analysis

#

How to solve Single Unit Material Balances

#

How to solve Multiple-Unit Balances

Groups - WFW The Balance Equation Mass Balance Analysis Tie Component

Cosmetics

Paints and Dyes

Glass

Drugs and Pharma

Food and Beverage

Beer

Lumactod

Almeda

Banag

Betancor

Ng

Abis

Castro

Dizon

Gutierrez

Caramat

Yu

Cua

Reyes

Moya

Ureta

Hadi

Sy, JL

Dy

Rubber

Pulp and Paper

Soaps

Cement

Plastics

Baladiang

Mirasol

Pabua

Co

Alunan

Petallo

Sy, S

Roco

Dizon

Tan

Victor

Joves

Suarez

Lim

Garcia

Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

Groups - WFX The Balance Equation

Cosmetics

Mass Balance Analysis Tie Component

Paints and Dyes

Glass

Drugs and Pharma

Food and Beverage

Beer

Jeresano

Trinidad

Maranon

Santiago

Castillo

Crisostomo

Conol

Piedad

Canicosa

Carrido

Perez, G

Barcelona

Vergara

Paet

Catulong Sobremonte

Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

Rubber

Pulp and Paper

Soaps

Cement

Plastics

Go

Dignos

Perez, E

Cudiamat

Baliton

Reyes

Saban

Quilantang

Dallarte

Caballero

Ng

Tan

Solis

Li

Rivera

View more...
Outline !

The Balance Equation

!

Mass Balance Analysis

!

Tie Component

!

Degree of Freedom Analysis

!

Mass Balance Calculations – Single Unit

!

Multiple Unit Balances

The Balance Equation The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

The Balance Equation The Balance Equation

!

Conservation Laws !

Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

!

!

Mass (except in nuclear reactions) !

Moles are NOT conserved when there are reactions!

!

Volume is generally NOT conserved

Energy (1st Law of Thermodynamics)

For any conserved property, In – Out + Generation = Accumulation !

IN: sum of all ﬂow rates into the system

!

OUT: OUT: sum of all ﬂow rates going out of the system

!

GEN: formation or consumption within the system

! ACC:

net rate of change of the property property in the system (zero at steady-state)

The Balance Equation The Balance Equation

!

!

Mass Balance Analysis Tie Component

!

!

Multiple-Unit Balances

! !

!

Terms are in rate form Usually for continuous processes

Integral Balance !

Degree of Freedom Analysis Mass Balance Calculations – Single Unit

Di"erential Balance

Describes two points in time Terms are in amounts Usually for batch processes

Basic Rules ! ! !

For total mass, GEN = 0 For non-reacting species, GEN = 0 For steady-state, ACC = 0

The Balance Equation The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

!

If there is no chemical reaction and the system is in steady-state:

Mass Balance Analysis The Balance Equation

1. Draw the ﬂowchart of the problem.

Mass Balance Analysis

2. Write the values and units of all known variables, preferably at the location of the stream.

Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

3. Assign algebraic symbols to unknown stream variables. 4. Select a convenient basis if needed. 5. Determine the appropriate set of equations. 6. Solve algebraically then numerically.

Mass Balance Analysis The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

General Material Balance Equation The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

!

Ex. Three students rent a room the night before an exam and pay the desk clerk P6000. A new clerk comes on duty and ﬁnds that the discount rate for students should have been P5500. The new clerk gives the bellhop P500 to return to the students, but the bellhop, not having change and being slightly dishonest, returns only P100 to each student and keeps the remaining P200. Now each student paid P2000 – P100 = P1900, and 3 # P1900 = P5700 paid in total. The bellhop kept P200 for a total of P5900. What happened to the other P100?

Material Balance Calculations - Mixing The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

!

Recall: Algebra Problems ! A

manufacturer of soft drinks advertises its orange soda as “naturally ﬂavored,” although it contains only 5% orange juice. A new federal regulation stipulates that to be called “natural,” a drink must contain at least 10% fruit juice. How much pure orange juice must this manufacturer add to 900 gal of orange soda to conform to the new regulation?

Material Balance Calculations - Mixing The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

!

Recall: Algebra Problems ! A

pharmacist, needs 100 L of a 50% alcohol solution. He has on hand a 30% alcohol solution and an 80% alcohol solution, which he can mix. How many liters of each will be required to make the 100 L of a 50% alcohol solution?

Material Balance Calculations - Mixing The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

Tie Component The Balance Equation

!

Enters in one input stream only

Mass Balance Analysis

!

Exits in one output stream only

!

Is easily traced, and can be used in relating other components

!

Ex. Find the mass M4

Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

Tie Component The Balance Equation

!

Ex. You buy 100 kg of cucumbers that contain 99% water. A few days later they are found to be 98% water. What is the remaining weight of the cucumbers?

!

Ex. Drying: A batch dryer can handle 5000 kg feed per hour. 5000 kg of a material containing 30% moisture is to be dried. How much moisture evaporates per hour if the dried material contains 5% moisture?

!

Ex. Evaporation: An evaporator has a capacity to evaporate 30,000 kg of water per hour. We want to concentrate a NaCl-water solution from 10% to 40%. If the evaporator is to operate at its rated capacity, what must the feed rate be? How much product is obtained per hour?

Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

Degree of Freedom Analysis The Balance Equation

!

Concept: DF = #Unknowns - #Indep.Eqn !

Mass Balance Analysis

!

Tie Component

!

Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

DF = 0 ! solvable DF > 0 ! underspeciﬁed, need more equations DF < 0 ! overspeciﬁed, possible redundant or inconsistent equations

!

Independent equation: Cannot be derived algebraically from other equations

!

Use to check if enough given information is available to solve the problem

!

Requires a complete ﬂowchart

Degree of Freedom Analysis The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

!

Example

Degree of Freedom Analysis The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

!

Sources of Equations !

Mass balance: total and component

!

Energy balance

!

Process speciﬁcations

!

Physical properties

!

Physical constraints

!

Stoichiometry

General Steps The Balance Equation Mass Balance Analysis

Read

Tie Component

Draw

Degree of Freedom Analysis Mass Balance Calculations – Single Unit

DFA

Basis

MB

Multiple-Unit Balances Check

• Identify the streams and equipment involved • Define the system. Sketch a BFD or PFD with all streams • Identify known and unknown parameters and relations • Choose and clearly state a convenient basis • Write overall and component balances • Consider the given conditions and common sense

Mass Balance Calculations The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

Mass Balance Calculations The Balance Equation

!

Mass Balance Analysis

Since the process is at steady-state condition and no chemical reaction is involved, the material balance equation becomes Input = Output

Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

!

This balance equation can be applied to: !

The total mass entering and leaving the process

!

Mass of individual component entering and leaving the process.

Mass Balance Calculations The Balance Equation

!

m1 = m2 + m3

Mass Balance Analysis Tie Component

!

Multiple-Unit Balances

Component A Balance m A1 = m A2 + m A3

Degree of Freedom Analysis Mass Balance Calculations – Single Unit

Total Mass (Overall Material Balance)

m1x A1 = m2x A2 + m3x A3 !

Component B Balance mB1 = mB2 + mB3 m1xB1 = m2xB2 + m3xB3

Mass Balance Calculations The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis

!

For the given process, 3 material balance equations can be written:

Total Balance:

m1 = m2 + m3

(E1)

A-Balance:

m A1 = m A2 + m A3

(E2)

B-Balance:

mB1 = mB2 + mB3

(E3)

Mass Balance Calculations – Single Unit Multiple-Unit Balances

Are these material balances independent equations?

Mass Balance Calculations The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis

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For the given process, 3 material balance equations can be written:

Total Balance:

m1 = m2 + m3

(E1)

A-Balance:

m A1 = m A2 + m A3

(E2)

B-Balance:

mB1 = mB2 + mB3

(E3)

Mass Balance Calculations – Single Unit Multiple-Unit Balances

Are these material balances independent equations?

Mass Balance Calculations The Balance Equation

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Mass Balance Analysis

Two methanol-water mixtures are contained in separate tanks.

Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

Example. Mixing of Methanol-Water Mixtures

The ﬁrst mixture contains 40.0 wt% methanol and the second contains 70.0 wt% methanol. If 200 kg of the ﬁrst mixture is combined with 150 kg of the second, what are the mass and composition of the product?

Mass Balance Calculations The Balance Equation

Step 1. Draw a ﬂowchart to visually organize the data.

Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

0.40 0.60

Mass Balance Calculations The Balance Equation

Step 2. Determine the degrees of freedom (DF)

Mass Balance Analysis

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Number of unknowns (U): 3 unknowns !

m3, xM3, xW3

Tie Component Degree of Freedom Analysis

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Number of independent equations (V):

!

DF = U – V = 3 – 3 = 0 , the problem is solvable

Mass Balance Calculations – Single Unit Multiple-Unit Balances

Mass Balance Calculations The Balance Equation

Step 3. Write down the equations

Mass Balance Analysis

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Material Balances (Steady-State, Non-Reactive Process):

Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit

Total Balance:

m1 + m2 = m3

Methanol-Balance: m1xM1 + m2xM2 = m3xM3 Water-Balance:

m1xW1 + m2xW2 = m3xW3

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choose only 2 equations since one of them is no longer independent

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Physical Constraint (applied to mixture 3):

Multiple-Unit Balances

xM3 + xW3 = 1.00

Mass Balance Calculations The Balance Equation

Step 4. Solve the unknowns ( m 3 , x M3 , x W3 )

Mass Balance Analysis

! Always

Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

start with the equation with the least number of unknowns if possible and minimize solving equations simultaneously.

Mass Balance Calculations The Balance Equation Mass Balance Analysis

Step 4. Solve the unknowns ( m 3 , x M3 , x W3 ) !

m3 = (200 kg) + (150 kg) = 350 kg

Tie Component Degree of Freedom Analysis

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CH3OH balance: (200 kg)(0.40) + (150 kg)(0.70) = (350 kg)x M3

Mass Balance Calculations – Single Unit Multiple-Unit Balances

Total balance:

xM3 = 0.529 !

Physical constraint: xW3 = 1.00 – xM3 = 1 – 0.529 xW3 = 0.471

Homework The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

Three hundred gallons of a mixture containing 75.0 wt% ethanol and 25% water (mixture speciﬁc gravity = 0.877) and a quantity of a 40.0 wt% ethanol-60% water mixture (SG=0.952) are blended to produce a mixture containing 60.0 wt % ethanol. Determine the required volume of the 40% mixture. Ans.: 207 gal

Questions?

Steady-State Material Balance The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

Input = Output !

In a mass balance set-up involving N components: ! ! ! !

N+1 relationships are available (one overall mass balance and N component balances) Maximum number of independent equations is equal to N Physical constraints (mass fractions) Process speciﬁcations

Steady-State Material Balance The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

! ! ! ! !

Total Mass (Overall Material Balance) M1 + M2 + M3 = M4 + M5 Component A Balance M1x A1 + M2x A2 = M4x A4 + M5x A5 Component B Balance M1xB1 + M2xB2 + M3xB3 = M4xB4 + M5xB5 Component C Balance M2xC2 + M3xC3 = M4xC4 + M5xC5 Component D Balance M = M

Steady-State Material Balance The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

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Sum of the mass fractions of the components of each stream is one. x A1 + xB1 = 1 x A2 + xB2 + xC2 = 1 xB3 + xC3 + xD3 = 1 x A4 + xB4 + xC4 + xD4 = 1 x + x + x = 1

Steady-State Material Balance The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

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Methods of Solution: ! Arithmetic ! !

Stream-to-stream analysis Tie component

! Algebraic ! Assign

! ! !

method

solution symbols to unknowns

Combination of arithmetic and algebraic methods Graphical technique Computer solutions using programs or packaged softwares

Single-Unit Balance: Drying The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

" Removal

of water or another liquid from a solid by vaporization

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Ex. Drying:

A batch dryer can handle 5000 kg feed per hour. 5000 kg of a material containing 30% moisture is to be dried. How much moisture evaporates per hour if the dried material contains 5% moisture?

Single-Unit Balance: Evaporation The Balance Equation

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Unit operation where part of the solvent in a solution is vaporized to concentrate the solution

!

Ex. Evaporation:

Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

An evaporator has a capacity to evaporate 30,000 kg of water per hour. We want to concentrate a NaCl-water solution from 10% to 40%. If the evaporator is to operate at its rated capacity, what must the feed rate be? How much product is obtained per hour?

Single-Unit Balance: Distillation The Balance Equation

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Mass transfer due to di "erence in volatility !

Mass Balance Analysis

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Tie Component

!

Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

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Vapor is richer in the more volatile component as it ascends from plate to plate Product withdrawn from the top = distillate Bottom plate = bottoms Side products can be withdrawn from any other plate

Ex. Distillation:

We desire to produce two ethyl alcohol-water mixtures each containing 90% and 50% alcohol by mole from a dilute mixture containing 20% mole alcohol by distillation. 98% of the ethyl alcohol in the feed is to be recovered in these two products, that is, only 2% of the ethyl alcohol in the feed may go into the bottoms. If the bottoms contains 0.6% alcohol, calculate the amount of the di "erent streams per 1000 kmol of feed stream.

Single-Unit Balance: Crystallization The Balance Equation

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Mass Balance Analysis

! Attained !

Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

Unit operation in which a solution is brought to a concentration where it cannot hold all the soluble component (solute) in solution

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by cooling a hot solution and/or by evaporating some of the solvent Solute precipitates out as crystals

Ex. Crystallization:

One thousand kilograms of a 64% NaNO3 solution in water is at a temperature of 100°C. This is sent to a crystallizer where it is cooled to 30°C. At this temperature, the solubility of NaNO3 is 96 parts per 100 parts of water (the solubility of NaNO 3 is maximum amount of NaNO3 that can stay in solution; the rests crystallizes out). Calculate the amount of crystals that precipitates out if a. b.

No water evaporates during cooling 5% of the original water evaporates during cooling

Single-Unit Balance: Solid-Liquid Extraction The Balance Equation

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Valuable component (solute) in a solid can be separated by soaking the solid in a solvent where the solute is soluble and allowing the solute to di"use out

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Ex. Solid-Liquid Extraction:

Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

A copra batch contains 44% oil and is to be extracted with n-hexane. The extracted meal contains 0.07 kg oil/kg oil-free matter, and the mass fraction of n-hexane is 0.05. The extract contains 49% n-hexane. If 10,000 kg of copra are to be processed, how many kilograms of nhexane is required?

Single-Unit Balance: Practice The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

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Ex. An aqueous solution of NaOH contains 20% NaOH by mass. It is desired to produce and 8% NaOH solution by diluting a stream of the 20% solution with a stream of pure water. Calculate the ratios [L H2O/kg feed] and [kg product/kg feed].

Single-Unit Balance: Practice The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

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Ex. A liquid mixture containing 45% benzene (B) and 55% toluene (T) is fed to a distillation column. The overhead product contains 95% B, and the bottom product stream contains 8% of the benzene fed to the column. The volumetric ﬂow rate of the feed stream is 2000L/h and the speciﬁc gravity of the mixture is 0.872. Determine the mass ﬂow rate of overhead product and the mass ﬂow rate and mass fractions of the bottom product.

Multiple-Unit Balances The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

Overall balance Mixing point

Unit balance Splitting point

Multiple-Unit Balances The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

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Ex. How will we solve for the ﬂow rates and compositions of (1), (2) and (3)?

Multiple-Unit Balances: Multiple-effect Evaporation The Balance Equation Mass Balance Analysis

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Evaporator unit – “e"ect” ! !

electricity or steam Steam side: !

Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

Heat exchangers: steam enters tubes/coils, transfers heat to the solution, condenses, and exits as liquid water

Mass of steam in = mass of condensed water out !

Solution side: ! !

Liquid solution boils Water converted to vapor – can be used as source of heat for another evaporator (multiple-e "ect)

Mass feed = mass product + mass water evaporated

Multiple-Unit Balances: Multiple-effect Evaporation The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

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Ex. A triple-e"ect evaporator concentrates a 10% caustic soda to 50% NaOH. Assuming equal amounts of evaporation in each e "ect, calculate the concentration of the solution leaving each e"ect.

Multiple-Unit Balances The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

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Ex. NaCl crystals are to be produced from 1,000 kg of a solution containing 10% NaCl, 1% KOH, and 89% H 2O. The solution is ﬁrst sent to an evaporator where 790 kg of the water from the solution is removed. The hot concentrated solution is then sent to a crystallizer where the solution is cooled to 20°C and NaCl crystals precipitate. Each kg of crystals carries with it 0.1 kg of adhering mother liquor. The wet crystals are then treated in a drier where 95% of the water is removed. Calculate: a. b. c.

the composition of the concentrated solution from the evaporator. the composition of the ﬁnal crystal product. the percentage recovery of NaCl from the original solution.

Data: At 20°C, the solubility of NaCl in H 2O is 36 g/100g H 2O. KOH is non-volatile and very soluble in H2O. Assume that no water evaporates in cooling the solution. Assume that the presence of KOH does not a "ect the solubility of NaCl in H 2O.

Multiple-Unit Balances The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

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Ex. Acetone is used in the manufacture of many chemicals and also as a solvent. In its latter role, many restrictions are placed on the release of acetone vapor to the environment. An acetone recovery system is illustrated below. All of the concentrations shown of both the gases and liquids are speciﬁed in weight percent in this special case to make the calculations simpler. Calculate A, F , W , B, and D in kilograms per hour. Assume that G = 1400 kg/hr.

! Answers:

A = 1336.7 F = 221.05 W = 157.7 D = 34.91 B = 186.1

Summary The Balance Equation Mass Balance Analysis Tie Component Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

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By now you should have learned #

The Balance Concept

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How to analyze and solve material balance equations

#

How to analyze solutions using Degree of Freedom Analysis

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How to solve Single Unit Material Balances

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How to solve Multiple-Unit Balances

Groups - WFW The Balance Equation Mass Balance Analysis Tie Component

Cosmetics

Paints and Dyes

Glass

Drugs and Pharma

Food and Beverage

Beer

Lumactod

Almeda

Banag

Betancor

Ng

Abis

Castro

Dizon

Gutierrez

Caramat

Yu

Cua

Reyes

Moya

Ureta

Hadi

Sy, JL

Dy

Rubber

Pulp and Paper

Soaps

Cement

Plastics

Baladiang

Mirasol

Pabua

Co

Alunan

Petallo

Sy, S

Roco

Dizon

Tan

Victor

Joves

Suarez

Lim

Garcia

Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

Groups - WFX The Balance Equation

Cosmetics

Mass Balance Analysis Tie Component

Paints and Dyes

Glass

Drugs and Pharma

Food and Beverage

Beer

Jeresano

Trinidad

Maranon

Santiago

Castillo

Crisostomo

Conol

Piedad

Canicosa

Carrido

Perez, G

Barcelona

Vergara

Paet

Catulong Sobremonte

Degree of Freedom Analysis Mass Balance Calculations – Single Unit Multiple-Unit Balances

Rubber

Pulp and Paper

Soaps

Cement

Plastics

Go

Dignos

Perez, E

Cudiamat

Baliton

Reyes

Saban

Quilantang

Dallarte

Caballero

Ng

Tan

Solis

Li

Rivera

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