Thermoeconomics

CHAPTER

THERMOECONOMICS

5

5.1 INTRODUCTION Thermodynamics principles describe the flow, conservation, and conversion of energy, and hence have implications for energy management and economics. The economics of processes always involve matter, energy, entropy, and information, and the consideration of economics leads to certain structures with minimum overall costs. Thermodynamic formulations impose directions and limits on the probability of processes; they also imply the use of scarce resources and compare the efficiencies of conversion between different kinds of energies, which may be a necessary step in net energy analyses and energy policy discussions. Thermal systems involve significant work and/or heat interactions with their surroundings and appear in almost every industrial plant. Consequently, the design of thermal systems requires the application of principles from thermodynamics, fluid mechanics, heat transfer, and engineering economics. Thermoeconomics combines thermodynamic principles with economic analysis and brings some fundamental changes in the economic evaluation, design, and maintenance of processes. The process engineer should minimize the input cost of a process by reducing exergy loss due to thermodynamic imperfections. Taking the perspective of exergy that measures the departure of the state of the system from that of the environment, thermodynamic analysis considers the interrelations among energy, economy, and ecology. Such considerations may have a positive impact on sustainable development and environmental protection. For example, a thermodynamic analysis of a solar desalination unit shows that the thermoeconomic evaluation of the system is closely related to a complete economic analysis of the possible improvements leading to a unit in which fewer irreversible processes occur.

5.2 THERMODYNAMIC COST Thermoeconomics assigns costs to exergy-related variables by using the exergy cost theory and exergy cost balances, and mass, energy, exergy, and cost considerations can be unified by a single formulation. There are two main groups of thermoeconomic methods: (1) cost accounting methods, such as exergy cost theory for a rational price assessment, and (2) optimization by minimizing the overall cost, under a proper set of financial, environmental, and technical constraints, to identify the optimum design and operating conditions. Cost accounting methods use average costs as a basis for a rational price assessment, while optimization methods employ marginal costs to minimize the costs of the Nonequilibrium Thermodynamics. https://doi.org/10.1016/B978-0-444-64112-0.00005-8 Copyright © 2019 Elsevier B.V. All rights reserved.

267

268

CHAPTER 5 THERMOECONOMICS

products of a system or a component. Extended exergy accounting considers nonenergetic costs, such as financial, labor, and environmental remediation costs, as functions of the technical and thermodynamic parameters of systems. Since exergy is a measure of thermodynamic work and available power within the system, it is a true and rational basis for assigning monetary costs. Therefore, the exergy costing is the main aspect of thermoeconomics. The cost of fuel is associated with economic value for its exergy. Combining exergy and economic analysis helps optimizing design and operation of thermal systems. Consider a heat exchanger; the average temperature difference DTlm between hot and cold streams is a measure of irreversibility, which vanishes as DTlm / 0. The cost of fuel increases with increasing DTlm, while the capital cost decreases. As seen in Fig. 5.1, the total cost consisting of fuel and capital costs between points “a” and “b” would be optimum. For minimizing capital cost, the optimum would be toward point “b,” and for minimizing fuel cost, the optimum approaches “a.” Some concerns in thermoeconomics evaluations are: • • •

Costs of fuel and equipment change with time and location. Optimization of an individual process does not guarantee an overall optimum for the system due to interactions among various processes. For the whole system, often several design variables should be considered and optimized simultaneously.

5.2.1 THERMODYNAMIC ANALYSIS AND THERMOECONOMICS Thermodynamic analysis can lead to a better understanding of the systems overall performance, and eventually to identifying the sources of losses due to irreversibilities in each process in the system. This will not guarantee that economical and useful process modifications or operational changes would be undertaken; the relationship between energy efficiency and capital cost must be based on an analysis of the overall plant system. Mainly, thermodynamic analysis methods of pinch analysis, exergy analysis, second law analysis, and equipartition principles are combined to analyze process and energy systems. These will enable engineers to modify existing systems or design new systems with complete

FIGURE 5.1 Annual cost optimization. Annual cost

Total cost a Capital cost

Fuel ΔTlm (NTU )

b

5.2 THERMODYNAMIC COST

269

objectives and targets by taking into consideration of environment, economics, society, and natural resources (Demirel, 2016). To account for the environmental impact in a more systematic way, a resource-based quantifier, called “extended exergy,” estimates the resource-based value of a commodity. Consider a separation process with outputs containing hot streams with various chemicals having conditions considerably different from environmental temperatures and concentrations. To achieve zero environmental impact, there is a real (exergetic) cost of the zero-impact that corresponds to the extended exergy ideally required to bring the conditions of effluents to both thermal and chemical equilibrium conditions with the surroundings. If an acceptable level of pollutant or the “tolerable environmental impact limit” for a certain pollutant is specified, then the environmental cost may be quantified.

5.2.2 EXERGY COST For any process or subsystem i, the specific cost of exergy c in $/kW-unit time for a stream is
_ c ¼ C_ Ex (5.1) _ are the cost rate and the rate of exergy transfer for a stream, respectively. However, where C_ and Ex the cost of a product and other exiting streams would include the contribution of fixed capital investment C_FCI and the annual operating cost of process C_OP . This will be called the total cost: C_P ¼ C_OP þ C_FCI . Then the cost rate balance is ! ! X X _ _ ci Exi ¼ ci Exi þ C_P (5.2) i

out

i

in

For example, consider the exergy costing on a boiler and turbine system shown in Fig. 5.2. The cost rate balance for the boiler (control volume 1) relates the total cost of producing high-pressure steam to the total cost of the entering streams plus the cost of the boiler C_B, and from Eq. (5.2) we have _ HP þ cEG Ex _ EG ¼ cF Ex _ F þ cA Ex _ A þ cW Ex _ W þ C_B cHP Ex

(5.3)

FIGURE 5.2

Exhaust gases

A two-unit system of a boiler and turbine.

Fuel

Boiler

Turbine Air Feedwater Exhausted steam

Work

270

CHAPTER 5 THERMOECONOMICS

where the symbol HP denotes the high-pressure steam, EG the exhaust gas, while F, A, and W are the fuel, air, and water, respectively. All the cost estimations are based on exergy as a measure of the true values of work, heat, and other interactions between a system and its surroundings. By neglecting the costs of air and water, and if the combustion products are discharged directly into the surroundings with negligible cost, Eqs. (5.1) and (5.3) yield the specific cost of high-pressure steam (product)   _ F Ex C_B cHP ¼ cF (5.4) þ _ HP _ HP Ex Ex 
 _ F Ex _ HP 1 due to the inevitable exergy loss is in the boiler, and hence cHP > cF . The ratio Ex Similarly, the cost rate balance for the turbine (control volume 2) is _ LP ¼ cHP Ex _ HP þ C_T cE W_ E þ cLP Ex where cE ; cLP ; and C_T are the specific costs of electricity, low-pressure exhaust steam, and the total _ LP are the work produced by the turbine and the exergy cost of the turbine, respectively; W_ E and Ex transfer rate of low-pressure steam, respectively. If the specific costs of low and high-pressure steams are the same cLP ¼ cHP , we have   _ HP  Ex _ LP Ex C_T (5.5) cE ¼ cHP þ W_ E W_ E
  _ HP  Ex _ LP , Eq. (5.5) becomes Using the exergetic efficiency of turbine hex ¼ W_ E Ex cE ¼

cHP C_T þ hex W_ E

(5.6)

As hex < 1, the specific cost of electricity (product) will be higher than that of high-pressure steam.

EXAMPLE 5.1 COST OF POWER GENERATION A turbine produces 30 MW of electricity per year. The average cost of the steam is $0.017/(kWh) of exergy (fuel). The total cost of the unit (fixed capital investment and operating costs) is $1.1  105. If the turbine exergetic efficiency increases from 84% to 89%, after an increase of 2% in the total cost of the unit, evaluate the change of the unit cost of electricity (Moran and Shapiro, 2000). Solution: Assume that heat transfer effects between the turbine and surroundings are negligible. Also, kinetic and potential energy effects are disregarded. From Eq. (5.6), we have cE ð84%Þ ¼

cHP C_T 0:017 1:1  105 þ þ ¼ ¼ $0:0239=ðkWhÞ 0:84 hex W_ E 30  106

cE ð89%Þ ¼

0:017 ð1:02Þ1:1  105 þ ¼ $0:0228=ðkWhÞ 0:89 30  106

The reduction in the unit cost of electricity after the increase in efficiency is about 4.4%. This simple example shows the positive effect of exergetic efficiency on the unit cost of electricity. The exergetic efficiency may be increased by minimizing the throttling of large thermodynamic driving forces, which are changes in pressure, temperature, and composition. The effect of exergetic efficiency would increase for larger steam mass flow rates. At lower temperature levels, friction losses would have more negative effects on the unit cost of electricity.

5.2 THERMODYNAMIC COST

271

EXAMPLE 5.2 COST OF POWER AND PROCESS STEAM GENERATION In a steam power plant, the boiler uses natural gas as fuel, which enters the boiler with an exergy rate of 110 MW (Fig. 5.2). The steam exits the boiler at 6000 kPa and 673.15K, and exhausts from the turbine at 700 kPa and 433.15K. The mass flow rate of steam is 32.5 kg/s. The unit cost of the fuel is $0.0144/kWh of exergy, and the specific cost of electricity is $0.055/kWh. The fixed capital and operating costs of the boiler and turbine are $1150/h and $100/h, respectively. The exhaust gases from the boiler are discharged into the surroundings with negligible cost. The environmental temperature is 298.15K. Determine the cost rates of the steam produced by the reboiler (HP) and discharged steam (LP) from the turbine. Solution: Assume that heat transfer effects between the boiler and turbine and surroundings are negligible. In addition, kinetic and potential energy effects are neglected. The environmental temperature is 298.15K. At this temperature, we have the reference values of enthalpy and entropy: Ho ¼ 2547.2 kJ/mole and So ¼ 8.5580 kJ/(mole K), m_ ¼ 32.5 kg/s. The cost data: _ F ¼ 110,000 kW. cE ¼ $0.055/(kWh), cF ¼ $0.01144/(kWh), Ex C_B ¼ $1150/h, C_T ¼ $110/h. Enthalpy and entropy values after the boiler: P1 ¼ 6000 kPa, T1 ¼ 673.15K, H1 ¼ 3177.2 kJ/kg, S1 ¼ 6.5408 kJ/(mole K) After the turbine: P2 ¼ 700 kPa, T2 ¼ 433.15K, H2 ¼ 2798.2 kJ/kg, S2 ¼ 7.8279 kJ/(mole K) The work produced: _ 1  H2 Þ ¼ 32:5 ð3177:2  2798:2Þ ¼ 12317:50 kW W_ ¼ mðH
The cost of electricity: C_E ¼ cE W_ ¼ $677:43 h Using the reference values for enthalpy and entropy, the rate of exergy of stream leaving the boiler (1) and turbine (2) is: _ 1 ¼ m½H _ 1  Ho  To ðS1  So Þ ¼ 40; 021:42 kW Ex _ 2 ¼ m½H _ 2  Ho  To ðS2  So Þ ¼ 15; 232:08 kW Ex The cost rate balance for the boiler yields the specific cost of steam produced by the boiler: _

_ 1 ¼ cF Ex _ F þ CB /c1 ¼ cF ExF þ CB ¼ $0:0683=ðkWhÞ c1 Ex _1 _1 Ex Ex
_ _ The cost of boiler steam: C1 ¼ c1 Ex1 ¼ $2733:46 h The cost rate balance for the turbine, Eq. (5.5), yields the specific cost of steam exhausted by the turbine: _

_

_

CE CT 1 c2 ¼ c1 Ex _  Ex _ þ Ex _ ¼ $0:1415=ðkWhÞ Ex 2

2

2

The cost of exhausted steam:
_ 2 ¼ $2156:53 h C_2 ¼ c2 Ex

EXAMPLE 5.3 THERMOECONOMIC CONSIDERATION OF A REFRIGERATION SYSTEM A refrigeration heat exchanger provides an opportunity to study the trade-off between the cost of availability loss and the capital cost of the exchanger. Explore the optimization opportunities of a refrigeration system.

272

CHAPTER 5 THERMOECONOMICS

Solution: We may need to supply a refrigeration flow to the condenser of a distillation column that returns reflux as a condensate at a certain temperature. The refrigeration temperature must be less than the condensing temperature, and the temperature difference of the refrigerant and the condenser DT is an important parameter. A larger difference results in a smaller and hence less expensive condenser; however, to get refrigerant at low temperature, the power required for a unit of refrigeration increases due to the higher fuel costs to operate the refrigeration compressor. This creates a typical optimization problem involving the value of DT and the annual cost of fuel and the condenser (see Fig. 5.1). From the second law consideration, the annual cost of fuel C_F is given by cF Fw Fb ty To qDT C_F ¼ T2 where cF is the cost of unit of fuel, Fw is the units of shaft work required by the refrigeration system to deliver a unit of availability, Fb is the units of fuel fired in the plant boiler per unit of shaft work produced, ty is the operating time per year, To is the absolute temperature of the refrigeration system condenser, or of ambient, q is the heat or refrigeration duty per unit time, and DT is the temperature difference. The annual capital cost C_a of the heat exchanger, assuming that the exchanger is large and the cost is directly proportional to its area A, is given by ce Fi A ce Fi q C_a ¼ ¼ Pt Pt UDT where ce is the purchase cost per unit of the heat exchanger area, Fi is the installation cost factor, Pt is the allowable payout time in years, and U is the overall heat transfer coefficient. The differential of total annual cost of fuel and capital with respect to DT is set equal to zero, and is used to determine the optimum DT 1=2  ce F i DTopt ¼ T cF Fw Fb ty Pt U The equation above shows that the optimum temperature DTopt is proportional to the temperature level at which the heat transfer occurs, which is well known for the refrigeration systems. The value of refrigeration increases as the temperature decreases; hence the smaller values of DT are used as the refrigeration temperature decreases. For above-ambient systems, the larger values of DT should be used at higher temperatures, even though the value of heat increases with increasing temperature.

5.2.3 CUMULATIVE EXERGY CONSUMPTION Exergy analysis evaluates the level of irreversibility, and hence identifies the possibilities of improvements for a process. It may play a primary role in minimizing the consumption of natural resources within the context of ecological economy. Exergy analysis is, however, a thermodynamic approach, not an economic one. Still, the partition of production costs between the useful products of a complex process can be managed by means of exergy. All useful products out of an industrial production line are the results of a complicated network of interconnected processes, which need the supply of raw materials, fuels, and other energies extracted from natural resources. The quality of the natural resources can be evaluated and expressed by means of exergy. The analysis of cumulative exergy consumption related to the entropy generated by each process provides an insight into the possibilities of improving the technological network of production (Valero et al., 2006; Tsataronis, 2007). The total consumption of natural resources involved in the production of a product can be expressed by the overall index of cumulative exergy consumption rj, P Exloss;kj rj ¼ k Pj

5.2 THERMODYNAMIC COST

where Exloss,kj is the exergy consumption, expressed as Exloss;j ¼ To

P

273

DSkj , of the n natural resources

k

(k ¼ 1,2 .,n) P for the product of j, and Pj is the final product flow rate from an industrial plant. The terms To and DSkj indicate the environmental temperature, and the sum of entropy changes over the k

consumption of sources for product j, respectively. As the value of rj is related to a unit of the product leaving the system, it depends on the assumption of the system boundary. Usually, the system of production processes is analyzed without considering employees and local levels of consumption, and hence rj can be used for comparison of production processes in various countries. The exergy consumption index for a certain resource k can be determined separately for a final product flow rate Pj rkj ¼

Exloss;kj Pj

Exergy consumption of resources that are renewable should also be considered. The overall cumulative exergy consumption index may help in assessing various energy utilization options for a specified product, such as the relationships between the amounts of raw materials and the products, or the cost of raw materials and the alternative production technologies available. In analyzing the production of materials and energy flows, the values of rj can be compared with the exergy of the product. The ratio of the specific exergy of the product ex to rj is called the cumulative degree of thermodynamic perfection h for a certain production network exj hj ¼ rj Here, rj is the cumulative exergy consumption index for a specified product j. For the manufacturing of major products, we have h < 1. Sometimes, for a byproduct we may have h > 1, if the exergy of the product is greater than the exergy of the substituted product of a certain process. It would not be useful to calculate h for certain products, such as cars, and airplanes, because their usefulness results mainly from their system features, not from the chemical composition of their components. However, the calculation of r may be beneficial for all kinds of products, because the values of r can be used to compare various design variables and production technologies. Cumulative exergy consumption can be calculated by the balance equations; the rkj for the useful products equals the sum of cumulative exergy consumption of all raw materials and semifinished products in the production network. For the link j of the network and for the natural resources k, the balance equations are X X ðlÞ ðlÞ ðlÞ ðtÞ ðlÞ rkj ¼ wit rki þ Exloss;kj (5.7) aij  fij i

t

where k is the index of a natural resource, i and j are the indices of technological network, l and t are the indices of the production technologies of the products i and j, aij is the coefficient of the gross consumption of the semifinished product i per unit of the complex useful product containing a unit of the major product j, fij is the coefficient of the by-production of the useful product i, wit is the fraction of the manufacturing technology t of the product i, and Exloss,kj is the immediate gross exergy consumption of the neutral resources k per unit of the useful product j. Eq. (5.7) describes a complex process producing more than one useful product. A complex process is usually related to a major product, which determines the capacity and location of production. A

274

CHAPTER 5 THERMOECONOMICS

useful product substituting the major product is called the byproduct. The coefficient of by-production is expressed in terms of the substitution ratio ziu which is the ratio of the unit of the major product i substituted by the unit of the byproduct u, and given by ðiÞ

fij

ðlÞ

fuj

¼ ziu

(5.8)

where fuj is the coefficient of production of byproduct u per unit of the major product j. Cumulative exergy consumption for the byproduct u is given by ðiÞ

ðlÞ

fij ri ¼ fuj ru

(5.9)

From Eqs. (5.8) and (5.9), we obtain ru ¼ ri ziu

5.2.4 CUMULATIVE DEGREE OF THERMODYNAMIC PERFECTION The cumulative degree of thermodynamic perfection for real byproducts u is defined by exu exu exu h hu ¼ ¼ ¼ hi ¼ i ru ri ziu bi ziu hz;iu

(5.10)

where hi is the cumulative degree of thermodynamic perfection for the major product i substituted by the product u, and hz,iu is the exergetic substitution efficiency defined by exi ziu hz;iu ¼ (5.11) exu If hz,iu < hi, the value hu > 1, results from Eq. (5.10). Some typical values of the cumulative degree of thermodynamic perfection h are given in Table 5.1. For small values of ru that result from the substitution ratio ziu, the cumulative exergy consumption will be large for the major product j of the specified technology. Eq. (5.7) can be transformed as follows X rkm ¼ ða  fnm Þrkn þ Exc;km (5.12) n nm ðlÞ

ðlÞ

where anm ¼ wit aij ; and fnm ¼ fij . For technology l, every subscript j corresponds to some subscripts m, while for technology t, every subscript i corresponds to some subscripts n. Eq. (5.12) can also be formulated for semifinished products, which are consumed in other parts of the technological network.

5.2.5 CUMULATIVE EXERGY LOSS The difference between cumulative exergy consumption r and exergy consumption of a natural resource represents the cumulative exergy loss Excl involved in all parts of a manufacturing technological network Excl ¼ r  Exc

5.2 THERMODYNAMIC COST

275

Table 5.1 Cumulative Degrees of Thermodynamic Perfection for Some Production Technologies (Szargut, 1990) Product

Specific Exergy (MJ/kg)

Aluminum

32.9

Iron Cement

8.2 0.635

Copper

2.11

Glass

0.174

Ammonia gas

20.03

Paper

16.5

Zinc

5.19

Sulphuric acid

1.66

h%

Production Technology

9.6 13.2 44.0 10.3 6.2 3.2 2.6 1.5 0.8 0.5 45.4 64.8 41.5 18.7 27.5 74.3 7.6 6.7 2.7 4.2 18.3

Bayer process and Hall cell 50% bauxite ore Electrolytic method from Al2O3 From hematite ore in the earth From raw materials with dry method Medium rotary kiln with wet method From ore containing Cu2S, smelting and refining Hydrometallurgical method Electrolytic From raw material From panels Steam reforming of naphtha Steam reforming of natural gas Semicombustion of natural gas From timber Integrated plant with fuels from waste products From waste paper From ore with vertical retort Electrothermic method From ZnS, metallurgical method From ZnS, electrolytic method Frasch process with sulfur combustion

The components of Excl provide information for improving the technological network. The difference (r  Exc)n defines the constituent exergy consumption of a semifinished product n, and results from the thermodynamic imperfection of the constituent technological network. In complex processes, raw materials and semifinished products are partially used for the manufacturing of byproducts. Hence, the coefficient of net consumption Cnm of semifinished products and raw materials per unit of the major product should be determined by X X Cnm ¼ anm  fum Cnu ¼ anm  fum zpu Cnp (5.13) u

p

where p is the index of the major product substituted by the byproduct u, and zpu is the substitution ratio of the product p by the product u. In a process substituted by the utilization of the byproduct the coefficient anm can be negative if the consumption of the semifinished product n is greater than in the principal process considered. The constituent exergy consumptions are calculated from the coefficients Cnm   Excc;nm ¼ Cnm rn  Exc;n ; nsm Some consumption can be negative due to the elimination of the constituent exergy consumptions in the substituted process.

276

CHAPTER 5 THERMOECONOMICS

5.2.6 LOCAL GROSS EXERGY CONSUMPTION Local gross exergy consumption Exgl,m represents the sum of internal and external exergy consumed in the used technology for the major product and byproduct, and it can be calculated from the following steady-state exergy balance X X anm Exn ¼ Exm þ fum Exu þ Exgl;m (5.14) n

u

Here the energy exchanged with surroundings should be treated as one of the useful product exergies represented by Exm, Exn, or Exu. The local net exergy loss refers to the complex of useful products containing a unit of the major product, and results from the following difference X Exnl;nm ¼ Exgl;m  fum Exll;u (5.15) u

where Exll,u is the local exergy consumption due to the byproduct u. The local exergy consumption due to the byproduct results not only from the local net exergy consumption in the substituted process but also from the difference between the exergy of the byproduct and the substituted major product.

5.2.7 EXHAUSTION OF NONRENEWABLE RESOURCES The utilization of domestic natural nonrenewable resources is inevitable, and analyzing these resources helps to assess the profitability of importing raw materials, fuels, and semifinished products as well as utilizing of secondary raw materials. In analyzing the exhaustion of nonrenewable natural resources, the balance equations of Eq. (5.12) should be modified if domestic nonrenewable resources are of interest. In this case, imported raw materials, fuels, and semifinished products should be considered separately X X ekm ¼ ðann  fum Þekn þ arm ekr þ Exc;km (5.16) n

r

where ekm, ekn, and ekr are the exhaustion of the domestic nonrenewable natural resources k per unit of the products m, n, and r, respectively, arm is the coefficient of gross consumption of imported raw material, fuel, or semifinished product r, and Exc,km is the gross consumption of the domestic nonrenewable natural resources k within the link m. The index ekr should be determined by assuming that the import is economical, and the unit value of the exported and imported products is considered with the same exhaustion of nonrenewable natural resources P En ekn ðdÞ n P ekr ¼ ek Dr ¼ Dr (5.17) En Dn n ðdÞ ek

where is the exhaustion of nonrenewable natural resources per unit of the monetary values of exported products, Dr and Dn are the specified monetary value of the imported product r and exported product n, respectively, and En is the export of the product n. Introducing Eq. (5.17) into Eq. (5.16), the balance equations become X ekm ¼ ½ðann  fum Þ þ dnm ekn þ Exc;km (5.18) n

5.3 ECOLOGICAL COST P

277

Dr arm

r where dnm ¼ En P : ED n

n

n

In the analysis of material production, the utilization efficiency of nonrenewable domestic natural resources can be defined as ex h0 ¼ (5.19) e Usually, e > ex and h0 < 1, but for secondary raw materials h0 [ 1, and for imported raw materials and fuels, we usually have h0 > 1. For secondary raw materials, the exhaustion of nonrenewable natural resources is related to the consumption of exergy for processing and transportation, and usually it is much smaller than the exergy of the materials under consideration. The inequality h0 [ 1 suggests that the utilization of secondary raw materials may be beneficial, since they substitute the semifinished products requiring a large amount of exergy for production. We may have ex > e for imported raw materials, fuels, and semifinished products if the exported goods are more advanced than the imported ones (Tsataronis, 2007).

5.2.8 EXERGY DESTRUCTION NUMBER The use of an augmentation device results in an improved heat-transfer coefficient, thus reducing exergy destruction due to convective heat transfer; however, exergy destruction due to frictional effects may increase. The exergy destruction number NEx is the ratio of the nondimensional exergy destruction number of the augmented system to that of the unaugmented one NEx ¼

Exa Exs

(5.20)

where subscripts a and s denote the augmented and unaugmented cases, respectively, and Ex is the nondimensional exergy destruction number, which is defined by exfd (5.21) Ex ¼ _ o Cp mT Here exfd is the flow-exergy destruction, or irreversibility, and To is the reference temperature. The system will be thermodynamically advantageous only if the NEx is less than unity. The exergy destruction number is widely used in second lawebased thermoeconomic analysis of thermal processes, compressors, chemical processes, power generation, and energy storage systems.

5.3 ECOLOGICAL COST The production, conversion, and utilization of energy may lead to ecological cost that includes environmental problems, such as air and water pollution, impact on the use of land and rivers, thermal pollution due to mismanagement of waste heat, and global climate change (Valero et al., 2006). As an energy conservation equation, the first law of thermodynamics is directly related to the energy management impact on the environment. One of the links between the principles of thermodynamics

278

CHAPTER 5 THERMOECONOMICS

and the environment is exergy, because it is mainly a measure of the departure of the state of a system from that of the equilibrium state of the environment. Performing an exergy analysis on Earth’s natural processes may reveal disturbances due to large-scale changes and could form a sound base for ecological planning for sustainable development. Some of the major disturbances are: 1. Chaos due to the destruction of order is a form of environmental damage. 2. Resource degradation leads to exergy loss. 3. Uncontrollable waste exergy emission can cause a change in the environment. Exergy analysis may be an important tool to interrelate energy management, the environment, and sustainable development to improve economic and environmental assessments. Ecological cost analysis may minimize the depletion of nonrenewable natural resources. Determining the exhaustion of nonrenewable natural resources connected with the extraction of raw materials and fuels from natural resources is not enough to fully understand the ecological impact of production processes. The influence of waste product discharge into the environment should also be considered. Waste products may be harmful to agriculture, plant life, human health, and industrial activity.

5.3.1 INDEX OF ECOLOGICAL COST The exhaustion of nonrenewable natural resources is called the index of ecological cost. To determine the domestic ecological cost ceco, the impact of imported materials and fuels is considered ceco;m ¼

X

anm  fnm þ dnm þ

X  Exc;sm xns ceco;n

n

þ

s

X

Exc;sm

s

X yks þ zs

!

X Exc;km þ

k

(5.22)

k

where s is the index of harmful waste product, dnm is defined in Eq. (5.18), Exc,km is the immediate gross consumption of the nonrenewable domestic natural resource k per unit of complex useful products containing a unit of the major product m, Exc,sm is the exergy of harmful waste product s, xns is the destruction coefficient of the product n per unit of the exergy of waste product s, yks is the destruction coefficient of the nonrenewable natural resources k per unit of the exergy of waste product s, and zs is the multiplier of exergy consumption to eliminate the results of human health deterioration per unit of exergy of the waste product s. The destruction coefficients x and y are xns ¼

dExc;n dExck ; yks ¼ Exc;s Exc;s

where dExc,n is the number of units of the destroyed useful product and dExc,k is the exergy decrease of the damaged natural resources. The coefficient xns should also consider the reduction of agricultural and forest production. The global ecological cost can be calculated. The degree of the negative impact of the process on natural resources can be characterized by means of the ecological efficiency he Exc he ¼ (5.23) ceco

5.4 AVAILABILITY

279

Usually, he < 1, but sometimes values of he > 1 can appear if the restorable natural resources are used for the process. The transition from one form of exergy to another, for example, from chemical to structural, may create economic value.

5.3.2 SUSTAINABILITY Sustainability is maintaining or improving the material and social conditions for human health and the environment over time without exceeding the ecological capabilities that support them. Demand for energy as a driver of technology and development continues to grow worldwide even as extraction of fossil fuels becomes more difficult and expensive. Producing, storing, and converting energy have a permanent impact on Earth’s climate and environment. There are two sustainability impacts that are resource depletion and environmental change. The dimensions of sustainability are economic, environmental, and societal and the sustainability metrics are (Demirel, 2016): • • • •

Material intensity (nonrenewable resources of raw materials, solvents/unit mass of products). Energy intensity (nonrenewable energy/unit mass of products). Potential environmental impact (pollutants and emissions/unit mass of products). Potential chemical risk (toxic emissions/unit mass of products).

The global warming potential (GWP) is a measure of how much a given mass of a chemical substance contributes to global warming over a given period. Greenhouse gases (GHGs) such as carbon dioxide, methane, and ozone absorb energy in the atmosphere and slow or prevent the loss of heat to outer space. In this way, GHGs act like a blanket and make Earth warmer, a process called the “greenhouse effect.” The GWP of carbon dioxide is defined as 1.0, while water has a GWP of 0. Chlorofluorocarbon-12 has a GWP of 8500, while chlorofluorocarbon-11 has a GWP of 5000. Various hydrochlorofluorocarbons and hydrofluorocarbons have global warming potentials ranging from 93 to 12,100. These values are calculated over a 100-year period. Life Cycle Analysis (LCA) is a standardized technique that tracks all material, energy, and pollutant flows of a systemdfrom raw material extraction, manufacturing, transport, and construction to operation and end-of-life disposal. LCA can help determine environmental impact from the boundaries “cradle to grave” or “cradle to gate” and facilitate comparisons of processes and techniques. Cradle to gate scenario estimates energy use (mainly natural gas and electricity), GHG emissions, for feedstock production and conversion processes. LCA may provide a well-established and comprehensive framework to compare renewable energy technologies with fossil-based and nuclear energy technologies (Matzen et al., 2015; Matzen and Demirel, 2016).

5.4 AVAILABILITY One of the important definitions in finite-time thermodynamics is the definition of finite-time availability A given by

 Z tf A ¼ Wmax ¼ max Aðti Þ  Aðtf Þ  To Stot dt ti

Here, ti and tf are the initial and final times of the irreversible process and To is the environmental temperature. The equation above represents the second law of thermodynamics in equality form by

280

CHAPTER 5 THERMOECONOMICS

subtracting the work equivalent of the entropy produced, which is the decrease in availability in the process. The maximization is carried out with the constraints imposed on the process. Availability depends on the variables of the system as well as the variables of the environment X A ¼ U þ P o V  To S  moi Ni Here, the temperature, pressure, and chemical potential are estimated at ambient conditions. For an optimal control problem, one must specify: (1) control variables such as, volume, rate, voltage, and limits on the variables; (2) equations that show the time evolution of the system which are usually differential equations describing heat transfer and chemical reactions; (3) constraints imposed on the system such as conservation equations; and (4) objective function, which is usually in integral form for the required quantity to be optimized. The value of process time may be fixed or may be part of the optimization. The potential work of any system is given by Ees ¼ E  To S þ Po V 

X mio Ni i

where m and N are the chemical potential and number of moles of substance i, E is the total energy including all kinetic and potential energy in addition, to internal energy, and the indices o denote the reference state representing the environment of the system. The term Ees is the essential energy in the form essential for work (power) production, so that Ees shows the essergy (essential energy). The corresponding flow of essergy jes, excluding kinetic and potential energy for any uniform mixture of substances, is X mio Ni jes ¼ H  To S  i

5.5 THERMODYNAMIC OPTIMUM Thermoeconomics formulates an economic balance through exergy cost and optimization (Nguyen and Demirel, 2011). The minimization of entropy generation plays only a secondary role in thermoeconomics, mainly because economic performance is always expressed in economic values of money and price. Therefore, the thermodynamic optimization problem may not be expressed in terms of the problem of the minimization of irreversibility. For example, the problem of minimum overall exergy consumption may not be equivalent to the problem of minimum dissipation because of the disregarded exergy of the outgoing flows and changing prices of exergy unit. This problem mainly belongs to the areas of energy management and the cost of energy. Industrial systems consist of various resource consumption processes and supporting processes to supply and remove resources. The supporting processes may involve exergy loss and exergy transfer between resources, new resource upgrading, postconsumption recovery, and the dispersion and degradation of resources released to the environment. The contemporary theory of optimization can be used for analyzing these systems. The first approach is to optimize the system by adjusting the design and operating parameters through governing equations that describe internal changes, and by imposing control through system boundaries. The second approach aims to predict system behavior under a set

5.5 THERMODYNAMIC OPTIMUM

281

FIGURE 5.3

Hot utility

Principle of pinch technology. Hot utility

Investment cost Size

Δ T min

Cold utility Hot utility

Optimum

Cold utility

Δ Tmin Cold utility

Operating cost Exergy loss

of specified external conditions with governing equations derived from certain variational or extremum principles. Thermodynamic analysis (Demirel, 2013b) may be in line with those of economic analysis when the thermodynamic cost optimum, not the maximum thermodynamic efficiency, is considered with process specifications. Fig. 5.3 shows pinch technology in terms of optimum hot and cold utilities by accounting for the investment costs and exergy cost. With an optimum approach temperature DTmin, the total cost may be optimized.

EXAMPLE 5.4 MINIMIZATION OF ENTROPY PRODUCTION Discuss entropy production minimization in fluid flow systems. Solution: For a fixed design, the minimization of the rate of entropy production may yield optimal solutions in some economic sense. Such a minimization comes with certain set of constraints. For a single force-flow system assuming that linear dependency of the flow with respect to driving force, the local rate of volumetric entropy production is Z s ¼ LX 2 dV V

where L is the phenomenological coefficient and is not a function of the driving force X. The minimization problem is the optimization of the system with a finite size V, and the solution is the homogeneous distribution of the force over the system. Assuming a steady-state heat-transfer operation with no momentum and mass transfer, since X ¼ V(1/T) ¼ 1/ T2V(T), the expression of total entropy production is Z Z  2 VT VT s¼ ðqÞdV ¼ k dV 2 T VT V where the heat flux is obtained from the Fourier law q ¼ kVTand k is the thermal conductivity assumed as a constant. The entropy production is a function of the temperature field. Then, the minimization problem is to obtain the temperature distribution T(x) corresponding to a minimum entropy production s using the following Euler-Lagrange equation X d  vs  s ¼0 (a) dx vT

282

CHAPTER 5 THERMOECONOMICS

Minimizing the entropy production function with the constraint, expressed as q ¼ kVT, Eq. (a) becomes " # X 1 vT 2 vT  2 ¼0 T vx vx

(b)

For a heat exchanger, a characteristic direction related to the temperature field is the direction Z(x) normal to the heattransfer area, and Eq. (b) yields

 v 1 vT ¼0 vx T vx ZðxÞ and we obtain   VT ¼ constant T ZðxÞ The equation above shows that by keeping the driving force VT=T uniformly distributed along the space variables, entropy production will be minimum. For an optimum design, we may consider VT DT z ¼ constant T T

(c)

DT ¼ ðp  1ÞTc

(d)

with DT ¼ Th  Tc or, and the temperature gradient is a function of the temperatures Th and Tc of hot and cold streams, respectively Th ¼ pTc where p is a constant with value p > 1. The following expression also produces a constant C ds UðTh  Tc Þ2 ¼ ¼ C ¼ constant da Th Tc where d is a small change, U is the overall heat-transfer coefficient, and a is the heat-exchanger area   1 1 dq  ; and da ¼ ds ¼ dq Tc Th UðTh  Tc Þ From Eqs. (c) and (d), we have h i1=2 ð2 þ C=UÞ þ ð2 þ C=UÞ2  4 p¼ 2 The energy balance is C_h dTh ¼ C_c dTc

(e)

where the terms C_h and C_c are the products of heat capacity and hot and cold streams flow rates, respectively. From Eqs. (d) and (e), we have C_c Th dTh ¼p¼ ¼ dTc C_h Tc which are the matching conditions to minimize the entropy production in any heat exchanger. For example, for a specified heat-exchanger area and hot stream input and output temperatures Ti and To, respectively, the minimum entropy production is obtained when C_c Ti ¼ ¼ constant C_h To We can extend this approach for a network of heat exchangers.

5.5 THERMODYNAMIC OPTIMUM

283

5.5.1 EQUIPARTITION AND OPTIMIZATION IN SEPARATION SYSTEMS Thermodynamic cost analysis relates the thermodynamic limits of separation systems to finite rate processes and considers the environmental impact through the depletion of natural resources within the exergy loss concept. Still, economic analysis and thermodynamic analysis approaches may not be parallel. For example, it is estimated that a diabatic column has a lower exergy loss (39%) than that of adiabatic distillation; however, this may not lead to a gain in the economic sense, yet it is certainly a gain in the thermodynamic sense. The minimization of entropy production is not always an economic criterion; sometimes, existing separation equipment may be modified for an even distribution of forces or an even distribution of entropy production (Demirel, 2013b). Thermodynamic analysis requires careful interpretation and application.

EXAMPLE 5.5 ECONOMICS OF EQUIPARTITION PRINCIPLE IN EXTRACTION PROCESS Since the minimization of entropy production is not always an economic criterion, it is necessary to relate the overall entropy production and its distribution to the economy of the process. To do this, we may consider various processes with different operating configurations. For example, by modifying an existing design, we may attain an even distribution of forces and hence an even distribution of entropy production. Solution: Consider a simple mixer for extraction. In minimal entropy production, size V, time t, and duty J are specified, and the average driving force is also fixed. We can also define the flow rate Q and the input concentration c of the solute, and at steady state, output concentration is determined. The only unknown variables are the solvent-flow rate and composition, and one of them is a decision variable; specifying the flow rate will determine the solvent composition. Cocurrent and countercurrent flow configurations of the extractor can now be compared with the same initial specifications (V, t, J, Q, c). Cocurrent operation will yield a larger entropy production P2 than the countercurrent operation P1, and investigating the implications of this on the decision variable is important. For a steady-state and adiabatic operation, for processes 1 and 2 with the solvent-flow rates of Q1 and Q2, we have (Tondeur and Kvaalen, 1987) Q1 Ds1 ¼ DS þ P1

(a)

Q2 Ds2 ¼ DS þ P2

(b)

where DS is the total entropy change, and Ds1 and Ds2 are the changes in specific entropies of the solvent. Subtracting Eq. (b) from Eq. (a), we have Q1 Ds1  Q2 Ds2 ¼ P1  P2 < 0

(c)

J ¼ Q1 Dc1 ¼ Q2 Dc2

(d)

The load is defined as

where Dc is the concentration change of the solute in the solvent throughout the process. Combining Eqs. (c) and (d), we obtain Ds1 Ds2 < (e) Dc1 Dc2 The specific entropy of a solvent increases with the solute concentration, and if the input solvent is the same, Inequality (e) yields Dc1 > Dc2 , and hence Eq. (c) shows that Q1 < Q2. This means that the solvent-flow rate is smaller in the less dissipative operation, and the solvent at the outlet is more concentrated. That is, the operating conditions of solvent determine the less dissipative operation. Whether this optimum is an overall economic optimum will depend mainly on the cost of the technology. We can also compare the two processes with the same total entropy production, the same size and duration, and the same phenomenological coefficients. Process 1 has only equipartitioned forces; therefore, the duties of these processes will be different. The total entropy productions for the processes are expressed as

284

CHAPTER 5 THERMOECONOMICS

P1 ¼ Pav1 ¼ LðXav1 Þ2 ðVtÞ Z Z L$X2 dVdt > Pav2 ¼ LðXav2 Þ2 ðVtÞ P2 ¼

(f) (g)

Since P1 ¼ P2, combining Eqns. (f) and (g) yields X2av1 > X2av2 , and hence jJ1 j > jJ2 j. That is the flow rate for equipartitioned process 1 is larger than that of process 2 at a given size, duration, and entropy production. In another operating configuration, we can compare the respective size and durations for specified duty and entropy production. Eqs. (e) and (f) are still valid, and we have P1 > Pav2 and J1 ¼ J2 , which yield P1 Pav2 > J1 J2 and thus Xav1 > Xav2 and. ðVtÞ1 < ðVtÞ2 This result indicates that for a given flow and entropy production, the equipartitioned configuration is smaller in size for a specified operational time. Alternatively, it requires less contact time for a given size, and thus a higher throughput. To determine an economic optimum, we assume the operating costs are a linear function of the solvent entropy change and entropy production, and the investment costs are a linear function of the space and time of the process. The total cost is Z Z   aLX 2 þ cs dVdt þ b (h) CT ¼ aP þ b þ csVt ¼ where s is the amortization rate, and a, b, and c are the constants related to the costs. The integral in Eq. (h) is subject to the constraint of a specified flow Z Z J¼

LXdVdt

The variational technique minimizes the total cost, and the Euler equation for variable X is given by  v  aLX 2 þ cs þ lLX ¼ 0 vX

(i)

where l is a Lagrange multiplier. Eq. (i) yields 2aLX þ lL ¼ 0 l X ¼  ¼ constant 2a The obtained value of X that minimizes the total cost subject to J is a uniform distribution. This illustrates the economic impact of the uniform distribution of driving forces in a transport process.

EXAMPLE 5.6 THERMOECONOMICS OF EXTRACTION PROCESS Consider a steady-state operation in which the forces are uniformly distributed; the investment cost Ci of a transfer unit is assumed to be linearly related to size V, and operating costs Co are linearly related to exergy consumption Cv ¼ Ci  Cif ¼ aV

(a)

Co ¼ Cof þ bDEx

(b)

where Cv is the variable part of the investment cost, Cif is a fixed investment cost, Cof is a fixed operating cost, and a and b are the cost parameters. Exergy loss DExc is expressed as DExc ¼ DExm þ To sav

(c)

5.5 THERMODYNAMIC OPTIMUM

285

Here, To is a reference temperature (dead state), and DExm is a thermodynamic minimum value. The total flow J ¼ LVXav can be written by using Eq. (a)   LXav Cv J¼ a Eliminating the constant (average) force Xav between Eq. (c) and the total entropy production sav, ¼ JXav, we obtain sav ¼

aJ 2 LCv

(d)

Substituting Eq. (d) into Eq. (b) and the latter into Eq. (c), a relationship between the operating and investment costs is obtained abTo J 2 Co ¼ þ Cof þ bDExm LCv The optimal size is obtained by minimizing the total operating and investments costs, which are linearly amortized with the amortization rate s: CT ðCi Þ ¼ sCi þ Co . The minimum of CT is obtained as dCT =dCi ¼ 0, and we have bsav bsav s ¼ ¼ ðCv Þopt aVopt To

(e)

According to the equation above, the quantity bTosav, which is related to irreversible dissipation and sVopt, should be equal in any transfer unit. Generally, operating costs are linearly related to dissipation, while investment costs are linearly related to the size of equipment. The optimum size distribution of the transfer units is obtained when amortization cost is equal to the cost of lost energy due to irreversibility. The cost parameters a and b may be different from one transfer unit to another; when a ¼ b, then sav/Vopt is a constant, and the optimal size distribution reduces to equipartition of the local rate of entropy production. The optimal size of a transfer unit can be obtained from Eq. (e)  ðCv Þopt ¼ Ci;opt  Cif ¼ J Vopt ¼ J

abTo Ls

1=2

 1=2 bTo aLs

Distributing the entropy production as evenly as possible along space and time would allow for the design and operation of an economical separation process. Dissipation equations show that both the driving forces and flows play the same role in quantifying the rate of entropy production. Therefore, the equipartition of entropy production principle may point out that the uniform distribution of driving forces is identical to the uniform distribution of flows.

EXAMPLE 5.7 HOT FLUID FLOW RATE EFFECT ON OPTIMIZATION Consider two heat exchangers 1 and 2 operating at steady-state and constant pressure with the same heat duty. The total entropy change of the cold fluid is the same for both heat exchangers and determined by the specified heat duty qs. There is no heat loss to the environment. The overall entropy balances for the heat exchangers are m_ 1 Ds1  P1 ¼ DS

(a)

m_ 2 Ds2  P2 ¼ DS

(b)

_ the mass flow rate, Ds the specific entropy change of the fluid between output and input, P the total entropy where mis production, and DS the total entropy change of the cold fluid. The heat duty is based on the enthalpy changes of the hot fluid Dh

286

CHAPTER 5 THERMOECONOMICS

Constant pressure line

h

inlet Δ h2

Δ h1

2 1

Outlet 2

Outlet 1 s

FIGURE 5.4 Mollier diagram for fluids 1 and 2 at constant pressure curve. q ¼ m_ 1 Dh1 ¼ m_ 2 Dh2

(c)

If we assume that P1 < P2, and subtracting Eq. (b) from Eq. (a), we have m_ 1 Ds1  m_ 2 Ds2 ¼ P1  P2 < 0

(d)

Since the hot fluid becomes colder, Ds < 0, and we have m_ 1 jDs1 j > m_ 2 jDs2 j From Eqs. (c) and (d), we find Dh1 Dh2 < (e) Ds1 Ds2 On an enthalpy versus entropy diagram (Mollier diagram), Fig. 5.4, Eq. (e) shows the slopesof chords
 to the constant pressure curve between input and output conditions. The constant pressure curves are convex v2 h vs2 >0. If the input conditions are the same for both exchangers, Inequality (e) and Fig. 5.4 show that jDh1 j > jDh2 j and because of Eq. (d), we have m_ 1 < m_ 2 . Therefore,
 exchanger 1, having the smallest entropy production, requires a smaller flow rate of hot fluid. The condition v2 h v2 s > 0 is always satisfied for pure fluids. For mixtures, however, this condition may not always be satisfied and should be verified.

5.5.2 OPTIMAL DISTILLATION COLUMNS Diabatic and isoforce column operations reduce exergy losses as well as the amount of utility required considerably. These modifications may lead to a mass transfer unit more reversible but may require more transfer units and hence more column height and heat-transfer area, which will increase capital costs. For example, in a diathermal distillation column with heat exchangers at every stage, it is possible to adjust the flow ratio of the phases and thus the slopes of operating lines, and the driving forces along the column. This also affects the driving force distribution and entropy production

5.5 THERMODYNAMIC OPTIMUM

287

(Demirel, 2004; 2006; 2013a). The task of a process engineer is to decide the target cost or costs to be optimized in a new design or in an existing operation. Energy saving in distillation systems has attracted considerable innovative approaches incorporating the principles of thermodynamics, such as pinch analysis, exergy analysis, and the equipartition principle. Thermodynamic analysis considers the critical interrelations among energy cost, thermodynamic cost, and ecological cost. Thermodynamic analysis is becoming popular for other separation systems, such as supercritical extraction, desalination processes, hybrid vapor permeation-distillation, and cryogenic air separation. For example, the energy requirement analysis of common cycles used in supercritical extraction has utilized exergy losses and an optimum extraction pressure, which produces a minimum in exergy loss for specified temperature and separation pressure. Distillation columns should be optimized considering both capital cost and operating (energy) cost. The heuristics of using a reflux ratio of 1.03e1.3 times the minimum reflux ratio is in line with both the capital cost and operating cost for binary distillation systems (Alhajji and Demirel, 2015; 2016).

EXAMPLE 5.8 OPTIMAL DISTILLATION COLUMN: DIABATIC CONFIGURATION Consider a distillation column made of N distinct elements. The heat and mass transfer area can be defined separately. Defining an investment cost Ci for element i as a linear function of the size Ai, we have Ci ¼ ai Ai þ bi where a is proportionality cost factor and b is a fixed cost (including the heat-transfer area). We assume that the average operating cost Ci is a linear function of the exergy loss in the element i per year Ci;av ¼ gi ½Exi þ T0 Pi  þ di where gi is a proportionality cost factor and di a fixed cost, while To the reference temperature (environmental temperature). Here, the exergy loss is split into a thermodynamic minimum Exi (for example, minimum separation work), and an irreversible contribution ToPi, where Pi is the entropy produced in element i. In the diabatic distillation column, each element is small enough that equipartition of entropy production may be approximately achieved by adjusting the heat flows and thus the liquid and vapor flow rates. We assume that each element performs a specified duty of Jio. The total cost function Ct for all N elements is Ct ¼

N  X

Ci;av þ sCi



i

where s is the yearly amortization rate. Using the variational approach, we minimize a Lagrangian as U¼

N  X

N  X li ðJi  Jio Þ ¼ 0 Ci;av þ sCi þ

i

i

This should satisfy the following conditions. vU vAi

v2 U vA2i Jo , Xav ¼ LA

¼ 0 and

Using

> 0 (for all i) J2

Pi becomes Pi ¼ Li Ai i . So, U is  N

X g To J 2 sai Ai þ sbi þ gi Exi þ di þ i i þ li ðJi  Jio Þ U¼ Li Ai i

The derivative with respect to Aj yields a system of independent equations gj To Jj2 vU ¼ saj  ¼0 vAj Li A2j

288

CHAPTER 5 THERMOECONOMICS

or s¼

gj To Pj aj Aj

(a)

and we have v2 U 2s ¼ >0 Aj vA2j Since s is a constant depending only on economic conditions, the right of Eq. (a) must be independent of the element j, and thus an invariant throughout the process. The product sjaAj is the annual cost related to irreversible energy waste in terms of exergy loss. These two quantities should be equal in any element. We may conclude that under these assumptions, the optimal size distribution of the elements requires the equipartition of the ratio gTo P=ðaAÞ on all the elements, and the cost of exergy loss is equal to the amortized proportional investment cost in that element. The equipartition principle is mainly used to investigate binary distillation columns and should be extended to multicomponent and nonideal mixtures. One should also account for the coupling between driving forces since heat and mass transfer coupling may be considerable and should not be neglected especially in diabatic columns.

5.5.3 THERMOECONOMICS OF LATENT HEAT STORAGE Latent heat storage is a popular research area with industrial and domestic applications, such as energy recovery of air conditioning, and underfloor electric heating by using a phase changing material. Fig. 5.5 shows the charging and discharging operations with appropriate valves, and temperature profiles for countercurrent latent heat storage with subcooling and sensible heating. An optimum latent heat storage system performs exergy storage and recovery operations by destroying as little as possible of the supplied exergy (Demirel and Ozturk, 2006; Demirel, 2007). A charging fluid heats the phase changing material, which may initially be at a subcooled temperature Tsc, and may eventually reach a temperature Tsh after sensible heating. Therefore, the latent heat storage system undergoes a temperature difference of Tsh  Tsc as shown in Fig. 5.6. Heat available for storage would be qc ¼ UAðDTlm Þc ¼ m_ c Cpc ðTci  Tco Þ where U is the overall heat-transfer coefficient, A is the heat-transfer area, m_ c is the charging fluid flow rate, Tci and Tco are Solar Energy . Exse, cse Air . Exa, ca

Solar Air Heaters

1

. Exp2c, cp2c . Exp1, cp1

Latent Heat Storage 2 . Exp2d, cp2d

FIGURE 5.5 Units of the latent heat storage system.

. Ex3a, c3a

Solar Energy . Ex3p, c3p

Greenhouse 3

5.5 THERMODYNAMIC OPTIMUM

. Ex1p, c1p

. Ex2pc, Tci

c2pc Tco

TscLatent Heat Storage . Ex2pd, c2pd

Tdo

Tsh

289

FIGURE 5.6 Approximate temperature profiles for a latent heat storage unit.

. Ex3a, c3a

Tdi

ðTci  Tsc Þ  ðTco  Tsh Þ Tci  Tco   (5.24) ¼ Tci  Tsc NTUc ln Tco  Tsh

 where NTU ¼ UA m_ c Cpc ¼ ðTci  Tco Þ DTlm is the number of transfer units. Eq. (5.24) relates the value of NTU with temperature. Heat lost by the charging fluid will be gained by the phase changing material qs ðDTlm Þc ¼

qc ¼ qs ¼ ms ½Cps ðTl  Tsi Þ þ DHm þ Cpl ðTsh  Th Þ where DHm is the heat of melting, Tl and Th are the lowest and highest melting points of the phase changing material, and Cps and Cpl denote the specific heats of solid and liquid states of the phase _ of the charging fluid is changing material, respectively. The net rate of exergy Ex  

  _ co  Ex _ ci ¼ m_ c Cpc ðTci  Tco Þ  T0 ln Tci _ c ¼ Ex DEx Tco Exergy stored by the phase changing material is   _ s ¼ q_s 1  T0 Ex Ts where Ts is an average temperature of storage, which may be approximated by ðTsc þ Tsh Þ=2. The first and second law efficiencies are actual heat stored Tci  Tco ¼ maximum energy gain Tci  Ts   T0 ðTci  Tco Þ 1  exergy of PCM T sh  ¼

¼ Tci exergy of charge fluid ðTci  Tco Þ  T0 ln Tco h¼

hex

If it is assumed that the phase changing material is totally melted and heated to a temperature Tsh, recovered heat is estimated by qd ¼ UADðTlm Þd ¼ m_ d Cpd ðTdi  Tdo Þ

290

CHAPTER 5 THERMOECONOMICS

The net exergy change of the charging fluid would be  

  _ di  Ex _ do ¼ m_ d Cpd ðTdi  Tdo Þ  T0 ln Tdi _ d ¼ Ex DEx Tdo The first and second law efficiencies are h¼

hex

Tdo  Tdi Tdi  Tsl

  Tdo Tdo  Tdi  T0 ln exergy given to discharge fluid Tdi   ¼ ¼ T exergy of PCM 0 ðTdo  Tdi Þ 1  Tsl

All the temperatures are time dependent, and the charging and discharging cycles need to be monitored over the time of operation. Structural theory facilitates the evaluation of exergy cost and the incorporation of thermoeconomic functional analysis. Structural theory is a common formulation for the various thermoeconomic methods. It provides costing equations from a set of modeling equations for the components or units of a system (Fig. 5.5). Structural theory needs a productive structure displaying how the resource consumption is distributed among the components of a system. The flows entering a component in the productive structure are considered fuels F and flows leaving a component are products P. The components are subsystems with control volumes as well as mixers and splitters. Therefore, the productive structure is a graphical representation of the fuel and product distribution. For any component j, or a subsystem, the unit exergy consumption exc is expressed on a fuel/product basis by excj ¼

Fj Fj ¼ _ j pj Ex

For linear modeling, the average costs of fuels and products are defined by  ¼ Cjm

vFo vFo  ; Cjp ¼ vmj vpj

where Fo is the fuel to the overall system expressed as a function of the flow mj or product Pj, respectively, and the other related parameters. The total annual production cost CT in $/(kWh) is CT ¼

N X j¼1

_ j¼ cj Ex

N X

CjF

j¼1

where ci is the specific cost of product i in $/(kWh), CjF is the cost of fuel, and Exj is the rate of exergy as a product of component j in kW and is expressed in terms of NTU using Eq. (5.24)  

_ j ¼ m_ j Cp NTUj DTlmj  T0 ln Tj1 Ex (5.25) To The optimum total production cost rate with respect to NTU is obtained from

PROBLEMS

291

dCT ¼0 dNTU Including an ecovector to account for the exergoeconomic costs or environmental impact can extend the thermoeconomic approach. An ecovector is a set of environmental burdens of an operation and can be associated with input flows; it includes information about natural resources, the exergy of these resources, and monetary costs. The external environmental costs associated with the environmental burdens may also be added into the ecovector. Extended exergy accounting includes the exergetics flowcharts for nonenergetic costs of labor and environmental remediation expenditures.

PROBLEMS 5.1 In a steam power generation plant, the boiler uses natural gas as fuel, which enters the boiler with an exergy rate of 110 MW. The steam exits the boiler at 6000 kPa and 673.15K, and exhausts from the turbine at 700 kPa and 433.15K. The mass flow rate of steam is 20.2 kg/s. The unit cost of the fuel is $0.016/kWh of exergy, and the specific cost of electricity is $0.05/kWh. The fixed capital and operating costs of the boiler and turbine are $1200/h and $90/h, respectively. The exhaust gases from the boiler are discharged into the surroundings with negligible cost. The environmental temperature is 298.15K. Determine the cost rate of process steam discharged from the turbine. Exhaust gases Fuel

Boiler

Turbine

Work

Air Feedwater Exhausted steam

5.2 In a steam power generation plant, the boiler uses a fuel, which enters the boiler with an exergy rate of 85 MW. The steam exits the boiler at 6000 kPa and 673.15K, and exhausts from the turbine at 700 kPa and 433.15K. The mass flow rate of steam is 19.5 kg/s. The unit cost of the fuel is $0.017/kWh of exergy, and the specific cost of electricity is $0.06/kWh. The fixed capital and operating costs of the boiler and turbine are $1150/h and $75/h, respectively. The exhaust gases from the boiler are discharged into the surroundings with negligible cost. The environmental temperature is 298.15K. Determine the cost rate of process steam discharged from the turbine. 5.3 A turbine produces 55 MWh of electricity per year. The annual average cost of the steam is $0.017/(kWh) of exergy (fuel). The total cost of the unit (fixed capital investment and operating

292

CHAPTER 5 THERMOECONOMICS

costs) is $2.6  105. If the turbine exergetic efficiency increases from 80% to 88%, after an increase of 3% in the total cost of the unit, evaluate the change of the unit cost of electricity. 5.4 A turbine produces 60 MWh of electricity per year. The annual average cost of the steam is $0.0175/kWh of exergy (fuel). The total cost of the unit (fixed capital investment and operating costs) is $2.5  105. If the turbine exergetic efficiency decreases from 90% to 80% after a deterioration of the turbine with use, evaluate the change of the unit cost of electricity. 5.5 Thermal analysis of the Aspen Plus simulator produces column grand composite curves of temperature-enthalpy and stage-enthalpy curves for rigorous distillation column simulations. These types of calculations are performed for RADFRAC columns. Using the following input summary for a RADFRAC column, construct the temperature-enthalpy, stage-enthalpy curves, and the stage exergy loss profiles. Assess the thermodynamic performance of the column. COMPONENTS C3 C3H8

/

IC4 C4H10-2

/ / IC5 C5H12-2 / NC5 C5H12-1 / NC4 C4H10-1

NC6 C6H14-1 FLOWSHEET BLOCK RADFRAC IN=FEED OUT=DIST BOTTOM PROPERTIES PENG-ROB PROPERTIES NRTL-2 STREAM FEED SUBSTREAM MIXED PRES=4.4 VFRAC=0. MOLE-FLOW=100. MOLE-FLOW C3 5. / IC4 10. / NC4 30. / IC5 20. / NC5 15. / NC6 20. BLOCK RADFRAC PARAM NSTAGE=28 COL-CONFIG CONDENSER=TOTAL FEEDS FEED 14 PRODUCTS DIST 1 L / BOTTOM 28 L P-SPEC 1 4.4 / 24 4.4 COL-SPECS D:F=0.44 MOLE-RR=1.8 T-EST 1 308. / 28 367.

5.6 Hydraulic analysis of the Aspen Plus simulator produces “thermodynamic ideal minimum flow” and actual flow curves for rigorous distillation column simulations. These types of calculations are performed for RADFRAC columns. Using the input summary given in Problem 5.5 construct the stage-flow curves. Assess the thermodynamic performance of the column. 5.7 Using the following input summary for RADFRAC columns, construct the column grand composite curves and stage exergy profiles with the property methods of Peng-Robinson. Discuss the results.

PROBLEMS

IN-UNITS ENG COMPONENTS ETHAN-01 C2H6O-2 1-PRO-01 C3H8O-1

/ /

ISOBU-01 C4H10O-3

/

N-BUT-01 C4H10O-1 FLOWSHEET BLOCK D1 IN=FEED OUT=DIS1 BOT1 BLOCK D2 IN=DIS1 OUT=DIS2 BOT2 STREAM FEED SUBSTREAM MIXED PRES=20. VFRAC=0. MOLE-FLOW=100. MOLE-FRAC ETHAN-01 0.25 / 1-PRO-01 0.5 / ISOBU-01 0.1 / N-BUT-01 0.15 BLOCK D1 RADFRAC PARAM NSTAGE=41 HYDRAULIC=YES COL-CONFIG CONDENSER=TOTAL FEEDS FEED 19 PRODUCTS DIS1 1 L / BOT1 41 L P-SPEC 1 20. COL-SPECS DP-COL=0. MOLE-D=74.7 MOLE-RR=3.65 SC-REFLUX DEGSUB=0. REPORT STDVPROF TARGET HYDANAL BLOCK D2 RADFRAC PARAM NSTAGE=23 HYDRAULIC=YES COL-CONFIG CONDENSER=TOTAL FEEDS DIS1 12 PRODUCTS DIS2 1 L / BOT2 23 L P-SPEC 1 20. COL-SPECS DP-COL=0. MOLE-D=25. MOLE-RR=3.64 SC-REFLUX DEGSUB=0.

5.8 Use the following economic data and prepare a discounted cash flow diagram. Assess the feasibility of the investment on the latent heat storage system: Economic data used for the seasonal heat storage system. Fixed capital investments for the components: FCI1 þ FCI2 þ FCI3 ¼ US$200000 þ US$200000 þ US$200000 ¼ US$600000 Cost of land: L ¼ US$50000 Working capital: WC ¼ 0.1 (US$600000) ¼ US$120000 Yearly revenues or savings: R ¼ US$150000 COP ¼ C1p þ Ccp þ Cdp þ C3p ¼ C1pf þ Ccpf þ Cdpf þ C3pf ¼ US$50000 Taxation rate: t ¼ 30% Salvage value of the whole seasonal storage system: S ¼ US$50000 Useful life of the system: n ¼ 15 years; Depreciation over 10 years Discount rate i ¼ 5.5%

293

294

CHAPTER 5 THERMOECONOMICS

REFERENCES Alhajji, M., Demirel, Y., 2016. Int. J. Energy Environ. Eng. 7, 45. Alhajji, M., Demirel, Y., 2015. Int. J. Energy Res. 39, 1925. Demirel, Y., 2006. Sep. Sci. Technol. 41, 791. Demirel, Y., 2004. Sep. Sci. Technol. 39, 3897. Demirel, Y., 2006. Int. J. Exergy 3, 345. Demirel, Y., Ozturk, H.H., 2006. Int. J. Energy Res. 30, 1001. Demirel, Y., 2016. Energy: Production, Conversion, Storage, Conservation, and Coupling, second ed. Springer, London. Demirel, Y., 2007. In: Paksoy, H.O. (Ed.), Thermal Energy Storage for Sustainable Energy Consumption, vol. 2007. Springer, p. 133. Demirel, Y., 2013a. Chem. Eng. Process Techniq 1005, 1. Demirel, Y., 2013b. Arabian J. Sci. Eng. 38, 221. Matzen, M., Demirel, Y., 2016. J. Cleaner Produc. 139, 1068. Matzen, M., Alhajji, M., Demirel, Y., 2015. Adv. Chem. Eng. 5, 128. Moran, M.J., Shapiro, H.N., 2000. Fundamentals of Engineering Thermodynamics, fourth ed. Wiley, New York. Nguyen, N., Demirel, Y., 2011. Energy 36, 4838. Szargut, J., 1990. In: Sieniutcyz, S., Salamon, P. (Eds.), Finite-time Thermodynamics and Thermoeconomics. Taylor & Francis, New York. Tondeur, D., Kvaalen, E., 1987. Ind. Eng. Chem. Res. 26, 50. Tsataronis, G., 2007. Energy 32, 249. Valero, A., Serra, L., Uche, J., 2006. Energy Sources Technol. 128, 1.

FURTHER READING AESF, 2010. The Environmental Cost of Energy, Applied Energy Studies Foundation. Demirel, Y., 2018. Energy conservation. In: Dincer, I. (Ed.), Comprehensive Energy Systems. Elsevier, Amsterdam. Finnveden, G., Arushanyan, Y., Branda˜o, M., 2016. Resources 5, 23. Gładysz, P., Ziebik, A., 2015. J. Power Technol. 95, 23. Matzen, M., Alhajji, M., Demirel, Y., 2015. Energy 93, 343. Nguyen, N., Demirel, Y., 2013. J. Sustain. Bioenergy Systems 3, 209.