Unformatted text preview: DESIGN AND PLANNING
UNDER UNCERTAINTY
Prof. Miguel Bagajewicz
CHE 4273 1 TwoStage
TwoStage
Stochastic Optimization Models
Philosophy
Philosophy
• Maximize the Expected Value of the objective over all possible realizations of
Expected
uncertain parameters.
• Typically, the objective is Profit or Net Present Value.
Profit Net
• Sometimes the minimization of Cost is considered as objective.
Cost Uncertainty
• Typically, the uncertain parameters are: market demands, availabilities,
market
prices, process yields, rate of interest, inflation, etc.
• In TwoStage Programming, uncertainty is modeled through a finite number
of independent Scenarios.
Scenarios
• Scenarios are typically formed by random samples taken from the probability
random
distributions of the uncertain parameters.
2 Characteristics of TwoStage
TwoStage
Stochastic Optimization Models FirstStage Decisions
• Taken before the uncertainty is revealed. They usually correspond to structural
decisions (not operational).
• Also called “Here and Now” decisions.
• Represented by “Design” Variables.
• Examples:
−To build a plant or not. How much capacity should be added, etc.
−To place an order now.
−To sign contracts or buy options.
−To pick a reactor volume, to pick a certain number of trays and size
the condenser and the reboiler of a column, etc 3 Characteristics of TwoStage
Characteristics
TwoStage
Stochastic Optimization Models
SecondStage Decisions
• Taken in order to adapt the plan or design to the uncertain parameters
realization.
• Also called “Recourse” decisions.
• Represented by “Control” Variables.
• Example: the operating level; the production slate of a plant.
• Sometimes first stage decisions can be treated as second stage decisions.
In such case the problem is called a multiple stage problem. 4 Example: Vinyl Chloride Plant HCl recycle Air or O2
Oxychlorination
Ethylene Cl2 Light ends EDC
purification
Direct
chlorination EDC
pyrolysis VCM
purification VCM EDC recycle
Heavy ends 5 Example: Vinyl Chloride Plant
Example:
Consider the following forecasts:
Forecasted prices of raw materials product
Year Forecasted excess demand over current capacity Ethylene Chlorine Oxygen VCM $/ton $/ton $/ft3 $/ton 2004 492.55 212.21 0.00144 499.19 2004 3602 2005 499.39 214.14 0.00144 506.19 2005 5521 2006 506.22 216.07 0.00143 513.18 2006 7355 2007 513.06 218.00 0.00142 520.18 2007 9551 2008 519.90 219.93 0.00141 527.18 2008 11888 2009 526.73 221.86 0.00140 529.17 2009 14322 2010 533.57 223.79 0.00139 535.17 2010 16535 2011 540.41 225.72 0.00138 543.17 2011 18972 Std. Dev 24.17 10.56 0.00010 26.15 Year VCM
lbmol/hr Consider building (in 2004) for three capacities to satisfy excess demand at 2004,
2006 and 2011. Plants will operate under capacity until 2006 or 2011 in the last
two cases. These are 3 different first stage decisions.
6 Example: Vinyl Chloride Plant
The different investment costs are:
Plant Capacity 4090
MMlb/yr 6440
MMlb/yr 10500
MMlb/yr TCI $47,110,219 $68,886,317 $77,154,892 Consider the following calculation procedure
Random Number
Generation Total Product
Cost Income from
selling VCM Random numbers are Raw material
Cost Gross Income Cash Flow Net Profit obtained for each year for
raw materials and product
prices using sampling from a Net Present
Worth normal distribution. This can
be done in Excel Risk &
Probability
7 Example: Vinyl Chloride Plant
Probability vs. Net Present Worth Histograms and Risk Curves are 0.35
0.3
Probability 0.25 Notice the asymmetry in the distributions. 0.2
0.15
0.1
0.05
0
4500 Risk at Different Capacity 1500
0 6 1500
NPW ($10 ) 6.44E9 lb/yr 1 Cummulative Probability 3000 4.09E9 lb/yr 3000 1.05E10 lb/yr 0.8
0.6 The risk curves show a 36% chance of losing 0.4 money for the 10.5 billion lbs/year capacity, 0.2 31.7% for the 6.44 billion lbs/yr capacity and 0
6.00E+09 4.00E+09 2.00E+09 0.00E+00 2.00E+09 4.00E+09 Net Present Worth ($) 6.44E9 lb/yr 4.09E9 lb/yr 6.00E+09 41% chance for the 4.09 billion lbs/year capacity.
Expected Profits are: 24%, 25% and 20%. 1.05E10 lb/yr 8 Capacity Planning
GIVEN: Process Network Forecasted Data DETERMINE: Set of Processes
Set of Chemicals B
A Demands & Availabilities
Costs & Prices
Capital Budget Network Expansions 2 C 3 D 1 Timing
Sizing
Location Production Levels OBJECTIVES: Maximize Net Present Value 9 Capacity Investment Planning
Design Variables: to be decided before the uncertainty reveals
to
x= {Yit , Eit , Qit } Y: Decision of building process i in period t
E: Capacity expansion of process i in period t
Q: Total capacity of process i in period t Control Variables: selected after uncertain parameters become known.
Assume them known for the time being!!!! ys = { S jlt , P jlt , Wit }
S: Sales of product j in market l at time t and scenario s
P: Purchase of raw mat. j in market l at time t and scenario s
and
W: Operating level of of process i in period t and scenario s 10 Example
Project Staged in 3 Time Periods of 2, 2.5, 3.5 years Chemical 5 Chemical 1 Process 2 Chemical 6 Process 5 Process 1 Chemical 8 Chemical 2 Chemical 7 Process 3
Chemical 4 Chemical 3
Process 4 11 Example
One feasible (not necessarily optimal) solution could be
Chemical 5 Period 1
2 years Process 1 Chemical 1 Process 2 Chemical 6 Chemical 2 Period 2
2.5 years Chemical 5
Chemical 1 Process 1 Process 2 Chemical 6 Chemical 2
Chemical 7
Process 3
Chemical 3 Period 3
3.5 years Process 5 Chemical 8 Chemical 4 Same flowsheet different production rates
12 MODEL
SETS VARIABLES I : Processes i,=1,…,NP
J : Raw materials and Products, j=1,…,NC
T: Time periods. T=1,…,NT
L: Markets, l=1,..NM
Yit: An expansion of process I in period t takes place (Yit=1), does not take place (Yit=0)
Eit: Expansion of capacity of process i in period t.
Qit: Capacity of process i in period t.
Wit: Utilized capacity of process i in period t.
Pjlt : Amount of raw material/intermediate product j consumed from market l in period t
Sjlt : Amount of intermediate product/product j sold in market l in period t ηij : Amount of raw material/intermediate product j used by process i
µij : Amount of product/intermediate product j consumed by process i
γjlt : Sale price of product/intermediate product j in market l in period t
Γjlt : Cost of product/intermediate product j in market l in period t
δ : Operating cost of process i in period t
PARAMETERS it
αit : Variable cost of expansion for process i in period t
βit : Fixed cost of expansion for process i in period t
Lt : Discount factor for period t
L
U
E it , E it :Lower and upper bounds on a process expansion in period t
a L , a U : Lower and upper bounds on availability of raw material j in market l in period t
jlt
jlt
d L , d U : Lower and upper bounds on demand of product j in market l in period t
jlt
jlt
CI t : Maximum capital available in period t NEXPt: maximum number of expansions in period t 13 MATHEMATICAL PROGRAMMING MODEL
OBJECTIVE FUNCTION Max NPV = NT ∑
t =1 NP
NP NT NM NC
Lt ∑∑ (γ jlt S jlt − Γjlt Pjlt ) − ∑ δitWit − ∑∑ (α it Eit + βitYit ) l =1 j =1
i =1
i =1 t =1 DISCOUNTED REVENUES INVESTMENT Yit: An expansion of process I in period t takes place (Yit=1), does not take place (Yit=0)
Eit: Expansion of capacity of process i in period t.
Wit: Utilized capacity of process i in period t.
Pjlt : Amount of raw material/interm. product j consumed from market l in period t
Sjlt : Amount of intermediate product/product j sold in market l in period t
I : Processes i,=1,…,NP
J : Raw mat./Products, j=1,…,NC
T: Time periods. T=1,…,NT
L: Markets, l=1,..NM γjlt : Sale price of product/intermediate product j in market l in period t
Γjlt : Cost of product/intermediate product j in market l in period t
δit : Operating cost of process i in period t
αit : Variable cost of expansion for process i in period t
βit : Fixed cost of expansion for process i in period t
Lt : Discount factor for period t 14 MODEL
LIMITS ON EXPANSION L
L
Yit Eit ≤ Eit ≤ Yit Eit i =1,K, NP t =1,K, NT TOTAL CAPACITY IN
EACH PERIOD Qit = Qi ( t −1) + E it i =1,K, NP t =1,K, NT LIMIT ON THE NUMBER
OF EXPANSIONS
LIMIT ON THE CAPITAL
INVESTMENT NT ∑Y
t =1 it NP ∑ (α
i =1 it ≤ NEXPi
E it + β it Yit ) ≤ CI t i =1,K, NP
t =1,K, NT Yit: An expansion of process I in period t takes place (Yit=1), does not take place (Yit=0)
Eit: Expansion of capacity of process i in period t.
Qit: Capacity of process i in period t. I : Processes i,=1,…,NP
J : Raw mat./Products, j=1,…,NC
T: Time periods. T=1,…,NT
L: Markets, l=1,..NM NEXPt: maximum number of expansions in period t
αit : Variable cost of expansion for process i in period t
βit : Fixed cost of expansion for process i in period t
L
U
E it , E it : Lower and upper bounds on a process expansion in period t
15 MODEL
UTILIZED CAPACITY IS
LOWER THAN TOTAL
CAPACITY Wit ≤ Qit
NM l =1 a L ≤ Pjlt ≤ a U
jlt
jlt BOUNDS
NONNEGATIVITY NP ∑P MATERIAL BALANCE i =1,K, NP t =1,K, NT jlt NM NP i =1 l =1 i =1 + ∑ ηijWit = ∑ S jlt + ∑ µ ijWit d L ≤ S jlt ≤ d U
jlt
jlt Eit , Qit , Wit , Pjlt , S jlt ≥ 0 ∀i, j , l , t i =1,K, NP t =1,K, NT j = 1,..., NC , t = 1,..., NT , l = 1,..., NM INTEGER
VARIABLES Yit ∈{0,1}
i =1,K, NP t =1,K, NT Yit: An expansion of process I in period t takes place (Yit=1), does not take place (Yit=0)
Eit: Expansion of capacity of process i in period t.
Qit: Capacity of process i in period t.
Wit: Utilized capacity of process i in period t.
Pjlt : Amount of raw material/intermediate product j consumed from market l in period t
Sjlt : Amount of intermediate product/product j sold in market l in period t I : Processes i,=1,…,NP
J : Raw mat./Products, j=1,…,NC
T: Time periods. T=1,…,NT
L: Markets, l=1,..NM a L , a U : Lower and upper bounds on availability of raw material j in market l in period t
jlt
jlt
d L , d U : Lower and upper bounds on demand of product j in market l in period t
jlt
jlt
16 MODEL
NM ∑P MATERIAL BALANCE l =1 k NP jlt NM NP i =1 l =1 i =1 i =1,K, NP t =1,K, NT + ∑ η ijWit ≤ ∑ S jlt + ∑ µ ijWit C A
i
D p B NM ∑P
l =1 Blt NM + ηkAWkt = ∑ S Clt + µ iDWit
l =1 Reference Component is C ηkA ,µ iD
“Stoichiometric” Coefficients 17 TwoStage Stochastic Formulation
TwoStage
Let us leave it linear
because as is it is
complex enough.!!! LINEAR MODEL SP
T
Max ∑ ps qs ys − cT x
s Technology matrix s.t. Recourse
Function FirstStage
Cost Ax = b FirstStage Constraints Ts x +Wys = hs SecondStage Constraints x≥0
First stage variables x∈ X Second Stage Variables ys ≥ 0
Recourse matrix (Fixed Recourse)
Sometimes not fixed (Interest rates in Portfolio Optimization) Complete recourse: the
recourse cost (or profit) for
every possible uncertainty
realization remains finite,
independently of the firststage
decisions (x).
Relatively complete recourse:
the recourse cost (or profit) is
feasible for the set of feasible
firststage decisions. This
condition means that for every
feasible firststage decision,
there is a way of adapting the
plan to the realization of
uncertain parameters.
We also have found that one
can sacrifice efficiency for
certain scenarios to improve
risk management. We do not
know how to call this yet. 18 Capacity Planning Under Uncertainty
GIVEN: Process Network Forecasted Data DETERMINE: Set of Processes
Set of Chemicals B
A Demands & Availabilities
Costs & Prices
Capital Budget Network Expansions 2 C 3 D 1 Timing
Sizing
Location Production Levels OBJECTIVES: Maximize Expected Net Present Value
Minimize Financial Risk
19 Process Planning Under Uncertainty
Design Variables: to be decided before the uncertainty reveals
to
x= {Yit , Eit , Qit } Y: Decision of building process i in period t
E: Capacity expansion of process i in period t
Q: Total capacity of process i in period t Control Variables: selected after the uncertain parameters become known
ys = { Sjlts , Pjlts , Wits} S: Sales of product j in market l at time t and scenario s
P: Purchase of raw mat. j in market l at time t and scenario s
and
W: Operating level of of process i in period t and scenario s 20 MODEL
LIMITS ON EXPANSION L
L
Yit Eit ≤ Eit ≤ Yit Eit i =1,K, NP t =1,K, NT TOTAL CAPACITY IN
EACH PERIOD Qit = Qi ( t −1) + E it i =1,K, NP t =1,K, NT LIMIT ON THE NUMBER
OF EXPANSIONS
LIMIT ON THE CAPITAL
INVESTMENT NT ∑Y
t =1 it NP ∑ (α
i =1 it ≤ NEXPi
E it + β it Yit ) ≤ CI t i =1,K, NP
t =1,K, NT Yit: An expansion of process I in period t takes place (Yit=1), does not take place (Yit=0)
Eit: Expansion of capacity of process i in period t.
Qit: Capacity of process i in period t. I : Processes i,=1,…,NP
J : Raw mat./Products, j=1,…,NC
T: Time periods. T=1,…,NT
L: Markets, l=1,..NM NEXPt: maximum number of expansions in period t
αit : Variable cost of expansion for process i in period t
βit : Fixed cost of expansion for process i in period t
L
U
E it , E it : Lower and upper bounds on a process expansion in period t
21 MODEL
UTILIZED CAPACITY IS
LOWER THAN TOTAL
CAPACITY
NM ∑P MATERIAL BALANCE l =1 a L ≤ Pjlts ≤ aU
jlts
jlts BOUNDS
NONNEGATIVITY Wits ≤ Qit
NP jlts NM NP i =1 l =1 i =1 + ∑ ηijWits ≤ ∑ S jlts + ∑ µ ijWits d L ≤ S jlts ≤ d U
jlts
jlts Eit , Qit ,Wits , Pjlts , S jlts ≥ 0
Yit ∈{0,1} i =1,K, NP t =1,K, NT ∀i, j , l , t i =1,K, NP t =1,K, NT , ∀s j = 1,..., NC , t = 1,..., NT , l = 1,..., NM , ∀s INTEGER
VARIABLES Yit: An expansion of process I in period t takes place (Yit=1), does not take place (Yit=0)
Eit: Expansion of capacity of process i in period t.
Qit: Capacity of process i in period t.
Wit: Utilized capacity of process i in period t.
Pjlt : Amount of raw material/intermediate product j consumed from market l in period t
Sjlt : Amount of intermediate product/product j sold in market l in period t I : Processes i,=1,…,NP
J : Raw mat./Products, j=1,…,NC
T: Time periods. T=1,…,NT
L: Markets, l=1,..NM a L , aU :Lower and upper bounds on availability of raw material j in market l in period t, scenario s
jlts
jlts
d L , d U : Lower and upper bounds on demand of product j in market l in period t, scenario s
jlts
jlts
22 MODEL
OBJECTIVE FUNCTION
NP NT NM NC NP NT Max NPV = ∑ ps ∑ Lt ∑∑ (γ jlts S jlts − Γjlts Pjlts ) − ∑ δ itsWits − ∑∑ (α it Eit + β itYit ) t =1 l =1 j =1
s
i =1 i =1 t =1 DISCOUNTED REVENUES INVESTMENT Yit: An expansion of process I in period t takes place (Yit=1), does not take place (Yit=0)
Eit: Expansion of capacity of process i in period t.
Wit: Utilized capacity of process i in period t.
Pjlt : Amount of raw material/interm. product j consumed from market l in period t
Sjlt : Amount of intermediate product/product j sold in market l in period t
I : Processes i,=1,…,NP
J : Raw mat./Products, j=1,…,NC
T: Time periods. T=1,…,NT
L: Markets, l=1,..NM γjlt : Sale price of product/intermediate product j in market l in period t
Γjlt : Cost of product/intermediate product j in market l in period t
δit : Operating cost of process i in period t
αit : Variable cost of expansion for process i in period t
βit : Fixed cost of expansion for process i in period t
Lt : Discount factor for period t 23 Example
Uncertain Parameters: Demands, Availabilities, Sales Price, Purchase Price
Total of 400 Scenarios
Project Staged in 3 Time Periods of 2, 2.5, 3.5 years
Chemical 5 Chemical 1 Process 2 Chemical 6 Process 5 Process 1 Chemical 8 Chemical 2 Chemical 7 Process 3
Chemical 4 Chemical 3
Process 4 24 Example – Solution with Max ENPV
Period 1
2 years
Chemical 5 5.27 kton/yr
Chemical 1 5.27 kton/yr Process 1 10.23 kton/yr
Chemical 7 19.60 kton/yr
Process 3
Chemical 3 22.73 kton/yr 19.60 kton/yr 25 Example – Solution with Max ENPV
Period 2
2.5 years
Chemical 5 4.71 kton/yr
Chemical 1 4.71 kton/yr Process 1 10.23 kton/yr 20.87 kton/yr
Process 3
Chemical 3 41.75 kton/yr Chemical 7 22.73 kton/yr Process 5 22.73 kton/yr Chemical 8 20.87 kton/yr Process 4 22.73 kton/yr Chemical 4
20.87 kton/yr 26 Example – Solution with Max ENPV
Period 3
3.5 years 14.95 kton/yr
Chemical 5 29.49 kton/yr
Chemical 1 44.44 kton/yr Process 2 Process 1 Chemical 6 29.49 kton/yr
80.77 kton/yr 80.77 kton/yr
Chemical 2
29.49 kton/yr 21.88 kton/yr
Chemical 7
Process 5 Process 3
Chemical 3 43.77 kton/yr Chemical 8 21.88 kton/yr 22.73 kton/yr 22.73 ton/yr Process 4 22.73 kton/yr Chemical 4
21.88 kton/yr 27 Example – Solution with Max ENPV
Risk
1
.0 PP solution 0.9
0.8
0.7
0.6
0.5
0.4 E[ NPV ] = 1140 M$ 0.3
0.2
0.1
0.0
250 500 750 1
000 1
250 1
500 1
750 2000 2250 2500 2750 3000 3250 NPV (M$) 28 Example – Risk Management Solutions
Risk
1
.0
0.9
0.8 Ω Ω increases 0.7 PP 0.6 500
600 0.5 700
800 0.4 900
1
000 0.3 1 00
1
1
200 0.2 1
300
1
400 0.1 1
500 0.0
250 500 750 1
000 1
250 1
500 1
750 2000 2250 2500 2750 3000 3250 NPV (M$) 29 Example – Risk Management Solutions
Risk
1
.0
0.9 Ω = 9 00
Ω = 1 100 ENPV = 908 0.8 PP
ENPV =1140 ENPV = 1074
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
250 500 750 1
000 1
250 1
500 1
750 2000 2250 2500 2750 3000 3250 NPV (M$) 30 Example – Risk Management Solutions
NPV PDF f (ξ)
0.0026
0.0024
0.0022 Ω = 9 00 0.0020
0.001
8
0.001
6
0.001
4
0.001
2
0.001
0
0.0008 Ω = 1100 PP 0.0006
0.0004
0.0002
0.0000
0 500 1
000 1
500 2000 2500 3000 NPV ( ξ , M $ ) 31 Example – Solution with Min DRisk(Ω=900)
DRisk(
Period 1
2 years  Same Flowsheet
 Different Capacities Chemical 5
5.59 kton/yr
Chemical 1
5.59 kton/yr Process 1
10.85 kton/yr
Chemical 7
19.30 kton/yr
Process 3 Chemical 3 22.37 kton/yr 19.30 kton/yr 32 Example – Solution with Min DRisk(Ω=900)
DRisk(
Period 2
2.5 years  Same Flowsheet
 Different Capacities Chemical 5
4.99 kton/yr
Chemical 1
4.99 kton/yr Process 1
10.85 kton/yr 20.85 kton/yr
Process 3
Chemical 3 Chemical 7 22.37 kton/yr Process 5
22.43 kton/yr Chemical 8
20.85 kton/yr 41.70 kton/yr
Process 4
22.37 kton/yr Chemical 4
20.85 kton/yr 33 Example – Solution with Min DRisk(Ω=900)
DRisk(
Period 3
3.5 years  Same Flowsheet
 Different Capacities
 No Expansion of Process 1
2.39 kton/yr
Chemical 5
5.15 kton/yr
Chemical 1
7.54 kton/yr Process 1 Process 2 Chemical 6
5.15 kton/yr 10.85 kton/yr 10.85 kton/yr
Chemical 2
5.15 kton/yr 21.77 kton/yr
Chemical 7
Process 5 Process 3
Chemical 3
43.54 kton/yr Chemical 8
21.77 kton/yr 22.37 kton/yr 22.77 ton/yr Process 4
22.37 kton/yr Chemical 4
21.77 kton/yr 34 RECENT RESULTS
Gas Commercialization in Asia
Network of Alternatives
Suppliers
Australia
Indonesia
Iran
Kazakhstan
Malaysia
Qatar
Russia “Transportation”
Methods
Pipeline Markets
China LNG India CNG Japan GTL S. Korea Ammonia Thailand Methanol United States
35 Gas Commercialization in Asia
Some solutions
Processing Facilities 1.0 OV @ 95%: 1.42 Time
Period
T1
T2
T3
T4
T5
T6 OV @ 95%: 1.75 0.9
0.8
0.7
0.6 MalaGTL
Ships: 4 & 2
ENPV:4.570
DR@ 4: 0.157
DR@ 3.5: 0.058 0.5
0.4
0.3 IndoGTL
Ships: 5 & 3
ENPV:4.633
DR@ 4: 0.190
DR@ 3.5: 0.086 0.2
0.1 VaR @ 5%: 1.82
1 2 VaR @ 5%: 1.49
3 4 5 6 7 FCI
3.00
0.00
1.89
0.00
0.00
0.00 Cap Flow Feed 0.00
4.57
4.57
7.49
7.49
7.49 0.00
4.47
4.57
7.32
7.49
7.49 0.0
297.9
304.9
488.2
499.6
499.6 Transportation to:
China 8 9 10 Time
Period
T1
T2
T3
T4
T5
T6 Flow Ships Flow Avrg.
Ships 0.00
1.16
0.00
0.42
0.00
0.00 0.00
0.98
0.00
0.35
0.00
0.00 0.00
2.79
3.66
5.58
6.00
6.00 0.00
3.49
4.57
6.97
7.49
7.49 0.00
3.95
3.66
6.00
6.00
6.00 4.0
4.0
6.0
6.0
6.0 Indo (GTL) FCI
3.00
0.00
1.90
0.00
0.00
0.00 Cap Flow Feed 0.00
4.43
4.43
7.18
7.18
7.18 0.00
4.25
4.43
7.09
7.18
7.18 0.0
283.1
295.5
472.6
479.0
479.0 Thai Ships Ships Processing Facilities 0.0
0 Mala (GTL) Transportation to:
China Thai Ships Ships Flow Ships Flow Avrg.
Ships 0.00
1.12
0.00
0.44
0.00
0.00 0.00
0.76
0.00
0.30
0.00
0.00 0.00
3.88
4.94
7.56
8.00
8.00 0.00
3.48
4.43
6.79
7.18
7.18 0.00
5.00
4.94
8.00
8.00
8.00 5.0
5.0
8.0
8.0
8.0 36 Conclusions
Risk is usually assessed after a design/plan has been selected but it cannot be
Risk
managed during the optimization stage (even when stochastic optimization
including uncertainty has been performed).
If Risk is to be managed, the decision maker has two simultaneous objectives:
If •
• Maximize Expected Profit.
Minimize Risk Exposure There are some good ways of obtaining good solutions (next lecture).
There 37 ...
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This note was uploaded on 08/31/2011 for the course CHE 4273 taught by Professor Staff during the Spring '10 term at Oklahoma State.
 Spring '10
 staff
 Chemical Engineering, pH

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