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Simultaneous optimisation of heat and power integration of evaporation–crystallisation systems: a case study of distiller waste from Solvay process

Abstract

This paper addresses the synthesis of combined evaporation–crystallisation systems for the recovery of valuable materials from waste in line with sustainable development and circular economy concepts. The primary focus of this work is the utilisation of distiller waste from the Solvay process, which comprises sodium chloride (NaCl), calcium chloride (\({\text {CaCl}}_2\)) and water (\(\hbox{H}_2\)O). The superstructure optimisation of a heat and power integrated evaporation–crystallisation system is performed by solving the proposed Mixed-Integer Nonlinear Programming (MINLP) model. The superstructure extends our recent research to include the partial crystallisation of NaCl and the production of concentrated \({\text {CaCl}}_2\) solution. To address the considered case study, a thermodynamic model for multi-component electrolytic systems is developed. A three-step solution strategy is proposed to circumvent a problem with nonlinearities and to solve the overall MINLP model. The optimal design of a heat-integrated evaporation–crystallisation system with mechanical vapour compression is presented and the main conclusions are highlighted.

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Abbreviations

CEPCI:

Chemical Engineering Plant Cost Index

GAMS:

General Algebraic Modeling System

GCC:

Grand Composite Curves

HEN:

Heat Exchanger Network

MEE:

Multiple Effect Evaporation

MEECRY:

Multiple Effect Evaporation/Crystallisation

MINLP:

Mixed Integer Nonlinear Programming

MP:

Mathematical Programming

MVC:

Mechanical Vapour Compression

NLP:

Nonlinear Programming

PA:

Pinch Analysis

TAC:

Total Annualised Cost

TEC:

Total Energy Cost

TVC:

Thermal Vapour Compression

U:

Upper bound

bpe:

Boiling point elevation

cnd:

Condensate

cry:

Crystals

CU:

Cold utility

cv:

Compressor vapour

f:

Feed

fs:

Fresh steam

fv:

Flash vessel

hs:

Heating steam

HU:

Hot utility

l:

Liquid

ml:

Mother liquor

sat:

Saturated

v:

Vapour

comp:

Compressor

evap:

Evaporator

flsh:

Flash vessel

hen:

Heat exchanger network

in:

Inlet stream

l:

Liquid

lin:

Linear

nonlin:

Nonlinear

out:

Outlet stream

sat:

Saturator-saturated

superheat:

Superheated

v:

Vapour

c :

Component

e :

Evaporator

i :

Hot stream

j :

Cold stream

k :

Temperature location within HEN

nok :

Number of stages in HEN superstructure

\({\text {CP}}\) :

Set of cold streams

\({\text {C}}\) :

Set of components

\({\text {EV}}\) :

Set of evaporators

\({\text {HP}}\) :

Set of hot streams

\({\text {ST}}\) :

Set of HEN stages

\(\eta \) :

Compressor efficiency, %/100

\(\varGamma \) :

Upper bound on driving force for the heat exchange, \(^{\circ }\hbox{C}\)

\({\text {LMTD}}_{i,CU}\) :

Logarithmic mean temperature difference between hot stream i and cold utility, \(^{\circ }{\text {C}}\)

\({\text {LMTD}}_{i,j,k}\) :

Logarithmic mean temperature difference between hot stream i and cold stream j at temperature location k\(^{\circ }{\text {C}}\)

\({\text {LMTD}}_{j,HU}\) :

Logarithmic mean temperature difference between cold stream j and hot utility, \(^{\circ }{\text {C}}\)

\({\text {match}}_{i,j,k}\) :

Binary parameter for resricting the number of heat exchange matches, –

\(a_{\text {comp}},b_{\text {comp}}\) :

Cost coefficients for compressor investment, –

\(a_{\text {evap}},b_{\text{evap}}\) :

Cost coefficients for evaporator investment, –

\(a_{\text {exc}},b_{\text {exc}}\) :

Cost coefficients for heat exchanger investment, –

\(a_{\text {flsh}},b_{\text {flsh}}\) :

Cost coefficients for flash vessel investment, –

\(a_{{\text {w}},e}\) :

Water activity of the solution

\(A_{i,CU}\) :

Cooler heat exchange area parameter, \(\hbox{m}^2\)

\(A_{i,j,k}\) :

Heat exchanger area parameter, \(\hbox{m}^2\)

\(A_{j,HU}\) :

Heater heat exchange area parameter, \(\hbox{m}^2\)

AF :

Annualisation factor for the investment, 1/y

\(c_p(t)\) :

Specific heat capacity, J/(kg K)

\(c_{\text {comp}},d_{\text {comp}}\) :

Cost coefficients for compressor investment, –

\(C_{\text{CU}}\) :

Cold utility cost for HEN requirements, $/(kW \(^{\circ }\text {C}\))

\(C_{\text {el}}\) :

Electricity cost, $/kWh

\(c_{\text {flsh}},d_{\text {flsh}}\) :

Cost coefficients for flash vessel investment, –

\(C_{\text {fs}}\) :

Fresh steam cost, $/kg

\(C_{\text{HU}}\) :

Hot utility cost for HEN requirements, $/(kW \(degr{\text{C}}\))

\(c_{p,0}\) :

Specific heat capacity of water, J/(kg K)

\(e_{\text{flsh}},f_{\text{flsh}}\) :

Cost coefficients for flash vessel investment, –

\(f_c\) :

Conversion factor of annual plant operating hours

H :

Plant operating hours per year, h

\(h_{\text{sol}}\) :

Specific enthalpy of electrolytic solution J/kg

\(h_{\text{CU}}\) :

Individual heat transfer coefficient for cold utility, kW/(\(\hbox{m}^2\,^{\circ }\text {C}\))

\(h_{\text{HU}}\) :

Individual heat transfer coefficient for hot utility, kW/(\(\hbox{m}^2\,^{\circ }\text {C}\))

\(h_{i}\) :

Individual heat transfer coefficient for hot stream i, kW/(\(\hbox{m}^2\,^{\circ }\text {C}\))

\(h_{j}\) :

Individual heat transfer coefficient for cold stream j, kW/(\(\hbox{m}^2\,^{\circ }\text {C}\))

\(m_{\text {f}}\) :

Feed mass flow rate, kg/s

\(p_{evap}(t)\) :

Evaporation pressure at system temperature, bar

\(p_{w}^{*}(t)\) :

Water vapour pressure at system temperature, bar

\(q_{{\text {cry, NaCl}}}\) :

Specific heat of crystallisation, kJ/kg

\(t\) :

Temperature of the process stream, \(^{\circ }\hbox{C}\)

\(t_{\text{ref}}\) :

Reference temperature, \(^{\circ }\hbox{C}\)

\(U_e\) :

Overall heat transfer coefficient within the evaporator e, kW/(\(\hbox{m}^{2}\,^{\circ }\hbox{C}\))

\(U_{i,{\text{CU}}}\) :

Overall heat transfer coefficient between hot stream i and cold utility, kW/(\(\hbox{m}^2\,^{\circ }\text {C}\))

\(U_{i,j}\) :

Overall heat transfer coefficient between hot stream i and cold stream j, kW/(\(\hbox{m}^2\,^{\circ }\text {C}\))

\(U_{j,{\text{HU}}}\) :

Overall heat transfer coefficient between cold stream j and hot utility, kW/(\(\hbox{m}^2\,^{\circ }\text {C}\))

\(x_{{\text {f}},c}\) :

Mass fraction of component c in the feed stream, –

\({c_i}\) :

Mass fraction of component i

\(\varDelta t_{{\text {bpe}},e}\) :

Boiling point elevation at the evaporator e, \(^{\circ }\hbox{C}\)

\(\varDelta t_{e}\) :

Mean temperature difference at the evaporator e, \(^{\circ }\hbox{C}\)

\(\varDelta t_{i,\text {CU}}\) :

Temperature approach between hot stream i and cold utility, \(^{\circ }\hbox{C}\)

\(\varDelta t_{i,j,k}\) :

Temperature approach between hot stream i and cold stream j at temperature location k, \(^{\circ }\hbox{C}\)

\(\varDelta t_{j,{\text {HU}}}\) :

Temperature approach between cold stream j and hot utility, \(^{\circ }\hbox{C}\)

\(\rho _e\) :

Condensate density at the flashing stage e , kg/(\(\hbox{m}^3\))

\({\text {ecc}}_j\) :

Heat content of cold stream j, kW

\({\text {ech}}_i\) :

Heat content of hot stream i, kW

\(\varepsilon _{{\text {cry}},e}\) :

Mass fraction of crystals in the suspension leaving the last evaporation stage e, –

\(a_{{\text {w}},e}\) :

Water activity of the evaporator e, –

\(A_{e}\) :

Heat transfer area of the evaporator e, \(\hbox{m}^2\)

\(CR_{e}\) :

Compression ration at the compression stage e

\(f_i\) :

Heat capacity flow rate of hot stream i, kW/\(^{\circ }\hbox{C}\)

\(f_j\) :

Heat capacity flow rate of cold stream j, kW/\(^{\circ }\hbox{C}\)

\(h_{{\text {as}},e}\) :

Specific enthalpy of additional heating steam at evaporator e, kJ/kg

\(h_{{\text {cnd}},e}\) :

Specific enthalpy of hot condensate leaving evaporator e , kJ/kg

\(h_{{\text {cnd}},e}^{(\text {hen,out})}\) :

Specific enthalpy of codensate after secondary steam condensation within HEN e, kJ/kg

\(h_{{\text {cnd}},e}^{({\text {out}})}\) :

Specific enthalpy of the condensate after heat integration, kJ/kg

\(h_{\text {cnd}}^{(\text {colect,in})}\) :

Specific enthalpy of condensate at the collection point corresponding to the last evaporation stage, kg/s

\(h_{{\text {cry}},e}^{({\text {out}})}\) :

Specific enthalpy of crystals within the suspension, kJ/kg

\(h_{{\text {cv}},e}^{({\text {in}})}\) :

Specific enthalpy of compressor inlet stream, kJ/kg

\(h_{{\text {cv}},e}^{(\text {out,ideal})}\) :

Specific enthalpy of compressor outlet stream (ideal compression), kJ/kg

\(h_{{\text {cv}},e}^{({\text {out}})}\) :

Specific enthalpy of compressor outlet stream, kJ/kg

\(h_{{\text {cv}},e}^{(\text {sat})}\) :

Specific enthalpy of re-compressed steam after saturation, kJ/kg

\(h_{\text {fs}}\) :

Specific enthalpy of fresh heating steam entering evaporator \(e=1\) , kJ/kg

\(h_{{\text {fv}},e}^{({\text {in}})}\) :

Specific entalpy of condensate entering flash vessel e, kJ/kg

\(h_{{\text {fv}},e}^{(\text {l})}\) :

Specific enthalpy of condensate leaving flash vessel e, kJ/kg

\(h_{{\text {fv}},e}^{(\text {v})}\) :

Specific enthalpy of vapour leaving flash vessel e, kJ/kg

\(h_{\text {f}}\) :

Specific enthalpy of the system feed, J/kg

\(h_{\text {f}}^{({\text {out}})}\) :

Specific enthalpy of the system feed after preheating, kJ/kg

\(h_{{\text {hs}},e}\) :

Specific enthalpy of heating steam entering evaporator e, kJ/kg

\(h_{{\text {l}},e}^{({\text {in}})}\) :

Specific enthalpy of evaporator liquid inlet stream, kJ/kg

\(h_{{\text {l}},e}^{({\text {out}})}\) :

Specific enthalpy of evaporator liquid outlet stream, kJ/kg

\(h_{{\text {ml}},e}^{({\text {out}})}\) :

Specific enthalpy of mother liquor within the suspension, kJ/kg

\(h_{{\text {v}},e}\) :

Specific enthalpy of the evaporator e vapour outlet stream, kJ/kg

\(m_{{\text {as}},e}\) :

Mass flow rate of additional heating steam for evaporator \(e\lnot 1\) , kg/s

\(m_{{\text {cnd}},e}\) :

Mass flow rate of condensate leaving the evaporator e, kg/s

\(m_{{\text {cnd}},e}^{(\text {flsh})}\) :

Mass flow rate of condensate leaving the evaporator e and directed to flash vessel e, kg/s

\(m_{{\text {cnd}},e}^{(\text {hen})}\) :

Mass flow rate of condensate leaving the evaporator e and directed to heat integration at stage e, kg/s

\(m_{{\text {cnd}},e}^{(\text {sat})}\) :

Mass flow rate of condensate leaving the evaporator e and directed to de-saturation unit (saturator e), kg/s

\(m_{\text {cnd}}^{(\text {colect})}\) :

Mass flow rate of condensate at the collection point corresponding to the last evaporation stage, kg/s

\(m_{{\text {cry}},e}^{({\text {out}})}\) :

Mass flow rate of crystals within the suspension from evaporator e, kg/s

\(m_{{\text {cv}},e+1}^{(\text {comp})}\) :

Mass flow rate of vapour at stage e directed to additional compression at compression stage \(e-1\), kg/s

\(m_{{\text {cv}},e}^{(\text {evap})}\) :

Mass flow rate of re-compressed vapour from compressor e for heating evaporator e, kg/s

\(m_{{\text {cv}},e}^{({\text {in}})}\) :

Mass flow rate of compressor inlet at stage e, kg/s

\(m_{{\text {cv}},e}^{({\text {out}})}\) :

Mass flow rate of compressor outlet at stage e, kg/s

\(m_{{\text {cv}},e}^{(\text {sat})}\) :

Mass flow rate of saturated vapour leaving de-superheater, kg/s

\(m_{\text {cv}}^{(\text {hen})}\) :

Mass flow rate of recompressed vapour from the first effect \(e=1\) directed to HEN for heat integration, kg/s

\(m_{\text {fs}}\) :

Mass flow rate of fresh heating steam entering first evaporator, kg/s

\(m_{{\text {fv}},e}^{({\text {in}})}\) :

Mass flow rate of condensate entering flash vessel e, kg/s

\(m_{{\text {fv}},e}^{(\text {l,out})}\) :

Mass flow rate of condensate leaving flash vessel e, kg/s

\(m_{{\text {fv}},e}^{(\text {v,out})}\) :

Mass flow rate of vapour leaving flash vessel e, kg/s

\(m_{{\text {hs}},e}\) :

Mass flow rate of heating steam entering evaporator e, kg/s

\(m_{{\text {l}},e}^{({\text {in}})}\) :

Mass flow rate of liquid inlet stream to evaporator e, kg/s

\(m_{{\text {l}},e}^{({\text {out}})}\) :

Mass flow rate of liquid outlet stream from evaporator e, kg/s

\(m_{{\text {ml}},e}^{({\text {out}})}\) :

Mass flow rate of mother liquor leaving the last evaporation stage, kg/s

\(m_{{\text {v}},e}\) :

Mass flow rate of vapour outlet stream from evaporator e, kg/s

\(m_{{\text {v}},e}^{(\text {bar})}\) :

Mass flow rate of vapour from evaporator e directed to barometric condenser, kg/s

\(m_{{\text {v}},e}^{(\text {comp})}\) :

Mass flow rate of vapour from evaporator e directed to vapour re-compression, kg/s

\(m_{{\text {v}},e}^{(\text {evap})}\) :

Mass flow rate of vapour from evaporator e to evaporator \(e+1\) , kg/s

\(m_{{\text {v}},e}^{(\text {hen})}\) :

Mass flow rate of vapour from evaporator e for heat integration within the HEN, kg/s

\(m_{fv,e}^{(\text {l,out})}\) :

Mass flow rate of condensate leaving flash vessel e, kg/s

\(N_{e}\) :

Compressor power at stage e, kJ/s

\(p_{\text {fs}}\) :

Fresh steam pressure at the first evaporator \(e=1\), bar

\(p_{{\text {fv}},e}\) :

Compressor e outlet pressure, bar

\(p_{e}\) :

Operating pressure of the evaporator e, bar

\(q_{{\text {tr}},e}\) :

Heat transferred in the evaporator e , kJ/s

\(q_{i,{CU}}\) :

Heat exchanged between hot stream i and cold utility, kW

\(q_{i,j,k}\) :

Heat exchanged between hot stream i and cold stream j within stage k, kW

\(q_{j,{HU}}\) :

Heat exchanged between cold stream j and cold utility, kW

\(s_{{\text {v}},e}\) :

Specific entropy of vapour leaving evaporator e, kJ/(kg \(^{\circ }\hbox{C}\))

\(t_{\text {fs}}\) :

Fresh heating steam temperature at the first evaporation stage, \(^{\circ }\hbox{C}\)

\(t_{{\text {fv}},e}\) :

Flash vessel operating temperature at the stage e, \(^{\circ }\hbox{C}\)

\(t_{\text {f}}\) :

Feed stream temperature, \(^{\circ }\hbox{C}\)

\(t_{\text {f}}^{({\text {out}})}\) :

Feed stream temperature after preheating, \(^{\circ }\hbox{C}\)

\(t_{\text {ref}}\) :

Reference temperature for the enthalpy calculation, \(^{\circ }\hbox{C}\)

\(t_{{\text {sat}},e}\) :

Water saturation temperature at the evaporation stage e, \(^{\circ }\hbox{C}\)

\(t_{e}\) :

Operating temperature of the evaporator e, \(^{\circ }\hbox{C}\)

\(t_{e}^{({\text {in}})}\) :

Evaporator e inlet temperature, \(^{\circ }\hbox{C}\)

\(t_{i,k}\) :

Temperature of hot stream i at temperature location k, \(^{\circ }\hbox{C}\)

\(t_{i}^{({\text {in}})}\) :

Inlet temperature of hot stream i, \(^{\circ }\hbox{C}\)

\(t_{i}^{({\text {out}})}\) :

Outlet temperature of hot stream i, \(^{\circ }\hbox{C}\)

\(t_{j,k}\) :

Temperature of cold stream j at temperature location k, \(^{\circ }\hbox{C}\)

\(t_{j}^{({\text {in}})}\) :

Inlet temperature of cold stream j, \(^{\circ }\hbox{C}\)

\(t_{j}^{({\text {out}})}\) :

Outlet temperature of cold stream j, \(^{\circ }\hbox{C}\)

\(V_{{\text {fv}},e}\) :

Flash vessel e volume, \(\hbox{m}^3\)

\(x_{{\text {cry}},e,c}^{({\text {out}})}\) :

Mass fraction of component c in the crystals at the last evaporation stage e, –

\(x_{{\text {l}},e,c}^{({\text {in}})}\) :

Mass fraction of component c in the evaporator e inlet stream, –

\(x_{{\text {l}},e,c}^{({\text {out}})}\) :

Mass fraction of component c in the evaporator e outlet stream, –

\(x_{{\text {ml}},e,c}^{({\text {out}})}\) :

Mass fraction of component c in the mother liquor at the last evaporation stage e, –

\(z_{{\text {comp}},e}\) :

Binary variable for the existence of compressor e

\(z_{{\text {fv}},e}\) :

Binary variable for the existence of flash vessel e

\(z_{i,{\text {CU}}}\) :

Binary variables for the existence of match between hot stream i and cold utility, –

\(z_{i,j,k}\) :

Binary variables for the existence of match between hot stream i and cold stream j within stage k, –

\(z_{j,{\text {HU}}}\) :

Binary variables for the existence of match between cold stream j and hot utility, –

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Acknowledgements

The authors gratefully acknowledge the support from the programme for Scientific and Technological Cooperation between the Republic of Slovenia and Bosnia and Herzegovina in the period 2019–2020, the Ministry of Education, Science and Sport of the Republic of Slovenia and the Federal Ministry of Education and Science of Bosnia and Herzegovina (Project: “SISUMP”, No: 01-1282-1/20). Also, the support from the internal call for financing/co-financing projects important for the Federation of Bosnia and Herzegovina (Project No: 01-6211-1-IV/19) and the support from the Slovenian Research Agency (P2-0032 and J7-1816). Professor Elvis Ahmetović and Assistant Professor Nidret Ibrić would like to express a posthumous acknowledgement to their teacher and colleague Professor Emeritus Midhat Suljkanović, who recently passed away, for his great contribution and support in their professional developments, and inspiration related to chemical engineering education and research, especially in evaporation and crystallisation processes.

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Correspondence to Nidret Ibrić.

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Ibrić, N., Ahmetović, E. & Kravanja, Z. Simultaneous optimisation of heat and power integration of evaporation–crystallisation systems: a case study of distiller waste from Solvay process. Optim Eng 22, 1853–1895 (2021). https://doi.org/10.1007/s11081-021-09641-z

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