## Heat Pump Optimization Problem

## Main.MilkPasteurization History

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- Inlet temperature (degC)

- Inlet temperature (degC)

- Outlet temperature (degC)

- Outlet temperature (degC)

- Pasteurization Temperature (degC), target temperature

- Pasteurization Temperature (degC), target temperature

- Variables
- compressor work (kW)

- Variables
- compressor work (kW)

- heat exchanger areas (m^2)

- heat exchanger areas (m^2)

t1 = m.Var(value = ti) #Temperature 1 (degC) t2 = m.Var(value = ti) #Temperature 2 (degC) te = m.Var(value = ti) #Evaporator Temperature (degC) tc = m.Var(value = ti) #Condenser Temperature (degC) Ctot = m.Var(value = 100000) #Total Cost

t1 = m.Var(value = ti) # Temperature 1 (degC) t2 = m.Var(value = ti) # Temperature 2 (degC) te = m.Var(value = ti) # Evaporator Temperature (degC) tc = m.Var(value = ti) # Condenser Temperature (degC) Ctot = m.Var(value = 100000) # Total Cost

- exchanger), size of the compressor, and temperatures t1, t2, te and tc
- that result in the minimum total cost

- exchanger), size of the compressor, and temperatures t1, t2, te and tc
- that result in the minimum total cost

- areas of the heat exchangers (evaporator, condensers, and regenerative exchanger),
- size of the compressor, and temperatures t1, t2, te and tc that result in the
- minimum total costOptimize

- areas of the heat exchangers (evaporator, condensers, and regenerative
- exchanger), size of the compressor, and temperatures t1, t2, te and tc
- that result in the minimum total cost

**Solution Help**

See GEKKO documentation and additional example problems.

(:source lang=python:) from gekko import GEKKO

- Initialize Gekko model

m = GEKKO()

- Optimal heat pump and regenerative exchanger that minimizes the total present
- worth of costs (capital cost and operating cost). Specifically, determine the
- areas of the heat exchangers (evaporator, condensers, and regenerative exchanger),
- size of the compressor, and temperatures t1, t2, te and tc that result in the
- minimum total costOptimize
- Parameters

U = 0.6 Ureg = 0.5 mdotm = 4 cpm = 3.75 cpw = 1.00 intrate = 0.09 nper = 6

- Inlet temperature (degC)

ti = 7

- Outlet temperature (degC)

to = 4

- Pasteurization Temperature (degC), target temperature

tp = 73

- Variables
- compressor work (kW)

W = m.Var()

- heat exchanger areas (m^2)

Ae = m.Var(value = 10, lb = 1.0, ub = 1000) Areg = m.Var(value = 10, lb = 1.0, ub = 1000) Afc = m.Var(value = 10, lb = 1.0, ub = 1000) Aac = m.Var(value = 10, lb = 1.0, ub = 1000)

t1 = m.Var(value = ti) #Temperature 1 (degC) t2 = m.Var(value = ti) #Temperature 2 (degC) te = m.Var(value = ti) #Evaporator Temperature (degC) tc = m.Var(value = ti) #Condenser Temperature (degC) Ctot = m.Var(value = 100000) #Total Cost

- evaporator

Qe = m.Var(value = 1) dapp1e = m.Var(value = 1) dapp2e = m.Var(value = 1)

- regenerative heat exchanger

Qreg = m.Var(value = 1) dapp1reg = m.Var(value = 1) dapp2reg = m.Var(value = 1)

- fore condenser

Qfc = m.Var(value = 1) dapp1fc = m.Var(value = 1) dapp2fc = m.Var(value = 1)

- after condenser

Qac = m.Var(value = 1) dapp1ac = m.Var(value = 1) dapp2ac = m.Var(value = 1)

- temps of approach

dappe = m.Var(lb = 0.1) # >= 10 dappc = m.Var(lb = 0.1) # >= 10 dappreg1 = m.Var(lb = 0.1) # >= 10 dappreg2 = m.Var(lb = 0.1) # >= 10

- ratios

ratio_e = m.Var(value = .05, lb = 0.001, ub = 0.999) ratio_reg = m.Var(value = .05, lb = 0.000, ub = 1.0) ratio_fc = m.Var(value = .05, lb = 0.001, ub = 0.999) ratio_ac = m.Var(value = .05, lb = 0.001, ub = 0.999)

- Log-mean temperature differences

delte = m.Var(value = 1) deltreg = m.Var(value = 1) deltfc = m.Var(value = 1) deltac = m.Var(value = 1)

- Coefficient Of Performance

COP = m.Var(value = 1)

- Intermediates
- Cost elec year at 4 hours per day

Celyear= m.Intermediate(W * 4.0 * 365.0 * 0.07)

- Cost total over time
- Multiplier is about 4.5 (instead of 6.0) because of the
- interest rate on unspent capital

mult = m.Intermediate((((1.0+intrate)**nper)-1.0) / (intrate*(1.0+intrate)**nper)) Celtot = m.Intermediate(Celyear * mult)

- Cost equipment

Cequip = m.Intermediate(Ae*200.0 + Afc*200.0 + Aac*200.0 + Areg*200 + W*240.0)

- temperatures in Kelvin

TeK = m.Intermediate(te + 273) TcK = m.Intermediate(tc + 273)

- Equations

m.Equations([

# Total cost Ctot == Celtot + Cequip, # overall energy balance on regen heat exchanger # mdotm and cpm cancel from each term t1 + t2 == tp + ti, # log-mean temperature difference # lmtd(thi,tco,tho,tci) # evaporator Qe == mdotm * cpm *(t2 - to), dapp1e == to - te, dapp2e == t2 - te, # regenerative heat exchanger Qreg == mdotm * cpm * (tp-t2), dapp1reg == t2 - ti, dapp2reg == tp - t1, # fore condenser Qfc == mdotm * cpm * (tp - t1), dapp1fc == tc - tp, dapp2fc == tc - t1, # after condenser Qac == W + mdotm * cpm * (ti - to), dapp1ac == tc - 35.0, dapp2ac == tc - 30.0, # approach temperatures dappe == to - te, # Temp app evap dappc == tc - tp, # Temp app fore cond dappreg1 == t2 - ti, # Temp app reg1 dappreg2 == tp - t1, # Temp app reg2 # ratios ratio_e * dapp2e == dapp1e, ratio_reg * dapp2reg == dapp1reg, ratio_fc * dapp2fc == dapp1fc, ratio_ac * dapp2ac == dapp1ac, # Log-mean Temperature Difference delte * m.log(ratio_e) == (dapp1e - dapp2e), #deltreg * m.log(ratio_reg) = (dapp1reg - dapp2reg) deltfc * m.log(ratio_fc) == (dapp1fc - dapp2fc), deltac * m.log(ratio_ac) == (dapp1ac - dapp2ac), # Average temperature difference #delte = (dapp1e + dapp2e) / 2 deltreg == (dapp1reg + dapp2reg) / 2, #deltfc = (dapp1fc + dapp2fc) / 2 #deltac = (dapp1ac + dapp2ac) / 2 # coefficient of performance # 75% of Carnot Efficiency COP * (TcK-TeK) == 0.75 * TeK, # Work to compressor W * COP == Qe, # heat transferred for each exchanger Ae * U * delte == Qe, Areg * Ureg * deltreg == Qreg, Afc * U * deltfc == Qfc, Aac * U * deltac == Qac ])

- objective function

m.Obj(Ctot)

- minimize ctot

m.options.SOLVER = 1 m.options.IMODE = 3

m.solve()

print('Minimum cost: ' + str(Ctot[0])) (:sourceend:)

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(:title Heat Pump Optimization Problem:) (:keywords heat pump, nonlinear, optimization, engineering optimization, two-bar optimization, engineering design, interior point, active set, differential, algebraic, modeling language, university course:) (:description Engineering design and optimization of a heat pump system for pasteurization of milk. Optimization principles are used to design the system.:)

In the pasteurization of milk the temperature is raised to 73°C, held for 20 sec., and then cooled. The milk arrives at a temperature of 7°C and is delivered from the pasteurizing process for packaging at a temperature of 4°C.

We will consider using a heat pump with a regenerative heat exchanger to do this. One possible cycle is shown in the figure below. The incoming milk is preheated in a regenerative heat exchanger and then heated further in the fore-condenser of the heat pump. As it exits the fore-condenser, the temperature of the milk is 73°C. Thereafter the milk is cooled as it flows through the other side of the regenerative heat exchanger and then through the evaporator of the heat pump.

Find the optimal heat pump and regenerative exchanger that minimizes the total present worth of costs (capital cost and operating cost). Specifically, determine the areas of the heat exchangers (evaporator, condensers, and regenerative exchanger), size of the compressor, and temperatures t_{1}, t_{2}, t_{e} and t_{c} that result in the minimum total cost. Run the problem with no constraints on temperatures of approach (Case 1); then rerun with constraints that all temperatures of approach (evaporator, condensers, regenerative heat exchanger) are at least 10 °C (Case 2).

This problem is based on a problem from W. Stoecker, Design of Thermal Systems, 3rd ed.