Material and Energy Balances

1
Chapter 3

• Material and Energy Balances

2
Mathematical Modeling

• Weve been doing conceptual designs in chapter 2.
  
• How do engineers determine if a new plant design
  is going to work?
• Usually by running a pilot plant and using
  mathematical models (balances) to predict the
  performance of the full-sized unit
• These mathematical models are covered in this
  chapter and in chapters 4 and 5

3
Illustrate Material Balances with Desalinization

• Comparison of Seawater to Potable water
• Define the objective of desalination Produce
  potable water.
• Do we care if we produce pure salt?

NO
4
Single Unit --First Idea for Desalinator

• Consider this process
• First of all whats wrong with it?
• Not realistic impossible to produce absolutely
  pure water
• Difficult to operatehow do we get the salt out
• Only advantage easy to calculate flows

5
Concept of System and Surroundings

• Many will remember in thermodynamics we defined a
  system and surroundings
• Generally enclose a system with a dotted box
• The law of matter conservation says
• In all of our work rate in rate out

accumulation
6
As an Example, Apply to our Water Illustration
Feed
10 kg/hr
Surroundings
System
How fast will water accumulate ?
4 kg/hr
If we didnt want water to accumulate, what does
mass conservation have to say about the feed and
purge stream flows ?
7
Back to the Desalinator
seawater
100 kg/min
H
O
2
separator
H2O 96.5 kg/min Salt 3.5 kg/min
salt

• Assuming no accumulation, what are the flow
  rates of water and salt out the unit, what would
  you answer?
• How did you get this?
• Mentally you said
• And

8
A more realistic separator

• This is a more realistic case
• But now the flow rate calculations of the two
  outlet streams arent so obvious
• So lets figure out how to calculate these flows

9
A more complicated Desalinator
seawater

• What facts do we know?
• Everything about (Total flow rate, conc)
• Concentration
• Concentration

1
Fs,2 0.27FT,2
2
Fw,3 FT,3
3
10
A more complicated Desalinator

• Balance on salt 3.5 kg/min Fs,2 0.27 FT,2
• Overall balance 100 kg/min FT,2 FT,3
• Solving FT,2 13 kg/min FT,3 87 kg/min

11
Using water balance instead of salt balance

• Balance on water 96.5 kg/min 0.73 FT,2 FT,3
• Overall 100 kg/min FT,2 FT,3
• Solving FT,2 13 and FT,3 87 (as before)

12
A more complicated Desalinator independent
equations

• Note number of independent equations number of
  components (W ,S) in this case two
• To show this
• Water balance 96.5 kg/min 0.73 FT,2 FT,3 (1)
• Overall balance 100 kg/min FT,2 FT,3 (2)
• Subtracting (1) from (2) we get
• 3.5 kg/min 0.27 FT,2 which is the salt balance
  equation

13
Playing With Models

• So now we can begin to ask ourselves, how does
  FT,3 vary with the salt content of stream 2
• Logically, what would it be?
• Generalizing let s be the salt content of stream
  2 (i.e. s Fs,2/FT,2)
• Reformulating our equations, we get
• Balance on salt 3.5 kg/min Fs,2 s FT,2
• Overall balance 100 kg/min FT,2 Fw,3
• Solving Fw,3 100 - 3.5/ s

14
Playing With Models
Asymptote is
96.5 kg/min
With F30, s 0.035
15
Multiple units--Desalinator
Brine coating
Ice cube
slush
ice
ice
sea water
brine
water
brine (1 wt )
slusher
melter
2
1
separator
4
5

100 kg/min
brine
3
s
salt (
wt fraction)
H
O
2

• Ice retains some brine (salt solution) on surface
  of ice (1)
• Let W water (water ice)
• Let S salt (brine is a salt solution)
• So we can solve any system that has two unknowns

16
Multiple units--Desalinator

• Count the unknowns Fw,5 Fs,5 Fs,3 Fw,3
• Count the knowns Fw,1 and Fs,1
• Have only two equations cant solve
• So pick another boundary
• Do we really have to move the boundary as shown
  in next slide?

No, Why?
17
Multiple units--Desalinator

• Write our two mass balance equations
• 96.5 kg/min Fw,3 Fw,4 (1)
• 3.5 kg/min Fs,3 Fs,4 (2)
• Work on the given 1 brine in steam 4 an
  example of how to translate flowsheet information
  into mathematics
• All brine contains s wt fraction salt
• Flow rate of salt in any stream given by sFT

18
Multiple units--Desalinator

• So Fs,4 0.01(FT,4) s
• But FT,4 Fw,4 Fs,4
• Fs,4 0.01(Fw,4 Fs,4) s (3)
• Also s Fs,3/(Fs,3 Fw,3) (4)
• So we have 4 equations and 4 unk (Fw,4 Fs,4
  Fw,3 and Fs,3)

19
Multiple units--Desalinator

• Solving to get the desired result (which is Fw,4)
• Eliminate Fw,3 and Fs,3
• Eliminate Fs,3 using (1) and (2) and (4)
• Eliminate Fs,4 using (3)

20
Multiple units--Desalinator

• Finally,
• Fw,4 -1.01s 101.05 3.535/ s
• With salt content in stream 4 ( stream 5)
• But
• So e s/100

21
Slusher Results
22
Exploring the Brine Content Stream 4 (or 5)

• So e s/100
• Since we know that e 0.0005 (maximum for
  potable water), we find that s 0.05 (or 5 wt)
• So with 1 brine, the saltiest the brine stream
  we can get is 5 wt and Fw,4 30 kg/min
• If we let ß Retained brine in stream 4, then we
  find
• Fw,4 100 - 7000 ß

23
Reducing Brine Stream is Key

• Checking our equation, at ß0.01, we find that
  Fw,4 100 7000(0.01)
• Fw,4 30 kg/min
• So we see that reducing the amount of retained
  brine in the water is key to increasing the
  amount of potable water produced.
• How are we going to do this?

Wash the ice with water
24
Water Wash Desalinator

• Where can we get the wash water?
• Does this make the process more complicated?
• The trick is to note that when we draw overall
  balance around the water wash desalinator as
  shown, details of recycle stream arent important

By recycle
Sure
25
Water Wash Desalinator

• Same analysis, but now the recycle stream lets us
  specify s 0.07 and

Fw,3 50 kg/min
26
Water Wash Desalinator

• So what has the recycle stream done?
• It allows us to run stream 3 (the brine stream)
  at s gt 0.05 and still get very low salt
  concentrations in the potable water
• Remember without water recycle
• e s/100