PART 6 HEAT INTEGRATION AND MATHEMATICAL PROGRAMMING

Transcription

1 PART EAT INTEGRATION AND MATEMATIAL PROGRAMMING

2 PIN TENOLOGY SORTOMINGS It is a two-step procedure, where utility usage is determined first and the design is performed later. Trade-offs are optimized (through Super-targeting) using an approximation to the value of area (Vertical heat transfer) and using minimum number of exchangers. T N min = (S-P) above pinch + (S-P) below pinch

3 PIN TENOLOGY SORTOMINGS apital Investment trade-offs between area and number of units is resolved in favor of minimizing the number of units. All exchangers are subject to the same ΔT min (RAT). There is no shell counting No procedure is given for handling splitting away from the Pinch It is not systematic: After pinch matches are placed one is left using common sense built expertise Larger problems offer combinatorial challenges

4 PIN TENOLOGY SORTOMINGS Vertical Transfer takes place in regions close to the pinch, while non-vertical transfer takes place away from it to minimize the number of units. T T Options allowing criss-cross at the pinch may offer some advantages.

5 ALTERNATIVES Alternative models need to: Treat all the trade-offs simultaneously through costs (energy, area, fixed costs for units/shells) and move away from two step procedures and relaxations. Let the true economics determine the design Remove limitations on heat transfer (RAT), and possibly introduce exchanger minimum approximation temperatures (EMAT) to specific pairs of streams when appropriate or desired. Do not exhibit limitations like lack of control of splitting or isothermal mixing. Be systematic and potentially automatic. WE LOOK INTO MATEMATIAL PROGRAMMING FOR TIS PURPOSE

6 MATEMATIAL MODELS δ Mathematical model to calculate the minimum utility. q 1 q 2 q i δ 1 δ i Where S min st. j j1 j j Min q j 1,... n q i+1 δ i+1 q F cp ( T T ) F cp ( T T ) j i i j1 j k k j1 j i k q n δ n

7 MATEMATIAL MODEL s i,k,1 r i,1 p k,1 δ i,1 s i,k,2 r i,2 p k,2 Assume now that we do the same cascade for each hot stream, while we do not cascade the cold streams at all. In addition we consider heat transfer from hot to cold streams in each interval. s i,k,j The material balances for hot streams and utilities are: r i,j r i,j+1 δ i,j s i,k,,j+1 p k,j p k,j+1 i, U, i, j i, j 1 i, j i, k, j k r s j 1,... n r i,n δ i,j+1 δ i,n s i,k,n p k,n The heat balances for cold streams are: pk, j si, k, j j 1,... n i where r i,j and p k,j are the heat content of hot stream i and cold stream k in interval j.

8 MATEMATIAL MODEL s i,k,1 r i,1 p k,1 δ i,1 s i,k,2 r i,2 p k,2 r i,j r i,j+1 δ i,j s i,k,j s i,k,,j+1 p k,j p k,j+1 Although we have a simpler model to solve it, in this new framework, the minimum utility problem becomes: st. Min i, U, i U i, j i, j 1 i, j i, k, j k r s i U, j 1,... n p s k, j 1,... n k, j i, k, j i r i,n δ i,j+1 δ i,n si,k,n p k,n Note that the cascade equation for hot streams now includes the utility U as well. old streams include cooling water. More than one utility? Add it as another hot/cold stream

9 AN TIS GIVE A NETWORK? We use the following principle: Always satisfy cold stream requirements using hot process streams first. onsider the example given in Part I Δ=27 MW Δ=- MW T=14 T=2 Reactor 2 T=2 T=8 ΔT min =1 o Δ=2 MW Δ=- MW Reactor 1 T=2 T=18 T=25 T=4 Stream Type Supply T Target T Δ F*p ( o ) ( o ) (MW) (MW o -1 ) Reactor 1 feed old Reactor 1 product ot Reactor 2 feed old Reactor 2 product ot

10 ASADE The problem had the following characteristics Minimum heating utility Pinch Minimum cooling utility

11 TRANSSIPMENT We will look at the intervals above the pinch only U No arrows down here. We are at the pinch!!!

12 TRANSSIPMENT We start with the first hot stream (1). There is no cold stream in the first interval. Then we cascade everything down U

13 TRANSSIPMENT We now send to the cold stream and use utility cascaded to fulfill the units the cold stream 2 requires U

14 TRANSSIPMENT In interval there are two streams. We use 1 first and then 2 if needed U

15 TRANSSIPMENT We finish in interval 4 using 1 first with 1, then 2-1 followed by 2-2 and then U, which is now been augmented U We obtain the same utility!!!

16 TRANSSIPMENT U The flowsheet can be simplified. -The heater in 2 can be moved to higher temperatures and merged with the other, allowing removal of the split in 2. -The split in 2 and 1 cannot be removed simply by inspection The resulting suggested network is very complicated

17 TRANSSIPMENT U The PDM network is less complicated No splitting takes place because hot 1 cascades units to supply the 2.5 in the rd interval 2 2 We conclude that we need to minimize the number of matches between streams

18 OUNTING MATES s i,k,1 r i,1 p k,1 δ i,1 s i,k,2 r i,2 p k,2 We would like to have a model that would tell us the s i,k,j such that the number of units is minimum. We now introduce a way of counting matches between streams. Let Y i,k be a binary variable (can only take the value or 1). r i,j s i,k,j p k,j Then we can force Y i,k to be one using the following inequality δ i,j r i,j+1 s i,k,,j+1 p k,j+1 j s i, k, j Yi, k δ i,j+1 si,k,n indicating therefore that heat has been transferred from stream i to stream k in at least one interval. r i,n p k,n δ i,n

19 MATEMATIAL MODEL s i,k,1 r i,1 p k,1 δ i,1 s i,k,2 r i,2 p k,2 r i,j r i,j+1 δ i,j δ i,j+1 s i,k,j s i,k,,j+1 p k,j p k,j+1 The complete model would be: Min st. i, i k ik, i, j i, j 1 ri, j si, k, j i, j 1,... mi k p s k, j 1,... m k, j i, k, j I i j i s Y i, k i, k, j i, k Y r U,1 =Minimum utility from * previous model ( ) U, s i,k,n r i,n δ i,n p k,n The model can only be solved above and below the pinch separately. Why???

20 ANSWER We are minimizing the number of matches. We saw already that more than one exchanger can exist if all regions (above and below the pinch) are considered. Example: 1-1 has one match above the pinch and one match below the pinch. If solved ignoring the pinch (one region for the whole problem), then it would count these two exchangers as one match. When solved separately, it would give the right number. TUS, this model will ONLY give the right answer (Matches= Exchangers) if run in each region separately

21 GAMS MODEL st. z Min Y i, GAMS MODEL ik, i, j i, j 1 ri, j si, k, j i, j 1,... mi k p s k, j 1,... m k, j i, k, j I i j s Y i, k i, k, j i, k i i SETS I hot streams above pinch / U, 1,2 / K cold streams above pinch / 1,2 / J temperature intervals / J*J / ; SALAR GAMMA /1/; k TABLE R(I,J) load of hot stream I1 in interval K J1 J2 J J4 U ; TABLE P(K,J) load of cold stream K1 in interval J J J1 J2 J ; VARIABLES S(I,K,J) heat exchanged hot and cold streams D(I,J) heat of hot streams flowing between intervals Y(I,K) existence of match Z total number of matches ; POSITIVE VARIABLE S POSITIVE VARIABLE D BINARY VARIABLE Y ; EQUATIONS MINMAT objective function-number of matches SBAL1(I,J) heat balances of hot stream I in INTERVAL J ne 1 SBAL(I,J) heat balances of hot stream I in INTERVAL 1 SBAL(K,J) heat balances of cold stream J1 in K TINEQ1(I,K) heat transferred inequalities; MINMAT.. Z =E= SUM((I,K), Y(I,K)); SBAL1(I,J)$(ORD(J) NE 1).. D(I,J)-D(I,J-1)+ SUM(K,S(I,K,J)) =E= R(I,J); SBAL(I,J)$(ORD(J) EQ 1).. D(I,J)+SUM(K,S(I,K,J)) =E= R(I,J); SBAL(K,J).. SUM(I, S(I,K,J)) =E= P(K,J) ; TINEQ1(I,K).. SUM(J, S(I,K,J))-GAMMA*Y(I,K) =L= ; MODEL TSIP /ALL/ ; SOLVE TSIP USING MIP MINIMIZING Z; DISPLAY S.L, D.L, Y.L,Z.l;

22 SOLUTION TRANSSIPME VARIABLE S.L J1 J2 J U VARIABLE D.L J J1 J2 U VARIABLE Y.L 1 2 U VARIABLE Z.L = 4. EXEUTION TIME =.1 SEONDS U This result is different from the one given by the PDM. owever, it predicts correctly the number of exchangers.

23 TRANSSIPMENT U Splitting of 2 takes place because hot 1 and 2 give to 2 in the last interval. 2 values cannot be moved to a sequential arrangement because the cascade is zero. Finally, 2and 2 split at the last interval. With this solution, this cannot be changed. Some form of limiting splitting is needed

24 TRANSSIPMENT By ad-hoc rule 5 Matches- 8 exchangers BAD!!! PDM 4 Matches- 4 exchangers RIGT ANSWER Mathematical Programming 4 Matches- exchangers BETTER Some way of controlling splitting AND the number of units is needed In addition, area cost is not included

25 TRANSSIPMENT We conclude that the transshipment model - an calculate Minimum utility and predict the number of matches. - The number of matches are equal to the number of exchangers, but. - The model may (and usually does) give more exchangers, than matches. Some way of controlling splitting AND the number of units is needed as well adding area cost is needed. This is done in the next MILP model

26 TRANSPORTATION MODELS Streams are divided in small temperature intervals eat can be sent to any interval of lower temperature. m z, qˆ ijm q nm j z z n L ; Tn T ; ip jn U m z im, jn ˆ z q, ijm z q im, jn z, z im qim, jn z z nm j n L U Tn Tm jpim z, z ipjn q ˆijn qim, jn z L U mm ;Tn Tm z i ; jp m im ˆ z q, ijn n z, z jn qim, jn z z mm i m L U Tn Tm ip n jp m Barbaro and Bagajewicz (25)

27 TRANSPORTATION MODELS The model IS LINEAR!!!! ounts heat exchangers units and shells. Determines the area required for each exchanger unit or shell. ontrols the total number of units. Determines the flow rates in splits. andles non-isothermal mixing. Identifies bypasses in split situations when convenient. ontrols the temperature approximation (ΔTmin) when desired. an address areas or temperature zones. Allows multiple matches between two streams

28 TRANSPORTATION MODELS º º º 4 18 º º º º º º º º 7.2 º º.8 11 º º º º º º º º º SP1 MODEL STATISTIS SINGLE VARIABLES 1428 DISRETE VARIABLES 24 TIME TO REA A FEASIBLE SOLUTION TIME TO REA GLOBAL OPTIMALITY 4 s 2 s

29 Stages Superstructure Model Assumptions: - eating and cooling takes place at the end. - Isothermal mixing. Superstructure for stages, two hot (1,2) and two cold (1,2) streams

30 Stages Superstructure Model At each stage every hot/cold stream splits and matches with all other (cold/hot) stream. 1 and 1 are highlighted

31 Stages Superstructure Model Variables t ik, Temperature of hot stream i in stage k ( the same for all exits of heat exchangers) TIN i Inlet Temperature of hot stream i (parameter) TOUT j Outlet Temperature of cold stream j (parameter) qijk eat transferred from hot stream i to cold stream j in stage k. This heat corresponds to an exchanger!!! t jk, Temperature of cold stream j in stage k ( the same for all exits of heat exchangers) TOIUT i Outlet temperature of hot stream i (parameter) TIN j Inlet Temperature of cold stream j (parameter)

32 Stages Superstructure Model Overall eat balance for each stream. ( TIN TOUT ) F q qcu i P eat balance at each stage i i i ijk i kst jp ( TOUT TIN ) F q qhu j P j j j ijk j kst ip ( t t ) F q i P, k ST ik i( k1) i ijk jp ( t t ) F q j P, k ST jk j( k1) j ijk ip Assignment of Superstructure inlet temperatures TIN t i P i j i1 TIN t j P j( NOK 1) eat load for utilities Stage 1 exchangers involved in stage balance for hot stream 1 Stage 2 exchangers involved in stage balance for cold stream 2 NOK is the last stage

33 Stages Superstructure Model Equations Feasibility ik jk ot and cold utility load t t i P k ST i( k1), j( k1), t t j P k ST TOUT t j P j i j,1 TOUT t i P i( NOK 1) Temperatures decrease from stage to stage ( ti( NOK 1) TOUT i ) Fi qcui i P ( TOUT t ) F qhu j P j j1 j j Logical onstraints (to count exchangers) q z i P, j P, k ST ijk ijk qcu zcu i P i i z ijk, zcu, i zhu j are binary variables q ijk z 1 ijk qhu zhu j P j j

34 Stages Superstructure Model Equations alculation of approach temperatures dtijk ti, k t j, k (1 zijk ) ip, j P, k ST dtij ( k1) ti,( k1) tj,( k1) (1 zijk ) ip, j P, k ST dtcui ti,( NOK 1) TOUT U (1 zcui ) ip dthu j TOUT U t j,1 (1 zhu j ) j P Limiting temperature approach dt EMAT ip, j P, k ST ijk When z ijk =1, then EMAT dt t t making ti, k tj, k positive and bounded from below ijk i, k j, k Objective function (Utility costs+ Fixed EX installation costs) O MinU qcu U qhu F z F zcu F zhu i j ij ijk iu i ju j ip jp ip jp kst ip jp Notice that one could also add area cost, but the model will be nonlinear

35 Stages Superstructure Model Equations The model is linear One can add area cost but then more equations are needed and the model will be nonlinear. Using the linear model and varying the weight of the fixed costs, one can obtain solutions exhibiting different energy usage and evaluate them

36 Stages Superstructure Model List of parameters and variables